{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\nIn recent years, autonomous driving has achieved widespread attention in academic and industry communities. However, there are still plenty of problems in interactive high-conflict traffic scenarios such as ramp merging, narrow street passing, unprotected left turn, and so on. The autonomous agent is required to interact with other traffic participants and choose an appropriate strategy to pass through the intersections safely and efficiently.\n\nFor the intersection navigation, there are three distinct planning and control approaches: the rule-based method, end-to-end method, and behavior-aware method. The rule-based method is based on some classical models such as Intelligent Driver Model (IDM) \\cite{kesting2010enhanced}, Optimal Velocity Model (OVM) \\cite{bando1998analysis}, and so on. In addition, the behavior strategies based on the hand-crafted rules are designed by a case-to-case mechanism, which lacks negotiation skills for high-conflict traffic scenarios and generalization ability to new scenarios. End-to-end control approaches such as imitation learning have also been investigated to obtain the driving policy based on image inputs \\cite{huang2019learning, codevilla2018end}. However, Waymo \\cite{bansal2018chauffeurnet} claims that pure imitation learning is not sufficient even with 30 million examples, which would get stuck or collide in highly interactive scenarios. \n\n\nRecently, reinforcement learning (RL) is considered as a feasible method to address these issues. RL methods have been widely used in video games \\cite{shao2018learning, shao2018starcraft} and autonomous driving \\cite{li2019reinforcement, li2019deep}, the success shows its great potential in resolving complex decision-making problems. By combining with other traffic participants' intentions implicitly, many behavior-aware RL planning methods are proposed such as context and intention-aware partially observable Markov decision process (POMDP) planning, social-aware deep reinforcement learning planning, and so on. Usually, safety and efficiency are considered in the reward shaping. However, it is still a great challenge for the trade-off between safety and efficiency based on reward engineering. For example, a less aggressive agent will certainly spend more time on a similar task. \n\nDuring the RL exploration phase, an agent is required to explore as many as different cases in order to find the near-optimal strategy. However, some of those cases may cause critical harm, especially for some physical systems such as robots or autonomous vehicles. Some safe reinforcement learning (safe RL) methods focus on constraining the exploration of an RL agent in order to avoid unsafe conditions. An optimal correction value of original dangerous action will help with the enhancement in safety without losing much efficiency.\n\n\n\\begin{figure}[hbtp]\n    \\centering\n    \\includegraphics[width=15cm]{total_framework.png}\n    \\caption{The proposed multi-task safe RL framework}\n    \\label{Fig: Total framework}\n\\end{figure}\n\n\nIn this paper, the assumption that ego vehicle in unsignalized intersections\nhas motion planning information will be hold similarly with \\cite{Kai2020Multi,tram2018learning}. We aim to investigate the multi-task unsignalized intersection navigation problem in dense traffic including turning left, going straight, and turning right. To improve the safety and efficiency, a novel multi-task safe RL framework is proposed as shown in Fig.\\ref{Fig: Total framework}. Compared with the state of the art (SOTA) work \\cite{Kai2020Multi}, the designed safety layer model and actor-critic with social attention can improve negotiation skills of the ego vehicle when interacting with other traffic participants.  As for the original contributions, this paper:\n\\begin{itemize}\n\n\\item Proposes a multi-task safe RL framework combined with a social attention module, the framework enhances safety and efficiency, also brings better interpretability of state representation;\n\n\\item Proposes an innovative design of safety layer for collision avoidance in the intersection navigation problem;\n\n\\item Evaluates the methods with a set of experiments in SUMO and CARLA simulators separately, and the result shows that the proposed method has a better performance than competitive methods.\n\n\\end{itemize}\n\n\n\\section{Related Work}\n\\label{sec:Related Work}\n\n\n\\subsection{Intersection Navigation}\nIntersection navigation in dense traffic is one of the most challenging tasks for the autonomous vehicles under urban scenarios, since it is very common to be trapped in the trade-off between safety and efficiency. Recently, a series of RL methods are proposed to solve this problem. \\cite{isele2018navigating} proves the effectiveness of deep reinforcement learning in the intersection decision-making problem, and \\cite{tram2018learning} improves RL methods' performance in a similar task by introducing several useful skills to the deep Q-learning(DQN) baseline. \\cite{qiao2018pomdp} converts unsignalized intersections navigating task as a hierarchical RL problem, and the hierarchical design of high-level discrete decision and low-level continuous control gains a significant improvement. \\cite{bouton2017belief} focuses on the unpredictable characteristics of other traffic participants, introducing the concept of belief state, which improves the ego vehicle's safety and traffic efficiency. \\cite{bouton2019reinforcement} proposes a generic approach to enforce probabilistic guarantees on a RL agent, which constrains acceptable actions of ego vehicle and improves its training efficiency.\n\nIn unsignalized intersection navigation domain, there are other RL methods that resolve the task from different aspects. For example, by integrating social attention mechanism with decision-making progress, a RL agent successfully learns an interaction pattern in \\cite{leurent2019social}, which focuses on the social vehicles that are highly related to the ego vehicle's current state. It gains significant quantitative improvements compared with DQN baseline. In \\cite{Kai2020Multi}, unsignalized intersection navigation task is modeled as a multi-task RL problem, in which turning left, turning right, and going straight are considered as specific sub-tasks. Through a multi-task learning framework, the agent learns to handle three navigating tasks at the same time and shows a competitive performance with single-task agents.\n\n\\subsection{Safe Exploration}\n\nFor the RL tasks in which the safety of agents is particularly concerned, not only that long-term reward maximization is desired, but also damage avoidance is requested. \\cite{garcia2015comprehensive} summarizes two major approaches to deploy safe RL methods. The first is to modify the optimality criterion while the second is based on the modification of the exploration process through the incorporation of external knowledge or the guidance of a risk metric. \\cite{bouton2019reinforcement} constructs a safe RL method for intersection navigation using linear temporal logic, directly selects safe action within available actions. For more complicated conditions, \\cite{bouton2019safe} proposes a method to decompose the scene of intersection navigation, easing the training difficulty of the safe agent.\n\nThough modifications on optimality criterion bring fast convergence in agent training, methods that modify the exploration show a better potential. \\cite{isele2018safe} proposes a method by combing a prediction model along with RL training. The prediction model masks unsafe actions to improve the safety performance of an intelligent vehicle. \\cite{wen2020safe} proposes a method extending actor-critic frame with an additional risk network to estimate the safety constraint of current policy, while brings a substantial improvement in safety performance. \\cite{dalal2018safe} proposes a method to explicitly define a safety constraint in a certain RL environment, and uses a first-order model to estimate the constraint value under an action distribution. According to the constraint model, an analytical solution of optimal safe action can be given. This method is evaluated in a deterministic and non-inertial environment, and it has the potential to be extended into a non-deterministic and inertial problem.\n\n\n\n\\section{Methods}\n\n\\subsection{Definitions}\n\nHere we will use the definition of constrained Markov Decision Processes (CMDP) with a bounded safety signal. We denote by $[K]$ the set $\\{1,..,K\\}$, and by $[x]^+$ the operation $\\max\\{x, 0\\}$, similarly $[x]^-$ for the operation $\\min\\{x, 0\\}$, where $x\\in\\mathbb{R}$. A CMDP is a tuple $(\\mathcal{S}, \\mathcal{A}, \\mathcal{P}, \\mathcal{R}, \\gamma, \\mathcal{C})$, where\n$\\mathcal{S}$ is a state space, $\\mathcal{A}$ is an action space, $\\mathcal{P}: \\mathcal{S} \\times \\mathcal{A} \\times \\mathcal{S} \\to [0, 1]$ is a transition kernel, $\\mathcal{R}: \\mathcal{S} \\times \\mathcal{A} \\to R$ is a reward function, $\\gamma \\in (0, 1)$ is a discount factor, and $\\mathcal{C} = \\{c_i : \\mathcal{S} \\times \\mathcal{A} \\to \\mathcal{S} \\ |\\  i \\in [K]\\}$ is a set of immediate-constraint functions. Based on that, we also define a set of safety signals $\\overline{\\mathcal{C}} = \\{ \\overline{c}_i : \\mathcal{S} \\to \\mathcal{R} | i \\in [K]\\}$. These are per-state observations of the immediate-constraint values. Policy $\\pi : \\mathcal{S} \\to \\mathcal{A}$ refers a stationary mapping from states to actions.\n\nTherefore, a safe RL problem considering an explicit safety constraint can be defined in form of an optimization problem, if all safety signals $\\overline{c}_i (\\cdot)$ are upper bounded by corresponding constants $C_i \\in \\mathbb{R}$\n\\begin{equation}\n\\begin{array}{c}\n\\max \\limits_{\\theta} \\mathbb{E}\\left[\\sum_{t=0}^{\\infty} \\gamma^{t} R\\left(s_{t}, \\pi_{\\theta}\\left(s_{t}\\right)\\right)\\right] \\\\\n\\text { s.t. } \\quad \\bar{c}_{i}\\left(s_{t}\\right) \\leq C_{i},  \\forall i \\in[K]\n\\end{array}\n\\end{equation}\n\n\\noindent where $r=R\\left(s_{t}, \\pi_{\\theta}\\left(s_{t}\\right)\\right) $ refers to the reward at timestep $t$, $\\bar{c}_{i}\\left(s_{t}\\right)$ refers to the constraint value of given state $s_t$, $C_{i}$ refers to the upper limit of the constraint value of a given state, $\\pi_{ \\theta }$ is a parametrized policy.\n\n\n\\subsection{Safety Layer Deployment}\n\nSafety problem in unsignalized intersection scenario is mainly considered as collision risk with other vehicles. All vehicles in the intersection must interact with each other to navigate their own target route. An intelligent agent constructed by deep reinforcement learning will explore all available strategies in order to learn the most proper strategy to solve the problem. In this paper, the safety model takes a two-stage approach to generate a safe action, as shown in Fig.\\ref{Fig: Total framework}. Firstly, a safety layer formed by a neural network will implicitly predict safety constraint according to current state. Secondly, the safety model will predict a modification on action conducted by the RL model analytically.\n\nTo estimate an impending collision of subsequent timesteps accurately, safety constraint requires a delicate design. Designing of safety constraint is quantization of collision events essentially and can be variable. Besides geometry overlapping, collision can be measured by other means, such as Time to Collision(TTC). During the driving task, a safe RL agent is supposed to satisfy safety constraints.\n\nIn this paper, we deploy a neural network trained by offline data for the constraint value estimation. Since the constraint value is defined with domain knowledge, the safety layer is basically a linearization of the dynamics of the state transformation. More specifically, the original safe constraint scalar $\\bar{c}_{i}\\left(s_{t}\\right)$ of a certain timestep can be estimated by the state and action of previous timestep. Therefore, the safety layer model is supposed to estimates the marginal effect of action on the safety constraint value. That is to say that the output of the neural network is the first order derivative of safe constraint value with respect to the action value. Therefore the first-order linearization model can be described as:\n\n\\begin{equation}\n\\bar{c}_{i}\\left(s^{\\prime}\\right) \\triangleq c_{i}(s, a) \\approx \\bar{c}_{i}(s)+f\\left(s ; \\omega_{i}\\right)^{\\top} a\n\\end{equation}\n\n\\noindent where $\\omega_i$ are weights of the neural network, subscript $i$ refers to the index of constraints if there are multiple ones, $s$ and $s^{\\prime}$ refer to the state of two continuous timesteps. We denote $\\bar{c}_{i}\\left(s^{\\prime}\\right)$ as $\\bar{c_i}'$ for simplicity.\n\n\nSafety model $f(s; \\omega_i)$ takes $s$ as input and outputs a vector which shares the same dimension with $a$. Based on such an assumption, the safety layer model will be trained by solving\n\\begin{equation}\n\\mathop{\\arg\\min}_{\\omega_{i}} \\sum_{\\left(s, a, s^{\\prime}\\right) \\in D}\\left(\\bar{c}_{i}\\left(s^{\\prime}\\right)-\\left(\\bar{c}_{i}(s)+f\\left(s ; \\omega_{i}\\right)^{\\top} a\\right)\\right)^{2}\n\\label{eq3}\n\\end{equation}\n\\noindent where $D$ refers to a dataset for model training. With the prediction of slope value $f(s; \\omega_i)$ from the safety neural network, the modification of exploration for the original RL can be defined as an optimal problem: \n\\begin{equation}\n\\begin{aligned}\n\\mathop{\\arg\\min}_{a} \\frac{1}{2}\\left\\|a-\\pi_{\\theta}(s)\\right\\|^{2} \\\\\n\\text { s.t. } c_{i}(s, a) \\leq C_{i}, \\forall i \\in[K]  \n\\end{aligned}\n\\end{equation}\n\n\\noindent An analytical proof has been given by \\cite{dalal2018safe}, and the optimal solution is derived by\n\\begin{equation}\na^{*}=\\pi_{\\theta}(s)-\\lambda_{i^*} f\\left(s ; \\omega_{i^{*}} \\right)\n\\label{eq5}\n\\end{equation}\n\n\\noindent where \n\\begin{equation}\n\\lambda_{i}=\\left[\\frac{f\\left(s ; \\omega_{i}\\right)^{\\top} \\pi_{\\theta}(s)+\\bar{c}_{i}(s)-C_{i}}{f\\left(s ; \\omega_{i}\\right)^{\\top} f\\left(s ; \\omega_{i}\\right)}\\right]^{+},  \\\\\ni^* = \\mathop{\\arg\\max}_{i} {\\lambda_i}\n\\label{eq6}\n\\end{equation}\n\n\nThe first-order safety above assumes that the constraints values yield a linear relation to the given action, which means the marginal effect of an action value is monotonous at any given time. The structure diagram of the safety layer model is shown in Fig.\\ref{Fig: Total framework}.\n\n\nAccording to the safe exploration method introduced above, a complete procedure can be summarized as follows. At each timestep, the safety layer will predict $f\\left(s ; \\omega_{i}\\right)$ using current state. With the constraint value $\\bar{c}_i-C_i$ of current state from the environment, a corrective action will be calculated by (\\ref{eq5}) and (\\ref{eq6}). Therefore it is critical to define constraints by which the unsafe conditions are reflected accurately. In this paper, we consider collisions between ego and social vehicles as an unsafe condition. Here we propose a safety constraint inspired by the TTC index, which is a critical index in autonomous driving research. \n\nThe original TTC describes vehicles cruising in a certain lane. Here we use the velocity projected on the relative location vector as an approximation. The instantaneous TTC can be approximately calculated using the relative position vector divided by the relative velocity projection\n\\begin{equation}\nTTC_{fix} = -1 \\cdot \\dfrac{ | R_i | }{\\left [{ | V_i | \\cdot \\cos{<R_i, V_i>}} \\right]^-}\n\\end{equation}\n\n\\noindent where $R_i$ refers to the relative position vector, $V_i$ refers to the relative velocity vector. $|\\cdot |$ refers to the 2-norm of a vector. The geometric relationship is shown in Fig.\\ref{Fig: safety constraint definition}, $\\rm \\textbf{F}_{ego}$ and $\\rm {\\textbf{F}}_{i}$ refer to coordinate frame of ego vehicle and social vehicle respectively, $R_i$ and $V_i$ indicate the relative location vector and relative velocity vector respectively. In order to avoid collision, $TTC_{fix}$ is supposed to be larger than a threshold value, which is as large as several timesteps of simulation. Therefore we deploy the upper limit of constraint $C_i = \\frac{t_0}{\\eta}$ and $\\bar{c}_i=TTC_{fix}$, so that ego vehicle is safe when\n\n\\begin{equation} \n\\frac{\\mu \\cdot t_0}{\\eta} \\leq TTC_{fix}, 0 \\leq \\eta \\leq 1\n\\label{eq8}\n\\end{equation}\n\n\\noindent where $\\eta$ refers to a discount factor, and $t_0$ refers to the timestep length of simulation. So the practical constraint can be written as\n\n\\begin{equation}\n\\bar{c}_i-C_i=\\mu \\cdot t_0 - \\eta \\cdot TTC_{fix} \\leq 0, 0 \\leq \\eta \\leq 1\n\\label{eq:constraint definition}\n\\end{equation}\n\n\\noindent Given by this constraint value definition, a potential collision revealed in constraint formation will cause the $\\bar{c}_i-C_i \\geq 0$.\n\n\\begin{figure}[thpb]\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{safe_constraint.png}\n    \\caption{Design of safety constraint inspired by TTC}\n    \\label{Fig: safety constraint definition}\n\\end{figure}\n\n\n\\subsection{Multi-task Reinforcement Learning}\n\nAs shown in Algorithm \\ref{alg1}, our proposed method is composed of safety model and RL model and trained separately. \\cite{Kai2020Multi} proposes a multi-task deep Q learning framework which can handle three different unsignalized intersection navigating tasks at the same time. In order to combine with the continuous action safety layer, we select TD3 (Twin Delayed Deep Deterministic policy gradient) \\cite{fujimoto2018addressing} algorithm, which is an actor-critic method, to train our multi-task agent. In particular, this technique is mainly used in the approximate value function, namely, critic, whose design is shown in Fig.\\ref{fig3}. Note that the encoders and decoder in the critic model are formed by Fully-Connected(FC) layers.\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=10cm]{critic}\n    \\caption{The Structure of the Critic Network}\n    \\label{fig3}\n\\end{figure}\n\nThe characteristics of this multi-task RL framework are mainly in three aspects: firstly, all the tasks that agent needs to learn are decomposed as a set of sub-tasks by domain knowledge, i.e. $G=\\{g_{s1},\u2026,g_{sn},g_{c1},\u2026g_{cn}\\}$, in which $g_{si}$ indicates a specific sub-task and $g_{ci}$ indicates a common sub-task. Secondly, a task code vector which is defined by different combinations of sub-tasks, is included in state representation to indicate the current task of the agent. Thirdly, a vectorized reward value is designed to separate the feedback transmission process of each sub-task to the model.\n\nIn the unsignalized intersection navigating task, the sub-tasks set is defined as $G_s= \\{ g_{sl},g_{sm},g_{sr},g_c \\} $, in which $g_{sl}$, $g_{sm}$ and $g_{sr}$ represent specific sub-task of turning left, going straight and turning right, $g_c$ is a additional dimension to improve the margin of state representation. Along with different combinations of $g_{s \\cdot}$ and $g_c$, the task vector $g$ can be defined as a 4-dimension vector, in which the first 3 dimensions imply agent's current specific sub-task using one-hot code, and the last dimension implies agent's common sub-task. For example, the task vector is $g=[1,0,0,1]$ for the turning left task,  $g=[0,1,0,1]$ for the going straight task, and $g=[0,0,1,1]$ for turning right task.\n\nReward is designed as a 4-dimension vector $r=[r_{sl},r_{sm},r_{sr},r_{c}]$ corresponding to the sub-tasks set $G_s$, in which the values of $r_{s \\cdot}$ and $r_c$ depend on the performance of sub-task $g_{s \\cdot}$ and $g_c$ separately. In this paper, the output of the critic network $Critic(s,a,g)$ is also a 4-dimension vector, which correlates with the design of sub-task set $G_s$ and current task vector $g$. Therefore the state-action value $Q(s,a,g)$ is calculated by the Hadamard product of critic's output $Critic(s,a,g)$ and task vector $g$\n\\begin{equation}\n    Q(s,a,g)=g^T Critic(s,a,g)\n\\end{equation} \nThe task vector $g$ filters the task-irrelevant values out of $Critic(s,a,g)$, and keeps task-relevant values in $Q(s,a,g)$, which helps to improve the convergence of critic model and prevents the model's preference for different tasks. \n\n\n\\begin{algorithm}\n  \\caption{Safe exploration TD3 with Multi-task framework}\n  \\label{alg1}\n  \\begin{algorithmic}[1]\n  \\STATE \\textbf{Step.1 Train Safety Model:}\n  \\STATE Initialize safety layer network $f\\left(s ; \\omega_i \\right)$ with random parameters $\\omega_i$, $i\\in[K]$  \\\\\n  \\STATE Collect datasets $D=\\left\\{(s,s',a,\\bar{c_i},\\bar{c_i}')\\right\\}$ through a random policy. \n  \\STATE Train the safety layer $f\\left(s ; \\omega_i \\right)$ using (\\ref{eq3})\n  \\\\ \\hspace*{\\fill} \\\\\n  \\STATE \\textbf{Step.2 Train Actor and Critic:}\n  \\STATE Initialize multi-task critic networks $Q_{\\phi_1}$, $Q_{\\phi_2}$, and social attention actor network $\\pi_\\theta$ with random parameters $\\phi_1$, $\\phi_2$,  $\\theta$\n  \\STATE Initialize target networks $\\phi_1' \\gets \\ \\phi_1$, $\\phi_2' \\gets \\  \\phi_2$, $\\theta' \\gets \\  \\theta$\n  \\STATE Initialize replay buffer $\\mathcal{B}$\n  \\FOR{$t=1$ \\textbf{to} $T$}\n  \\STATE Select action with exploration noise through actor $\\hat{a} = \\pi(s)+\\epsilon$,  $\\epsilon \\sim \\mathcal{N}(0,\\psi)$ \n  \\STATE Get safe action through safety layer model $a=\\hat{a}-\\lambda_{i^{*}} f\\left(s ; \\omega_{i^{*}} \\right)$, where $\\lambda_{i^{*}}$ is determined by (\\ref{eq6})\n  \\STATE Observe reward $r$ and new state $s'$\n  \\STATE Store transition tuple $(s,a,r,s')$ in $\\mathcal{B}$\n  \\\\ \\hspace*{\\fill} \\\\\n  \\STATE  Sample mini-batch of $N$ transitions $(s,a,r,s')$ from $\\mathcal{B}$\n  \\STATE $\\tilde{a} \\gets \\ \\pi_{\\theta'}(s)+\\epsilon$, $\\epsilon \\sim$ clip$(\\mathcal{N}(0,\\tilde{\\psi}), -l, l)$ \n  \\STATE $q \\gets \\ r \\ + \\ \\gamma$ $\\min_{j=1,2}$ $Q_{\\phi_j'}(s', \\tilde{a})$\n  \\STATE Update critics $\\phi_j \\gets \\min_{\\phi_j}$ $\\frac{1}{N} \\sum (q-Q_{\\phi_j}(s,a))^{2}$\n  \\STATE Update $\\theta$ by the deterministic policy gradient: \\\\ $\\nabla_\\theta J(\\theta) = $ $\\frac{1}{N} \\sum \\nabla_a Q_{\\phi_1}(s,a) \\lvert_{a=\\pi_\\theta (s)} \\nabla_\\theta \\pi_\\theta (s)$\n  \\STATE Update target networks: \\\\ $\\phi_j' \\gets \\tau \\phi_j + (1-\\tau) \\phi_j'$ \\\\ $\\theta' \\gets \\tau \\theta + (1-\\tau) \\theta'$\n  \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Attention Mechanism}\n\nThe way we employ the attention mechanism to resolve multi-task intersection navigating problem is similar to the social attention mechanism \\cite{leurent2019social}. The main purpose of employing this technique is enabling the RL agent to automatically capture dependencies between ego and social vehicles when making a decision, then acquire a better performance as well as better interpretability. In particular, this technique is only used in the policy model, namely, actor. The lower half of Fig.\\ref{Fig: Actor framework} shows the structure of the policy model. Obviously, the result of attention mechanism module directly affects the outcome of decision-making. Note that the encoders and decoder in actor model are also formed by Fully-Connected(FC) layers like the critic model.\n\nThe process of producing an attention tensor can be described as follows: First, the state representation of all the vehicles need to be decomposed into two parts: the state of ego vehicle and the state of social vehicles, then they are encoded separately and all vehicles' embeddings are given. The embeddings are fed into the social attention mechanism module, which is shown in the upper half of Fig.\\ref{Fig: Actor framework}. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=12cm]{actor_with_attention_framework.png}\n    \\caption{The Structure of the Actor Network}\n    \\label{Fig: Actor framework}\n\\end{figure}\n\nThere are three nonlinear projections in the module, which are $L_q \\in \\mathbb{R}^{d_x \\times d_k}$, $L_k \\in \\mathbb{R}^{d_x \\times d_k}$, and $L_v \\in \\mathbb{R}^{d_x \\times d_v}$. They are respectively responsible for the generation of query, key, and value vectors. Note that $d_x$ is the length of each vehicle's embedding, $d_k$ is the length of each query and key vector, $d_v$ is the length of each value vector, and the weights of $L_k$ and $L_v$ are shared between all vehicles. The query vector $Q^A=[q_e] \\in \\mathbb{R}^{1 \\times d_k}$ is calculated by processing the ego vehicle's embedding with $L_q$, the key vectors $K=[k_e,k_1\u2026k_n] \\in \\mathbb{R}^{(1+n) \\times d_k}$ and the value vectors $V=[v_e,v_1\u2026v_n] \\in \\mathbb{R}^{(1+n) \\times d_v}$ are calculated by processing both ego vehicle's embedding and social vehicles' embeddings with $L_k$ and $L_v$. The similarity between the query vector $Q^A$ and the key vectors $K$ can be accessed through their dot product $q_e k_i^T, i \\in [e,1,...,n]$. These similarities are then scaled by the inverse square-root-dimension $1\/ \\sqrt{d_k}$, and normalized with a softmax function $\\sigma$ across vehicles, the result stochastic matrix is called $attention \\ matrix$, in which the normalized values indicate ego vehicle's attention scores through different traffic participants, including itself. Finally, the product between the $attention \\ matrix$ and the value vectors $V$ is the $attention \\ tensor$ for forward propagation:\n\\begin{equation}\n    attention \\ tensor = \\sigma (\\frac{Q^AK^T}{\\sqrt{d_k}})V\n\\end{equation} \n\n\\section{Experiments}\n\nIn this paper, we deploy our method in two simulation environments, which are developed based on SUMO\\cite{SUMO2018} and CARLA\\cite{Dosovitskiy17} simulators. The SUMO simulator provides a highly portable interface for intelligent controller deployment, as well as the convenient environmental traffic flow generation. Meanwhile, the CARLA simulator takes a more delicate consideration on the dynamics of vehicles and provides a high-fidelity simulation. In CARLA, we deploy the proposed method into a more realistic autonomous driving pipeline for a further evaluation.\n\n\\subsection{SUMO Experiments}\n\n\\subsubsection{Experiment Setup}\n\nWe employ the SUMO simulator for intersection navigation tasks for clear comparison with related work \\cite{Kai2020Multi}, which provides the SOTA performance of the autonomous driving agent for the intersection scenario. An intersection navigation scenario is shown in Fig.\\ref{sumo_experiments}, the ego vehicle(cyan) is initially spawn on the west side of the intersection heading east, with north at the top. There is a 4-lane dual carriageway on the east-west direction while a 2-lane dual carriageway on north-south direction.\n\nIn SUMO, social vehicles are generated by a continuous traffic flow. We use the same parameters in \\cite{Kai2020Multi} in order to compare with the SOTA method. Social vehicles in the traffic are controlled by IDM from SUMO build-in algorithm. Default kinetics parameters are set to both ego and social vehicles so the maximum acceleration will be limited to a realistic value.\n\n\n\\begin{figure}[htb]\n    \\centering\n    \n    \\subfigure[left turn task]{\n    \\includegraphics[width=6cm]{sumo_left.png}\n   \n    }\n    \\subfigure[right turn task]{\n    \\includegraphics[width=6cm]{sumo_right.png}\n   \n    }\n\\caption{SUMO experiments}\n\\label{sumo_experiments}\n\\end{figure}\n\n\n\\subsubsection{Reinforcement Learning Setup}\n\n\\textbf{State representations:} We use a 33-dimension vector to represent state information, which can be written as $s = [s_e, s_1,\u2026s_5, g]$. As shown in TABLE \\ref{STATE REPRESENTATIONS}, the state vector contains 3 major parts: $s_e$ is a 4-dimension vector referring to the state of ego vehicle, which contains ego vehicle's speed and a 3-dimension one-hot code, indicating ego vehicle's current lane. $s_i, i=1,2,...,5$ refer to the state of 5 social vehicles, each of them is a 5-dimension vector which can be written as $s_i=[v_i, x_i, y_i, cos(\\alpha_i), sin(\\alpha_i)]$, in which $v_i$ indicates social vehicle's speed, $x_i,  y_i$ indicate social vehicle's Cartesian coordinates and $\\alpha_i$ indicates social vehicle's heading angle, note that $x_i$, $y_i$ and $\\alpha_i$ are all measured under ego vehicle's coordinate system as shown in Fig.\\ref{Fig: safety constraint definition}, $g$ is a 4-dimension task vector, where the first 3 dimensions indicate a specific sub-task like going straight, turning left or turning right using one-hot code, and the last dimension indicates a common sub-task like improving traffic efficiency. Note that if the number of social vehicles is larger than 5, only the nearest 5 vehicles are considered, if social vehicles' number is less than 5, then the state vector will be filled to 33-dimension using zero padding.\n\n\\begin{table}[h]\n\\caption{STATE REPRESENTATIONS}\n\\label{STATE REPRESENTATIONS}\n\\begin{center}\n\\begin{tabular}{c|c|c}\n\\toprule[2pt]\nState Component & Description & Feature Length \\\\\n\\hline\n$s_e$ & ego vehicle's state & 1*4  \\\\\n$s_i (i=1,2,...,5)$ & social vehicles' state & 5*5 \\\\\n$g$ & task vector & 1*4 \\\\\n\\hline\n$s$ & complete state representation & 33 \\\\\n\\bottomrule[2pt]\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn order to select task-relevant social vehicles for the state representation, a filter is designed to reserve vehicles with following rules:\n\\begin{itemize}\n\\item vehicle whose distance to ego vehicle is less than a certain threshold (the threshold is 75m in our experiment) will be reserved.\n\\item vehicles that are in front of ego vehicle. Specifically, as shown in Fig.\\ref{Fig: safety constraint definition}, the position vector of a social vehicle in ego coordinate system $\\rm {\\textbf{F}}_{ego}$ is $R_i=(x_i, y_i)$. The vehicles whose $x_i \\geq-5$ will be reserved.\n\\end{itemize}\n\n\n\\textbf{Action representation:} As we focus on improving autonomous vehicle's high-level decision-making performance on unsignalized intersection navigating tasks, only longitudinal control of ego vehicle is given by algorithm, the lateral control is assumed to be ideal. \nFor policy network, whose output is designed as a 2-dimension vector ${ \\left ( a^+, a^-\\right )}$. The action of RL module is calculated by $ a = a^+ - a^- $, then normalized to $[0,1]$. Since SUMO simulator uses target speed of vehicle as control command, $a$ is linearly mapped to $(0,9)m\/s$ to be transmitted to the simulator. Our experimental result shows that the separate design of the policy network's output can speed up the training process and stabilize RL agent's performance on interacting with other social vehicles.\n\n\\textbf{Reward design:} Inspired by \\cite{Kai2020Multi,tram2018learning}, the reward value is designed in a vector form. The reward function can be written as $r = [r_{sl}, r_{sm}, r_{sr}, r_c]$, with\n\n\\begin{equation}\n    r_{s \\cdot}=\\left\\{\n    \\begin{array}{rcl}\n    +30 & , & {current \\ task \\  success}\\\\\n    -650 & , & {current \\  task \\ collision}\\\\\n    0 & , & {not \\ current \\ task}\\\\\n    \\end{array} \\right. \n    \\label{eq: reward definition 1}\n\\end{equation}\n\nas specific sub-task reward function, which encourages the ego vehicle to reach the target point and punishes the ego vehicle for colliding with social vehicles, and\n\n\\begin{equation}\n    r_c=\\left\\{\n    \\begin{array}{rcl}\n    -0.15 & , & {t \\leq 0.5 \\cdot T_{max} }\\\\\n    -0.3 & , & {t \\geq 0.5 \\cdot T_{max}}\\\\\n    -50 & , & {time \\ exceed}\\\\\n    \\end{array} \\right.\n    \\label{eq: reward definition 2}\n\\end{equation}\n\nwhere $T_{max}$ refers to maximum time limit of one episode, by which common sub-task reward encourages ego vehicle to improve traffic efficiency.\n\n\\subsubsection{Network Architecture}\n\nGenerally, the safety model takes state vector of RL model as input and generates a safe action as output. Therefore the input and output dimensions of safety model are the same with RL model, which are 33 and 2 respectively. In this paper, safety model is constructed using neural networks. The actual input of network is the clipped state vector by the following rules. The input layer dimension of safety model for the single task is different from the one used in the multi-task framework. In the single route task, in order to estimate the approximate TTC between ego vehicle and the nearest social vehicle, the original state is clipped for a better convergency. The clipped state consists of the information of ego vehicle and kinetics of nearest social vehicles. Therefore, the first 9 dimensions of the original state vector are used as input of safety layer for the single task. For the multi-task experiment, the 3-dimension sub-task code is supposed to be considered. Therefore we concatenate the 3-dimension one-hot task code with the first 9 dimensions of state vector, constitute a 12-dimension tensor. Besides, the safety model has 3 hidden FC layers with 256 nodes each, connected with the ReLU activation function. Note that the $t_0$ value in (\\ref{eq8}) refers to timestep length of the simulation, which is 0.1s in our experiments.\n\nFor the RL model, all the encoders and decoders are formed by FC layers with different number of hidden layers and nodes. Vehicle encoder is formed by $64\\times64$ FC layers, task encoder is a single FC layer and decoder is formed by $256\\times256$ FC layers.\n\n\n\\subsubsection{Results and Analysis}\n\n\\paragraph{Training of the Safety Layer} \n\nAs analyzed in Section \\uppercase\\expandafter{\\romannumeral3}, the definition of safety constraint value is defined using (\\ref{eq:constraint definition}). In the SUMO experiments, we set $\\mu=1, \\eta=0.85$. The safety model is trained using supervised learning. Then the RL agent is trained with the deployment of the safety model to guarantee the safe exploration. Data tuples like ${\\left ( s, s', a, \\bar c, \\bar {c}' \\right) }$ are collected through random policy, which means we collect the state vector and constraint value of two continuous timesteps. Normalized posterior error defined by (\\ref{eq3}) is used as the training loss. We collect 4 million tuples of data for the training procedure. The training curves of different experiments are shown in Fig.\\ref{Fig: SUMO safety layer training curve}.\n\n\n\\begin{figure}[htb]\n\\centering\n\\subfigure[single-task training loss]{\n    \\includegraphics[width=6cm]{SUMO_Left_safety_layer_training_curve.png}\n}\n\\subfigure[multi-task training loss]{\n    \\includegraphics[width=6cm]{SUMO_Multi-task_safety_layer_training_curve.png}\n}\n\n\\caption{Training loss of safety layer}\n\\label{Fig: SUMO safety layer training curve}\n\\end{figure}\n\n\nThe training results show that the loss values of safety layer for single-task and multi-task eventually converge to a minimum value about 2 and 3, respectively. Since the data is collected by a random policy, the value of the converged loss refers to the minimal estimation error of constraint value.\n\n\\paragraph{Performance Metrics}\n\n\nIn our experiment, we use TD3 algorithm to train our agent. Two major conditions are compared, the first one is our proposed multi-task TD3 method, which integrates safety layer, social attention mechanism and multi-task framework together. It is trained on three interaction navigation tasks at the same time. The second one is single-task TD3 method which integrates only safety layer and social attention mechanism, which is trained on three tasks separately. Learning curves are shown in Fig.\\ref{learning_curves}, in which we compare the success rate and cumulative reward for both conditions. Note that only the most challenging turning left task is compared, since social vehicles will yield to ego vehicle with high probability in the going straight and turning right tasks, which makes it too simple to show significant differences. \n\n\n\\begin{figure}[htb]\n\\centering\n\\subfigure[single-task rewards]{\n    \\includegraphics[width=6cm]{single_task_Rewards.png}\n}\n\\subfigure[single-task success rate]{\n    \\includegraphics[width=6cm]{single_task_success_rate.png}\n}\n\\quad\n\\subfigure[multi-task rewards]{\n    \\includegraphics[width=6cm]{multi-task_Rewards.png}\n}\n\\subfigure[multi-task success rate]{\n    \\includegraphics[width=6cm]{multi-task_success_rate.png}\n}\n\\caption{Single\/Multi-task TD3 learning curves}\n\\label{learning_curves}\n\\end{figure}\n\nAs Fig.\\ref{learning_curves} shows, in turning left task, our proposed multi-task TD3 method reaches a competitive performance against the single-task TD3 method. The convergence of the learning curves also shows that our proposed multi-task TD3 method doesn't have a preference for different tasks and is capable of dealing with a composite unsignalized intersection navigating task.\n\n\nIn the proposed framework, the social attention module is capable of improving the interpretability of the decision-making. The attention weights can be visualized by plotting a color map on the vehicles, as shown in Fig.\\ref{Fig: Attention visualization}. In the Fig.\\ref{Fig: attention_1}, since the ego vehicle is not interacting with any environmental vehicle, the policy will assign the attention weights on the ego vehicle. Similarly, as shown in Fig.\\ref{Fig: attention_2}, the policy network puts attention on the encountering environmental vehicle.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[Attention on Ego Vehicle]{\n    \\includegraphics[width=6cm]{attention_1.png}\n    \\label{Fig: attention_1}\n}\n\\subfigure[Attention on Environmental Vehicle]{\n    \\includegraphics[width=6cm]{attention_2.png}\n    \\label{Fig: attention_2}\n}\n\n\\caption{Attention visualization results}\n\\label{Fig: Attention visualization}\n\\end{figure}\n\n\n\n\\paragraph{Comparison Results}\n\nTo prove the superiority of our proposed method, a comparative study is then employed in the testing setting. Success rate and total episode time is the 2 major indexes used to evaluate the performance of the agent. An agent should finish the route as fast as possible and try to prevent collisions as well. The comparison results which contains 1000 episodes of testing are shown in TABLE \\ref{COMPARISON RESULTS}. Our proposed single-task TD3 method outperforms all other methods in turning left task, which has the highest success rate and a relatively less episode time. On the other hand, our proposed multi-task TD3 method shows a competitive result against the single task version, outperforms in traffic efficiency while being with only a slight drop in success rate. Moreover, our proposed multi-task TD3 method fully exceeds the multi-task DQN method from \\cite{Kai2020Multi} in both success rate and traffic efficiency: in turning left task, our proposed method increases by 0.2\\% in success rate and 24\\% in traffic efficiency; in turning right task, our proposed method increases by 3.2\\% in success rate and 11.4\\% in traffic efficiency.\n\nBesides, a random policy is evaluated as a basic baseline method. IDM is compared as a rule-based baseline method. As the result shows, both single and multi-task methods we proposed exceed the performance of IDM in safety. IDM appears to sacrifice safety to perform well in efficiency, which is not acceptable under most circumstances. However in turning left task, IDM shows a tremendous gap in success rate compared with our proposed method, which shows superiority of our methods in dealing with safe interaction problem.\n\n\n\\begin{table}[htp]\n\\caption{COMPARISON RESULTS}\n\\label{COMPARISON RESULTS}\n\\begin{center}\n\\begin{tabular}{c|c|c c | c c}\n\\toprule[2pt]\n\\multirow{2}{*}{Method} & \\multirow{2}{*}{Framework} & \\multicolumn{2}{c|}{Success rate(\\%) $\\uparrow$} & \\multicolumn{2}{c}{Average time(s) $\\downarrow$} \\\\\n\\cline{3-6}\n& & left & right & left & right \\\\\n\\hline\n\\rule{0pt}{10pt}\nSafety+Attention & single-task & \\textbf{98.3} & 99.4 & 10.35 & \\textbf{5.35} \\\\[2pt]\nTD3(ours) & multi-task & 97.1 & \\textbf{99.5} & 9.40 & 6.62\\\\[3pt]\n\n\n\n\\hline\n\\rule{0pt}{10pt}\n\\multirow{2}{*}{SOTA DQN\\cite{Kai2020Multi}} & single-task & 95.5 & 96.9 & 14.29 & 8.09\\\\[2pt]\n & multi-task & 96.9 & 96.3 & 12.37 & 7.47 \\\\[3pt]\n \n\\hline\n\\multirow{1}{*}{Random Policy} & \\diagbox[width=6em] & 52.4 & 91.2 & 15.96 & 12.46 \\\\\n\\hline\n\\multirow{1}{*}{IDM} & \\diagbox[width=6em] & 73.4 & 97.3 & \\textbf{5.51} & 9.68 \\\\\n\\bottomrule[2pt]\n\\end{tabular}\n\\end{center}\n\n\n\\end{table}\n\n\n\n\\paragraph{Ablation Study} \n\nIn order to analyze the impact of the safety layer and attention mechanism on both single-task and multi-task performance more clearly, we have carried out a more detailed experimental analysis. Since the turning left task is the most challenging one, we deploy our ablation study merely in turning left task. The results are shown in TABLE \\ref{ABLATION RESULTS}, single-task and multi-task experiments are compared separately, each row in both parts of the table is correlated. The first row is the method deployed with safety layer and attention module, the second and third row refers to TD3 with attention and TD3 with safety respectively, the fourth row refers to TD3 without any additional module. The fifth row refers to a trained TD3 with attention module deployed a pre-trained safety layer in the evaluation phase.\n\nFirst, we analyze the impact of safety layer. As shown in the first, third and fifth rows of TABLE \\ref{ABLATION RESULTS}, the safety layer increases the success rate in single-task and multi-task experiments. On the other hand, attention mechanism is proved to be effective. The attention mechanism significantly reduces the average time of the task, with a slight loss in the success rate in multi-task experiments. And the combination of safety layer and attention module improves success rate and traffic efficiency at the same time in all experiments. \n\nThe major difference is that in single-task condition, attention mechanism outperforms the safety layer in both safety and efficiency, while in multi-task experiments, the safety layer and attention module increase safety and efficiency respectively. We speculate that it is because the multi-task navigation is not conducive to the convergence of the attention module. Finally, we deploy safety layer directly on an agent trained with attention module. The success rate increases without much loss of average time. According to experimental results, the safety layer method significantly improves the safety while not losing much efficiency. Meanwhile, it is obvious that the design of an appropriate safety constraint is critical in the problem.\n\n\n\\begin{table}[htb]\n\\caption{ABLATION RESULTS}\n\\label{ABLATION RESULTS}\n\\begin{center}\n \\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{c|c| c c }\n\\toprule[2pt]\nFramework & Method & Success rate(\\%) $\\uparrow$ & Average time(s) $\\downarrow$ \\\\\n\\hline\n\\multirow{5}{*}{single-task} \n& TD3 & 92.4 & 12.37 \\\\\n& TD3+Attention  & 94.5 & 13.08 \\\\\n& TD3+Safety & 93.9 & 14.56 \\\\\n& TD3+Safety+Attention & \\textbf{98.3} & \\textbf{10.35} \\\\\n& pre-trained+Safety & 94.9 & 13.13 \\\\\n\\hline\n\\multirow{5}{*}{multi-task} \n& TD3 & 90.9 & 16.48 \\\\\n& TD3+Attention & 91.2 & \\textbf{7.52} \\\\\n& TD3+Safety & 93.2 & 17.04 \\\\\n& TD3+Safety+Attention & \\textbf{97.1} & 9.40 \\\\\n& pre-trained+Safety & 91.7 & 7.73 \\\\\n\\bottomrule[2pt]\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\subsection{CARLA Experiments}\n\nThough the SUMO experiments validate the proposed method preliminarily, we wish to further study the effectiveness of safety exploration in a high-fidelity simulator. Compared to SUMO simulator, CARLA simulator\\cite{Dosovitskiy17} provides abundant adjustable settings for building a high-fidelity vehicles model. Therefore, we employ the CARLA simulator to build the RL environment. In CARLA experiments, some experimental details are various from the one we used in SUMO experiments, which will be further introduced below.\n\n\\subsubsection{Experiment Setup}\n\\label{subsubsec:carla experiment setup}\n\n\n\\begin{figure}[htp]\n    \\centering\n    \\includegraphics[width=8cm]{carla_environment.png}\n    \\caption{The CARLA experiments}\n    \\label{Fig: CARLA environment}\n\\end{figure}\n\n\nIn the CARLA simulator, the Town03 map provides a suitable intersection for the scenario construction, which is shown in Fig.\\ref{Fig: CARLA environment}.\n\nAs shown in the figure, the red bounding box refers to the junction area, and the red vehicle refers to the ego vehicle, which is driven by the RL policy. The target waypoint and the complete route of the ego vehicle are determined according to the task option. All three available routes of the ego vehicle are plotted with green color, as shown in Fig.\\ref{Fig: CARLA environment}. For the ego vehicle control, we adopted a common treatment \\cite{bouton2019safe} to reduce the dimension of action space. The output of the algorithm is mapped to the target speed of the ego vehicle. A PID controller is designed to obtain the vehicle control command according to the target speed.\n\nNote that the traffic flow in CARLA is slightly different from SUMO experiments. In the CARLA simulator, environmental vehicles are driven by the built-in Autopilot, which provides a set of tunable parameters for the behavior of vehicles. Meanwhile, the routes of all environmental vehicles are determined randomly since the target point of each vehicle controlled by Autopilot is determined recursively and randomly. CARLA provides random seed options to set the behavioral pattern of environmental vehicles implicitly. Meanwhile, the traffic flow behavior in CARLA is determined by a set of critical parameters, including upper-speed limit and collision detection probability(CDP). The CDP indicates the possibility of collision detection between the vehicle and a specific vehicle. We set all the environmental vehicles to fully detect each other, and set the CDP to the ego vehicle as an adjustable parameter.\n\nAt the beginning of the training, the CDP is set to 0.5, which means each environmental vehicle has a 50\\% probability to avoid collision with the ego vehicle. The CDP value will be linearly increased. When the agent is trained after 8000 episodes, the CDP is set to 1.0, which means the environmental vehicles will not detect the collision with the ego vehicle and bring the most challenging situation for the RL training. During the testing phase, the CDP is set to 1.0 permanently to make a solid evaluation.\n\n\n\\subsubsection{Reinforcement Learning Setting}\n\nIn CARLA experiments, the RL setting is very similar to the one in the SUMO experiments. The action space definition and the reward function are the same as the SUMO experiments, as shown in (\\ref{eq: reward definition 1}) and (\\ref{eq: reward definition 2}). For the state representation, we divide the velocity vector into the horizontal plane using two orthogonal vectors, which makes the state vector of each environmental vehicle be a vector of 6 dimensions $s_i=[x_i, y_i, v_{x, i}, v_{y, i}, cos(\\alpha_i), sin(\\alpha_i)]$. $v_{x, i}$ and $v_{y, i}$ refer to the component of the velocity vector in x and y directions.\nIn addition, the parameters in (\\ref{eq:constraint definition}) are tuned artificially, and be set to $\\mu=3, \\eta=0.9$. Such settings will keep a larger margin for the safe action correction. In order to make a more understandable deployment of safe action exploration, the safe action modification is only considered when the TTC is less than 4 seconds. Such setup is adopted in both the data collection phase as well as the front propagation phase of the safety layer, including in the RL training and testing.\n\n\n\\subsubsection{Results and Analysis}\n\n\\paragraph{Training of the Safety Layer} \n\n\\begin{figure}[htb]\n    \\centering\n    \\subfigure[single-task training loss]{\n        \\includegraphics[width=6cm]{left_training_loss.png}\n       \n    }\n    \\subfigure[multi-task training loss]{\n        \\includegraphics[width=6cm]{mt_training_loss.png}\n       \n    }\n\\caption{Training loss of safety layer}\n\\label{Fig: CARLA safety layer training curve}\n\\end{figure}\n\n\nThe training curves of the safety layer for the turning left task and multi-task are shown in Fig.\\ref{Fig: CARLA safety layer training curve}. Since the safety layer is a linear approximation of the safety constraint dynamics, the training of the safety layer network can be seen as a system recognition through supervised learning.  Although there are some fluctuations with the training data collected by a random policy, the training losses can converge to a small error. \n\n\n\n\n\n\\paragraph{Performance and Ablation Study}\n\n\\begin{figure}[htb]\n\\centering\n\\subfigure[single-task rewards]{\n    \\includegraphics[width=6cm]{Left_rewards.png}\n}\n\\subfigure[single-task success rate]{\n    \\includegraphics[width=6cm]{Left_success_rate.png}\n}\n\n\\quad\n\n\\subfigure[multi-task rewards]{\n    \\includegraphics[width=6cm]{Multi-task_rewards.png}\n}\n\\subfigure[multi-task success rate]{\n    \\includegraphics[width=6cm]{Multi-task_success_rate.png}\n}\n\\caption{Single\/Multi-task RL learning curves}\n\\label{Fig: CARLA RL learning curves}\n\\end{figure}\n\n\nIn CARLA experiments, the performance of the RL agent is evaluated using the same metrics, which are the success rate and average time.\nThe training curves of the CARLA experiments are shown in Fig.\\ref{Fig: CARLA RL learning curves}. The training curves are obtained by averaging multiple training sessions. In the CARLA experiments, five training sessions are deployed, and a different random seed value is used in each training session.\n\nFrom the training curves, we can see that the original TD3 agent is capable of converging to a stable performance. Meanwhile, the TD3 agent which deployed a safety layer slightly outperforms the TD3 baseline. Besides, the introduction of the social attention module brings a significant oscillation on the training process. The social attention mechanism is supposed to emphasize the interpretability of the RL methods. In our experiments, the combination of safety layer and attention module accelerates the converging speed of the training process.\n\nFurthermore, we analyze the ablation performances for the challenging turning left task and multi-task scenarios. In the testing phase, the environmental agents are controlled by the CARLA Autopilot, and the CDP value is set to 1, which brings the most challenging testing situation. According to the results in TABLE \\ref{Table: CARLA result}, we can see the safety layer improves the success rate for the random policy, and the TD3 agent deployed with a safety layer reaches better performances than the standard TD3 agents on both single-task and multi-task testing. The agent which combines both the attention module and safety layer reaches the best performance in the single-task testing. Meanwhile, in multi-task testing, such an agent reaches the best success rate in exchange for passing efficiency.\n\n\n\n\\begin{table}[htb]\n\\caption{CARLA EXPERIMENT RESULTS}\n\\begin{center}\n\\label{Table: CARLA result}\n \\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{c|c| c c }\n\\toprule[2pt]\nFramework & Method & Success rate(\\%) $\\uparrow$ & Average time(s) $\\downarrow$ \\\\\n\\hline\n\\multirow{5}{*}{single-task} \n& Random & 37.5 & 14.5 \\\\\n& Random+Safety & 39.4 & 15.2 \\\\\n& TD3 & 88.6 & 13.0 \\\\\n& TD3+Safety & 89.1 & 11.7 \\\\\n& TD3+Attention  & 81.8 & 11.6 \\\\\n& TD3+Safety+Attention & \\textbf{89.6} & \\textbf{10.5} \\\\\n\\hline\n\\multirow{5}{*}{multi-task} \n& Random & 63.1 & 13.4 \\\\\n& Random+Safety & 63.6 & 13.5 \\\\\n& TD3 & 88.1 & 11.2 \\\\\n& TD3+Safety & 88.5 & 8.8 \\\\\n& TD3+Attention & 88.0 & \\textbf{8.3} \\\\\n& TD3+Safety+Attention & \\textbf{91.2} & 9.64 \\\\\n\n\\bottomrule[2pt]\n\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nIn summary, according to the testing experiments, the effectiveness of the safety layer is proved. The safety layer constrains the action generated by the RL agent and provides an enhancement on safety with a slight sacrifice of efficiency. In our experiments, we also find that the attention mechanism provides a relatively more aggressive policy exploration.\n\n\\section{CONCLUSIONS}\n\nIn this paper, a multi-task RL framework is proposed to combine attention mechanism and safe exploration with TD3 algorithm. Under such a framework, efficiency and safety are both taken into consideration. A novel design of safety constraint is proposed to represent the collision constraint of the optimization model of the navigation problem. An attention mechanism is also deployed in the framework to improve the interpretability of the algorithm. In order to make an adequate validation for the proposed method, two sets of experiments are deployed in both SUMO and CARLA environments. The SUMO experiment results show that the method achieves very competitive results on the intersection navigation problem, the average time reduces by 24\\% while the success rate exceeds by 0.2\\% compared with the SOTA method. In the CARLA experiments, it is proved that the safety layer is capable of providing the action modification timely, effectively prevent the collision with environmental vehicles. Our proposed multi-task safe RL framework is capable of dealing with different intersection navigation tasks. The testing environment from CARLA experiments provides a convincible benchmark for the autonomous driving research.\n\n\n\n\n \\bibliographystyle{elsarticle-num} \n \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe technology advancements in end-to-end speech-to-text translation (ST) recently allowed to reduce the performance gap with classic cascade solutions combining separate automatic speech recognition (ASR) and machine translation (MT) components. \nHowever, despite its advantages in terms of architectural simplicity and reduced error propagation, direct ST still suffers from drawbacks related to its \nlimited data effectiveness \\cite{sperber2019attentionSHORT}. \nA general problem is that neural approaches are \\textit{per se} data-hungry and the publicly available ST corpora are still orders of magnitude smaller than those released for ASR and MT \\cite{mustc19}. \nThe data demand issue is exacerbated by the fact that, being a higher-level task than ASR and MT,  direct ST  requires higher abstraction capabilities to capture relevant features of the input (audio signals) and learn  the mapping into proper output  representations (texts in the target language). \nLearning this mapping  end-to-end is usually more complex and data demanding than exploiting the intermediate representations of separate, individually trained components.\n\nPrevious solutions to cope with data scarcity focused on two orthogonal aspects: improving the learning process and increasing the training material.\nOn the learning side, \\cite{kano2017structured,weiss2017sequence,berard2018end,anastasopoulos2018tied,bansal2018pre,di2019enhancing} exploited transfer learning \nfrom ASR and MT showing, for instance, that pre-training the ST encoder on ASR data can yield significant \nimprovements.\nOn the data side, the most promising approach is data augmentation, which has been experimented via knowledge distillation from a \nneural MT (NMT) model \\cite{liu2019end}, \nsynthesizing monolingual MT data in the source language\n\\cite{jia2018leveraging},\nmultilingual training \\cite{digangiASRU2019}, \nor \ntranslating monolingual ASR data into the target language \\cite{jia2018leveraging,digangi2019data,liu2018ustc}. \nNevertheless, despite \nsome claims of big industrial players operating in rich data conditions\n\\cite{jia2018leveraging},\ntop results at recent\nshared tasks\n\\cite{liu2018ustc} show that effectively exploiting the scarce training data available still remains a crucial issue to \nreduce the performance gap with cascade ST solutions. \n\nAlong this direction, we propose a general framework \nfor maximizing data exploitation and customizing  an existing ST model to each incoming translation request at inference time.\nIn a nutshell, given a generic model  $M_{g}$ and \nan ST data pool $D$, each translation request $r$\nis handled by a two-step process. First, a set of (\\textit{audio}, \\textit{translation}) pairs is retrieved  from $D$ based on the similarity between their  audio element and \n$r$.\nThen, the retrieved pairs are used to adapt $M_{g}$ via fine-tuning.\nThe \nunderlying \nintuition is that the similarity of the new samples with the input audio can be used at run-time to \noverfit $M_{g}$ \nto samples similar to $r$, and influence its behaviour towards a better translation.\n\nWe explore this idea in \ndifferent scenarios \nconsidering different language directions and experimenting \nin intra-, multi- and cross-domain adaptation.\nOur results show that, compared to \nstatic ST models, \ninstance-based on-the-fly adaptation \n yields variable but coherent improvements, with larger gains in cross-domain scenarios where the \nmismatch between the training and test domains makes \nST \nmore \nchallenging.\n\n\\section{Direct Speech Translation}\n\\label{sec:task}\n\nIn direct speech translation, a single neural model is trained end-to-end on the speech-to-text translation task. Given an input audio segment $\\mathbf{X}$ representing a speech in a source language $e$, and an output text $\\mathbf{Y}$ representing the translation of $\\mathbf{X}$ in a target language $f$, a direct ST model is trained by optimizing the log-likelihood \nfunction in Equation 1, \nwhere $B$ is the size of a batch, $l_b$ is the length of the target sequence at position $b$, and $\\mathbf{\\theta}$ is the vector of model's parameters.\n\n\\begin{equation}\n    L = -\\sum_{b=0}^B \\sum_{i=0}^{l_b} y_{ib} \\log(p(\\tilde{y}_{ib} \\vert \\mathbf{X}, y_{< i,b}; \\mathbf{\\theta}))\n\\end{equation}\n\nModels for this task have a sequence-to-sequence architecture \\cite{sutskever2014sequence} with at least one encoder that processes the audio input, and one decoder that generates the output, one token at a time, in an autoregressive manner. \nIn this work, we  use  S-Transformer \\cite{digangi2019adapting}, an adaptation of  Transformer~\\cite{vaswani2017attention} to the ST task. \nIn addition to the original Transformer,  S-Transformer elaborates the input spectrograms with\n\\textit{ad-hoc} layers. The input is first processed by two stacked 2D CNNs with stride (2, 2), which also reduce the input sequence length by a factor of $4$. Then, the output of the second CNN is fed to a stack of two 2D Self-Attention~\\cite{dong2018speech}. The goal of the 2D Self-Attention is to model the bi-dimensional dependencies along the spectrogram's time and frequency dimensions. \n2D Self-attention layers process the input with 2D CNNs and compute attention along both matrix directions. \nAll CNNs are followed by batch normalization~\\cite{ioffe2015batch} and ReLU nonlinearity. Moreover,  to  focus the  encoder  on  short-range dependencies, a distance penalty mechanism is added in  every  self-attention layer of the encoder. Given a position $i$ in the query vector, and a position $j$ in the key vector, with $i \\neq j$ we compute $\\textit{pen}=\\log(\\vert i - j \\vert)$ and subtract \\textit{pen} from the attention scores before softmax normalization.\n\n\\section{Instance-based Model Adaptation}\n\\label{sec:method}\n\nAlgorithm~\\ref{algo:uniad} illustrates our \ninstance-based model adaptation procedure. Its goal is to improve the performance of a pre-trained ST model $M_{g}$ by fine-tuning it at inference time on (\\textit{audio}, \\textit{translation}) pairs in which the audio is similar to the input translation request $r$. These pairs are retrieved from a data pool $D$, which can either be \nthe same training set used for $M_{g}$ or a new dataset.\nIn the former case, instance-based\nadaptation aims to maximize the exploitation of the training data.\nIn the latter case, the goal is to exploit newly available data to also cover new domains.\nOur experiments ($\\S\\ref{sec:experiments}$) will address both the scenarios.\n\n\\begin{algorithm}[h!]\n\\small\n\\caption{Instance-based Model Adaptation (IMA)}\\label{algo:uniad}\n\n\\begin{algorithmic}[1]\n\\LineComment $M_{g}$: generic ST model \n\\LineComment $M_{r}$: adapted ST model \n\\LineComment $D$: ST data pool\n\\LineComment $r$: translation request\n\\LineComment $\\tau$: similarity threshold\n\\LineComment $D_{r}$: $\\{(a_1, t_1), ..., (a_n, t_n)\\}$ retrieved (audio, translation) pairs\n\\LineComment $t^{*}$: translated segment\n\\Procedure{IMA($M_{g}$, $D$,  $r$, $\\tau$)}{}\n    \\LineComment Local copy of the generic model\n    \\State $M'_{g}$:=$M_{g}$\n    \\LineComment Instance selection\n    \\State $D_{r}$:={\\bf Retrieve}($r$, $D$, $\\tau$)\n    \\If{$D_{r}$  $\\not=$  $\\varnothing$}\n        \\LineComment Model optimization\n        \\State $M_{r}$:={\\bf Adapt}($M'_{g}$, $D_{r}$)\n    \\Else \n        \\State $M_{r}$:=$M'_{g}$\n    \\EndIf\n    \\LineComment Translate the segment with the adapted ST model\n    \\State $t^{*}$:={\\bf Translate}($M_{r}$, ${r}$)\n    \n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\noindent\n\\textbf{Data pool.}\n$D$ consists of \\textit{(audio, translation)} pairs, in which \nthe audio element is also used as a \nretrieval key for\nthe pair. \nFor our experiments, the audio segments are stored \neither  \\textit{i)} as a spectrogram $S$ with $N$ time frames and $k$ features (Raw Features in Table \\ref{tab:intra}), or \\textit{ii)} as a function $E(S)$ obtained by processing $S$ with the model's encoder (Encoder Features). The generated segments are stored in order to be retrieved during translation.\n\n\\noindent\n\\textbf{Similarity.}\nThe similarity between the query audio segment $r$ and the \naudio segments in $D$\nis computed as the cosine similarity between the pairs of vectors \n$(z_r, z_1),  \\dots, (z_r, z_n) \\in R^{k}$, \nwhere $k$ is the number of features of the chosen segment representations. Each $z_i$ is obtained by summing all the time frames of its sequence along the time axis.\nThe advantage of this similarity is its applicability in a direct ST scenario, where no intermediate transcription step is involved.\n\n\\noindent\n\\textbf{Retrieval.} \nThe retrieval procedure receives as argument the translation request $r$, \nthe data pool\n$D$ and a similarity threshold $\\tau$. It returns the set of (\\textit{audio}, \\textit{translation}) pairs ($D_{r}$) for which the similarity of the audio element with $r$ is above $\\tau$.\n\n\\noindent\n\\textbf{Adaptation.} \nIf $D_{r}$ is not empty, the generic model $M_g$ is  fine-tuned for $e$ epochs on the top $n$ samples to obtain the adapted model $M_r$ used to translate $r$. In our experiments, $e$ and $n$ are fixed hyperparameters. After translating $r$, the adapted model is discarded \nso that, for the next input query, the process restarts from the initial generic  model $M_g$.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Datasets}\n\\label{ssec:dataset}\n\nWe use two datasets. One is MuST-C \\cite{mustc19}, a multilingual ST corpus containing English speech (TED Talks) translated into 8 European languages. Data size ranges from 385 hours for English$\\rightarrow$Portuguese to 504 hours for English$\\rightarrow$Spanish. \nThe other corpus is How2 \\cite{sanabria2018how2}, a multimedia corpus for English$\\rightarrow$Portuguese also including  ST data (300 hours). \nIn both corpora, the speech segments are in the form of log MEL filterbanks with time width 25ms and step of 10ms. \n\nA comparison between the target side of the En-Pt section of MuST-C and How2  shows that they have a different level of \ntext repetitiveness (How2 has a repetition rate \\cite{cettolo2014repetition} that is 40\\% higher) and vocabulary overlap (27\\% of the MuST-C terms appear in How2, while 48\\% of the How2 terms are also in MuST-C).\nOther differences in terms \nbackground noise \nand number of non-native speakers (both higher in MuST-C) suggest  \nthat the How2 data are in general easier to handle \nfor ST training\/adaptation.\nDepending on the selected test set, we hence expect variable \ngains over the static ST models. \n\n\n\\subsection{Settings}\nWe trained S-Transformer on all the datasets with the following hyper-parameters: 2D CNNs have kernels of size $3\\times 3$ and stride (2, 2), 2D self-attentions have internal 2D CNNs with $4$ output channels (and thus $4$ heads in multi-head attention), and $64$ output channels in the last layer. Transformer layers have size $512$ with $8$ heads in multi-head attention and $1024$ units in the hidden feed-forward sub-layers. Dropout is set to 0.1 after each layer.\nFor training, we used the Adam optimizer \\cite{kingma2014adam} with noam decay \\cite{vaswani2017attention} using initial learning rate 0.0003, $4000$ warm-up steps and maximum learning of $0.001$. The loss we used is cross-entropy with label smoothing \\cite{szegedy2016rethinking} set to $0.1$. The batch size is of 4 segments, but we trained on 4 GPUs NVIDIA K80 and accumulated gradients for $16$ batches.\nTarget texts are split at character level. The results are \ncomputed using the BLEU score \\cite{papineni2002bleu} at word level.\n\n\\subsection{Experiments}\n\\label{ssec:experiments}\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{l|c|c|c}\n  \\multicolumn{4}{c}{\\textbf{\\textit{Intra-Domain}}}\\\\\\hline\n& \\textbf{Baseline} & \\textbf{ Raw Features} & \\textbf{Encoder Features} \\\\\\hline\nDe & 17.0 & 16.9 & \\textbf{17.3} \\\\\nEs & 21.5 & 21.5 & \\textbf{22.0} \\\\\nFr & 27.0 & 27.1 & \\textbf{27.4}\\\\\nIt & 17.5 & 17.8 & \\textbf{18.0}\\\\\nNl & 21.8 & 21.9 & \\textbf{22.0} \\\\\nPt & 21.5 & 21.4 & \\textbf{21.7} \\\\\nRo & 16.4 & 16.4 & \\textbf{16.8}\\\\\nRu  & 12.2 & 12.3 & \\textbf{12.4}\\\\\\hline\nHow2 & 39.4 & 39.9 & \\textbf{40.1} \\\\\n\\end{tabular}\n\\caption{BLEU results on MuST-C and How2 in the intra-domain scenario. The retrieval is based on either MEL filterbanks or the encoder's output representations. }\n\n\\label{tab:intra}\n\\end{table}\n\nWe evaluate our instance-based model adaptation approach in three scenarios. In the first scenario (``\\textbf{intra-domain}''), the data pool used for retrieval ($D$) is the same \ncorpus used to train the initial ST model $M_{g}$.\nThese experiments aim to evaluate whether\ninstance adaptation helps \nto make better use of the training data.\nIn the second scenario (``\\textbf{multi-domain}''), \n$M_{g}$ is trained on data from two domains ($D1$+$D2$) and the goal is \nto maximize \nperformance on both. In this case, \nthe adaptation is performed using as a data pool either the \ndomain-specific material from the same domain of the query $r$, or the whole data from the two domains.\nIn the last scenario (``\\textbf{cross-domain}''),\n$M_{g}$ is trained  on data from one domain only, and it has to be adapted to a new domain. We consider two variants of this scenario. \nIn the first variant, \nan in-domain data pool from the same domain of the test set is available for retrieval.\nIn the second variant, \nthe data pool contains only the original, out-of-domain training data. \nThe latter variant, in which \n$M_{g}$ has to be adapted to unseen test data by only exploiting out-of-domain material, represents the hardest condition from an on-field \ndeployment standpoint.\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{l|c|c|c}\n &\\textbf{1 Epoch} & \\textbf{3 Epochs} & \\textbf{5 Epochs} \\\\\\hline\nDe  & 15.8 & 13.1 & \\textit{10.0} \\\\\nEs & 21.0 & 19.8 & \\textit{19.0} \\\\\nFr &  26.3 & 22.5 & \\textit{18.8} \\\\\nIt &  17.0 & 15.0 & \\textit{13.2} \\\\\nNl &  21.5 & 20.1 & \\textit{18.0} \\\\\nPt & 21.2 & 19.6 & \\textit{17.7} \\\\\nRo &  15.8 & 13.7 & \\textit{11.4}\\\\\nRu &  11.9 & 9.7 & \\textit{7.3} \\\\\n\\end{tabular}\n\\caption{BLEU results on MuST-C running the adaptation for 1, 3 and 5 epochs on the \nleast similar pair retrieved from $D$.}\n\n\\label{tab:firstlast}\n\\end{table}\n\nFor each setting, we perform hyperparameter search in the validation set, then\nthe best selection is applied on the test set. \nWe perform \ninstance-based \nadaptation with the Adam optimizer \\cite{kingma2014adam} and choose the best set of hyperparameters among learning rates=$\\{1, 2, 3\\}\\times 10^{-\\{3,4,5\\}}$, number of retrieved samples = $\\{1, 5, 10\\}$, and number of tuning epochs = $\\{1, 3, 5\\}$. Additionally, we filter out the retrieved samples whose cosine similarity score is below a threshold \n$\\tau$.\nAfter an initial exploration, we found out that a threshold \n$\\tau=0.5$\nallows the systems to keep the best performance while reducing the tuning time. As an additional note, we found that the SGD optimizer does not work as well as the Adam optimizer, particularly for the \nmulti\/cross-domain adaptation experiments.\n\n\\section{Results}\n\\label{sec:results}\n\n\n\n\\textbf{Intra-domain.}\nThe results of the intra-domain experiments are \nshown\nin Table \\ref{tab:intra}. In general, the performance  \non MuST-C is lower than \non How2.\nAs pointed out in $\\S\\ref{ssec:dataset}$, despite the smaller \nsize of the training corpus, \nthe higher repetitiveness of How2 creates\na favourable evaluation condition. \nInstance-based adaptation, however, \n provides \nsmall but coherent  improvements on all the language pairs and on both corpora (from 0.2 to 0.5 for MuST-C and 0.7 for How2). \nSince the Encoder Features are slightly better than the Raw Features, they will be used in the rest of the experiments.\nTo better understand the\neffectiveness of our approach, Table \\ref{tab:firstlast} \nshows the impact of adapting on the \nleast similar pair retrieved from the pool, for different numbers of epochs and for each  language direction of MuST-C. These results \nare always worse\nthan the \nbaseline and, by  increasing the number of epochs, they \ndeteriorate  up to  $-7.5$ BLEU points on Fr with 5 epochs.\nThis suggests that  instance-based adaptation \nis sensitive to the quality (i.e. the similarity) of the \nretrieved material and that our approach is able to identify pairs that are useful to the model, \nresulting in variable performance \ngains in all the experiments.\n\n\n\\noindent\\textbf{Multi-domain.}\nTo evaluate \ninstance-based adaptation \nin the \nmulti-domain scenario,\nwe trained our initial model ($M_{g}$) on the concatenation of the En-Pt data from MuST-C and the How2 data. The results presented in \nlines 1-3\nof Table \\ref{tab:cross-domain} \nindicate that using more data is beneficial for both the generic (+1.2 on the MuST-C baseline \nreported in Table \\ref{tab:intra}\nand +1.6 on \nHow2) and the instance-based adaptation (+2.1 for MuST-C and +2.4 for How2). \nThis can be explained by the fact that, when a model has been trained on larger and more diverse data,  \nit is \nstronger due to its higher generalization capability.\nIn this case, instance-based adaptation can account for the domain shift without \nperformance loss in the initial domains.\n\n\\noindent\n\\textbf{Cross-domain.}\nAs mentioned in $\\S$\\ref{ssec:experiments}, we also run our domain-adaptation experiments by training the ST model in one domain and testing it on the other. The similar pairs can be retrieved either from\nthe same \ndomain of the test set \nor from\nthe training data only.\nIn general, when \ntraining and test \ndata\ncome from different domains (Table \\ref{tab:cross-domain}, line 4), \nthe \nnon-adapted models show a significant drop in performance (-11.6 BLEU points for the MuST-C test set and -25.3 for How2). \nRetrieving from the same domain (line 5) helps with gains over the static model of 1.2 BLEU points for the MuST-C test set and +7.5 for How2. These results are promising but still far from the baseline values reported in Table \\ref{tab:intra}. \nHowever, it\nis important to remark that \nour baselines have access to the in-domain data in advance, so they work in a more favorable condition. For the sake of comparison, we fine-tuned the \nbaseline models on the incoming pool of in-domain data, but this results in models with performance comparable to the baselines for the new domain without pre-training.\nRetrieving similar pairs from a different domain (line 6) is extremely difficult, in particular considering the differences between the two datasets (see $\\S$ \\ref{ssec:dataset}).\nAlso in this case, however,\ninstance selection is able to leverage the training data to produce translations that are slightly better than those obtained from the static system ($+0.6$ on MuST-C and $+0.3$ on How2).\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{l|c|c|c|c|c}\n& \\textbf{Train}  & \\textbf{Test}  &  \\textbf{Pool}  & \\footnotesize{\\textbf{D1: How2 }}&  \\footnotesize{\\textbf{D1: MuST-C}} \\\\\n &  &   &    & \\footnotesize{\\textbf{D2: MuST-C}}&  \\footnotesize{\\textbf{D2: How2}} \\\\\\hline\n  \\multicolumn{6}{c}{\\textbf{\\textit{Multi-Domain}}}\\\\\\hline\n1 & D1 + D2 & D1  & -  &  22.7 & 41.0  \\\\\n2 &  D1 + D2 & D1  & D1  &  \\textbf{23.6} & \\textbf{41.8}  \\\\\n3 &  D1 + D2 & D1  & D1+D2  &  \\textbf{23.5} & \\textbf{41.8}  \\\\\\hline \\hline\n  \\multicolumn{6}{c}{\\textbf{\\textit{Cross-Domain}}}\\\\\\hline\n4 & D1  & D2  & - & 9.90 & 14.1   \\\\\n5& D1  & D2  & D2 & \\textbf{11.1} & \\textbf{21.6}   \\\\\n6& D1  & D2  & D1 & 10.5 & 14.4  \\\\\n\\end{tabular}\n\\caption{Results on mixed- and cross-domain experiments.\n\\label{tab:cross-domain}\n\\end{table}\n\n\\section{Related works and open issues}\n\\label{sec:relworks}\n\nThe idea of instance-based adaptation exploiting information retrieval dates back to \\cite{mahajan1999improved}, in which it was \ndeveloped to dynamically customize a language model for ASR. In statistical MT, it was applied for the same purpose in \\cite{eck-etal-2004-language,zhao-etal-2004-language} and later, in  \\cite{Hildebrand:2005SHORT}, for domain adaptation.\nMore recently, different variants of the approach have been proposed for neural MT \\cite{farajian2017multi,li-etal-2018-one,zhang-etal-2018-guiding,nonparametric19} and  MT-related tasks  \\cite{chatterjee-etal-2017-online}. \nHowever, differently from ST, all the previously explored translation scenarios involve managing \\textit{textual data} for \\textit{domain adaptation} purposes. These aspects mark the main differences with our work, which, to the best of our knowledge, is the first attempt to apply instance-based adaptation to cope with data paucity in a speech-related task.\n\nOn this front, it is worth remarking that the challenges posed by speech input data can not be addressed with the mere application of  previous text-based techniques. Indeed, differently from MT that only deals with \\textit{what} a sentence says in terms of content, the ST (or ASR) input has a more complex nature. \nTogether with the conveyed meaning, it also provides  information about the acoustic properties of the spoken utterances (e.g. speaker's voice, recording conditions) describing \\textit{how} meaning is expressed. \nThis adds additional challenges to instance-based adaptation, where fine-tuning can exploit the retrieval of ``similar'' instances from the  point of view of the audio (e.g. a similar voice), the content (a similar meaning), or both. This paper provides a first exploration along this direction, in which the two aspects are not decoupled. A strand of future works will focus on better understanding and balancing their contribution, as well as dynamically leveraging the notion of similarity (e.g. by a similarity-informed setting of the model's hyper-parameters).\n\nThe deeper exploration of different domain-adaptation strategies represents another promising strand of research. In principle, besides maximizing data exploitation in scarce resource conditions, instance-based adaptation \nwould allow to simultaneously  manage multiple domains with one single ST system. This is a crucial feature from the industrial standpoint, where training and maintaining domain-dedicated models is costly and time-consuming. We demonstrated the feasibility of the approach with initial experiments but several technical \naspects still remain to be explored \n(e.g. whether to ``reset'' the model after each update to preserve its performance on all the domains\nor to keep the updated one so to favour knowledge transfer across domains when processing new translation requests).\n\n\\section{Conclusions}\nWe proposed a method to maximize data exploitation in the scarce resource conditions posed by end-to-end ST. \nThe method is based on \nfine-tuning \nat inference time \na pre-trained model on a set of instances retrieved from the original training data or from an external corpus based on their similarity with the input audio.\nWe evaluated our approach in different data conditions (different languages, in\/out-of-domain adaptation) reporting coherent improvements over \ngeneric ST systems and highlighting promising \nresearch directions for the future.\n\n\n\n\n\\section*{Acknowledgements}\nThis  work  is  part  of  a  project  financially  supported  by  an Amazon AWS ML Grant.\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\chapter{Wavefunction integrals}\n\\label{app:wavefunctions}\n\nIn this appendix we detail the evaluation of the explicit momentum space wavefunction integrals that played a key role in our disccussion of the classical limit in chapters~\\ref{chap:pointParticles} and~\\ref{chap:impulse}.\n\nWe begin with the single particle momentum space wavefunction normalisation in \\eqn~\\eqref{eqn:LinearExponential}. To determine this we must compute the following integral:\n\\begin{equation}\nm^{-2}\\int \\df(p)\\;\\exp\\biggl[-\\frac{2 p\\cdot u}{m\\xi}\\biggr]\\,.\n\\end{equation}\nLet us parametrise the on-shell phase space in a similar manner to equation~\\eqref{eqn:qbRapidityParametrisation}, but appropriate for timelike momenta, writing\n\\begin{equation}\np^\\mu = E_p\\bigl(\\cosh\\zeta,\\,\\sinh\\zeta\\sin\\theta\\cos\\phi,\\,\\sinh\\zeta\\sin\\theta\\sin\\phi,\n\\sinh\\zeta\\cos\\theta\\bigr)\\\n\\label{eqn:Parametrization}               \n\\end{equation}\nso that\n\\begin{equation}\n\\begin{aligned}\n\\df(p) &= (2\\pi)^{-3}\\dd E_p d\\zeta d\\Omega_2\\,\\del(E_p^2-m^2)\\Theta(E_p)\\,\nE_p^3 \\sinh^2\\zeta\n\\\\&=\n(2\\pi)^{-3}\\dd E_p d\\zeta d\\theta d\\phi \\,\\del(E_p^2-m^2)\\Theta(E_p)\\,\nE_p^3 \\sinh^2\\zeta\\sin\\theta\\,.\n\\end{aligned}\n\\end{equation}\nPerforming the $E_p$ integration, we obtain\n\\begin{equation}\n\\df(p)  \\rightarrow \\frac{m^2}{2(2\\pi)^{3}} \nd\\zeta d\\theta d\\phi\\,\\sinh^2\\!\\zeta\\sin\\theta\\,,\n\\label{Measure}\n\\end{equation}\nalong with $E_p=m$ in the integrand.  The integral must be a Lorentz-invariant\nfunction of $u$; as the only available Lorentz invariant is $u^2=1$, we conclude\nthat the result must be a function of $\\xi$ alone.  We can compute it in the rest\nframe of $u$, where our desired integral is\n\\begin{equation}\n\\frac12 \\! \\int_0^\\infty \\!\\dd\\zeta\\,\\sinh^2\\!\\zeta\\!\\int_0^\\pi\\! \\dd\\theta\\,\\sin\\theta\\!\n\\int_0^{2\\pi} \\!\\dd\\phi\\, \\exp\\biggl[-\\frac{2\\cosh\\zeta}{\\xi}\\biggr]\n=\\frac1{2(2\\pi)^2} \\xi\\, K_1(2\/\\xi)\\,,\\label{eqn:normalisationIntegral}\n\\end{equation}\nwhere $K_1$ is a modified Bessel function of the second kind.  The normalisation\ncondition~(\\ref{eqn:WavefunctionNormalization}) then yields\n\\begin{equation}\n\\frac{2\\sqrt2\\pi}{\\xi^{1\/2} K_1^{1\/2}(2\/\\xi)}\n\\end{equation}\nfor the wavefunction's normalisation.\n\nNext, we compute $\\langle p^\\mu\\rangle$.  Lorentz invariance implies that the expectation\nvalue must be proportional to $u^\\mu$;\nagain computing in the rest frame, we find that\n\\begin{equation}\n\\langle p^\\mu\\rangle = m u^\\mu \\,\\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,.\n\\end{equation}\nThe phase-space measure fixes $\\langle p^2\\rangle = m^2$, so we conclude that\n\\begin{equation}\n\\begin{aligned}\n\\frac{\\sigma^2(p)}{\\langle p^2\\rangle} =\n1-\\frac{\\langle p\\rangle^2}{\\langle p^2\\rangle} &= 1-\\frac{K_2^2(2\/\\xi)}{K_1^2(2\/\\xi)}\n\\\\&=-\\frac32\\xi+\\Ord(\\xi^2)\\,,\n\\end{aligned}\n\\end{equation}\nwhere as $\\xi\\rightarrow 0$ we have used the asymptotic approximations\n\\[\nK_1(2\/\\xi) &= \\sqrt{\\frac{\\pi \\xi}{4}}\\, \\exp\\left[-\\frac{2}{\\xi}\\right]\\left(1 + \\frac{3\\xi}{16} + \\mathcal{O}(\\xi^2)\\right),\\\\ K_2(2\/\\xi) &= \\sqrt{\\frac{\\pi \\xi}{4}}\\, \\exp\\left[-\\frac{2}{\\xi}\\right] \\left(1 + \\frac{15\\xi}{16} + \\mathcal{O}(\\xi^2)\\right).\n\\]\nFor the single particle case it remains to compute the double expectation value in equation~\\eqref{eqn:doubleMomExp}. We can again apply Lorentz invariance, which dictates that\n\\begin{equation}\n\\langle p^\\mu p^\\nu \\rangle = A u^\\mu u^\\nu + B \\eta^{\\mu\\nu}\\,.\\label{eqn:doubleExpVal}\n\\end{equation}\nContracting with the velocity leads to\n\\begin{equation}\nA + B = \\frac{\\mathcal{N}^2}{m^2}\\! \\int\\! \\df(p)\\, (u\\cdot p)^2\\, \\exp\\left[-\\frac{2 u\\cdot p}{m \\xi}\\right] = m^2 + \\frac32 m^2 \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,,\n\\end{equation}\nwhere we used the result in equation~\\eqref{eqn:normalisationIntegral}. Combining this constraint with the trace of~\\eqref{eqn:doubleExpVal} is then enough to determine the coefficients as\n\\begin{equation}\nA = m^2 + 2 m^2 \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,, \\qquad B = -\\frac{m^2}{2} \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,,\n\\end{equation}\nyielding the result in equation~\\eqref{eqn:doubleMomExp}.\n\nOur discussion of point-particle scattering in section~\\ref{sec:classicalLimit} hinged upon the integral \n\\begin{equation}\nT(\\qb) = \\frac{\\Norm^2}{\\hbar m^3}\\exp\\biggl[-\\frac{\\hbar \\qb\\cdot u}{m\\xi}\\biggr]\\int \\df(p)\\;\n\\del(2 p\\cdot\\qb\/m+\\hbar\\qb^2\/m)\\,\\exp\\biggl[-\\frac{2 p\\cdot u}{m\\xi}\\biggr]\\,.\n\\end{equation}\nThis integral is dimensionless, and can depend only on two Lorentz invariants,\n$\\qb\\cdot u$ and $\\qb^2$, along with $\\xi$.  \nIt is convenient to write it as a function of two\ndimensionless variables built out of these invariants,\n\\begin{equation}\n\\omega \\equiv \\frac{\\qb\\cdot u}{\\sqrt{-\\qb^2}}\\,,\n\\qquad \\tau \\equiv \\frac{\\hbar\\sqrt{-\\qb^2}}{2m}\\,.\n\\end{equation}\nWe again work in the rest frame of $u^\\mu$, and without loss of generality,\nchoose the $z$-axis of the $p$ integration to lie along the direction of\n$\\v{\\qb}$.  The only components that appear in the integral are then\n$\\qb^0$ and $\\qb^z$; after integration, we can obtain the dependence on\n$\\tau$ and $\\omega$ via the replacements\n\\begin{equation}\n\\qb^0\\rightarrow \\frac{2 m\\omega\\tau}{\\hbar}\\,,\n\\qquad \\qb^z \\rightarrow \\frac{2m\\sqrt{1+\\omega^2}\\tau}{\\hbar}\\,;\n\\end{equation}\nhence,\n\\begin{equation}\n-\\!\\qb^2 \\rightarrow \\frac{4m^2\\tau^2}{\\hbar^2}\\,.\n\\end{equation}\nUsing the measure~(\\ref{Measure}), we find that\n\\begin{equation}\n\\begin{aligned}\nT_1(\\wn q) &= \\frac1{2\\pi\\,\\hbar m\\,\\xi K_1(2\/\\xi)} \\exp\\biggl[-\\frac{\\hbar \\qb^0}{m\\xi}\\biggr]\n\\\\&\\qquad\\times \\int_0^\\infty \\!\\d\\zeta\\,\\sinh^2\\!\\zeta\\! \\int_0^\\pi\\! \\d\\theta\\,\\sin\\theta\\!\n\\int_0^{2\\pi}\\! \\d\\phi\\, \\exp\\biggl[-\\frac{2\\cosh\\zeta}{\\xi}\\biggr]\n\\\\& \\qquad\\qquad\\qquad \\times \\del(2 \\qb^0 \\cosh\\zeta - 2\\qb^z \\sinh\\zeta \\cos\\theta\n+\\hbar\\qb^2\/m)\\,.\n\\end{aligned}\n\\end{equation}\nThe $\\phi$ integral is trivial, and we can use the delta function to do the\n$\\theta$ integral:\n\\newcommand{\\textrm{cosech}}{\\textrm{cosech}}\n\\begin{equation}\n\\begin{aligned}\nT_1(\\wn q) &= \\frac1{2\\hbar m\\,\\qb^z\\,\\xi K_1(2\/\\xi)} \\exp\\biggl[-\\frac{\\hbar \\qb^0}{m\\xi}\\biggr]\n\\int_0^\\infty d\\zeta\\;\\sinh\\zeta\\;\n\\exp\\biggl[-2\\frac{\\cosh\\zeta}{\\xi}\\biggr]\n\\\\& \\qquad\\qquad\\qquad\\times \n\\Theta\\bigl(1+ \\qb^0 \\coth\\zeta\/\\qb^z +\\hbar\\qb^2\\textrm{cosech}\\,\\zeta\/(2 m\\qb^z)\\bigr)\\,\n\\\\& \\qquad\\qquad\\qquad\\times \n\\Theta\\bigl(1-\\qb^0 \\coth\\zeta\/\\qb^z -\\hbar\\qb^2\\textrm{cosech}\\,\\zeta\/(2 m\\qb^z)\\bigr)\\,.\n\\end{aligned}\n\\end{equation}\nIn the $\\hbar\\rightarrow 0$ limit, the first theta function will have no effect,\neven with $\\qb^2<0$.  Changing variables to $w=\\cosh\\zeta$, \nthe second theta function will impose the constraint\n\\begin{equation}\nw \\ge \\frac{\\qb^z\\sqrt{1-\\hbar^2\\qb^2\/(4m^2)}}{\\sqrt{-\\qb^2}} -\\frac{\\hbar\\qb^0}{2m}\\,.\n\\end{equation}\nIn terms of $\\omega$ and $\\tau$,\nthis constraint is\n\\begin{equation}\nw \\ge \\sqrt{1+\\omega^2}\\sqrt{1+\\tau^2}-\\omega\\tau\\,.\n\\end{equation}\nUp to corrections of $\\Ord(\\hbar)$, the right-hand side is greater than 1,\nand so becomes the lower limit of integration.  The result for the integral is then\n\\begin{equation}\n\\begin{aligned}\nT(\\qb) &=\\frac1{8 m^2\\sqrt{1+\\omega^2}\\tau\\,K_1(2\/\\xi)} \n\\exp\\biggl[-\\frac{2\\omega\\tau}{\\xi}\\biggr]\\\\\n&\\qquad\\qquad\\qquad \\times\\exp\\biggl[-\\frac2{\\xi}\\Bigl(\\sqrt{1+\\omega^2}\\sqrt{1+\\tau^2}-\\omega\\tau\\Bigr)\\biggr]\n\\\\&=\n\\frac1{4 \\hbar m\\sqrt{(\\qb\\cdot u)^2-\\qb^2}\\,K_1(2\/\\xi)} \\\\\n&\\qquad\\qquad\\qquad \\times \\exp\\biggl[-\\frac2{\\xi}\\frac{\\sqrt{(\\qb\\cdot u)^2-\\qb^2}}{\\sqrt{-\\qb^2}}\n\\sqrt{1-\\hbar^2\\qb^2\/(4m^2)}\\biggr]\n\\,,\n\\end{aligned}\n\\end{equation}\nwhich is the result listed in equation~\\eqref{eqn:TIntegral}.\n\\chapter{Worldline perturbation theory}\n\\label{app:worldlines}\n\nThroughout part~\\ref{part:observables} we computed on-shell classical observables from scattering amplitudes. We justified our results by their agreement with iterative solutions to the appropriate classical equations of motion, as shown in detail in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}. To provide a flavour of these methods, let us calculate the LO current for gravitational radiation due to the electromagnetic scattering of Reissner--Nordstr\\\"{o}m black holes, which we studied using the double copy in section~\\ref{sec:inelasticBHscatter}. \n\nClassically, the emitted gravitational radiation will satisfy the linearised Einstein--Maxwell field equation, which for the trace-reversed perturbation in de--Donder gauge is\n\\begin{equation}\n\\partial^2\\bar h^{\\mu\\nu}(x) = T^{\\mu\\nu}(x) = \\frac{\\kappa}{2}\\left(T_{\\rm pp}^{\\mu\\nu}(x) + T_{\\rm EM}^{\\mu\\nu}(x)\\right).\n\\end{equation}\nThe electromagnetic field strength tensor $T^{\\rm EM}_{\\mu\\nu}$ is that in \\eqn~\\eqref{eqn:EMfieldStrength}, while in gravity\n\\begin{equation}\nT_{\\rm pp}^{\\mu\\nu}(x) = \\sum_{\\alpha=1}^n m_\\alpha \\int\\! \\frac{d\\tau_\\alpha}{\\sqrt{g}}\\, u^\\mu_\\alpha(\\tau)u^\\nu_\\alpha(\\tau)\\, \\delta^{(D)}(x-x_\\alpha(\\tau))\\label{eqn:ppGravTensor}\n\\end{equation}\nfor $n$ particles, where $g = -\\det(g_{\\mu\\nu})$.  We are seeking perturbative solutions, and therefore expand worldline quantities in the coupling:\n\\[\nx_\\alpha^\\mu(\\tau_\\alpha) &= b_\\alpha^\\mu+u_\\alpha\\tau_\\alpha + \\Delta^{(1)}x_\\alpha^\\mu(\\tau_\\alpha) + \\Delta^{(2)} x_\\alpha^\\mu(\\tau_\\alpha) +\\cdots\\,,\\\\\nv_\\alpha^\\mu(\\tau_\\alpha) &= u_\\alpha + \\Delta^{(1)}v_\\alpha^\\mu(\\tau_\\alpha) + \\Delta^{(2)} v_\\alpha^\\mu(\\tau_\\alpha) +\\cdots \\,.\n\\]\nHere $\\Delta^{(i)}x_\\alpha^\\mu$ will indicate quantities entering at $\\mathcal{O}(\\tilde g^{2i})$, for couplings $\\tilde g = e$ or $\\kappa\/2$. We also work with boundary conditions $x^\\mu_\\alpha(\\tau_\\alpha\\rightarrow-\\infty) = b^\\mu_\\alpha + u^\\mu_\\alpha\\tau_\\alpha$, so $v_\\alpha^\\mu(\\tau_\\alpha\\rightarrow-\\infty) = u^\\mu_\\alpha$. At leading order we can treat the contributions to the field equation separately, splitting the problem into gravitational radiation sourced by either the particle worldlines or the electromagnetic field. \n\nOf course we are interested in electromagnetic interactions of two particles, and thus require the leading order solution to the Maxwell equation in Lorenz gauge, $\\partial^2A^\\mu(x) = eJ^\\mu_\\textrm{pp}(x)$, where the (colour-dressed) current appears in~\\eqref{eqn:YangMillsEOM} This is a very simple calculation, and once quickly finds that the field sourced by particle 2, say, is\n\\begin{equation}\nF^{\\mu\\nu}_2(x) = ie \\! \\int\\! \\dd^4\\wn q\\, \\del(u_2\\cdot\\wn q) e^{i\\wn q\\cdot b_2} e^{-i\\wn q\\cdot x} \\frac{\\wn q^\\mu u_2^\\nu - u_2^\\mu \\wn q^\\nu}{\\wn q^2}\\,;\\label{eqn:LOfieldStrength}\n\\end{equation}\nsee \\cite{Kosower:2018adc} for details. The key point is that we can now obtain the leading deflections simply by solving the classical equation of motion, the Lorentz force, which yields\n\\begin{equation}\n\\frac{d \\Delta^{(1)} v^{\\mu}_1}{d\\tau} = \\frac{ie^2}{m_1}\\! \\int \\! \\dd^4 \\wn q\\, \\del(u_2\\cdot \\wn q)  e^{-i\\wn q \\cdot (b_1 - b_2)} \\frac{e^{-i\\wn q\\cdot u_1 \\tau}}{\\wn q^2} \\Big(\\wn q^\\mu\\, u_1\\cdot u_2 - u_2^\\mu\\, u_1\\cdot \\wn q\\Big)\\,.\n\\end{equation}\nWe now have all the data required to determine the full LO gravitational current. Focusing on the radiation sourced from the point-particle worldlines, we can expand~\\eqref{eqn:ppGravTensor} to find\n\\begin{multline}\nT^{\\mu\\nu}_{\\textrm{pp},(1)}(\\wn k) = -2e^2\\!\\int\\!\\dd^4\\wn q\\, \\del((\\wn k - \\wn q)\\cdot u_1)\\del(\\wn q\\cdot u_2) \\frac{e^{i(\\wn k - \\wn q)\\cdot b_1}e^{i\\wn q\\cdot b_2}}{\\wn q^2 (\\wn q\\cdot u_1)} \\bigg[u_1\\cdot u_2\\, \\wn q^{(\\mu} u_1^{\\nu)} \\\\ - \\frac12 u_1\\cdot u_2\\, \\frac{\\wn k\\cdot \\wn q \\, u_1^\\mu u_1^\\nu}{\\wn q\\cdot u_1}  - \\wn q\\cdot u_1\\, u_1^{(\\mu} u_2^{\\nu)} + \\frac12 \\wn k\\cdot u_2 \\, u_1^\\mu u_1^\\nu \\bigg] +(1\\leftrightarrow 2)\\,.\n\\end{multline}\nIt is then convenient to relabel $\\wn q = \\wn w_2$, and introduce a new momentum $\\wn w_1 = \\wn k - \\wn w_2$. In these variables, the analogous contribution from the EM stress tensor in~\\eqref{eqn:LOfieldStrength} to the gravitational radiation is\n\\begin{multline}\nT^{\\mu\\nu}_{\\textrm{pp},(1)}(x) = -e^2\\!\\int\\!\\dd^4\\wn w_1\\dd^4\\wn w_2\\, \\del(\\wn w_1\\cdot u_1)\\del(\\wn w_2\\cdot u_2) \\frac{e^{i\\wn w_1\\cdot b_1} e^{i\\wn w_2\\cdot b_2}}{\\wn w_1^2 \\wn w_2^2} e^{-i(\\wn w_1 + \\wn w_2)\\cdot x}\\\\ \\times\\bigg[2 \\wn w_1\\cdot u_2\\, u_1^{(\\mu} \\wn w_2^{\\nu)}  - \\wn w_1\\cdot \\wn w_2\\, u_1^\\mu u_2^\\nu - u_1\\cdot u_2 \\, \\wn w_1^\\mu \\wn w_2^\\nu + (1\\leftrightarrow 2)\\bigg]\\,. \n\\end{multline}\nSumming the two pieces and restoring classical momenta $p_\\alpha = m_\\alpha u_\\alpha$ yields\n\\begin{multline}\nT^{\\mu\\nu}_{(1)}(\\wn k) = -\\frac{e^2\\kappa}{4} \\!\\int\\!\\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(\\wn w_1\\cdot p_1) \\del(\\wn w_2\\cdot p_2) \\del^{(4)}(\\wn k- \\wn w_1 - \\wn w_2) \\, e^{i\\wn w_1\\cdot b_1} e^{i\\wn w_2\\cdot b_2}\\\\ \\times \\bigg[\\frac{Q_{12}^\\mu P_{12}^\\nu + Q_{12}^\\nu P_{12}^\\mu}{\\wn w_1^2 \\wn w_2^2} + (p_1\\cdot p_2)\\left(\\frac{Q_{12}^\\mu Q_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} - \\frac{P_{12}^\\mu P_{12}^\\nu}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2}\\right)\\bigg]\\,,\\label{eqn:classicalRN}\n\\end{multline}\nwhere we have adopted the gauge invariant functions defined in~\\eqref{eqn:gaugeInvariants}. Upon contraction with a graviton field strength tensor this current then agrees (for  $b_2 = 0$, and up to an overall sign) with equation~\\eqref{eqn:RNradKernel}, the LO gravitational radiation kernel for electromagnetic scattering of Reissner--Nordstr\\\"{o}m black holes.\n\\chapter{Wavefunction integrals}\n\\label{app:wavefunctions}\n\nIn this appendix we detail the evaluation of the explicit momentum space wavefunction integrals that played a key role in our disccussion of the classical limit in chapters~\\ref{chap:pointParticles} and~\\ref{chap:impulse}.\n\nWe begin with the single particle momentum space wavefunction normalisation in \\eqn~\\eqref{eqn:LinearExponential}. To determine this we must compute the following integral:\n\\begin{equation}\nm^{-2}\\int \\df(p)\\;\\exp\\biggl[-\\frac{2 p\\cdot u}{m\\xi}\\biggr]\\,.\n\\end{equation}\nLet us parametrise the on-shell phase space in a similar manner to equation~\\eqref{eqn:qbRapidityParametrisation}, but appropriate for timelike momenta, writing\n\\begin{equation}\np^\\mu = E_p\\bigl(\\cosh\\zeta,\\,\\sinh\\zeta\\sin\\theta\\cos\\phi,\\,\\sinh\\zeta\\sin\\theta\\sin\\phi,\n\\sinh\\zeta\\cos\\theta\\bigr)\\\n\\label{eqn:Parametrization}               \n\\end{equation}\nso that\n\\begin{equation}\n\\begin{aligned}\n\\df(p) &= (2\\pi)^{-3}\\dd E_p d\\zeta d\\Omega_2\\,\\del(E_p^2-m^2)\\Theta(E_p)\\,\nE_p^3 \\sinh^2\\zeta\n\\\\&=\n(2\\pi)^{-3}\\dd E_p d\\zeta d\\theta d\\phi \\,\\del(E_p^2-m^2)\\Theta(E_p)\\,\nE_p^3 \\sinh^2\\zeta\\sin\\theta\\,.\n\\end{aligned}\n\\end{equation}\nPerforming the $E_p$ integration, we obtain\n\\begin{equation}\n\\df(p)  \\rightarrow \\frac{m^2}{2(2\\pi)^{3}} \nd\\zeta d\\theta d\\phi\\,\\sinh^2\\!\\zeta\\sin\\theta\\,,\n\\label{Measure}\n\\end{equation}\nalong with $E_p=m$ in the integrand.  The integral must be a Lorentz-invariant\nfunction of $u$; as the only available Lorentz invariant is $u^2=1$, we conclude\nthat the result must be a function of $\\xi$ alone.  We can compute it in the rest\nframe of $u$, where our desired integral is\n\\begin{equation}\n\\frac12 \\! \\int_0^\\infty \\!\\dd\\zeta\\,\\sinh^2\\!\\zeta\\!\\int_0^\\pi\\! \\dd\\theta\\,\\sin\\theta\\!\n\\int_0^{2\\pi} \\!\\dd\\phi\\, \\exp\\biggl[-\\frac{2\\cosh\\zeta}{\\xi}\\biggr]\n=\\frac1{2(2\\pi)^2} \\xi\\, K_1(2\/\\xi)\\,,\\label{eqn:normalisationIntegral}\n\\end{equation}\nwhere $K_1$ is a modified Bessel function of the second kind.  The normalisation\ncondition~(\\ref{eqn:WavefunctionNormalization}) then yields\n\\begin{equation}\n\\frac{2\\sqrt2\\pi}{\\xi^{1\/2} K_1^{1\/2}(2\/\\xi)}\n\\end{equation}\nfor the wavefunction's normalisation.\n\nNext, we compute $\\langle p^\\mu\\rangle$.  Lorentz invariance implies that the expectation\nvalue must be proportional to $u^\\mu$;\nagain computing in the rest frame, we find that\n\\begin{equation}\n\\langle p^\\mu\\rangle = m u^\\mu \\,\\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,.\n\\end{equation}\nThe phase-space measure fixes $\\langle p^2\\rangle = m^2$, so we conclude that\n\\begin{equation}\n\\begin{aligned}\n\\frac{\\sigma^2(p)}{\\langle p^2\\rangle} =\n1-\\frac{\\langle p\\rangle^2}{\\langle p^2\\rangle} &= 1-\\frac{K_2^2(2\/\\xi)}{K_1^2(2\/\\xi)}\n\\\\&=-\\frac32\\xi+\\Ord(\\xi^2)\\,,\n\\end{aligned}\n\\end{equation}\nwhere as $\\xi\\rightarrow 0$ we have used the asymptotic approximations\n\\[\nK_1(2\/\\xi) &= \\sqrt{\\frac{\\pi \\xi}{4}}\\, \\exp\\left[-\\frac{2}{\\xi}\\right]\\left(1 + \\frac{3\\xi}{16} + \\mathcal{O}(\\xi^2)\\right),\\\\ K_2(2\/\\xi) &= \\sqrt{\\frac{\\pi \\xi}{4}}\\, \\exp\\left[-\\frac{2}{\\xi}\\right] \\left(1 + \\frac{15\\xi}{16} + \\mathcal{O}(\\xi^2)\\right).\n\\]\nFor the single particle case it remains to compute the double expectation value in equation~\\eqref{eqn:doubleMomExp}. We can again apply Lorentz invariance, which dictates that\n\\begin{equation}\n\\langle p^\\mu p^\\nu \\rangle = A u^\\mu u^\\nu + B \\eta^{\\mu\\nu}\\,.\\label{eqn:doubleExpVal}\n\\end{equation}\nContracting with the velocity leads to\n\\begin{equation}\nA + B = \\frac{\\mathcal{N}^2}{m^2}\\! \\int\\! \\df(p)\\, (u\\cdot p)^2\\, \\exp\\left[-\\frac{2 u\\cdot p}{m \\xi}\\right] = m^2 + \\frac32 m^2 \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,,\n\\end{equation}\nwhere we used the result in equation~\\eqref{eqn:normalisationIntegral}. Combining this constraint with the trace of~\\eqref{eqn:doubleExpVal} is then enough to determine the coefficients as\n\\begin{equation}\nA = m^2 + 2 m^2 \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,, \\qquad B = -\\frac{m^2}{2} \\xi \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,,\n\\end{equation}\nyielding the result in equation~\\eqref{eqn:doubleMomExp}.\n\nOur discussion of point-particle scattering in section~\\ref{sec:classicalLimit} hinged upon the integral \n\\begin{equation}\nT(\\qb) = \\frac{\\Norm^2}{\\hbar m^3}\\exp\\biggl[-\\frac{\\hbar \\qb\\cdot u}{m\\xi}\\biggr]\\int \\df(p)\\;\n\\del(2 p\\cdot\\qb\/m+\\hbar\\qb^2\/m)\\,\\exp\\biggl[-\\frac{2 p\\cdot u}{m\\xi}\\biggr]\\,.\n\\end{equation}\nThis integral is dimensionless, and can depend only on two Lorentz invariants,\n$\\qb\\cdot u$ and $\\qb^2$, along with $\\xi$.  \nIt is convenient to write it as a function of two\ndimensionless variables built out of these invariants,\n\\begin{equation}\n\\omega \\equiv \\frac{\\qb\\cdot u}{\\sqrt{-\\qb^2}}\\,,\n\\qquad \\tau \\equiv \\frac{\\hbar\\sqrt{-\\qb^2}}{2m}\\,.\n\\end{equation}\nWe again work in the rest frame of $u^\\mu$, and without loss of generality,\nchoose the $z$-axis of the $p$ integration to lie along the direction of\n$\\v{\\qb}$.  The only components that appear in the integral are then\n$\\qb^0$ and $\\qb^z$; after integration, we can obtain the dependence on\n$\\tau$ and $\\omega$ via the replacements\n\\begin{equation}\n\\qb^0\\rightarrow \\frac{2 m\\omega\\tau}{\\hbar}\\,,\n\\qquad \\qb^z \\rightarrow \\frac{2m\\sqrt{1+\\omega^2}\\tau}{\\hbar}\\,;\n\\end{equation}\nhence,\n\\begin{equation}\n-\\!\\qb^2 \\rightarrow \\frac{4m^2\\tau^2}{\\hbar^2}\\,.\n\\end{equation}\nUsing the measure~(\\ref{Measure}), we find that\n\\begin{equation}\n\\begin{aligned}\nT_1(\\wn q) &= \\frac1{2\\pi\\,\\hbar m\\,\\xi K_1(2\/\\xi)} \\exp\\biggl[-\\frac{\\hbar \\qb^0}{m\\xi}\\biggr]\n\\\\&\\qquad\\times \\int_0^\\infty \\!\\d\\zeta\\,\\sinh^2\\!\\zeta\\! \\int_0^\\pi\\! \\d\\theta\\,\\sin\\theta\\!\n\\int_0^{2\\pi}\\! \\d\\phi\\, \\exp\\biggl[-\\frac{2\\cosh\\zeta}{\\xi}\\biggr]\n\\\\& \\qquad\\qquad\\qquad \\times \\del(2 \\qb^0 \\cosh\\zeta - 2\\qb^z \\sinh\\zeta \\cos\\theta\n+\\hbar\\qb^2\/m)\\,.\n\\end{aligned}\n\\end{equation}\nThe $\\phi$ integral is trivial, and we can use the delta function to do the\n$\\theta$ integral:\n\\newcommand{\\textrm{cosech}}{\\textrm{cosech}}\n\\begin{equation}\n\\begin{aligned}\nT_1(\\wn q) &= \\frac1{2\\hbar m\\,\\qb^z\\,\\xi K_1(2\/\\xi)} \\exp\\biggl[-\\frac{\\hbar \\qb^0}{m\\xi}\\biggr]\n\\int_0^\\infty d\\zeta\\;\\sinh\\zeta\\;\n\\exp\\biggl[-2\\frac{\\cosh\\zeta}{\\xi}\\biggr]\n\\\\& \\qquad\\qquad\\qquad\\times \n\\Theta\\bigl(1+ \\qb^0 \\coth\\zeta\/\\qb^z +\\hbar\\qb^2\\textrm{cosech}\\,\\zeta\/(2 m\\qb^z)\\bigr)\\,\n\\\\& \\qquad\\qquad\\qquad\\times \n\\Theta\\bigl(1-\\qb^0 \\coth\\zeta\/\\qb^z -\\hbar\\qb^2\\textrm{cosech}\\,\\zeta\/(2 m\\qb^z)\\bigr)\\,.\n\\end{aligned}\n\\end{equation}\nIn the $\\hbar\\rightarrow 0$ limit, the first theta function will have no effect,\neven with $\\qb^2<0$.  Changing variables to $w=\\cosh\\zeta$, \nthe second theta function will impose the constraint\n\\begin{equation}\nw \\ge \\frac{\\qb^z\\sqrt{1-\\hbar^2\\qb^2\/(4m^2)}}{\\sqrt{-\\qb^2}} -\\frac{\\hbar\\qb^0}{2m}\\,.\n\\end{equation}\nIn terms of $\\omega$ and $\\tau$,\nthis constraint is\n\\begin{equation}\nw \\ge \\sqrt{1+\\omega^2}\\sqrt{1+\\tau^2}-\\omega\\tau\\,.\n\\end{equation}\nUp to corrections of $\\Ord(\\hbar)$, the right-hand side is greater than 1,\nand so becomes the lower limit of integration.  The result for the integral is then\n\\begin{equation}\n\\begin{aligned}\nT(\\qb) &=\\frac1{8 m^2\\sqrt{1+\\omega^2}\\tau\\,K_1(2\/\\xi)} \n\\exp\\biggl[-\\frac{2\\omega\\tau}{\\xi}\\biggr]\\\\\n&\\qquad\\qquad\\qquad \\times\\exp\\biggl[-\\frac2{\\xi}\\Bigl(\\sqrt{1+\\omega^2}\\sqrt{1+\\tau^2}-\\omega\\tau\\Bigr)\\biggr]\n\\\\&=\n\\frac1{4 \\hbar m\\sqrt{(\\qb\\cdot u)^2-\\qb^2}\\,K_1(2\/\\xi)} \\\\\n&\\qquad\\qquad\\qquad \\times \\exp\\biggl[-\\frac2{\\xi}\\frac{\\sqrt{(\\qb\\cdot u)^2-\\qb^2}}{\\sqrt{-\\qb^2}}\n\\sqrt{1-\\hbar^2\\qb^2\/(4m^2)}\\biggr]\n\\,,\n\\end{aligned}\n\\end{equation}\nwhich is the result listed in equation~\\eqref{eqn:TIntegral}.\n\\chapter{Worldline perturbation theory}\n\\label{app:worldlines}\n\nThroughout part~\\ref{part:observables} we computed on-shell classical observables from scattering amplitudes. We justified our results by their agreement with iterative solutions to the appropriate classical equations of motion, as shown in detail in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}. To provide a flavour of these methods, let us calculate the LO current for gravitational radiation due to the electromagnetic scattering of Reissner--Nordstr\\\"{o}m black holes, which we studied using the double copy in section~\\ref{sec:inelasticBHscatter}. \n\nClassically, the emitted gravitational radiation will satisfy the linearised Einstein--Maxwell field equation, which for the trace-reversed perturbation in de--Donder gauge is\n\\begin{equation}\n\\partial^2\\bar h^{\\mu\\nu}(x) = T^{\\mu\\nu}(x) = \\frac{\\kappa}{2}\\left(T_{\\rm pp}^{\\mu\\nu}(x) + T_{\\rm EM}^{\\mu\\nu}(x)\\right).\n\\end{equation}\nThe electromagnetic field strength tensor $T^{\\rm EM}_{\\mu\\nu}$ is that in \\eqn~\\eqref{eqn:EMfieldStrength}, while in gravity\n\\begin{equation}\nT_{\\rm pp}^{\\mu\\nu}(x) = \\sum_{\\alpha=1}^n m_\\alpha \\int\\! \\frac{d\\tau_\\alpha}{\\sqrt{g}}\\, u^\\mu_\\alpha(\\tau)u^\\nu_\\alpha(\\tau)\\, \\delta^{(D)}(x-x_\\alpha(\\tau))\\label{eqn:ppGravTensor}\n\\end{equation}\nfor $n$ particles, where $g = -\\det(g_{\\mu\\nu})$.  We are seeking perturbative solutions, and therefore expand worldline quantities in the coupling:\n\\[\nx_\\alpha^\\mu(\\tau_\\alpha) &= b_\\alpha^\\mu+u_\\alpha\\tau_\\alpha + \\Delta^{(1)}x_\\alpha^\\mu(\\tau_\\alpha) + \\Delta^{(2)} x_\\alpha^\\mu(\\tau_\\alpha) +\\cdots\\,,\\\\\nv_\\alpha^\\mu(\\tau_\\alpha) &= u_\\alpha + \\Delta^{(1)}v_\\alpha^\\mu(\\tau_\\alpha) + \\Delta^{(2)} v_\\alpha^\\mu(\\tau_\\alpha) +\\cdots \\,.\n\\]\nHere $\\Delta^{(i)}x_\\alpha^\\mu$ will indicate quantities entering at $\\mathcal{O}(\\tilde g^{2i})$, for couplings $\\tilde g = e$ or $\\kappa\/2$. We also work with boundary conditions $x^\\mu_\\alpha(\\tau_\\alpha\\rightarrow-\\infty) = b^\\mu_\\alpha + u^\\mu_\\alpha\\tau_\\alpha$, so $v_\\alpha^\\mu(\\tau_\\alpha\\rightarrow-\\infty) = u^\\mu_\\alpha$. At leading order we can treat the contributions to the field equation separately, splitting the problem into gravitational radiation sourced by either the particle worldlines or the electromagnetic field. \n\nOf course we are interested in electromagnetic interactions of two particles, and thus require the leading order solution to the Maxwell equation in Lorenz gauge, $\\partial^2A^\\mu(x) = eJ^\\mu_\\textrm{pp}(x)$, where the (colour-dressed) current appears in~\\eqref{eqn:YangMillsEOM} This is a very simple calculation, and once quickly finds that the field sourced by particle 2, say, is\n\\begin{equation}\nF^{\\mu\\nu}_2(x) = ie \\! \\int\\! \\dd^4\\wn q\\, \\del(u_2\\cdot\\wn q) e^{i\\wn q\\cdot b_2} e^{-i\\wn q\\cdot x} \\frac{\\wn q^\\mu u_2^\\nu - u_2^\\mu \\wn q^\\nu}{\\wn q^2}\\,;\\label{eqn:LOfieldStrength}\n\\end{equation}\nsee \\cite{Kosower:2018adc} for details. The key point is that we can now obtain the leading deflections simply by solving the classical equation of motion, the Lorentz force, which yields\n\\begin{equation}\n\\frac{d \\Delta^{(1)} v^{\\mu}_1}{d\\tau} = \\frac{ie^2}{m_1}\\! \\int \\! \\dd^4 \\wn q\\, \\del(u_2\\cdot \\wn q)  e^{-i\\wn q \\cdot (b_1 - b_2)} \\frac{e^{-i\\wn q\\cdot u_1 \\tau}}{\\wn q^2} \\Big(\\wn q^\\mu\\, u_1\\cdot u_2 - u_2^\\mu\\, u_1\\cdot \\wn q\\Big)\\,.\n\\end{equation}\nWe now have all the data required to determine the full LO gravitational current. Focusing on the radiation sourced from the point-particle worldlines, we can expand~\\eqref{eqn:ppGravTensor} to find\n\\begin{multline}\nT^{\\mu\\nu}_{\\textrm{pp},(1)}(\\wn k) = -2e^2\\!\\int\\!\\dd^4\\wn q\\, \\del((\\wn k - \\wn q)\\cdot u_1)\\del(\\wn q\\cdot u_2) \\frac{e^{i(\\wn k - \\wn q)\\cdot b_1}e^{i\\wn q\\cdot b_2}}{\\wn q^2 (\\wn q\\cdot u_1)} \\bigg[u_1\\cdot u_2\\, \\wn q^{(\\mu} u_1^{\\nu)} \\\\ - \\frac12 u_1\\cdot u_2\\, \\frac{\\wn k\\cdot \\wn q \\, u_1^\\mu u_1^\\nu}{\\wn q\\cdot u_1}  - \\wn q\\cdot u_1\\, u_1^{(\\mu} u_2^{\\nu)} + \\frac12 \\wn k\\cdot u_2 \\, u_1^\\mu u_1^\\nu \\bigg] +(1\\leftrightarrow 2)\\,.\n\\end{multline}\nIt is then convenient to relabel $\\wn q = \\wn w_2$, and introduce a new momentum $\\wn w_1 = \\wn k - \\wn w_2$. In these variables, the analogous contribution from the EM stress tensor in~\\eqref{eqn:LOfieldStrength} to the gravitational radiation is\n\\begin{multline}\nT^{\\mu\\nu}_{\\textrm{pp},(1)}(x) = -e^2\\!\\int\\!\\dd^4\\wn w_1\\dd^4\\wn w_2\\, \\del(\\wn w_1\\cdot u_1)\\del(\\wn w_2\\cdot u_2) \\frac{e^{i\\wn w_1\\cdot b_1} e^{i\\wn w_2\\cdot b_2}}{\\wn w_1^2 \\wn w_2^2} e^{-i(\\wn w_1 + \\wn w_2)\\cdot x}\\\\ \\times\\bigg[2 \\wn w_1\\cdot u_2\\, u_1^{(\\mu} \\wn w_2^{\\nu)}  - \\wn w_1\\cdot \\wn w_2\\, u_1^\\mu u_2^\\nu - u_1\\cdot u_2 \\, \\wn w_1^\\mu \\wn w_2^\\nu + (1\\leftrightarrow 2)\\bigg]\\,. \n\\end{multline}\nSumming the two pieces and restoring classical momenta $p_\\alpha = m_\\alpha u_\\alpha$ yields\n\\begin{multline}\nT^{\\mu\\nu}_{(1)}(\\wn k) = -\\frac{e^2\\kappa}{4} \\!\\int\\!\\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(\\wn w_1\\cdot p_1) \\del(\\wn w_2\\cdot p_2) \\del^{(4)}(\\wn k- \\wn w_1 - \\wn w_2) \\, e^{i\\wn w_1\\cdot b_1} e^{i\\wn w_2\\cdot b_2}\\\\ \\times \\bigg[\\frac{Q_{12}^\\mu P_{12}^\\nu + Q_{12}^\\nu P_{12}^\\mu}{\\wn w_1^2 \\wn w_2^2} + (p_1\\cdot p_2)\\left(\\frac{Q_{12}^\\mu Q_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} - \\frac{P_{12}^\\mu P_{12}^\\nu}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2}\\right)\\bigg]\\,,\\label{eqn:classicalRN}\n\\end{multline}\nwhere we have adopted the gauge invariant functions defined in~\\eqref{eqn:gaugeInvariants}. Upon contraction with a graviton field strength tensor this current then agrees (for  $b_2 = 0$, and up to an overall sign) with equation~\\eqref{eqn:RNradKernel}, the LO gravitational radiation kernel for electromagnetic scattering of Reissner--Nordstr\\\"{o}m black holes.\n\\chapter{Introduction}\n\\label{chap:intro}\n\nThe dawn of gravitational wave astronomy, heralded by the binary black hole and neutron star mergers detected by the LIGO and VIRGO collaborations~\\cite{Abbott:2016blz,Abbott:2016nmj,Abbott:2017oio,Abbott:2017vtc,TheLIGOScientific:2017qsa}, has opened a new observational window on the universe. Future experiments offer the tantalising prospect of unprecedented insights into the physics of black holes, as well as neutron star structure, extreme nuclear matter and general relativity itself. Theorists have a critical role to play in this endeavour: to access such insights, an extensive bank of theoretical waveform templates are required for both event detection and parameter extraction \\cite{Buonanno:2014aza}.\n\nThe vast majority of accessible data in a gravitational wave signal lies in the inspiral regime. This is the phase preceding the dramatic merger, in which the inspiralling pair coalesce and begin to influence each other's motion. The two bodies remain well separated, and one therefore can tackle their dynamics perturbatively: we can begin by treating them as point particles, and then increase precision by calculating corrections at higher orders in a given approximation. For an inspiral with roughly equal mass black holes the most directly applicable perturbative series is the non-relativistic \\textit{post--Newtonian} (PN) expansion, where one expands in powers of the bodies' velocities $v$. Meanwhile the \\textit{post--Minkowskian} (PM) expansion in powers of Newton's constant $G$ is fully relativistic, and thus more naturally suited to scattering interactions; however, it makes crucial contributions to precision inspiral calculations \\cite{Antonelli:2019ytb}. Finally, when one black hole is far heavier than the other a \\textit{self-force} expansion can be taken about the test body limit, expanding in the mass ratio of the black holes but keeping $v$ and $G$ to all orders. \n\nAlthough simple in concept, the inherent non-linearity of general relativity (GR) makes working even with these approximations an extremely difficult task. Yet future prospects in gravitational wave astronomy require perturbative calculations at very high precision~\\cite{Babak:2017tow}. This has spawned interest in new techniques for solving the two-body problem in gravity and generating the required waveforms. Such techniques would complement methods based on the `traditional' Arnowitt--Deser--Misner Hamiltonian formalism~\\cite{Deser:1959zza,Arnowitt:1960es,Arnowitt:1962hi,Schafer:2018kuf}, direct post--Newtonian solutions in harmonic gauge \\cite{Blanchet:2013haa}, long-established effective-one-body (EOB) methods introduced by Buonnano and Damour~\\cite{Buonanno:1998gg,Buonanno:2000ef,Damour:2000we,Damour:2001tu}, numerical-relativity approaches~\\cite{Pretorius:2005gq,Pretorius:2007nq}, and the effective field theory approach pioneered by Goldberger and Rothstein~\\cite{Goldberger:2004jt,Porto:2016pyg,Levi:2018nxp}.\n\nRemarkably, ideas and methods from quantum field theory (QFT) offer a particularly promising avenue of investigation. Here, interactions are encoded by scattering amplitudes. Utilising amplitudes allows a powerful armoury of modern on-shell methods\\footnote{See \\cite{Cheung:2017pzi,Elvang:2015rqa} for an introduction.} to be applied to a problem, drawing on the success of the NLO (next-to-leading order) revolution in particle phenomenology. An appropriate method for extracting observables relevant to the problem at hand is also required. The relevance of a scattering amplitude --- in particular, a loop amplitude --- to the classical potential, for example, is well understood from work on gravity as an effective field theory~\\cite{Iwasaki:1971,Duff:1973zz,Donoghue:1993eb,Donoghue:1994dn,Donoghue:1996mt,Donoghue:2001qc,BjerrumBohr:2002ks,BjerrumBohr:2002kt,Khriplovich:2004cx,Holstein:2004dn,Holstein:2008sx}. There now exists a panoply of techniques for applying modern amplitudes methods to the computation of the classical gravitational potential \\cite{Neill:2013wsa,Bjerrum-Bohr:2013bxa,Bjerrum-Bohr:2014lea,Bjerrum-Bohr:2014zsa,Bjerrum-Bohr:2016hpa,Bjerrum-Bohr:2017dxw,Cachazo:2017jef,Cheung:2018wkq,Caron-Huot:2018ape,Cristofoli:2019neg,Bjerrum-Bohr:2019kec,Cristofoli:2020uzm,Kalin:2020mvi,Cheung:2020gbf}, generating results directly applicable to gravitational wave physics \\cite{Damour:2016gwp,Damour:2017zjx,Bjerrum-Bohr:2018xdl,Bern:2019nnu,Brandhuber:2019qpg,Bern:2019crd,Huber:2019ugz,Cheung:2020gyp,AccettulliHuber:2020oou,Cheung:2020sdj,Kalin:2020fhe,Haddad:2020que,Kalin:2020lmz,Bern:2020uwk,Huber:2020xny,Bern:2021dqo}. By far the most natural relativistic expansion from an amplitudes perspective is the post--Minkowskian expansion (in the coupling constant): indeed, amplitudes methods have achieved the first calculations of the 3PM~\\cite{Bern:2019nnu} and 4PM~\\cite{Bern:2021dqo} potential. Furthermore, it is possible to use analytic continuation to obtain bound state observables directly from the scattering problem \\cite{Kalin:2019rwq,Kalin:2019inp}. \n\nThe gravitational potential is a versatile tool; however it is also coordinate, and thus gauge, dependent. The conservative potential also neglects the radiation emitted from interactions, leading to complications at higher orders. Amplitudes and physical observables, meanwhile, are on-shell and gauge-invariant, and should naturally capture all the physics of the problem. Direct maps between amplitudes and classical physics are well known to hold in certain regimes: for example, the eikonal exponentation of amplitudes in the extreme high energy limit has long been used to derive scattering angles \\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe,Amati:1992zb,Kabat:1992tb,Amati:1993tb,Muzinich:1995uj,DAppollonio:2010krb,Melville:2013qca,Akhoury:2013yua,DAppollonio:2015fly,Ciafaloni:2015vsa,DAppollonio:2015oag,Ciafaloni:2015xsr,Luna:2016idw,Collado:2018isu,KoemansCollado:2019ggb,DiVecchia:2020ymx,DiVecchia:2021ndb}. Furthermore, calculating amplitudes in the high energy regime exposes striking universal features in gravitational scattering \\cite{Bern:2020gjj,Parra-Martinez:2020dzs,DiVecchia:2020ymx}. Meanwhile, a careful analysis of soft limits of amplitudes with massless particles can extract data about both classical radiation \\cite{Laddha:2018rle,Laddha:2018myi,Sahoo:2018lxl,Laddha:2018vbn,Laddha:2019yaj,A:2020lub,Sahoo:2020ryf,Bonocore:2020xuj} and radiation reaction effects \\cite{DiVecchia:2021ndb}. Calculations with radiation can also be accomplished in the eikonal formalism \\cite{Amati:1990xe}, but have proven particularly natural in classical worldline approaches, whereby one applies perturbation theory directly to point-particle worldlines rather than quantum states \\cite{Goldberger:2016iau,Goldberger:2017frp, Goldberger:2017vcg,Goldberger:2017ogt,Chester:2017vcz,Li:2018qap, Shen:2018ebu,Plefka:2018dpa,Plefka:2019hmz,PV:2019uuv,Almeida:2020mrg,Prabhu:2020avf,Mougiakakos:2021ckm}. Using path integrals to develop a worldline QFT enables access to on-shell amplitudes techniques in this context \\cite{Mogull:2020sak,Jakobsen:2021smu}, and this method has been used to calculate the NLO current due to Schwarzschild black hole bremsstrahlung.\n\nWe know how to extract information about classical scattering and radiation from quantum amplitudes in a gauge invariant manner --- but only in specific regimes. It is therefore natural to seek a more generally applicable, on-shell mapping between amplitudes and classical observables: this will form the topic of the first part of this thesis. We will construct general formulae for a variety of on-shell observables, valid in any quantum field theory and for any two-body scattering event. In this context we will also systematically study how to extract the classical limit of an amplitude, developing in the process a precise understanding of how to use quantum amplitudes to calculate observables for classical point-particles. We will show that by studying appropriate observables, expressed directly in terms of amplitudes, we can handle the nuances of the classical relationship between conservative and dissipative physics in a single, systematic approach, avoiding the difficulties surrounding the Abraham--Lorentz--Dirac radiation reaction force in electrodynamics \\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz}. First presented in ref.~\\cite{Kosower:2018adc}, the formalism we will develop has proven particularly useful for calculating on-shell observables for black hole processes involving classical radiation \\cite{Luna:2017dtq,Bautista:2019tdr,Cristofoli:2020hnk,A:2020lub,delaCruz:2020bbn,Mogull:2020sak,Gonzo:2020xza,Herrmann:2021lqe} and spin \\cite{Maybee:2019jus,Guevara:2019fsj,Arkani-Hamed:2019ymq,Moynihan:2019bor,Huang:2019cja,Bern:2020buy,Emond:2020lwi,Monteiro:2020plf}. Wider applications also exist to other aspects of classical physics, such as the Yang--Mills--Wong equations \\cite{delaCruz:2020bbn,Wong:1970fu} and hard thermal loops \\cite{delaCruz:2020cpc}.\n\nEven the most powerful QFT techniques require a precise understanding of how to handle the numerous subtleties involved in taking the classical limit and accurately calculating observables. The reader may therefore wonder whether the philosophy of applying quantum amplitudes to classical physics really offers any fundamental improvement --- after all,  there are many concurrent advances in our understanding of the two-body problem arising from alternative calculational methods. For example, information from the self-force approximation can provide extraordinary simplifications directly at the level of classical calculations \\cite{Bini:2020flp,Bini:2020hmy,Bini:2020nsb,Bini:2020uiq,Bini:2020wpo,Damour:2020tta,Bini:2020rzn}. Aside from the fact that amplitudes methods have achieved state-of-the-art precision in the PM approximation \\cite{Bern:2019nnu,Bern:2021dqo,Herrmann:2021lqe}, such a sweeping judgement would be premature, as we have still yet to encounter two unique facets of the amplitudes programme: the double copy, and the treatment of spin effects.\n\n\\subsection{The double copy}\n\nAn important insight arising from the study of scattering amplitudes is that amplitudes in perturbative quantum gravity are far simpler than one would expect, and in particular are closely connected to the amplitudes of Yang--Mills (YM) theory. This connection is called the double copy, because gravitational amplitudes are obtained as a product of two Yang--Mills quantities. One can implement this double copy in a variety of ways: the original statement, by Kawai, Lewellen and Tye~\\cite{Kawai:1985xq} presents a tree-level gravitational (closed string) amplitude as a sum over terms, each of which is a product of two tree-level colour-ordered Yang--Mills (open string) amplitudes, multiplied by appropriate Mandelstam invariants. More recently, Bern, Carrasco and Johansson~\\cite{Bern:2008qj,Bern:2010ue} demonstrated that the double copy can be understood very simply in terms of a diagrammatic expansion of a scattering amplitude. They noted that any tree-level $m$-point amplitude in Yang--Mills theory could be expressed as a sum over the set of cubic diagrams $\\Gamma$,\n\\begin{equation}\n\\mathcal{A}_{m} = g^{m-2}\\sum_{\\Gamma}\\frac{n_i c_i}{\\Delta_i}\\,,\n\\end{equation}\nwhere $\\Delta_i$ are the propagators, $n_i$ are gauge-dependent kinematic numerators, and $c_i$ are colour factors which are related in overlapping sets of three by Jacobi identities,\n\\begin{equation}\nc_\\alpha \\pm c_\\beta \\pm c_\\gamma = 0\\,.\n\\end{equation}\nThe colour factors are single trace products of $SU(N)$ generators $T^a$, normalised such that $\\textrm{tr}(T^aT^b) = \\delta^{ab}$. Remarkably, BCJ found that gauge freedom makes it possible to always choose numerators satisfying the same Jacobi identities \\cite{Bern:2010yg}. This fundamental property is called \\textit{colour-kinematics duality}, and has been proven to hold for tree-level Yang--Mills theories \\cite{Bern:2010yg}.\n\nWhen colour-kinematics duality holds, the double copy then tells us that\n\\begin{equation}\n\\mathcal{M}_m = \\left(\\frac{\\kappa}{2}\\right)^{m-2}\\sum_{\\Gamma}\\frac{n_i\\tilde{n}_i}{\\Delta_i}\\,\n\\end{equation}\nis the corresponding $m$-point gravity amplitude, obtained by the replacements\n\\begin{equation}\ng\\mapsto\\frac{\\kappa}{2}\\,,\\quad c_i\\mapsto \\tilde{n}_i\\,.\n\\end{equation}\nHere $\\kappa = \\sqrt{32\\pi G}$ is the appropriate gravitational coupling, and $\\tilde{n}_i$ is a distinct second set of numerators satisfying colour-kinematics duality. The choice of numerator determines the resulting gravity theory. To obtain gravity the original numerators are chosen, and thus amplitude numerators for gravity are simply the square of kinematic numerators in Yang--Mills theory, provided that colour-kinematics duality holds. \n\nOne complication is that regardless of the choice of $\\tilde{n}_i$, the result is not a pure theory of gravitons, but instead is a factorisable graviton multiplet. This can easily be seen in pure Yang--Mills theory, where the tensor product $A^\\mu \\otimes A^\\nu \\sim \\phi_\\textrm{d} \\oplus B^{\\mu\\nu} \\oplus h^{\\mu\\nu}$ leads to a scalar dilaton field, antisymmetric Kalb--Ramond axion and traceless, symmetric graviton respectively. In 4 dimensions the axion has only one degree of freedom, so its field strength $H^{\\mu\\nu\\rho} = \\partial^{[\\mu}B^{\\nu\\rho]}$ can be written as\n\\begin{equation}\nH^{\\mu\\nu\\rho}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\partial_\\sigma \\zeta,\\label{eqn:axionscalar}\n\\end{equation}\nwith $\\zeta$ representing the single propagating pseudoscalar degree of freedom. There any many possible ways to deal with the unphysical axion and dilaton modes and isolate the graviton degrees of freedom \\cite{Bern:2019prr} --- we will see some such methods in the course of the thesis. However, the main point here is that Einstein gravity amplitudes can be determined exclusively by gauge theory data.\n\nThis modern formulation of the double copy is particularly exciting as it has a clear generalisation to loop level; one simply includes integrals over loop momentum and appropriate symmetry factors. A wealth of non-trivial evidence  supports this conjecture --- for reviews, see \\cite{Carrasco:2015iwa,Bern:2019prr}. The work of BCJ suggests that gravity may be simpler than it seems, and also more closely connected to Yang--Mills theory than one would guess after inspecting their Lagrangians. Here our simple presentation of colour-kinematics duality was only for tree level, massless gauge theory. However, the double copy can be applied far more generally: it forms bridges between a veritable web of theories, for both massless and massive states \\cite{Johansson:2014zca,Johansson:2015oia,Johansson:2019dnu,Haddad:2020tvs}.\n\nSince perturbation theory is far simpler in Yang--Mills theory than in standard approaches to gravity, the double copy has revolutionary potential for gravitational physics. Indeed, it has proven to be the key tool enabling state-of-the-art calculations of the PM potential from amplitudes \\cite{Bern:2019crd,Bern:2019nnu,Bern:2021dqo}. Furthermore, it has also raised the provocative question of whether exact solutions in general relativity satisfy similar simple relationships to their classical Yang--Mills counterparts, extending the relationship beyond perturbation theory. First explored in \\cite{Monteiro:2014cda}, many exact classical double copy maps are now known to hold between classical solutions of gauge theory and gravity \\cite{Luna:2015paa,Luna:2016hge,Adamo:2017nia,Bahjat-Abbas:2017htu,Carrillo-Gonzalez:2017iyj,Lee:2018gxc,Berman:2018hwd,Carrillo-Gonzalez:2018pjk,Adamo:2018mpq,Luna:2018dpt,CarrilloGonzalez:2019gof,Cho:2019ype,Carrillo-Gonzalez:2019aao,Bah:2019sda,Huang:2019cja,Alawadhi:2019urr,Borsten:2019prq,Kim:2019jwm,Banerjee:2019saj,Bahjat-Abbas:2020cyb,Moynihan:2020gxj,Adamo:2020syc,Alfonsi:2020lub,Luna:2020adi,Keeler:2020rcv,Elor:2020nqe,Alawadhi:2020jrv,Casali:2020vuy,Adamo:2020qru,Easson:2020esh,Chacon:2020fmr,Emond:2020lwi,White:2020sfn,Monteiro:2020plf,Lescano:2021ooe}, even when there is gravitational radiation present~\\cite{Luna:2016due}. \n\nTo emphasise that the classical double copy has not been found simply for esoteric exact solutions, let us briefly consider the original Kerr--Schild map constructed in \\cite{Monteiro:2014cda}. Kerr--Schild spacetimes are a particularly special class of solutions possessing sufficient symmetry that their metrics can be written\n\\begin{equation}\ng_{\\mu\\nu} = \\eta_{\\mu\\nu} + \\varphi k_\\mu k_\\nu\\,,\\label{eqn:KSmetric}\n\\end{equation}\nwhere $\\varphi$ is a scalar function and $k_\\mu$ is null with respect to both the background and full metric, and satisfies the background geodesic equation:\n\\begin{equation}\ng^{\\mu\\nu} k_\\mu k_\\nu = \\eta^{\\mu\\nu} k_\\mu k_\\nu = 0\\,, \\qquad k\\cdot\\partial k_\\mu = 0\\,.\\label{eqn:KSvector}\n\\end{equation}\nThe symmetries of this class of spacetime ensure that the (mixed index placement) Ricci tensor is linearised. It was proposed in \\cite{Monteiro:2014cda} that for such spacetimes there then exists a single copy gauge theory solution,\n\\begin{equation}\nA^a_\\mu = \\varphi\\, c^a k_\\mu\\,.\\label{eqn:singleKScopy}\n\\end{equation}\nwhere $c^a$ is a classical colour charge. This incarnation of the double copy is therefore enacted by replacing copies of the classical colour with the null vector $k_\\mu$, in analogue to the BCJ amplitude replacement rules.\n\nThe crucial importance of the Kerr--Schild double copy is that it encompasses both Schwarzschild and Kerr black holes. Both (exterior) spacetime metrics can be written in the compact Kerr--Schild form, with respective data \\cite{Monteiro:2014cda}\n\\begin{equation}\n\\varphi_\\textrm{Schwz}(r) = \\frac{2GM}{r}\\,, \\quad k^\\mu = \\left(1, \\frac{\\v{x}}{r}\\right)\\\\\n\\end{equation}\nfor Schwarzschild, where $r^2 = \\v{x}^2$; and\n\\begin{equation}\n\\varphi_\\textrm{Kerr}(\\tilde r, \\theta) = \\frac{2GM\\tilde r}{\\tilde r^2 + a^2 \\cos^2\\theta}\\,, \\quad k^\\mu = \\left(1,\\frac{\\tilde r x + ay}{\\tilde r^2 + a^2}, \\frac{\\tilde ry - ax}{\\tilde r^2 + a^2},\\frac{z}{r}\\right)\\label{eqn:blackholesKSforms}\n\\end{equation}\nfor Kerr, where the parameter $a$ is the radius of the Kerr singularity about the $z$ axis. This key parameter is the norm of a pseudovector $a^\\mu$ which fully encodes the spin of the black hole, the \\textit{spin vector}. It is important to note that in the Kerr case $(\\tilde r, \\theta)$ are not the usual polar coordinates, instead satisfying\n\\begin{equation}\n\\frac{x^2 +y^2}{\\tilde r^2 + a^2} + \\frac{z^2}{\\tilde{r}^2} = 1\\label{eqn:KerrKSradial}\n\\end{equation}\nand $z = \\tilde r \\cos\\theta$. The corresponding gauge theory single copies are then given by \\eqn~\\eqref{eqn:singleKScopy}. The Schwarzchild single copy is simply a Coulomb charge. The Kerr single copy meanwhile is a disk of uniform charge rotating about the $z$, axis whose mass distribution exhibits a singularity at $x^2 + y^2 = a^2$ \\cite{Monteiro:2014cda}. We will refer to this unique charged particle by its modern name, $\\rootKerr$ \\cite{Arkani-Hamed:2019ymq}.\n\nThe $\\rootKerr$ solution was first explored by Israel in \\cite{Israel:1970kp}, and will be of great interest for us in the second part of the thesis: its double copy relation to Kerr ensures that the structure and dynamics of Kerr in gravity are precisely mirrored by the behaviour of $\\rootKerr$ in gauge theory, where calculations are often simpler. This is particularly important in the context of the second key area in which applying amplitudes ideas to  black hole interactions can offer a significant computational and conceptual advantage: spin.\n\n\n\\subsection{Spin}\n\nThe astrophysical bodies observed in gravitational wave experiments spin. The spins of the individual bodies in a compact binary coalescence event influence the details of the outgoing gravitational radiation \\cite{Buonanno:2014aza}, and moreover contain information on the poorly-understood formation channels of the binaries \\cite{Mandel:2018hfr}. Measurement of spin is therefore one of the primary physics outputs of gravitational wave observations.\n\nAny stationary axisymmetric extended body has an infinite tower of mass-multipole moments $\\mathcal{I}_\\ell$ and current-multipole moments $\\mathcal{J}_\\ell$, which generally depend intricately on its internal structure and composition. In the point-particle limit it is thus the multipole structure of the body which accurately identifies to an observer whether that object is a neutron star, black hole or other entity. Incorporating spin multipoles into the major theoretical platform for these experiments, the EOB formalism \\cite{Buonanno:1998gg,Buonanno:2000ef}, is well established in the PN approximation \\cite{Damour:2001tu,Damour:2008qf,Barausse:2009aa,Barausse:2009xi,Barausse:2011ys,Damour:2014sva,Bini:2017wfr,Khalil:2020mmr}, and has also been extended to the PM approximation by means of a gauge-invariant spin holonomy \\cite{Bini:2017xzy}. This has been used to compute the dipole (or spin-orbit) contribution to the conservative potential for two spinning bodies through 2PM order \\cite{Bini:2018ywr}. Calculating higher-order PN spin corrections has been a particular strength of the effective field theory treatment of PN dynamics \\cite{Goldberger:2004jt,Porto:2005ac,Porto:2006bt,Porto:2008tb,Levi:2011eq,Levi:2015msa,Levi:2016ofk,Levi:2020kvb,Levi:2020uwu,Levi:2020lfn}, while self-force data has also driven independent progress in this approximation \\cite{Siemonsen:2019dsu,Antonelli:2020aeb,Antonelli:2020ybz}. EFT progress in the handling of spin has also recently been extended to the PM series, yielding the first calculation of finite-size effects beyond leading-order \\cite{Liu:2021zxr}; moreover, these results can be mapped to bound observables by analytic continuation \\cite{Kalin:2019rwq,Kalin:2019inp}.\n\nA common feature of all of these calculations is that they are significantly more complicated than the spinless examples considered previously, and moreover are nearly unanimously restricted to the special case where the spins of the bodies are aligned with each other.\n\nThe black hole case is special. For a Kerr black hole, every multipole is determined by only the mass $m$ and spin vector $a^\\mu$, through the simple relation due to Hansen \\cite{Hansen:1974zz},\n\\begin{equation}\n\\mathcal I_\\ell+i\\mathcal J_\\ell = m\\left(ia\\right)^\\ell \\,.\\label{eqn:multipoles}\n\\end{equation}\nThis distinctive behaviour is a precise reflection of the \\textit{no-hair theorem} \\cite{Israel:1967wq,Israel:1967za,Carter:1971zc}, which ensures that higher multipoles are constrained by the dipole. This simple multipole structure is also reflected in the dynamics of spinning black holes --- for example, remarkable all-spin results are known for black hole scattering at leading order in both the PN and PM approximations \\cite{Vines:2016qwa,Siemonsen:2017yux,Vines:2017hyw}; aligned-spin black hole scattering was also considered at 2PM order for low multipoles in \\cite{Vines:2018gqi}.\n\nMoreover, over the last few years it has become increasingly apparent that an on-shell expression of the no-hair theorem is that black holes correspond to \\textit{minimal coupling} in classical limits of quantum scattering amplitudes for massive spin~$s$ particles and gravitons.  Amplitudes for long-range gravitational scattering of spin 1\/2 and spin 1 particles were found in \\cite{Ross:2007zza,Holstein:2008sx} to give the universal spin-orbit (pole-dipole level) couplings in the post-Newtonian corrections to the gravitational potential.  Further similar work in \\cite{Vaidya:2014kza}, up to spin 2, suggested that the black hole multipoles \\eqref{eqn:multipoles} up to order $\\ell=2s$ are faithfully reproduced from tree-level amplitudes for minimally coupled spin~$s$ particles.\n\nSuch amplitudes for arbitrary spin $s$ were computed in \\cite{Guevara:2017csg}, by adopting the representation of minimal coupling for arbitrary spins presented in \\cite{Arkani-Hamed:2017jhn} using the massive spinor-helicity formalism---see also~\\cite{Conde:2016vxs,Conde:2016izb}. Those amplitudes were shown in \\cite{Guevara:2018wpp,Bautista:2019tdr} to lead in the limit $s\\to\\infty$ to the two-black-hole aligned-spin scattering angle found in \\cite{Vines:2017hyw} at first post--Minkowskian order and to all orders in the spin-multipole expansion, while in \\cite{Chung:2018kqs} they were shown to yield the contributions to the interaction potential (for arbitrary spin orientations) at the leading post--Newtonian orders at each order in spin. Meanwhile in \\cite{Bern:2020buy,Kosmopoulos:2021zoq} amplitudes for arbitrary spin fields were combined with the powerful effective theory matching techniques of \\cite{Cheung:2018wkq} to yield the first dipole-quadrapole coupling calculation at 2PM order. Methods from heavy quark effective theory \\cite{Damgaard:2019lfh,Aoude:2020onz,Haddad:2020tvs} and quantum information \\cite{Aoude:2020mlg} have also proven applicable to spinning black hole scattering, the former leading to the first amplitudes treatment of tidal effects on spinning particles \\cite{Aoude:2020ygw}.\n\nTo replicate the behaviour of Kerr black holes, the massive spin $s$ states in amplitudes must be minimally coupled to the graviton field, by which we mean that the high energy limit is dominated by the corresponding helicity configuration of massless particles \\cite{Arkani-Hamed:2017jhn}. This has been especially emphasised in \\cite{Chung:2018kqs}, where, by matching at tree-level to the classical effective action of Levi and Steinhoff \\cite{Levi:2015msa}, it was shown that the theory which reproduces the infinite-spin limit of minimally coupled graviton amplitudes is an effective field theory (EFT) of spinning black holes. It can be explicitly shown that any deviation from minimal coupling adds further internal structure to the effective theory \\cite{Chung:2019duq,Chung:2019yfs,Chung:2020rrz}, departing the special black hole case. \n\nApplying amplitudes methods to the scattering of any spinning object, black hole or otherwise, we face the familiar problem of requiring an appropriate observable and precise understanding of the classical limit. When the spins of scattering objects are not aligned there no longer exists a well defined scattering plane, and thus the most common observable calculated from classical potentials, the scattering angle, becomes meaningless. We shall therefore apply the methods developed in Part~\\ref{part:observables} to quantum field theories of particles with spin, setting up observables in terms of scattering amplitudes which can fully specify the dynamics of spinning black holes. When the spins are large these methods are known to exactly reproduce established 1PM results \\cite{Guevara:2019fsj,Vines:2017hyw}. \n\nAfter systematically dealing with the classical limit of quantum spinning particles, we will apply insights from amplitudes to the classical dynamics of Kerr and its single copy, $\\rootKerr$. We will utilise the fact that the beautiful relationship between Kerr black holes and minimally coupled amplitudes goes far deeper than simply being a powerful calculational tool. Amplitudes can explain and reveal structures in general relativity that are obscured by geometrical perspectives: for example, the double copy. \n\nAnother key example in the context of spin is the fact, first noted by Newman and Janis in \\cite{Newman:1965tw}, that the Kerr metric can be obtained from Schwarzschild by means of a complex coordinate transformation. This is easy to see when the metrics are in Kerr--Schild form: take the data for Scwharzschild in \\eqn~\\eqref{eqn:blackholesKSforms}. Under the transformation $z \\rightarrow z + ia$,\n\\[\nr^2 \\rightarrow &\\, r^2 + 2iaz - a^2 \\\\ &\\equiv  \\tilde r^2 - \\frac{a^2 z^2}{\\tilde r^2} + 2ia\\tilde{r}\\cos\\theta = (\\tilde r +i a\\cos\\theta)^2\\,,\n\\]\nwhere the Kerr radial coordinate $\\tilde r$ is defined in \\eqn~\\eqref{eqn:KerrKSradial}. Hence under $z \\rightarrow z + ia$ we have that $r \\rightarrow \\tilde r + ia\\cos\\theta$, and moreover,\n\\[\n\\varphi_{\\rm Schwz}(r) \\rightarrow &\\, 2GM\\Re\\left\\{\\frac1{r}\\right\\}\\bigg|_{r\\rightarrow \\tilde r + ia\\cos\\theta}\\\\ &= \\frac{2GM \\tilde r}{(\\tilde r^2 + a^2 \\cos^2\\theta)} \\equiv \\varphi_{\\rm Kerr}(\\tilde r,\\theta)\\,.\n\\]\nIn other words, the Kerr solution looks like a complex translation of the Schwarzschild solution \\cite{Newman:2002mk}. Clearly the same shift holds in the gauge theory single copies. This is but one example of complex maps between spacetimes; further examples were derived by Talbot \\cite{Talbot:1969bpa}, encompassing the Kerr--Newman and Taub--NUT solutions.\n\nA closely related way to understand these properties of classical solutions is to consider their Weyl curvature spinor $\\Psi$. For example, with appropriate coordinates the NJ shift applies exactly to the spinor:\n\\[\n\\Psi^\\text{Kerr}(x) = \\Psi^\\text{Schwarzschild}(x + i a) \\,,\\label{eqn:NJshift}\n\\]\nSimilarly, in the electromagnetic $\\rootKerr$~case~\\cite{Newman:1965tw} it is the Maxwell spinor $\\maxwell$ that undergoes a shift:\n\\[\n\\maxwell^{\\sqrt{\\text{Kerr}}}(x) = \\maxwell^\\text{Coulomb}(x + i a) \\,.\n\\]\nTherefore, $\\rootKerr$~is a kind of complex translation of the Coulomb solution. \n\nAlthough these complex maps are established classically, there is no geometric understanding for \\textit{why} such a complex map holds. However, this is not the case from the perspective of amplitudes --- it was explicitly shown in \\cite{Arkani-Hamed:2019ymq} that the Newman--Janis shift is a simple consequence of the the exponentiation of minimally coupled amplitudes in the large spin limit. The simplicity of minimally coupled amplitudes has since been utilised to explain a wider range of complex mappings between spacetime and gauge theory solutions \\cite{Moynihan:2019bor,Huang:2019cja,Moynihan:2020gxj,Kim:2020cvf}, culminating in a precise network of relationships constructed from the double copy, Newman--Janis shifts and electric-magnetic duality \\cite{Emond:2020lwi}. These investigations have relied on the on-shell observables we will introduce in Part~\\ref{part:observables} \\cite{Kosower:2018adc}.\n\nInspired by the insights offered by amplitudes, we will adopt the complex Newman--Janis shift as a starting point for investigating the classical dynamics of these unique spinning objects. In particular, we will show that interacting effective actions for Kerr and $\\rootKerr$, in the vein of Levi and Steinhoff \\cite{Levi:2015msa}, can be interpreted as actions for a complex worldsheet. We will also apply the power of the massive spinor helicity representations of ref.~\\cite{Arkani-Hamed:2017jhn} to classical dynamics, rapidly deriving on-shell scattering observables for Kerr and $\\rootKerr$ from spinor equations of motion. Although working purely classically, our methodology and philosophy will be entirely drawn from amplitudes-based investigations.\n\n\\section{Summary}\n\nTo summarise, the structure of this thesis is as follows. Part~\\ref{part:observables} is dedicated to the construction of on-shell observables which are well defined in both classical and quantum field theory. We begin in chapter~\\ref{chap:pointParticles} by setting up single particle quantum wavepackets which describe charged scalar point-particles in the classical limit. In chapter~\\ref{chap:impulse} we then turn to descriptions of point-particle scattering by considering our first observable, the impulse, or the total change in the momentum of a scattering particle. We derive general expressions for this observable in terms of amplitudes, and undertake a careful examination of the classical limit, extracting the rules needed to pass from the quantum to the classical regime. In chapter~\\ref{chap:radiation} we introduce the total radiated momentum and demonstrate momentum conservation and the automatic handling of radiation reaction effects in our formalism. We introduce spin in part~\\ref{part:spin}, which is concerned with spinning black holes. In chapter~\\ref{chap:spin} we construct on-shell observables in QFT for spinning particles, reproducing results for Kerr black holes after considering in detail the classical limit of amplitudes with finite spin. We then return to classical dynamics in chapter~\\ref{chap:worldsheet}, using insights from structures in on-shell amplitudes to uncover worldsheet effective actions for $\\rootKerr$ and Kerr particles. We finish by discussing our results in~\\ref{chap:conclusions}. Results in chapters~\\ref{chap:pointParticles}, \\ref{chap:impulse} and~\\ref{chap:radiation} were published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}, chapter~\\ref{chap:spin} is based on \\cite{Maybee:2019jus}, and chapter~\\ref{chap:worldsheet} appeared in \\cite{Guevara:2020xjx}. \n\n\\subsection{Conventions}\n\nIn all the work that follows, our conventions for Fourier transforms are\n\\begin{equation}\nf(x) = \t\\int\\!\\frac{\\d^n q}{(2\\pi)^n}\\, \\tilde{f}(q) e^{-i q\\cdot x}\\,, \\qquad \\tilde{f}(q) = \\int\\! d^4x\\, f(x) e^{i q\\cdot x}\\,.\n\\end{equation}\nWe will consistently work in relativistically natural units where $c=1$, however we will always treat $\\hbar$ as being dimensionful. We work in the mostly minus metric signature $(+,-,-,-)$, where we choose $\\epsilon_{0123} = +1$ for the Levi--Civita tensor. We will occasionally find it convenient to separate a Lorentz vector $x^\\mu$ into its time component $x^0$ and its spatial components $\\v{x}$, so that $x^\\mu = (x^0,x^i) = (x^0, \\v{x})$, where $i=1,2,3$. \n\nFor a given tensor $X$ of higher rank, total symmmetrisation and antisymmetrisation respectively of tensor indices are represented as usual by\n\\begin{equation}\n\\begin{aligned}\nX^{(\\mu_1} \\dots X^{\\mu_n)} &= \\frac1{n!}\\left(X^{\\mu_1} X^{\\mu_2} \\dots X^{\\mu_n} + X^{\\mu_2} X^{\\mu_1} \\dots X^{\\mu_n} + \\cdots\\right)\\\\\nX^{[\\mu_1} \\dots X^{\\mu_n]} &= \\frac1{n!}\\left(X^{\\mu_1} X^{\\mu_2} \\dots X^{\\mu_n} - X^{\\mu_2} X^{\\mu_1} \\dots X^{\\mu_n} + \\cdots\\right).\n\\end{aligned}\n\\end{equation}\nFinally, our definition of the amplitude will consistently differ by a phase factor relative to the standard definition used for the double copy. Here, in either gauge theory or gravity\n\\begin{equation}\ni\\mathcal{A}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) = \\sum \\left(\\text{Feynman diagrams}\\right)\\,,\n\\end{equation}\nwhereas in the convention used in the original work of BCJ~\\cite{Bern:2008qj,Bern:2010ue} the entire left hand side is defined as the amplitude.\n\\part{Classical observables from quantum field theory}\t\n\\label{part:observables}\n\n\\chapter{Point-particles}\n\\label{chap:pointParticles}\n\nOn-shell amplitudes in quantum field theory are typically calculated on a basis of plane wave states: the context of the calculation is the physics of states with definite momenta, but indefinite positions. However, to capture the physics of a black hole in quantum mechanics, or indeed any other classical point-particle, this is clearly not sufficient: we need states which are well localised. We also need quantum states that accurately correspond to point-particles when the ``classical limit'' is taken. These requirements motivate the goals of this first chapter. We will precisely specify what we mean by the classical limit, and explicitly construct localised wavepackets which describe single, non-spinning point-particles in this limit. The technology that we develop will provide the foundations for our construction of on-shell observables for interacting black holes in later chapters.\n\nTo ensure full generality we will consider charged particles, studying the classical limits of states in an $SU(N)$ gauge group representation. Such point-particles are described by the Yang--Mills--Wong equations in the classical regime~\\cite{Wong:1970fu}. For gravitational physics one could have in mind an Einstein--Yang--Mills black hole, but there are more interesting perspectives available. YM theory, treated as a classical field theory, shares many of the important physical features of gravity, including non-linearity and a subtle gauge structure. In this respect the YM case has always served as an excellent toy model for gravitational dynamics. But, as we discussed in the previous chapter, our developing understanding of the double copy has taught us that the connection between Yang--Mills theory and gravity is deeper than this; detailed aspects\nof the perturbative dynamics of gravity, including gravitational radiation, can be deduced from Yang--Mills theory and the double copy. Understanding non--trivial gauge, or \\textit{colour}, representation states will thus play a key role in our later calculations of black hole observables. \n\nThis chapter is based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}, in collaboration with Leonardo de la Cruz, David Kosower, Donal O'Connell and Alasdair Ross.\n\n\\section{Restoring $\\hbar$}\n\\label{sec:RestoringHBar}\n\nTo extract the classical limit of a quantum mechanical system describing the physics of point-particles we are of course going to need to be careful in our treatment of Planck's constant, $\\hbar$. A straightforward and pragmatic approach to restoring all factors of $\\hbar$ in an expression is dimensional analysis: we denote the dimensions of mass and length by $[M]$ and $[L]$ respectively.\n\nWe may choose the dimensions of an $n$-point scattering amplitude in four dimensions to be $[M]^{4-n}$ even when $\\hbar \\neq 1$. This is consistent with choosing the dimensions of creation and annihilation operators so that\n\\begin{equation}\n[a_i(p), a^{\\dagger j}(p')] = 2E_p (2\\pi)^3 \\delta^{(3)}(\\v{p} - \\v{p}')\\,\\delta_i{ }^j\\,,\\label{eqn:ladderCommutator}\n\\end{equation}\nHere the indices label the representation $R$ of any Lie group. We define single-particle momentum eigenstates in this representation by\n\\begin{equation}\n|p^i \\rangle = a^{\\dagger i}(p) |0\\rangle\\,.\\label{eqn:singleParticleStateDef}\n\\end{equation}\nSince the vacuum state is taken to be dimensionless, the dimension of $|p^i\\rangle$ is thus $[M]^{-1}$. We further define $n$-particle asymptotic states as tensor products of these normalised single-particle states. In order to avoid an unsightly splatter of factors of $2\\pi$, it is convenient to define\n\\begin{equation}\n\\del^{(n)}(p) \\equiv (2\\pi)^n \\delta^{(n)}(p)\n\\label{eqn:delDefinition}\n\\end{equation}\nfor the $n$-fold Dirac $\\delta$ distribution. With these conventions the state normalisation is\n\\begin{equation}\n\\langle p'_i | p^j \\rangle = 2 E_p \\, \\del^{(3)} (\\v{p}-\\v{p}') \\delta_i{ }^j\\,.\n\\label{eqn:MomentumStateNormalization}\n\\end{equation}\nWe define the amplitudes in four dimensions on this plane wave basis by\n\\begin{multline}\n\\langle p'_1 \\cdots p'_m  | T | p_1 \\cdots p_n \\rangle = \\Ampl(p_1 \\cdots p_n \\rightarrow p'_1 \\cdots p'_m) \\\\ \\times \\del^{(4)}(p_1 + \\cdots p_n - p'_1 - \\cdots - p'_m)\\,.\n\\label{eqn:amplitudeDef}\n\\end{multline}\nThe scattering matrix $S$ and the transition matrix $T$ are both dimensionless, leading to the initially advertised dimensions for amplitudes.\n\nLet us now imagine restoring the $\\hbar$'s in a given amplitude. When $\\hbar = 1$, the amplitude has dimensions of $[M]^{4-n}$.  When $\\hbar \\neq 1$, the dimensions of the momenta and masses in the amplitude are unchanged. Similarly there is no change to the dimensions of polarisation vectors. However, we must remember that the dimensionless coupling in electrodynamics is $e\/\\sqrt{\\hbar}$. Similarly, in gravity a factor of $1\/\\sqrt{\\hbar}$ appears, as the appropriate coupling with dimensions of inverse mass is $\\kappa = \\sqrt{32 \\pi G\/ \\hbar}$. We will see shortly that the situation is a little more intricate in Yang-Mills theory, as the colour factors can carry dimensions of $\\hbar$. However we will establish conventions such that the coupling has the same scaling as the QED\/gravity case.  The algorithm to restore the dimensions of any amplitude in electrodynamics, chromodynamics or gravity is then simple: each factor of a coupling is multiplied by an additional factor of $1\/\\sqrt{\\hbar}$. For example, an $n$-point, $L$-loop amplitude in scalar QED is proportional to $\\hbar^{1-n\/2-L}$. \n\nThis conclusion, though well-known, may be surprising in the present context because it seems na\\\"{i}vely that as $\\hbar \\rightarrow 0$, higher multiplicities and higher loop orders are \\textit{more\\\/} important. However, when restoring powers of $\\hbar$ one must distinguish between the momentum $p^\\mu$ of a particle and its wavenumber, which has dimensions of $[L]^{-1}$. This distinction will be important for us, so we introduce a notation for the wavenumber $\\barp$ associated with a momentum $p$:\n\\begin{align}\n\\wn p \\equiv p \/ \\hbar.\n\\label{eqn:notationWavenumber}\n\\end{align}\nIn the course of restoring powers of $\\hbar$ by dimensional analysis, we will first treat the momenta of all particles as genuine momenta. We will also treat any mass as a mass, rather than the associated (reduced) Compton wavelength $\\ell_c = \\hbar\/m$.\n\nAs we will, the approach to the classical limit --- for observables that make sense classically --- effectively forces the wavenumber scaling upon certain momenta. Examples include the momenta of massless particles, such as photons or gravitons.  In putting the factors of $\\hbar$ back into the couplings, we have therefore not yet made manifest all of the physically relevant factors of $\\hbar$ in an amplitude. This provides one motivation for this part of the thesis: we wish to construct on-shell observables which are both classically and quantum-mechanically sensible. \n\n\\section{Single particle states}\n\\label{sec:stateSetup}\n\nLet us take a generic single particle state expanded on the plane wave basis of \\eqref{eqn:singleParticleStateDef}:\n\\begin{equation}\n|\\psi\\rangle = \\sum_i \\int\\! \\dd^4 p \\, \\delp(p^2 - m^2) \\, \\psi_i(p) \\, |p^i\\rangle\\,,\\label{eqn:InitialState}\n\\end{equation}\nHere $\\dd p$ absorbs a factor of $2 \\pi$; more generally $\\dd^n p$ is defined by\n\\begin{equation}\n\\dd^n p \\equiv \\frac{d^n p}{(2 \\pi)^n}\\,.\n\\label{eqn:ddxDefinition}\n\\end{equation}\nWe restrict the integration to positive-energy solutions of the delta functions of $p^2-m^2$, as indicated by the $(+)$ superscript in $\\delp$, as well as absorbing a factor of $2\\pi$, just as for $\\del(p)$:\n\\begin{equation}\n\\delp(p^2-m^2) \\equiv 2\\pi\\Theta(p^0)\\delta(p^2-m^2)\\,.\n\\label{eqn:delpDefinition}\n\\end{equation}\nWe will find it convenient to further abbreviate the notation for on-shell integrals (over Lorentz-invariant phase space), defining\n\\begin{equation}\n\\df(p) \\equiv \\dd^4 p \\, \\delp(p^2-m^2)\\,.\n\\label{eqn:dfDefinition}\n\\end{equation}\nWe will generally leave the mass implicit, along with the designation of the integration variable as the first summand when the argument is a sum. Note that the right-hand side of~\\eqref{eqn:MomentumStateNormalization} is the appropriately normalised delta function for this measure, \n\\begin{equation}\n\\label{eqn:norm1}\n\\int \\df(p') \\, 2 E_{p'} \\, \\del^{(3)} (\\v{p}-\\v{p}') f(p') = f(p)\\,.\n\\end{equation}\nThus for any function $f(p_1')$, we define\n\\begin{equation}\n\\Del(p-p') \\equiv  2 E_{p'} \\del^{(3)} (\\v{p}-\\v{p}')\\,.\n\\end{equation}\nThe argument on the left-hand side is understood as a function of four-vectors. This leads to a notationally clearer version of \\eqn~\\eqref{eqn:norm1}:\n\\begin{equation}\n\\int \\df(p') \\, \\Del(p - p') f(p') = f(p)\\,,\n\\end{equation}\nand of \\eqn~\\eqref{eqn:MomentumStateNormalization}:\n\\begin{equation}\n\\langle p'_i | p^j \\rangle = \\Del(p-p') \\delta_i{ }^j\\,.\n\\end{equation}\n\nThe full state $|\\psi\\rangle$ is a non-trivial representation, of a Lie group associated with symmetries which constrain the description of our particle. The kinematic data, however, should be independent of these symmetries, and thus a singlet of $R$. The full state is thus a tensor product of momentum and representation states: \n\\begin{align}\n\\ket{\\psi}= \\sum \\ket{\\psi_{\\text{mom}}} \\otimes  \\ket{\\psi_{R}}.\n\\end{align}\nWe will make this explicit by splitting the wavefunctions $\\psi_i(p)$, writing\n\\begin{equation}\n\\sum_i\\psi_i(p)| p^i\\rangle = \\sum_i \\varphi(p) \\chi_i |p^i\\rangle = \\varphi(p) |p\\, \\chi\\rangle\\,.\\label{eqn:wavefunctionSplit}\n\\end{equation}\n\nIn these conventions, \\eqn~\\eqref{eqn:InitialState} becomes\n\\begin{equation}\n| \\psi \\rangle = \\int \\! \\df(p)\\;\n\\varphi(p) | p \\, \\chi \\rangle\\,.\n\\label{eqn:InitialStateSimple}\n\\end{equation}\nUsing this simplified notation, the normalisation condition is\n\\begin{equation}\n\\begin{aligned}\n1 &= \\langle \\psi | \\psi \\rangle \\\\\n&= \\sum_{i,j} \\! \\int \\! \\df(p) \\df(p') \\varphi^*(p_1') \\varphi(p_1) \\chi^{*i} \\chi_j\\, \\Del(p_1 - p_1') \\, \\delta_i{ } ^j\\\\\n&= \\sum_i \\! \\int \\! \\df(p)\\; |\\varphi(p)|^2 |\\chi_i|^2\\,.\n\\end{aligned}\n\\end{equation}\nWe can obtain this normalisation by requiring that both wavefunctions $\\phi(p)$ and $\\chi_i$ be normalised to unity:\n\\begin{equation}\n\\int \\! \\df(p)\\; |\\varphi(p)|^2 = 1\\,, \\qquad \\sum_i \\chi^{i*} \\chi_i = 1\\,.\n\\label{eqn:WavefunctionNormalization}\n\\end{equation}\n\nSince $|\\psi\\rangle$ is expanded on a basis of momentum eigenstates, it is trivial to measure the momentum of the state with the momentum operator $\\mathbb P^\\mu$:\n\\begin{equation}\n\\langle \\psi|\\mathbb{P}^\\mu |\\psi\\rangle = \\int \\! \\df(p)\\; p^\\mu |\\varphi(p)|^2\\label{eqn:momentumExp}\n\\end{equation} \nBut how do we measure the physical charges associated with the representation states $|\\chi\\rangle$?\n\n\\subsection{Review of the theory of colour}\n\\label{sec:setup}\n\nFor a physical particle state, there are two distinct interpretations for the representation $R$: it could be the irreducible representation of the little group for a particle of non-zero spin; or it could be the representation of an internal symmetry group of the theory. In this first part of the thesis we are only interested in scalar particle states. We will therefore restrict to the second option for the time being, returning to little group representations in part~\\ref{part:spin}. \n\nOur ultimate goal is to extract, from QFT, long-range interactions between point particles, mediated by a classical field. We will therefore take $R$ to be any representation of an $SU(N)$ gauge group. The classical dynamics of the corresponding Yang--Mills field $A_\\mu = A_\\mu^a T^a$, coupled to several classical point-like particles, are then described by the Yang--Mills--Wong equations:. \n\\begin{subequations}\n\t\\label{eqn:classicalWong}\n\t\\begin{gather}\n\t\\frac{\\d p_\\alpha^\\mu }{\\d \\tau_\\alpha} = g\\, c^a_\\alpha(\\tau_\\alpha)\\, F^{a\\,\\mu\\nu}\\!(x_\\alpha(\\tau_\\alpha))\\,  v_{\\alpha\\, \\nu}(\\tau_\\alpha)\\,,  \\label{eqn:Wong-momentum} \n\t\\\\\n\t\\frac{\\d c^a_\\alpha}{\\d \\tau_\\alpha}= g f^{abc} v^\\mu_\\alpha(\\tau_\\alpha) A_\\mu^b(x_\\alpha(\\tau_\\alpha))\\,c^c_\\alpha(\\tau_\\alpha)\\,,\n\t\\label{eqn:Wong-color}\n\t\\\\\n\tD^\\mu F_{\\mu\\nu}^a(x) = J^a_\\nu(x) = g \\sum\\limits_{\\alpha= 1}^N \\int\\!\\d\\tau_\\alpha\\,  c^a_\\alpha(\\tau_\\alpha)  v^\\mu(\\tau_\\alpha)\\, \\delta^{(4)}(x-x(\\tau_\\alpha))\\,, \\label{eqn:YangMillsEOM}\n\t\\end{gather}\n\\end{subequations}\nThese equations describe particles, following worldlines $x_\\alpha(\\tau_\\alpha)$ and with velocities $v_\\alpha$, that each carry colour charges $c^a$ which are time-dependent vectors in the adjoint-representation of the gauge group. \n\nLet us review the emergence of of these non-Abelian colour charges from quantum field theory by restricting our attention to scalars $\\pi_\\alpha$ in any representation $R_\\alpha$ of the gauge group, coupled to the Yang--Mills field. The action is\n\\begin{equation}\nS = \\int\\!\\d^4x\\, \\left(\\sum_{\\alpha}\\left[ (D_\\mu \\pi_\\alpha)^\\dagger D^\\mu \\pi_\\alpha - \\frac{m_\\alpha^2}{\\hbar^2} \\pi_\\alpha^\\dagger \\pi_\\alpha\\right] - \\frac14 F^a_{\\mu\\nu} F^{a\\,\\mu\\nu}\\right), \\label{eqn:scalarAction}\n\\end{equation}\nwhere  $D_\\mu = \\partial_\\mu + i g A_\\mu^a T^a_R$. The generator matrices (in a representation $R$) are $T^a_R = (T_R^a)_i{ }^j$, and satisfy the Lie algebra \n$[T_R^a, T_R^b]_i{ }^j = if^{abc} (T_R^c)_i{ }^j$. \n\nLet us consider only a single massive scalar. At the classical level, the colour charge can be obtained from the Noether current $j^a_\\mu$ associated with the global part of the gauge symmetry. The colour charge is explicitly given by\n\\begin{equation}\n\\int\\!\\d^3x\\, j^a_0(t,\\v{x}) = i\\!\\int\\! \\d^3x\\, \\Big(\\pi^\\dagger T^a_R\\, \\partial_0 \\pi - (\\partial_0 \\pi^\\dagger) T^a_R\\, \\pi\\Big)\\,.  \\label{eqn:colourNoetherCharge}\n\\end{equation}\nNotice that a direct application of the Noether procedure has led to a colour charge with dimensions of action, or equivalently, of angular momentum. It is now worth dwelling on dimensional analysis in the context of the Wong equations~\\eqref{eqn:classicalWong}, since they motivate us to make certain choices which may, at first, seem surprising. The Yang--Mills field strength\n\\begin{equation}\nF^a_{\\mu\\nu}  =\\partial_\\mu A^a_\\nu - \\partial_\\nu A^a_\\mu - gf^{abc} A^b_\\mu A^c_\\nu\\,\\label{eqn:fieldStrength}\n\\end{equation}\nis obviously an important actor in these classical equations. Classical equations should contain no factors\\footnote{An equivalent point of view is that any factors of $\\hbar$ appearing in an equation which has classical meaning should be absorbed into parameters of the classical theory.} of $\\hbar$, so we choose to maintain this precise expression for the field strength when $\\hbar \\neq 1$. By inspection it follows that $[g A^{a}_\\mu] = L^{-1}$. We can develop this further; since the action of \\eqn~\\eqref{eqn:scalarAction} has dimensions of angular momentum, the Yang--Mills field strength must have dimensions of $\\sqrt{M\/L^3}$. Thus, from \\eqn~\\eqref{eqn:fieldStrength},\n\\begin{equation}\n[A^a_\\mu] = \\sqrt{\\frac{M}{L}}\\,, \\qquad [g] = \\frac1{\\sqrt{ML}}\\,.\\label{eqn:YMdims}\n\\end{equation}\nThis conclusion about the dimensions of $g$ is in contrast to the situation in electrodynamics, where $[e] = \\sqrt{ML}$. Put another way, in electrodynamics the dimensionless fine structure constant is $e^2 \/ 4\\pi \\hbar$ while in our conventions the Yang--Mills analogue is $\\hbar g^2 \/ 4\\pi\\,$! It is possible to arrange matters such that the YM and EM cases are more similar, but we find the present conventions to be convenient in perturbative calculations.\n\nContinuing with our discussion of dimensions, note that the Yang--Mills version of the Lorentz force, \\eqn~\\eqref{eqn:Wong-momentum}, demonstrates that the quantity $g c^a$ must have the same dimension as the electric charge. This is consistent with our observation above that the colour has dimensions of angular momentum.\n\nAt first our assignment of dimensions of $g$ may seem troubling; the fact that $g$ has dimensions of $1\/\\sqrt{ML}$ implies that the dimensionless coupling at each vertex is $g \\sqrt \\hbar$, so factors of $\\hbar$ associated with the coupling appear with the opposite power to the case of electrodynamics (and gravity). However, because the colour charges are dimensionful the net power of $\\hbar$ turns out to be the same. The classical limit of this aspect of the theory is clarified by the dimensionful nature of the colour --- to see how this works we must quantise. \n\nDimensional analysis demonstrates that the field $\\pi$ has dimensions of $\\sqrt{M\/L}$, so its mode expansion is\n\\begin{equation}\n{\\pi}_i(x) = \\frac1{\\sqrt{\\hbar}} \\int\\!\\df(p)\\, \\left(a_i(p) e^{-ip\\cdot x\/\\hbar} + b_i^\\dagger(p) e^{ip\\cdot x\/\\hbar}\\right).\n\\end{equation}\nThe ladder operators are normalised as in equation~\\eqref{eqn:ladderCommutator}, with the index $i$ again labelling the representation $R$. After quantisation, the colour charge of \\eqn~\\eqref{eqn:colourNoetherCharge} becomes a Hilbert space operator,\n\\begin{equation}\n\\begin{aligned}\n\\C^a &= i\\!\\int\\! \\d^3x\\, \\Big(\\pi^\\dagger T^a_R\\, \\partial_0 \\pi - (\\partial_0 \\pi^\\dagger) T^a_R\\, \\pi\\Big) \\\\\n&= \\hbar\\int\\!\\df(p)\\, \\left( a^\\dagger(p) \\,T^a_R \\, a(p) + b^\\dagger(p)\\, T^a_{\\bar R} \\, b(p)\\right),\\label{eqn:colourOp}\n\\end{aligned}\n\\end{equation}\nwhere we have used that the generators of the conjugate representation $\\bar R$ satisfy $T^a_{\\bar R} = - T^a_R$. The overall $\\hbar$ factor guarantees that the colour has dimensions of angular momentum, as we require. It is important to note that these global colour operators inherit the usual Lie algebra of the generators, modified by factors of $\\hbar$, so that\n\\begin{equation}\n[\\C^a, \\C^b] = i\\hbar f^{abc} \\C^c\\,.\\label{eqn:chargeLieAlgebra}\n\\end{equation}\n\nActing with the colour charge operator of \\eqn~\\eqref{eqn:colourOp} on momentum eigenstates (as defined in \\eqn~\\eqref{eqn:singleParticleStateDef}), we immediately see that\n\\begin{equation}\n\\C^a|p^i\\rangle = \\hbar\\, (T^a_R)_j{ }^i|p^j\\rangle\\,, \\qquad \\langle p_i|\\C^a = \\hbar\\, \\langle p_j|(T^a_R)_i{ }^j\\,.\n\\end{equation}\nThus inner products yield generators scaled by $\\hbar$:\n\\begin{equation}\n\\langle p_i|\\C^a|p^j\\rangle \\equiv (\\newT^a)_i{ }^j = \\hbar\\, (T^a_R)_i{ }^j\\,.\n\\end{equation}\nThe $(C^a)_i{ }^j$ are simply rescalings of the usual generators $T^a_R$ by a factor of $\\hbar$, and thus satisfy the rescaled Lie algebra in \\eqn~\\eqref{eqn:chargeLieAlgebra}; since this rescaling is important for us, it is useful to make the distinction between the two. \n\nWe can now finally act with the colour operator on the single particle state of equation~\\eqref{eqn:InitialStateSimple}:\n\\begin{equation}\n\\C^a|\\psi\\rangle = \\int\\!\\df(p)\\, (\\newT^a)_i{ }^j\\, \\varphi(p) \\chi_j|p^i\\rangle\\,,\n\\end{equation}\nallowing us to define the colour charge of the particle as\n\\begin{equation}\n\\langle \\psi |\\C^a| \\psi \\rangle =  \\chi^{i*} (\\newT^a)_{i}{ }^{j}\\, \\chi_j\\,. \\label{eqn:colourCharge}\n\\end{equation}\n\nAs a final remark on the rescaled generators, let us write out the covariant derivative in the representation $R$. In terms of $\\newT^a$, \nthe $\\hbar$ scaling of interactions is precisely the same as in QED (and in perturbative gravity):\n\\begin{equation}\nD_\\mu = \\partial_\\mu + i \\, g A^a_\\mu T^a = \\partial_\\mu + \\frac{ig}{\\hbar}\\, A^a_\\mu \\newT^a \\,;\\label{eqn:covDerivative}\n\\end{equation}\nfor comparison, the covariant derivative in QED consistent with our discussion in section~\\ref{sec:RestoringHBar} is $\\partial_\\mu + i e A_\\mu\/ \\hbar$. Thus we have arranged that factors of $\\hbar$ appear in the same place in YM theory as in electrodynamics, provided that the colour is measured by $C^a$. This ensures that the basic rules for obtaining the classical limits of amplitudes will be the same; in practical calculations one restores $\\hbar$'s in colour factors and works using $C^a$'s everywhere. However, it is worth emphasising that unlike classical colour charges, the factors $C^a$ do not commute.\n\n\\section{Classical point-particles}\n\\label{sec:PointParticleLimit}\n\nFor the states in equation~\\eqref{eqn:InitialStateSimple} to have a well defined point-particle limit, for any operator $\\mathbb{O}$ they must, at a bare minimum, satisfy the following two constraints in the classical limit \\cite{Yaffe:1981vf}:\n\\[\n\\langle \\psi |\\mathbb{O} | \\psi \\rangle &= \\textrm{finite} \\,, \\\\\n\\langle \\psi |\\mathbb{O} \\, \\mathbb{O}| \\psi \\rangle &= \\langle \\psi |\\mathbb{O} | \\psi \\rangle\\langle \\psi | \\mathbb{O}| \\psi \\rangle + \\textrm{negligible} \\,.\\label{eqn:classicalConstraints}\n\\]\nFurthermore the classical limit is not necessarily injective: distinct quantum states $|\\psi\\rangle$ and $|\\psi'\\rangle$ may yield the same classical limit. Classical physics should of course be independent of the details of quantum states, and therefore we also require that in the limit, the overlap\n\\begin{equation}\n\\langle \\psi' | \\psi \\rangle = \\langle \\psi | \\psi \\rangle + \\textrm{negligible}\\,.\\label{eqn:classicalOverlap}\n\\end{equation}\nSimilarly, the expectation values above should remain unchanged in the limit if taken over distinct but classicaly equivalent states \\cite{Yaffe:1981vf}.\n\nOur goal in this section is to choose suitable momentum and colour wavefunctions, $\\varphi(p)$ and $\\chi$ respectively, which ensure that the observables in equations~\\eqref{eqn:momentumExp} and~\\eqref{eqn:colourCharge} meet these crucial requirements.\n\n\\subsection{Wavepackets}\n\\label{subsec:Wavefunctions}\n\nClassical point particles have well defined positions and momenta. Heuristically, we therefore require well localised quantum states. We will take the momentum space wavefunctions $\\varphi(p)$ to be wavepackets, characterised by a smearing or spread in momenta\\footnote{Evaluating positions and uncertainties therein in\trelativistic field theory is a bit delicate, and we will not consider the question in this thesis.}.   \n\nLet us ground our intuition by first examining nonrelativistic wavefunctions.  An example of a minimum-uncertainty wavefunction in momentum space (ignoring normalisation) for a particle of mass $m$ growing sharper in the $\\hbar\\rightarrow 0$ limit has the form\n\\begin{equation}\n\\exp\\left( -\\frac{\\v{p}\\mskip1mu{}^2}{2 \\hbar m \\lcomp\/ \\lpack^2}\\right)\n= \\exp\\left( -\\frac{\\v{p}\\mskip1mu{}^2}{2m^2 \\lcomp^2\/\\lpack^2}\\right),\n\\label{eqn:NonrelativisticMomentumSpaceWavefunction}\n\\end{equation}\nwhere $\\lcomp$ is the particle's Compton wavelength, and where $\\lpack$ is an additional parameter with dimensions of length. We can obtain the conjugate in position space by Fourier transforming:\n\\begin{equation}\n\\exp\\left( -\\frac{(\\v{x}-\\v{x}_0)^2}{2 \\lpack^2}\\right).\n\\end{equation}\nThe precision with which we know the particle's location is given by $\\lpack$, which we could take as an intrinsic measure of the wavefunction's spread.\n\nThis suggests that in considering relativistic wavefunctions, we should also take the dimensionless parameter controlling the approach to the classical limit in momentum space to be the square of the ratio of the Compton wavelength $\\lcomp$ to the intrinsic spread~$\\lpack$,\n\\begin{equation}\n\\xi \\equiv \\biggl(\\frac{\\lcomp}{\\lpack}\\biggr){\\vphantom{\\frac{\\lcomp}{\\lpack}}}^2\\,.\\label{eqn:defOfXi}\n\\end{equation}\nWe therefore obtain the classical result by studying the behaviour of expectation values as $\\xi\\rightarrow 0$; or alternatively, in the region where\n\\begin{equation}\n\\ell_c \\ll \\ell_w\\,.\\label{eqn:ComptonConstraint1}\n\\end{equation}\nTowards the limit, the wavefunctions must be sharply peaked around the classical value for the momenta, $\\pcl = m \\ucl$, with the classical four-velocity $\\ucl$ normalised to $\\ucl^2 = 1$.  We can express this requirement through the conditions\\footnote{The integration measure for $p$ enforces $\\langle p^2\\rangle = m^2$.}\n\\begin{equation}\n\\begin{aligned}\n\\langle p^\\mu\\rangle &= \\int \\df(p)\\; p^\\mu\\, |\\varphi(p)|^2 = \nm \\uapprox^\\mu f_{p}(\\xi)\\,,\n\\\\ f_{p}(\\xi) &= 1+\\Ord(\\xi^{\\beta'})\\,,\n\\\\ \\uapprox\\cdot \\ucl &= 1+\\Ord(\\xi^{\\beta''})\\,,\n\\\\ \\spread(p)\/m^2 &=\n\\langle \\bigl(p-\\langle p\\rangle\\bigr){}^2\\rangle\/m^2\n\\\\&= \\bigl(\\langle p^2\\rangle-\\langle p\\rangle{}^2\\bigr)\/m^2\n= c_\\Delta \\xi^\\beta\\,,\n\\end{aligned}\n\\label{eqn:expectations}\n\\end{equation}\nwhere $c_\\Delta$ is a constant of order unity, and the $\\beta$'s are simple rational exponents. For the simplest wavefunctions, $\\beta=1$. This spread around the classical value is not necessarily positive, as the difference $p^\\mu-\\langle p^\\mu\\rangle$ may be spacelike, and the expectation of its Lorentz square possibly negative. For that reason, we should resist the usual temptation of taking its square root to obtain a variance.\n\nThese constraints are the specific statement of those in equation~\\eqref{eqn:classicalConstraints} for momentum space wavepackets. What about the vanishing overlap between classically equivalent states,~\\eqref{eqn:classicalOverlap}? To determine a constraint on the wavepackets we need a little more detail of their functional form. Now, because of the on-shell condition $p^2=m^2$ imposed by the phase-space integral over the wavepacket's momenta, the only Lorentz invariant built out of the momentum is constant, and so the wavefunction cannot usefully depend on it. This means the wavefunction must depend on at least one four-vector parameter. The simplest wavefunctions will depend on exactly one four-vector, which we can think of as the (classical) 4-velocity $\\ucl$ of the corresponding particle.  It can depend only on the dimensionless combination $p\\cdot \\ucl\/m$ in addition to the parameter $\\xi$.  The simplest form will be a function of these two in the combination $p\\cdot\\ucl\/(m\\xi)$, so that large deviations from $m \\ucl$ will be suppressed in a classical quantity. The wavefunction will have additional dependence on $\\xi$ in its normalisation.\n\nThe difference between two classically equivalent wavepackets must therefore come down to a characteristic mismatch $q_0$ of their momentum arguments --- without loss of generality, classically equivalent wavepackets are then specified by wavefunctions $\\varphi(p)$ and $\\varphi(p + q_0)$. As one nears the classical limit, both wavefunctions must represent the particle: that is they should be sharply peaked, and in addition their overlap should be $\\Ord(1)$, up to corrections of $\\Ord(\\xi)$. Requiring the overlap to be $\\Ord(1)$ is equivalent to requiring that $\\varphi(p+q_0)$ does not differ much from $\\varphi(p)$, which in turn requires that the derivative at $p$ is small, or that\n\\begin{equation}\n\\frac{q_0\\cdot\\ucl}{m\\xi} \\ll 1\\,.\n\\label{eqn:qConstraint1}\n\\end{equation}\nIf we scale $q$ by $1\/\\hbar$, this constraint takes the following form:\n\\begin{equation}\n\\qb_0\\cdot\\ucl\\,\\lpack \\ll \\sqrt{\\xi}.\n\\label{eqn:qbConstraint1}\n\\end{equation}\nWe have replaced the momentum by a wavenumber. We will see in the next chapter that this constraint is the fundamental constraint forcing the classical scaling advertised in equation~\\eqref{eqn:notationWavenumber} upon certain momenta in scattering amplitudes.\n\nLet us remain in the single particle case and finally examine an explicit example wavefunction satisfying our constraints. We will take a linear exponential,\n\\begin{equation}\n\\varphi(p) = \\Norm m^{-1}\\exp\\biggl[-\\frac{p\\cdot u}{\\hbar\\lcomp\/\\lpack^2}\\biggr]\n= \\Norm m^{-1}\\exp\\biggl[-\\frac{p\\cdot u}{m\\xi}\\biggr]\\,,\n\\label{eqn:LinearExponential}\n\\end{equation}\nwhich shares some features with relativistic wavefunctions discussed in ref.~\\cite{AlHashimi:2009bb}. In spite of the linearity of the exponent in $p$, this function gives rise to the Gaussian of \\eqn~\\eqref{eqn:NonrelativisticMomentumSpaceWavefunction} in the nonrelativistic limit (in the rest frame of $u$).\n\nThe normalisation condition~(\\ref{eqn:WavefunctionNormalization}) requires\n\\begin{equation}\n\\Norm = \\frac{2\\sqrt2\\pi}{\\xi^{1\/2} K_1^{1\/2}(2\/\\xi)}\n\\,,\n\\end{equation}\nwhere $K_1$ is a modified Bessel function of the second kind. For details of this computation and following ones, see appendix~\\ref{app:wavefunctions}.  An immediate corollary is that the overlap\n\\begin{equation}\n\\int \\! \\df(p) \\, \\varphi^*(p + q_0) \\varphi(p) = \\exp\\left[-\\frac{u\\cdot q_0}{m\\xi}\\right] \\equiv \\eta_1(q_0;p)\\,.\\label{eqn:wavefunctionOverlap}\n\\end{equation}\nClearly, for this result to vanish in the limit $\\xi=0$ we must rescale $q_0 = \\hbar \\wn q_0$, which then explicitly recovers the constraint~\\eqref{eqn:qbConstraint1}.\n\nWe can compute the momentum expectation value of the wavepacket straightforwardly, obtaining\n\\begin{equation}\n\\langle p^\\mu\\rangle = m u^\\mu \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,.\n\\end{equation}\nAs we approach the classical region, where $\\xi\\rightarrow 0$, the wavefunction indeed becomes sharply peaked, as\n\\begin{equation}\n\\langle p^\\mu \\rangle \\rightarrow m u^\\mu \\left(1 + \\frac34 \\xi\\right) + \\Ord(\\xi^2)\\,.\n\\end{equation}\nMoreover, the spread of the wavepacket\n\\begin{equation}\n\\frac{\\sigma^2(p)}{\\langle p^2\\rangle} = 1 - \\frac{K_2^2(2\/\\xi)}{K_1^2(2\/\\xi)} \\rightarrow -\\frac32 \\xi + \\Ord(\\xi^2)\\,.\n\\end{equation}\n\nFinally, a similar calculation yields\n\\begin{equation}\n\\langle p^\\mu p^\\nu \\rangle = m^2 u^\\mu u^\\nu \\left(1 +  \\frac{2\\xi \\, K_2(2\/\\xi)}{K_1(2\/\\xi)}\\right)  - \\frac{m^2}{2}  \\frac{\\xi\\, K_2(2\/\\xi)}{K_1(2\/\\xi)}\\, \\eta^{\\mu\\nu}\\,,\\label{eqn:doubleMomExp}\n\\end{equation}\nso in the classical region our wavepackets explicitly satisfy\n\\[\n\\langle p^\\mu p^\\nu \\rangle &\\rightarrow m^2 u^\\mu u^\\nu \\left(1 + 2 \\xi\\right)  - \\frac{m^2}{2} \\xi\\, \\eta^{\\mu\\nu}  + \\mathcal{O}(\\xi^2)\\\\ \n&= \\langle p^\\mu \\rangle \\langle p^\\nu \\rangle + \\mathcal{O}(\\xi)\\,.\n\\]\nFrom these results, we see that the conditions in equation~\\eqref{eqn:expectations} are explicitly satisfied, with $c_\\Delta = -3\/2$ and rational exponents $\\beta = \\beta' = \\beta'' = 1$.\n\n\\subsection{Coherent colour states}\n\\label{sec:classicalSingleParticleColour}\n\nWe have seen that the classical point-particle picture emerges from sharply peaked quantum wavepackets. To understand colour, governed by the Yang--Mills--Wong equations in the classical arena, a similar picture should emerge for our quantum colour operator in \\eqn~\\eqref{eqn:colourOp}. We define the classical limit of the colour charge in equation~\\eqref{eqn:colourCharge} to be\n\\begin{equation}\nc^a \\equiv \\langle \\psi |\\C^a| \\psi \\rangle\\,.\n\\end{equation}\nSince the colour operator in \\eqref{eqn:colourOp} explicitly involves a factor of $\\hbar$, another parameter must be large so that the colour expectation $\\langle \\psi |\\C^a | \\psi \\rangle$ is much bigger than $\\hbar$ in the classical region. For states in irreducible representation $R$ the only new dimensionless parameter available is the size of the representation, $n$, and indeed we will see explicitly in the case of $SU(3)$ that we indeed need $n$ large in this limit.\n\nCoherent states are the key to the classical limit very generally~\\cite{Yaffe:1981vf}, and we will choose a coherent state to describe the colour of our particle. The states adopted previously to describe momenta can themselves be  understood as coherent states for a ``first-quantised'' particle --- more specifically they are states for the restricted Poincar\\'e group \\cite{Kaiser:1977ys, TwarequeAli:1988tvp, Kowalski:2018xsw}. By ``coherent\" we mean in the sense of the definition introduced by Perelomov \\cite{perelomov:1972}, which formalises the notion of coherent state for any Lie group and hence can be utilised for both the kinematic and the colour parts.\n\nTo construct explicit colour states we will use the Schwinger boson formalism.  For $SU(2)$, constructing irreducible representations from Schwinger bosons is a standard textbook exercise \\cite{Sakurai:2011zz}. One simply introduces the Schwinger bosons --- that is, creation $a^{\\dagger i}$ and annihilation $a_i$ operators, transforming in the fundamental two-dimensional representation so that $i = 1,2$. The irreducible representations of $SU(2)$ are all symmetrised tensor powers of the fundamental, so the state\n\\[\na^{\\dagger i_1} a^{\\dagger i_2} \\cdots a^{\\dagger i_{2j}} \\ket{0} ,\n\\]\nwhich is automatically symmetric in all its indices, transforms in the spin $j$ representation. \n\nFor groups larger than $SU(2)$, the situation is a little more complicated because the construction of a general irreducible representation requires both symmetrisation and \nantisymmetrisation over appropriate sets of indices. This leads to expressions which are involved already for $SU(3)$ \\cite{Mathur:2000sv,Mathur:2002mx}. We content \nourselves with a  brief discussion of the $SU(3)$ case, which captures all of the interesting features of the general case.\n\nOne can construct all irreducible representations from tensor products only of fundamentals \\cite{Mathur:2010wc,Mathur:2010ey}; however, for our treatment of $SU(3)$ it is helpful to instead make use of the fundamental and antifundamental, and tensor these together to generate representations.  Following \\cite{Mathur:2000sv}, we introduce two sets of ladder operators $a_i$ and $b^i$ , $i=1, 2, 3$, which transform in the $\\mathbf{3}$ and $\\mathbf{3}^*$ respectively.  The colour operator can then be written as\n\\begin{equation}\n\\C^e= \\hbar \\left( a^\\dagger \\frac{\\lambda^e}{2} a -\nb^\\dagger \\frac{\\bar{\\lambda}^e}{2} b  \\right), \\quad e=1, \\dots, 8\\,, \\label{eqn:charge-SU3}\n\\end{equation}\nwhere $\\lambda^e$ are the Gell--Mann matrices and $\\bar\\lambda^e$ are their conjugates. The operators $a$ and $b$ satisfy the usual commutation relations\n\\begin{align}\n[a_i, a^{\\dagger j}]= \\delta_{i}{ }^{j}\\,, \\quad   [b^i, b^{\\dagger}_j]= \\delta^{i}{ }_{j}\\,, \\quad \n[a_i, b^j]= 0\\,, \\quad  [a^{\\dagger i}, b^{\\dagger }_j]= 0\\,.\n\\end{align}\nBy virtue of these commutators, the colour operator \\eqref{eqn:charge-SU3} obeys the commutation relation \\eqref{eqn:chargeLieAlgebra}.\n\nThere are two Casimir operators given by the number operators\\footnote{Here we define $a^{\\dagger}  \\cdot a \\equiv \\sum_{i=1}^3 a^{\\dagger i} a_i$ and\n\t$|\\varsigma|^2\\equiv \\sum_{i=1}^3 |\\varsigma_i|^2$.}\n\\begin{equation}\n\\mathcal{N}_1\\equiv a^\\dagger \\cdot a\\,, \\qquad  \\mathcal{N}_2\\equiv b^\\dagger \\cdot b\\,,\n\\end{equation}\nwith eigenvalues $n_1$ and $n_2$ respectively, so we label irreducible representations by $[n_1, n_2]$. \nNa\\\"ively, the states we are looking for are constructed by acting on the vacuum state as follows:\n\\begin{align}\n\\left(a^{\\dagger i_1} \\cdots  a^{\\dagger i_{n_1}} \\right)\n\\left(b_{j_1}^{\\dagger} \\cdots  b_{j_{n_2}}^{\\dagger} \\right)\n\\ket{0}.\\label{eqn:states-reducible}\n\\end{align}\nHowever, these states are $SU(3)$ reducible and thus cannot be used in our construction of coherent states. We write the irreducible states schematically by acting with a Young projector $\\mathcal{P}$ which appropriately (anti-) symmetrises upper and lower indices, thereby subtracting traces:\n\\begin{align}\n\\ket{\\psi}_{[n_1, n_2]} \\equiv \\mathcal{P} \\left( \\left(a^{\\dagger i_1} \\cdots  a^{\\dagger i_{n_1}} \\right)\n\\left(b^{\\dagger}_{j_1} \\cdots  b_{j_{n_2}}^{\\dagger} \\right)\\ket{0} \\right).\n\\label{eqn:YPstate}\n\\end{align}\nIn general these operations will lead to involved expressions for the states, but we can understand them from their associated Young tableaux (Fig.~\\ref{fig:SU3-YT}). Each double box column represents an operator $b_i^{\\dagger}$ and each single column box represents the operator $a^{\\dagger i}$, and thus for a mixed representation we have $n_2$ double columns and $n_1$ single columns.\n\n\\begin{figure}\n\t\\centering \n\t\\begin{ytableau}\n\t\tj_1 & j_2  & \\dots  &j_{n_2} & i_1 & i_2 & \\cdots & i_{n_1}  \\cr    &  &  &  \n\t\\end{ytableau}\n\t\\caption{Young tableau of $SU(3)$.}\n\t\\label{fig:SU3-YT}\n\\end{figure}\n\nHaving constructed the irreducible states, one can define a coherent state parametrised by two triplets of complex numbers $\\varsigma_i$ and $ \\varrho^i$, $i=1,2, 3$.  These are normalised according to\n\\begin{equation}\n|\\varsigma|^2 = |\\varrho|^2 = 1\\,, \\qquad \\varsigma \\cdot \\varrho=0\\,.\n\\end{equation}\nWe won't require fully general coherent states, but instead their projections onto the $[n_1,n_2]$ representation, which are\n\\begin{equation}\n\\ket{\\varsigma \\,\\varrho}_{[n_1,n_2]}\\equiv  \\frac{1}{\\sqrt{(n_1! n_2!)}} \\left( \\varrho \\cdot b^\\dagger\\right)^{n_2} \\left(\\varsigma \\cdot a^\\dagger \\right)^{n_1} \\ket{0}.\n\\label{eqn:restricted-coherent}\n\\end{equation}\n\nThe square roots ensure that the states are normalised to unity\\footnote{Note that the Young projector in equation~\\eqref{eqn:YPstate} is no longer necessary since the constraint $\\xi \\cdot \\zeta = 0$ removes all the unwanted traces.}. With this normalisation we can write the identity operator as  \n\\begin{equation}\n\\mathbb{I}_{[n_1,n_2]} = \\int \\d \\mu(\\varsigma,\\varrho) \\Big(\\ket{\\varsigma \\,\\varrho}\\bra{\\varsigma \\,\\varrho}\\Big)_{[n_1,n_2]},\\label{eqn:Haar}\n\\end{equation}\nwhere $\\int \\d \\mu(\\varsigma,\\varrho)$ is the $SU(3)$ Haar measure, normalised such that $\\int \\d \\mu(\\varsigma,\\varrho)=1$. Its precise form is irrelevant for our purposes.\n\nWith the states in hand, we can return to the expectation value of the colour operator $\\C^a$ in \\eqn~\\eqref{eqn:colourOp}. The size of the representation, that is $n_1$ and $n_2$, must be large compared to $\\hbar$ in the classical regime so that the final result is finite. To see this let us compute this expectation value explicitly.\nBy definition we have \n\\begin{equation}\n\\langle \\varsigma \\,\\varrho|\\mathbb{C}^e|\\varsigma \\,\\varrho \\rangle_{[n_1,n_2]} =  \\frac{\\hbar}{2} \\left(\\langle \\varsigma \\,\\varrho|a^\\dagger \\lambda^e a|\\varsigma \\,\\varrho \\rangle_{[n_1,n_2]} -\n\\langle \\varsigma \\,\\varrho|b^\\dagger \\bar{\\lambda}^e b|\\varsigma \\,\\varrho\\rangle_{[n_1,n_2]} \\right).\n\\end{equation}\nAfter a little algebra we find that \n\\begin{equation}\n\\langle \\varsigma \\,\\varrho|\\C^e|\\varsigma \\,\\varrho \\rangle = \\frac{\\hbar}{2} \\left( n_1   \\varsigma^{*} \\lambda^e \\varsigma - n_2    \\varrho^*\\bar \\lambda^e \\varrho\\right).\n\\end{equation}\nWe see that a finite charge requires a scaling limit in which we take $n_1$, $n_2$ large  as $\\hbar \\to 0$, keeping the product $\\hbar n_\\alpha$  fixed for at least one value of $\\alpha=1,2$. The classical charge is therefore the finite c-number\n\\begin{equation}\nc^a =  \\langle \\varsigma \\,\\varrho|\\C^a|\\varsigma \\,\\varrho \\rangle_{[n_1, n_2]}  = \\frac{\\hbar}{2} \\left( n_1   \\varsigma^{*}\\lambda^a \\varsigma - n_2 \\varrho^{*} \\bar\\lambda^a \\varrho\\right). \\label{eqn:clas-charge-SU3}\n\\end{equation}\n\nThe other feature we must check is the expectation value of products. Using the result above, a similar calculation for two pairs of charge operators in a large representation leads to\n\\begin{multline}\n\\langle \\varsigma\\,\\varrho|\\C^a\\C^b | \\varsigma\\,\\varrho\\rangle_{[n_1, n_2]}  = \\langle \\varsigma\\,\\varrho|\\C^a| \\varsigma\\,\\varrho\\rangle_{[n_1,n_2]}  \\langle \\varsigma\\,\\varrho|\\C^b | \\varsigma\\,\\varrho\\rangle_{[n_1,n_2]} \\\\ + \\hbar \\left( \\hbar n_1 \\, \\varsigma^*\\lambda^a\\cdot \\lambda ^b \\varsigma - \\hbar n_2\\, \\varrho^* \\bar\\lambda^a\\cdot \\bar\\lambda^b \\varrho \\right).\n\\end{multline}\nThe finite quantity in the classical limit $\\hbar \\to 0, \\, n_\\alpha \\to \\infty$ is the product $\\hbar n_\\alpha$. The term inside the brackets on the second line is itself finite, but comes with a lone $\\hbar$ coefficient, and thus vanishes in the classical limit. Thus,\n\\[\n\\langle \\varsigma\\,\\varrho|\\C^a \\C^b|\\varsigma\\,\\varrho \\rangle _{[n_1,n_2]} = c^a c^b + \\mathcal O(\\hbar)\\,. \\label{eqn:factorisation-charges}\n\\]\nThis is in fact a special case of a more general construction discussed in detail by Yaffe~\\cite{Yaffe:1981vf}. Similar calculations can also be used to demonstrate  that the overlap $\\langle \\chi' | \\chi \\rangle$ is very strongly peaked about $\\chi = \\chi'$, as required by equation~\\eqref{eqn:classicalOverlap}. We have thus constructed explicit colour states which ensure the correct classical behaviour of the colour charges. \n\nFor the remainder of the thesis we will only need to make use of the finiteness and factorisation properties, so we will avoid further use of the explicit form of the representation states. Henceforth we write $\\chi$ for the parameters of a general colour state $\\ket{\\chi}$ with these properties, and $\\d \\mu(\\chi)$ for the Haar measure of the $SU(N)$ colour group.\n\n\\section{Multi-particle wavepackets}\n\nHaving set up appropriate wavepackets for a single particle, we can now consider multiple particles, and thus generalise the generic states adopted in equation~\\eqref{eqn:InitialStateSimple}. We will take two distinguishable scalar particles, associated with distinct quantum fields $\\pi_\\alpha$ with $\\alpha = 1, 2$. The action is therefore as given in \\eqn~\\eqref{eqn:scalarAction}. Both fields $\\pi_\\alpha$ must be in representations $R_\\alpha$ which are large, so that a classical limit is available for the individual colours.\n\nIn anticipation of considering scattering processes in the next section, we will now take our state to be at some initial time in the far past, where we assume that our two particles both have well-defined positions, momenta and colours. In other words, particle $\\alpha$ has a wavepacket $\\varphi_\\alpha(p_\\alpha)$ describing its momentum-space distribution, and a coherent colour wavefunction $\\chi_\\alpha$, as described in the previous section. Then the appropriate generalisation of the multi-particle state is\n\\[\n|\\Psi\\rangle &= \\int\\!\\df(p_1)\\df(p_2)\\, \\varphi_1(p_1) \\varphi_2(p_2)\\, e^{ib\\cdot p_1\/\\hbar}\\, |{p_1}\\, \\chi_1 ; \\, {p_2}\\, \\chi_2 \\rangle \\\\\n&= \\int\\!\\df(p_1)\\df(p_2) \\, \\varphi_1(p_1) \\varphi_2(p_2)\\, e^{ib\\cdot p_1\/\\hbar}\\, \\chi_{1i}\\, \\chi_{2j} |{p_1}^i ; \\, {p_2}^j \\rangle\\,,\\label{eqn:inState}\n\\]\nwhere the displacement operator insertion accounts for the particles' spatial separation.\n\nWe measure observables for multi-particle states by acting with operators which are simply the sum of the individual operators for each of the scalar fields. For example, acting with the colour operator~\\eqref{eqn:colourOp} on the state $|{p_1}\\, \\chi_1 ; \\, {p_2}\\, \\chi_2 \\rangle$ we have\n\\[\n\\C^a |{p_1}\\,\\chi_1 ;&\\, {p_2}\\, \\chi_2 \\rangle = |{p_1}^{i'} \\, {p_2}^{j'} \\rangle \\, \\left( (C^a_{1})_{i'}{}^i \\delta_{j'}{}^j + \\delta_{i'}{}^i (C^a_{2})_{j'}{}^j \\right) \\chi_{1i}\\, \\chi_{2j} \\\\\n&= \\int \\! d\\mu(\\chi'_1) d\\mu(\\chi'_2) \\, \\ket{{p_1} \\, \\chi'_1 ; \\, {p_2}\\, \\chi'_2} \\,\\langle \\chi'_1\\, \\chi'_2| C^a_{1} \\otimes 1 + 1 \\otimes C^a_{2} |\\chi_1\\, \\chi_2 \\rangle\\\\\n&= \\int \\! d\\mu(\\chi'_1) d\\mu(\\chi'_2) \\, \\ket{{p_1} \\, \\chi'_1 ; \\, {p_2}\\, \\chi'_2} \\,\\langle\\chi'_1\\, \\chi'_2| C^a_{1+2}  |\\chi_1\\, \\chi_2\\rangle\\,,\\label{eqn:charge2particleAction}\n\\]\nwhere $C^a_\\alpha$ is the colour in representation $R_\\alpha$ and we have written $C^a_{1+2}$ for the colour operator on the tensor product of representations $R_1$ and $R_2$.\nIn the classical regime, using the property that the overlap between states sets $\\chi'_\\alpha=\\chi_\\alpha $ in the classical limit, it follows that\n\\[\n\\bra{p_1\\,\\chi_1; \\, p_2 \\,\\chi_2} C^a_{1+2} \\ket{p_1\\, \\chi_1 ; \\, p_2\\, \\chi_2 } = c_1^a + c_2^a \\,,\n\\]\nso the colours simply add. A trivial similar result holds for the momenta of the two particles.\n\nSuppose that at some later time the two particles described by our initial state interact --- for example, two black holes scattering elastically. When does a point-particle description  remain appropriate? This will be the crucial topic of the next chapter. We will take the initial separation $b^\\mu$ to be the transverse impact parameter for the scattering of two point-like objects with momenta $p_{1,2}$. (The impact parameter is transverse in the sense that $p_\\alpha \\cdot b = 0$ for $\\alpha = 1, 2$.) At the quantum level, the particles are individually described by the wavefunctions in section~\\ref{sec:PointParticleLimit}. We would expect the point-particle description to be valid when the separation of the two scattering particles is always very large compared to their (reduced) Compton wavelengths, so the point-particle description will be accurate provided that\n\\begin{equation}\n\\sqrt{-b^2} \\gg \\lcomp^{(1,2)}\\,.\n\\end{equation}\nThe impact parameter and the Compton wavelengths are not the only scales we must consider, however --- the spread of the wavepackets, $\\lpack$, is another intrinsic scale. As we will discuss, the quantum-mechanical expectation values of observables are well approximated by the corresponding classical ones when the packet spreads are in the `Goldilocks' zone, $\\lcomp\\ll \\lpack\\ll \\sqrt{-b^2}$. These inequalities will have powerful ramifications on the behaviour of scattering amplitudes in the classical limit. To see this however, we need an on-shell scattering observable.\n\n\\chapter{The impulse}\n\\label{chap:impulse}\n\nAt a gravitational wave observatory we are of course interested in the gravitational radiation emitted by the source of interest. However, gravitational waves also carry information about the potential experienced by, for example, a black hole binary system. This observation motivates our interest in an on-shell observable related to the potential. We choose to explore the \\textit{impulse\\\/} on a particle during a scattering event: at the classical level, this is simply the total change in the momentum of one of the particles --- say particle~1 --- during the collision.\n\nIn this chapter we will begin by examining the change in momentum during a scattering event, without accompanying radiation, extracting the classical values from a fully relativistic quantum-mechanical computation. We examine scattering events in which two widely separated particles are prepared in the state~\\eqref{eqn:inState} at $t \\rightarrow -\\infty$, and then shot at each other with impact parameter $b^\\mu$. We will use this observable as a laboratory to explore certain conceptual and practical issues in approaching the classical limit. Using the explicit wavepackets we have constructed in chapter~\\ref{chap:pointParticles}, we will carefully analyse the small-$\\hbar$ region to understand how scattering amplitudes encode classical physics. We will see that the appropriate treatment is one where point-particles have momenta which are fixed as we take $\\hbar$ to zero, whereas for massless particles and momentum transfers between massive particles, it is the wavenumber which we should treat as fixed in the limit.\n\nOur formalism is quite general, applying in both gauge theory and gravity; for simplicity, we will nonetheless continue to focus on the scattering of two massive, stable quanta of scalar fields described by the Lagrangian in equation~\\eqref{eqn:scalarAction}. We will generalise to higher spin fields in part~\\ref{part:spin}. We will always restrict our attention to scattering processes in which quanta of fields 1 and 2 are both present in the final state. This will happen, for example, if the particles have separately conserved quantum numbers.  We also always assume that no new quanta of fields 1 and 2 can be produced during the collision, for example because the centre-of-mass energy is too small.\n\nIn the first section of this chapter we construct expressions for the impulse in terms of on-shell scattering amplitudes, providing a formal definition of the momentum transfer to a particle in quantum field theory. In \\sect{sec:classicalLimit}, we derive the Goldilocks zone in which the point-particle limit of our wavepackets remains valid, deriving from first principles the behaviour of scattering amplitudes in the classical limit. In \\sect{sec:examples} we apply our formalism explicitly, deriving the NLO impulse in scalar Yang--Mills theory, a result which is analagous to well-established post--Minkowskian results for the scattering of Schwarzschild black holes \\cite{Portilla:1979xx,Portilla:1980uz,Westpfahl:1979gu,Westpfahl:1985tsl}\n\nThis chapter continues to be based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}.\n\n\\section{Impulse in quantum field theory}\n\\label{sec:QFTsetup}\n\nTo define the observable, we place detectors at asymptotically large distances pointing at the collision region. The detectors measure only the momentum of particle 1. We assume that these detectors cover all possible scattering angles. Let $\\mathbb{P}_\\alpha^\\mu$ be the momentum operator for particle~$\\alpha$; the expectation of the first particle's outgoing momentum $\\outp1^\\mu$ is then\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle &= \n{}_\\textrm{out}{\\langle}\\Psi|  \\mathbb{P}_1^\\mu |\\Psi\\rangle_\\textrm{out} \\\\\n&= {}_\\textrm{out}{\\langle}\\Psi|  \\mathbb{P}_1^\\mu U(\\infty,-\\infty)\\,\n|\\Psi\\rangle_\\textrm{in} \\\\\n&= {}_\\textrm{in}{\\langle} \\Psi | \\, U(\\infty, -\\infty)^\\dagger   \\mathbb{P}_1^\\mu U(\\infty, -\\infty) \\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $U(\\infty, -\\infty)$ is the time evolution operator from the far past to the far future.  This evolution operator is just the $S$ matrix, so the expectation value is simply\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle &= \n{}_\\textrm{in}{\\langle} \\Psi | S^\\dagger   \\mathbb{P}_1^\\mu S\\, \n| \\Psi \\rangle_\\textrm{in}\\,.\n\\end{aligned}\n\\end{equation}\nWe can insert a complete set of states and rewrite the expectation value as\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle \n&=\\sum_X \\int \\df(\\finalk_1)\\, \\df(\\finalk_2) \\, d\\mu(\\zeta_1) \\, d\\mu(\\zeta_2)\\; \\finalk_1^\\mu \n\\; \\bigl| \\langle \\finalk_1\\, \\zeta_1; \\finalk_2\\, \\zeta_2; X |S| \\Psi\\rangle\\bigr|^2\\,,\n\\end{aligned}\n\\label{eqn:p1Expectation}\n\\end{equation}\nwhere we can think of the inserted states as the final state of a scattering process. In this equation, $X$ refers to any other particles which may be created. The intermediate state containing $X$ also necessarily contains exactly one particle each corresponding to fields~1 and~2. Their momenta are denoted by $\\finalk_{1,2}$ respectively, while $\\d\\mu(\\zeta_\\alpha)$ is the $SU(N)$ Haar measure for their coherent colour states, as introduced in~\\eqref{eqn:Haar}. The sum over $X$ is a sum over all states, including $X$ empty, and includes phase-space integrals for $X$ non-empty. The expression~(\\ref{eqn:p1Expectation}) already hints at the possibility of evaluating the momentum in terms of on-shell scattering amplitudes.\n\nThe physically interesting quantity is rather the change of momentum of the particle during the scattering, so we define\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle =  \\langle \\Psi |S^\\dagger \\,   \\mathbb{P}^\\mu_1 \\, S |\\Psi\\rangle - \\langle \\Psi | \\,   \\mathbb{P}^\\mu_1 \\,  |\\Psi\\rangle\\,.\n\\end{equation}\nThis impulse is the difference between the expected outgoing and\nthe incoming momenta of particle 1. It is an on--shell observable, defined in both the quantum and the classical theories. Similarly, we can measure the impulse imparted to particle 2. In terms of the momentum operator, $\\mathbb{P}_2^\\mu$, of quantum field 2, this impulse is evidently\n\\begin{equation}\n\\langle \\Delta p_2^\\mu \\rangle =  \\langle \\Psi |S^\\dagger \\,  \\mathbb{P}^\\mu_2 \\, S |\\Psi\\rangle - \\langle \\Psi | \\,   \\mathbb{P}^\\mu_2 \\,  |\\Psi\\rangle.\n\\end{equation}\n\nReturning to the impulse on particle 1, we proceed by writing the scattering matrix in terms of the transition matrix $T$ via $S = 1 + i T$, in order to make contact with the usual scattering amplitudes. The no-scattering (unity) part of the $S$ matrix cancels in the impulse, leaving behind only delta functions that identify the final-state momenta with the initial-state ones in the wavefunction or its conjugate. Using unitarity we obtain the result\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle \n= \\langle \\Psi | \\, i [   \\mathbb{P}_1^\\mu, T ] \\, | \\Psi \\rangle + \\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\label{eqn:defl1}\n\\end{equation}\n\n\\subsection{Impulse in terms of amplitudes}\n\nHaving established a general expression for the impulse, we turn to expressing it in terms of scattering amplitudes. It is convenient to work on the two terms in equation~\\eqref{eqn:defl1} separately. For ease of discussion, we define\n\\begin{equation}\n\\begin{aligned}\n\\ImpA \\equiv \\langle \\Psi | \\, i [ \\mathbb{P}_1^\\mu, T ] \\, | \\Psi \\rangle\\,, \\qquad\n\\ImpB \\equiv \\langle \\Psi | \\, T^\\dagger [ \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,,\n\\end{aligned}\n\\end{equation}\nso that the impulse is $\\langle \\Delta p_1^\\mu \\rangle = \\ImpA + \\ImpB$. Using equation~\\eqref{eqn:inState} to expand the wavepacket in the first term, $\\ImpA$, we find\n\\[\n\\hspace*{-4mm}\\ImpA &= \n\\int \\! \\df(\\initialk_1)\\df(\\initialk_2)\n\\df(\\initialkc_1)\\df(\\initialkc_2)\\;\ne^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar} \\, \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialkc_1) \n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialkc_2) \n\\\\ &\\hspace{40mm}\n\\times i (\\initialkc_1\\!{}^\\mu - \\initialk_1^\\mu) \\, \n\\langle \\initialkc_1\\, \\chi_1'; \\initialkc_2 \\, \\chi_2'| \\,T\\, |\n\\initialk_1 \\, \\chi_1; \\initialk_2\\, \\chi_2 \\rangle\n\\\\&= \\int \\! \\df(\\initialk_1)\\df(\\initialk_2)\n\\df(\\initialkc_1)\\df(\\initialkc_2)\\;\ne^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar} \\, \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialkc_1) \n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialkc_2) \n\\\\ &\\qquad\\qquad\n\\times i \\int \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;(\\finalk_1^\\mu-\\initialk_1^\\mu)\n\\\\ &\\qquad\\qquad\\qquad\\qquad \n\\times \\langle \\initialkc_1 \\, \\chi_1'; \\initialkc_2 \\, \\chi_2'| \\finalk_1 \\, \\zeta_1 ;\\finalk_2\\, \\zeta_2 \\rangle\n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2\\, \\zeta_2 | \\,T\\, |\\initialk_1 \\, \\chi_1; \\initialk_2 \\, \\chi_2\\rangle\n\\,,\n\\label{eqn:defl2}\n\\]\nwhere in the second equality we have re-inserted the final-state momenta $\\finalk_\\alpha$ in\norder to make manifest the phase independence of the result. We label the states in the incoming wavefunction by $\\initialk_{1,2}$, those in the conjugate ones by $\\initialkc_{1,2}$. Let us now introduce the momentum shifts $q_\\alpha = \\initialkc_\\alpha-\\initialk_\\alpha$, and then change variables in the integration from the $p_\\alpha'$ to the $q_\\alpha$. In these variables, the matrix element is\n\\begin{equation}\n\\begin{aligned}\n\\langle p_1'\\, \\chi'_1;\\,p_2'\\, \\chi'_2|T| p_1\\,\\chi_1;\\,p_2\\,\\chi_2\\rangle &= \\langle \\chi'_1\\, \\chi'_2|\\mathcal{A}(p_1,p_2 \\rightarrow p_1',p_2')|\\chi_1\\,\\chi_2\\rangle\\\\ &\\qquad\\qquad\\qquad\\qquad \\times\n\\del^{(4)}(\\initialkc_1+\\initialkc_2-\\initialk_1-\\initialk_2) \n\\\\&\\equiv\n\\langle \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \\initialk_1 + q_1\\,, \\initialk_2 + q_2)\\rangle\n\\del^{(4)}(q_1 + q_2)\\,,\\label{eqn:defOfAmplitude}\n\\end{aligned}\n\\end{equation}\nyielding\n\\begin{equation}\n\\begin{aligned}\n\\ImpA &= \\int \\! \\df(\\initialk_1) \\df(\\initialk_2)\n\\df(q_1+\\initialk_1)\\df(q_2+\\initialk_2)\\;\n\\\\&\\qquad\\times \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialk_1 + q_1)\n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialk_2+q_2) \n\\, \\del^{(4)}(q_1 + q_2)\n\\\\&\\qquad\\times \n\\, e^{-i b \\cdot q_1\/\\hbar} \n\\,i q_1^\\mu  \\, \\langle\\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q_1, \\initialk_2 + q_2)\\rangle\\,\n\\,,\n\\end{aligned}\n\\label{eqn:impulseGeneralTerm1a}\n\\end{equation}\nwhere the remaining expectation value is solely over the representation states $\\chi_\\alpha$. Note that we are implicitly using the condition~\\eqref{eqn:classicalOverlap}, in anticipation of the classical limit, for clarity of presentation. Now, recall the shorthand notation introduced earlier for the phase-space measure,\n\\begin{equation}\n\\df(q_1+p_1) = \\dd^4 q_1\\; \\del\\bigl((p_1 + q_1)^2 - m_1^2\\bigr)\n\\Theta(p_1^0 + q_1^0)\\,.\n\\end{equation} \nWe can perform the integral over $q_2$ in \\eqn~\\eqref{eqn:impulseGeneralTerm1a} using the four-fold delta function. Further relabeling $q_1 \\rightarrow q$, we obtain\n\\[\n\\ImpA&= \\int \\! \\df(\\initialk_1)\\df(\\initialk_2) \\dd^4 q  \\; \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\, e^{-i b \\cdot q\/\\hbar}\\\\\n&\\qquad \\times \\Theta(\\initialk_1^0+q^0) \\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_1^*(\\initialk_1 + q)\n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialk_2-q)\n\\\\& \\qquad\\qquad \\times  \n\\,  i q^\\mu  \\, \\langle\\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\\rangle\\,.\n\\label{eqn:impulseGeneralTerm1}\n\\]\nUnusually for a physical observable, this contribution is linear in the amplitude. We emphasise that the incoming and outgoing momenta of this amplitude do \\textit{not\\\/} correspond to the initial- and final-state momenta of the scattering process, but rather both correspond to the initial-state momenta, as they appear in the wavefunction and in its conjugate.  The momentum $q$ looks like a momentum transfer if we examine the amplitude alone, but for the physical scattering process it represents a difference between the momentum within the wavefunction and that in the conjugate.  Inspired by our discussion in section~\\ref{sec:PointParticleLimit}, we will refer to it as a `momentum mismatch'.  As indicated on the first line of \\eqn~\\eqref{eqn:defl2}, we should think of this term as an interference of a standard amplitude with an interactionless forward scattering. Recalling that in equation~\\eqref{eqn:wavefunctionSplit} we defined ${\\psi_i}_\\alpha(p_\\alpha) = \\varphi_\\alpha(p_\\alpha) {\\chi_i}_\\alpha$, we can write this diagrammatically as\n\\begin{equation}\n\\begin{aligned}\n\\ImpA & = \n\\int \\! \\df(\\initialk_1)\\df(\\initialk_2) \\dd^4 q \\, \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\\\\n& \\qquad \\times  \\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0) \\, e^{-i b \\cdot q\/\\hbar}  \\,  iq^\\mu \\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\n\\vertex (b) ;\n\\vertex [above left=1 and 0.66 of b] (i1) {$\\psi_1(p_1)$};\n\\vertex [above right=1 and 0.33 of b] (o1) {$\\psi_1^*(p_1+q)$};\n\\vertex [below left=1 and 0.66 of b] (i2) {$\\psi_2(p_2)$};\n\\vertex [below right=1 and 0.33 of b] (o2) {$\\psi_2^*(p_2-q)$};\n\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- (o2);\n\\draw[postaction={decorate}] (b) -- (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- (b);\n\\draw[postaction={decorate}] (i2) -- (b);\n\\end{scope}\n\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\filldraw [fill=allOrderBlue] (b) circle [radius=10pt];\n\\end{feynman}\n\\end{tikzpicture} \n\\!\\!\\!\\!.\n\\end{aligned}\n\\end{equation}\n\nTurning to the second term, $\\ImpB$, in the impulse, we again introduce a complete set of states labelled by momenta $\\finalk_1$, $\\finalk_2$ and $X$ so that\n\\begin{equation}\n\\begin{aligned}\n\\ImpB &= \\langle \\Psi | \\, T^\\dagger [ \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\n\\\\&= \\sum_X \\int \\! \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2) \n\\\\& \\qquad\\qquad \\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\rangle \n\\langle \\finalk_1  \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X| [  \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\end{aligned}\n\\end{equation}\nAs above, we can now expand the wavepackets. We again label the momenta in the incoming wavefunction by $\\initialk_{1,2}$, and those in the conjugate ones by $\\initialkc_{1,2}$:\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X \\int \\!\\prod_{\\alpha = 1, 2}  \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \\df(\\initialkc_\\alpha)\n\\; \\varphi_\\alpha(\\initialk_\\alpha) \\varphi^*_\\alpha(\\initialkc_\\alpha) \ne^{i b \\cdot (\\initialk_1 - \\initialkc_1) \/ \\hbar}\n(\\finalk_1^\\mu - \\initialk_1^\\mu) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X) \n\\del^{(4)}(\\initialkc_1+\\initialkc_2 - \\finalk_1 - \\finalk_2 -\\finalk_X)\\\\\n&\\hspace*{15mm}\\times \\langle \\Ampl^*(\\initialkc_1\\,, \\initialkc_2 \\rightarrow \\finalk_1 \\,, \\finalk_2 \\,, \\finalk_X) \n\\Ampl(\\initialk_1\\,,\\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:forcedef2}\n\\end{equation}\nIn this expression we again absorb the representation states $\\chi_\\alpha$ into an expectation value over the amplitudes, while $\\finalk_X$ denotes the total momentum carried by particles in $X$. The second term in the impulse can thus be interpreted as a weighted cut of an amplitude; the lowest order contribution is a weighted two-particle cut of a one-loop amplitude. \n\nIn order to simplify $\\ImpB$, let us again define the momentum shifts $q_\\alpha = \\initialkc_\\alpha-\\initialk_\\alpha$, and change variables in the integration from the $\\initialkc_\\alpha$ to the $q_\\alpha$, so that\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X  \\int \\!\\prod_{\\alpha = 1,2} \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \n\\df(q_\\alpha+\\initialk_\\alpha)\n\\; \\varphi_\\alpha(\\initialk_\\alpha) \\varphi^*_\\alpha(\\initialk_\\alpha+q_\\alpha) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X)\\, \n\\del^{(4)}(q_1+q_2)\\, e^{-i b \\cdot q_1 \/ \\hbar} (\\finalk_1^\\mu - \\initialk_1^\\mu)\\\\\n&\\hspace*{5mm}\\times \\langle\\Ampl^*(\\initialk_1+q_1\\,, \\initialk_2+q_2 \\rightarrow \\finalk_1 \\,,\\finalk_2 \\,, \\finalk_X)\n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned} \\label{eqn:impulseGeneralTerm2a}\n\\end{equation}\nWe can again perform the integral over $q_2$ using the four-fold delta function, and relabel $q_1 \\rightarrow q$ to obtain\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X  \\int \\!\\prod_{\\alpha = 1,2} \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \n\\dd^4 q\\;\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\\\\n&\\hspace*{5mm}\\times \\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2)\\, \\varphi^*_1(\\initialk_1+q) \\varphi^*_2(\\initialk_2-q) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X) \\, e^{-i b \\cdot q \/ \\hbar} (\\finalk_1^\\mu - \\initialk_1^\\mu)\n\\\\ &\\hspace*{5mm}\\times \\langle \\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \\finalk_1 \\,,\\finalk_2 \\,, \\finalk_X)\n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\\rangle\n\\,.\n\\end{aligned} \n\\label{eqn:impulseGeneralTerm2b}\n\\end{equation}\nThe momentum $q$ is again a momentum mismatch. The momentum transfers $\\xfer_\\alpha\\equiv r_\\alpha-p_\\alpha$ will play an important role in analysing the classical limit, so it is convenient to change variables to them from the final-state momenta $\\finalk_\\alpha$,\n\\begin{equation}\n\\begin{aligned}\n\\ImpB &= \\sum_X  \\int \\!\\prod_{\\alpha = 1,2}  \\df(\\initialk_i) \\dd^4 \\xfer_\\alpha\n\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\qquad\\times\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2)\\\\\n&\\qquad \\times\\varphi^*_1(\\initialk_1+q) \\varphi^*_2(\\initialk_2-q) e^{-i b \\cdot q \/ \\hbar}\\,\\xfer_1^\\mu \\; \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \n\\\\ &\\qquad\\qquad\\times \n\\langle \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1 \\,,\\initialk_2+\\xfer_2 \\,, \\finalk_X)\n\\\\ &\\qquad\\qquad\\qquad\\qquad\\qquad \\times \n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2 \\,, \\finalk_X) \\rangle\\,.\n\\end{aligned} \n\\label{eqn:impulseGeneralTerm2}\n\\end{equation}\nDiagrammatically, this second contribution to the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\usetikzlibrary{decorations.markings}\n\\ImpB&= \n\\sum_X {\\int} \\! \\prod_{\\alpha = 1,2}  \\df(\\initialk_\\alpha) \\dd^4 \\xfer_\\alpha\n\\dd^4 q\\; \\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\\,e^{-i b \\cdot q \/ \\hbar}\\,\\xfer_1^\\mu\n\\\\&\\qquad\\times\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\\\\n& \\hspace*{10mm} \\times \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \\!\\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\begin{scope}\n\\vertex (ip1) ;\n\\vertex [right=3 of ip1] (ip2);\n\\node [] (X) at ($ (ip1)!.5!(ip2) $) {};\n\\begin{scope}[even odd rule]\n\n\\vertex [above left=0.66 and 0.5 of ip1] (q1) {$ \\psi_1(p_1)$};\n\\vertex [above right=0.66 and 0.33 of ip2] (qp1) {$ \\psi^*_1(p_1 + q)$};\n\\vertex [below left=0.66 and 0.5 of ip1] (q2) {$ \\psi_2(p_2)$};\n\\vertex [below right=0.66 and 0.33 of ip2] (qp2) {$ \\psi^*_2(p_2 - q)$};\n\\diagram* {\n\t(ip1) -- [photon]  (ip2)\n};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (q1) -- (ip1);\n\\draw[postaction={decorate}] (q2) -- (ip1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip2) -- (qp1);\n\\draw[postaction={decorate}] (ip2) -- (qp2);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.34 with {\\arrow{Stealth}},\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip1) to [out=90, in=90,looseness=1.2] node[above left] {{$p_1 + w_1$}} (ip2);\n\\draw[postaction={decorate}] (ip1) to [out=270, in=270,looseness=1.2]node[below left] {$p_2 + w_2$} (ip2);\n\\end{scope}\n\n\\node [] (Y) at ($(X) + (0,1.4)$) {};\n\\node [] (Z) at ($(X) - (0,1.4)$) {};\n\\node [] (k) at ($ (X) - (0.65,-0.25) $) {$\\finalk_X$};\n\n\\filldraw [color=white] ($ (ip1)$) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip1) $) circle [radius=8pt];\n\n\\filldraw [color=white] ($ (ip2) $) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip2) $) circle [radius=8pt];\n\n\\end{scope}\n\\end{scope}\n\\filldraw [color=white] ($  (Y) - (3pt, 0) $) rectangle ($ (Z) + (3pt,0) $) ;\n\\draw [dashed] (Y) to (Z);\n\\end{feynman}\n\\end{tikzpicture} .\n\\end{aligned}\n\\end{equation}\n\n\\section{Point-particle scattering}\n\\label{sec:classicalLimit}\n\nThe observable we have discussed  --- the impulse --- is designed to be well-defined in both the quantum and the classical theories. As we approach the classical limit, the quantum expectation values should reduce to the classical impulse, ensuring that we are able to explore the $\\hbar \\rightarrow 0$ limit. Here we explore this limit, and its ramifications on scattering amplitudes, in detail.\n\n\\subsection{The Goldilocks inequalities}\n\nWe have already discussed in section~\\ref{sec:RestoringHBar} how to make explicit the factors of $\\hbar$ in the observables, and in section~\\ref{sec:PointParticleLimit} we selected wavefunctions which have the desired classical point-particle limit, which we established was the region where\n\\begin{equation}\n\\ell_c \\ll \\ell_w\\,.\\label{eqn:comptonConstraint}\n\\end{equation}\nAt this point, we could in principle perform the full quantum calculation, using the specific wavefunctions we chose, and expand in the $\\xi\\!\\rightarrow\\! 0$ limit at the end.  However, having established previously the detailed properties of our wavefunctions $\\varphi_\\alpha$, it is far more efficient to neglect the details and simply use the fact that they allow us to approach the limit as early as possible in calculations.  This will lead us to impose stronger constraints on our choice than the mere existence of a suitable classical limit.\n\nHeuristically, the wavefunctions for the scattered particles must satisfy two separate conditions. As discussed in the single particle case, the details of the wavepacket should not be sensitive to quantum effects. At the same time, now that we aim to describe the scattering of point-particles the spread of the wavefunctions should not be too large, so that the interaction with the other particle cannot peer into the details of the quantum wavepacket.\n\nTo quantify this discussion let us examine $\\ImpA$ in~(\\ref{eqn:impulseGeneralTerm1}) more closely. It has the form of an amplitude integrated over the on-shell phase space for both of the incoming momenta, subject to additional $\\delta$ function constraints --- and then weighted by a phase $e^{-ib\\cdot q\/\\hbar}$ dependent on the momentum mismatch $q$, and finally integrated over all $q$. As one nears the classical limit~\\eqref{eqn:comptonConstraint}, the wavefunction and its conjugate should both represent the particle. The amplitude will vary slowly on the scale of the wavefunction when one is close to the limit. This is therefore precisely the same constraint as we had in the single particle case, and we immediately have\n\\begin{equation}\n\\qb\\cdot\\ucl_\\alpha\\,\\lpack \\ll \\sqrt{\\xi}\\,,\n\\label{eqn:qbConstraint}\n\\end{equation}\nwhere we have scaled $q$ by $1\/\\hbar$, replacing the momentum by a wavenumber.\n\nWe next examine another rapidly varying factor that appears in all our integrands, the delta functions in $q$ arising from the on-shell constraints on the conjugate momenta $\\initialkc_\\alpha$. These delta functions, appearing in equations~(\\ref{eqn:impulseGeneralTerm1} and~\\ref{eqn:impulseGeneralTerm2}), take the form\n\\begin{equation}\n\\del(2p_\\alpha\\cdot q+q^2) = \\frac1{\\hbar m_\\alpha}\\del(2\\qb\\cdot u_\\alpha+\\lcomp \\qb^2)\\,.\n\\label{eqn:universalDeltaFunction}\n\\end{equation}\nThe integration over the initial momenta $\\initialk_\\alpha$ and the initial wavefunctions will smear out these delta functions to sharply peaked functions whose scale is of the same order as the original wavefunctions.  As $\\xi$ gets smaller, this function will turn back into a delta function imposed on the $\\qb$ integration. To see this, let us consider an explicit wavefunction integral similar to $\\ImpA$, but with a simpler integrand:\n\\defT_1{T_1}\n\\begin{equation}\nT_1 = \\int \\df(p_1)\\,\\varphi(p_1)\\varphi^*(p_1+q)\\,\\del(2 p_1\\cdot q+q^2)\\,.\\label{eqn:deltaFunctionIntegral}\n\\end{equation}\nWith $\\varphi$ chosen to be the linear exponential~(\\ref{eqn:LinearExponential}), this integral simplifies to\n\\begin{equation}\nT_1 = \\frac{1}{\\hbar m_1} \\eta_1(\\qb;p_1)\\,\\int \\df(p_1)\\,\n\\del(2 p_1\\cdot\\qb\/m_1+\\hbar\\qb^2\/m_1)\\,|\\varphi(p_1)|^2\\,,\n\\end{equation}\nwhere $\\eta_1(\\qb;p_1)$ is the overlap defined in equation~\\eqref{eqn:wavefunctionOverlap} and we have also replaced $q\\rightarrow\\hbar \\qb$.\n\nThe remaining integrations in $T_1$ are relegated in appendix~\\ref{app:wavefunctions}, but yield\\footnote{The wavenumber transfer is necessarily spacelike.}\n\\begin{multline}\nT_1 = \\frac1{4 \\hbar m_1\\sqrt{(\\qb\\cdot u)^2-\\qb^2}\\,K_1(2\/\\xi)} \n\\\\\\times \\exp\\biggl[-\\frac2{\\xi}\\frac{\\sqrt{(\\qb\\cdot u)^2-\\qb^2}}{\\sqrt{-\\qb^2}}\n\\sqrt{1-\\hbar^2\\qb^2\/(4m_1^2)}\\biggr]\n\\,.\\label{eqn:TIntegral}\n\\end{multline}\nNotice that our result depends on two dimensionless ratios in addition to its dependence on $\\xi$,\n\\begin{equation}\n\\lcomp \\sqrt{-\\qb^2}\n\\qquad\n\\textrm{and}\\qquad\n\\frac{\\qb\\cdot u}{\\sqrt{-\\qb^2}}\\,.\n\\end{equation}\nLet us call $1\/\\sqrt{-\\qb^2}$ a `scattering length' $\\lscatt$.  In terms of this length, our two dimensionless ratios are\n\\begin{equation}\n\\frac{\\lcomp}{\\lscatt}\n\\qquad\n\\textrm{and}\\qquad\n{\\qb\\cdot u}\\,\\lscatt\\,.\\label{eqn:dimensionlessRatios}\n\\end{equation}\n\nAs we approach the $\\hbar,\\xi\\rightarrow 0$ limit, we may expect $T_1$ to be concentrated in a small region in $\\qb$.  Towards the limit, the dependence on the magnitude is just given by the prefactor.  To understand the behaviour in the boost and angular degrees of freedom, we may note that \n\\begin{equation}\n\\frac1{K_1(2\/\\xi)} \\sim \\frac2{\\sqrt{\\pi}\\sqrt{\\xi}} \\exp\\biggl[\\frac2{\\xi}\\biggr]\\,,\\label{eqn:BesselLimit}\n\\end{equation}\nand that $\\hbar\\sqrt{\\xi}$ is of order $\\xi$, so that overall $T_1$ has the form\n\\begin{equation}\n\\frac1{\\xi}\\exp\\biggl[-\\frac{f(\\qb)}{\\xi}\\biggr]\\,.\n\\label{eqn:LimitForm}\n\\end{equation}\nThis will yield a delta function so long as $f(\\qb)$ is positive. To figure out its argument, we recall that $\\qb^2<0$, and parametrise the wavenumber as\n\\begin{equation}\n\\qb^\\mu = \\Eqb\\bigl(\\sinh\\zeta,\\,\\cosh\\zeta\\sin\\theta\\cos\\phi,\n\\,\\cosh\\zeta\\sin\\theta\\sin\\phi,\n\\,\\cosh\\zeta\\cos\\theta\\bigr)\\,,\\label{eqn:qbRapidityParametrisation}\n\\end{equation}\nwith rapidity $\\zeta$ running over $[0,\\infty]$, $\\theta$ over $[0,\\pi]$, and $\\phi$ over $[0,2\\pi]$. Working in the rest frame of $u^\\mu$, the exponent in \\eqn~\\eqref{eqn:TIntegral} (including the term from \\eqn~\\eqref{eqn:BesselLimit}) is\n\\begin{equation}\n-\\frac2{\\xi}\\Bigl(\\cosh\\zeta\\sqrt{1+\\hbar^2 \\Eqb^2\/(4m^2)}-1\\Bigr)\\,,\n\\end{equation}\nso that the delta function will ultimately localise \n\\begin{equation}\n\\cosh\\zeta \\rightarrow \\frac1{\\sqrt{1+\\hbar^2 \\Eqb^2\/(4m^2)}} = \n1-\\frac{\\hbar^2 \\Eqb^2}{8m^2}+\\Ord(\\hbar^4)\n\\end{equation}\nto zero. Thus in terms of the Lorentz--invariant dimensionless ratios in equation~\\eqref{eqn:dimensionlessRatios}, we find that the delta function is\n\\begin{equation}\n\\delta\\!\\left(\\qb\\cdot u\\, \\lscatt + \\frac{\\ell_c^2}{4 \\lscatt^2} \\, \\frac1{\\qb\\cdot u\\, \\lscatt}\\right). \\label{eqn:deltaArgument}\n\\end{equation}\nThe direction-averaging implicit in the integration over $\\initialk_1$ has led to a constraint on two positive quantities built out of the ratios.\n\nRecall that to arrive at this expression we absorbed a factor of $\\hbar\\sim\\sqrt{\\xi}$. Since the argument of the delta function is a polynomial in the dimensionless ratios, both must be independently constrained to be of this order:\n\\begin{subequations}\n\\begin{align}\n{\\qb\\cdot u}\\,\\lscatt &\\lesssim \\sqrt{\\xi}\\,,\\label{eqn:deltaConstraint1}\n\\\\\\frac{\\lcomp}{\\lscatt} &\\lesssim \\sqrt{\\xi}\\,.\\label{eqn:deltaConstraint2}\n\\end{align}\n\\end{subequations}\nIf we had not already scaled out a factor of $\\hbar$ from $q$, these constraints would make it natural to do so. \n\nCombining constraint~\\eqref{eqn:deltaConstraint1} with that in \\eqn~\\eqref{eqn:qbConstraint}, we obtain the constraint $\\lpack\\ll\\lscatt$. Then including constraint~\\eqref{eqn:ComptonConstraint1}, or $\\xi\\ll 1$, we obtain our first version of the `Goldilocks' inequalities,\n\\begin{equation}\n\\lcomp \\ll \\lpack \\ll \\lscatt\\,.\n\\label{eqn:Goldilocks1}\n\\end{equation}\nAs we shall see later in the explicit evaluation of $\\ImpA$, $\\lscatt\\sim \\sqrt{-b^2}$; this follows on dimensional grounds.  This gives us the second version of the `Goldilocks' requirement,\n\\begin{equation}\n\\lcomp\\ll \\lpack\\ll \\sqrt{-b^2}\\,. \n\\label{eqn:Goldilocks2}\n\\end{equation}\n\nNote that the constraint following from~\\eqref{eqn:deltaConstraint2} is weaker, $\\lpack\\lesssim\\lscatt$. Indeed, we should not expect a similar strengthening of this restriction; the sharp peaking of the wavefunctions alone will not force the left-hand side to be much smaller than the right-hand side. This means that we should expect $\\qb\\cdot u$ to be smaller than, but still of order, $\\sqrt{\\xi}\/\\lscatt$.  If we compare the two terms in the argument to the delta function~(\\ref{eqn:universalDeltaFunction}), we see that the second term\n\\begin{equation}\n\\lcomp \\qb^2 \\sim \\frac{\\lcomp}{\\lscatt} \\frac1{\\lscatt} \\ll \\frac{\\sqrt{\\xi}}{\\lscatt}\\,,\\label{eqn:neglectq2}\n\\end{equation}\nso that $\\lcomp \\qb^2 \\ll \\qb\\cdot u_\\alpha$, and the second term should be negligible. In our evaluation of $T_1$, we see that the integral is sharply peaked about the delta function\n\\begin{equation}\n\\delta(\\wn q\\cdot u)\\,.\\label{eqn:universalDeltaFunction2}\n\\end{equation}\nWe are thus free to drop the $\\wn q^2$ correction in the classical limit. There is one important caveat to this simplification, which we will mention below.\n\\begin{figure}[t]\n\t\\center\n\t\\includegraphics[width = 0.75\\textwidth]{Goldilocks}\n\t\\vspace{-3pt}\n\t\\caption{Heuristic depiction of the Goldilocks inequalities.\\label{Goldilocks}}\n\\end{figure}\n\n\\subsection{Taking the limit of observables}\n\nIn computing the classical observable, we cannot simply set $\\xi=0$. Indeed, we don't even want to fully take the $\\xi\\rightarrow 0$ limit. Rather, we want to take the leading term in that limit. This term may in fact be proportional to a power of $\\xi$.  To understand this, we should take note of one additional length scale in the problem, namely the classical radius of the point particle.  In electrodynamics, this is $\\lclass=e^2\/(4\\pi m)$. However,\n\\begin{equation}\n\\lclass = \\frac{\\hbar e^2}{4\\pi\\hbar m} = \\alpha\\lcomp\\,,\n\\end{equation}\nwhere $\\alpha$ is the usual, dimensionless, electromagnetic coupling. Dimensionless ratios of $\\lclass$ to other length scales will be the expansion parameters in classical observables; but as this relation shows, they too will vanish in the $\\xi\\rightarrow 0$ limit. There are really three dimensionless parameters we must consider: $\\xi$; $\\lpack\/\\lscatt$; and $\\lclass\/\\lscatt$.  We want to retain the full dependence on the latter, while considering only effects independent of the first two.\n\nUnder the influence of a perturbatively weak interaction (such as electrodynamics or gravity) below the particle-creation threshold, we expect a wavepacket's shape to be distorted slightly, but not radically changed by the scattering.  We would expect the outgoing particles to be characterised by wavepackets similar to those of the incoming particles. However, using a wavepacket basis of states for the state sums in \\sect{sec:QFTsetup} would be cumbersome, inconvenient, and computationally less efficient than the plane-wave states we used.  We expect the narrow peaking of the wavefunction to impose constraints on the momentum transfers as they appear in higher-order corrections to the impulse $\\ImpB$ in equation~\\eqref{eqn:impulseGeneralTerm2}; but we will need to see this narrowness indirectly, via assessments of the spread as in \\eqn~\\eqref{eqn:expectations}, rather than directly through the presence of wavefunction (or wavefunction mismatch) factors in our observables. We can estimate the spread $\\spread(\\finalk_\\alpha)$ in a final-state momentum $\\finalk_\\alpha$ as follows:\n\\begin{equation}\n\\begin{aligned}\n\\spread(\\finalk_\\alpha)\/m_\\alpha^2 &= \n\\langle\\bigl(\\finalk_\\alpha-\\langle\\finalk_\\alpha\\rangle\\bigr)^2\\rangle\/m_\\alpha^2\n\\\\&= \\bigl(\\langle \\finalk_\\alpha^2\\rangle-\\langle\\finalk_\\alpha\\rangle{}^2\\bigr)\/m_\\alpha^2\n\\\\&= 1-\\bigl(\\langle\\initialk_\\alpha\\rangle+\\expchange\\bigr){}^2\/m_\\alpha^2\n\\\\&= \\spread(p_\\alpha)\/m_\\alpha^2 -\\langle\\Delta p_\\alpha\\rangle\\cdot \n\\bigl(2\\langle\\initialk_\\alpha\\rangle+\\expchange\\bigr)\/m_\\alpha^2\\,.\n\\end{aligned}\n\\end{equation}\nSo long as $\\expchange\/m_\\alpha\\lesssim\\spread(\\initialk_\\alpha)\/m_\\alpha^2$, the second term will not greatly increase the result, and the spread in the final-state momentum will be of the same order as that in the initial-state momentum.   Whether this condition holds depends on the details of the wavefunction. Even if it is violated, so long as $\\expchange\/m_\\alpha \\lesssim c'_\\Delta \\xi^{\\beta'''}$ with $c'_\\Delta$ a constant of $\\Ord(1)$, then the final-state momentum will have a narrow spread towards the limit.  (It would be broader than the initial-state momentum spread, but that does not affect the applicability of our results.)\n\nThe magnitude of $\\expchange$ can be determined perturbatively.  The leading-order value comes from $\\ImpA$, with $\\ImpB$ contributing yet-smaller corrections.  As we shall see, these computations reveal $\\expchange\/m_\\alpha$ to scale like $\\hbar$, or $\\sqrt{\\xi}$, and be numerically much smaller.\n\nThis in turn implies that for perturbative consistency, the `characteristic' values of momentum transfers $w_\\alpha$ inside the definition of $I^\\mu_{(2)}$ must also be very small compared to $m_\\alpha\\sqrt{\\xi}$. This constraint is in fact much weaker than implied by the leading-order value of $\\expchange$. Just as for $q_0$ in \\eqn~\\eqref{eqn:qConstraint1}, we should scale these momentum transfers by $1\/\\hbar$, replacing them by wavenumbers $\\xferb_\\alpha$.  The corresponding scattering lengths $\\tilde\\lscatt^\\alpha = \\sqrt{-w_{\\alpha}^2}$ must again satisfy $\\tilde\\lscatt^\\alpha \\gg \\lpack$. If we now examine the energy-momentum-conserving delta function \nin \\eqn~\\eqref{eqn:impulseGeneralTerm2},\n\\begin{equation}\n\\del^{(4)}(w_1+w_2 + \\finalk_X)\\,,\\label{eqn:radiationScalingDeltaFunction}\n\\end{equation}\nwe see that all other transferred momenta $\\finalk_\\alpha$ must likewise be small compared to $m_\\alpha\\sqrt{\\xi}$: all their energy components must be positive and hence no cancellations are possible inside the delta function. The typical values of these momenta should again by scaled by $1\/\\hbar$ and replaced by wavenumbers. We will see in the next chapter that $I_{(2)}^\\mu$ encodes radiative effects, and the same constraint will force these momenta for emitted radiation to scale as wavenumbers. \n\nWhat about loop integrations?  As we integrate the loop  momentum over all values, it is a matter of taste how we scale it.  If it is the momentum of a (virtual) massless line, however, unitarity considerations suggest that as the natural scaling is to remove a factor of $\\hbar$ in the real contributions to the cut in equation~\\eqref{eqn:impulseGeneralTerm2}, we should likewise do so for virtual lines.  More generally, we should scale those differences of the loop momentum with external legs that correspond to massless particles, and replace them by wavenumbers.  Moreover, unitarity considerations also suggest that we should choose the loop momentum to be that of a massless line in the loop, if there is one.\n\nIn general, we may not be able to approach the $\\hbar\\rightarrow0$ limit of each contribution to an observable separately, because they may contain terms which are singular, having too many inverse powers of $\\hbar$.  We find that such singular terms meet one of two fates: they are multiplied by functions which vanish in the regime of validity of the limit; or they cancel in the sum over all contributions.  We cannot yet offer a general argument that such troublesome terms necessarily disappear in one of these two\nmanners.  We can treat independently contributions whose singular terms ultimately cancel in the sum, so long as we expand each contribution in a Laurent series in $\\hbar$.\n\nFor theories with non-trivial internal symmetries, when identifying singular terms (in both parts of the impulse) it is essential not to forget factors coming from the classical limit of the representation states $\\chi_\\alpha$. Since the scattering particles remain well separated at all times there is no change to the story in chapter~\\ref{chap:pointParticles}. However, it is important to keep in mind that in our conventions colour factors in scattering amplitudes carry dimensions of $\\hbar$, and in particular satisfy the Lie algebra in equation~\\eqref{eqn:chargeLieAlgebra}. There is subsequently an independent Laurent series available when evaluating colour factors, and this can remove terms which appear singular in the kinematic expansion.\n\nFull impulse integrand factors that appear uniformly in all contributions --- that is, factors which appear directly in a final expression after cancellation of terms singular in the $\\hbar\\rightarrow 0$ limit --- can benefit from applying two simplifications to the integrand: setting $p_\\alpha$ to $m_\\alpha\\ucl_\\alpha$, as prescribed by equation~\\eqref{eqn:universalDeltaFunction2}, and truncating at the lowest order in $\\hbar$ or $\\xi$.  For other factors, we must be careful to expand in a Laurent series.  As mentioned above, a consequence of equation~\\eqref{eqn:neglectq2} is that inside the on-shell delta functions $\\del(2p_\\alpha\\cdot \\qb\\pm \\hbar \\qb^2)$ we can neglect the $\\hbar \\qb^2$ term; this is true so long as the factors multiplying these delta functions are not singular in $\\hbar$. If they are indeed nonsingular (after summing over terms), we can safely neglect the second term inside such delta functions, and replace them by $\\del(2p_\\alpha\\cdot \\qb)$.  A similar argument allows us to neglect the $\\hbar \\qb^0$ term inside the positive-energy theta functions; the $\\qb$ integration then becomes independent of them.  Similar arguments, and caveats, apply to the squared momentum-transfer terms $\\hbar \\xferb_\\alpha^2$ appearing inside on-shell delta functions in higher-order contributions, along with the energy components $\\xferb_\\alpha^0$ appearing inside positive-energy theta functions.  They can be neglected so long as the accompanying factors are not singular in $\\hbar$.  If accompanying factors \\textit{are} singular as $\\hbar\\rightarrow 0$, then we may need to retain such formally suppressed $\\hbar \\qb^2$ or $\\hbar \\xferb_\\alpha^2$ terms inside delta functions.\nWe will see an example of this in the calculation of the NLO contributions to the impulse in section~\\ref{sec:examples}.\n\n\\subsubsection{Summary}\n\nFor ease of future reference, let us collect the rules we have derived for calculating classical scattering observables from quantum field theory. We have all together established that, in the classical limit, we must apply the following constraints when evaluating amplitudes in explicit calculations:\n\\begin{itemize}\n\t\\item The momentum mismatch $q = p'_1 - p_1$ scales as a wavenumber, $q = \\hbar \\wn q$.\n\t\\item The momentum transfers $w_\\alpha$ in $I_{(2)}^\\mu$ scale as wavenumbers.\n\t\\item Massless loop momenta scale as wavenumbers.\n\t\\item The $\\hbar \\wn q^2$ factors in on--shell delta functions can be dropped, but only when there are no terms singular in $\\hbar$.\n\t\\item Any amplitude colour factors are evaluated using the commutation relation~\\eqref{eqn:chargeLieAlgebra}.\n\\end{itemize}\nWe derived these rules using the impulse, but they hold for any on-shell observable constructed in the manner of section~\\ref{sec:QFTsetup}. Furthermore, it will be convenient to introduce a notation to allow us to manipulate integrands under the eventual approach to the $\\hbar\\rightarrow0$ limit; we will use large angle brackets for the purpose,\n\\begin{multline}\n\\Lexp f(p_1,p_2,\\cdots) \\Rexp = \\int\\! \\df(p_1) \\df(p_2) |\\varphi(p_1)|^2 |\\varphi(p_2)|^2\\\\ \\times \\langle \\chi_1\\,\\chi_2|f(p_1,p_2,\\cdots)|\\chi_1\\,\\chi_2\\rangle\\,,\n\\label{eqn:angleBrackets}\n\\end{multline}\nwhere the integration over both $\\initialk_1$ and $\\initialk_2$ is implicit.  Within the angle brackets, we have approximated $\\varphi_\\alpha(p\\pm q)\\simeq \\varphi_\\alpha(p)$ and $\\chi'_\\alpha \\simeq  \\chi_\\alpha$. Then, relying on our detailed study of the momentum and colour wavefunctions in sections~\\ref{sec:PointParticleLimit} and~\\ref{sec:classicalLimit}, to evaluate the integrals and representation expectation values implicit in the large angle brackets we can simply set $p_\\alpha\\simeq m_\\alpha \\ucl_\\alpha$, and replace quantum colour charges $C_\\alpha^a$ with their (commuting) classical counterparts $c^a_\\alpha$.\n\n\\subsection{The classical impulse}\n\\label{sec:classicalImpulse}\n\nWe have written the impulse in terms of two terms, $\\langle \\Delta p_1^\\mu \\rangle = \\ImpA + \\ImpB$, and expanded these in terms of wavefunctions in equations~\\eqref{eqn:impulseGeneralTerm1} and ~\\eqref{eqn:impulseGeneralTerm2}. We will now discuss the classical limit of these terms in detail, applying the rules gathered above.\n\nWe begin with the first and simplest term in the impulse, $\\ImpA$, given in \\eqn~\\eqref{eqn:impulseGeneralTerm1}, and here recast in the notation of \\eqn~\\eqref{eqn:angleBrackets} in preparation:\n\\begin{multline}\n\\ImpAcl =  \\Lexp i\\!\\int \\!\\dd^4 q  \\; \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\\\\n \\times  e^{-i b \\cdot q\/\\hbar} \n\\, q^\\mu  \\, \\Ampl(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\\,\\Rexp\\,.\n\\label{eqn:impulseGeneralTerm1recast}\n\\end{multline}\nRescale $q \\rightarrow \\hbar\\qb$; drop the $q^2$ inside the on-shell delta functions;\nand also remove the overall factor of $\\tilde g^2$ and accompanying $\\hbar$'s from the amplitude, to obtain the leading-order (LO) contribution to the classical impulse,\n\\begin{multline}\n\\DeltaPlo \\equiv \\ImpAclsup{(0)} =  \\frac{i\\tilde g^2}{4} \\Lexp \\hbar^2\\! \\int \\!\\dd^4 \\qb  \\; \n\\del(\\qb\\cdot p_1) \\del(\\qb\\cdot p_2) \n\\\\\\times  \ne^{-i b \\cdot \\qb} \n\\, \\qb^\\mu  \\, \\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow \np_1 + \\hbar\\qb, p_2 - \\hbar\\qb)\\,\\Rexp\\,.\n\\label{eqn:impulseGeneralTerm1classicalLO}\n\\end{multline}\nWe denote by $\\AmplB^{(L)}$ the reduced $L$-loop amplitude, that is the $L$-loop amplitude with a factor of the (generic) coupling $\\tilde g\/\\sqrt{\\hbar}$ removed for every interaction: in the gauge theory case, this removes a factor of $g\/\\sqrt{\\hbar}$, while in the gravitational case, we would remove a factor of $\\kappa\/\\sqrt{\\hbar}$.  In general, this rescaled fixed-order amplitude depends only on $\\hbar$-free ratios of couplings; in pure electrodynamics or gravitational theory, it is independent of couplings.  In pure electrodynamics, it depends on the charges of the scattering particles. While it is free of the powers of $\\hbar$ discussed in section~\\ref{sec:RestoringHBar}, it will in general still scale with an overall power of $\\hbar$ thanks to dependence on momentum mismatches or transfers. As we shall see in the next section, additional inverse powers of $\\hbar$ emerging from $\\AmplB$ will cancel the $\\hbar^2$ prefactor and yield a nonvanishing result.\n\nAs a reminder, while this contribution to a physical observable is linear in an amplitude, it arises from an expression involving wavefunctions multiplied by their conjugates.  This is reflected in the fact that both the `incoming' and `outgoing' momenta in the amplitude here are in fact initial-state momenta. Any phase which could be introduced by hand in the initial state would thus cancel out of the observable.\n\nThe LO classical impulse is special in that only the first term~(\\ref{eqn:impulseGeneralTerm1}) contributes. In general however, it is only the sum of the two terms in \\eqn~\\eqref{eqn:defl1} that has a well-defined classical limit.  We may write this sum as\n\\begin{multline}\n\\Delta p_1^\\mu = \n\\Lexp i\\hbar^{-2}\\!\n\\int \\!\\dd^4 q  \\; \\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2)\\\\ \\times \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0) \\; e^{-i b \\cdot q\/\\hbar} \\; \\impKer \\Rexp \\,,\n\\label{eqn:partialClassicalLimitNLO}\n\\end{multline}\nwhere the \\textit{impulse kernel\\\/} $\\impKer$ is defined as\n\\begin{equation}\n\\begin{aligned}\n\\impKer \\equiv&\\,  \\hbar^2 q^\\mu \\, \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\n\\\\& -i \\hbar^2 \\sum_X \\int \\!\\prod_{\\alpha = 1,2}  \\dd^4 \\xfer_\\alpha\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\hphantom{-} \\times \\xfer_1^\\mu\\, \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2 \\,, \\finalk_X)\n\\\\ &\\hspace*{25mm}\\times \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \\initialk_1+\\xfer_1 \\,,\\initialk_2+\\xfer_2 \\,, \\finalk_X)\\,.\n\\end{aligned}\n\\label{eqn:FullImpulse}\n\\end{equation}\nThe prefactor in \\eqn~\\eqref{eqn:partialClassicalLimitNLO} and the normalization of $\\impKer$ are chosen so that the latter is $\\Ord(\\hbar^0)$ in the classical limit. At leading order, \nthe only contribution comes from the tree-level four-point amplitude in the first term, and after passing to the classical limit, we recover \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} as expected. At next-to-leading order (NLO), both terms contribute. The contribution from\nthe first term is from the one-loop amplitude, while that from the second term has $X=\\emptyset$, so that both the amplitude and conjugate inside the integral are tree level four-point amplitudes.\n\nFocus on the NLO contributions, and pass to the classical limit. As discussed in section~\\ref{subsec:Wavefunctions} we may neglect the $q^2$ terms in the delta functions present in \\eqn~\\eqref{eqn:partialClassicalLimitNLO} so long as any singular terms in the impulse\nkernel cancel. We then rescale $q \\rightarrow \\hbar\\qb$; and remove an overall factor of $\\tilde g^4$ and accompanying $\\hbar$''s from the amplitudes. In addition, we may rescale $\\xfer\\rightarrow \\hbar\\xferb$. However, since singular terms may be present in the individual summands of the impulse kernel --- in general, they will cancel against singular terms emerging from the loop integration in the first term in \\eqn~\\eqref{eqn:FullImpulse} ---  \nwe are not entitled to drop the $w^2$ inside the on-shell delta functions. We obtain\n\\begin{equation}\n\\DeltaPnlo= \\frac{i\\tilde g^4}{4}\\Lexp \\int \\!\\dd^4 \\qb  \\; \\del(\\initialk_1 \\cdot \\qb ) \n\\del(\\initialk_2 \\cdot \\qb) \n\\; e^{-i b \\cdot \\qb} \\;  \\impKerCl \\Rexp \\,,\n\\label{eqn:classicalLimitNLO}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl &= \\hbar \\qb^\\mu \\, \\AmplB^{(1)}(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + \\hbar\\qb  , \\initialk_2 -  \\hbar\\qb)\n\\\\&\\hphantom{=} \n-i \\hbar^3 \\int \\! \\dd^4 \\xferb \\; \n\\del(2p_1\\cdot \\xferb+ \\hbar\\xferb^2)\\del(2p_2\\cdot \\xferb- \\hbar\\xferb^2)  \\; \\xferb^\\mu \\;\n\\\\&\\hspace*{15mm}\\times \n\\AmplB^{(0)}(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+ \\hbar \\xferb\\,, \\initialk_2- \\hbar\\xferb)\n\\\\&\\hspace*{15mm}\\times \n\\AmplB^{(0)*}(\\initialk_1+ \\hbar\\qb\\,, \\initialk_2-  \\hbar\\qb \\rightarrow \n\\initialk_1+\\hbar\\xferb  \\,,\\initialk_2- \\hbar\\xferb) \\,.\n\\label{eqn:impKerClDef}\n\\end{aligned}\n\\end{equation}\nOnce again, we will see in the next section that additional inverse powers of $\\hbar$ will arise from the amplitudes, and will yield a finite and nonvanishing answer\nin the classical limit.\n\n\\section{Examples}\n\\label{sec:examples}\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\n\nTo build confidence in the formalism we have developed, let us use it to conduct explicit calculations of the classical impulse. We will work in the context of scalar Yang--Mills theory, as defined by the Lagrangian in equation~\\eqref{eqn:scalarAction}, using the double copy where our interest is in perturbative gravity.\n\nBefore we begin to study the impulse at leading and next-to-leading order, note that it is frequently convenient to write amplitudes in Yang--Mills theory in colour-ordered form; for example, see~\\cite{Ochirov:2019mtf} for an application to amplitudes with multiple different external particles. The full amplitude $\\mathcal{A}$ is decomposed onto a basis of colour factors times partial amplitudes $A$. The colour factors are associated with some set of Feynman topologies. Once a basis of independent colour structures is chosen, the corresponding partial amplitudes must be gauge invariant. Thus,\n\\[\n\\mathcal{A}(p_1,p_2 \\rightarrow p_1',p_2') = \\sum_D \\mathcal{C}(D)\\, A_D(p_1,p_2 \\rightarrow p_1',p_2')\\,,\\label{eqn:colourStripping}\n\\]\nwhere $\\mathcal{C}(D)$ is the colour factor of diagram $D$ and $A_D$ is the associated partial amplitude. Expectation values of the representation states $\\chi_\\alpha$ can now be taken as being purely over the colour structures.\n\n\\subsection{Leading-order impulse}\n\\label{sec:LOimpulse}\n\\newcommand{\\tree}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .125 and .275 of v1] (o1);\n\t\\vertex [below right = .125 and .275 of v2] (o2);\n\t\\vertex [above left = .125 and .275 = of v1] (i1);\n\t\\vertex [below left = .125 and .275 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\subsubsection{Gauge theory}\n\nWe begin by computing in YM theory the impulse, $\\DeltaPlo$, on particle 1 at leading order. At this order, only $\\ImpA$ contributes, as expressed in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. To evaluate the impulse, we must first compute the $2\\rightarrow 2$ tree-level scattering amplitude. The reduced amplitude $\\AmplB^{(0)}$ is\n\\begin{equation}\ni\\AmplB^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = \\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\\begin{feynman}\n\\vertex (v1);\n\\vertex [below = 0.97 of v1] (v2);\n\\vertex [above left=0.5 and 0.66 of v1] (i1) {$p_1$};\n\\vertex [above right=0.5 and 0.33 of v1] (o1) {$p_1+\\hbar \\qb$};\n\\vertex [below left=0.5 and 0.66 of v2] (i2) {$p_2$};\n\\vertex [below right=0.5 and 0.33 of v2] (o2) {$p_2-\\hbar \\qb$};\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o2);\n\\diagram*{(v1) -- [gluon] (v2)};\n\\end{feynman}\t\n\\end{tikzpicture}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!= i \\newT_1\\cdot\\newT_2 \\frac{4 p_1 \\cdot p_2 + \\hbar^2 \\qb^2}\n{\\hbar^2 \\qb^2}\\,.\n\\label{eqn:ReducedAmplitude1}\n\\end{equation}\nClearly, the colour decomposition of the amplitude is trivial:\n\\begin{equation}\\label{eqn:treeamp}\n\\bar{A}_{\\scalebox{0.5}{\\tree}} =  \\frac{4 p_1\\cdot p_2 +\\hbar \\barq^2}{\\hbar^2 \\barq^2}\\,, \\qquad \\mathcal{C}\\!\\left(\\tree\\right) = \\newT_1\\cdot\\newT_2\\,.\n\\end{equation}\nWe can neglect the second term in the numerator, which is subleading in the classical limit.\n\nSubstituting this expression into \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}, we obtain\n\\begin{equation}\n\\DeltaPlo = i g^2 \\Lexp \\int \\!\\dd^4 \\qb  \\; \\del(\\qb\\cdot p_1) \\del(\\qb\\cdot p_2)\\, e^{-i b \\cdot \\qb} \\newT_1\\cdot\\newT_2 \\frac{p_1 \\cdot p_2}{\\qb^2}\\, \\qb^\\mu\\,\\Rexp\\,.\n\\label{eqn:impulseClassicalLOa}\n\\end{equation}\nAs promised, the leading-order expression is independent of $\\hbar$. Evaluating the $p_{1,2}$ integrals, in the process applying the simplifications explained in section~\\ref{sec:classicalLimit}, namely replacing $p_\\alpha\\rightarrow m_\\alpha\\ucl_\\alpha$, we find that\n\\begin{equation}\n\\DeltaPlo = i g^2 c_1\\cdot c_2 \\int \\!\\dd^4 \\qb  \\; \\del(\\qb\\cdot \\ucl_1) \\del(\\qb\\cdot \\ucl_2) \ne^{-i b \\cdot \\qb} \\frac{\\ucl_1 \\cdot \\ucl_2}{\\qb^2} \\, \\qb^\\mu\\,.\n\\label{eqn:impulseClassicalLO}\n\\end{equation}\nNote that evaluating the double angle brackets has also replaced quantum colour factors with classical colour charges. Replacing the classical colour with electric charges $Q_\\alpha$ yields the result for QED; this expression then has intriguing similarities to quantities that arise in the high-energy\nlimit of two-body scattering~\\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe,Amati:1992zb,Kabat:1992tb,Amati:1993tb,Muzinich:1995uj,DAppollonio:2010krb,Melville:2013qca,Akhoury:2013yua,DAppollonio:2015fly,Ciafaloni:2015vsa,DAppollonio:2015oag,Ciafaloni:2015xsr,Luna:2016idw,Collado:2018isu}. The eikonal approximation used there is known to\nexponentiate, and it would be interesting to explore this connection further. \n\nNote that it is natural that the Yang--Mills LO impulse is a simple colour dressing of its QED counterpart, since at leading order the gluons do not self interact.\n\nIt is straightforward to perform the integral over $\\qb$ in \\eqn~\\eqref{eqn:impulseClassicalLO} to obtain an explicit expression for the leading order impulse. To do so, we work in the rest frame of particle 1, so that $\\ucl_1 = (1, 0, 0, 0)$. Without loss of generality we can orientate the spatial coordinates in this frame so that particle 2 is moving along the $z$ axis, with proper velocity $\\ucl_2 = (\\gamma, 0, 0, \\gamma \\beta)$. We have introduced the standard Lorentz gamma factor $\\gamma = \\ucl_1 \\cdot \\ucl_2$ and the velocity parameter $\\beta$ satisfying $\\gamma^2 ( 1- \\beta^2 ) = 1$. In terms of these variables, the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPlo &= i g^2 c_1\\cdot c_2 \\int \\!\\dd^4 \\qb  \\;\n\\del(\\qb^0) \\del(\\gamma \\qb^0 - \\gamma \\beta \\qb^3) \\;\ne^{-i b \\cdot \\qb} \\frac{\\gamma}{\\qb^2}\n\\, \\qb^\\mu \\\\\n&= -i \\frac{g^2 c_1 \\cdot c_2 }{4\\pi^2 |\\beta|}\\int \\! d^2 \\qb  \\;\ne^{i \\v{b} \\cdot \\v{\\qb}_\\perp} \\frac{1}{\\v \\qb_\\perp^2}\n\\, \\qb^\\mu \\, ,\n\\end{aligned}\n\\end{equation}\nwhere $\\qb^0 = \\qb^3 = 0$ and the non-vanishing components of $\\qb^\\mu$ in the $xy$ plane of our corrdinate system are $\\v \\qb_\\perp$. It remains to perform the two dimensional integral over $\\v \\qb_\\perp$, which is easily done using polar coordinates. Let the magnitude of $\\v \\qb_\\perp$ be $\\chi$ and orient the $x$ and $y$ axes so that $\\v b \\cdot \\v \\qb_\\perp = | \\v b| \\chi \\cos \\theta$. Then the non-vanishing components of $\\qb^\\mu$ are $\\qb^\\mu = (0, \\chi \\cos \\theta, \\chi \\sin \\theta, 0)$ and the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPlo &= -i \\frac{g^2 c_1\\cdot c_2 }{4\\pi^2 |\\beta|}\\int_0^\\infty d \\chi \\; \\chi \\int_{-\\pi}^\\pi d \\theta   \\;\ne^{i | \\v b| \\chi \\cos \\theta} \\frac{1}{\\chi^2}\n\\, (0, \\chi \\cos \\theta, \\chi \\sin \\theta, 0) \\\\\n&= -i \\frac{g^2 c_1\\cdot c_2 }{4\\pi^2 |\\beta|}\\int_0^\\infty d \\chi \\; \\int_{-\\pi}^\\pi d \\theta   \\;\ne^{i | \\v b| \\chi \\cos \\theta} \n\\, (0, \\cos \\theta, \\sin \\theta, 0) \\\\\n&= \\frac{g^2 c_1\\cdot c_2 }{2\\pi |\\beta|}\\int_0^\\infty d \\chi \\; \nJ_1 ( |\\v b| \\chi) \\; \\hat{\\v b} \\\\ \n&= \\frac{g^2 c_1\\cdot c_2 }{2\\pi |\\beta|} \\; \\frac{\\hat{\\v b}}{| \\v b|} \\, ,\\label{eqn:LOimpulseIntegral}\n\\end{aligned}\n\\end{equation}\nwhere $\\hat {\\v b}$ is the spatial unit vector in the direction of the impact parameter. To restore manifest Lorentz invariance, note that\n\\begin{equation}\n\\frac{1}{| \\beta|} = \\frac{\\gamma}{\\sqrt{\\gamma^2 - 1}}\\,, \n\\quad \\frac{\\hat{\\v b}}{|\\v b|} = - \\frac{b^\\mu}{b^2}\\,.\n\\end{equation}\n(Recall that $b^\\mu$ is spacelike, so $-b^2>0$.) With this input, we may write the impulse as\n\\begin{equation}\n\\DeltaPlo  \n= -\\frac{g^2 c_1\\cdot c_2}{2\\pi} \\frac{\\gamma}{\\sqrt{\\gamma^2 - 1}} \\frac{b^\\mu}{b^2}\\,.\n\\end{equation}\n\nStripping away the colour and adopting the QED coupling $e$, in the non-relativistic limit this should match a familiar formula: the expansion of the Rutherford scattering angle $\\theta(b)$ as a function of the impact parameter. To keep things simple, we consider Rutherford scattering of a light particle (for example, an electron) off a heavy particle (a nucleus). Taking particle 1 to be the moving light particle, particle 2 is very heavy and we work in its rest frame. Expanding the textbook Rutherford result to order $e^2$, we find\n\\begin{equation}\n\\theta(b) = 2 \\tan^{-1} \\frac{e^2}{4 \\pi m v^2 b} \\simeq \\frac{e^2}{2 \\pi m v^2 b},\n\\end{equation}\nwhere $v$ is the non-relativistic velocity of the particle. To recover this simple result from equation~\\eqref{eqn:impulseClassicalLO}, recall that in the non-relativistic limit $\\gamma \\simeq 1 + v^2\/2$. The scattering angle, at this order, is simply $\\Delta v\/v$.  We will make use of this frame in later sections as well.\n\nWe note in passing that the second term in the numerator \\eqn~\\eqref{eqn:ReducedAmplitude1} is a quantum correction.  It will ultimately be suppressed by $\\lcomp^2\/b^2$, and in addition would contribute only a contact interaction, as it leads to a $\\delta^{(2)}(b)$ term in the impulse.\n\n\\subsubsection{Gravity}\n\nRather than compute gravity amplitudes using the Feynman rules associated with the Einstein--Hilbert action, we can easily just apply the double copy where we have knowledge of their gauge theory counterparts. The generalisation of the traditional BCJ gauge theory replacement rules \\cite{Bern:2008qj,Bern:2010ue} to massive matter states was developed by Johansson and Ochirov \\cite{Johansson:2014zca}.  In our context the colour-kinematics replacement is simple: the amplitude only has a $t$-channel diagram, making the Jacobi identity trivial. Thus by replacing the colour factor with the desired numerator we are guaranteed to land on a gravity amplitude, provided we replace $g\\rightarrow\\frac{\\kappa}{2}$, where $\\kappa = \\sqrt{32\\pi G}$ is the coupling in the Einstein--Hilbert Lagrangian.\n\nA minor point before double-copying is to further rescale\\footnote{We choose this normalisation as it simplifies the colour replacements in the double copy.} the (dimensionful) colour factors as $\\tilde{\\newT}^a$ = $\\sqrt{2}\\newT^a$, such that\n\\begin{equation}\n\\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = \\frac{g^2}{\\hbar^3}\\frac{2p_1 \\cdot p_2 + \\mathcal{O}(\\hbar)}{\\wn q^2} \\tilde{\\newT}_1 \\cdot \\tilde{\\newT}_2\\,.\\label{eqn:scalarYMamp}\n\\end{equation}\nThen replacing the colour factor with the (rescaled) scalar numerator from equation~\\eqref{eqn:treeamp}, we immediately obtain the gravity tree amplitude\\footnote{The overall sign is consistent with the replacements in \\cite{Bern:2008qj,Bern:2010ue} for our amplitudes' conventions.}\n\\begin{equation}\n\\mathcal{M}^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = -\\frac{4}{\\hbar^3}\\left(\\frac{\\kappa}{2}\\right)\\frac{(p_1 \\cdot p_2)^2 + \\mathcal{O}(\\hbar)}{\\wn q^2} \\,.\n\\end{equation}\nThis is not quite an amplitude in Einstein gravity: the interactions suffer from dilaton pollution, as can immediately be seen by examining the amplitude's factorisation channels:\n\\[\n\\lim\\limits_{\\wn q^2 \\rightarrow 0} \\left(\\wn q^2 \\hbar^3 \\mathcal{M}^{(0)}\\right) &= -4\\left(\\frac{\\kappa}{2}\\right)^2\\, p_1^\\mu p_1^{\\tilde{\\mu}} \\left(\\mathcal{P}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} + \\mathcal{D}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\right) p_2^\\nu p_2^{\\tilde{\\nu}}\\,,\n\\]\nwhere\n\\begin{equation}\n\\mathcal{P}^{(D)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} = \\eta_{\\mu(\\nu}\\eta_{\\tilde{\\nu})\\tilde{\\mu}} - \\frac{1}{D-2}\\eta_{\\mu\\tilde{\\mu}}\\eta_{\\nu\\tilde{\\nu}} \\qquad \\text{and} \\qquad\n\\mathcal{D}^{(D)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} = \\frac{1}{D-2}\\eta_{\\mu\\tilde{\\mu}}\\eta_{\\nu\\tilde{\\nu}}\\label{eqn:gravityProjectors}\n\\end{equation}\nare the $D$-dimensional de-Donder gauge graviton and dilaton projectors respectively. The pure Einstein gravity amplitude can now just be read off as the part of the amplitude contracted with the graviton projector. We find that\n\\begin{equation}\n\\mathcal{M}^{(0)}_{\\rm GR}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\hbar^3\\,\\wn q^2} \\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right).\n\\end{equation}\nFollowing the same steps as those before~\\eqref{eqn:impulseClassicalLO}, we find that the LO impulse for massive scalar point-particles in general relativity (such as Schwarzschild black holes) is\n\\begin{equation}\n\\Delta p_1^{\\mu,(0)} = -2i m_1 m_2 \\left(\\frac{\\kappa}{2}\\right)^2\\! \\int \\!\\dd^4 \\qb  \\; \\del(2\\qb\\cdot \\ucl_1) \\del(2\\qb\\cdot \\ucl_2) \ne^{-i b \\cdot \\qb} \\frac{(2\\gamma^2 - 1)}{\\qb^2} \\, \\qb^\\mu\\,.\n\\end{equation}\nIntegrating as in equation~\\eqref{eqn:LOimpulseIntegral} yields the well known 1PM result~\\cite{Westpfahl:1979gu,Portilla:1980uz}\n\\begin{equation}\n\\Delta p_1^{\\mu,(0)} = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}} (2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2}\\,.\n\\end{equation}\n\n\\subsection{Next-to-leading order impulse}\n\\label{sec:nloQimpulse}\n\nAt the next order in perturbation theory, a well-defined classical impulse is only obtained by combining all terms in the impulse $\\langle \\Delta p_1^\\mu \\rangle$ of order $\\tilde g^4$. As we discussed in section~\\ref{sec:classicalImpulse}, both $\\ImpA$ and $\\ImpB$ contribute. We found in \\eqn~\\eqref{eqn:classicalLimitNLO} that the impulse is a simple integral over an impulse kernel $\\impKerCl$, defined in \\eqn~\\eqref{eqn:impKerClDef}, which has a well-defined classical limit. \n\nThe determination of the impulse kernel at this order requires us to compute the four-point one-loop amplitude along with a cut amplitude; that is, an integral over a term quadratic in the tree amplitude. We will compute the NLO impulse in scalar Yang--Mills theory. As the one-loop amplitude in gauge theory is simple, we compute using on-shell renormalised perturbation theory in Feynman gauge.\n\n\\subsubsection{Purely Quantum Contributions}\n\\label{sec:PurelyQuantum}\n\nThe contributions to the impulse in the quantum theory can be divided into three classes, according to the prefactor in the charges they carry. For simplicity of counting, let us momentarily restrict to Abelian gauge theory, with charges $Q_\\alpha$. There are then three classes of diagrams: $\\Gamma_1$, those proportional to $Q_1^3 Q_2$; $\\Gamma_2$, those to $Q_1^2 Q_2^2$; and $\\Gamma_3$, those to $Q_1 Q_2^3$. The first class can be further subdivided into $\\Gamma_{1a}$, terms which would be proportional to $Q_1 (Q_1^2+n_s Q_3^2) Q_2$ were we to add $n_s$ species of a third scalar with charge $Q_3$, and into $\\Gamma_{1b}$, terms which would retain the simple $Q_1^3 Q_2$ prefactor.  Likewise, the last class can be further subdivided into $\\Gamma_{3a}$, terms which would be proportional to $Q_1 (Q_2^2+n_s Q_3^2) Q_2$, and into $\\Gamma_{3b}$, those whose prefactor would remain simply $Q_1 Q_2^3$.\n\nClasses $\\Gamma_{1a}$ and $\\Gamma_{3a}$ consist of gauge boson self-energy corrections along with renormalisation counterterms.  They appear only in the 1-loop corrections to the four-point amplitude, in the first term in the impulse kernel $\\impKerCl$. As one may intuitively expect, they give no contribution in the classical limit. Consider, for example, the self-energy terms, focussing on internal scalars of mass $m$ and charge $Q_i$. We define the self-energy via\n\\begin{equation}\n\\hspace{-10pt}Q_i^2 \\Pi(q^2) \\left( q^2 \\eta^{\\mu\\nu} - q^\\mu q^\\nu \\right) \\equiv \\scalebox{0.9}{\\feynmandiagram [inline = (a.base), horizontal=a to b, horizontal=c to d] { a -- [photon, momentum'=\\(q\\)] b -- [fermion, half left] c -- [fermion, half left] b -- [draw = none] c -- [photon] d};\n\\,  + \\!\\!\\!\\! \\feynmandiagram[inline = (a.base), horizontal=a to b]{a -- [photon, momentum'=\\(q\\)] c -- [out=45, in=135, loop, min distance=2cm]c -- [photon] b};\n\\!\\!\\!\\! + \\, \\feynmandiagram[inline = (a.base), layered layout, horizontal=a to b] { a -- [photon, momentum'=\\(q\\)] b [crossed dot] -- [photon] c};} \\,,\n\\label{eqn:SelfEnergyContributions}\n\\end{equation}\nwhere we have made the projector required by gauge invariance manifest, but have not included factors of the coupling. We have extracted the charges $Q_i$ for later convenience. The contribution of the photon self-energy to the reduced 4-point amplitude is\n\\begin{equation}\n\\AmplB_\\Pi = {Q_1 Q_2 Q_i^2} \\frac{(2p_1 + \\hbar \\qb) \\cdot (2p_2 -\\hbar \\qb)}\n{\\hbar^2 \\qb^2} \\Pi(\\hbar^2 \\qb^2)\\,.\n\\end{equation}\nThe counterterm is adjusted to impose the renormalisation condition that $\\Pi(0) = 0$, \nrequired in order to match the identification of the gauge coupling with its classical counterpart. As a power series in the dimensionless ratio $q^2 \/ m^2 = \\hbar^2 \\qb^2 \/ m^2$, which is of order $\\lcomp^2 \/ b^2$,\n\\begin{equation}\n\\Pi(q^2) = \\hbar^2 \\Pi'(0) \\frac{\\qb^2}{m^2} \n+ \\mathcal{O}\\biggl(\\frac{\\lcomp^4}{b^4} \\biggr)\\,.\n\\end{equation}\nThe renormalisation condition is essential in eliminating possible contributions of $\\Ord(\\hbar^0)$. One way to see that $\\AmplB_\\Pi$ is a purely quantum correction is to follow the powers of $\\hbar$. As $\\Pi(q^2)$ is of order $\\hbar^2$, $\\AmplB_\\Pi$ is of order $\\hbar^0$. This gives a contribution of $\\Ord(\\hbar)$ to the impulse kernel~(\\ref{eqn:impKerClDef}), which in turn gives a contribution of $\\Ord(\\hbar)$ to the impulse, as can be seen in \\eqn~\\eqref{eqn:classicalLimitNLO}.\n\nAlternatively, one can consider the contribution of these graphs to $\\Delta p \/ p$. Counting each factor of $\\qb$ as of order $b$, and using $\\Pi(q^2) \\sim \\lcomp^2 \/ b^2$, it is easy to see that these self-energy graphs yield a contribution to $\\Delta p \/ p$ of order $\\alpha^2 \\hbar^3 \/ (mb)^3 \\sim (\\lclass^2 \/ b^2) \\,( \\lcomp \/ b)$.\n\nThe renormalisation of the vertex is similarly a purely quantum effect. Since the classes $\\Gamma_{1b}$ and $\\Gamma_{3b}$ consisted of vertex corrections, wavefunction renormalisation, and their counterterms, they too give no contribution in the classical limit.\n\nThese conclusions continue to hold in the non--Abelian theory, with charges promoted to colour factors $C_\\alpha$. The different colour structures present in each class of diagram introduces a further splitting of topologies, but one that does not disrupt our identification of quantum effects.\n\n\\subsubsection{Classical colour basis}\n\\label{sec:colourDecomp}\n\\newcommand{\\boxy}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.25 of v1] (v2);\n\t\\vertex [above = 0.25 of v1] (v3);\n\t\\vertex [right = 0.25 of v3] (v4);\n\t\\vertex [above left = 0.15 and 0.15 of v3] (o1);\n\t\\vertex [below left = 0.15 and 0.15 of v1] (i1);\n\t\\vertex [above right = 0.15 and 0.15 of v4] (o2);\n\t\\vertex [below right = 0.15 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (v3);\n\t\\draw (v3) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v4);\n\t\\draw (v4) -- (o2);\n\t\\draw (v1) -- (v2);\n\t\\draw (v3) -- (v4);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\crossbox}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.3 of v1] (v2);\n\t\\vertex [above = 0.3 of v1] (v3);\n\t\\vertex [right = 0.3 of v3] (v4);\n\t\\vertex [above left = 0.125 and 0.125 of v3] (o1);\n\t\\vertex [below left = 0.125 and 0.125 of v1] (i1);\n\t\\vertex [above right = 0.125 and 0.125 of v4] (o2);\n\t\\vertex [below right = 0.125 and 0.125 of v2] (i2);\n\t\\vertex [above right = 0.1 and 0.1 of v1] (g1);\n\t\\vertex [below left = 0.1 and 0.1 of v4] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (v2);\n\t\\draw (v3) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v3) -- (v4);\n\t\\draw (v4) -- (o2);\n\t\\draw (v4) -- (g2);\n\t\\draw (g1) -- (v1);\n\t\\draw (v2) -- (v3);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\newcommand{\\triR}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left = 0.25 and 0.17of v1] (v2);\n\t\\vertex [above right = 0.25 and 0.17 of v1] (v3);\n\t\\vertex [below right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [below left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [above right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [above left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v2) -- (v1);\n\t\\draw (v3) -- (v1);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\newcommand{\\triL}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below left = 0.25 and 0.17of v1] (v2);\n\t\\vertex [below right = 0.25 and 0.17 of v1] (v3);\n\t\\vertex [above right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [above left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [below right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [below left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v2) -- (v1);\n\t\\draw (v3) -- (v1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\nonAbL}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.15 of v1] (g1);\n\t\\vertex [below left = 0.2 and 0.175 of g1] (v2);\n\t\\vertex [below right = 0.2 and 0.175 of g1] (v3);\n\t\\vertex [above right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [above left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [below right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [below left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v1) -- (g1);\n\t\\draw (v2) -- (g1);\n\t\\draw (v3) -- (g1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\nonAbR}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above = 0.15 of v1] (g1);\n\t\\vertex [above left = 0.2 and 0.175 of g1] (v2);\n\t\\vertex [above right = 0.2 and 0.175 of g1] (v3);\n\t\\vertex [below right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [below left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [above right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [above left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v1) -- (g1);\n\t\\draw (v2) -- (g1);\n\t\\draw (v3) -- (g1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\nThis leaves us with contributions of class $\\Gamma_2$; these appear in both terms in the impulse kernel. These contributions to the 1-loop amplitude in the first term take the form\n\\begin{equation}\n\\begin{aligned}\ni \\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') &= \\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}] \n\\begin{feynman}\n\\vertex (b) ;\n\\vertex [above left=1 and 0.66 of b] (i1) {$p_1$};\n\\vertex [above right=1 and 0.33 of b] (o1) {$p_1+q$};\n\\vertex [below left=1 and 0.66 of b] (i2) {$p_2$};\n\\vertex [below right=1 and 0.33 of b] (o2) {$p_2-q$};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- (o2);\n\\draw[postaction={decorate}] (b) -- (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- (b);\n\\draw[postaction={decorate}] (i2) -- (b);\n\\end{scope}\t\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\draw 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(v4);\n\t\\draw [postaction={decorate}] (v4) -- (o2);\n\t\\diagram*{(v3) -- [gluon] (v1)};\n\t\\diagram*{(v2) -- [gluon] (v4)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.9 of v1] (v2);\n\t\\vertex [above = 0.97 of v1] (v3);\n\t\\vertex [right = 0.9 of v3] (v4);\n\t\\vertex [above left = 0.5 and 0.5 of v3] (i2);\n\t\\vertex [below left = 0.5 and 0.5 of v1] (i1);\n\t\\vertex [above right = 0.5 and 0.5 of v4] (o2);\n\t\\vertex [below right = 0.5 and 0.5 of v2] (o1);\n\t\\vertex [above right = 0.45 and 0.485 of v1] (g1);\n\t\\vertex [below left = 0.4 and 0.4 of v4] (g2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (v4);\n\t\\draw [postaction={decorate}] (v4) -- (o2);\n\t\\diagram*{(v4) -- [gluon] (v1)};\n\n\t\\filldraw [color=white] (g1) circle [radius=6.4pt];\n\t\\diagram*{(v2) -- [gluon] (v3)};\n\t\\end{feynman}\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left = 0.94 and 0.45 of v1] (v2);\n\t\\vertex [above right = 0.94 and 0.45 of v1] (v3);\n\t\\vertex [below right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [below left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [above right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [above left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(v1) -- [gluon] (v2)};\n\t\\diagram*{(v1) -- [gluon] (v3)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below left = 0.94 and 0.45 of v1] (v2);\n\t\\vertex [below right = 0.94 and 0.45 of v1] (v3);\n\t\\vertex [above right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [above left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [below right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [below left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(v2) -- [gluon] (v1)};\n\t\\diagram*{(v3) -- [gluon] (v1)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} } \\\\ &\\hspace{-10mm}+\\scalebox{1.1}{\\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above = 0.55 of v1] (g1);\n\t\\vertex [above left = 0.45 and 0.45 of g1] (v2);\n\t\\vertex [above right = 0.45 and 0.45 of g1] (v3);\n\t\\vertex [below right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [below left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [above right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [above left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(g1) -- [gluon] (v1)};\n\t\\diagram*{(v2) -- [gluon] (g1)};\n\t\\diagram*{(g1) -- [gluon] (v3)};\n\n\t\\end{feynman}\t\n\t\\end{tikzpicture} +\t\\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.55 of v1] (g1);\n\t\\vertex [below left = 0.45 and 0.45 of g1] (v2);\n\t\\vertex [below right = 0.45 and 0.45 of g1] (v3);\n\t\\vertex [above right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [above left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [below right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [below left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(g1) -- [gluon] (v1)};\n\t\\diagram*{(g1) -- [gluon] (v2)};\n\t\\diagram*{(v3) -- [gluon] (g1)};\n\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\t\\begin{feynman}\n\t\\vertex (v1) ;\n\t\\vertex [above left= 0.6 and 1 of v1] (i1);\n\t\\vertex [above right= 0.6 and 1 of v1] (o1);\n\t\\vertex [below = 0.7 of v1] (v2);\n\t\\vertex [below left= 0.6 and 1 of v2] (i2);\n\t\\vertex [below right= 0.6 and 1 of v2] (o2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (o2);\n\t\\diagram*{(v1) -- [gluon, half left] (v2)};\n\t\\diagram*{(v2) -- [gluon, half left] (v1)};\n\t\\end{feynman}\n\t\\end{tikzpicture}}.\n\\end{aligned}\n\\end{equation}\nIn each contribution, we count powers of $\\hbar$ following the rules in section~\\ref{subsec:Wavefunctions}, replacing $\\ell\\rightarrow\\hbar\\ellb$ and $q\\rightarrow\\hbar \\qb$.  In the final double-seagull contribution, we will get four powers from the loop measure, and four inverse powers from the two photon propagators.  Overall, we will not get enough inverse powers to compensate the power in front of the integral in \\eqn~\\eqref{eqn:impKerClDef}, and thus the seagull will die in the classical limit. \n\nWe will refer to the remaining topologies as the box $B$, cross box $C$, triangles $T_{\\alpha\\beta}$, and non-Abelian diagrams $Y_{\\alpha\\beta}$, respectively. Applying the colour decomposition of equation~\\eqref{eqn:colourStripping}, the 1-loop amplitude contributing classically to the linear part of the impulse is\n\\begin{multline}\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') = \\mathcal{C}\\!\\left(\\boxy \\right) B + \\mathcal{C}\\!\\left(\\crossbox \\right) C + \\mathcal{C}\\!\\left(\\triR \\right) T_{12} \\\\ + \\mathcal{C}\\!\\left(\\triL \\right) T_{21} + \\mathcal{C}\\!\\left(\\nonAbR \\right) Y_{12} + \\mathcal{C}\\!\\left(\\nonAbL \\right) Y_{21}\\,.\n\\end{multline}\nA first task is to choose a basis of independent colour structures. The complete set of colour factors can easily be calculated:\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{C}\\!\\left(\\boxy \\right) = \\newT_1^a \\newT_2^a \\newT_1^b \\newT_2^b\\,, \\qquad\n\\mathcal{C}\\!\\left(\\crossbox \\right) = \\newT_1^a \\newT_2^b \\newT_1^b \\newT^a_2\\,,\\\\\n\\mathcal{C}\\!\\left(\\nonAbL \\right) = \\hbar\\, \\newT_1^a f^{abc} \\newT_2^b \\newT_2^c\\,, \\qquad \n\\mathcal{C}\\!\\left(\\nonAbR \\right) = \\hbar\\, \\newT_1^a \\newT_1^b f^{abc} \\newT_2^c\\,,\\\\\n\\mathcal{C}\\!\\left(\\triL \\right) = \\frac12\\, \\mathcal{C}\\!\\left(\\boxy \\right) + \\frac12\\, \\mathcal{C}\\!\\left(\\crossbox \\right) = \\mathcal{C}\\!\\left(\\triR \\right).\n\\end{gathered}\n\\end{equation}\nAt first sight, we appear to have a basis of four independent colour factors: the box, cross box and the two non-Abelian triangles. However, it is very simple to see that the latter are in fact both proportional to the tree colour factor of \\eqn~\\eqref{eqn:treeamp}; for example, \n\\[\n\\mathcal{C}\\!\\left(\\nonAbL \\right) = \\frac{\\hbar}{2}\\, \\newT_1^a f^{abc} [\\newT_2^b, \\newT_2^c] &= \\frac{i\\hbar^2}{2} f^{abc} f^{bcd} \\newT_1^a \\newT_2^d\\\\\n& = \\frac{i\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right),\n\\]\nwhere we have used \\eqn~\\eqref{eqn:chargeLieAlgebra}. Moreover, similar manipulations demonstrate that the cross-box colour factor is not in fact linearly independent:\n\\[\n\\mathcal{C}\\!\\left(\\crossbox \\right) &= \\newT_1^a \\newT_1^b \\left( \\newT_2^a \\newT_2^b - i\\hbar f^{abc} \\newT_2^c\\right)\\\\\n&= (\\newT_1 \\cdot \\newT_2) (\\newT_1 \\cdot \\newT_2 ) - \\frac{i\\hbar}{2} [\\newT_1^a, \\newT_2^b] f^{abc} \\newT_2^c\\\\\n& = \\mathcal{C}\\!\\left(\\boxy \\right) + \\frac{\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right).\n\\]\nThus at 1-loop the classically significant part of the amplitude has a basis of two colour structures: the box and tree. Hence the decomposition of the 1-loop amplitude into partial amplitudes and colour structures is\n\\begin{multline}\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') =  \\mathcal{C}\\!\\left(\\boxy \\right) \\bigg[B + C + T_{12} + T_{21}\\bigg] \\\\ + \\frac{\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right) \\bigg[C + \\frac{ T_{12}}{2} + \\frac{T_{21}}{2} + iY_{12} + iY_{21}\\bigg]\\,.\\label{eqn:1loopDecomposition}\n\\end{multline}\nThis expression for the amplitude is particularly useful when taking the classical limit. The second term is proportional to two powers of $\\hbar$, while the only possible singularity in $\\hbar$ at one loop order is a factor $1\/\\hbar$ in the evaluation of the kinematic parts of the diagrams. Thus, it is clear that the second line of the expression must be a quantum \ncorrection, and can be dropped in calculating the classical impulse. Perhaps surprisingly, these terms include the sole contribution from the non-Abelian triangles $Y_{\\alpha\\beta}$, and thus we will not need to calculate these diagrams. We learn that classically, the 1-loop scalar YM amplitude has a basis of only one colour factor:\n\\[\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') &=  \\mathcal{C}\\!\\left(\\boxy \\right) \\bigg[B + C + T_{12} + T_{21}\\bigg] + \\mathcal{O}(\\hbar)\\,.\\label{eqn:OneLoopImpulse}\n\\]\nMoreover, the impulse depends on precisely the same topologies as in QED \\cite{Kosower:2018adc}.\n\n\\subsubsection{Triangles}\n\\label{sec:Triangles}\n\nLet us first examine the two (colour stripped) triangle diagrams in \\eqn~\\eqref{eqn:OneLoopImpulse}.  They are related by swapping particles~1 and~2. The first diagram is\n\\begin{equation}\ni T_{12} = \n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (i2) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (o1) {$p_2$};\n\\vertex [below=2.5 of i2] (o2) {$p_2 - q$};\n\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of i2] (v2);\n\\vertex [above right=1.1 and 1.25 of o1] (v3);\n\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (i2);\n\\draw [postaction={decorate}] (o1) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (o2);\n\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\ell\\)] (v1);\n\t(v2) -- [gluon] (v3);\n};\n\\end{feynman}\n\\end{tikzpicture}\n= -2 \\!\\int \\!\\dd^D \\ell\\, \\frac{(2p_1 + \\ell) \\cdot (2 p_1 + q + \\ell)}\n{\\ell^2 (\\ell - q)^2 (2p_1 \\cdot \\ell + \\ell^2 + i \\epsilon)}\\,.\n\\end{equation}\nIn this integral, we use a dimensional regulator in a standard way ($D=4-2\\varepsilon$)\nin order to regulate potential divergences. We have retained an explicit $i \\epsilon$ in the massive scalar propagator, because it will play an important role below.\n\nTo extract the classical contribution of this integral to the amplitude, we recall from section~\\ref{subsec:Wavefunctions} that we should set $q =  \\hbar\\qb$ and $\\ell =  \\hbar\\ellb$, and therefore that the components of $q$ and $\\ell$ are all small compared \nto $m$. Consequently, the triangle simplifies to\n\\begin{equation}\nT_{12} = \\frac{4 i m_1^2}{\\hbar} \\!\\int\\! \\dd^4 \\bar \\ell \\, \\frac{1}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2 (p_1 \\cdot \\bar \\ell + i \\epsilon)}\\,.\n\\label{eqn:triangleIntermediate1}\n\\end{equation}\nHere, we have taken the limit $D\\rightarrow 4$, as the integral is now free of divergences.\nNotice that we have exposed one additional inverse power of $\\hbar$. Comparing to the definition of $\\impKerCl$ in \\eqn~\\eqref{eqn:impKerClDef}, we see that this inverse power of $\\hbar$ will cancel against the explicit factor of $\\hbar$ in $\\ImpAclsup{(1)}$, signalling a classical contribution to the impulse.\n\nAt this point we employ a simple trick which simplifies the loop integral appearing in \\eqn~\\eqref{eqn:triangleIntermediate1}, and which will be of great help in simplifying the more complicated box topologies below. The on-shell condition for the outgoing particle 1 requires that $p_1 \\cdot \\qb = - \\hbar \\qb^2\/2$, so replace $\\ellb \\rightarrow \\ellb' = \\qb - \\ellb$ in $T_{12}$:\n\\begin{equation}\n\\begin{aligned}\nT_{12} &= -\\frac{4 i m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell'\\, \\frac{1}{\\bar \\ell'^2 (\\bar \\ell' - \\bar q)^2 (p_1 \\cdot \\bar \\ell' + \\hbar \\qb^2 - i \\epsilon)} \\\\\n&= -\\frac{4 i m_1^2}{\\hbar} \\!\\int \\!\\dd^4 \\bar \\ell' \\,\\frac{1}{\\bar \\ell'^2 (\\bar \\ell' - \\bar q)^2 (p_1 \\cdot \\bar \\ell' - i \\epsilon)} + \\mathcal{O}(\\hbar^0)\\,,\n\\end{aligned}\n\\end{equation}\nBecause of the linear power of $\\hbar$ appearing in \\eqn~\\eqref{eqn:impKerClDef}, the second term\nis in fact a quantum correction. We therefore neglect it, and write\n\\begin{equation}\nT_{12}= -\\frac{4 i m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell\\, \\frac{1}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2 (p_1 \\cdot \\bar \\ell - i \\epsilon)} \\, ,\n\\end{equation}\nwhere we have dropped the prime on the loop momentum: $\\ell' \\rightarrow \\ell$. Comparing with our previous expression, \\eqn~\\eqref{eqn:triangleIntermediate1}, for the triangle, the net result of these replacements has simply been to introduce an overall sign while, crucially, also switching the sign of the $i \\epsilon$ term. Symmetrising over the two expressions for $T_{12}$, we learn that\n\\begin{equation}\nT_{12}= \\frac{2 m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell\\, \\frac{\\del(p_1 \\cdot \\bar \\ell)}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2} \\,,\n\\end{equation}\nusing the identity\n\\begin{equation}\n\\frac{1}{x-i \\epsilon} - \\frac{1}{x+i \\epsilon} = i \\del(x)\\,.\n\\label{eqn:deltaPoles}\n\\end{equation}\n\nThe second triangle contributing to the amplitude, $T_{21}$, can be obtained from $T_{12}$ simply by interchanging the labels 1 and 2:\n\\begin{equation}\nT_{21} = \\frac{2 m_2^2}{\\hbar}\\! \\int \\! \\dd^4 \\bar \\ell \\,\n\\frac{\\del(p_2 \\cdot \\bar \\ell)}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2}\\,.\n\\end{equation}\nThese triangles contribute to the impulse kernel via\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl \\big|_\\mathrm{triangle} &= \\hbar\\qb^\\mu\\, \\mathcal{C}\\!\\left(\\boxy \\right) (T_{12} + T_{21}) \n\\\\&= 2\\left(\\newT_1\\cdot \\newT_2 \\right)^2 \\bar q^\\mu\\! \\int \\! \\frac{\\dd^4 \\bar \\ell}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2} \\left(m_1^2\\del(p_1 \\cdot \\bar \\ell) + m_2^2\\del(p_2 \\cdot \\bar \\ell) \\right).\n\\end{aligned}\n\\end{equation}\nRecall that we must integrate over the wavefunctions in order to obtain the classical impulse from the impulse kernel. As we have discussed in section~\\ref{subsec:Wavefunctions}, because the inverse power of $\\hbar$ here is cancelled by the linear power present explicitly in \\eqn~\\eqref{eqn:classicalLimitNLO}, we may evaluate the wavefunction integrals by replacing the $p_\\alpha$ by their classical values $m_\\alpha \\ucl_\\alpha$. The result for the contribution to the kernel is\n\\begin{equation}\n\\impKerTerm1 \\equiv\n{2 (c_1\\cdot c_2)^2} \\qb^\\mu \\!\\int \\! \\dd^4 \\ellb\\;\n\\frac{1}{\\ellb^2 (\\ellb - \\qb)^2} \n\\biggl(m_1{\\del(\\ucl_1 \\cdot \\ellb)} \n+ m_2{\\del(\\ucl_2 \\cdot \\ellb)} \\biggr)\\,.\n\\label{eqn:TriangleContribution}\n\\end{equation}\nOne must still integrate this expression over $\\qb$, as in \\eqn~\\eqref{eqn:classicalLimitNLO}, to\nobtain the contribution to the impulse.\n\n\\subsubsection{Boxes}\n\\label{sec:Boxes}\n\nThe one-loop amplitude also includes boxes and crossed boxes, and the NLO contribution to the impulse includes as well a term quadratic in the tree amplitude which we can think of as the cut of a one-loop box. Because of the power of $\\hbar$ in front of the first term in \\eqn~\\eqref{eqn:impKerClDef}, we need to extract the contributions of all of these quantities at order $1\/\\hbar$. However, as we will see, each individual diagram also contains singular terms of order $1\/\\hbar^2$.  We might fear that these terms pose an obstruction to the \nvery existence of a classical limit of the observable in which we are interested. As we will see, this fear is misplaced, as these singular terms cancel completely, leaving a well-defined classical result. It is straightforward to evaluate the individual contributions, but making the cancellation explicit requires some care. We begin with the colour-stripped box:\n\\begin{equation}\n\\begin{aligned}\n\\hspace*{-7mm}i B &= \\hspace*{-2mm}\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - q$};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\ell\\)] (v1);\n\t(v2) -- [gluon] (v4);\n};\n\\end{feynman}\n\\end{tikzpicture} \n\\hspace*{-9mm}\n= \\int \\! \\dd^D \\ell \\;\n\\frac{(2 p_1 \\tp \\ell) \\td (2p_2 \\tm \\ell)\\,(2 p_1 \\tp q\\tp \\ell) \\td (2 p_2\\tm q\\tm \\ell)}\n{\\ell^2 (\\ell \\tm q)^2 (2 p_1 \\cdot \\ell \\tp \\ell^2 \\tp i \\epsilon)\n\t(-2p_2 \\cdot \\ell \\tp \\ell^2 \\tp i \\epsilon)}\n=\\hspace*{-8mm}\n\\\\[-2mm]\n\\\\&\\hspace*{-5mm} \\frac{1}{\\hbar^{2+2\\varepsilon}} \\!\\!\\int \\! \\dd^D \\ellb\\;\n\\frac{\\bigl[4 p_1\\td p_2\\tm 2\\hbar(p_1\\tm p_2)\\td\\ellb\\tm \\hbar^2\\ellb^2\\bigr]\n\t\\bigl[4 p_1\\td p_2\\tm 2\\hbar(p_1\\tm p_2)\\td(\\ellb\\tp \\qb)\\tm \\hbar^2(\\ellb\\tp \\qb)^2\\bigr]}\n{\\ellb^2 (\\ellb - \\qb)^2 (2 p_1 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)\n\t(-2p_2 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)}\\,,\\hspace*{-8mm}\n\\end{aligned}\n\\end{equation}\nwhere as usual, we have set $q = \\hbar \\qb$, $\\ell = \\hbar \\ellb$. We get four powers of $\\hbar$ from changing variables in the measure, but six inverse powers from the propagators\\footnote{We omit fractional powers of $\\hbar$ in this counting as they will disappear when we take $D \\rightarrow 4$.}. We thus encounter an apparently singular $1\/\\hbar^2$ leading behaviour.  We must extract both this singular, $\\Ord(1\/\\hbar^2)$, term \nas well as the terms contributing in the classical limit, which here are $\\Ord(1\/\\hbar)$.\nConsequently, we must take care to remember that the on-shell delta functions enforce $\\qb \\cdot p_1 = - \\hbar \\qb^2 \/ 2$ and $\\qb \\cdot p_2 = \\hbar \\qb^2 \/ 2$. \n\nPerforming a Laurent expansion in $\\hbar$, truncating after order $1\/\\hbar$, and separating different orders in $\\hbar$, we find that the box's leading terms are given by\n\\begin{equation}\n\\begin{aligned}\nB &= B_{-1}+B_0\\,,\n\\\\ B_{-1} &= \\frac{4 i}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\int \\frac{\\dd^D \\ellb}{\\ellb^2 (\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb - i \\epsilon)} \\,,\n\\\\ B_{0} &= -\\frac{2i }{\\hbar^{1+2\\varepsilon}} p_1 \\cdot p_2 \n\\int \\frac{\\dd^D \\ellb}{\\ellb^2 (\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb - i \\epsilon)}\n\\\\& \\hspace*{30mm}\\times\n\\biggl[ 2{(p_1 - p_2)\\cdot \\ellb}\n+ \\frac{ (p_1 \\cdot p_2) \\ellb^2}{(p_1 \\cdot \\ellb + i \\epsilon)} \n- \\frac{(p_1 \\cdot p_2) \\ellb^2}{(p_2 \\cdot \\ellb - i \\epsilon)}\\biggl]\\,.\n\\end{aligned}\n\\label{eqn:BoxExpansion}\n\\end{equation}\nNote that pulling out a sign from one of the denominators has given the appearance of\nflipping the sign of one of the denominator $i\\epsilon$ terms.  We must also bear in mind that the integral in $B_{-1}$ is itself \\textit{not\\\/} $\\hbar$-independent, so that we will later need to expand it as well.\n\nSimilarly, the crossed box is\n\\begin{align}\ni C &= \n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - q$};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- node [above] {$\\scriptstyle{p_1 + \\ell}$} (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\diagram*{(v3) -- [gluon] (v2);};\n\\filldraw [color=white] ($ (v3) !.5! (v2) $) circle [radius = 4.3pt];\n\\diagram*{\t(v1) -- [gluon] (v4);};\n\\end{feynman}\n\\end{tikzpicture} \n\\\\\n&=  \\int \\! \\dd^D \\ell\\, \\frac{(2 p_1 + \\ell) \\cdot (2p_2 - 2q + \\ell)(2 p_1 +q+ \\ell) \\cdot (2 p_2 -q + \\ell)}{\\ell^2 (\\ell - q)^2 (2 p_1 \\cdot \\ell + \\ell^2 + i \\epsilon)(2p_2 \\cdot (\\ell-q) + (\\ell-q)^2 + i \\epsilon)}\\nonumber\n\\\\\n&= \\frac{1}{\\hbar^{2+2\\varepsilon}} \\!\\! \\int \\! \\dd^D \\ellb\\, \n\\frac{(2 p_1 + \\hbar\\ellb) \\cdot (2p_2 - 2\\hbar\\qb + \\hbar\\ellb)\\,\n\t(2 p_1 +\\hbar\\qb+ \\hbar\\ellb) \\cdot (2 p_2 -\\hbar\\qb + \\hbar\\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2 (2 p_1 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)\n\t(2p_2 \\cdot (\\ellb-\\qb) + \\hbar(\\ellb-\\qb)^2 + i \\epsilon)}\\,.\\nonumber\n\\end{align}\nUsing the on-shell conditions to simplify $p_\\alpha\\cdot \\qb$ terms in the denominator\nand numerator, and once again expanding in powers of $\\hbar$, truncating after order $1\/\\hbar$, and separating different orders in $\\hbar$,  we find\n\\begin{equation}\n\\begin{aligned}\nC &= C_{-1}+C_0\\,,\n\\\\ C_{-1} &= -\\frac{4i}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int \\!\\frac{\\dd^D \\ellb}{\\ellb^2(\\ellb - \\qb)^2} \n\\frac{1}\n{(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb + i \\epsilon)} \n\\\\ C_{0} &= -\\frac{2i}{\\hbar^{1+2\\varepsilon}} p_1 \\cdot p_2\n\\!\\int \\!\\frac{\\dd^D \\ellb}{\\ellb^2(\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb + i \\epsilon)} \n\\\\& \\qquad \\times\n\\biggl[2  (p_1 + p_2) \\cdot \\ellb\n- \\frac{(p_1 \\cdot p_2) \\ellb^2}{(p_1 \\cdot \\bar \\ell + i \\epsilon)} \n- \\frac{(p_1 \\cdot p_2) [(\\ellb - \\qb)^2 - \\qb^2]}\n{(p_2 \\cdot \\ellb + i \\epsilon)}\\biggr]\\,.\\hspace*{-20mm}\n\\end{aligned}\n\\label{eqn:CrossedBoxExpansion}\n\\end{equation}\nComparing the expressions for the $\\Ord(1\/\\hbar^2)$ terms in the box and the crossed box, \n$B_{-1}$ and $C_{-1}$ respectively, we see that there is only a partial cancellation of \nthe singular, $\\mathcal{O}(1\/\\hbar^2)$, term in the reduced amplitude $\\AmplB^{(1)}$. The impulse kernel, \\eqn~\\eqref{eqn:impKerClDef}, does contain another term, which is quadratic in the tree-level reduced amplitude  $\\AmplB^{(0)}$.  We will see below that taking this additional contribution into account leads to a complete cancellation of the singular term; \nbut the classical limit does not exist for each of these terms separately.\n\n\\subsubsection{Cut Box}\n\\label{sec:CutBoxes}\n\nIn order to see the cancellation of the singular term we must incorporate the term in the impulse kernel which is quadratic in tree amplitudes. As with the previous loop diagrams, let us begin by splitting the colour and kinematic information as in equation~\\eqref{eqn:colourStripping}. Then the quadratic term in~\\eqref{eqn:impKerClDef} can be written as\n\\begin{equation}\n\\impKerCl \\big|_\\textrm{non-lin} = \\mathcal{C}\\!\\left({\\scalebox{1}{\\tree}} \\right)^\\dagger \\mathcal{C}\\!\\left({\\scalebox{1}{\\tree}}\\right) \\cutbox^\\mu\\,,\\label{eqn:cutBoxColDecomp}\n\\end{equation}\nwhere the kinematic data $\\cutbox^\\mu$ can be viewed as proportional to the cut of the one-loop box, weighted by the loop momentum $\\hbar \\xferb^\\mu$:\n\\begin{equation}\n\\cutbox^\\mu = -i\\hbar^2\\int \\! \\dd^4 \\xferb \\, \\xferb^\\mu \\, \\del(2 p_1 \\cdot \\xferb + \\hbar \\xferb^2) \\del(2p_2 \\cdot \\xferb - \\hbar \\xferb^2) \\times\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + \\hbar \\qb$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - \\hbar\\qb$};\n\\node [] (cutTop) at ($ (i1)!.5!(o1) $) {};\n\\node [] (cutBottom) at ($ (i2)!.5!(o2) $) {};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\filldraw [color=white] ($  (cutTop) - (3pt, 0) $) rectangle ($ (cutBottom) + (3pt,0) $) ;\n\\draw [dashed] (cutTop) -- (cutBottom);\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\hbar\\xferb\\)] (v1);\n\t(v2) -- [gluon] (v4);\n};\n\\end{feynman}\n\\end{tikzpicture}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!;.\n\\end{equation}\nNote that an additional factor of $\\hbar$ in the second term of \\eqn~\\eqref{eqn:impKerClDef} will be multiplied into \\eqn~\\eqref{CombiningBoxes} below, as it parallels the factor in the first term of \\eqn~\\eqref{eqn:impKerClDef}. Evaluating the Feynman diagrams, we obtain\n\\begin{multline}\n\\cutbox^\\mu = -i\\frac{1}{\\hbar^2}\\! \\int \\! \\dd^4 \\xferb \\, \n\\del(2 p_1 \\cdot \\xferb + \\hbar \\xferb^2) \\del(2p_2 \\cdot \\xferb - \\hbar \\xferb^2) \\,\n\\frac{\\xferb^\\mu}{\\xferb^2 (\\xferb - \\qb)^2} \\\\\n\\quad \\times (2 p_1 + \\hbar\\xferb ) \\cdot (2p_2 - \\xferb \\hbar)\\,\n(2 p_1 + \\hbar\\qb + \\hbar\\xferb ) \\cdot (2 p_2 - \\hbar\\qb  -  \\hbar\\xferb)\\, .\\label{eqn:cutBoxFull}\n\\end{multline}\nAs in the previous subsection, expand in $\\hbar$, and truncate after order $1\/\\hbar$,\nso that\n\\begin{align}\n\\cutbox^\\mu &= \\cutbox_{-1}^\\mu + \\cutbox_{0}^\\mu\\,,\\nonumber\n\\\\ \\cutbox_{-1}^\\mu &=  -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\frac{\\dd^4 \\ellb \\; \\ellb^\\mu}{\\ellb^2 (\\ellb - \\qb)^2}\n\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb) \\,, \\label{eqn:CutBoxExpansion}\n\\\\ \\cutbox_{0}^\\mu &=  -\\frac{2i}{\\hbar} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\frac{\\dd^4 \\ellb \\; \\ellb^\\mu}{\\ellb^2 (\\ellb - \\qb)^2}\\,\n{\\ellb^2} \\Big(\\del'(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)\n- \\del(p_1 \\cdot \\ellb) \\del'(p_2 \\cdot \\ellb) \\Big)\\,.\\nonumber\n\\end{align}\nWe have relabelled $\\xferb\\rightarrow\\ellb$ in order to line up terms more transparently with corresponding ones in the box and crossed box contributions.\n\nFinally, it is easy to see that the cut box colour factor in~\\eqref{eqn:cutBoxColDecomp} is simply\n\\begin{equation}\n\\mathcal{C}\\!\\left(\\tree \\right)^\\dagger \\mathcal{C}\\!\\left(\\tree \\right) = (\\newT_2\\cdot \\newT_1) (\\newT_1 \\cdot \\newT_2) = \\mathcal{C}\\!\\left(\\boxy \\right)\\,.\n\\end{equation}\nThus there is only one relevant colour structure in the NLO momentum impulse, that of the box. This will be important in the following.\n\n\\subsubsection{Combining Contributions}\n\\label{sec:CombiningTerms}\n\nWe are now in a position to assemble the elements computed in the three previous subsections in order to obtain the NLO contributions to the impulse kernel $\\impKerCl$, and thence the NLO contributions to the impulse using \\eqn~\\eqref{eqn:classicalLimitNLO}. Let us begin by examining the singular terms. We must combine the terms from the box, crossed box, and cut box. We can simplify the cut-box contribution $\\cutbox_{-1}^\\mu$ by exploiting the linear change of variable $\\ellb' = \\qb - \\ellb$:\n\\begin{align}\n\\cutbox_{-1}^\\mu \n&= -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \\nonumber\n\\!\\int\\! \\dd^4 \\ellb' \\;\\frac{ (\\qb^\\mu - \\ellb'^\\mu)}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\del(p_1 \\cdot \\ellb'-p_1\\cdot\\qb\n\\del(p_2 \\cdot \\ellb'-p_2\\cdot \\qb)\n\\\\&= -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\dd^4 \\ellb' \\;\\frac{ (\\qb^\\mu - \\ellb'^\\mu)}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\del(p_1 \\cdot \\ellb'+\\hbar\\qb^2\/2)\n\\del(p_2 \\cdot \\ellb'-\\hbar\\qb^2\/2)\\nonumber\n\\\\&= -\\frac{2i}{\\hbar^2} (p_1 \\cdot p_2)^2 \\qb^\\mu\n\\! \\int\\! \\dd^4 \\ellb \\;\\frac{\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2} +\\Ord(1\/\\hbar)\\,,\\label{eqn:CutSingular}\n\\end{align}\nwhere we have used the on-shell conditions to replace $p_1\\cdot\\qb\\rightarrow -\\hbar\\qb^2\/2$\nand $p_2\\cdot\\qb\\rightarrow \\hbar\\qb^2\/2$, and where the last line arises from averaging over the two equivalent expressions for $\\cutbox_{-1}^\\mu$.\n\n\nWe may similarly simplify the singular terms from the box and cross box. Indeed, using the identity~\\eqref{eqn:deltaPoles} followed by the linear change of variable, we have\n\\begin{equation}\n\\begin{aligned}\nB_{-1} + C_{-1} &= \n-\\frac{4 }{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int \\! \\frac{\\dd^D \\ellb}{\\ellb^2 (\\bar \\ell - \\bar q)^2} \n\\frac{1}{(p_1 \\cdot \\ellb + i \\epsilon)} \\del(p_2 \\cdot \\ellb)\n\\\\&= \\frac{4 Q_1^2 Q_2^2}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int\\! \\frac{\\dd^D \\ellb'}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\frac{\\del(p_2 \\cdot \\ellb'-\\hbar\\qb^2\/2) }{(p_1 \\cdot \\ellb'+\\hbar\\qb^2\/2 - i \\epsilon)} \n\\\\&= \\frac{2i}{\\hbar^2} (p_1 \\cdot p_2)^2\n\\!\\int\\! \\frac{\\dd^4 \\ellb\\;\\del(p_1 \\cdot \\ellb)\\del(p_2 \\cdot \\ellb) }\n{\\ellb^2 (\\ellb - \\qb)^2} + \\Ord(1\/\\hbar)\\,,\n\\end{aligned}\n\\label{eqn:BoxSingular}\n\\end{equation}\nwhere we have averaged over equivalent forms, and then used \\eqn~\\eqref{eqn:deltaPoles} a second time in obtaining the last line. At the very end, we took $D\\rightarrow 4$.\n\nCombining \\eqns{eqn:CutSingular}{eqn:BoxSingular}, we find that the potentially singular contributions to the impulse kernel in the classical limit are\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl &\\big|_\\textrm{singular} =\n\\hbar\\qb^\\mu\\, \\mathcal{C}\\!\\left(\\boxy \\right) (B_{-1}+C_{-1}) +\\hbar\\, \\mathcal{C}\\!\\left(\\boxy \\right) \\cutbox^\\mu_{-1} \n\\\\&\\hspace*{-6mm} = \\frac{2i}{\\hbar} (p_1 \\cdot p_2)^2\\qb^\\mu \\left(\\newT_1\\cdot \\newT_2 \\right)^2  \\Bigg[\n\\int \\frac{\\dd^4 \\ellb\\;\\del(p_1 \\cdot \\ellb)\\del(p_2 \\cdot \\ellb) }\n{\\ellb^2 (\\ellb - \\qb)^2}\n\\\\ &\\hspace{50mm} - \\!\\int \\dd^4 \\ellb \\;\\frac{\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2}\\Bigg]  +\\Ord(\\hbar^0)\n\\\\&\\hspace*{-6mm}=\\Ord(\\hbar^0)\\,.\n\\end{aligned}\n\\label{CombiningBoxes}\n\\end{equation}\nSince all terms have common colour factors the dangerous terms cancel, leaving only well-defined contributions.\n\n\\defZ{Z}\nIt remains to extract the $\\Ord(1\/\\hbar)$ terms from the box, crossed box, and cut box contributions, and to combine them with the triangles~(\\ref{eqn:TriangleContribution}), which are of this order.  In addition to $B_0$ from \\eqn~\\eqref{eqn:BoxExpansion}, $C_0$ from \\eqn~\\eqref{eqn:CrossedBoxExpansion}, and $\\cutbox^\\mu_0$ from \\eqn~\\eqref{eqn:CutBoxExpansion}, we must also include the $\\Ord(1\/\\hbar)$ terms left implicit in \\eqns{eqn:CutSingular}{eqn:BoxSingular}.  In the former contributions, we can now set $p_\\alpha\\cdot \\qb = 0$, as the $\\hbar$ terms in the on-shell delta functions would give rise to contributions of $\\Ord(\\hbar^0)$ to the impulse kernel, which in turn will give contributions of $\\Ord(\\hbar)$ to the impulse.  In combining all these terms, we make use of summing over an expression and the expression after the linear change of variables;\nthe identity~(\\ref{eqn:deltaPoles}); and the identity\n\\begin{equation}\n\\del'(x) = \\frac{i}{(x-i\\epsilon)^2} - \\frac{i}{(x+i\\epsilon)^2}\\,.\n\\end{equation}\nOne finds that\n\\begin{equation}\n\\begin{aligned}\n&\\hbar\\qb^\\mu (B_0 + C_0) +\\bigl[\\hbar\\qb^\\mu (B_{-1} + C_{-1})\\bigr]\\big|_{\\Ord(\\hbar^0)}\n= Z^\\mu\n\\\\& + 2  (p_1 \\cdot p_2)^2 \\qb^\\mu\n\\!\\int \\!\\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\biggl(\\del(p_2 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(p_1 \\td \\ellb \\tp i \\epsilon )^2} \n+ \\del(p_1 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(p_2 \\td \\ellb \\tm i \\epsilon )^2}\\biggr) \n\\,,\n\\\\ &\\hbar\\cutbox^\\mu_0 +\\bigl[\\hbar\\cutbox^\\mu_{-1}\\bigr]\\big|_{\\Ord(\\hbar^0)}\n= - Z^\\mu \n\\\\& -\\!2 i (p_1 \\cdot p_2)^2 \n\\! \\int\\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\ellb^\\mu \\, \\ellb \\td (\\ellb \\tm \\qb)\\,\n\\bigl( \\del^\\prime(p_1 \\td \\ellb) \\del(p_2 \\td \\ellb) - \\del^\\prime(p_2 \\td \\ellb)\n\\del(p_1 \\td \\ellb)\\bigr) \\,,\n\\end{aligned}\n\\end{equation}\nwhere we have now taken $D\\rightarrow4$, and where the quantity $Z^\\mu$ is\n\\begin{multline}\nZ^\\mu = i (p_1 \\cdot p_2)^2 \\qb^\\mu \n\\!\\int \\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2} \n\\;(2 \\ellb \\cdot \\qb - \\ellb^2 )\n\\bigl( \\del^\\prime(p_1 \\td \\ellb) \\del(p_2 \\td \\ellb) \\\\ - \\del^\\prime(p_2 \\td \\ellb)\n\\del(p_1 \\td \\ellb)\\bigr) \\, .\n\\end{multline}\n\nFinally, we integrate over the external wavefunctions. The possible singularity in $\\hbar$ has cancelled, so as discussed in section~\\ref{subsec:Wavefunctions}, we perform the integrals by replacing the momenta $p_\\alpha$ with their classical values $m_\\alpha \\ucl_\\alpha$, and replace the quantum colour factors with classical colour charges. The box-derived contribution is therefore\n\\begin{multline}\n\\impKerTerm2 \\equiv\n2 (c_1\\cdot c_2)^2 \\gamma^2 \\qb^\\mu\n\\! \\int \\!\\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\biggl(m_2\\del(\\ucl_2 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(\\ucl_1 \\td \\ellb \\tp i \\epsilon )^2} \n\\\\ + m_1\\del(\\ucl_1 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(\\ucl_2 \\td \\ellb \\tm i \\epsilon )^2}\\biggr) \\,,\n\\end{multline}\nwhile that from the cut box is\n\\begin{multline}\n\\impKerTerm3 \\equiv\n-2 i (c_1\\cdot c_2)^2 \\, \\gamma^2 \n\\! \\int\\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\ellb^\\mu \\, \\ellb \\td (\\ellb \\tm \\qb)\\\\\n\\times\\Big( m_2\\del^\\prime(\\ucl_1 \\td \\ellb)  \\del(\\ucl_2 \\td \\ellb) - m_1\\del^\\prime(\\ucl_2 \\td \\ellb) \\del(\\ucl_1 \\td \\ellb)\\Big)\\,.\n\\end{multline}\nIn both contributions we have dropped the $Z^\\mu$ term which cancels\nbetween the two. The full impulse kernel is given by the sum $\\impKerTerm1+\\impKerTerm2+\\impKerTerm3$, and the impulse by\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPnlo &= \\frac{i g^4}{4}\\hbar\\! \\int \\! \\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \ne^{-i \\qb\\cdot b} \n\\left( \\impKerTerm1 + \\impKerTerm2 + \\impKerTerm3 \\right)\n\\\\&= \\frac{ig^4}{2} (c_1\\cdot c_2)^2\\!\n\\int \\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb - \\qb)^2}\\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) e^{-i \\qb\\cdot b}\n\\\\&\\hphantom{=}\\times\\biggl[\n\\qb^\\mu \\biggl( \\frac{\\del(\\ucl_1 \\cdot \\ellb)}{m_2}\n+ \\frac{\\del(\\ucl_2 \\cdot \\ellb)}{m_1} \\biggr)\n\\\\&\\hphantom{=} \\hphantom{\\times\\biggl[}\n+\\gamma^2\\qb^\\mu \\biggl(\\frac{\\del(\\ucl_2 \\td \\ellb)}{m_1}\n\\frac{\\ellb \\td (\\ellb -\\qb) }{(\\ucl_1 \\td \\ellb + i \\epsilon )^2} \n+ \\frac{\\del(\\ucl_1 \\td \\ellb)}{m_2}\n\\frac{\\ellb \\td (\\ellb -\\qb) }{(\\ucl_2 \\td \\ellb - i \\epsilon )^2}\\biggr) \n\\\\&\\hphantom{=} \\hphantom{\\times\\biggl[} -\ni \\gamma^2\\ellb^\\mu \\, \\ellb \\td (\\ellb - \\qb)\\,\n\\biggl( \\frac{\\del^\\prime(\\ucl_1 \\td \\ellb) \\del(\\ucl_2 \\td \\ellb)}{m_1}\n- \\frac{\\del^\\prime(\\ucl_2 \\td \\ellb) \\del(\\ucl_1 \\td \\ellb)}{m_2}\\biggr)\\biggr]\\,.\n\\end{aligned}\n\\label{eqn:NLOImpulse}\n\\end{equation}\nIt was shown in \\cite{delaCruz:2020bbn} that this result is precisely reproduced by applying worldline perturbation theory to iteratively solve the Yang--Mills--Wong equations in equation~\\eqref{eqn:classicalWong}. Moreover, our the final result for the impulse in non-Abelian gauge theory is in fact identical to QED \\cite{Kosower:2018adc} (in which context this calculation was first performed), but with the charge to colour replacement $Q_1 Q_2 \\rightarrow c_1 \\cdot c_2$. This is a little peculiar, as it is natural to expect the non-linearity of the Yang--Mills field to enter at this order (and it does so in the quantum theory). The origin of the result is the colour basis decomposition in equation~\\eqref{eqn:1loopDecomposition}, and in particular the fact that the non-Abelian triangle diagrams only contribute to the $\\hbar^2$ suppressed second colour structure.\n\nWith the relevant Yang--Mills amplitude at hand, one may of course wonder about the prospect of double copying to obtain the NLO impulse in gravity. The construction of colour-kinematics dual numerators at loop level following our methods is highly non-trivial; however, recent progress with massive particles may now make this problem tractable \\cite{Carrasco:2015iwa}. It is also interesting to compare our methods to those of Shen~\\cite{Shen:2018ebu}, who implemented the double copy at NLO wholly within the classical worldline formalism following ground-breaking work of Goldberger and Ridgway~\\cite{Goldberger:2016iau}. Shen found it necessary to include vanishing terms involving structure constants in his work. Similarly, in our context, some colour factors are paired with kinematic numerators proportional to $\\hbar$. It would be interesting to use the tools developed in these chapters to explore the double copy construction of Shen~\\cite{Shen:2018ebu} from the perspective of amplitudes.\n\nThe agreement of~\\eqref{eqn:NLOImpulse} with worldline perturbation theory offers a strong check on our formalism, and is of greater importance than the evaluation of the remaining integrals, which also arise in the classical theory. Their evaluation is surprisingly intricate; however, we can gain some interesting insights into the physics just from considering momentum conservation.\n\n\\subsubsection{On-Shell Cross Check}\n\nAs we have seen, careful inclusion of boxes, crossed boxes as well as cut boxes are necessary to determine the impulse in the classical regime. This may seem to be at odds with other work on the classical limit of amplitudes, which often emphasises the particular importance of triangle diagrams to the classical potential at next to leading order. However, in the context of the potential, the partial cancellation between boxes and crossed boxes is well-understood~\\cite{Donoghue:1996mt}, and it is because of this fact that triangle diagrams are particularly important. The residual phase is known to exponentiate so that it does not effect classical physics. Meanwhile, the relevance of the subtraction of iterated (cut) diagrams has long been a topic of discussion~\\cite{Sucher:1994qe,BjerrumBohr:2002ks,Neill:2013wsa}.\n\nNevertheless, in the case of the impulse it may seem that the various boxes play a more significant role, as they certainly contribute to the classical result for the impulse. In fact, it is easy to see that these terms must be included to recover a physically sensible result. The key observation is that the final momentum, $\\finalk_1^\\mu$, of the outgoing particle after a classical scattering process must be on shell, $\\finalk_1^2 = m_1^2$.\n\nWe may express the final momentum in terms of the initial momentum and the impulse, so that\n\\begin{equation}\n\\finalk_1^\\mu = p_1^\\mu + \\Delta p_1^\\mu\\,.\n\\end{equation}\nThe on-shell condition is then\n\\begin{equation}\n(\\Delta p_1)^2 + 2 p_1 \\cdot \\Delta p_1 = 0\\,.\n\\end{equation}\nAt order $g^2$, this requirement is satisfied trivially. At this order $(\\Delta p_1)^2$ is negligible, while\n\\begin{equation}\np_1 \\cdot \\Delta p_1 = i m_1 g^2 c_1\\cdot c_2 \\!\\int \\! \\dd^4 \\qb \\,\n\\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \n\\, e^{-i \\qb \\cdot b} \\, \\qb \\cdot \\ucl_1 \\frac{\\ucl_1 \\cdot \\ucl_2 }{\\qb^2} = 0\\,,\n\\end{equation}\nusing our result for the LO impulse in \\eqn~\\eqref{eqn:impulseClassicalLO}.\n\nThe situation is less trivial at order $g^4$, as neither $p_1 \\cdot \\Delta p_1$ nor $(\\Delta p_1)^2$ vanish. In fact, at this order we may use \\eqn~\\eqref{eqn:impulseClassicalLO} once again to find that\n\\begin{multline}\n(\\Delta p_1)^2 = - g^4  (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n\\times \\int \\! \\dd^4 \\qb \\,\\dd^4 \\qb' \\, \\del(\\qb \\cdot \\ucl_1) \n\\del(\\qb \\cdot \\ucl_2) \\del(\\qb' \\cdot \\ucl_1) \\del(\\qb' \\cdot \\ucl_2) \n\\, e^{-i (\\qb + \\qb') \\cdot b} \\,  \\frac{\\qb \\cdot \\qb'}{\\qb^2 \\, \\qb'^2}\\,.\n\\label{eqn:DeltaPsquared}\n\\end{multline}\nMeanwhile, to evaluate $p_1 \\cdot \\Delta p_1$ we must turn to our NLO result for the impulse, \\eqn~\\eqref{eqn:NLOImpulse}. Thanks to the delta functions present in the impulse, we \nfind a simple expression:\n\\begin{multline}\n2 p_1 \\cdot \\Delta p_1 = g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2\\! \\int \\! \\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \\, e^{-i \\qb\\cdot b} \\\\\n\\times \\int \\! \\dd^4 \\ellb\\; \\ellb \\cdot \\ucl_1 \\, \\del'(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2) \\, \\frac{\\ellb \\cdot (\\ellb - \\qb)}{\\ellb^2 (\\ellb - \\qb)^2}\\,.\n\\label{eqn:pDotDeltaP}\n\\end{multline}\nTo simplify this expression, it may be helpful to imagine working in the restframe of the timelike vector $u_1$. Then, the $\\ellb$ integral involves the distribution $\\ellb_0 \\, \\del'(\\ellb_0)$, while $\\qb_0 = 0$. Thus the $\\ellb_0$ integral has the form\n\\begin{equation}\n\\int \\! \\dd \\ellb_0 \\, \\ellb_0 \\, \\del'(\\ellb_0) \\, f(\\ellb_0{}^2) = -\\!\\int \\! \\dd \\ellb_0 \\, \\del(\\ellb_0) \\, f(\\ellb_0{}^2)\\,.\n\\end{equation}\nUsing this observation, we may simplify equation~\\eqref{eqn:pDotDeltaP} to find\n\\begin{equation}\n\\begin{aligned}\n\\vspace{-2mm}2 p_1 \\cdot \\Delta p_1 &= -g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n& \\,\\,\\, \\times \n\\int \\! \\dd^4 \\qb\\, \\dd^4 \\ellb \\; \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \n\\del(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2)  \ne^{-i \\qb\\cdot b}  \\frac{\\ellb \\cdot (\\ellb - \\qb)}{\\ellb^2 (\\ellb - \\qb)^2}  \\\\\n&=  g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n& \\,\\,\\,\\times\n\\int \\! \\dd^4 \\ellb \\, \\dd^4 \\qb' \\;\n\\del(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2) \\del(\\qb' \\cdot \\ucl_1) \n\\del(\\qb' \\cdot \\ucl_2) \\, e^{-i (\\ellb+\\qb')\\cdot b} \n\\frac{\\ellb \\cdot \\qb'}{\\ellb^2\\, \\qb'^2}\\,,\n\\end{aligned}\n\\end{equation}\nwhere in the last line we set $\\qb' = \\qb - \\ellb$. This expression is equal but opposite to \\eqn~\\eqref{eqn:DeltaPsquared}, and so the final momentum is on-shell as it must be.\n\nIt is worth remarking that the part of the NLO impulse that is relevant in this cancellation arises solely from the cut boxes. One can therefore view this phenomenon as an analogue of the removal of iterations of the tree in the potential.\n\n\\section{Beyond next-to-leading-order}\n\\label{sec:NNLO}\n\nWe have worked in this chapter under the premise of studying conservative scattering. Yet the LO and NLO impulse are only conservative in the sense that momentum is simply exchanged from particle 1 to particle 2 at these orders. However, beyond these lowest orders in perturbation theory physics does not clearly distinguish between conservative and dissipative behaviour: we will see shortly that at NNLO momentum can be radiated away, and moreover back-reacts on the impulse. To complete our on-shell formalism we must therefore incorporate radiation --- the interplay between the impulse and the radiated momentum forms the subject of our next chapter.\n\\chapter{Radiation: emission and reaction}\n\\label{chap:radiation}\n\n\\section{Introduction}\n\nGravitational wave astronomy relies on extracting measurable data from radiation. In this chapter we will therefore apply the methods developed for the impulse to construct a second on-shell and quantum-mechanical observable, the total emitted radiation.\n\nThese two observables are not independent. Indeed, the relation between them goes to the heart of one of the difficulties in traditional approaches to classical field theory with point sources. In two-particle scattering in classical electrodynamics, for example, momentum is transferred from one particle to the other via the electromagnetic field, as described by the Lorentz force. But the energy-momentum lost by point-particles to radiation is not accounted for by the Lorentz force. Conservation of momentum is restored by taking into account an additional force, the Abraham--Lorentz--Dirac (ALD) force~\\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz,LandauLifshitz}; see e.g. refs.~\\cite{Higuchi:2002qc,Galley:2006gs,Galley:2010es,Birnholtz:2013nta,Birnholtz:2014fwa,Birnholtz:2014gna} for more recent treatments. Inclusion of this radiation reaction force is not without cost: rather, it leads to the celebrated issues of runaway solutions or causality violations in the classical electrodynamics of point sources.\n\nUsing quantum mechanics to describe charged-particle scattering in should cure these ills. Indeed, we will see explicitly that a quantum-mechanical description will conserve energy and momentum in particle scattering automatically. First, in section~\\ref{sec:radiatedmomentum} we will set up expressions for the total radiated momentum in quantum field theory, and show that when combined with the impulse of the previous chapter, momentum is automatically conserved to all orders in perturbation theory. We will apply our previous investigation of the classical limit in section~\\ref{sec:classicalradiation}, introducing the radiation kernel and discussing how it relates to objects familiar from classical field theory. In section~\\ref{sec:LOradiation} we explicitly compute the radiation kernel at leading order in gauge and gravitational theories, and using its form in QED explicitly show that our impulse formalism from chapter~\\ref{chap:impulse} reproduces the predictions of the classical Abraham--Lorentz--Dirac force. We discuss the results of this and the previous chapter in section~\\ref{sec:KMOCdiscussion}.\n\nThis chapter continues to be based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}.\n\n\\section{The momentum radiated during a collision}\n\\label{sec:radiatedmomentum}\n\nA familiar classical observable is the energy radiated by an accelerating particle, for example during a scattering process. More generally we can compute the four-momentum radiated. In quantum mechanics there is no precise prediction for the energy or the momentum radiated by localised particles; we obtain a continuous spectrum if we measure a large number of events. However we can compute the expectation value of the four-momentum radiated during a scattering process. This is a well-defined observable, and as we will see it is on-shell in the sense that it can be expressed in terms of on-shell amplitudes.\n\nTo define the observable, let us again surround the collision with detectors which measure outgoing radiation of some type. We will call the radiated quanta `messengers'. Let $\\mathbb{K}^\\mu$ be the momentum operator for whatever field is radiated; then the expectation of the radiated momentum is\n\\begin{equation}\n\\begin{aligned}\n\\langle k^\\mu \\rangle = {}_\\textrm{out}{\\langle} \\Psi |  \\mathbb{K}^\\mu S \\, | \\Psi \\rangle_\\textrm{in} = {}_\\textrm{in}{\\langle} \\Psi | \\, S^\\dagger   \\mathbb{K}^\\mu S\\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $|\\Psi\\rangle_\\textrm{in}$ is again taken as the wavepacket in equation~\\eqref{eqn:inState}. Once again we can anticipate that the radiation will be expressed in terms of amplitudes. Rewriting $S = 1 + i T$, the expectation value becomes\n\\begin{align}\n\\Rad^\\mu \\equiv \\langle k^\\mu \\rangle &= {}_\\textrm{in}{\\langle} \\Psi | \\, S^\\dagger   \\mathbb{K}^\\mu S \\, | \\Psi \\rangle_\\textrm{in}\n= {}_\\textrm{in}\\langle \\Psi | \\, T^\\dagger   \\mathbb{K}^\\mu T \\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{align}\nbecause $ \\mathbb{K}^\\mu |\\Psi \\rangle_\\textrm{in} = 0$ since there are no quanta of radiation in the incoming state. \n\nWe can insert a complete set of states $|X ; k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2\\rangle$ containing at least one radiated messenger of momentum $k$, and write the expectation value of the radiated momentum as follows:\n\\begin{equation}\n\\begin{aligned}\n\\hspace{-2mm}\\Rad^\\mu = \\sum_X \\int\\! \\df(k) \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\nk_X^\\mu \\bigl|\\langle k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\,| \\, T \\,\n| \\Psi \\rangle\\bigr|^2\\,.\n\\end{aligned}\n\\label{eqn:radiationTform}\n\\end{equation}\nIn this expression, $X$ can again be empty, and $k_X^\\mu$ is the sum of the explicit messenger momentum $k^\\mu$ and the momenta of any messengers in the state $X$. Notice that we are including explicit integrals for particles 1 and 2, consistent with our assumption that the number of these particles is conserved during the process. The state $| k \\rangle$ describes a radiated messenger; the phase space integral over $k$ implicitly includes a sum over its helicity.\n\nExpanding the initial state, we find that the expectation value of the radiated momentum is given by\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu &= \\sum_X \\int\\! \\df(k) \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\,\nk_X^\\mu \\\\ \n& \\times\\bigg| \\int\\! \\df(\\initialk_1)\\df(\\initialk_2)  e^{i b \\cdot \\initialk_1\/\\hbar} \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \\del^{(4)}(\\initialk_1 + \\initialk_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X) \\\\\n& \\hspace{40mm} \\times \\langle \\zeta_1 \\, \\zeta_2 \\, X|\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X) |\\chi_1\\, \\chi_2 \\rangle \\bigg|^2\n\\,,\n\\label{eqn:ExpectedMomentum}\n\\end{aligned}\n\\end{equation}\nwhere we have accounted for any representation states in $X$, with the appropriate Haar measure implicity contained in the external sum. We can again introduce momentum transfers, $q_\\alpha=\\initialkc_\\alpha-\\initialk_\\alpha$, and trade the integrals over $\\initialkc_\\alpha$ for integrals over the $q_\\alpha$.  One of the four-fold $\\delta$ functions will again become $\\del^{(4)}(q_1+q_2)$, and we can use it to perform the $q_2$ integrations.  We again relabel $q_1\\rightarrow q$. The integration leaves behind a pair of on-shell $\\delta$ functions and positive-energy $\\Theta$ functions, just as in \\eqns{eqn:impulseGeneralTerm1}{eqn:impulseGeneralTerm2}:\n\\begin{equation}\n\\begin{aligned}\n\\hspace{-8pt}\\Rad^\\mu =&\\, \\sum_X \\int\\! \\df(k) \\prod_{\\alpha=1,2}\\df(\\finalk_\\alpha) \\df(\\initialk_\\alpha) \\dd^4 q\\;\n\\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \n\\varphi_1^*(\\initialk_1+q) \\varphi_2^*(\\initialk_2-q) \\,\n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1{}^0+q^0)\\Theta(\\initialk_2{}^0-q^0)\n\\\\&\\times \nk_X^\\mu \\, e^{-i b \\cdot q\/\\hbar} \n\\,\\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 - k - \\finalk_X)%\n\\\\&\\times \n\\langle \\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X)\n \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2b}\n\\end{equation}\nRepresentation states have been absorbed into an expectation value as in equation~\\eqref{eqn:defOfAmplitude}. We emphasise that this is an all-orders expression: the amplitude $\\Ampl(\\initialk_1,\\squeeze \\initialk_2 \\squeeze\\rightarrow \\squeeze \\finalk_1,\\squeeze \\finalk_2,\\squeeze k,\\squeeze \\finalk_X)$ includes all loop corrections, though of course it can be expanded in perturbation theory. The corresponding real-emission contributions are present in the sum over states $X$.   If we truncate the amplitude at a fixed order in perturbation theory, we should similarly truncate the sum over states. Given that the expectation value is expressed in terms of an on-shell amplitude, it is also appropriate to regard this observable as a fully on-shell quantity.\n\nIt can be useful to represent the observables diagrammatically. Two equivalent expressions for the radiated momentum are helpful:\n\\begin{multline}\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{positioning}\n\\Rad^\\mu =\n\\sum_X  \\mathlarger{\\int}\\! \\df(k)\\df(\\finalk_1)\\df(\\finalk_2)\\;\nk_X^\\mu \n\\\\ \\times\\left| \\mathlarger{\\int}\\! \\df(\\initialk_1) \\df(\\initialk_2)\\; \ne^{i b \\cdot \\initialk_1\/\\hbar} \\, \n\\del^{(4)}\\!\\left(\\sum p\\right) \\hspace*{-13mm}\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([xshift=-5cm,yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\vertex (b) ;\n\\vertex [above left=of b] (i1) {$\\psi_1(\\initialk_1)$};\n\\vertex [above right=of b] (o1) {$\\finalk_1$};\n\\vertex [above right =0.2 and 1.4 of b] (k) {$k$};\n\\vertex [below right =0.2 and 1.4 of b] (X) {$\\finalk_X$};\n\\vertex [below left=1 and 1 of b] (i2) {${\\psi_2(\\initialk_2)}$};\n\\vertex [below right=1 and 1 of b] (o2) {$\\finalk_2$};\n\\diagram* {(b) -- [photon, photonRed] (k)};\n\\diagram*{(b) -- [boson] (X)};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- node [right=4pt] {}(o2);\n\\draw[postaction={decorate}] (b) -- node [left=4pt] {} (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.42 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- node [left=4pt] {} (b);\n\\draw[postaction={decorate}] (i2) --node [right=4pt] {} (b);\n\\end{scope}\t\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\filldraw [fill=allOrderBlue] (b) circle [radius=10pt];\t\n\\end{feynman}\n\\end{tikzpicture}\n\\right|^2,\n\\label{eqn:RadiationPerfectSquare}\n\\end{multline}\nwhich is a direct pictorial interpretation of equation~\\eqref{eqn:ExpectedMomentum}, and\n\\begin{equation}\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{positioning}\n\\begin{aligned}\n\\Rad^\\mu &=\n\\sum_X \\mathlarger{\\int}\\! \\df(k) \\prod_{\\alpha = 1, 2} \\df(\\finalk_\\alpha)\n \\df(\\initialk_\\alpha) \\df(\\initialkc_\\alpha)\\; k_X^\\mu  \\, e^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar}   \\\\\n& \\times \\del^{(4)}(\\initialk_1 + \\initialk_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X)\\, \\del^{(4)}(\\initialkc_1 + \\initialkc_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X) \\\\\n& \\hspace{50mm}\\times\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\begin{scope}\n\\vertex (ip1) ;\n\\vertex [right=2 of ip1] (ip2);\n\\node [] (X) at ($ (ip1)!.5!(ip2) $) {};\n\\begin{scope}[even odd rule]\n\\begin{pgfinterruptboundingbox}\n\\path[invclip] ($  (X) - (4pt, 30pt) $) rectangle ($ (X) + (4pt,30pt) $) ;\n\\end{pgfinterruptboundingbox} \n\n\\vertex [above left=0.66 and 0.33 of ip1] (q1) {$ \\psi_1(\\initialk_1)$};\n\\vertex [above right=0.66 and 0.33 of ip2] (qp1) {$ \\psi^*_1(\\initialkc_1)$};\n\\vertex [below left=0.66 and 0.33 of ip1] (q2) {$ \\psi_2(\\initialk_2)$};\n\\vertex [below right=0.66 and 0.33 of ip2] (qp2) {$ \\psi^*_2(\\initialkc_2)$};\n\n\\diagram* {(ip1) -- [photon, out=30, in=150, photonRed]  (ip2)};\n\\diagram*{(ip1) -- [photon, out=330, in=210]  (ip2)};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (q1) -- (ip1);\n\\draw[postaction={decorate}] (q2) -- (ip1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip2) -- (qp1);\n\\draw[postaction={decorate}] (ip2) -- (qp2);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.38 with {\\arrow{Stealth}},\n\tmark=at position 0.74 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip1) to [out=90, in=90,looseness=1.7] node[above left] {{$ \\finalk_1$}} (ip2);\n\\draw[postaction={decorate}] (ip1) to [out=270, in=270,looseness=1.7]node[below left] {${\\finalk_2}$} (ip2);\n\\end{scope}\n\n\\node [] (Y) at ($(X) + (0,1.5)$) {};\n\\node [] (Z) at ($(X) - (0,1.5)$) {};\n\\node [] (k) at ($ (X) - (0.35,-0.55) $) {$k$};\n\\node [] (x) at ($ (X) - (0.35,0.55) $) {$\\finalk_X$};\n\n\\filldraw [color=white] ($ (ip1)$) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip1) $) circle [radius=8pt];\n\n\\filldraw [color=white] ($ (ip2) $) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip2) $) circle [radius=8pt];\n\n\\end{scope} \n\\end{scope}\n\\draw [dashed] (Y) to (Z);\n\\end{feynman}\n\\end{tikzpicture},\n\\end{aligned}\n\\end{equation}\nwhich demonstrates that we can think of the expectation value as the weighted cut of a loop amplitude. As $X$ can be empty, the lowest-order contribution arises from the weighted cut of a two-loop amplitude.\n\n\\subsection{Conservation of momentum}\n\\label{sect:allOrderConservation}\n\nThe expectation of the radiated momentum is not independent of the impulse. In fact the relation between these quantities is physically rich. In the classical electrodynamics of point~particles, for example, the impulse is due to a total time integral of the usual Lorentz force,~\\eqref{eqn:Wong-momentum}. However, when the particles emit radiation the point-particle approximation leads to well-known issues. This is a celebrated problem in classical field theory.  Problems arise because of the singular nature of the point-particle source. In particular, the electromagnetic field at the position of a point charge is infinite, so to make sense of the Lorentz force acting on the particle the traditional route is to subtract the particle's own field from the full electromagnetic field in the force law. The result is a well-defined force, but conservation of momentum is lost.\n\nConservation of momentum is restored by including another force, the Abraham--Lorentz--Dirac (ALD) force~\\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz}, acting on the particles. This gives rise to an impulse on particle 1 in addition to the impulse due to the Lorentz force. The Lorentz force exchanges momentum between particles 1 and 2, while the radiation reaction impulse,\n\\begin{equation}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^2 Q_1^2}{6\\pi m_1}\\int_{-\\infty}^\\infty\\! d\\tau \\left(\\frac{d^2p_1^\\mu}{d\\tau^2} + \\frac{p_1^\\mu}{m_1^2}\\frac{dp_1}{d\\tau}\\cdot\\frac{dp_1}{d\\tau}\\right),\n\\label{eqn:ALDclass}\n\\end{equation}\naccounts for the irreversible loss of momentum due to radiation. Of course, the ALD force is a notably subtle issue in the classical theory.\n\nIn the quantum theory of electrodynamics there can be no question of violating conservation of momentum, so the quantum observables we have defined must already include all the effects which would classically be attributed to both the Lorentz and ALD forces. This must also hold for the counterparts of these forces in any other theory. In particular, it must be the case that our definitions respect conservation of momentum; it is easy to demonstrate this formally to all orders using our definitions. Later, in section~\\ref{sec:ALD}, we will indicate how the radiation reaction is included in the impulse more explicitly.\n\nOur scattering processes involve two incoming particles. Consider, then,\n\\begin{equation}\n\\begin{aligned}\n\\langle \\Delta p_1^\\mu \\rangle + \\langle \\Delta p_2^\\mu \\rangle &= \n\\langle \\Psi | i [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle \n+ \\langle \\Psi | T^\\dagger [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle \\\\\n&= \\bigl\\langle \\Psi \\big| i \\bigl[ \\textstyle{\\sum_\\alpha}   \\mathbb{P}_\\alpha^\\mu, T \\bigr] \n\\big| \\Psi \\bigr\\rangle \n+ \\langle \\Psi | T^\\dagger [  \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle\\,,\n\\end{aligned}\n\\end{equation}\nwhere the sum $\\sum   \\mathbb{P}_\\alpha^\\mu$ is now over all momentum operators in the theory, not just those for the two initial particles. The second equality above holds because $  \\mathbb{P}_\\alpha^\\mu | \\Psi \\rangle = 0$ for $\\alpha \\neq 1,2$; only quanta of fields 1 and 2 are present in the incoming state. Next, we use the fact that the total momentum is time independent, or in other words\n\\begin{equation}\n\\Bigl[ \\sum  \\mathbb{P}_\\alpha^\\mu, T \\Bigr] = 0\\,,\n\\end{equation}\nwhere the sum extends over all fields. Consequently,\n\\begin{equation}\n\\langle \\Psi | i [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle =  \n\\bigl\\langle \\Psi \\big| i \\bigl[ \\textstyle{\\sum_\\alpha}   \\mathbb{P}_\\alpha^\\mu, T \\bigr] \\big| \n\\Psi \\bigr\\rangle = 0\\,.\n\\label{eqn:commutatorVanishes}\n\\end{equation}\nThus the first term $\\langle \\Psi | i [   \\mathbb{P}_1^\\mu, T ] | \\Psi \\rangle$ in the impulse~(\\ref{eqn:defl1}) describes only the exchange of momentum between particles~1 and~2; in this sense it is associated with the classical Lorentz force (which shares this property) rather than with the classical ALD force (which does not). The second term in the impulse, on the other hand, includes radiation. To make the situation as clear as possible, let us restrict attention to the case where the only other momentum operator is $  \\mathbb{K}^\\mu$, the momentum operator for the messenger field. Then we know that $[   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu +   \\mathbb{K}^\\mu, T] = 0$, and conservation of momentum at the level of expectation values is easy to demonstrate:\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle + \\langle \\Delta p_2^\\mu \\rangle = \n- \\langle \\Psi | T^\\dagger [   \\mathbb{K}^\\mu, T ] | \\Psi \\rangle = \n- \\langle \\Psi | T^\\dagger    \\mathbb{K}^\\mu T  | \\Psi \\rangle = \n- \\langle k^\\mu \\rangle = - \\Rad^\\mu\\,,\n\\end{equation}\nonce again using the fact that there are no messengers in the incoming state.\n\nIn the classical theory, radiation reaction is a subleading effect, entering for two-body scattering at order $e^6$ in perturbation theory in electrodynamics. This is also the case in the quantum theory. To see why, we again expand the operator product in the second term of \\eqn~\\eqref{eqn:defl1} using a complete set of states:\n\\begin{multline}\n\\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle = \\sum_X \\int \\! \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\n\\\\ \\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\rangle \n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X | [  \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\end{multline}\nThe sum over $X$ is over all states, including an implicit integral over their momenta and a sum over any other quantum numbers. The inserted-state momenta of particles 1 and~2 (necessarily present) are labeled by $\\finalk_\\alpha$, and the corresponding integrations over these momenta by $\\df(\\finalk_\\alpha)$.  These will ultimately become integrations over the final-state momenta in the scattering. To make the loss of momentum due to radiation explicit at this level, we note that\n\\begin{multline}\n\\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu +    \\mathbb{P}_2^\\mu, T] \\, |\\Psi \\rangle \n= -\\sum_X \\int \\! \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\n\\\\\\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X\\rangle  \n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X | \\,   \\mathbb{P}_X^\\mu T \\,  |\\Psi \\rangle\\,,\n\\end{multline}\nwhere $ \\mathbb{P}_X$ is the sum over momentum operators of all quantum fields other than the scalars~1 and 2. The sum over all states $X$ will contain, for example, terms where the state $X$ includes messengers of momentum $k^\\mu$ along with other massless particles. We can further restrict attention to the contributions of the messenger's momentum to $\\mathbb{P}_X^\\mu$. This contribution produces a net change of momentum of particle 1 given by\n\\begin{multline}\n-\\sum_X \\int \\!  \\df(k) \\df(\\finalk_1)\\df(\\finalk_2)\\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\; k^\\mu \\, \n\\\\\\times\\langle \\Psi | \\, T^\\dagger | k ; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X\\rangle\n\\langle k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X| \\, T \\,  |\\Psi \\rangle \n= - \\langle k^\\mu \\rangle\\,,\n\\end{multline} \nwith the help of equation~\\eqref{eqn:radiationTform}. Thus we explicitly see the net loss of momentum due to radiating messengers. In any theory this quantity is suppressed by factors of the coupling $\\tilde g$ because of the additional state. The lowest order case corresponds to $X = \\emptyset$; as there are two quanta in $|\\psi \\rangle$, we must compute the modulus squared of a five-point tree amplitude. The term is proportional to $\\tilde g^6$, where $\\tilde g$ is the coupling of an elementary three-point amplitude; as far as the impulse is concerned, it is a next-to-next-to-leading order (NNLO) effect. Other particles in the state $X$, and other contributions to its momentum, describe higher-order effects.\n\n\\section{Classical radiation}\n\\label{sec:classicalradiation}\n\\defK^\\mu{K^\\mu}\n\nFollowing our intensive study of the classical limit of the impulse in the previous chapter, the avenue leading to the classical limit of $R^\\mu$ is clear: provided we work with the wavefunctions of chapter~\\ref{chap:pointParticles} in the the Goldilocks zone $\\ell_c \\ll \\ell_w \\ll \\lscatt$, we can simply adopt the rules of section~\\ref{sec:classicalLimit}. In particular the radiated momentum $k$ will scale as a wavenumber in the classical region. This is enforced by the energy-momentum-conserving delta function in \\eqn~\\eqref{eqn:ExpectedMomentum2b}, rewritten in terms of momentum transfers $w_\\alpha = r_\\alpha - p_\\alpha$:\n\\begin{equation}\n\\del^{(4)}(w_1+w_2 + k + \\finalk_X)\\,.\n\\end{equation}\nThe arguments given after equation~\\eqref{eqn:radiationScalingDeltaFunction} then ensure that the typical values of all momenta in the argument should again by scaled by $1\/\\hbar$ and replaced by wavenumbers.\n\nWith no new work required on the formalities of the classical limit, let us turn to explicit expressions for the classical radiated momentum in terms of amplitudes. Recall that our expressions for the total emitted radiation in section~\\ref{sec:radiatedmomentum} depended on $q$, which represents a momentum mismatch rather than a momentum transfer. However, we expect the momentum transfers to play an important role in the classical limit, and so it is convenient to change variables from the $r_\\alpha$ to make use of them:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu &= \\sum_X \\int\\! \\df(k) \\prod_{\\alpha=1,2} \\df(\\initialk_\\alpha) \\dd^4\\xfer_\\alpha\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1{}^0+q^0)\\Theta(\\initialk_2{}^0-q^0)\\,\\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \n\\\\&\\qquad\\times \\varphi_1^*(\\initialk_1+q) \\varphi_2^*(\\initialk_2-q) \\, k_X^\\mu \\, e^{-i b \\cdot q\/\\hbar}  \\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X)\n\\\\&\\qquad\\qquad\\times \n\\langle\\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\qquad\\times \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2}\n\\end{equation}\nWe can now recast this expression in the notation of \\eqn~\\eqref{eqn:angleBrackets}:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class &= \\sum_X\\, \\Lexp \\int\\! \\df(k) \\prod_{\\alpha=1,2} \\dd^4\\xfer_\\alpha\\,\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\\,  k_X^\\mu \n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X) \\Theta(\\initialk_1{}^0+q^0)%\n\\\\& \\times \\Theta(\\initialk_2{}^0-q^0)\\, e^{-i b \\cdot q\/\\hbar} \\, \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\qquad\\times \n\\Ampl(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\\,\\Rexp\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2recast}\n\\end{equation}\nWe will determine the classical limit of this expression using precisely the same logic as in the preceding chapter. Let us again focus on the leading contribution, with $X=\\emptyset$.  Once again, rescale $q \\rightarrow \\hbar\\qb$, and drop the $q^2$ inside the on-shell delta functions. Here, remove an overall factor of $\\tilde g^6$ and accompanying $\\hbar$'s from the amplitude and its conjugate. In addition, rescale the momentum transfers $\\xfer\\rightarrow \\hbar\\xferb$ and the radiation momenta, $k\\rightarrow\\hbar\\wn k$. At leading order there is no sum, so there will be no hidden cancellations, and we may drop the $\\xfer_\\alpha^2$ inside the on-shell delta functions to obtain\n\\def\\bar k{\\bar k}\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class &= \n\\tilde g^6 \\Lexp \\hbar^4\\! \\int\\! \\df(\\bar k) \\prod_{\\alpha=1,2} \\dd^4\\xferb_\\alpha\\dd^4 \\qb\\, \\del(2\\xferb_\\alpha\\cdot p_\\alpha)\n\\del(2\\qb\\cdot p_1) \\del(2\\qb\\cdot p_2) \\, e^{-i b \\cdot \\qb}\n\\\\& \\qquad \\times \\bar k^\\mu \\, \\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb, \\initialk_2-\\hbar \\qb \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2\\,, \\hbar\\bar k)\n\\\\& \\qquad\\times \n\\AmplB^{(0)}(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2\\,, \\hbar\\bar k)\\,\\del^{(4)}(\\xferb_1+\\xferb_2+ \\bar k)\\,\\Rexp\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2classicalLO}\n\\end{equation}\nWe will make use of this expression below to verify that momentum is conserved as expected.\n\nOne disadvantage of this expression for the leading order radiated momentum is that it is no longer in a form of an \nintegral over a perfect square, such as shown in \\eqn~\\eqref{eqn:RadiationPerfectSquare}. Nevertheless we can recast \\eqn~\\eqref{eqn:ExpectedMomentum2recast} in such a form.\nTo do so, perform a change of variable, including in the (momentum space) wavefunctions. To begin, it is helpful to write \\eqn~\\eqref{eqn:ExpectedMomentum2recast} as\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class =&\\, \\sum_X \\prod_{\\alpha=1,2}  \\int \\! \\df(\\initialk_\\alpha)\\,  |\\varphi_\\alpha(\\initialk_\\alpha)|^2 \\int \\! \\df(k) \\df( \\xfer_\\alpha+\\initialk_\\alpha) \\df(q_\\alpha+\\initialk_\\alpha) \\; \n\\\\& \\times \\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X) \\del^{(4)}(q_1 + q_2) \\, e^{-i b \\cdot q_1\/\\hbar} \\, k_X^\\mu \\,   %\n\\\\&\\qquad\\times \n\\langle \\Ampl^*(\\initialk_1+q_1\\,, \\initialk_2 + q_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\times \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\rangle\\,\n\\,.\n\\end{aligned}\n\\end{equation}\n\\def\\tilde\\initialk{\\tilde\\initialk}\n\\def\\tilde\\xfer{\\tilde\\xfer}\n\\def\\tilde q{\\tilde q}\n\\noindent We will now re-order the integration and perform a change of variables. Let us define $\\tilde\\initialk_\\alpha=\\initialk_\\alpha - \\tilde\\xfer_\\alpha$, $\\tilde q_\\alpha = q_\\alpha + \\tilde\\xfer_\\alpha$, and $\\tilde\\xfer_\\alpha = - \\xfer_\\alpha$, changing variables from $\\initialk_\\alpha$ to $\\tilde\\initialk_\\alpha$, from $q_\\alpha$ to $\\tilde q_\\alpha$, and from $\\xfer_\\alpha$ to $\\tilde\\xfer_\\alpha$:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class =&\\, \\sum_X \\prod_{\\alpha=1,2}  \\int \\! \\df(\\tilde\\initialk_\\alpha)  \\df(k) \\df(\\tilde\\xfer_\\alpha+\\tilde\\initialk_\\alpha) \\df (\\tilde q_\\alpha+\\tilde\\initialk_\\alpha) |\\varphi_\\alpha(\\tilde\\initialk_\\alpha+\\tilde\\xfer_\\alpha)|^2 \\; \n\\\\& \\times \\del^{(4)}(\\tilde \\xfer_1+ \\tilde \\xfer_2- k- \\finalk_X) \\del^{(4)}(\\tilde q_1 + \\tilde q_2 - k - \\finalk_X)\\, e^{-i b \\cdot (\\tilde q_1 - \\tilde\\xfer_1)\/\\hbar} \\, k_X^\\mu\n\\\\&\\qquad\\times \n\\langle\\Ampl^*(\\tilde\\initialk_1+ \\tilde q_1\\,, \\tilde\\initialk_2 + \\tilde q_2 \\rightarrow \n\\tilde\\initialk_1\\,, \\tilde\\initialk_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\times \\Ampl(\\tilde\\initialk_1 + \\tilde\\xfer_1\\,, \\tilde\\initialk_2 + \\tilde\\xfer_2\\rightarrow \n\\tilde\\initialk_1\\,, \\tilde\\initialk_2\\,, k\\,, \\finalk_X)\n\\rangle\\,\n\\,.\n\\end{aligned}\n\\end{equation}\nAs the $\\tilde\\xfer_\\alpha$ implicitly carry a factor of $\\hbar$, just as argued in \\sect{subsec:Wavefunctions} for the momentum mismatch $q$, we may neglect the shift in the wavefunctions.  Dropping the tildes, and associating the $\\xfer_\\alpha$ integrals with $\\Ampl$ and the $q_\\alpha$ integrals with $\\Ampl^*$, our expression is revealed as an integral over a perfect square,\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class\n& = \\sum_X \\prod_{\\alpha=1,2} \\Lexp \\int \\! \\df(k) \\, k_X^\\mu\n\\biggl | \\int \\! \\df(\\xfer_\\alpha + \\initialk_\\alpha)  \\; \n\\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X)\\\\\n& \\hspace*{25mm} \\times   e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X) \\biggr|^2 \\Rexp\n\\,.\n\\label{eqn:radiatedMomentumClassicalAllOrder}\n\\end{aligned}\n\\end{equation}\nThe perfect-square structure allows us to define a \\textit{radiation kernel\\\/},\n\\begin{equation}\n\\begin{aligned}\n\\RadKer(k, \\finalk_X)\n&\\equiv \\hbar^{3\/2} \\prod_{\\alpha = 1, 2}  \\int \\! \\df( \\initialk_\\alpha +  \\xfer_\\alpha)  \\; \n\\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X) \\\\\n& \\qquad \\qquad \\times  e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X), \\\\\n= & \\hbar^{3\/2}\\prod_{\\alpha = 1, 2} \\int \\! \\dd^4 \\xfer_\\alpha\\; \n\\del(2 p_\\alpha \\cdot \\xfer_\\alpha + \\xfer_\\alpha^2)\\, \\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X) \\\\\n& \\quad\\times \\Theta(\\initialk_\\alpha^0+\\xfer_\\alpha^0)\\, e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X)\\,,\n\\label{eqn:defOfR}\n\\end{aligned}\n\\end{equation}\nso that\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class &=  \\sum_X \\hbar^{-3}\\Lexp \\int \\! \\df(k) \\, k_X^\\mu\n\\left |\\RadKer(k, \\finalk_X) \\right|^2 \\Rexp\n\\,.\n\\label{eqn:radiatedMomentumClassical}\n\\end{aligned}\n\\end{equation}\nThe prefactor along with the normalization of $\\RadKer$ are again chosen so that the classical limit of the radiation kernel will be of $\\Ord(\\hbar^0)$. Let us now focus once more on the leading contribution, with $X=\\emptyset$.  As usual, rescale $\\xfer \\rightarrow \\hbar\\xferb$, and remove an overall factor of $\\tilde g^6$ and accompanying $\\hbar$'s from the amplitude and its conjugate. Then the LO radiation kernel is\n\\begin{equation}\n\\begin{aligned}\n\\RadKerCl(\\wn k) \n& \\equiv \\hbar^2 \\prod_{\\alpha = 1, 2} \\int \\! \\dd^4 \\xferb_\\alpha \\, \\del(2p_\\alpha \\cdot \\xferb_\\alpha + \\hbar\\xferb_\\alpha^2) \\,\n\\del^{(4)}( \\xferb_1+  \\xferb_2- \\wn k)\ne^{i b \\cdot  \\xferb_1}  \n\\\\& \\hspace{70pt} \\times  \\AmplB^{(0)}( \\initialk_1 +  \\hbar \\xferb_1,  \\initialk_2 + \\hbar \\xferb_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, \\hbar \\wn k)\\,,\n\\label{eqn:defOfRLO}\n\\end{aligned}\n\\end{equation}\nensuring that the leading-order momentum radiated is simply\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu, (0)}_\\class &= \\tilde g^6 \\Lexp \\int \\! \\df(\\wn k) \\, \\wn k^\\mu \\left | \\RadKerCl(\\wn k) \\right|^2 \\Rexp\\,.\n\\label{eqn:radiatedMomentumClassicalLO}\n\\end{aligned}\n\\end{equation}\n\n\\subsubsection{Conservation of momentum}\n\\label{sect:classicalConservation}\n\nConservation of momentum certainly holds to all orders, as we saw in \\sect{sect:allOrderConservation}.  However, it is worth making sure that we have not spoiled this critical physical property in our previous discussion, or indeed in our discussion of the classical impulse in \\sect{sec:classicalImpulse}. One might worry, for example, that there is a subtlety with the order of limits.\n\nThere is no issue at LO and NLO for the impulse, because\n\\begin{equation}\n\\DeltaPlo + \\DeltaPloTwo = 0 ,\\quad \\DeltaPnlo + \\DeltaPnloTwo = 0.\n\\end{equation}\nThese follow straightforwardly from the definitions of the observables, \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} and \\eqn~\\eqref{eqn:classicalLimitNLO}. The essential point is that the amplitudes entering into these orders in the impulse conserve momentum for two particles. At LO, for example, using \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} the impulse on particle 2 can be written as\n\\begin{multline}\n\\DeltaPloTwo=  \\frac{i\\tilde g^2}{4} \\Lexp \\hbar^2\\! \\int \\!\\dd^4 \\qb_1  \\dd^4 \\qb_2 \\; \n\\del(\\qb_1\\cdot p_1) \\del(\\qb_1\\cdot p_2)  \\del^{(4)}(\\qb_1 + \\qb_2)\n\\\\\\times  \ne^{-i b \\cdot \\qb_1} \n\\, \\qb_2^\\mu  \\, \\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow \np_1 + \\hbar\\qb_1, p_2 + \\hbar\\qb_2)\\,\\Rexp.\n\\end{multline}\nIn this equation, conservation of momentum at the level of the four point amplitude $\\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow p_1 + \\hbar\\qb_1, p_2 + \\hbar\\qb_2)$ is expressed by the presence of the four-fold delta function $\\del^{(4)}(\\qb_1 + \\qb_2)$. Using this delta function, we may replace $\\qb_2^\\mu$ with $- \\qb_1^\\mu$ and then integrate over $\\qb_2$, once again using the delta function. The result is manifestly $-\\DeltaPlo$,  \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. A similar calculation goes through at NLO.\n\nIn this sense, the scattering is conservative at LO and at NLO. At NNLO, however, we must take radiative effects into account.  This backreaction is entirely described by the quadratic part of the impulse, $\\ImpB$. As indicated in \\eqn~\\eqref{eqn:commutatorVanishes}, $\\ImpA$ is always conservative. From our perspective here, this is because it involves only four-point amplitudes.  Thus to understand conservation of momentum we need to investigate $\\ImpB$. The lowest order case in which a five point amplitude can enter $\\ImpB$ is at NNLO. Let us restrict attention to this lowest order case, taking the additional state $X$ to be a messenger.\n\nFor $\\ImpB$ the lowest order term inolving one messenger is, in the classical regime,\n\\begin{equation}\n\\hspace*{-3mm}\\begin{aligned}\n\\ImpBclsup{(\\textrm{rad})} =&\\, \\tilde g^6\n\\Lexp \\hbar^{4}\\!\\int \\! d\\Phi(\\wn k)  \\prod_{\\alpha = 1,2} \\dd^4\\xferb_\\alpha\\, \n\\dd^4 \\qb_1 \\dd^4 \\qb_2 \\;\\del(2 \\xferb_\\alpha\\cdot p_\\alpha + \\xferb_\\alpha^2)\n\\\\&\\times  \\del(2 \\qb_1\\cdot p_1) \\del(2 \\qb_2\\cdot p_2)\\,\ne^{-i b \\cdot \\qb_1}\\,\\xferb_1^\\mu\\,\n\\del^{(4)}(\\xferb_1+\\xferb_2 + \\bar k)\\, \\del^{(4)}(\\qb_1+\\qb_2)\n\\\\&\\quad\\times \\AmplB^{(0)}(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2, \\hbar\\bar k)\n\\\\&\\qquad\\times \n\\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb_1\\,, \\initialk_2 + \\hbar \\qb_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2, \\hbar\\bar k)\n\\,\\Rexp\\,.\n\\end{aligned} \n\\label{eqn:nnloImpulse}\n\\end{equation}\nTo see that this balances the radiated momentum, we use \\eqn~\\eqref{eqn:ExpectedMomentum2classicalLO}. The structure of the expressions are almost identical; conservation of momentum holds because the factor $\\bar k^\\mu$ in \\eqn~\\eqref{eqn:ExpectedMomentum2classicalLO} is balanced by $\\xferb_1^\\mu$ in \\eqn~\\eqref{eqn:nnloImpulse} and $\\xferb_2^\\mu$ in the equivalent expression for particle 2.\n\nThus conservation of momentum continues to hold in our expressions once we have passed to the classical limit, at least through NNLO. At this order there is non-zero momentum\nradiated, so momentum conservation is non-trivial from the classical point of view. We will see by explicit calculation in QED that our classical impulse correctly incorporates the impulse from the ALD force in addition to the Lorentz force.\n\n\\subsection{Perspectives from classical field theory}\n\\defx{x}\n\nBefore jumping into examples, it is useful to reflect on the total radiated momentum, expressed in terms of amplitudes, by digressing into classical field theory. To do so we must classically describe the distribution and flux of energy and momentum in the radiation field itself. Although our final conclusions also hold in YM theory and gravity, let us work in electrodynamics for simplicity. Here the relevant stress-energy tensor is\n\\begin{equation}\nT^{\\mu\\nu}(x) = F^{\\mu\\alpha}(x) F_\\alpha{}^\\nu(x) + \\frac 14 \\eta^{\\mu\\nu} F^{\\alpha\\beta}(x) F_{\\alpha\\beta}(x) \\,.\\label{eqn:EMfieldStrength}\n\\end{equation}\nIn particular, the (four-)momentum flux through a three dimensional surface $\\partial \\Omega$ with surface element $\\d\\Sigma_\\nu$ is\n\\begin{equation}\nK^\\mu = \\int_{\\partial \\Omega}\\!\\! \\d \\Sigma_\\nu T^{\\mu\\nu}(x)\\,.\n\\end{equation}\nWe are interested in the total momentum radiated as two particles scatter. At each time $t$, we therefore surround the two particles with a large sphere. The instantaneous flux of momentum is measured by integrating over the surface area of the sphere; the total momentum radiated is then the integral of this instantaneous flux over all times. It is straightforward to determine the momentum radiated by direct integration over these spheres using textbook methods --- see appendix D of \\cite{Kosower:2018adc}.\n\nA simpler but more indirect method is the following. We wish to use the Gauss theorem to write\n\\begin{equation}\nK^\\mu = \\int_{\\partial \\Omega}\\!\\! \\d \\Sigma_\\nu T^{\\mu\\nu}(x) =  \\int \\! \\d^4x \\, \\partial_\\nu T^{\\mu\\nu}(x)\\,.\n\\end{equation}\nHowever, the spheres surrounding our particle are not the boundary of all spacetime: they do not include the timelike future and past boundaries. To remedy this, we use a trick due to Dirac~\\cite{Dirac:1938nz}. \n\nThe radiation we have in mind is causal, so we solve the Maxwell equation with retarded boundary conditions. We denote these fields by $F^{\\mu\\nu}_\\textrm{ret}(x)$.\nWe could equivalently solve the Maxwell equation using the advanced Green's function. If we wish to determine precisely the same fields $F^{\\mu\\nu}_\\textrm{ret}(x)$ but using the advanced Green's function, we must add a homogeneous solution of the Maxwell equation. Fitting the boundary conditions in this way requires subtracting the incoming radiation field $F^{\\mu\\nu}_\\textrm{in}(x)$ which is present in the advanced solution (but not in the retarded solution) and adding the outgoing radiation field (which is present in the retarded solution, but not the advanced solution.) In other words,\n\\begin{equation}\nF^{\\mu\\nu}_\\textrm{ret}(x) - F^{\\mu\\nu}_\\textrm{adv}(x) = - F^{\\mu\\nu}_\\textrm{in}(x) + F^{\\mu\\nu}_\\textrm{out}(x)\\,.\n\\end{equation}\nNow, the radiated momentum $K^\\mu$ in which we are interested is described by $F^{\\mu\\nu}_\\textrm{out}(x)$. The field $F^{\\mu\\nu}_\\textrm{in}(x)$ transports the same total amount of momentum in from infinity, ie it transports momentum $-K^\\mu$ out. Therefore the difference between the momenta transported out to infinity by the retarded and by the advanced fields is simply $2 K^\\mu$. This is useful, because the contributions of the point-particle sources cancel in this difference.\n\nThe relationship between the momentum transported by the retarded and advanced field is reflected at the level of the Green's functions themselves. \nThe difference in the Green's function takes an instructive form:%\n\\begin{equation}\n\\begin{aligned}\n\\tilde G_\\textrm{ret}(\\bar k) - \\tilde G_\\textrm{adv}(\\bar k) &= \n\\frac{(-1)}{(\\bar k^0 + i \\epsilon)^2 - \\v{\\bar k}^2} \n- \\frac{(-1)}{(\\bar k^0 - i \\epsilon)^2 - \\v{\\bar k}^2} \n\\\\&=  i     \\left( \\Theta(\\bar k^0) - \\Theta(-\\bar k^0) \\right) \\del(\\bar k^2)\\,.\n\\end{aligned}\n\\end{equation}\nIn this equation, $\\v{\\bar k}$ denotes the spatial components of wavenumber four-vector $\\bar k$. This difference is a homogeneous solution of the wave equation since it is supported \non $\\bar k^2 = 0$. The two terms correspond to positive and negative angular frequencies. As we will see, the relative sign ensures that the momenta transported to infinity add.\n\nWith this in mind, we return to the problem of computing the momentum radiated and write\n\\begin{equation}\n2 K^\\mu = \\int_{\\partial \\Omega} \\!\\!\\d \\Sigma_\\nu \\Big(T^{\\mu\\nu}_\\textrm{ret}(x) -T^{\\mu\\nu}_\\textrm{adv}(x) \\Big)\\,.\n\\end{equation}\nIn this difference, the contribution of the sources at timelike infinity cancel, so we may regard the surface $\\partial \\Omega$ as the boundary of spacetime. Therefore,\n\\begin{equation}\n2K^\\mu = \\int \\! d^4 x \\, \\partial_\\nu \\!\\left(T^{\\mu\\nu}_\\textrm{ret}(x) -T^{\\mu\\nu}_\\textrm{adv}(x) \\right) =- \\int \\! d^4 x \\left( F^{\\mu\\nu}_\\textrm{ret}(x) - F^{\\mu\\nu}_\\textrm{adv}(x)\\right) J_\\nu(x)\\,,\n\\end{equation}\nwhere the last equality follows from the equations of motion. We now pass to momentum space, noting that\n\\begin{equation}\nF^{\\mu\\nu}(x) = -i\\! \\int \\! \\dd^4 \\bar k \\left( \\bar k^\\mu \\tilde A^\\nu(\\bar k) - \\bar k^\\nu \\tilde A^\\mu(\\bar k) \\right) e^{-i \\bar k \\cdot x}\\,.\n\\end{equation}\nUsing conservation of momentum, the radiated momentum becomes\n\\begin{equation}\n\\begin{aligned}\n2K^\\mu \n&= i\\! \\int \\! \\dd^4 \\bar k \\; \\bar k^\\mu \\left( \\tilde A^\\nu_\\textrm{ret}(\\bar k) - \\tilde A^\\nu_\\textrm{adv}(\\bar k) \\right) \\tilde J_\\nu^*(\\bar k), \\\\\n&= -\\int \\! \\dd^4 \\bar k \\; \\bar k^\\mu \\left(\\Theta(\\bar k^0) - \\Theta(-\\bar k^0)\\right) \\del(\\bar k^2) \\tilde J^\\nu(\\bar k) \\tilde J_\\nu^*(\\bar k)\\,.\n\\label{eqn:momentumMixed}\n\\end{aligned}\n\\end{equation}\nThe two different $\\Theta$ functions arise from the outgoing and incoming radiation fields. Setting $k'^\\mu = - k^\\mu$ in the second term, and then dropping the prime, it is easy to see that the two terms add as anticipated. We arrive at a simple general result for the momentum radiated:\n\\begin{equation}\n\\begin{aligned}\nK^\\mu &= -\\int \\! \\dd^4 \\wn k \\,  \\Theta(\\wn k^0)\\del(\\wn k^2) \\, \\wn k^\\mu \\, \\tilde J^\\nu(\\wn k) \\tilde J_\\nu^*(\\wn k) \\\\\n&= -\\int \\! \\df( \\wn k) \\, \\bar k^\\mu \\, \\tilde J^\\nu(\\bar k) \\tilde J_\\nu^*(\\bar k) \\,.\n\\label{eqn:classicalMomentumRadiated}\n\\end{aligned}\n\\end{equation}\nIt is now worth pausing to compare this general classical formula for the radiated momentum to the expression we derived previously in \\eqn~\\eqref{eqn:radiatedMomentumClassical}. Evidently the radiation kernel we defined in \\eqn~\\eqref{eqn:defOfR} is related to the classical current $\\tilde J^\\mu(\\bar k)$. This fact was anticipated in ref.~\\cite{Luna:2017dtq}. Indeed, if we introduce a basis of polarisation vectors $\\varepsilon^{h}_\\mu(\\bar k)$ associated with the wavevector $\\bar k$ with helicity $h$, we may write the classical momentum radiated as\n\\begin{equation}\nK^\\mu = \\sum_h \\int \\! \\df(\\bar k) \\, \\bar k^\\mu \\, \n\\left| \\varepsilon^{h} \\cdot \\tilde J(\\bar k) \\right|^2\\,,\n\\label{eqn:classicalMomentumRadiated1}\n\\end{equation}\nwhere here we have written the sum over helicities explicitly. Similar expressions hold in classical YM theory and gravity \\cite{Goldberger:2016iau}.\n\n\\section{Examples}\n\\label{sec:LOradiation}\n\\def\\varepsilon{\\varepsilon}\n\nAt leading-order the amplitude appearing in the radiation kernel in equation~\\eqref{eqn:defOfRLO} is a five-point, tree amplitude (figure~\\ref{fig:5points}) that can be readily computed. In Yang--Mills theory,\n\\newcommand{\\treeLa}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [above right = .05 and .125 of v1] (g1);\n\t\\vertex [below right = 0.125 and 0.125 of g1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (g1);\n\t\\draw (g1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeLb}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [above left = .05 and .125 of v1] (g1);\n\t\\vertex [below left = 0.125 and 0.125 of g1] (g2);\n\t\\draw (i1) -- (g1);\n\t\\draw (g1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeLc}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [below right = 0.15 and 0.25 of v1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (v1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeYM}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [below = .15 of v1] (g1);\n\t\\vertex [right = 0.275 of g1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\[\n\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) &=\n\\sum_D \\mathcal{C}^a(D) \\bar{A}^{(0)}_D(p_1 + w_1, p_2 + w_2\\rightarrow p_1, p_2; k, h) \\\\\n&= \\Big[\\mathcal{C}^a\\!\\left(\\treeLa \\right)\\!A_{\\scalebox{0.5}{\\treeLa}} \n+ \\mathcal{C}^a\\!\\left(\\treeLb \\right)\\! A_{\\scalebox{0.5}{\\treeLb}} \n\\\\ & \\qquad\\qquad + \\mathcal{C}^a\\!\\left(\\treeLc \\right)\\! A_{\\scalebox{0.5}{\\treeLc}}\n+ (1\\leftrightarrow 2) \\Big]\n- i\\,\\mathcal{C}^a\\!\\left(\\treeYM \\right)\\! A_{\\scalebox{0.5}{\\treeYM}}\\,.\n\\]\nExplicitly, the colour factors are given by\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{C}^a\\!\\left(\\treeLa \\right) = (\\newT_1^a \\cdot \\newT_1^b) \\newT_2^b\\,, \\qquad\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) = (\\newT_1^b \\cdot \\newT_1^a) \\newT_2^b\\,,\\\\\n\\mathcal{C}^a\\!\\left(\\treeLc \\right) = \\frac12\\mathcal{C}^a\\!\\left(\\treeLa \\right) + \\frac12\\mathcal{C}^a\\!\\left(\\treeLb \\right), \\qquad\n\\mathcal{C}^a\\!\\left(\\treeYM \\right) = i\\hbar f^{abc} \\newT_1^b \\newT_2^c\\,,\\label{eqn:radColourFactors}\n\\end{gathered}\n\\end{equation}\nwith the replacement $1\\leftrightarrow2$ for diagrams with gluon emission from particle 2. Just as in the 4-point case at  1-loop, this is an overcomplete set for specifying a basis, because \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) = (\\newT_1^a\\cdot \\newT_1^b) \\newT_2^b +  i\\hbar f^{bac} \\newT_1^c \\newT_2^b = \\mathcal{C}^a\\!\\left(\\treeLa \\right) +  \\mathcal{C}^a\\!\\left(\\treeYM \\right).\\label{eqn:JacobiSetUp}\n\\end{aligned}\n\\end{equation}\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tikzpicture}[decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left=1 and 0.66 of v1] (i1) {$\\initialk_1+\\xfer_1$};\n\t\\vertex [above right=1 and 0.8 of v1] (o1) {$\\initialk_1$};\n\t\\vertex [right=1.2 of v1] (k) {$k$};\n\t\\vertex [below left=1 and 0.66 of v1] (i2) {$\\initialk_2+\\xfer_2$};\n\t\\vertex [below right=1 and 0.8 of v1] (o2) {$\\initialk_2$};\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o2);\n\t\\diagram*{(v1) -- [photon] (k)};\n\t\\filldraw [color=white] (v1) circle [radius=10pt];\n\t\\draw [pattern=north west lines, pattern color=patternBlue] (v1) circle [radius=10pt];\n\t\\end{feynman}\t\n\t\\end{tikzpicture} \n\t\\caption[The amplitude appearing in the leading-order radiation kernel.]{The amplitude $\\Ampl^{(0)}(\\initialk_1+\\xfer_1\\,,\\initialk_2+\\xfer_2\\rightarrow\n\t\t\\initialk_1\\,,\\initialk_2\\,,k)$ appearing in the radiation kernel at leading order.}\n\t\\label{fig:5points}\n\\end{figure}\nHence the full basis of colour factors is only 3 dimensional, and the colour decomposition of the 5-point tree is\n\\begin{multline}\n\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) = \\mathcal{C}^a\\!\\left(\\treeLa \\right)\\Big(A_{\\scalebox{0.5}{\\treeLa}} + A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}}\\Big) \\\\ \n+ \\frac12\\mathcal{C}^a\\!\\left(\\treeYM \\right)\\Big(-iA_{\\scalebox{0.5}{\\treeYM}} + 2 A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}}\\Big) + (1\\leftrightarrow2)\\,.\n\\end{multline}\nGiven that the second structure is $\\mathcal{O}(\\hbar)$, it would appear that we could again neglect the second term as a quantum correction. However, this intuition is not quite correct, as calculating the associated partial amplitude shows:\n\\begin{multline}\n-iA_{\\scalebox{0.5}{\\treeYM}} + 2  A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}} = -\\frac{4\\,\\varepsilon^{h}_\\mu(\\wn k)}{\\hbar^2} \\bigg[\\frac{2p_1\\cdot p_2}{{\\wn w_2^2\\, p_1\\cdot\\wn k}}\\, \\frac{p_1^\\mu}{\\hbar} + \\frac{1}{\\hbar\\, \\wn w_1^2 \\wn w_2^2}\\Big(2 p_2\\cdot\\wn k\\, p_1^\\mu \\\\- p_1\\cdot p_2 \\,(\\wn w_1^\\mu - \\wn w_2^\\mu) - 2p_1\\cdot\\wn k\\, p_2^\\mu\\Big) + \\mathcal{O}(\\hbar^0) \\bigg]\\,,\\label{eqn:radSingular}\n\\end{multline}\nwhere we have used $p_1\\cdot\\wn w_2 = p_1\\cdot\\wn k + \\hbar\\wn w_1^2\/2$ on the support of the on-shell delta functions in the radiation kernel~\\eqref{eqn:defOfRLO}. The partial amplitude appears to be singular, as there is an extra power of $\\hbar$ downstairs. However, this will cancel against the extra power in the colour structure, yielding a classical contribution. Meanwhile in the other partial amplitude the potentially singular terms cancel trivially, and the contribution is classical:\n\\begin{multline}\nA_{\\scalebox{0.5}{\\treeLa}} + A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}} = \\frac{2}{\\hbar^2} \\frac{\\varepsilon^{h}_\\mu(\\wn k)}{\\wn w_2^2\\, p_1\\cdot\\wn k}\\bigg[ 2p_1\\cdot p_2\\,\\wn w_2^\\mu +  \\frac{p_1\\cdot p_2}{p_1\\cdot\\wn k} p_1^\\mu(\\wn w_1^2 - \\wn w_2^2) \\\\ - 2p_1\\cdot\\wn k\\, p_2^\\mu + 2p_2\\cdot \\wn k\\, p_1^\\mu + \\mathcal{O}(\\hbar)\\bigg]\\,.\n\\end{multline}\nSumming all colour factors and partial amplitudes, the classical part of the 5-point amplitude is\n\\begin{align}\n&\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) = \\sum_D \\mathcal{C}^a(D) \\bar{A}^{(0)}_D(p_1 + w_1, p_2 + w_2\\rightarrow p_1, p_2; k, h) \\nonumber\\\\\n&= - \\frac{4\\varepsilon_\\mu^h(\\wn k)}{\\hbar^2} \\bigg\\{ \\frac{\\newT_1^a (\\newT_1\\cdot \\newT_2)}{\\wn w_2^2 \\, \\wn k \\cdot p_1} \\bigg[-(p_1\\cdot p_2)\\left(\\wn w_2^\\mu - \\frac{\\wn k\\cdot\\wn w_2}{\\wn k\\cdot p_1} p_1^\\mu\\right) + \\wn k\\cdot p_1 \\, p_2^\\mu  - \\wn k\\cdot p_2\\, p_1^\\mu\\bigg] \\nonumber \\\\ &\\qquad + \\frac{if^{abc}\\,\\newT_1^b \\newT_2^c}{\\wn w_1^2 \\wn w_2^2}\\bigg[2\\wn k\\cdot p_2\\, p_1^\\mu\n - p_1\\cdot p_2\\, \\wn w_1^\\mu + p_1\\cdot p_2 \\frac{\\wn w_1^2}{\\wn k\\cdot p_1}p_1^\\mu\\bigg] + (1\\leftrightarrow 2)\\bigg\\}\\,,\n\\end{align}\nwhere we have used that $\\wn w_1^2 - \\wn w_2^2 = -2\\wn k\\cdot \\wn w_2$ since the outgoing radiation is on-shell. Finally, we can substitute into the radiation kernel in \\eqn~\\eqref{eqn:defOfRLO} and take the classical limit. Averaging over the wavepackets sets $p_\\alpha = m_\\alpha u_\\alpha$ and replaces quantum colour charges with their classical counterparts, yielding\n\\[\n&\\mathcal{R}^{a,(0)}_\\text{YM}(\\wn k) = -\\int\\!\\dd^4\\,\\wn w_1 \\dd^4 \\wn w_2 \\, \\del^{(4)}(\\wn k - \\wn w_1 - \\wn w_2) \\del(u_1\\cdot\\wn w_1) \\del(u_2\\cdot\\wn w_2)\\, e^{ib\\cdot\\wn w_1} \\varepsilon_\\mu^{h}\\\\ \n&\\times \\bigg\\{ \\frac{c_1\\cdot c_2}{m_1} \\frac{c_1^a}{\\wn w_2^2 \\, \\wn k \\cdot u_1} \\left[-(u_1\\cdot u_2)\\left(\\wn w_2^\\mu - \\frac{\\wn k\\cdot\\wn w_2}{\\wn k\\cdot u_1} u_1^\\mu\\right) + \\wn k\\cdot u_1 \\, u_2^\\mu - \\wn k\\cdot u_2\\, u_1^\\mu\\right]\\\\\n& \\qquad + \\frac{if^{abc}\\,c_1^b c_2^c}{\\wn w_1^2 \\wn w_2^2}\\left[2\\wn k\\cdot u_2\\, u_1^\\mu - u_1\\cdot u_2\\, \\wn w_1^\\mu + u_1\\cdot u_2 \\frac{\\wn w_1^2}{\\wn k\\cdot u_1}\\,u_1^\\mu\\right] + (1\\leftrightarrow 2)\\bigg\\}\\,.\\label{eqn:LOradKernel}\n\\]\nOur result is equal to the leading order current $\\tilde{J}^{\\mu,(0)}_a$ obtained in \n\\cite{Goldberger:2016iau} by iteratively solving the Wong equations in \\eqn~\\eqref{eqn:Wong-momentum} and \\eqn~\\eqref{eqn:Wong-color} for timelike particle worldlines.\n\n\\subsection{Inelastic black hole scattering}\n\\label{sec:inelasticBHscatter}\n\nLet us turn to an independent application of our LO Yang--Mills radiation kernel~\\eqref{eqn:LOradKernel}. By returning to the colour-kinematics structure of the underlying amplitude, we can readily use the double copy to calculate results for gravitational wave emission from black hole scattering.\n\nTo apply the double copy we need the overcomplete set of colour factors in equation~\\eqref{eqn:radColourFactors}. This is because the object of fundamental interest is now the Jacobi identity that follows from~\\eqref{eqn:JacobiSetUp},\n\\begin{equation}\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) - \\mathcal{C}^a\\!\\left(\\treeLa \\right) = \\mathcal{C}^a\\!\\left(\\treeYM \\right),\n\\end{equation}\nwith an identical identity holding upon exchanging particles 1 and 2. Unlike our example in section~\\ref{sec:LOimpulse}, this is a non-trivial relation, and we must manipulate the numerators of each topology into a colour-kinematics dual form. This can be readily achieved by splitting the topologies with four-point seagull vertices, and adding their kinematic information to diagrams with the same colour structure. It is simple to verify that, in the classical limit, a basis of colour-kinematics dual numerators for this amplitude is\n\\begin{align}\n\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeLb}} &= 4(p_1\\cdot p_2)\\, p_1\\cdot\\varepsilon^{h}_k + 2\\hbar\\Big(p_1\\cdot \\wn k\\, (p_1 + p_2)^\\mu + p_1\\cdot p_2\\, (\\wn w_1 - \\wn w_2)^\\mu\\Big)\\cdot\\varepsilon^h_k + \\mathcal{O}(\\hbar^2)\\nonumber\\\\\n\\hspace{-7mm}\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeLa}} &= 4(p_1\\cdot p_2)\\, p_1\\cdot\\varepsilon^{h}_k + 2\\hbar\\Big(p_1\\cdot \\wn k \\, (p_1 - p_2)^\\mu + 2 p_2\\cdot\\wn k\\, p_1^\\mu\\Big)\\cdot\\varepsilon^h_k + \\mathcal{O}(\\hbar^2)\\label{eqn:vecnums}\\\\\n\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeYM}} &= 2\\hbar\\Big(2 p_1\\cdot \\wn k\\, p_2^\\mu - 2p_2\\cdot\\wn k\\, p_1^\\mu + p_1\\cdot p_2\\, (\\wn w_1 - \\wn w_2)^\\mu\\Big)\\cdot\\varepsilon^h_k +\\mathcal{O}(\\hbar^2)\\,, \\nonumber\n\\end{align}\nwhere $\\varepsilon^h_k \\equiv \\varepsilon^h_\\mu(\\wn k)$. It is crucial that we keep $\\mathcal{O}(\\hbar)$ terms, as we know from equation~\\eqref{eqn:radSingular} that when the YM amplitude is not written on a minimal basis of colour factors there are terms which are apparently singular in the classical limit. The factors of $\\sqrt{2}$ are to account for the proper normalisation of colour factors involved in the double copy --- see discussion around equation~\\eqref{eqn:scalarYMamp}.\n\nWith a set of colour-kinematics dual numerators at hand, we can now double copy by replacing colour factors with these numerators, leading to\n\\begin{multline}\n\\hbar^{\\frac{7}{2}}  {\\mathcal{M}}^{(0)}_\\textrm{JNW}(\\wn k) = \\left(\\frac{\\kappa}{2}\\right)^3 \\bigg[ \\frac{1}{\\hbar\\,\\wn w_2^2} \\bigg(\\frac{n^\\mu_{\\scalebox{0.5}{\\treeLa}} n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{n^\\mu_{\\scalebox{0.5}{\\treeLb}}n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar \\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2}n^\\mu_{\\scalebox{0.5}{\\treeYM}} n^\\nu_{\\scalebox{0.5}{\\treeYM}} \\bigg] e_{\\mu\\nu}^h(\\wn k)\\,,\\label{eqn:JNWamplitude}\n\\end{multline}\nwhere we have used the outer product of the polarisation vectors (of momentum $\\wn k$) from the numerators,\n\\begin{equation}\ne_{\\mu\\nu}^{h} = \\frac12\\left(\\varepsilon^{h}_\\mu \\varepsilon^{h}_\\nu + \\varepsilon^{h}_\\nu \\varepsilon^{h}_\\mu - P_{\\mu\\nu}\\right) + \\frac12\\left(\\varepsilon^{h}_\\mu \\varepsilon^{h}_\\nu - \\varepsilon^{h}_\\nu \\varepsilon^{h}_\\mu\\right) + \\frac12 P_{\\mu\\nu}\\,.\\label{eqn:polarisationOuterProd}\n\\end{equation}\nHere $P_{\\mu\\nu} = \\eta_{\\mu\\nu} - \\left(\\wn k_\\mu \\wn r_\\nu + \\wn k_\\nu \\wn r_\\mu\\right)\/(\\wn k\\cdot \\wn r)$ is a transverse projector with reference momentum $\\wn r$. Since the amplitude is symmetric in its numerators, we can immediately restrict attention to the initial symmetric and traceless piece --- the polarisation tensor for a graviton. It now simply remains to Laurent expand in $\\hbar$ to retrieve the parts of the amplitude which contribute to the classical radiation kernel for gravitational radiation. \n\nRather than doing so at this point, it is more pertinent to note that our result is not yet an amplitude in Einstein gravity. As in our 4-point discussion in the previous chapter, this result is polluted by dilaton interactions. In particular, the corresponding current is for the scattering of JNW naked singularities in Einstein--dilaton gravity \\cite{Goldberger:2016iau}. A convenient way to remove the dilaton states is to use a scalar ghost \\cite{Johansson:2014zca}, as shown explicitly for the classical limit of this amplitude in ref.~\\cite{Luna:2017dtq}. We introduce a new massles, adjoint-representation scalar $\\chi$, minimally coupled to the YM gauge field, and use the double copy of its amplitude to remove the pollution. The ghost couples to the scalar fields in our action~\\eqref{eqn:scalarAction} via the interaction term\n\\begin{equation}\n\\mathcal{L}_{\\chi \\textrm{int}} = -2g \\sum_{\\alpha=1,2} \\Phi_\\alpha^\\dagger \\chi \\Phi_\\alpha\\,.\\label{eqn:adjointScalar}\n\\end{equation}\nOn the same 5-point kinematics as our previous YM amplitude, the equivalent numerators for the topologies in~\\eqref{eqn:vecnums} with interactions mediated by the new massless scalar are\n\\[\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeLb}} &= -4 (p_1 - \\hbar\\wn w_2) \\cdot\\varepsilon^h_k \\\\\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeLa}} &= -4 p_1 \\cdot\\varepsilon^h_k \\\\\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeYM}} &= -2\\hbar (\\wn w_1 - \\wn w_2)\\cdot\\varepsilon^h_k\\,. \\label{eqn:scalarnums}\n\\]\nSince $\\chi$ is in the adjoint representation, the appropriate colour factors are again those in equation~\\eqref{eqn:radColourFactors} (there is now no seagull topology). The numerators hence trivially satisfy colour-kinematics duality. We can therefore double copy to yield the ghost amplitude,\n\\begin{multline}\n\\hbar^{\\frac72} {\\mathcal{M}}^{(0)}_\\textrm{ghost}(\\wn k) = \\left(\\frac{\\kappa}{2}\\right)^3 \\bigg[\\frac1{\\hbar\\,\\wn w_2^2}\\bigg(\\frac{\\tilde n^\\mu_{\\scalebox{0.5}{\\treeLa}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{\\tilde n^\\mu_{\\scalebox{0.5}{\\treeLb}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar\\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2} \\tilde n^\\mu_{\\scalebox{0.5}{\\treeYM}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeYM}}\\bigg] e^h_{\\mu\\nu}(\\wn k)\\,,\\label{eqn:ghostAmp}\n\\end{multline}\nThis is an amplitude for a ghost in the sense that we can now write \\cite{Luna:2017dtq}\n\\begin{equation}\n\\mathcal{M}^{(0)}_\\textrm{Schwz}(\\wn k) = \\mathcal{M}^{(0)}_\\textrm{JNW} - \\frac1{D-2}\\mathcal{M}^{(0)}_\\textrm{ghost}\\,,\n\\end{equation}\nwhere the appearance of the spacetime dimension comes from matching the ghost to the dilaton propagator. Expanding the numerators, in $D=4$ one finds that\n\\begin{multline}\n\\hbar^2 \\bar{\\mathcal{M}}^{(0)}_\\textrm{Schwz}(\\wn k) =   \\bigg[\\frac{{P}_{12}^\\mu {P}_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} + \\frac{p_1\\cdot p_2}{2\\wn w_1^2 \\wn w_2^2}\\left({Q}^\\mu_{12} {P}_{12}^\\nu + {Q}_{12}^\\nu {P}_{12}^\\mu\\right) \\\\ + \\frac14\\left((p_1\\cdot p_2)^2 - \\frac12\\right)\\left(\\frac{{Q}^\\mu_{12} {Q}^\\nu_{12}}{\\wn w_1^2 \\wn w_2^2} - \\frac{{P}^\\mu_{12} {P}^\\nu_{12}}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2}\\right)\\bigg]e_{\\mu\\nu}^{h} + \\mathcal{O}(\\hbar)\\,,\\label{eqn:ampSchwarzschild}\n\\end{multline}\nwhere\n\\begin{subequations}\n\t\\begin{gather}\n\tP_{12}^\\mu = (\\wn k\\cdot p_1)\\, p_2^\\mu - (\\wn k\\cdot p_2)\\, p_1^\\mu\\,,\\label{eqn:gaugeinvariant1}\\\\\n\tQ_{12}^\\mu = (\\wn w_1 - \\wn w_2)^\\mu - \\frac{\\wn w_1^2}{\\wn k\\cdot p_1} p_1^\\mu + \\frac{\\wn w_2^2}{\\wn k\\cdot p_2} p_2^\\mu\\,,\\label{eqn:gaugeinvariant2}\n\t\\end{gather}\\label{eqn:gaugeInvariants}\n\\end{subequations}\nare two gauge invariant functions of the kinematics. Substituting the amplitude into the LO radiation kernel~\\eqref{eqn:defOfRLO} yields the LO current for the scattering of two Schwarzschild black holes. Note that whereas we took the classical limit before double copying, this result was first obtained in ref.~\\cite{Luna:2017dtq} by only taking the classical limit (via a large mass expansion) once the gravity amplitudes were at hand.\n\n\\subsubsection{Reissner--Nordstr\\\"{o}m black holes}\n\nIn our previous calculation, the adjoint massless scalar introduced in equation~\\eqref{eqn:adjointScalar} merely acted as a useful computational trick to remove internal dilaton pollution. However, if we consider other black hole species it can be promoted to a far more fundamental role. For example, let us consider gravitational radiation emitted by the scattering of two Reissner--Nordstr\\\"{o}m black holes. RN black holes are solutions to Einstein--Maxwell theory rather than vacuum general relativity, and thus have non-zero electric (or magnetic) charges $Q_\\alpha$. At leading order their gravitational interactions are the same as for Schwarzschild black holes, but the total current due to gravitational radiation will be different, precisely because of terms sourced from electromagnetic interactions, mediated by a massless vector field.\n\nThis is in contrast to our previous example, where we ``squared'' the vector numerators to obtain tensor interactions, in the sense that $A^\\mu \\otimes A^\\mu \\sim H_{\\mu\\nu}$. To obtain electromagnetic interactions in the gravity amplitude guaranteed by the double copy we need to use numerators in a vector and scalar representation respectively, such that we have $A^\\mu \\otimes \\phi \\sim \\tilde{A}^\\mu$. This is exactly what the sets of numerators in equations~\\eqref{eqn:vecnums} and~\\eqref{eqn:scalarnums} provide.\n\nThe double copy does not require that one square numerators, merely that colour data is replaced with numerators satisfying the same Jacobi identities. Thus the double copy construction\n\\begin{multline}\n\\hbar^{\\frac72}  {\\mathcal{M}}^{(0)}(\\wn k) = \\frac{\\kappa}{2} e^2 Q_1 Q_2\\bigg[ \\frac{1}{\\hbar\\,\\wn w_2^2} \\bigg(\\frac{n^\\mu_{\\scalebox{0.5}{\\treeLa}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{n^\\mu_{\\scalebox{0.5}{\\treeLb}}\\tilde n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar \\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2}n^\\mu_{\\scalebox{0.5}{\\treeYM}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeYM}} \\bigg] e_{\\mu\\nu}^h\\label{eqn:RNamp}\n\\end{multline}\nis guaranteed to be a well-defined gravity amplitude. Note that we have altered the coupling replacement in the double copy appropriately.  \n\nA consequence of using kinematic numerators from alternative sets is that the amplitude is asymmetric in its Lorentz indices; specifically, for the scalar diagrams there are no seagull vertex terms, while the scalar boson triple vertex term is manifestly different to the pure vector case. Since the graviton polarisation tensor is symmetric and traceless, the graviton amplitude is obtained by symmetrising over the Lorentz indices in~\\eqref{eqn:RNamp}. Substituting the result into equation~\\eqref{eqn:defOfRLO} yields the LO gravitational radiation kernel due to electromagnetic interactions,\n\\begin{multline}\n\\mathcal{R}_{\\rm RN,grav}^{(0)} = \\frac{e^2\\kappa}{4} Q_1 Q_2 \\!\\int\\! \\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(p_1\\cdot\\wn w_1) \\del(p_2\\cdot\\wn w_2) \\del^{(4)}(\\wn w_1 + \\wn w_2 - \\wn k) e^{ib\\cdot\\wn w_1} \\\\\\times \\bigg[\\frac{{Q}_{12}^\\mu {P}_{12}^\\nu + {Q}_{12}^\\nu {P}_{12}^\\mu}{\\wn w_1^2 \\wn w_2^2} + (p_1\\cdot p_2) \\bigg(\\frac{{Q}_{12}^\\mu {Q}_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} - \\frac{{P}_{12}^\\mu {P}_{12}^\\nu}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2} \\bigg)\\bigg] e_{(\\mu\\nu)}^{h}\\,.\\label{eqn:RNradKernel}\n\\end{multline}\nWe verify this result in appendix~\\ref{app:worldlines}, by calculating the corresponding classical energy-momentum tensor from perturbative solutions to the field equations of Einstein--Maxwell theory.\n\nTo obtain an amplitude for graviton emission we symmetrised the result from the double copy. This rather conveniently restricted to a graviton amplitude. However, because we used double copy numerators from different theories there is also a non-zero contribution from the antisymmetric part of the polarisation tensor, $e_{[\\mu\\nu]}^{h}$. For gravity, this corresponds to the axion mode $B_{\\mu\\nu}$, and thus $\\mathcal{M}^{[\\mu\\nu]}$ should correspond to axion emission. It is a simple matter to show that\n\\begin{equation}\n\\hbar^{\\frac52} \\mathcal{M}^{(0)}_{\\rm RN, axion} = -\\frac{e^2\\kappa}{\\wn w_1^2 \\wn w_2^2} Q_1 Q_2 \\left({Q}^{\\nu}_{12} {P}^\\mu_{12} - {Q}^\\mu_{12} {P}^\\nu_{12}\\right) e_{[\\mu\\nu]}^{h} + \\mathcal{O}(\\hbar)\\,.\\label{eqn:axion}\n\\end{equation}\nAs well as being antisymmetric, for an on-shell state with momentum $\\wn k^\\mu$ the axion polarisation tensor must satisfy $e^h_{[\\mu\\nu]}(\\wn k) \\wn k^\\mu = 0$ . Thus it has the explicit form $e^h_{[\\mu\\nu]}(\\wn k) = \\epsilon_{\\mu\\nu\\rho\\sigma} \\wn k^\\rho \\xi^\\sigma\/(\\wn k \\cdot \\xi)$, where $\\xi^\\sigma$ is an unspecified reference vector. We will find it convenient to take $\\xi^\\sigma = p_1^\\sigma + p_2^\\sigma$. Expanding the amplitude and then including this choice we have\n\\[\n\\hbar^{\\frac72} \\bar{\\mathcal{M}}^{(0)}_{\\rm RN,axion} &= -\\frac{2e^2\\kappa}{\\wn w_1^2 \\wn w_2^2} Q_1 Q_2 \\Big[\\left((\\wn k\\cdot p_1)\\, p_2^\\mu  - (\\wn k\\cdot p_2)\\, p_1^\\mu\\right)(\\wn w_1 - \\wn w_2)^\\nu \\\\ & \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad - \\left(\\wn w_1^2 - \\wn w_2^2\\right) p_2^\\mu p_1^\\nu \\Big] \\epsilon_{\\mu\\nu\\rho\\sigma} \\frac{\\wn k^\\rho \\xi^\\sigma}{\\wn k \\cdot \\xi}\\\\\n&= \\frac{4 e^2\\kappa}{\\wn w_1^2 \\wn w_2^2}Q_1 Q_2\\, \\epsilon_{\\mu\\nu\\rho\\sigma} p_1^\\mu p_2^\\nu \\wn w_1^\\rho \\wn w_2^\\sigma\\,,\n\\]\nleading to a LO axion radiation kernel\n\\begin{multline}\n\\mathcal{R}_{\\rm RN,axion}^{(0)} = e^2\\kappa Q_1 Q_2 \\! \\int\\! \\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(p_1\\cdot\\wn w_1) \\del(p_2\\cdot\\wn w_2) \\del^{(4)}(\\wn w_1 + \\wn w_2 - \\wn k) \\\\ \\times \\frac{e^{ib\\cdot\\wn w_1}}{\\wn w_1^2 \\wn w_2^2} \\epsilon_{\\mu\\nu\\rho\\sigma} u_1^\\mu u_2^\\nu \\wn w_1^\\rho \\wn w_2^\\sigma\\,.\n\\end{multline}\nThe electromagnetic scattering is only able to radiate axions due to their coupling with the vector bosons. It is not possible to couple axions to a scalar particle in the absence of spin, as to do so breaks diffeomorphism invariance and axion gauge symmetry \\cite{Goldberger:2017ogt}. The amplitude of \\eqn~(\\ref{eqn:axion}) is therefore purely due to the antisymmetrisation of the triple gauge vertex; given that we have an isolated single vertex responsible, we can verify that the radiation is indeed axionic by using Einstein--Maxwell axion-dilaton coupled gravity. The action is\n\\begin{equation}\n\\hspace{-3mm} S=\\frac1{2\\kappa}\\int\\! \\d^4x \\,{\\sqrt{g}}\\left[R - \\frac{1}{2}(\\partial_\\mu\\phi_\\textrm{d})^2 - \\frac{1}{2}e^{4\\phi_\\textrm{d}}(\\partial_\\mu\\zeta)^2 -  e^{-2\\phi_\\textrm{d}}F^2\\! - \\zeta F^{\\mu\\nu}{F}^*_{\\mu\\nu}\\right].\n\\end{equation}\nHere ${F}^*_{\\mu\\nu}$ is the dual electromagnetic field strength, $\\phi_\\textrm{d}$ is the dilaton field, $\\zeta$ is the axion pseudoscalar defined in \\eqn~\\eqref{eqn:axionscalar}, and $g=-\\det(g_{\\mu\\nu})$. Treating the axion-photon interaction in the final term perturbatively and integrating by parts gives\n\\begin{equation}\n\\mathcal{L}_{\\rm int} \\sim A_\\lambda\\partial_{[\\alpha}A_{\\beta]}H^{\\lambda\\alpha\\beta}\\,.\n\\end{equation}\nAllocating momenta to the axion-photon vertex in the same way as in the scattering amplitudes, this interaction term corresponds to the Feynman rule\n\\hspace{-0.5cm}\n\\begin{tabular}[h!]{cccc}\n\t\\begin{minipage}[c]{0.03\\textwidth}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{0.18\\textwidth}\n\t\t\\centering\n\t\t\\scalebox{0.8}{\n\t\t\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={zigzag}]\n\t\t\\tikzset{zigzag\/.style={decorate, decoration=zigzag}}\n\t\t\\begin{feynman}\n\t\t\\vertex (v1);\n\t\t\\vertex [above = 1.44 of v1] (v2) {$\\mu \\nu$};\n\t\t\\vertex [below right = 1 and 1.5 of v1] (v3);\n\t\t\\vertex [below left = 1 and 1.5 of v1] (v4);\n\t\t\\vertex [left = 0.06 of v1] (a1) ;\n\t\t\\vertex [right = 0.06 of v1] (a2) ;\n\t\t\\vertex [above = 1.44 of a1] (a3) ;\n\t\t\\vertex [above = 1.44 of a2] (a4) ;\n\t\t\\vertex [right = 0.2 of v1] (a5) ;\n\t\t\\vertex [above = 1.44 of a5] (a6) ;\n\t\t\\vertex [below right = 1 and 1.5 of v1] (v33) {$\\sigma$};\n\t\t\\vertex [below left = 1 and 1.5 of v1] (v44) {$\\rho$};\n\t\t\\draw[zigzag] (a1) -- (a3);\n\t\t\\draw[zigzag] (a2) -- (a4);\n\t\t\\diagram*{(a5) -- [white, scalar, momentum' = {[arrow style=black,thick]\\(k\\)}] (a6)};\n\t\t\\diagram*{(v3) -- [photon, momentum = \\(q_2\\)] (v1)};\n\t\t\\diagram*{(v4) -- [photon, momentum = \\(q_1\\)] (v1)};\n\t\t\\filldraw [color=black] (v1) circle [radius=2pt];\n\t\t\\end{feynman}\n\t\t\\end{tikzpicture}}\n\t\\end{minipage}&\n\t\\hspace{-0.7cm}\n\t\\begin{minipage}[c]{0.8\\textwidth}\n\t\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t=-2i\\kappa\\,\\big\\{(q_2-q_1)_\\mu  k_{[\\rho} \\eta_{\\sigma]\\nu} -& (q_2-q_1)_\\nu k_{[\\rho}\\eta_{\\sigma]\\mu} \\\\ &+ \\eta_{\\mu[\\rho}\\eta_{\\sigma]\\nu} \\left[q_1\\cdot k - q_2\\cdot k\\right]\\big\\}.\n\t\t\\end{aligned}\n\t\t\\end{equation}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{0.03\\textwidth}\n\t\\end{minipage}\n\t\\vspace{6pt}\n\\end{tabular}\n\\noindent Constructing a five-point amplitude for massive two external scalars with this axion emission vertex and taking the classical limit then precisely reproduces the results of \\eqn~\\eqref{eqn:axion}, verifying that the antisymmetric double copy artefact is indeed axionic.\n\n\\subsection{Momentum conservation and radiation reaction}\n\\label{sec:ALD}\n\nLet us return to gauge theory, and in particular the YM radiation kernel in equation~\\eqref{eqn:LOradKernel}. An immediate corollary of this result is the LO radiation in classical electrodynamics; replacing colour with electric charges and ignoring the structure constant terms yields\n\\begin{equation}\n\\begin{aligned}\n\\RadKerCl_\\text{EM}(\\bar k)&=\\int \\! \\dd^4 \\xferb_1 \\dd^4 \\xferb_2 \\;\n\\del(\\ucl_1\\cdot\\xferb_1) \\del(\\ucl_2\\cdot\\xferb_2) \n\\del^{(4)}(\\bar k - \\xferb_1 - \\xferb_2) \\, e^{i\\xferb_1 \\cdot b} \n\\\\& \\hphantom{\\rightarrow}\\times\\biggl\\{\n\\frac{1}{m_1} \\frac{Q_1^2Q_2^{\\vphantom{2}}}{\\xferb_2^2} \n\\biggl[-\\ucl_2\\cdot\\varepsilon^h_k + \\frac{(\\ucl_1\\cdot \\ucl_2)(\\xferb_2\\cdot\\varepsilon^h_k)}{\\ucl_1\\cdot\\bar k} \n+ \\frac{(\\ucl_2\\cdot\\bar k)(\\ucl_1\\cdot\\varepsilon^h_k)}{\\ucl_1\\cdot\\bar k} \n\\\\&\\hspace*{40mm} \n- \\frac{(\\bar k\\cdot\\xferb_2)(\\ucl_1\\cdot \\ucl_2)(\\ucl_1\\cdot\\varepsilon^h_k)}{(\\ucl_1\\cdot\\bar k)^2}\\bigg] \n+ (1 \\leftrightarrow 2)\\biggr\\} \\,.\n\\label{eqn:Rcalculation}\n\\end{aligned}\n\\end{equation}\nIt is a simple calculation to see that the LO current which solves the Maxwell field equation has precisely the same expression, up to an overall sign~\\cite{Kosower:2018adc}.\n\nNow, we have already seen that conservation of momentum holds exactly (in \\sect{sect:allOrderConservation}) and in our classical expressions (in \\sect{sect:classicalConservation}). Let us ensure that there is no subtlety in these discussions by explicit calculation.\n\nTo do so, we calculate the part of the NNLO impulse $\\ImpBclsup{(\\textrm{rad})}$ which encodes radiation reaction, defined in \\eqn~\\eqref{eqn:nnloImpulse}. The two amplitudes appearing in equation~\\eqref{eqn:nnloImpulse} are in common with the amplitudes relevant for the radiated momentum, equation~\\eqref{eqn:defOfRLO}, though they are evaluated at slightly different kinematics. It will be convenient to change the sign of $\\xferb_\\alpha$ here; with that change, the amplitudes are\n\\begin{multline}\n\\AmplB^{(0)}(\\initialk_1, \\initialk_2 \\rightarrow \\initialk_1-\\hbar\\xferb_1\\,, \\initialk_2-\\hbar\\xferb_2 \\, , \\bar k) = \\frac{4Q_1^2 Q_2^{\\vphantom{2}}}{\\hbar^{2}\\, \\xferb_2^2} \n\\bigg[-p_2\\td\\varepsilon^h_k + \\frac{(p_1\\td p_2)(\\xferb_2\\td\\varepsilon^h_k)}{p_1\\cdot\\bar{k}} \n\\\\+ \\frac{(p_2\\cdot\\bar k)(p_1\\td\\varepsilon^h_k)}{p_1\\td\\bar k} \n- \\frac{(\\bar k\\td\\xferb_2)(p_1\\td p_2)(p_1\\td\\varepsilon^h_k)}{(p_1\\cdot\\bar k)^2}\\bigg] + ( 1 \\leftrightarrow 2)\n\\end{multline}\nand\n\\begin{multline}\n\\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb_1\\,, \\initialk_2 + \\hbar \\qb_2 \\rightarrow \n\\initialk_1-\\hbar\\xferb_1\\,, \\initialk_2-\\hbar\\xferb_2 \\,, \\bar k)\n= \\frac{4Q_1^2 Q_2^{\\vphantom{2}}}{\\hbar^{2}\\, \\xferb_2'^2} \n\\bigg[\\tm p_2\\td\\varepsilon^{h*}_k\n\\\\\\tp \\frac{(p_1\\td p_2)(\\xferb'_2\\td\\varepsilon^{h*}_k)}{p_1\\cdot\\bar k} \n\\tp \\frac{(p_2\\td\\bar k)(p_1\\td\\varepsilon_k^{h*})}{p_1\\cdot\\bar k} \n\\tm \\frac{(\\bar k\\td\\xferb'_2)(p_1\\td p_2)(p_1\\td\\varepsilon^{h*}_k)}{(p_1\\cdot\\bar k)^2}\\bigg] \n + ( 1 \\leftrightarrow 2)\\,,\n\\end{multline}\nwhere we find it convenient to define $\\xferb_\\alpha' = \\qb_\\alpha + \\xferb_\\alpha$ (after the change of sign). \n\nWe can now write the impulse contribution as\n\\begin{multline}\n\\ImpBclsup{(\\textrm{rad})} = -e^6 \\Lexp \\int \\! d\\Phi(\\bar k) \n\\prod_{\\alpha = 1,2} \\int \\dd^4\\xferb_\\alpha\\, \\dd^4 \\xferb'_\\alpha \\; \\xferb_1^\\mu \\; \n\\\\ \\times \\mathcal{X}^*(\\xferb'_1, \\xferb'_2, \\bar k) \\mathcal{X}(\\xferb_1, \\xferb_2, \\bar k)\n\\Rexp\\,  ,\n\\label{eqn:impulseNNLO}\n\\end{multline}\nwhere \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{X}(\\xferb_1, \\xferb_2, \\bar k) &= {4}  \\, \\del(2 \\xferb_1\\cdot p_1)\n\\del(2 \\xferb_2\\cdot p_2) \\del^{(4)}(\\bar k - \\xferb_1 - \\xferb_2) \n\\, e^{i b \\cdot  \\xferb_1}\n\\\\ \n& \\quad \\times \n\\biggl\\{Q_1^2 Q_2^{\\vphantom{2}} \\frac{\\varepsilon_{\\mu}^h(\\wn k) }{\\xferb_2^2}\n\\bigg[-p_2^\\mu + \\frac{p_1\\cdot p_2 \\, \\xferb_2^\\mu}{p_1\\cdot\\bar k} \n+ \\frac{p_2\\cdot\\bar k \\, p_1^\\mu}{p_1\\cdot\\bar k} \n\\\\&\\hspace*{35mm}\n- \\frac{(\\bar k\\cdot\\xferb_2)(p_1\\cdot p_2) \\, p_1^\\mu}{(p_1\\cdot\\bar k)^2}\\bigg] \n+ (1 \\leftrightarrow 2)\\biggr\\}\\,.\n\\end{aligned}\n\\label{eqn:X1}  \n\\end{equation}\nThis expression is directly comparable to those for radiated momentum: \\eqn~\\eqref{eqn:impulseNNLO}, and the equivalent impulse contribution to particle 2, balance the radiated momentum \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO} using $\\xferb_1^\\mu + \\xferb_2^\\mu = \\bar k^\\mu$, provided that the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation}, is related to integrals over $\\mathcal{X}$. Indeed this relationship holds: the integrations present in the radiation kernel are supplied by the $\\xferb_\\alpha$ and $\\xferb'_\\alpha$ integrals in \\eqn~\\eqref{eqn:impulseNNLO}; these integrations disentangle in the sum of impulses on particles 1 and 2 when we impose $\\xferb_1^\\mu + \\xferb_2^\\mu = \\bar k^\\mu$, and then form the square of the radiation kernel.\n\nIt is interesting to compare this radiated momentum with the situation in traditional formulations of classical physics, where one must include the ALD radiation reaction force \nby hand in order to enforce momentum conservation. Because the situation is simplest when only one particle is dynamical, let us take the mass $m_2$ to be very large compared to \n$m_1$ in the remainder of this section, and work in particle 2's rest frame.  In this frame, it does not radiate, and the only radiation reaction is on particle 1 --- the radiated momentum is precisely balanced by  the impulse on particle 1 due to the ALD force. We can therefore continue our discussion with reference to our expression for radiated momentum, \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO}, and the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation}. In this situation we may also simplify the kernels by dropping the $(1 \\leftrightarrow 2)$ instruction: these terms will be dressed by an inverse power of $m_2$, and so are subdominant when $m_2 \\gg m_1$. \n\nWe will soon compute the impulse due to the ALD force directly from its classical expression in \\eqn~\\eqref{eqn:ALDclass}. But in preparation for that comparison there is one step which we must take. Classical expressions for the force---which involve only the particle's momentum and its derivatives---do not involve any photon phase space. So we must perform the integration over $\\df(\\wn k)$ which is present in \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO}. \n\nTo organise the calculation, we integrate over the $\\qb_1$ variables in the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation} using the four-fold delta function, so that we may write the radiated momentum as\n\\begin{align}\n\\hspace*{-3mm}\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{m_1^2}\\! \\int\\!\\dd^4\\qb\\, \\dd^4\\qb'\\;\ne^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb')) \n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2} \\phInt\\,,\\label{eqn:rrmidstage}\n\\end{align}\nwhere we renamed the remaining variables, $\\xferb_2\\rightarrow \\qb$ and $\\xferb'_2\\rightarrow \\qb'$, in order to match the notation used later. After some algebra we find\n\\begin{multline}\n\\phInt = \\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu \\,\n\\left[ 1 + \\frac{(\\ucl_1\\cdot \\ucl_2)^2(\\qb\\cdot \\qb')}{\\wn E^2} \n+ \\frac{(\\ucl_2\\cdot\\bar k)^2}{\\wn E^2}  \\right. \\\\ \n\\left. - \\frac{(\\ucl_1\\cdot \\ucl_2)(\\ucl_2\\cdot\\bar k)\\,\\bar k\\cdot(\\qb+\\qb')}{\\wn E^3} + \\frac{(\\ucl_1\\cdot \\ucl_2)^2(\\bar k\\cdot\\qb)(\\bar k\\cdot\\qb')}{\\wn E^4}  \\right].\n\\label{eqn:phaseSpaceIntegral}\n\\end{multline}\nThe quantity $\\wn E$ is defined to be $\\wn E = \\ucl_1 \\cdot \\wn k$; in view of the delta function, the integral is constrained so that $\\wn E = \\ucl_1 \\cdot \\qb$. This quantity is the wavenumber of the photon in the rest frame of particle 1, and is fixed from the point of view of the phase space integration. As a result, the integrals are simple: there are two delta functions (one explicit, one in the phase space measure) which can be used to perform the $\\bar k^0$ integration and to fix the magnitude of the spatial wavevector. The remaining integrals are over angles. The relevant results were calculated in appendix~C of ref.~\\cite{Kosower:2018adc}, and are\n\\begin{equation}\n\\begin{gathered}\n\\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu = \\frac{\\wn E^2}{2\\pi} u_1^\\mu \\Theta(\\wn E)\\,,\\\\\n\\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu \\bar k^\\nu \\bar k^\\rho = \\frac{\\wn E^4}{\\pi}\\left(u_1^\\mu u_1^\\nu u_1^\\rho - \\frac12 u_1^{(\\mu} \\eta^{\\nu\\rho)}\\right) \\Theta(\\wn E)\\,.\n\\end{gathered}\n\\end{equation}\nThe radiated momentum then takes a remarkably simple form after the phase space integration:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{3\\pi m_1^2} &\\int\\!\\dd^4\\qb \\,\n\\dd^4\\qb'\\; e^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb'))\n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2}\n\\\\ &\\times\\Theta(\\ucl_1\\cdot\\qb)\\,\\left[(\\ucl_1\\cdot\\qb)^2 \n+ \\qb\\cdot\\qb'(\\ucl_1\\cdot \\ucl_2)^2\\right] \\ucl_1^\\mu \\,.\n\\label{eqn:radTheta}\n\\end{aligned}\n\\end{equation}\n\nThe $\\Theta$ function is a remnant of the photon phase space volume, so it will be convenient to remove it. The delta functions in the integrand in \\eqn~\\eqref{eqn:radTheta} constrain the components of the vectors $\\qb$ and $\\qb'$ which lie in the two dimensional space spanned by $u_1$ and $u_2$. Let us call the components of $q$ and $q'$ in this plane to be $q_\\parallel$ and $q'_\\parallel$. Then the delta functions set $q_\\parallel = q'_\\parallel$. As a result, the integrand (ignoring the $\\Theta$ function) is symmetric in $q_\\parallel \\rightarrow - q_\\parallel$. Consequently we may symmetrise to find\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{6\\pi m_1^2} &\\int\\!\\dd^4\\qb\\, \\dd^4\\qb'\\;\ne^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb')) \n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2} \n\\\\ &\\hspace*{10mm}\\times\\left[(\\ucl_1\\cdot\\qb)^2 \n+ \\qb\\cdot\\qb'(\\ucl_1\\cdot \\ucl_2)^2\\right]\\ucl_1^\\mu \\,.\n\\label{eqn:rrResult}\n\\end{aligned}\n\\end{equation}\n\nIt is now remarkably simple to see that this expression is equal but opposite to the impulse obtained from the classical ALD force in \\eqn~\\eqref{eqn:ALDclass}. Working in perturbation theory, the lowest order contribution to $dp_1 \/ d\\tau$ is of order $e^2$, due to the (colour-stripped) LO Lorentz force~\\eqref{eqn:Wong-momentum}. We can determine this explicitly using the methods of appendix~\\ref{app:worldlines}: with particle 2 kept static, one finds\n\\begin{equation}\n\\frac{d p^{\\mu,(0)}_1}{d \\tau_1} =i e^2 Q_1 Q_2 \\int \\!\\dd^4 \\qb \\, \\del(\\qb \\cdot u_2) \\, e^{- i \\qb \\cdot (b + u_1 \\tau_1)} \\,  \\frac{\\qb^\\mu \\, u_1 \\cdot u_2 - u_2^\\mu \\, \\qb \\cdot u_1}{\\qb^2}\\,.\n\\label{eqn:LOforce}\n\\end{equation}\nTherefore $\\Delta {p^\\mu_1}_{\\rm ALD}$ is at least of order $e^4$. However, this potential contribution to the ALD impulse vanishes. To see this, observe that the acceleration due to the LO Lorentz force gives rise to an ALD impulse of\n\\begin{equation}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^4 Q_1^3 Q_2}{6\\pi m_1}\\int\\!\\dd^4\\qb\\, \\del(\\qb\\cdot u_1) \\del(\\qb\\cdot u_2) \\, e^{-i\\wn{q}\\cdot b} \\, \\qb\\cdot u_1 \\, \\big(\\cdots\\big) = 0\\,.\n\\end{equation}\nAn alternative point of view on the same result is to perform the time integral in equation~\\eqref{eqn:ALDclass}, noting that the second term in the ALD force is higher order. The impulse is then proportional to $f^\\mu(+\\infty) - f^\\mu(-\\infty)$, the difference in the asymptotic Lorentz forces on particle 1. But at asymptotically large times the two particles are infinitely far away, so the Lorentz forces must vanish. Since this second argument does not rely on perturbation theory we may ignore the first term in the ALD force law.\n\nThus, the first non-vanishing impulse due to radiation reaction is of order $e^6$. Since we only need the leading order Lorentz force to evaluate the ALD impulse, we can anticipate that the result will be very simple. Indeed, integrating the ALD force, we find that the impulse on particle 1 due to radiation reaction is\n\\begin{multline}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^6 Q_1^4 Q_2^2}{6\\pi m_1^2} u_1^\\mu \\!\\int\\! \\dd^4\\qb\\,\\dd^4\\qb'\\, \\del(\\qb\\cdot u_2) \\del(\\qb'\\cdot u_2)  \\del(u_1\\cdot(\\qb-\\qb')) \\, e^{-ib\\cdot(\\qb-\\qb')} \\\\ \\times\\frac{1}{\\qb^2 \\qb'^2} \\left[(\\qb\\cdot u_1)^2 + \\qb\\cdot\\qb'(u_1\\cdot u_2)^2 \\right].\n\\label{eqn:classicalRadiationImpulse}\n\\end{multline}\nThis is precisely the expression~\\eqref{eqn:rrResult} we found using our quantum mechanical approach.\n\n\n\n\\section{Discussion}\n\\label{sec:KMOCdiscussion}\n\nIn order to apply on-shell scattering amplitudes to the calculation of classically observable quantities for black holes, one needs a definition of the observables in the quantum theory.  One also needs a path and clear set of rules for taking the classical limit of the quantum observables.  In this first part of the thesis we have constructed one such path. Our underlying motivation is to understand the dynamics of classical general relativity through the double copy. In particular, we are interested in the relativistic two-body problem which is so central to the physics of the compact binary coalescence events observed by LIGO and Virgo. Consequently, we focused on observables in two-body events.\n\nWe have shown how to construct two observables relevant to this problem: the momentum transfer or impulse~(\\ref{eqn:defl1}) on a particle; and the momentum emitted as radiation~(\\ref{eqn:radiationTform}) during the scattering of two charged but spinless point particles.  We have shown how to restore $\\hbar$'s and classify momenta in \\sect{sec:RestoringHBar}; in \\sect{sec:PointParticleLimit}, how to choose suitable wavefunctions for localised single particle states; and established in section~\\ref{sec:classicalLimit} the conditions under which the classical limit is simple for point-particle scattering. With these formalities at hand we were able to further provide simplified leading and next-to-leading-order expressions in terms of on-shell scattering amplitudes for the impulse in \\eqns{eqn:impulseGeneralTerm1classicalLO}{eqn:classicalLimitNLO}, and for the radiated momentum in \\eqn~\\eqref{eqn:radiatedMomentumClassical}.  These expressions apply directly to both gauge theory and gravity. In sections~\\ref{sec:examples} and \\ref{sec:LOradiation}, we used explicit expressions for amplitudes in QED, Yang--Mills theory and perturbative Einstein gravity to obtain classical results. We have been careful throughout to ensure that our methods correctly incorporate conservation of momentum, without the need to introduce an analogue of the Abraham--Lorentz--Dirac radiation reaction.\n\nOther momentum observables should be readily accessible by similar derivations: for example the total radiated angular momentum is of particular current interest \\cite{Damour:2020tta}, and is moreover accessible from worldline QFT \\cite{Jakobsen:2021smu}; it would be very interesting to understand how this observable fits into our formalism. Higher-order corrections, to the extent they are unambiguously defined in the classical theory, require the harder work of computing two- and higher-loop amplitudes, but the formalism of these chapters will continue to apply.\n\nOur setup has features in common with two related, but somewhat separate, areas of current interest. One area is the study of the potential between two massive bodies. The second is the study of particle scattering in the eikonal. Diagrammatically, the study of the potential is evidently closely related to the impulse of chapter~\\ref{chap:impulse}. To some extent this is by design: we wished to construct an on-shell observable related to the potential. But we have also been able to construct an additional observable, the radiated momentum, which is related to the gravitational flux.\n\nIt is interesting that classical physics emerges in the study of the high-energy limit of quantum scattering~\\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe}, see also refs.~\\cite{Damour:2016gwp,Damour:2017zjx}. \nIndeed the classical centre-of-momentum scattering angle can be obtained from the eikonal function (see, for example ref.~\\cite{DAppollonio:2010krb}). This latter function must therefore be related as well to the impulse, even though we have not taken any high-energy limit. Indeed, the impulse and the scattering angle are equivalent at LO and NLO, \nbecause no momentum is radiated at these orders. Therefore the scattering angle completely determines the change in momentum of the particles (and vice versa). The connection to the eikonal function should be interesting to explore.\n\nAt NNLO, on the other hand, the equivalence between the angle and the impulse fails. This is because of radiation: knowledge of the angle tells you where the particles went,\nbut not how fast. In this respect the impulse is more informative than the angle. Eikonal methods are still applicable in the radiative case~\\cite{Amati:1990xe},\nso they should reproduce the high-energy limit of the expectation value of the radiated momentum. Meanwhile at low energies, methods based on soft theorems could provide a bridge between the impulse and the radiated momentum~\\cite{Laddha:2018rle,Laddha:2018myi,Sahoo:2018lxl}. Indeed, a first step in these directions was recently made in \\cite{A:2020lub}. Radiation reaction physics can also be treated in this regime \\cite{DiVecchia:2021ndb}, and we look forward to future progress in understanding how these references overlap with our formalism.\n\nThe NLO scattering angle is, in fact, somewhat simpler than the impulse: see ref.~\\cite{Luna:2016idw} for example. Thanks to the exponentiation at play in the eikonal limit, it is the triangle diagram which is responsible for the NLO correction. But the impulse contains additional contributions, as we discussed in \\sect{sec:nloQimpulse}. Perhaps this is because the impulse must satisfy an on-shell constraint, unlike the angle.\n\nThe focus of our study of inelastic scattering has been the radiation kernel introduced in~\\eqref{eqn:defOfR}. Equivalent to a classical current, this object has proven to be especially versatile in the application of amplitudes methods to gravitational radiation. It has played a direct role in studies of the Braginsky--Thorne memory effect \\cite{Bautista:2019tdr} and gravitational shock waves \\cite{Cristofoli:2020hnk}; derivations of the connections between amplitudes, soft limits and classical soft theorems \\cite{A:2020lub}; and calculations of Newman--Penrose spinors for long-range radiation in split-signature spacetimes \\cite{Monteiro:2020plf}. The kernel is also closely related to progress in worldline QFT \\cite{Mogull:2020sak,Jakobsen:2021smu}. However, the reader may object that we have not in fact calculated the total emitted radiation. This was recently achieved in ref.~\\cite{Herrmann:2021lqe}, using integral evaluation techniques honed in $\\mathcal{N}=8$ supergravity \\cite{Parra-Martinez:2020dzs}. To calculate the full observable the authors of \\cite{Herrmann:2021lqe} took a similar approach to our treatment of the non-linear impulse contributions at 1-loop, treating the product of radiation kernels as a cut of a 2-loop amplitude. Their application of the formalism presented here has led to state-of-the-art results for post--Minkowskian bremsstrahlung, already partially recovered from PM effective theory \\cite{Mougiakakos:2021ckm}. \n\nIn these two chapters we restricted attention to spinless scattering. In this context, for colourless particles such as astrophysical black holes the impulse (or equivalently, the angle) is the only physical observable at LO and NLO, and completely determines the interaction Hamiltonian between the two particles~\\cite{Damour:2016gwp,Damour:2017zjx}. The situation is richer in the case of arbitrarily aligned spins --- then the change in spins of the particles is an observable which is not determined by the scattering angle. Fully specifying the dynamics of black holes therefore requires including spin in our formalism: this is the topic of the second part of the thesis.\n\\part{Spinning black holes}\n\\label{part:spin}\n\n\\chapter{Observables for spinning particles}\n\\label{chap:spin}\n\n\\section{Introduction}\n\nTo begin this part of the thesis we will continue in the vein of previous chapters and use quantum field theory, now for particles with non-zero spin, to calculate observables for spinning point-particles. Our focus will be the leading-order scattering of black holes, however the formalism is applicable more widely \\cite{Maybee:2019jus}. In chapter~\\ref{chap:intro} we discussed at some length how scattering amplitudes have been applied to the dynamics of spinning Kerr black holes, and more fundamentally how minimally coupled amplitudes behave as the on-shell avatar of the no-hair theorem. Here we remove the restriction to the aligned-spin configuration in the final results of \\cite{Guevara:2018wpp,Bautista:2019tdr}, and the restriction to the non-relativistic limit in the final results of \\cite{Chung:2018kqs}.  We use on-shell amplitudes to directly compute relativistic classical observables for generic spinning-particle scattering, reproducing such results for black holes obtained by classical methods in \\cite{Vines:2017hyw}, thereby providing more complete evidence for the correspondence between minimal coupling to gravity and classical black holes.\n\nWe will accomplish this by relaxing the restriction to scalars in previous chapters. In addition to the momentum impulse $\\Delta p^\\mu$, there is now another relevant on-shell observable, the change $\\Delta s^\\mu$ in the spin (pseudo-)vector $s^\\mu$, which we will call the \\textit{angular impulse}.  We introduce this quantity in \\sect{sec:GRspin}, where we also review classical results from \\cite{Vines:2017hyw} for binary black hole scattering at 1PM order. \nIn \\sect{sec:QFTspin} we consider the quantum analogue of the spin vector, the Pauli--Lubanski operator; manipulations of this operator allow us to write expressions for the angular impulse akin to those for the impulse of chapter~\\ref{chap:impulse}. Obtaining the classical limit requires some care, which we discuss before constructing example gravity amplitudes in \\sect{sec:amplitudes} from the double copy. \n\nRather than working at this stage with massive spinor representations valid for any quantum spin $s$, in the vein of \\cite{Arkani-Hamed:2017jhn}, we will ground our intuition in explicit field representations of the Poincar\\'{e} group --- we will work with familiar spin 1\/2 Dirac fermions and massive, spin 1 bosons. Explicit representations can indeed be chosen for any generic spin $s$ field; the applications of these representations to black hole physics was studied in ref.~\\cite{Bern:2020buy}. However, for higher spins the details become extremely involved. In \\sect{sec:KerrCalcs} we show that substituting familiar low spin examples into our general formalism exactly reproduces the leading terms of all-multipole order expressions for the impulse and angular impulse of spinning black holes \\cite{Vines:2017hyw}. Finally, we discuss how our results further entwine Kerr black holes and scattering amplitudes in \\sect{sec:angImpDiscussion}.\n\nThis chapter is based on work conducted in collaboration with Donal O'Connell and Justin Vines, published in \\cite{Maybee:2019jus}.\n\n\\section{Spin and scattering observables in classical gravity}\\label{sec:GRspin}\n\nBefore setting up our formalism for computing the angular impulse, let us briefly review aspects of this observable in relativistic classical physics. \n\n\\subsection{Linear and angular momenta in asymptotic Minkowski space}\n\nTo describe the incoming and outgoing states for a weak scattering process in asymptotically flat spacetime we can use special relativistic physics, working as in Minkowski spacetime. There, any isolated body has a constant linear momentum vector $p^\\mu$ and an antisymmetric tensor field $J^{\\mu\\nu}(x)$ giving its total angular momentum about the point $x$, with the $x$-dependence determined by $J^{\\mu\\nu}(x')=J^{\\mu\\nu}(x)+2p^{[\\mu}(x'-x)^{\\nu]}$, or equivalently $\\nabla_\\lambda J^{\\mu\\nu}=2p^{[\\mu}\\delta^{\\nu]}{}_\\lambda$.\n\nRelativistically, centre of mass (CoM) position and intrinsic and orbital angular momenta are frame-dependent concepts, but a natural inertial frame is provided by the direction of the momentum $p^\\mu$, giving the proper rest frame.  We define the body's proper CoM worldline to be the set of points $r$ such that $J^{\\mu\\nu}(r)p_\\nu=0$, i.e.\\ the proper rest-frame mass-dipole vector about $r$ vanishes, and we can then write\n\\begin{equation}\\label{eqn:Jmunu}\nJ^{\\mu\\nu}(x)=2p^{[\\mu}(x-r)^{\\nu]}+S^{\\mu\\nu},\n\\end{equation}\nwhere $r$ can be any point on the proper CoM worldline, and where $S^{\\mu\\nu}=J^{\\mu\\nu}(r)$ is the intrinsic spin tensor, satisfying\n\\begin{equation}\nS^{\\mu\\nu} p_\\nu=0.\\label{eqn:SSC}\n\\end{equation}\nEquation \\eqref{eqn:SSC} is often called the ``covariant'' or Tulczyjew--Dixon spin supplementary condition (SSC) \\cite{Fokker:1929,Tulczyjew:1959} in its (direct) generalization to curved spacetime in the context of the Mathisson--Papapetrou--Dixon equations \\cite{Mathisson:1937zz,Mathisson:2010,Papapetrou:1951pa,Dixon1979,Dixon:2015vxa} for the motion of spinning extended test bodies.\nGiven the condition \\eqref{eqn:SSC}, the complete information of the spin tensor $S^{\\mu\\nu}$ is encoded in the momentum $p^\\mu$ and the spin pseudovector \\cite{Weinberg:1972kfs},\n\\begin{equation}\ns_\\mu = \\frac{1}{2m}\\epsilon_{\\mu\\nu\\rho\\sigma} p^\\nu S^{\\rho\\sigma} = \\frac{1}{2m}\\epsilon_{\\mu\\nu\\rho\\sigma} p^\\nu J^{\\rho\\sigma}(x),\\label{eqn:GRspinVec}\n\\end{equation}\nwhere $\\epsilon_{0123} = +1$ and $p^2=m^2$.  Note that $s\\cdot p=0$; $s^\\mu$ is a spatial vector in the proper rest frame.\nGiven \\eqref{eqn:SSC}, the inversion of the first equality of \\eqref{eqn:GRspinVec} is\n\\begin{equation}\nS_{\\mu\\nu} =\\frac{1}{m} \\epsilon_{\\mu\\nu\\lambda\\tau} p^\\lambda s^\\tau.\\label{eqn:SSCintrinsicSpin}\n\\end{equation}\nThe total angular momentum tensor $J^{\\mu\\nu}(x)$ can be reconstructed from $p^\\mu$, $s^\\mu$, and a point $r$ on the proper CoM worldline, via \\eqref{eqn:SSCintrinsicSpin} and \\eqref{eqn:Jmunu}.\n\n\n\n\n\\subsection{Scattering of spinning black holes in linearised gravity}\n\nFollowing the no-hair property emphasised by equation~\\eqref{eqn:multipoles} of chapter~\\ref{chap:intro}, the full tower of gravitational multipole moments of a spinning black hole, and thus also its (linearised) gravitational field, are uniquely determined by its monopole $p^\\mu$ and dipole $J^{\\mu\\nu}$.  This is reflected in the scattering of two spinning black holes, in that the net changes in the holes' linear and angular momenta depend only on their incoming linear and angular momenta.  It has been argued in \\cite{Vines:2017hyw} that the following results concerning two-spinning-black-hole scattering, in the  1PM approximation to GR, follow from the linearised Einstein equation and a minimal effective action description of spinning black hole motion, the form of which is uniquely fixed at 1PM order by general covariance and appropriate matching to the Kerr solution.\n\n\n\n\nConsider two black holes with incoming momenta $p_1^\\mu=m_1 u_1^\\mu$ and $p_2^\\mu=m_2 u_2^\\mu$, defining the 4-velocities $u^\\mu=p^\\mu\/m$ with $u^2=1$, and incoming spin vectors $s_1^\\mu=m_1 a_1^\\mu$ and $s_2^\\mu=m_2 a_2^\\mu$, defining the rescaled spins $a^\\mu=s^\\mu\/m$ (with units of length, whose magnitudes measure the radii of the ring singularities).  Say the holes' zeroth-order incoming proper CoM worldlines are orthogonally separated at closest approach by a vectorial impact parameter $b^\\mu$, pointing from 2 to 1, with $b\\cdot u_1 =b\\cdot u_2=0$.  Then, according to the analysis of \\cite{Vines:2017hyw}, the net changes in the momentum and spin vectors of black hole 1 are given by\n\\begin{alignat}{3}\n\\begin{aligned}\n\\Delta p_1^{\\mu} &= \\textrm{Re}\\{\\mathcal Z^\\mu\\}+O(G^2),\n\\\\\n\\Delta s_1^{\\mu} &= - u_1^\\mu a_1^\\nu\\, \\textrm{Re}\\{\\mathcal Z_\\nu\\} - \\epsilon^{\\mu\\nu\\alpha\\beta} u_{1\\alpha} a_{1\\beta}\\, \\textrm{Im}\\{\\mathcal Z_\\nu\\}+O(G^2),\n\\end{aligned}\\label{eqn:KerrDeflections}\n\\end{alignat}\nwhere\n\\begin{equation}\n\\mathcal Z_\\mu = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}}\\Big[(2\\gamma^2 - 1)\\eta_{\\mu\\nu} - 2i\\gamma \\epsilon_{\\mu\\nu\\alpha\\beta} u_1^\\alpha u_2^\\beta\\Big]\\frac{ b^\\nu + i\\Pi^\\nu{ }_\\rho (a_1+a_2)^\\rho}{[b + i\\Pi(a_1+a_2)]^2}\\,,\n\\end{equation}\nwith $\\gamma = u_1\\cdot u_2$ the relative Lorentz factor, and with\n\\begin{equation}\n\\begin{aligned}\n\\Pi^\\mu{ }_\\nu &= \\epsilon^{\\mu\\rho\\alpha\\beta} \\epsilon_{\\nu\\rho\\gamma\\delta} \\frac{{u_1}_\\alpha {u_2}_\\beta u_1^\\gamma u_2^\\delta}{\\gamma^2 - 1}\\\\ &= \\delta^\\mu{ }_\\nu +\\frac1{\\gamma^2 - 1}\\bigg(u_1^\\mu({u_1}_\\nu - \\gamma {u_2}_\\nu) + u_2^\\mu({u_2}_\\nu - \\gamma {u_1}_\\nu)\\bigg) \\label{eqn:projector}\n\\end{aligned}\n\\end{equation}\nthe projector into the plane orthogonal to both incoming velocities.\nThe analogous results for black hole 2 are given by interchanging the identities $1\\leftrightarrow 2$.\n\nIf we take black hole 2 to have zero spin, $a_2^\\mu\\to0$, and if we expand to quadratic order in the spin of black hole 1, corresponding to the quadrupole level in 1's multipole expansion, then we obtain the results shown in \\eqref{eqn:JustinImpResult} and \\eqref{eqn:JustinSpinResult} below.  In the remainder of this chapter, developing necessary tools along the way, we show how those results can be obtained from classical limits of scattering amplitudes. In particular, we will consider one-graviton exchange between a massive scalar particle and a massive spin $s$ particle, with minimal coupling to gravity, with $s=1\/2$ to yield the dipole level, and with $s=1$ to yield the quadrupole level.\n\n\\section{Spin and scattering observables in quantum field theory}\n\\label{sec:QFTspin}\n\nWe have already established general formulae in quantum field theory for the impulse and radiated momentum; as the angular impulse is also on-shell similar methods should be applicable. A first task is to understand what quantum mechanical quantity corresponds to the classical spin pseudovector of equation~\\eqref{eqn:GRspinVec}. This spin vector is a quantity associated with a single classical body, and we therefore momentarily return to discussing single-particle states. \n\nParticle states of spin $s$ are irreducible representations of the little group. For massive particles in 4 dimensions the little group is isomorphic to $SU(2)$, and thus we can adopt the simplest coherent states considered in section~\\ref{sec:classicalSingleParticleColour}. The size of the representation is now determined by the spin quantum number $s$ associated with the states. For fractional spins the normalisation in \\eqn~\\eqref{eqn:ladderCommutator} of course generalises to an anticommutation relation.\n\nFor spinning particles we will thus adopt the wavepackets in equation~\\eqref{eqn:InitialStateSimple}, with the important distinction that the representation states now refer to the little group, not a gauge group. To make this distinction clear we will denote the little group states by $|\\xi\\rangle$ rather than $|\\chi\\rangle$ (not to be confused with the parameter~\\eqref{eqn:defOfXi}). The momentum space wavefunctions $\\varphi(p)$ remain entirely unchanged.\n\n\\subsection{The Pauli--Lubanski spin pseudovector}\n\nWhat operator in quantum field theory is related to the classical spin pseudovector of equation~\\eqref{eqn:GRspinVec}? We propose that the correct quantum-mechanical\ninterpretation is that the spin is nothing but the expectation value of the \\textit{Pauli--Lubanski} operator,\n\\begin{equation}\n\\W_\\mu = \\frac{1}{2}\\epsilon_{\\mu\\nu\\rho\\sigma} \\P^\\nu \\J^{\\rho\\sigma}\\,,\\label{eqn:PLvec}\n\\end{equation}\nwhere $\\P^\\mu$ and $\\mathbb{J}^{\\rho\\sigma}$ are translation and Lorentz generators respectively. In particular,\nour claim is that the expectation value\n\\begin{equation}\n\\langle s^\\mu \\rangle = \\frac1{m} \\langle \\mathbb{W}^\\mu\\rangle = \\frac1{2m} \\epsilon^{\\mu\\nu\\rho\\sigma} \\langle \\mathbb{P}_\\nu \\mathbb{J}_{\\rho\\sigma}\\rangle\n\\end{equation}\nof the Pauli--Lubanski operator on a single particle state~\\eqref{eqn:InitialStateSimple} is the quantum-mechanical generalisation of the classical spin pseudovector. Indeed, a simple comparison of equations~\\eqref{eqn:GRspinVec} and~\\eqref{eqn:PLvec} indicates a connection between the two quantities. We will provide abundant evidence for this link in the remainder of this chapter --- it is shown in greater detail in appendix~B of ref.~\\cite{Maybee:2019jus}.\n\nThe Pauli--Lubanski operator is a basic quantity in the classification of free particle states, although it receives less attention in introductory accounts of quantum field theory than it should. With the help of the Lorentz algebra\n\\[\n[\\J^{\\mu\\nu}, \\P^\\rho] &= i \\hbar (\\eta^{\\mu\\rho} \\P^\\nu - \\eta^{\\nu\\rho} \\P^\\mu) \\,, \\\\\n[\\J^{\\mu\\nu}, \\J^{\\rho\\sigma}] &= i \\hbar (\\eta^{\\nu\\rho} \\J^{\\mu\\sigma} - \\eta^{\\mu\\rho} \\J^{\\nu\\sigma} - \\eta^{\\nu\\sigma} \\J^{\\mu\\rho} + \\eta^{\\mu\\rho} \\J^{\\mu\\sigma}) \\, ,\n\\]\nit is easy to establish the important fact that the Pauli--Lubanski operator commutes with the momentum:\n\\[\n[\\P^\\mu, \\W^\\nu] = 0\\,.\\label{eqn:PWcommute}\n\\] \nFurthermore, as $\\W^\\mu$ is a vector operator it satisfies\n\\[\n[\\J^{\\mu\\nu}, \\W^\\rho] = i\\hbar (\\eta^{\\mu\\rho} \\W^\\nu - \\eta^{\\nu\\rho} \\W^\\mu) \\,.\n\\]\nIt then follows that the commutation relations of $\\W$ with itself are\n\\[\n[\\W^\\mu, \\W^\\nu] = i\\hbar \\epsilon^{\\mu\\nu\\rho\\sigma} \\W_\\rho \\P_\\sigma\\,.\n\\]\nOn single particle states this last commutation relation takes a particularly instructive form. Working in the rest frame of our massive particle state, evidently $W^0 = 0$. The remaining generators satisfy\\footnote{We normalise $\\epsilon^{123} = +1$, as usual.}\n\\[\n[ \\W^i, \\W^j] = i \\hbar m \\,\\epsilon^{ijk} \\W^k \\,,\n\\]\nso that the Pauli--Lubanski operators are nothing but the generators of the little group. Not only is this the basis for their importance, but also we will find that these commutation relations are directly useful in our computation of the change in a particle's spin during scattering.\n\nBecause the $\\mathbb{W}^\\mu$ commutes with the momentum, we have\n\\begin{equation}\n\\langle p'\\, j| \\W^\\mu |p\\, i \\rangle \\propto \\del_\\Phi(p-p')\\,.\n\\end{equation}\nWe define the matrix elements of $\\W$ on the states of a given momentum to be\n\\[\n\\langle p'\\, j| \\W^\\mu |p\\, i \\rangle \\equiv m \\s^\\mu_{ij}(p)\\, \\del_\\Phi(p-p') \\,,\\label{eqn:PLinnerProd}\n\\]\nso that the expectation value of the spin vector over a single particle wavepacket is\n\\[\n\\langle \\s^\\mu \\rangle = \\sum_{i,j} \\int d\\Phi(p) \\, | \\varphi(p) |^2 \\, \\xi^*_i \\s^\\mu_{ij} \\xi_j\\,.\n\\]\nThe matrix $\\s^\\mu_{ij}(p)$, sometimes called the spin polarisation vector, will be important below. These matrices inherit the commutation relations of the Pauli--Lubanski vector, so that in particular\n\\[\n[\\s^\\mu(p), \\s^\\nu(p) ] = \\frac{i\\hbar}{m} \\, \\epsilon^{\\mu\\nu\\rho\\sigma} \\s_\\rho(p) p_\\sigma \\,.\n\\]\n\nSpecialising now to a particle in a given representation, we may derive well known \\cite{Ross:2007zza,Holstein:2008sx,Bjerrum-Bohr:2013bxa,Guevara:2017csg} explicit expressions for the spin polarisation $s^\\mu_{ij}(p)$ by starting with the Noether current associated with angular momentum. Such derivations for the simple spin 1\/2 and 1 cases were given in appendix B of~\\cite{Maybee:2019jus} --- for a Dirac spin $1\/2$ particle, the spin polarisation is\n\\[\ns^\\mu_{ab}(p) = \\frac{\\hbar}{4m} \\bar{u}_a(p) \\gamma^\\mu \\gamma^5 u_b(p)\\,.\\label{eqn:spinorSpinVec}\n\\]\nMeanwhile, for massive vector bosons we have\n\\begin{equation}\ns_{ij}^\\mu(p) = -\\frac{i\\hbar}{m} \\epsilon^{\\mu\\nu\\rho\\sigma} p_\\nu \\varepsilon{^*_i}_\\rho(p) {\\varepsilon_j}_\\sigma(p)\\,. \\label{eqn:vectorSpinVec}\n\\end{equation}\nWe have these normalised quantities to be consistent with the algebraic properties of the Pauli--Lubanski operator.\n\n\\subsection{The change in spin during scattering}\n\nNow that we have a quantum-mechanical understanding of the spin vector, we move on to discuss the dynamics of the spin vector in a scattering process. Following the set-up of chapter~\\ref{chap:impulse} we consider the scattering of two stable, massive particles which are quanta of different fields, and are separated by an impact parameter $b^\\mu$. We will explicitly consider scattering processes mediated by vector bosons and gravitons. The relevant incoming two-particle state is therefore that in equation~\\eqref{eqn:inState}, but with little group states $\\xi_\\alpha$.\n\nThe initial spin vector of particle 1 is\n\\[\n\\langle s_1^\\mu \\rangle = \\frac1{m_1} \\langle \\Psi |\\W^\\mu_1 |\\Psi \\rangle\\,,\n\\]\nwhere $\\W^\\mu_1$ is the Pauli--Lubanski operator of the field corresponding to particle 1. Since the $S$ matrix is the time evolution operator from the far past to the far future, the final spin vector of particle 1 is\n\\[\n\\langle s_1'^\\mu \\rangle = \\frac1{m_1} \\langle \\Psi | S^\\dagger \\W^\\mu_1 S| \\Psi \\rangle\\,.\n\\]\nWe define the angular impulse on particle 1 as the difference between these quantities:\n\\begin{equation}\n\\langle \\Delta s_1^\\mu \\rangle = \\frac1{m_1}\\langle\\Psi|S^\\dagger \\mathbb{W}_1^\\mu S|\\Psi\\rangle - \\frac1{m_1}\\langle\\Psi|\\mathbb{W}_1^\\mu|\\Psi\\rangle\\,.\\label{eqn:defOfAngImp}\n\\end{equation}\nWriting $S = 1 + iT$ and making use of the optical theorem yields\n\\begin{equation}\n\\langle \\Delta s_1^\\mu\\rangle = \\frac{i}{m_1}\\langle\\Psi|[\\mathbb{W}_1^\\mu,{T}]|\\Psi\\rangle + \\frac{1}{m_1}\\langle\\Psi|{T}^\\dagger[\\mathbb{W}_1^\\mu,{T}]|\\Psi\\rangle\\,.\\label{eqn:spinShift}\n\\end{equation}\nJust as with equation~\\eqref{eqn:defl1}, it is clear that the second of these terms will lead to twice as many powers of the coupling constant for a given interaction. Therefore only the first term is able to contribute at leading order. We will be exclusively considering tree level scattering $\\mathcal{A}^{(0)}$, so the first term is the sole focus of our attention\\footnote{The expansion of the second term is very similar to that of the colour impulse in ref.~\\cite{delaCruz:2020bbn}.}.\n\nOur goal now is to express the leading-order angular impulse in terms of amplitudes. To that end we substitute the incoming state in equation~\\eqref{eqn:inState} into the first term of \\eqn~\\eqref{eqn:spinShift}, and the leading-order angular impulse is given by\n\\begin{multline}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = \\frac{i}{m_1}\\prod_{\\alpha=1,2} \\int\\! \\df(p_\\alpha') \\df(p_\\alpha) \\,\\varphi_\\alpha^*(p_\\alpha') \\varphi_\\alpha(p_\\alpha) e^{ib\\cdot(p_1-p'_1)\/\\hbar} \\\\ \\times \\left\\langle p_1'\\,\\xi'_1 ; p_2'\\, \\xi'_2\\left|\\W_1^\\mu\\, {T} -{T}\\, \\W_1^\\mu \\right|p_1 \\, \\xi_1; p_2\\, \\xi_2 \\right\\rangle.\n\\end{multline}\nScattering amplitudes can now be explicitly introduced by inserting a complete set of states between the spin and interaction operators, as in equation~\\eqref{eqn:p1Expectation}. In their first appearance this yields\n\\begin{multline}\n\\int\\! \\df(r_1) \\df(r_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\langle p'_1\\, \\xi'_1; p'_2\\, \\xi'_2|\\W_1^\\mu|r_1\\, \\zeta_1; r_2, \\zeta_2\\rangle \\langle r_1\\, \\zeta_1; r_2\\, \\zeta_2|T|p_1\\, \\zeta_1 ; p_2\\, \\zeta_2\\rangle \\\\\n =  m_1 \\langle \\xi'_1\\, \\xi'_2 |{\\s}^\\mu_1(p'_1)\\mathcal{A}(p_1, p_2 \\rightarrow p'_1, p'_2)|\\xi_1\\,\\xi_2\\rangle\\,  \\del^{(4)}(p'_1 + p'_2 - p_1 - p_2)\\,,\n\\end{multline}\nwhere, along with the definition of the scattering amplitude, we have used the definition of the spin polarisation vector~\\eqref{eqn:spinorSpinVec}. The result for the other ordering of $T$ and $\\W^\\mu_1$ is very similar. \n\nAn essential point is that under the little group state inner product above, the spin polarisation vector and amplitude do not commute: they are both matrices in the little group representation, and we have simply suppressed the explicit indices. This novel feature of the angular impulse will become extremely important. Substituting into the full expression for $\\langle\\Delta s_1^{\\mu,(0)}\\rangle$ and integrating over the delta functions, we find that the observable is\n\\begin{equation}\n\\begin{aligned}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = i \\prod_{\\alpha = 1, 2} &\\int \\!\\df(p_\\alpha') \\df(p_\\alpha) \\,\\varphi_\\alpha^*(p_\\alpha') \\varphi_\\alpha (p_\\alpha) \\del^{(4)}(p_1' + p_2' - p_1 - p_2)  \\\\ & \\times  e^{ib\\cdot(p_1-p'_1)\/\\hbar} \\langle\\xi'_1\\, \\xi'_2| \\s_{1}^\\mu(p_1') \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p'_1, p'_2) \\\\ &\\hspace{30mm} - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p'_1, p'_2) \\s_{1}^{\\mu}(p_1)|\\xi_1\\, \\xi_2\\rangle\\,.\n\\end{aligned}\n\\end{equation}\nWe now eliminate the delta function by introducing the familiar momentum mismatches $q_\\alpha = p'_\\alpha - p_\\alpha$ and performing an integral. The leading-order angular impulse becomes\n\\begin{equation}\n\\begin{aligned}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = i & \\int\\! \\df(p_1) \\df(p_2)\\, \\dd^4q\\,\\del(2p_1\\cdot q + q^2) \\del(2p_2\\cdot q - q^2) \\Theta(p_1^0 + q^0) \\\\ \\times& \\Theta(p_2^0 - q^0) \\varphi_1^*(p_1 + q) \\varphi_2^*(p_2 - q) \\varphi_1(p_1) \\varphi_2(p_2) e^{-ib\\cdot q\/\\hbar} \\\\\\times& \\langle \\xi'_1\\, \\xi'_2|\\s_{1}^{\\mu}(p_1 + q) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s^{\\mu}(p_1)|\\xi_1\\, \\xi_2\\rangle\\,.\\label{eqn:exactAngImp}\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Passing to the classical limit}\n\\label{sec:classicalLim}\n\nThe previous expression is an exact, quantum formula for the leading-order change in the spin vector during conservative two-body scattering. As a well-defined observable, we can extract the classical limit of the angular impulse by following the formalism introduced in part~\\ref{part:observables}.\n\nRecall that the basic idea is simple: the momentum space wavefunctions must localise the particles, without leading to a large uncertainty in their momenta. They therefore have a finite but small width $\\Delta x = \\ell_w$ in position space, and $\\Delta p = \\hbar\/\\ell_w$ in momentum space. This narrow width restricts the range of the integral over $q$ in equation~\\eqref{eqn:exactAngImp} so that $q \\lesssim \\hbar \/\\ell_w$. We therefore introduce the wavenumber $\\qb = q\/\\hbar$. We further found that our explicit choice of wavefunctions $\\varphi_\\alpha$ were very sharply peaked in momentum space around the value $\\langle p_\\alpha^\\mu \\rangle = m_\\alpha u_\\alpha^\\mu$, where $u_\\alpha^\\mu$ is a classical proper velocity. We neglect the small shift $q = \\hbar \\qb$ in the wavefunctions present in equation~\\eqref{eqn:exactAngImp}, and also the term $q^2$ compared to the dominant $2 p \\cdot q$ in the delta functions, arriving at\n\\[\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle =&\\, i \\!  \\int \\! \\df(p_1) \\df(p_2)\\, \\dd^4q\\,\\del(2p_1\\cdot q ) \\del(2p_2\\cdot q) |\\varphi_1(p_1)|^2 |\\varphi_2(p_2)|^2 \\\\&\\times e^{-ib\\cdot q\/\\hbar}\\, \\bigg\\langle\\s_{1}^{\\mu}(p_1 + q) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s_{1}^{\\mu}(p_1)\\bigg\\rangle   \\,.\n\\]\nNotice we have also dropped the distinction between the little group states, simply writing an expectation value as in~\\eqref{eqn:amplitudeDef}. This is permissible since we established coherent states suitable for the classical limit of $SU(2)$ states in section~\\ref{sec:classicalSingleParticleColour}. Adopting the notation for the large angle brackets from \\eqn~\\eqref{eqn:angleBrackets}, the angular impulse takes the form\n\\[\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle =  &\\, \\Lexp i\\! \\int \\!\\dd^4 q\\, \\del (2p_1 \\cdot q) \\del (2p_2\\cdot q) e^{-ib\\cdot q\/\\hbar}  \\\\ & \\times\\bigg(\\s^{\\mu}(p_1 +  \\hbar\\qb) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s_{1}^{\\mu}(p_1)\\bigg)\\Rexp\\,,\n\\label{eqn:intermediate}\n\\]\n\nAn important $\\hbar$ shift remaining is that of the spin polarisation vector $\\s_{1}^{\\mu}(p_1 + \\hbar\\wn q)$. This object is a Lorentz boost of $\\s_{1}^{\\mu}(p_1)$. In the classical limit $q$ is small, so the Lorentz boost $\\Lambda^{\\mu}{ }_\\nu p_1^\\nu = p_1^\\mu + \\hbar\\wn q^\\mu$ is infinitesimal. In the vector representation an infinitesimal Lorentz transformation is $\\Lambda^\\mu\\,_\\nu=\\delta^\\mu_\\nu + w^\\mu{ }_\\nu$, so for our boosted momenta $w^\\mu{ }_\\nu p_1^\\nu = \\hbar\\wn q^\\mu$. The appropriate generator is\n\\begin{equation}\nw^{\\mu\\nu} = -\\frac{\\hbar}{m_1^2}\\left(p_1^\\mu \\wn q^\\nu - \\wn q^\\mu p_1^\\nu\\right)\\,.\\label{eqn:LorentzParameters}\n\\end{equation} \nThis result is valid for particles of any spin as it is purely kinematic, and therefore can be universally applied in our general formula for the angular impulse. In particular, since $w_{\\mu\\nu}$ is explicitly $\\mathcal{O}(\\hbar)$ the spin polarisation vector transforms as\n\\begin{equation}\n\\s_{1\\,ij}^{\\mu}(p_1 + \\hbar\\wn q) = \\s_{1\\,ij}^{\\mu}(p_1) - \\frac\\hbar{m^2} p^\\mu \\qb \\cdot s_{ij}(p_1)\\,.\n\\label{eqn:infinitesimalBoost}\n\\end{equation}\nThe angular impulse becomes\n\\begin{multline}\n\\langle \\Delta s_1^{\\mu,(0)} \\rangle \\rightarrow \\Delta s_1^{\\mu,(0)}\\label{eqn:limAngImp}\n= \\Lexp i\\! \\int\\!\\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q)e^{-ib\\cdot\\wn q} \\\\ \\times \\bigg(-\\hbar^3 \\frac{p_1^\\mu}{m_1^2} \\qb \\cdot \\s_{1} (p_1) \\mathcal{A}^{(0)}(\\wn q) + \\hbar^2 \\big[\\s^\\mu_1(p_1), \\mathcal{A}^{(0)}(\\wn q)\\big] \\bigg)\\Rexp\\,.\n\\end{multline}\nThe little group states have manifested themselves in the appearance of a commutator. The formula appears to be of a non-uniform order in $\\hbar$, but fortunately this is not really the case: any terms in the amplitude with diagonal indices will trivially vanish under the commutator; alternatively, any term with a commutator will introduce a factor of $\\hbar$ through the algebra of the Pauli--Lubanski vectors.\nTherefore all terms have the same weight, $\\hbar^3$, independently of factors appearing in the amplitude. \nThe analogous formula for the leading order, classical momentum impulse was given in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}.\nWe will make use of both the momentum and angular impulse formulae below.\n\nThere is a caveat regarding the uncertainty principle in the context of our spinning particles. In the following examples we restrict to low spins: spin 1\/2 and spin 1. Consequently the expectation of the spin vector $\\langle s^\\mu \\rangle$ is of order $\\hbar$; indeed $\\langle s^2 \\rangle = s(s+1) \\hbar^2$. This requires us to face the quantum-mechanical distinction between $\\langle s^\\mu s^\\nu \\rangle$ and $\\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$. Because of the uncertainty principle, the uncertainty $\\sigma_1^2$ associated with the operator $s^1$, for example, is of order $\\hbar$, and therefore the difference between $\\langle s_1^2 \\rangle$ and $\\langle s_1 \\rangle^2$ is of order $\\hbar^2$. Thus the difference $\\langle s^\\mu s^\\nu \\rangle - \\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$ is of order $\\langle s^\\mu s^\\nu \\rangle$. We are therefore not entitled to replace $\\langle s^\\mu s^\\nu \\rangle$ by $\\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$ for any arbitrary states, and will make the distinction between these quantities below. As we showed explicitly (for $SU(3)$ states) in chapter~\\ref{chap:pointParticles}, this limitation can be overcome by studying very large spin representations. To elucidate the details of taking the classical limit of amplitudes with spin we will primarily work with the explicit spin 1\/2 and spin 1 fields in this chapter; however, we will comment on the all--spin generalisation of our results, studied in \\cite{Guevara:2019fsj}. The large spin limit will then play a crucial role in the next chapter.\n\nThe procedure for passing from amplitudes to a concrete expectation value is as follows. Once one has computed the amplitude, and evaluated any commutators, explicit powers of $\\hbar$ must cancel. We then evaluate the integrals over the on-shell phase space of the incoming particles simply by evaluating the momenta $p_\\alpha$ as $p_\\alpha = m_\\alpha u_\\alpha$. An expectation value over the spin wave functions $\\xi_\\alpha$ remains; these are always of the form $\\langle s^{\\mu_1} \\cdots s^{\\mu_n}\\rangle$ for various values of $s$. Only when the spin $s$ is large can we factorise this expectation value.\n\n\\section{Classical limits of amplitudes with finite spin}\n\\label{sec:amplitudes}\n\nWe have constructed a general formula for calculating the leading classical contribution to the angular impulse from scattering amplitudes. In the limit these amplitudes are Laurent expanded in $\\hbar$, with only one term in the expansion providing a non-zero contribution. How this expansion works in the case for charged scalar amplitudes was established in part~\\ref{part:observables}, but now we need to consider examples of amplitudes for particles with spin. The identification of the spin polarisation vector defined in \\eqn~\\eqref{eqn:PLinnerProd} will be crucial to this limit.\n\nWe will again look at the two lowest spin cases, considering tree level scattering of a spin $1\/2$ or spin $1$ particle off a scalar in Yang--Mills theory and gravity. Tree level Yang--Mills amplitudes will now be denoted by $\\mathcal{A}_{s_1-0}$, and those for Einstein gravity as $\\mathcal{M}_{s_1-0}$. To ensure good UV behaviour of our amplitudes, we adopt minimally coupled interactions between the massive states and gauge fields. This has the effect of restricting the classical value of the gyromagnetic ratio to $g_L =2$, for all values of $s$ \\cite{Chung:2018kqs,Ferrara:1992yc}.\n\n\\subsection{Gauge theory amplitudes}\n\nWe will continue to consider Yang--Mills theory minimally coupled to matter in some representation of the gauge group. The common Lagrangian is that in equation~\\eqref{eqn:scalarAction}. Our calculations will be in the vein of section~\\ref{sec:LOimpulse}, and we will always have the same $t$-channel colour factor. The amplitude for scalar-scalar, $\\mathcal{A}_{0-0}$, scattering is of course that in equation~\\eqref{eqn:treeamp}.\n\n\\subsubsection*{Spinor-scalar}\n\nWe can include massive Dirac spinors $\\psi$ in the Yang--Mills amplitudes by using a Lagrangian $\\mathcal{L} = \\mathcal{L}_0 + \\mathcal{L}_{\\textrm{Dirac}}$, where the Dirac Lagrangian \n\\begin{equation}\n\\mathcal{L}_\\textrm{Dirac} = \\bar{\\psi}\\left(i\\slashed{D} - m\\right)\\psi\\label{eqn:DiracL}\n\\end{equation}\nincludes a minimal coupling to the gauge field, and $\\mathcal{L}_0$ is the scalar Langrangian in equation~\\eqref{eqn:scalarAction}. The tree level amplitude for spinor-scalar scattering is then\n\\begin{equation}\ni\\mathcal{A}^{ab}_{1\/2-0} = \\frac{ig^2}{2 \\hbar q^2}\\bar{u}^a(p_1+q)\\gamma^\\mu u^{b}(p_1)(2p_2 - q)_\\mu\\, \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,,\n\\end{equation}\nwhere we have normalised the (dimensionful) colour factors consistent with the double copy. We are interested in the pieces of this amplitude that survive to the classical limit. To extract them we must set the momentum transfer as $q = \\hbar\\wn q$ and expand the amplitude in powers of $\\hbar$. \n\nThe subtlety here is the on-shell Dirac spinor product. In the limit, when $q$ is small, we can follow the logic of \\eqn~\\eqref{eqn:infinitesimalBoost} and interpret $\\bar{u}^a(p_1 + \\hbar\\wn q) \\sim \\bar{u}^a(p_1) + \\Delta\\bar{u}^a(p_1)$ as being infinitesimally Lorentz boosted, see also~\\cite{Lorce:2017isp}. One expects amplitudes for spin 1\/2 particles to only be able to probe up to linear order in spin (i.e. the dipole of a spinning body) \\cite{Vaidya:2014kza,Guevara:2017csg,Guevara:2018wpp}, so in deriving the infinitesimal form of the Lorentz transformation we expand to just one power in the spin. The infinitesimal parameters $w_{\\mu\\nu}$ are exactly those determined in \\eqn~\\eqref{eqn:LorentzParameters}, so in all the leading terms of the spinor product are\n\\begin{equation}\n\\bar{u}^a(p_1 + \\hbar\\wn q) \\gamma_\\mu u^b(p_1) = 2{p_1}_\\mu \\delta^{ab} + \\frac{\\hbar}{4m^2}\\bar{u}^a(p_1) p{_1}^\\rho \\wn q^\\sigma [\\gamma_\\rho,\\gamma_\\sigma] \\gamma_\\mu u^b(p_1) + \\mathcal{O}(\\hbar^2)\\,.\\label{eqn:spinorshift}\n\\end{equation}\nEvaluating the product of gamma matrices via the identity\n\\begin{equation}\n[\\gamma_\\mu, \\gamma_\\nu] \\gamma_\\rho = 2\\eta_{\\nu\\rho}\\gamma_\\mu - 2\\eta_{\\mu\\rho}\\gamma_\\nu -  2i\\epsilon_{\\mu\\nu\\rho\\sigma} \\gamma^\\sigma \\gamma^5\\,,\\label{eqn:3gammaCommutator}\n\\end{equation}\nwhere $\\epsilon_{0123} = +1$ and $\\gamma^5 = i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3$, the spinor product is just\n\\begin{multline}\n\\bar{u}^a(p_1 + \\hbar\\wn q) \\gamma_\\mu u^b(p_1) = 2{p_1}_\\mu\\delta^{ab}  + \\frac{\\hbar}{2m_1^2} \\bar{u}^a(p_1) p_1^\\rho \\wn q^\\sigma \\left(\\gamma_\\sigma\\eta_{\\mu\\rho} - \\gamma_\\rho\\eta_{\\mu\\sigma}\\right) u^b(p_1) \\\\ - \\frac{i\\hbar}{2m_1^2} \\bar{u}^a(p_1) p{_1}^\\rho \\wn q^\\sigma \\epsilon_{\\rho\\sigma\\mu\\delta} \\gamma^\\delta\\gamma^5 u^b(p_1) + \\mathcal{O}(\\hbar^2)\\,.\n\\end{multline}\nComparing with our result from \\eqn~\\eqref{eqn:spinorSpinVec}, the third term clearly hides an expression for the spin 1\/2 polarisation vector. Making this replacement and substituting the spinor product into the amplitude yields, for on-shell kinematics, only two terms at an order lower than $\\mathcal{O}(\\hbar^2)$:\n\\begin{equation}\n\\hbar^3\\mathcal{A}^{ab}_{1\/2-0} = \\frac{2g^2}{\\wn q^2} \\left((p_1\\cdot p_2)\\delta^{ab} - \\frac{i}{m_1}  \\epsilon( p{_1}, \\wn q, p_2, s^{ab}_1) + \\mathcal{O}(\\hbar^2)\\right) \\tilde{\\newT}_1\\cdot\\tilde{\\newT}_2\\,,\\label{eqn:spinorYMamp}\n\\end{equation}\nwhere here and below we adopt the short-hand notation ${s_1}^{\\mu}_{ab} = {s_1}^{\\mu}_{ab}(p_1)$ and\n\\[\n\\epsilon(a,b,c,d) = \\epsilon_{\\mu\\nu\\rho\\sigma} a^\\mu b^\\nu c^{\\rho} d^{\\sigma} \\,,\n\\qquad \\epsilon_\\mu(a,b,c) = \\epsilon_{\\mu\\nu\\rho\\sigma} a^\\nu b^\\rho c^\\sigma\\,.\n\\]\nUpon substitution into the impulse in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} or angular impulse in \\eqn~\\eqref{eqn:limAngImp} the apparently singular denominator in the $\\hbar\\rightarrow 0$ limit is cancelled. It is only these quantities, not the amplitudes, that are classically well defined and observable.\n\n\\subsubsection*{Vector-scalar}\n\nNow consider scattering a massive vector rather than spinor. The minimally coupled gauge interaction can be obtained by applying the Higgs mechanism to the Yang--Mills Lagrangian\\footnote{Regardless of minimal coupling, for vector states with masses generated in this way the classical value of $g_L=2$ \\cite{Chung:2018kqs,Ferrara:1992yc}.}, which when added to the scalar Lagrangian $\\mathcal{L}_0$ yields the tree-level amplitude\n\\begin{multline}\ni\\mathcal{A}^{ij}_{1-0} = -\\frac{ig^2}{2\\hbar q^2} \\varepsilon{_i^*}^\\mu(p_1+q) \\varepsilon^\\nu_j(p_1) \\left(\\eta_{\\mu\\nu}(2p_1+q)_\\lambda - \\eta_{\\nu\\lambda}(p_1-q)_\\mu \\right.\\\\\\left. - \\eta_{\\lambda\\mu}(2q+p_1)_\\nu \\right)(2p_2 - q)^\\lambda\\, \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,.\n\\end{multline}\nTo obtain the classically significant pieces of this amplitude we must once more expand the product of on-shell tensors, in this case the polarisation vectors. In the classical limit we can again consider the outgoing polarisation vector as being infinitesimally boosted, so $\\varepsilon{_i^*}^\\mu(p_1 + \\hbar\\wn q) \\sim \\varepsilon{_i^*}^\\mu(p_1) + \\Delta\\varepsilon{_i^*}^\\mu(p_1)$. \n\nHowever, from spin 1 particles we expect to be able to probe $\\mathcal{O}(s^2)$, or quadrupole, terms \\cite{Vaidya:2014kza,Guevara:2017csg,Guevara:2018wpp}. Therefore it is salient to expand the Lorentz boost to two orders in the Lorentz parameters $w_{\\mu\\nu}$, so under infinitesimal transformations we take\n\\begin{equation}\n\\varepsilon_i^\\mu(p) \\mapsto \\Lambda^\\mu{ }_\\nu\\, \\varepsilon_i^\\nu(p) \\simeq \\left(\\delta^\\mu{ }_\\nu - \\frac{i}{2} w_{\\rho\\sigma} (\\Sigma^{\\rho\\sigma})^\\mu{ }_\\nu - \\frac18 \\left((w_{\\rho\\sigma} \\Sigma^{\\rho\\sigma})^2\\right)^\\mu{ }_\\nu \\right) \\!\\varepsilon_i^\\nu(p)\\,,\n\\end{equation}\nwhere $(\\Sigma^{\\rho\\sigma})^\\mu{ }_\\nu = i\\left(\\eta^{\\rho\\mu} \\delta^\\sigma{ }_\\nu - \\eta^{\\sigma\\mu} \\delta^\\rho{}_\\nu\\right)$. Since the kinematics are again identical to those used to derive \\eqn~\\eqref{eqn:LorentzParameters}, we get\n\\begin{equation}\n\\varepsilon{_i^*}^\\mu(p_1 + \\hbar\\wn q)\\, \\varepsilon^\\nu_j(p_1) = \\varepsilon{_i^*}^\\mu \\varepsilon^\\nu_j - \\frac{\\hbar}{m_1^2} (\\wn q \\cdot \\varepsilon_i^*) p_1^\\mu \\varepsilon^\\nu_j - \\frac{\\hbar^2}{2 m_1^2} (\\wn q\\cdot\\varepsilon_i^*) \\wn q^\\mu \\varepsilon^\\nu_j + \\mathcal{O}(\\hbar^3)\\,,\\label{eqn:vectorS}\n\\end{equation}\nwhere now $\\varepsilon_i$ will always be a function of $p_1$, so in the classical limit $\\varepsilon_i^*\\cdot p_1 = \\varepsilon_i\\cdot p_1=0$. Using this expression in the full amplitude, the numerator becomes\n\\begin{multline}\nn_{ij} = 2(p_1\\cdot p_2)(\\varepsilon_i^*\\cdot\\varepsilon_j) - 2\\hbar (p_2\\cdot\\varepsilon_i^*)(\\wn q\\cdot\\varepsilon_j) + 2\\hbar (p_2\\cdot\\varepsilon_j)(\\wn q\\cdot\\varepsilon_i^*) \\\\ + \\frac{1}{m_1^2}\\hbar^2 (p_1\\cdot p_2)(\\wn q\\cdot\\varepsilon_i^*)(\\wn q\\cdot\\varepsilon_j) + \\frac{\\hbar^2}{2}\\wn q^2 (\\varepsilon_i^*\\cdot \\varepsilon_j) + \\mathcal{O}(\\hbar^3)\\,.\n\\end{multline}\nHow the spin vector enters this expression is not immediately obvious, and relies on Levi--Civita tensor identities. At $\\mathcal{O}(\\hbar)$, $\\epsilon^{\\delta\\rho\\sigma\\nu} \\epsilon_{\\delta\\alpha\\beta\\gamma} = -3!\\, \\delta^{[\\rho}{ }_\\alpha \\delta^{\\sigma}{ }_\\beta \\delta^{\\nu]}{ }_\\gamma$ leads to\n\\begin{align}\n\\hbar (p_2\\cdot\\varepsilon{_i^*})(\\wn q \\cdot \\varepsilon_j) - \\hbar (p_2 \\cdot \\varepsilon_j)(\\wn q \\cdot \\varepsilon_i^*) = \\frac{\\hbar}{m_1^2}&p_1^\\rho \\wn q^\\sigma p_2^\\lambda \\epsilon_{\\delta\\rho\\sigma\\lambda}\\epsilon^{\\delta\\alpha\\beta\\gamma}{\\varepsilon_i^*}_\\alpha {\\varepsilon_j}_\\beta {p_1}_\\gamma \\nonumber \\\\ &\\equiv  -\\frac{i}{m_1} \\epsilon(p_1, \\wn q, p_2, {s_1}_{ij})\\,,\n\\end{align}\nwhere again we are able to identify the spin 1 polarisation vector calculated in \\eqn~\\eqref{eqn:vectorSpinVec} and introduce it into the amplitude. There is also a spin vector squared contribution entering at $\\mathcal{O}(\\hbar^2)$; observing this is reliant on applying the identity $\\epsilon^{\\mu\\nu\\rho\\sigma} \\epsilon_{\\alpha\\beta\\gamma\\delta} = -4!\\, \\delta^{[\\mu}{ }_\\alpha \\delta^{\\nu}{ }_\\beta \\delta^{\\rho}{ }_\\gamma \\delta^{\\sigma]}{ }_\\delta$ and the expression in \\eqn~\\eqref{eqn:vectorSpinVec} to calculate\n\\begin{equation}\n\\sum_k \\left(\\wn q \\cdot s_1^{ik}\\right) (\\wn q \\cdot s_1^{kj}) = -\\hbar^2 (\\wn q \\cdot \\varepsilon_i^*) (\\wn q\\cdot \\varepsilon_j) - \\hbar^2 \\wn q^2 \\delta_{ij} + \\mathcal{O}(\\hbar^3)\\,.\n\\end{equation}\nThis particular relationship is dependent on the sum over helicities $\\sum_h {\\varepsilon^*_h}^\\mu \\varepsilon^\\nu_h = -\\eta^{\\mu\\nu} + \\frac{p_1^\\mu p_1^\\nu}{m_1^2}$ for massive vector bosons, an additional consequence of which is that $\\varepsilon_i^*\\cdot \\varepsilon_j = -\\delta_{ij}$. Incorporating these rewritings of the numerator in terms of spin vectors, the full amplitude is\n\\begin{multline}\n\\hbar^3\\mathcal{A}^{ij}_{1-0} = \\frac{2g^2}{\\wn q^2}\\left((p_1\\cdot p_2)\\delta^{ij} - \\frac{i}{m_1} \\epsilon(p_1, \\wn q, p_2, s_1^{ij}) + \\frac{1}{2 m_1^2}(p_1\\cdot p_2)(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) \\right.\\\\\\left. + \\frac{\\hbar^2 \\wn q^2}{4m_1^2}\\left(2p_1\\cdot p_2 + m_1^2\\right) + \\mathcal{O}(\\hbar^3)\\right) \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,.\n\\end{multline}\nThe internal sum over spin indices in the $\\mathcal{O}(s^2)$ term will now always be left implicit. In classical observables we can also drop the remaining $\\mathcal{O}(\\hbar^2)$ term, as this just corresponds to a quantum correction from contact interactions.\n\n\\subsection{Gravity amplitudes}\n\nRather than re-calculate these amplitudes in perturbative gravity, let us apply the double copy\\footnote{One can easily verify that direct calculations with graviton vertex rules given in \\cite{Holstein:2008sx} reproduce our results.}. Note that for massive states with spin this ability is reliant on our gauge theory choice of $g_L=2$, as was noted in \\cite{Goldberger:2017ogt}. Only with this choice is the gravitational theory consistent with the low energy spectrum of string theory \\cite{Goldberger:2017ogt,Chung:2018kqs}, of which the double copy is an intrinsic feature.\n\nFor amplitudes in the LO impulse the Jacobi identity is trivial, as we saw in section~\\ref{sec:LOimpulse}. We can therefore simply replace colour factors with the desired numerator. In particular, if we replace the colour factor in the previous spin $s$--spin 0 Yang--Mills amplitudes with the scalar numerator from \\eqn~\\eqref{eqn:scalarYMamp} we will obtain a spin $s$--spin 0 gravity amplitude, as the composition of little group irreps is simply $(\\mathbf{2s + 1})\\otimes\\mathbf{1}=\\mathbf{2s + 1}$. Using the scalar numerator ensures that the spin index structure passes to the gravity theory unchanged. Thus we can immediately obtain that the classically significant part of the spin $1\/2$--spin 0 gravity amplitude is\n\\begin{equation}\n\\hbar^3 \\mathcal{M}^{ab} = -\\left(\\frac{\\kappa}{2}\\right)^2\\frac{4}{\\wn q^2} \\bigg[(p_1\\cdot p_2)^2\\delta^{ab} - \\frac{i}{m_1}(p_1\\cdot p_2)\\, \\epsilon(p_1, \\wn q, p_2, s_1^{ab}) + \\mathcal{O}(\\hbar^2)\\bigg]\\,,\\label{eqn:spinorScalarGravAmp}\n\\end{equation}\nwhile that for spin 1--spin 0 scattering is\n\\begin{multline}\n\\hbar^3\\mathcal{M}^{ij} = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\wn q^2} \\left[(p_1\\cdot p_2)^2\\delta^{ij} - \\frac{i}{m_1}(p_1\\cdot p_2)\\, \\epsilon(p_1, \\wn q, p_2, s_1^{ij}) \\right.\\\\\\left. + \\frac{1}{2 m_1^2}(p_1\\cdot p_2)^2(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) + \\mathcal{O}(\\hbar^2)\\right]. \\label{eqn:vectorScalarGravAmp}\n\\end{multline}\nNotice that the $\\mathcal{O}(s)$ parts of these amplitudes are exactly equal, up to the different spin indices. This is a manifestation of gravitational universality: the gravitational coupling to the spin dipole should be independent of the spin of the field, precisely as we observe.\n\nWe have deliberately not labelled these as Einstein gravity amplitudes, because the gravitational modes in our amplitudes contain both gravitons and scalar dilatons. To see this, let us re-examine the factorisation channels in the $t$ channel cut, but now for the vector amplitude:\n\\[\n\\lim\\limits_{\\wn q^2 \\rightarrow 0} \\left(\\wn q^2 \\hbar^3 \\mathcal{M}^{ij}\\right) &= -4\\left(\\frac{\\kappa}{2}\\right)^2\\, \\bigg(p_1^\\mu p_1^{\\tilde{\\mu}} \\delta^{ij} - \\frac{i}{m_1}p_1^\\mu \\epsilon^{\\tilde{\\mu}\\rho\\sigma\\delta} {p_1}_\\rho \\wn q_\\sigma {s_1}_\\delta^{ij} \\\\ & \\qquad\\qquad\\qquad + \\frac{1}{2m_1^2} (\\wn q\\cdot s_1^{ik})(\\wn q\\cdot s_1^{kj}) p_1^\\mu p_1^{\\tilde{\\mu}}\\bigg) \\mathcal{P}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\, p_2^\\nu p_2^{\\tilde{\\nu}} \\\\ & - 4\\left(\\frac{\\kappa}{2}\\right)^2\\left(p_1^\\mu p_1^{\\tilde{\\mu}} \\delta^{ij} + \\frac{(\\wn q\\cdot s_1^{ik})(\\wn q\\cdot s_1^{kj})}{2 m_1^2} p_1^\\mu p_1^{\\tilde{\\mu}}\\right) \\mathcal{D}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\, p_2^\\nu p_2^{\\tilde{\\nu}}\\,,\n\\]\nwhere we have utilised the de-Donder gauge graviton and dilaton projectors from equation~\\eqref{eqn:gravityProjectors}. As for the scalar case, the pure Einstein gravity amplitude for classical spin 1--spin 0 scattering can just be read off as the part of the amplitude contracted with the graviton projector. We find that\n\\begin{multline}\n\\hbar^3\\mathcal{M}^{ij}_{1-0} = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\wn q^2} \\left[\\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right)\\delta^{ij} - \\frac{i}{m_1}(p_1\\cdot p_2)\\,\\epsilon(p_1, \\wn q, p_2, s_1^{ij}) \\right.\\\\\\left. + \\frac{1}{2 m_1^2}\\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right)(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) + \\mathcal{O}(\\hbar^2)\\right]. \\label{eqn:vectorGravAmp}\n\\end{multline}\nThe spinor--scalar Einstein gravity amplitude receives the same correction to the initial, scalar component of the amplitude. \n\nNote that dilaton modes are coupling to the scalar monopole and $\\mathcal{O}(s^2)$ quadrapole terms in the gravity amplitudes, but not to the $\\mathcal{O}(s)$ dipole component. We also do not find axion modes, as observed in applications of the classical double copy to spinning particles \\cite{Li:2018qap,Goldberger:2017ogt}, because axions are unable to couple to the massive external scalar.\n\n\\section{Black hole scattering observables from amplitudes}\n\\label{sec:KerrCalcs}\n\nWe are now armed with a set of classical tree-level amplitudes and formulae for calculating the momentum impulse $\\Delta p_1^\\mu$ and angular impulse $\\Delta s_1^\\mu$ from them. We also already have a clear target where the analogous classical results are known: the results for 1PM scattering of spinning black holes found in \\cite{Vines:2017hyw}. \n\nGiven our amplitudes only reach the quadrupole level, we can only probe lower order terms in the expansion of \\eqn~\\eqref{eqn:KerrDeflections}. Expanding in the rescaled spin $a_1^\\mu$, and setting $a_2^\\mu\\to0$, the momentum impulse is\n\\begin{multline}\n\\Delta p_1^{\\mu} = \\frac{2 G m_1m_2}{\\sqrt{\\gamma^2 - 1}} \\left\\{(2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2} + \\frac{2\\gamma}{b^4} \\Big( 2{b}^\\mu {b}^\\nu-b^2\\Pi^{\\mu\\nu}\\Big)\\, \\epsilon_{\\nu\\rho}(u_1, u_2)\\,  a_1^\\rho\\right.\n\\\\\n\\left. - \\frac{2\\gamma^2 - 1}{{b}^6} \\Big(4b^\\mu b^\\nu b^\\rho-3b^2 b^{(\\mu}\\Pi^{\\nu\\rho)}\\Big)a_{1\\nu}a_{1\\rho} + \\mathcal{O}(a^3) \\right\\}+\\mathcal O(G^2)\\,, \\label{eqn:JustinImpResult}\n\\end{multline}\nwhere $\\Pi^\\mu{}_\\nu$ is the projector into the plane orthogonal to $u_1^\\mu$ and $u_2^\\mu$ from \\eqref{eqn:projector}.\nMeanwhile the angular impulse to the same order is\n\\begin{multline}\n\\Delta s_1^{\\mu} =-u_1^\\mu a_{1\\nu}\\Delta p_1^\\nu -\\frac{2G m_1m_2}{\\sqrt{\\gamma^2-1}}\\left\\{\n2\\gamma \\epsilon^{\\mu\\nu\\rho\\sigma}  u_{1\\rho}\\, \\epsilon_{\\sigma} (u_1,u_2, b) \\frac{a_{1\\nu} }{b^2}\\right.\n\\\\\n\\left. - \\frac{2\\gamma^2 - 1}{b^4} \\epsilon^{\\mu\\nu\\kappa\\lambda} u_{1\\kappa}  \\Big( 2{b}_\\nu {b}_\\rho-b^2\\Pi_{\\nu\\rho} \\Big)a_{1\\lambda} a_1^\\rho \n+ \\mathcal{O}(a^3) \\right\\}+\\mathcal O(G^2)\\,. \\label{eqn:JustinSpinResult}\n\\end{multline}\nIn this section we demonstrate that both of these results can be recovered by using the classical pieces of our Einstein gravity amplitudes.\n\n\\subsection{Momentum impulse}\n\nTo calculate the momentum impulse we substitute $\\mathcal{M}_{1-0}$ into the general expression in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. Following the prescription in \\sect{sec:classicalLimit}, the only effect of the momentum integrals in the expectation value is to set $p_\\alpha \\rightarrow m_\\alpha u_\\alpha$ in the classical limit. This then reduces the double angle bracket to the single expectation value over the spin states:\n\\begin{equation}\n\\begin{aligned}\n\\Delta p_1^{\\mu,(0)} &= -i m_1 m_2 \\left(\\frac{\\kappa}{2}\\right)^2 \\!\\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot\\wn q) e^{-ib\\cdot\\wn q} \\frac{\\wn q^\\mu}{\\wn q^2} \\\\&\\times \\left\\langle \\frac12(2\\gamma^2 - 1) - \\frac{i \\gamma}{m_1} \\epsilon(u_1, \\wn q, u_2, s_1) + \\frac{2\\gamma^2 - 1}{4m_1^2} (\\wn q\\cdot s_1) (\\wn q \\cdot s_1) \\right\\rangle\\\\\n&\\equiv -4i m_1 m_2 \\pi G \\bigg((2\\gamma^2 - 1) I^\\mu  -  2i\\gamma u_1^\\rho u_2^\\nu \\epsilon_{\\rho\\sigma\\nu\\delta}\\, \\big\\langle a_1^\\delta \\big\\rangle I^{\\mu\\sigma} \\\\ &\\hspace{60mm} + \\frac{2\\gamma^2 - 1}{2} \\big\\langle {a_1}_\\nu {a_1}_\\rho \\big\\rangle I^{\\mu\\nu\\rho} \\bigg)\\,,\n\\end{aligned}\n\\end{equation}\nwhere we have rescaled $a^\\mu = s^\\mu\/m$ and defined three integrals of the general form\n\\begin{equation}\nI^{\\mu_1\\cdots \\mu_n} = \\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot \\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\wn q^2} \\wn q^{\\mu_1} \\cdots \\wn q^{\\mu_n}\\,.\\label{eqn:defOfI}\n\\end{equation}\n\nThe lowest rank integral of this type was evaluated in chapter~\\ref{chap:impulse}, with the result\n\\begin{equation}\nI^\\mu = \\frac{i}{2\\pi \\sqrt{\\gamma^2 - 1}} \\frac{{b}^\\mu}{b^2}\\,.\\label{eqn:I1result}\n\\end{equation}\nTo evaluate the higher rank examples, note that the results must lie in the plane orthogonal to the four velocities. This plane is spanned by the impact parameter $b^\\mu$, and the projector $\\Pi^\\mu{ }_\\nu$ defined in \\eqn~\\eqref{eqn:projector}. Thus, for example,\n\\begin{equation}\nI^{\\mu\\nu} = \\alpha_2 b^\\mu b^\\nu + \\beta_2 \\Pi^{\\mu\\nu}\\,.\n\\end{equation}\nGiven that we are working away from the threshold value $b = 0$, the left hand side is traceless and $\\beta_2 = - \\alpha_2\\, b^2 \/2$. Then contracting both sides with $b_\\nu$, one finds\n\\begin{equation}\n\\alpha_2 b^2\\, b^\\mu = 2\\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot \\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\wn q^2} \\wn q^{\\mu} (b\\cdot\\wn q) = \\frac{1}{\\pi \\sqrt{\\gamma^2 - 1}} \\frac{b^\\mu}{b^2}\\,,\n\\end{equation} \nwhere we have used the result of \\eqn~\\eqref{eqn:I1result}. Thus the coefficient $\\alpha_2$ is uniquely specified, and we find\n\\begin{equation}\nI^{\\mu\\nu} = \\frac{1}{\\pi b^4 \\sqrt{\\gamma^2 - 1}} \\left(b^\\mu b^\\nu - \\frac12 b^2 \\Pi^{\\mu\\nu} \\right)\\label{eqn:I2result}.\n\\end{equation}\nFollowing an identical procedure for $I^{\\mu\\nu\\rho}$, we can then readily determine that\n\\begin{equation}\nI^{\\mu\\nu\\rho} = -\\frac{4i}{\\pi b^6 \\sqrt{\\gamma^2 - 1} } \\left(b^\\mu b^\\nu b^\\rho - \\frac34 b^2 b^{(\\mu} \\Pi^{\\nu\\rho)} \\right).\\label{eqn:I3result}\n\\end{equation}\n\nSubstituting the integral results into the expression for the leading order classical impulse, and expanding the projectors from \\eqn~\\eqref{eqn:projector}, then leads to\n\\begin{multline}\n\\Delta p_1^{\\mu,(0)} = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}}\\left((2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2} + \\frac{2\\gamma}{b^4} ( 2 b^\\mu {b}^\\alpha-b^2\\Pi^{\\mu\\alpha}) \\epsilon_{\\alpha\\rho}(u_1, u_2) \\big\\langle a_1^\\rho \\big\\rangle  \\right.\n\\\\\n\\left.- \\frac{2\\gamma^2 - 1}{{b}^6} (4b^\\mu b^\\nu b^\\rho-3b^2 b^{(\\mu}\\Pi^{\\nu\\rho)})\\langle a_{1\\nu}a_{1\\rho}\\rangle\\right).\\label{eqn:QuantumImpRes}\n\\end{multline}\nComparing with \\eqn~\\eqref{eqn:JustinImpResult} we observe an exact match, up to the appearance of spin state expectation values, between our result and the $\\mathcal{O}(a^2)$ expansion of the result for spinning black holes from \\cite{Vines:2017hyw}. \n\n\\subsection{Angular impulse}\n\nOur expression, equation~\\eqref{eqn:limAngImp}, for the classical leading-order angular impulse naturally has two parts: one term has a commutator while the other term does not. For clarity we will handle these two parts separately, beginning with the term without a commutator, which we will call the direct term.\n\n\\subsubsection*{The direct term}\n\nSubstituting our $\\mathcal{O}(s^2)$ Einstein gravity amplitude, equation~\\eqref{eqn:vectorGravAmp}, into the direct part of the general angular impulse formula, we find\n\\begin{align}\n&\\Delta s_1^{\\mu,(0)}\\big|_{\\textrm{direct}} \\nonumber\n\\equiv\n\\Lexp i\\! \\int\\!\\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q)e^{-ib\\cdot\\wn q} \\bigg(-\\hbar^3 \\frac{p_1^\\mu}{m_1^2} \\qb \\cdot \\s_{1} (p_1) \\mathcal{M}_{1-0} \\bigg) \\Rexp \\\\\n&\\;= \\Lexp \\frac{i \\kappa^2}{m_1^2} \\int\\! \\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\qb^2}\\, p_1^\\mu \\wn q \\cdot \\s_1(p_1) \\\\ &\\qquad\\times\\bigg(\\bigg((p_1\\cdot p_2)^2  - \\frac{1}{2}m_1^2m_2^2\\bigg)  - \\frac{i}{m_1}(p_1\\cdot p_2) \\, \\epsilon(p_1, \\wn q, p_2,s_1) \\bigg)+ \\mathcal{O}(s^3) \\Rexp\\,.\\nonumber\n\\end{align}\nAs with the momentum impulse, we can reduce the double angle brackets to single, spin state, angle brackets by replacing $p_\\alpha \\rightarrow m_\\alpha u_\\alpha$, so that\n\\begin{equation}\n\\hspace{-3pt}\\Delta s^{\\mu,(0)}_1\\big|_{\\textrm{direct}}\\!\\! = 4\\pi G m_2\\, u_1^\\mu\\!  \\left(\\! i  \\left(2\\gamma^2 - 1\\right)\\! \\langle s_1^\\nu \\rangle I_\\nu + \\frac{2}{m_1} \\gamma \n\\, u_1^\\alpha u_2^\\gamma \\epsilon_{\\alpha\\beta\\gamma\\delta} \\langle s_{1\\nu}  s_1^{\\delta} \\rangle \\, I^{\\nu\\beta} \\!\\right),\n\\end{equation}\nwhere the integrals are again defined by \\eqn~\\eqref{eqn:defOfI}. We can now just substitute our previous evaluations of these integrals, equations~\\eqref{eqn:I1result} and~\\eqref{eqn:I2result}, to learn that\n\\begin{multline}\n\\Delta a_1^{\\mu,(0)}\\big|_{\\textrm{direct}} = -\\frac{2G m_2}{\\sqrt{\\gamma^2 -1}} u_1^\\mu \\left((2\\gamma^2 - 1)\\frac{{b}_\\nu}{b^2} \\spinExp{a_1^\\nu} \\right. \\\\ \\left. + \\frac{2\\gamma}{b^4} \\left( 2{b}^\\nu {b}^\\alpha-{b^2}\\Pi^{\\nu\\alpha}\\right) \\epsilon_{\\alpha\\rho} (u_1, u_2) \\spinExp{{a_1}_\\nu a_1^\\rho} \\right).\\label{eqn:linPiece}\n\\end{multline}\n\n\\subsubsection*{The commutator term}\n\nNow we turn to the commutator piece of \\eqn~\\eqref{eqn:limAngImp}. The scalar part of our Einstein gravity amplitude, equation~\\eqref{eqn:vectorGravAmp}, has diagonal spin indices, so its commutator vanishes. We encounter two non-vanishing commutators:\n\\[\n[\\s_1^\\mu, \\s_1^\\nu]&=\\frac{i \\hbar}{m_1} \\, \\epsilon^{\\mu\\nu}(s_{1} , p_{1}) \\,,\\\\\n[\\s_1^\\mu, \\qb \\cdot \\s_1 \\, \\qb\\cdot \\s_1] &= \\frac{2 i \\hbar}{m_1} \\, \\qb \\cdot s_1 \\, \\epsilon^{\\mu} (\\qb, s_{1}, p_1) + \\mathcal{O}(\\hbar^2) \\,,\n\\]\nomitting a term which is higher order. Using these expressions in the commutator term, the result is\n\\[\n&\\Delta s_1^{\\mu, (0)}|_\\textrm{com} = i\\, \\Lexp \\int\\! \\dd^4 \\qb \\, \\del(2p_1 \\cdot \\qb) \\del(2p_2 \\cdot \\qb) e^{-ib \\cdot \\qb} \\, \\hbar^2 [s^\\mu(p_1), \\mathcal{M}_{1-0}] \\Rexp \\\\\n&= i \\kappa^2 \\Lexp \\int\\! \\dd^4 \\qb \\, \\del(2p_1 \\cdot \\qb) \\del(2p_2 \\cdot \\qb) \\frac{e^{-ib \\cdot \\qb}}{\\qb^2} \\bigg( (p_1 \\cdot p_2) \\, \\epsilon(p_1, \\qb, p_2)_\\sigma\\epsilon^{\\mu\\nu\\rho\\sigma} s_{1 \\nu} \\frac{p_{1 \\rho}}{m_1^2}  \\\\\n& \\qquad\\qquad\\qquad\\qquad- \\frac{i}{m_1^3} \\left( (p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right) \\qb \\cdot \\s_1 \\, \\epsilon^{\\mu\\nu}(\\qb, p_1) s_{1\\nu} \\bigg) \\Rexp\\,.\n\\]\nAs is familiar by now, we evaluate the integrals over the momentum-space wavefunctions by setting $p_\\alpha = m_\\alpha u_\\alpha$, but expectation values over the spin-space wavefunctions remain. The result can be organised in terms of the integrals $I^\\alpha$ and $I^{\\alpha\\beta}$ defined in equation~\\eqref{eqn:defOfI}:\n\\begin{multline}\n\\Delta s_1^{\\mu, (0)}|_\\textrm{com} = 2\\pi i \\, G m_2 \\bigg( 4\\gamma \\epsilon^{\\mu\\nu\\rho\\sigma} \\langle s_{1\\,\\nu} \\rangle u_{1\\,\\rho} \\epsilon_{\\sigma\\alpha} (u_1, u_2) I^\\alpha \n\\\\ - \\frac{2 i}{m_1} (2\\gamma^2 - 1) \\epsilon^{\\mu\\nu\\rho\\sigma} u_{1\\, \\rho} \\langle s_{1\\, \\sigma} {s_{1}}^\\alpha \\rangle I_{\\alpha\\nu}\n\\bigg) \\, .\n\\end{multline}\nFinally, we perform the integrals using equations~\\eqref{eqn:I1result} and~\\eqref{eqn:I2result}, rescale the spin vector to $a_1^\\mu$ and combine the result with the direct contribution in \\eqn~\\eqref{eqn:linPiece}, to find that the angular impulse at $\\mathcal{O}(a^2)$ is\n\\[\n\\hspace{-4pt}\\Delta s_1^{\\mu,(0)}\n= -\\frac{2Gm_1 m_2}{\\sqrt{\\gamma^2 -1}}& \\bigg\\{(2\\gamma^2 - 1) u_1^\\mu \\frac{{b}_\\nu}{b^2} \\spinExp{a_1^\\nu} + \\frac{2\\gamma}{b^2} \\epsilon^{\\mu\\nu\\rho\\sigma} \\spinExp{a_{1\\,\\nu}} u_{1\\,\\rho} \\epsilon_{\\sigma}(u_1, u_2, b) \\\\ &- \\frac{2\\gamma}{b^2} u_1^\\mu \\left(\\eta^{\\nu\\alpha} - \\frac{2{b}^\\nu {b}^\\alpha}{b^2}\\right) \\epsilon_{\\alpha\\rho} (u_1, u_2) \\spinExp{{a_1}_\\nu a_1^\\rho} \\\\ & + \\frac{(2\\gamma^2 - 1)}{b^2} \\epsilon^{\\mu\\nu\\rho\\sigma} {u_1}_\\rho \\spinExp{{a_1}_\\sigma {a_1}_\\lambda} \\left(\\Pi^\\lambda{}_\\nu - \\frac{2{b}_\\nu {b}^\\lambda}{b^2}\\right)  \n \\bigg\\}\\,.\n\\label{eqn:QuantumAngImpRes}\n\\]\nThis final result agrees with the classical result of equation~\\eqref{eqn:JustinSpinResult}, modulo the remaining spin expectation values.\n\n\n\\section{Discussion}\n\\label{sec:angImpDiscussion}\n\nStarting from a quantum field theory for massive spinning particles with arbitrary long-range interactions (mediated e.g.\\ by gauge bosons or gravitons), we have followed a careful analysis of the classical limit $(\\hbar\\to0)$ for long-range scattering of spatially localised wavepackets.   We have thereby arrived at fully relativistic expressions for the angular impulse, the net change in the intrinsic angular momentum of the massive particles, due to an elastic two-body scattering process.  This, our central result of the chapter, expressed in terms of on-shell scattering amplitudes, is given explicitly at leading order in the coupling by  \\eqref{eqn:limAngImp}.  Our general formalism places no restrictions on the order in coupling, and the expression \\eqref{eqn:spinShift} for the angular impulse, like its analogues for the momentum and colour impulse found in earlier chapters, should hold at all orders. Since the publication of this formalism in ref.~\\cite{Maybee:2019jus}, general, fully classical formulae have been proposed in \\cite{Bern:2020buy,Kosmopoulos:2021zoq} for the angular impulse, expressed in terms of commutators of a modified eikonal phase. It would be very interesting to establish a firm relationship between this proposal and our results. Our 1PM calculations of the spin-squared parts of the fully covariant momentum and angular impulses have also been confirmed using post--Minkowskian EFT methods, and extended to next-to-leading-order in the coupling \\cite{Liu:2021zxr}, equivalent to a 1-loop computation.\n\nIn this chapter we applied our general results to the examples of a massive spin 1\/2 or spin 1 particle (particle 1) exchanging gravitons with a massive spin 0 particle (particle 2), imposing minimal coupling.  The results for the linear and angular impulses for particle 1, $\\Delta p_1^\\mu$ and $\\Delta s_1^\\mu$, due to its scattering with the scalar particle 2, are given by \\eqref{eqn:QuantumImpRes} and \\eqref{eqn:QuantumAngImpRes}.  These expressions are valid to linear order in the gravitational constant $G$, or to 1PM order, having arisen from the tree level on-shell amplitude for the two-body scattering process.  By momentum conservation (in absence of radiative effects at this order), $\\Delta p_2^\\mu=-\\Delta p_1^\\mu$, and the scalar particle has no intrinsic angular momentum, $s_2^\\mu=\\Delta s_2^\\mu=0$.  The spin 1\/2 case provides the terms through linear order in the rescaled spin $a_1^\\mu=s_1^\\mu\/m_1$, and the spin 1 case yields the same terms through linear order plus terms quadratic in $a_1^\\mu$.\n\nOur final results \\eqref{eqn:QuantumImpRes} and \\eqref{eqn:QuantumAngImpRes} from the quantum analysis are seen to be in precise agreement with the results \\eqref{eqn:JustinImpResult} and \\eqref{eqn:JustinSpinResult} from \\cite{Vines:2017hyw} for the classical scattering of a spinning black hole with a non-spinning black hole, through quadratic order in the spin --- except for the appearance of spin-state expectation values $\\langle a_1^\\mu\\rangle$ and $\\langle a_1^\\mu a_1^\\nu\\rangle$ in the quantum results replacing $a_1^\\mu$ and $a_1^\\mu a_1^\\nu$ in the classical result.  For any quantum states of a finite-spin particle, these expectation values cannot satisfy the  appropriate properties of their classical counterparts, e.g., $\\langle a^\\mu a^\\nu\\rangle \\ne \\langle a^\\mu\\rangle \\langle a^\\nu\\rangle$.  Furthermore, we know from section~\\ref{sec:classicalSingleParticleColour} that the intrinsic angular momentum of a quantum spin-$s$ particle scales like $\\langle s^\\mu\\rangle=m\\langle a^\\mu\\rangle\\sim s\\hbar$, and we would thus actually expect any spin effects to vanish in a classical limit where we take $\\hbar\\to0$ at fixed spin quantum number $s$.\n\nFor a fully consistent classical limit yielding non-zero contributions from the intrinsic spin we of course would need to take the spin $s\\to\\infty$ as $\\hbar\\to0$, so as to keep $\\langle s^\\mu\\rangle\\sim s\\hbar$ finite. However, the expansions in spin operators of the minimally coupled amplitudes and impulses, expressed in the forms we have derived here, are found to be universal, in the sense that going to higher spin quantum numbers $s$ continues to reproduce the same expressions at lower orders in the spin operators.  We have seen this explicitly here for the linear-in-spin level, up to spin 1.\nThis pattern was confirmed to hold for minimally coupled gravity amplitudes for arbitrary spin $s$ in ref.~\\cite{Guevara:2019fsj}. The authors showed that applying the formalism we have presented here to amplitudes in the limit $s\\to\\infty$ fully reproduces the full Kerr 1PM observables listed in \\eqref{eqn:KerrDeflections}, up to spin state expectation values. Using the coherent states in section~\\ref{sec:classicalSingleParticleColour}, one can indeed then take the limit where $\\langle a^\\mu a^\\nu\\rangle = \\langle a^\\mu\\rangle \\langle a^\\nu\\rangle$ and so forth. The precise forms of $1\/s$ corrections to the higher-multipole couplings were discussed in \\cite{Chung:2019duq}.\n\nOur formalism provides a direct link between gauge-invariant quantities, on-shell amplitudes and classical asymptotic scattering observables, with generic incoming and outgoing states, for relativistic spinning particles.  It is tailored to be combined with powerful modern techniques for computing relevant amplitudes, such as unitarity methods and the double copy.  Already, with our examples at the spin 1\/2 and spin 1 levels, we have explicitly seen that it produces new evidence (for generic spin orientations, and without taking the non-relativistic limit) for the beautiful correspondence between classical spinning black holes and massive spinning quantum particles minimally coupled to gravity, as advertised in chapter~\\ref{chap:intro}. Let us now turn to a striking application of this on-shell relationship.\n\\chapter{A worldsheet for Kerr}\n\\label{chap:worldsheet}\n\n\\section{NJ shifts from amplitude exponentiation}\n\\label{sec:NJintro}\n\nThe Newman--Janis (NJ) shift in equation~\\eqref{eqn:NJshift} is a remarkable exact property of the Kerr solution, relating it to the simpler non-spinning Schwarzschild solution via means of a complex translation.\nA partial understanding of this phenomenon is available in the context of minimally-coupled scattering amplitudes. \nRather than consider field representations of a definite spin as in the last chapter, here it is more instructive to follow Arkani--Hamed, Huang and Huang~\\cite{Arkani-Hamed:2017jhn}, and consider massive little group representations of arbitrary spin. Specifically, we will introduce spinors $\\ket{p_I}$ and $|p_I]$ with $SU(2)$ little group indices $I=1,2$, such that that the momentum is written\\footnote{We adopt the conventions of ref.~\\cite{Chung:2018kqs}.}\n\\[\np^\\mu = \\frac12 \\epsilon^{IJ} \\bra{p_J} \\sigma^\\mu | p_I] = \\frac12 \\bra{p^I} \\sigma^\\mu | p_I] \\,.\n\\]\nWe raise and lower the little group indices $I, J, \\ldots$ with two-dimensional Levi--Civita tensors, as usual. The $\\sigma^\\mu$ matrices\nare a basis of the Clifford algebra, and we use the common choice\n\\[\n\\sigma^\\mu = (1, \\sigma_x, \\sigma_y, \\sigma_z) \\,.\n\\]\nMinimally coupled three-point amplitudes take a particularly simple form when written in these spinor helicity representations. In particular, the amplitude for a spin $s$ particle of mass $m$ and charge $Q$ absorbing a photon with positive-helicity polarisation vector $\\varepsilon_k^+$ is simply given by \\cite{Chung:2018kqs,Arkani-Hamed:2019ymq}\n\\begin{equation}\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\\begin{feynman}\n\\vertex (v1);\n\\vertex [below left = 0.66 and 1 of v1] (i1);\n\\vertex [below left = 0.5 and 0.9 of v1] (i11) {$1^s$};\n\\vertex [below right= 0.66 and 1 of v1] (o1);\n\\vertex [below right = 0.5 and 0.9 of v1] (o11) {$2^s$};\n\\vertex [above = 1 of v1] (v2) ;\n\\vertex [above = 1 of v1] (v22) {$ {}_{+}$};\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (o1);\n\\diagram*{(v2) -- [photon,momentum=\\(k\\)] (v1)};\n\\filldraw [color=black] (v1) circle [radius=2pt];\n\\end{feynman}\t\n\\end{tikzpicture} = - \\frac{Q}{\\sqrt{\\hbar}}\\, \\varepsilon^{+}_k\\cdot(p_1 + p_2)\\,\\frac{\\langle p_1^I \\, p_2^J\\rangle^{\\odot 2s}}{m^{2s}}\\,,\n\\end{equation}\nwhere the exponent includes a prescription to symmetrise over the little group indices. We can deduce the classical limit of this generic amplitude in the same manner as in section~\\ref{sec:amplitudes} for specific field representations. The photon momentum $k$ behaves as a wavenumber in the limit, so the outgoing momentum $p_2$ can be viewed as an infinitesimal Lorentz boost, with generators~\\eqref{eqn:LorentzParameters}. In terms of spinors,\n\\begin{equation}\n|p_2^I\\rangle = |p_1^I\\rangle - \\frac{\\hbar}{2m_1} \\wn k \\cdot \\sigma \\,|p_1^I]\\,.\n\\end{equation}\nAs in our earlier examples we expand the amplitude in the spin polarisation vector, which in these variables is in general given by \\cite{Arkani-Hamed:2019ymq,Chung:2018kqs}\n\\begin{equation}\ns^\\mu_{IJ}(p) = \\frac{1}{m} s\\hbar \\langle p_I| \\sigma^\\mu|p_J]\\,.\n\\end{equation}\nFrom our single particle wavefunctions in chapter~\\ref{chap:pointParticles}, we know that as $\\hbar\\rightarrow0$ the spin $s$ must tend to infinity (since it is the size of the little group representation), such that the combination $s \\hbar$ appearing in state expectation values is finite. Thus, in the classical limit, the three-point amplitude is\n\\begin{equation}\n\\mathcal{A}_{3,+} = - \\frac{2Q}{\\sqrt{\\hbar}} (p_1\\cdot\\varepsilon_k^+) \\lim_{s\\rightarrow\\infty} \\left(\\mathbb{I} - \\frac{\\bar k \\cdot a}{2s}\\right)^{2s} = - \\frac{2Q}{\\sqrt{\\hbar}} (p_1\\cdot\\varepsilon_k^+) e^{-\\bar k \\cdot a}\\,,\n\\end{equation}\nwhere recall that $a^\\mu = s^\\mu\/m$. The denominator $\\hbar$ factor is a remnant of the correct normalisation for amplitude coupling constants from section~\\ref{sec:RestoringHBar} --- such factors will be unimportant in this chapter and thus uniformly neglected. \n\nSince $-2Q(p_1\\cdot\\varepsilon_k^+) = \\mathcal{A}_{3,+}^\\textrm{Coulomb}$ is the usual QED amplitude for a scalar of charge $Q$ and momentum $p$ absorbing a positive-helicity photon, we can conclude that\n\\[\n\\mathcal{A}_{3,+}^{\\sqrt{\\text{Kerr}}} =\ne^{-\\wn k \\cdot a} \\mathcal{A}_{3,+}^{\\text{Coulomb}} \\,.\n\\label{eq:rootKerrAmp}\n\\]\nSimilarly, the gravitational three-point amplitude for a massive particle is\n\\[\n\\mathcal{M}_{3,+}^{{\\text{Kerr}}} =\ne^{-\\wn k \\cdot a} \\mathcal{M}_{3,+}^{\\text{Schwarzschild}} \\,,\n\\]\nin terms of the ``Schwarzschild'' amplitude for a scalar particle\ninteracting with a positive-helicity graviton of momentum $k$.\nA straightforward way to establish\nthe connection of these amplitudes to spinning black holes\n\\cite{Guevara:2018wpp,Chung:2018kqs,Guevara:2019fsj}\nis to then use them to compute the impulse on a scalar probe at leading-order in the Kerr background \\cite{Arkani-Hamed:2019ymq}.\nThe calculation can be performed using the formalism of this thesis on the one hand, in particular equation~\\eqref{eqn:impulseGeneralTerm1classicalLO}; or\nusing classical equations of motion on the other.\nA direct comparison of the two approaches makes it evident\n\\cite{Guevara:2019fsj,Arkani-Hamed:2019ymq}\nthat the NJ shift of the background\nis captured by the exponential factors $e^{\\pm k\\cdot a}$.\n\nThis connection between the NJ shift and scattering amplitudes\nsuggests that the NJ shift should extend beyond the exact Kerr solution\nto the \\emph{interactions} of spinning black holes.\nIndeed, it is straightforward to scatter two Kerr particles\n(by which we mean massive particles with classical spin lengths\n$a_1$ and $a_2$)\noff one another using amplitudes.\nThe purpose of this chapter is to investigate\nthe classical interpretation of this fact.\nTo do so, we turn to the classical effective theory describing the worldline interactions of a Kerr particle \\cite{Porto:2005ac,Porto:2006bt,Porto:2008tb,Steinhoff:2015ksa,Levi:2015msa}.\nWe will see that the NJ property endows this worldline action\nwith a remarkable two-dimensional worldsheet structure.\nThe Newman--Janis story emerges via Stokes's\ntheorem on this worldsheet with boundary, and indeed persists\nfor at least the leading interactions.\nWe will see that novel equations of\nmotion, making use of the spinor-helicity formalism in a purely classical context, allow us to make the shift manifest in the leading interactions.\n\nOur effective action is constructed only from the information in the three-point amplitudes. At higher orders, information from four-point\nand higher amplitudes (or similar sources) is necessary to fully specify the effective action. Therefore our action is in principle supplemented\nby an infinite tower of higher-order operators. We may hope, however, that the worldsheet structure may itself constrain the allowed higher-dimension operators. \n\nAs further applications of our methods, we will use a generalisation of the Newman--Janis shift \\cite{Talbot:1969bpa} to introduce magnetic charges (in electrodynamics) and NUT parameters (in gravity) for the particles described by our equations of motion.\nAs an example, we compute the leading impulse on a probe particle with mass,\nspin and NUT charge moving in a Kerr--Taub--NUT background.\nThe charged generalisation of the NJ complex map can similarly be connected to the behaviour of three-point amplitudes in the classical limit \\cite{Moynihan:2019bor,Huang:2019cja,Chung:2019yfs,Moynihan:2020gxj,Emond:2020lwi,Kim:2020cvf}, and we will reproduce results recently derived from this perspective~\\cite{Emond:2020lwi},\nfurthermore calculating the leading angular impulse for the first time.\n\nThe material in this chapter is organised as follows. We begin our discussion in the context of electrodynamics, constructing the effective action for a $\\rootKerr$~\nprobe in an arbitrary electromagnetic background. In this case it is rather easy to understand how the worldsheet emerges. We discuss key\nproperties of the worldsheet, including the origin of the Newman--Janis shift, in this context. It turns out to be useful to perform the matching\nin a spacetime with ``split'' signature $(+,+,-,-)$, largely because the three-point amplitude does not exist on-shell in Minkowski space.\nThe structure of the worldsheet is particularly simple\nin split-signature spacetimes.\nIn section~\\ref{sec:gr} we turn to the gravitational case, showing that the worldsheet naturally describes the dynamics of a spinning Kerr particle.\nWe discuss equations of motion in section~\\ref{sec:spinorEOM},\nfocussing on the leading-order interactions which are not sensitive to\nterms in the effective action which we have not constrained. In this section, we will see how useful the methods of spinor-helicity are for capturing the\nchiral dynamics associated with the NJ shift, as well as magnetic charges.\n\nThis chapter is based on material first published in \\cite{Guevara:2020xjx}, in collaboration with Alfredo Guevara, Alexander Ochirov, Donal O'Connell and Justin Vines.\n\n\\section{From amplitude to action}\n\\label{sec:rootKerrEFT}\n\nWe begin by concentrating on the slightly simpler example of\nthe $\\rootKerr$~particle in electromagnetism.\nWe wish to construct an effective action\nfor a massive, charged particle with spin angular momentum $S^{\\mu\\nu}$.\nBuilding on the work of Porto, Rothstein, Levi and Steinhoff~\\cite{Porto:2005ac,Porto:2006bt,Porto:2008tb,Levi:2014gsa,Levi:2015msa},\nwe write the worldline action as\n\\[\nS = \\int\\!\\d\\tau \\bigg\\{ {-m}\\sqrt{u^2} - \\frac12 S_{\\mu\\nu} \\Omega^{\\mu\\nu} - Q A \\cdot u \\bigg\\} +  S_\\text{EFT} \\,,\n\\label{eq:fullSimpleAction}\n\\]\nwhere $u^\\mu$ and $\\Omega^{\\mu\\nu}$ are the linear\nand angular velocities,\\footnote{We will be fixing $\\tau$\nto be the proper time,\nso the velocity $u^\\mu = \\d r^\\mu\/\\d\\tau$ will satisfy $u^2=1  $.\nThe angular velocity can be defined through a body-fixed frame $e^a_\\mu(\\tau)$ on the worldline as\n\\[\n\\Omega^{\\mu\\nu}(\\tau) = e^\\mu_a(\\tau) \\frac{\\d~}{\\d\\tau} e^{a\\nu}(\\tau) \\,.\n\\label{eq:angMom}\n\\]\nThe tetrad allows us to pass from body-fixed frame indices $a, b, \\ldots$\nto Lorentz indices $\\mu,\\nu, \\ldots $, as usual.\nMore details on spinning particles in effective theory\ncan be found in recent reviews~\\cite{Porto:2016pyg,Levi:2018nxp}.}\nand $S_\\text{EFT}$ contains additional operators\ncoupling the spinning particle to the electromagnetic field. The momentum is defined as the canonical conjugate of the velocity,\n\\begin{equation}\np_\\mu = -\\frac{\\partial L}{\\partial u^\\mu} = m u_\\mu + {\\cal O}(A)\\,.\n\\end{equation}\nWe will continue to assume the spin tensor to be transverse\naccording to the Tulczyjew covariant spin supplementary condition in equation~\\eqref{eqn:SSC}.\nWe can therefore relate the spin angular momentum to the spin pseudovector $a^\\mu$ by\n\\[\na^{\\mu} = \\frac1{2p^2} \\epsilon^{\\mu\\nu\\rho\\sigma} p_\\nu S_{\\rho\\sigma} \\equiv \\frac{1}{2p^2} \\epsilon^\\mu(p,S)\\qquad \\Leftrightarrow \\qquad\nS_{\\mu\\nu} = \\epsilon_{\\mu\\nu}(p, a) \\,.\n\\label{eq:spinEquivs}\n\\]\nThe effective action~\\eqref{eq:fullSimpleAction} can be written independently of the choice of SSC, at the expense of introducing an additional term from minimal coupling \\cite{Yee:1993ya,Porto:2008tb,Steinhoff:2015ksa}. This has played an important role in recent work pushing the gravitational effective action beyond linear-in-curvature terms \\cite{Levi:2020kvb,Levi:2020uwu,Levi:2020lfn},\nbut for our present purposes a fixed SSC will suffice.\nNote that any differences in the choice of the spin tensor $S_{\\mu\\nu}$\nare projected out from the pseudovector $a^\\mu$ by definition,\nand it is the latter that will be central to our discussion.\n\nWe will only consider the effective operators in $S_\\text{EFT}$\nthat involve one power of the electromagnetic field $A_\\mu$,\nwhich can be fixed by the three-point amplitudes.\nSince these amplitudes are parity-even, \nthe possible single-photon operators are\n\\begin{multline}\nS_\\text{EFT} = Q\\sum_{n=1}^\\infty \\int\\!\\d\\tau \\, u^\\mu a^\\nu\n\\Big[ B_n  (a\\cdot \\partial)^{2n-2} F_{\\mu\\nu}^*(x)  \\\\ + C_n (a \\cdot \\partial)^{2n-1} F_{\\mu\\nu}(x) \\Big]_{x=r(\\tau)} \\,.\n\\label{eq:introEFT}\n\\end{multline}\nNotice that an odd number of spin pseudovectors is accompanied\nby the dual field strength\n\\[\nF^*_{\\mu\\nu} = \\frac12 \\epsilon_{\\mu\\nu\\rho\\sigma} F^{\\rho \\sigma} \\,,\n\\]\nwhile the plain field strength goes together with an even power of $a$.\nBy dimensional analysis,\nthe unknown constant coefficients $B_n$ and $C_n$ are dimensionless. \n\n\\subsection{Worldsheet from source}\n\nTo determine the unknown coefficients,\nwe choose to match our effective action\nto a quantity that can be derived directly from the three-point\n$\\rootKerr$~amplitude~\\eqref{eq:rootKerrAmp}.\nA convenient choice is the classical Maxwell spinor given by the amplitude for an incoming photon,\nwhich is~\\cite{Monteiro:2020plf}\n\\[\n\\phi(x) = -\\frac{\\sqrt{2}}{m} \\Re \\int\\!\\d\\Phi(\\wn k) \\, \\del(\\wn k \\cdot u) \\, \\ket{\\wn k} \\bra{\\wn k} \\, e^{- i \\wn k \\cdot x } \\mathcal{A}_{3,+} \\,.\n\\]\nIn this expression the integration is over on-shell massless phase space, cast in the notation of equation~\\eqref{eqn:dfDefinition}.\nThis Maxwell spinor is defined in (2,2) signature.\nIndeed, in Minkowski space the only solution of the zero-energy condition $\\wn k \\cdot u$ for\na massless, on-shell momentum\nis $\\wn k^\\mu=0$, so the three-point amplitude cannot exist on-shell for non-trivial kinematics.\nHowever, there is no such issue in (2,2) signature, which motivates analytically continuing from Minkowski space. (The spinor $\\ket{\\wn k}$ is constructed from the on-shell null momentum $\\wn k$ as usual in spinor-helicity.)\n\nIn fact, the Newman--Janis shift makes it extremely natural for us to analytically continue to split signature even in the classical sense, without any consideration of three-point amplitudes.\nThe Maxwell spinor for a static $\\rootKerr$~particle is explicitly\n\\[\n\\phi^{\\sqrt{\\text{Kerr}}}(x) = -\\frac{Q}{4\\pi} \\frac{1}{(x^2 + y^2 + (z + i a)^2)^{3\/2}} (x, y, z + ia) \\cdot \\boldsymbol{\\sigma} \\,.\n\\label{eq:phiKerrMink}\n\\]\nIn preparation for the analytic continuation $z= -iz'$,\nwe may choose to order the Pauli matrices\nas $\\boldsymbol{\\sigma}=(\\sigma_z, \\sigma_x, \\sigma_y)$.\nThen the spinor structure in equation~\\eqref{eq:phiKerrMink} becomes real,\nwhile the radial fall-off factor in the Maxwell spinor simplifies to\n\\[\n\\frac{1}{(x^2 + y^2 - (z - a)^2)^{3\/2}} \\,, \\nonumber\n\\]\nwhere we have dropped the prime sign of $z$.\nIn short, we have a real Maxwell spinor in (2,2) signature,\nand the spin $a$ is now a real translation in the timelike $z$ direction.\n\nWe now analytically continue the action~\\eqref{eq:introEFT}\nby choosing the spin direction to become timelike.\nIn doing so, we also continue the component of the EM field in \nthe spin direction, consistent with a covariant derivative\n$\\partial + i Q A$.\nIn split signature, it is convenient to rewrite the effective action\nansatz in terms of self- and anti-self-dual field strengths,\nwhich we define as\n\\[\nF^\\pm_{\\mu\\nu}(x) = F_{\\mu\\nu}(x) \\pm F^*_{\\mu\\nu}(x) \\,.\n\\]\nOur action then depends on a new set of unknown Wilson coefficients\n$\\tilde{B}_n$ and $\\tilde{C}_n$:\n\\[\nS_\\text{EFT} = Q \\sum_{n=0}^\\infty \\int\\!\\d\\tau \\, u^\\mu a^\\nu\n\\big[ \\tilde{B}_n (a \\cdot \\partial)^n F_{\\mu\\nu}^+(x)\n+ \\tilde{C}_n (a \\cdot \\partial)^n F_{\\mu\\nu}^-(x)\n\\big]_{x=r(\\tau)} \\,.\\label{eq:2,2actionUnmatched}\n\\]\nTo determine these coefficients we can match to the three-point amplitude by computing the Maxwell spinor for the radiation field sourced by the $\\rootKerr$~particle, which we assume to have constant spin $a^\\mu$ and constant proper velocity $u^\\mu$. In (2,2) signature the exponential factor in equation~\\eqref{eq:rootKerrAmp} also picks up a factor $-i$, so we match our action to\n\\[\n\\phi(x) = -\\frac{\\sqrt{2}}{m} \\Re\\! \\int\\! \\d\\Phi(\\wn k) \\, \\del(\\wn k \\cdot u) \\, \\ket{\\wn k} \\bra{\\wn k} \\, e^{- i \\wn k \\cdot x } \\mathcal{A}_{3,+}^\\text{Coulomb} e^{i \\wn k\\cdot a} \\,.\\label{eq:rKthreePointSpinor}\n\\]\nOur matching calculation hinges upon the field strength sourced by the particle, which is determined by the $\\rootKerr$~worldline current $\\tilde j^{\\mu}(\\wn k)$. In solving the Maxwell equation we impose retarded boundary conditions precisely as in~\\cite{Monteiro:2020plf}, placing our observation point $x$ in the future with respect to one time coordinate $t^0$, but choosing the proper velocity $u$ to point along the orthogonal time direction. It is useful to make use of the result\n\\[\n\\frac{1}{\\wn k^2_\\text{ret}} = -i \\sign \\wn k^0 \\del(\\wn k^2) + \\frac{1}{\\wn k^2_\\text{adv}} \\,,\n\\]\nwhere the $\\text{ret}$ and $\\text{adv}$ subscripts indicate retarded and advanced Green's functions respectively. Since the advanced Green's\nfunction has support for $t^0 < 0$, we may simply replace\n\\[\n\\frac{1}{\\wn k^2_\\text{ret}} = -i \\sign(\\wn k^0) \\del(\\wn k^2)  \\,.\\label{eq:2,2retardedProp}\n\\]\nThe field strength sourced by the current in split signature is therefore\n\\[\nF^{\\mu\\nu}(x)  &= 2\\!\\int\\!\\dd^4\\wn k\\, \\sign(\\wn k^0) \\del(\\wn k^2) \\, \\wn k^{[\\mu} \\tilde j^{\\nu]}\\,e^{-i\\wn k\\cdot x}\\\\\n&= 2\\!\\int\\!\\dd^4\\wn k\\, \\left(\\Theta(\\wn k^0) - \\Theta(-\\wn k^0)\\right) \\del(\\wn k^2) \\,  \\wn k^{[\\mu} \\tilde j^{\\nu]}\\,e^{-\\wn ik\\cdot x}\\,,\n\\]\nNotice that the appropriate integral measure is now precisely the invariant phase-space measure~\\eqref{eqn:dfDefinition}; substituting the worldline current for our $\\rootKerr$~effective action in equation~\\eqref{eq:2,2actionUnmatched}, evaluated on a leading-order trajectory, we thus have\n\\begin{multline}\nF^{\\mu\\nu}(x) = 4Q \\Re\\! \\int\\!\\d\\Phi(\\wn k)\\,\\del(\\wn k\\cdot u) \\bigg\\{\\wn k^{[\\mu} u^{\\nu]}\\Big[1 + ia\\cdot \\wn k \\sum_{n=0}^\\infty \\left(\\tilde B_n (ia\\cdot \\wn k)^{n} + \\tilde C_n (ia\\cdot \\wn k)^{n}\\right)\\Big]\\\\\n+ i\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\sum_{n=0}^\\infty\\left(\\tilde B_n(ia\\cdot \\wn k)^{n} - \\tilde C_n (ia\\cdot \\wn k)^{n}\\right)\\bigg\\} \\, e^{-i\\wn k\\cdot x}\\,.\\label{eq:unmatchedField}\n\\end{multline}\n\nTo match to the three-point $\\rootKerr$~amplitude, we need to compute the Maxwell spinor $\\phi$ and its conjugate, $\\tilde \\phi$. To do so, we introduce a basis of positive and negative helicity polarisation vectors $\\varepsilon_k^{\\pm}$. On the support of the delta function in~\\eqref{eq:unmatchedField}, manipulations using the spinorial form of the polarisation vectors given in \\cite{Monteiro:2020plf} then lead to\n\\[\n\\wn k^{[\\mu} u^{\\nu]} \\sigma_{\\mu\\nu} &= +\\frac{\\sqrt{2}}{2}\n \\varepsilon_k^+\\cdot u\\, |\\wn k\\rangle \\langle \\wn k|\\\\\n\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} a\\cdot \\wn k\\ \\varepsilon_k^+\\cdot u \\,|\\wn k\\rangle \\langle \\wn k|\\,.\n\\]\nThe latter equality relies upon the identity $\\wn k^{[\\mu} \\epsilon^{\\nu\\rho\\sigma\\lambda]} = 0$, and the fact that $\\sigma_{\\mu\\nu}$ is self-dual in this signature. With these expressions in hand, it is easy to see that the Maxwell spinor has a common spinorial basis, and takes the simple form\n\\begin{multline}\n\\phi(x) = 2\\sqrt{2} Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k\\rangle \\langle \\wn k| \\,\\varepsilon_k^+\\cdot u  \\\\ \\times \\left(1 + \\sum_{n=0}^\\infty 2 \\tilde C_n (ia\\cdot\\wn k)^{n+1}\\right).\\label{eq:unmatchedMaxwell}\n\\end{multline}\nRecall from equation~\\eqref{eq:2,2actionUnmatched} that the Wilson coefficients $\\tilde B_n$ and $\\tilde C_n$ were identified with self- and anti-self-dual field strengths, respectively. Since a positive-helicity wave is associated with an anti-self-dual field strength, it is no surprise that the Maxwell spinor should depend only on this part of the $\\rootKerr$~effective action. Fixing the $\\tilde B_n$ coefficients requires the dual spinor, which is given by\n\\begin{multline}\n\\tilde \\phi(x) = -2\\sqrt{2} Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k]\\, [\\wn k|\\, \\varepsilon_k^-\\cdot u \\\\ \\times \\left(1 + \\sum_{n=0}^\\infty 2 \\tilde B_n (ia\\cdot\\wn k)^{n+1}\\right)\\,.\n\\label{eq:unmatchedMaxwellDual}\n\\end{multline}\nHere we have used that\n\\[\n\\wn k^{[\\mu} u^{\\nu]} \\tilde\\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} \\varepsilon_k^-\\cdot u\\, |\\wn k]\\,[\\wn k|\\\\\n\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\tilde\\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} a\\cdot\\wn k\\ \\varepsilon_k^-\\cdot u \\,|\\wn k]\\,[\\wn k|\\,,\n\\]\nrecalling that $\\tilde \\sigma$ is anti-self-dual in split signature spacetimes.\n\nIt now only remains to match to the Maxwell spinors for the three-point~amplitude, as given in equation~\\eqref{eq:rKthreePointSpinor}. The scalar Coulomb amplitudes for photon absorption are just $\\mathcal{A}_\\pm = -2mQ\\, u\\cdot\\varepsilon_k^\\pm$, so for the $\\rootKerr$~three-point amplitude\n\\[\n\\phi(x) &= + 2\\sqrt{2}  Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k\\rangle \\langle\\wn k| \\, \\varepsilon_k^+\\cdot u \\, e^{i\\wn k\\cdot a}\\,,\\\\\n\\tilde\\phi(x) &= -2\\sqrt{2}  Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\,  |\\wn k]\\, [\\wn k|\\, \\varepsilon_k^-\\cdot u\\,e^{-i \\wn k\\cdot a}\\,.\n\\]\nExpanding the exponentials and matching to eqs.~\\eqref{eq:unmatchedMaxwell} and \\eqref{eq:unmatchedMaxwellDual} identifies\n\\begin{equation}\n\\tilde B_n = \\frac{(-1)^{n+1}}{2(n+1)!}\\,, \\qquad \\quad\n\\tilde C_n = \\frac{1}{2(n+1)!}\\,,\n\\end{equation}\nwhich upon substitution into equation~\\eqref{eq:2,2actionUnmatched} finally yields the $\\rootKerr$~effective action in split-signature:\n\\[\nS_\\text{EFT}  &= Q \\sum_{n=0}^\\infty \\int\\!\\d\\tau\\, u^\\mu a^\\nu\n\\bigg[ {-\\frac{(-a\\cdot \\partial)^n}{2(n+1)!}}  F_{\\mu\\nu}^+(x)\n+ \\frac{(a\\cdot \\partial)^n}{2(n+1)!} F_{\\mu\\nu}^-(x)\n\\bigg]_{x=r(\\tau)} \\\\ \n=& -\\frac{Q}{2} \\int\\!\\d\\tau\\, u^\\mu a^\\nu\n\\bigg[ \\left(\\frac{e^{-a \\cdot \\partial} - 1}{-a \\cdot \\partial}\\right) F_{\\mu\\nu}^+(x)\n- \\left(\\frac{e^{ a \\cdot \\partial} - 1}{a \\cdot \\partial}\\right) F_{\\mu\\nu}^-(x)\n\\bigg]_{x=r(\\tau)} \\,.\\label{eq:2,2actionMatchedCoefficients}\n\\]\n\nSo far, the Newman--Janis structure is hinted at by the translation\noperators $e^{\\pm a \\cdot \\partial}$ appearing in the effective action.\nWe can make this structure more manifest by writing the effective action equivalently as\n\\[\nS_\\text{EFT} & = -\\frac{Q}{2}\n\\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\, u^\\mu a^\\nu\n\\big[ e^{-\\lambda \n(a \\cdot \\partial)} F_{\\mu\\nu}^+(x)\n- e^{\\lambda (a \\cdot \\partial)} F_{\\mu\\nu}^-(x)\n\\big]_{x=r(\\tau)} \\\\ &\n= -\\frac{Q}{2} \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\, u^\\mu a^\\nu\n\\big[ F^+_{\\mu\\nu}(r - \\lambda a) - F^-_{\\mu\\nu}(r + \\lambda a) \\big] \\,.\n\\label{eq:seftStep}\n\\]\nOur effective action is now an integral over a two-dimensional region\n--- a worldsheet, rather than a worldline.\n\nTo see that this worldsheet is indeed connected to the Newman--Janis shift,\nlet us recover this shift for the Maxwell spinor.\nFirst, we can read off the worldsheet current~$J^\\mu$ from the action\n$S_\\text{EFT}-Q\\!\\int\\!\\d\\tau A_\\mu u^\\mu = -\\!\\int\\!\\d^4x A_\\mu J^\\mu$.\nThen, the gauge field $A_\\mu$ set up at a point~$x$ by this source\nmay be written as an integral of a Green's function $G(x-y)$ over the worldsheet:\n\\begin{align}\nA^\\mu(x)  = \\int\\!\\d^4y\\,G(x-y) J^\\mu(y) \\hphantom{+ \\frac{1}{2}\\!\\int_0^1\\!\\d\\lambda}&\\\\ \n=  Q\\!\\int\\!\\d\\tau\n\\bigg\\{ u^\\mu G(x-r)\n+ \\frac{1}{2}\\!\\int_0^1\\!\\d\\lambda \\Big(\n&\\big[ u^\\mu (a \\cdot \\partial)\n+ \\epsilon^{\\mu}(u, a, \\partial)\n\\big] G(x-r+\\lambda a) \\nonumber \\\\ \n- &\\big[ u^\\mu (a \\cdot \\partial)\n- \\epsilon^{\\mu} (u, a, \\partial)\n\\big] G(x-r-\\lambda a) \\Big)\\!\n\\bigg\\} \\,. \\nonumber\n\\end{align}\nThe field strength follows by differentiation, after which contraction with $\\sigma$ matrices yields the Maxwell spinor,\n\\[\n\\phi(x) = 2Q\\!\\int\\!\\d\\tau\\, \\sigma_{\\mu\\nu}\nu^{\\nu} \\partial^{\\mu}\\! \\left[ G(x-r) - \\int_0^1\\!\\d\\lambda\n\\,(a \\cdot \\partial) G(x-r-\\lambda a) \\right] ,\n\\label{eq:NJbyintegration}\n\\]\nwhere the first term comes from the non-spinning part of the action~\\eqref{eq:fullSimpleAction}.\nNow the $a\\cdot\\partial$ operator acting on the Green's function\ncan be understood as a derivative with respect to $\\lambda$.\nThis produces a $\\lambda$ integral of a total derivative,\nwhich reduces to the boundary terms.\nCancelling the first term in equation~\\eqref{eq:NJbyintegration}\nagainst the boundary contribution at $\\lambda = 0$, we find simply that\n\\[\n\\phi(x) = 2Q\\!\\int\\!\\d\\tau \\, \\sigma_{\\mu\\nu} u^\\nu \\partial^\\mu  G(x-r-a)  \\,.\n\\]\nThe Maxwell spinor depends only on the anti-self-dual part of the effective action, shifted by the spin length. The real translation in $(2,2)$ signature\nis a result of this real worldsheet structure. We will shortly see that this structure persists for interactions.\n\n\n\n\\subsection{Worldsheet for interactions}\n\nLet us analytically continue\nthe action~\\eqref{eq:seftStep} back to Minkowski space:\n\\[\nS_\\text{EFT} & = \\frac{Q}{2} \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\nu^\\mu a^\\nu \\big[ i F^+_{\\mu\\nu}(r + i \\lambda a)\n- i F^-_{\\mu\\nu}(r - i \\lambda a) \\big] \\\\ &\n= Q \\Re \\int_\\Sigma \\! \\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(r + i \\lambda a) \\, u^\\mu a^\\nu \\,.\n\\]\nHere the self-dual and anti-self-dual field strengths are\n\\[\nF^\\pm_{\\mu\\nu} = F_{\\mu\\nu} \\pm i F^*_{\\mu\\nu}\n= \\pm \\frac{i}{2} \\epsilon_{\\mu\\nu\\rho\\sigma} F^{\\pm\\,\\rho\\sigma} \\,,\n\\label{eq:rKwsaction}\n\\]\nand $\\Sigma$ is the worldsheet, with $\\tau$ running over $(-\\infty,\\infty)$, and $\\lambda$ over $[0,1]$.\n\nNow that we are back in Minkowski space,\nlet us turn to the Newman--Janis structure of interactions.\nSuppose that our spinning particle is moving\nunder the influence of an external electromagnetic field, generated by distant sources. The total interaction Lagrangian contains the worldsheet\nterm~\\eqref{eq:rKwsaction} as well as the usual worldline minimal coupling:\n\\[\nS_\\text{int} = -Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!\\!\\d\\tau A_\\mu(r) u^\\mu\n+ Q \\Re\\! \\int_\\Sigma\\!\\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(r + i \\lambda a) \\, u^\\mu a^\\nu + \\ldots \\,,\n\\label{eq:sint}\n\\]\nwhere $\\partial \\Sigma_\\text{n}$ is the ``near'' boundary of the worldsheet, at $\\lambda = 0$, as shown in figure~\\ref{rootKerrIntegration}. We will similarly refer to the boundary at $\\lambda = 1$ as the ``far'' boundary. The near boundary is the physical location of the object, while the far boundary is a timelike line embedded in the complexification of Minkowski space. We have also indicated the presence of unknown additional operators (involving at least two powers of the field strength) in the action by the ellipsis in equation~\\eqref{eq:sint}.\n\nIt is convenient to introduce a complex coordinate $z = r + i \\lambda a$ on the worldsheet. In terms of this coordinate, we may write the two-form\n\\[\nF^+(z) =\n\\frac{1}{2} F^+_{\\mu\\nu}(z) \\d z^\\mu \\wedge \\d z^\\nu = i F^+_{\\mu\\nu}(z) (u^\\mu + i \\lambda \\dot a^\\mu) a^\\nu \\, \\d \\tau \\wedge \\d \\lambda \\,,\n\\]\nwhere $\\dot a^\\mu = \\d a^\\mu \/ \\d \\tau$. Since in the absence of interactions the spin is constant, $\\dot a$ must be of order $F$.\nTherefore, we may rewrite our interaction action as\n\\[\nS_\\text{int} = -Q \\int_{\\partial \\Sigma_\\text{n}} \\!\\!\\! A_\\mu(r) \\d r^\\mu\n+ \\frac{Q}{2} \\Re\\! \\int_\\Sigma F^+_{\\mu\\nu}(z) \\, \\d z^\\mu \\wedge \\d z^\\nu + \\ldots \\,.\n\\label{eq:sintNicer}\n\\]\nIn doing so, we have redefined the higher-order operators indicated by the ellipsis.\n\n\\begin{figure}[t]\n\t\\center\n\t\\includegraphics[width = 0.5\\textwidth]{contour.pdf}\n\t\\vspace{-3pt}\n\t\\caption[Geometry of the Kerr worldsheet effective action.]{Geometry of the effective action: boundary $\\partial \\Sigma_\\text{n}$ of the  complex worldsheet (translucent plane) is  fixed to the particle worldline in real space (solid plane).\\label{rootKerrIntegration}}\n\\end{figure}\n\nWhen the electromagnetic fields appearing in the action~\\eqref{eq:sintNicer} are generated by external sources,\nboth $F$ and $F^*$ are closed two-forms,\nso we may introduce potentials $A$ and $A^*$\nsuch that $F = \\d A$ and $F^* = \\d A^*$.\nThe dual gauge potential $A^*$ is related to $A$ by duality,\nbut this relationship need not concern us here:\nwe only require that both potentials exist in the vicinity\nof the $\\rootKerr$~particle.\nHence we may also write $F^+ = \\d A + i\\, \\d A^* = \\d A^+$.\nThen the action~\\eqref{eq:sintNicer} becomes\n\\[\nS_\\text{int} &\n=-Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!A\n+ Q \\Re\\!\\int_\\Sigma\\!\\d A^+ + \\ldots \\\\ &\n=-Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!A\n+ Q \\Re\\!\\int_{\\partial \\Sigma_\\text{n}}\\!\\!A^+\n- Q \\Re\\!\\int_{\\partial \\Sigma_\\text{f}}\\!\\!A^+ + \\ldots\\,,\n\\]\nwhere the boundary consists of two disconnected lines (the far and near boundaries).\nThe orientation of the integration contour was set by $F^+$,\nas depicted in figure~\\ref{rootKerrIntegration}.\n\nNow, notice that on the near boundary $z = r(\\tau)$ is real.\nHence $\\Re A^+ = A$,\nso we are only left with the far-boundary contribution in the action:\n\\[\nS_\\text{int} = -Q \\Re\\!\\int_{\\partial \\Sigma_\\text{f}}\\!\\!A^+ + \\ldots\n=-Q \\Re\\!\\int\\!\\d \\tau \\, u^\\mu A_\\mu^+(r + i a) + \\ldots \\,.\n\\label{eq:rtKerrIntShift}\n\\]\nThus we explicitly see that the interactions\nof a $\\rootKerr$~particle can be described with a Newman--Janis shift.\nWe will exploit this fact explicitly in section~\\ref{sec:spinorEOM}.\nBefore we do, we turn to gravitational interactions.\n\n\n\n\\section{Spin and gravitational interactions}\n\\label{sec:gr}\n\nAs a step towards a worldsheet action for a probe Kerr in a non-trivial background, it is helpful to understand how to make the electromagnetic\neffective action~\\eqref{eq:rKwsaction} generally covariant. In a curved spacetime, we cannot simply add a vector $\\lambda a$ to a point $r$.\nTo see what to do, let us reintroduce translation operators as in equation~\\eqref{eq:seftStep}. The worldsheet EFT term in Minkowski space is\n\\[\nS_\\text{EFT} &= Q \\Re\\! \\int_\\Sigma \\! \\d\\tau \\d\\lambda\\, \ni\\, e^{i \\lambda  \\, a \\cdot \\partial} F^+_{\\mu\\nu}(x)\\, u^\\mu a^\\nu \\Big|_{x=r(\\tau)}  \\\\\n&= Q \\Re \\int_\\Sigma \\! \\d\\tau \\d\\lambda \\, i\n\\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\partial)^n  F^+_{\\mu\\nu}(x) \\, u^\\mu a^\\nu\\Big|_{x=r(\\tau)} \\,.\n\\]\nNow, it is clear that a minimal way to make this term generally covariant is to replace the partial derivatives $\\partial$ with covariant \nderivatives $\\nabla$, so in curved space we have\n\\[\nS_\\text{EFT} &= Q \\Re \\!\\int_\\Sigma \\! \\d\\tau \\d\\lambda \\,\n\\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\nabla)^n F^+_{\\mu\\nu}(x) \\, i u^\\mu a^\\nu  \\Big|_{x=r(\\tau)} \\,.\n\\label{eq:rKeftCurvedStep}\n\\]\n\nIt is therefore natural for us to consider a covariant translation operator\n\\[\ne^{i\\lambda\\,a \\cdot \\nabla} \\equiv \\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\nabla)^n \\,.\n\\label{eq:explicitTranslation}\n\\]\nThis operator generates translations along geodesics in the direction $a$.\nTo see\nwhy, note that the perturbative expansion of such a geodesic beginning at a point\n$x_0$ in the direction $a$ with parameter $\\ell$ is\n\\[\nx^\\mu(\\ell) = x_0^\\mu + \\ell \\, a^\\mu - \\frac{\\ell^2}{2} \\Gamma^\\mu_{\\nu\\rho}(x_0) a^\\nu a^\\rho + \\ldots \\,.\n\\]\nNow consider the perturbative expansion of a scalar function $f(x)$ along such a\ngeodesic. We have\n\\begin{align}\nf(x(\\ell)) &= f(x_0) + \\ell \\, a^\\mu \\partial_\\mu f(x_0) + \\frac{\\ell^2}2 a^\\mu a^\\nu \n\\left(\n\\partial_\\mu \\partial_\\nu f(x_0) - \\Gamma^\\alpha_{\\mu\\nu}(x_0) \\partial_\\alpha f(x_0)\n\\right) \n+ \\ldots \\nonumber\\\\\n&=\nf(x_0) + \\ell (a \\cdot \\nabla) f(x_0) + \\frac{\\ell^2}2 (a \\cdot \\nabla) (a \\cdot \\nabla) f(x_0)\n+ \\ldots \\\\\n&= e^{\\ell \\, a\\cdot \\nabla} f(x_0) \\,.\\nonumber\n\\end{align}\n\nA traditional point of view on equation~\\eqref{eq:rKeftCurvedStep} is that the\noperators only act on the two-form $F^+_{\\mu\\nu}$. However, we can alternatively\nthink of the operator acting on a scalar function $F^+_{\\mu\\nu} u^\\mu a^\\nu$, \nprovided we extend the definitions of the velocity $u$ and the spin $a$\nso that they become fields on the domain of the translation operator. We can simply\ndo this by parallel-transporting $u(r(\\tau))$ \nand $a(r(\\tau))$ along the geodesic beginning at $r(\\tau)$ in the direction $a(\\tau)$\n(using the Levi--Civita connection). We denote these geodesics by $z(\\tau, \\lambda)$; \nexplicitly,\n\\[\nz^\\mu(\\tau, \\lambda) = r^\\mu(\\tau) + i \\lambda a^\\mu(\\tau) + \\frac{\\lambda^2}{2} \\Gamma^\\mu_{\\nu\\rho}(r(\\tau)) a^\\nu a^\\rho + \\ldots \\,.\n\\]\n(Notice that the translation operator~\\eqref{eq:explicitTranslation} has parameter\n$i \\lambda$.)\nThe parallel-transported vectors, with initial conditions $a(z(\\tau, 0)) = a(\\tau)$ \nand $u(z(\\tau, 0)) = u(\\tau)$,\nhave the similar perturbative expansions\n\\[\nu^\\mu(z(\\tau, \\lambda)) &= u^\\mu(\\tau) - i \\lambda \\Gamma^\\mu_{\\nu\\rho} (r(\\tau)) \na^\\nu(\\tau) u^\\rho(\\tau) + \\ldots \\,,\\\\\na^\\mu(z(\\tau, \\lambda)) &= a^\\mu(\\tau) - i \\lambda \\Gamma^\\mu_{\\nu\\rho} (r(\\tau)) \na^\\nu(\\tau) a^\\rho(\\tau) + \\ldots \\,.\n\\]\nWe now view the translation operator in equation~\\eqref{eq:rKeftCurvedStep}\nas acting on the scalar quantity $F^+_{\\mu\\nu} u^\\mu a^\\nu$:\n\\[\ne^{i \\lambda a \\cdot \\nabla} F_{\\mu\\nu}(r(\\tau)) a^\\mu(\\tau) u^\\nu(\\tau) \n= F_{\\mu\\nu}(z(\\tau,\\lambda)) a^\\mu(z(\\tau,\\lambda)) u^\\nu(z(\\tau, \\lambda)) \\,.\n\\]\nExpanding perturbatively to first order in $\\lambda$, we have\n\\begin{align}\ne^{i \\lambda a \\cdot \\nabla} F_{\\mu\\nu}(r(\\tau)) a^\\mu(\\tau) u^\\nu(\\tau)\n&= \\left(F_{\\mu\\nu}(r(\\tau)) + i \\lambda a^\\rho \\left(\n\\partial_\\rho F_{\\mu\\nu} - \\Gamma^\\alpha_{\\mu\\rho} F_{\\alpha\\nu} \n-\\Gamma^\\alpha_{\\nu\\rho} F_{\\mu \\alpha} \n\\right)\\right) a^\\mu u^\\nu \\nonumber \\\\\n&= \\left( F_{\\mu\\nu}(r(\\tau)) + i \\lambda a^\\rho \\nabla_\\rho F_{\\mu\\nu}\\right)a^\\mu u^\\nu \\,.\n\\end{align}\nThe final expression is precisely the same as the picture in which \nthe derivatives act only on the field strength: these are equivalent points of\nview.\n\nThe worldsheet arises from interpreting the translation operators as genuine\ntranslations. In curved space, the operators replace the straight-line sum \n$r+i a \\lambda$ appearing in our action~\\eqref{eq:rKwsaction} with the \nnatural generalisation --- a geodesic in the direction $a$.\\footnote{In general, \nthese geodesics may become singular. We assume that such singularities do\nnot arise. If they were to arise, there would also be a divergence in the\ninterpretation of the EFT as an infinite sum of operators.}\nWe can express the curved-space effective action as\n\\[\nS_\\text{EFT} = Q \\Re \\!\\int_\\Sigma \\! \\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(z) \\, u^\\mu(z) a^\\nu(z) \\Big|_{z=z(\\tau,\\lambda)} \\,.\n\\label{eq:rkCovActionVelocity}\n\\]\nThe surface $\\Sigma$ is built up from the worldline of the particle, augmented by the geodesics in the direction $a$ for each $\\tau$.\n\nNote that, since we neglect higher-order interactions,\nwe may replace the velocity vector field $u(\\tau,\\lambda)$ \nin the action~\\eqref{eq:rkCovActionVelocity}\nwith the similarly defined momentum field $p(\\tau, \\lambda)$.\nIndeed, at $\\lambda = 0$ the difference adds another order in the gauge field,\nand this persists for $\\lambda \\neq 0$ after parallel translation along the geodesics.\nTherefore, up to $F^2$ operators that we are neglecting,\nthe $\\rootKerr$~action may be written as\n\\[\nS_\\text{EFT} = \\frac{Q}{m} \\Re\\!\\int_\\Sigma\\!\\d\\tau \\d\\lambda \\, iF^+_{\\mu\\nu}(z) \\, p^\\mu(z) a^\\nu(z) \\Big|_{z=z(\\tau,\\lambda)} \\,.\n\\label{eq:rkCovActionMomentum}\n\\]\n\nWe are now ready for the fully gravitational Kerr worldsheet action,\nwhich is naturally motivated as a classical double copy of this covariantised worldsheet action.\nRecalling that we should double-copy from non-Abelian gauge theory rather than electrodynamics, we promote the field strength to the Yang--Mills case:\n\\[\nQ F^+_{\\mu\\nu}(z) ~\\rightarrow~ c^A(z) F^{A+}_{\\mu\\nu}(z) \\,,\n\\]\nwhere $c^A(z(\\tau,\\lambda))$ is a vector in the colour space (generated by parallel transport from the classical colour vector of a particle,\nas described by the Yang--Mills--Wong equations~\\eqref{eqn:classicalWong}).\nThe double copy replaces colour by kinematics, so we anticipate a replacement of the form $c^A \\rightarrow u^\\mu$.\nMoreover, to replace $F^{A}_{\\mu\\nu}$ we need an object with three indices,\nantisymmetric in two of them,\nfor which the spin connection\n\\[\n\\omega_\\mu{}^{ab} = e^b_\\nu\\, \\nabla_\\mu e^{a\\nu} =\ne^b_\\sigma \\big( \\partial_\\mu e^{a\\sigma} + \\Gamma^\\sigma_{\\mu\\nu} e^{a\\nu} \\big)\\label{eq:spinConnection}\n\\]\nis the natural candidate.\nSince it is defined via a derivative of the (body-fixed) spacetime tetrad~$e_a^\\mu$, which is\na dimensionless quantity, on dimensional grounds the replacement should be\nof the form $F^{A}_{\\mu\\nu} \\to m \\omega_\\mu{}^{ab}$.\nIndeed, we find that the correct worldsheet action for Kerr is\n\\[\nS_\\text{EFT} = \\Re\\! \\int_\\Sigma \\d\\tau \\d \\lambda \\, i\\,u_\\mu(z) \\, \\omega^{+\\mu}{}_{ab}(z) \\, p^a(z) a^b(z) \\Big|_{z=z(\\tau,\\lambda)} \\,,\n\\label{eq:kerrws}\n\\]\nwhere $\\omega^+$ is a self-dual part of the spin connection,\ndefined explicitly by\n\\[\n\\omega^{+\\mu}{}_{ab}(x) = \\omega^{\\mu}{}_{ab}(x) + i\\,\\omega^{*\\mu}{}_{ab}(x) \\,, \\qquad\n\\omega^{*\\mu}{}_{ab}(x) = \\frac12 \\epsilon_{abcd}\\,\\omega^{\\mu\\,cd}(x) \\,.\n\\label{eq:omegaDefs}\n\\]\nIn writing these equations, we have extended the body-fixed frame $e_a = e_a^\\mu \\partial_\\mu$ of vectors to every point of the complex worldsheet. We do so by parallel transport.\nAs usual, the frame indices $a, b, \\cdots$ take values from 0 to 3,\nand $\\epsilon_{abcd}$ is the flat-space Levi--Civita tensor,\nwith $\\epsilon_{0123} = + 1$.\n\n\n\n\\subsection{Flat-space limit}\n\nWe will shortly prove that the worldsheet term~\\eqref{eq:kerrws} reproduces all single-curvature terms in the known effective action for a Kerr black hole in an arbitrary background~\\cite{Porto:2006bt,Porto:2008tb,Levi:2015msa}. But first we wish to show that the term is non-trivial even in\nflat space, and is in fact the standard kinetic term for a spinning particle in Minkowski space~\\cite{Porto:2005ac} in that context.\n\nIn flat space and Cartesian coordinates, the worldsheet effective term~\\eqref{eq:kerrws} is\n\\[\nS_\\text{EFT} = \\Re\\! \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\ni\\,u_\\mu(\\tau)\\,\\omega^{+\\mu}{}_{ab}(r + i \\lambda a)\\, p^a(\\tau) a^b(\\tau) \\,, \n\\label{eq:recoverSpinKineticStep}\n\\]\nsince the parallel transport of the vectors $u, p$ and $a$ is now trivial, and the geodesics reduce to straight lines. In flat space, \nthe frame $e^a_\\mu(\\tau, \\lambda)$ is also independent of $\\lambda$, since it is\ngenerated by parallel transport. \nThus, the spin connection is $\\lambda$-independent and the $\\lambda$ integral in equation~\\eqref{eq:recoverSpinKineticStep} becomes trivial.\n\nGiven the $\\lambda$ independence of the spin connection, we may write\n\\[\nS_\\text{EFT} & = \\Re\\! \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,i\\,\nu_\\mu(\\tau)\\,\\omega^{+\\mu}{}_{ab}(r(\\tau))\\, p^a(\\tau) a^b(\\tau) \\\\ &\n=-\\!\\int\\!\\d\\tau\\,\nu_\\mu(\\tau)\\,\\omega^{*\\mu}{}_{ab}(r(\\tau))\\, p^a(\\tau) a^b(\\tau) \\,.\n\\]\nRecalling the definitions of the dual spin connection~$\\omega^*$ and the spin pseudovector~$a$, eqs.~\\eqref{eq:omegaDefs} and \\eqref{eq:spinEquivs}, we equivalently have\n\\[\nS_\\text{EFT} = - \\frac12 \\int\\!\\d\\tau\\,\nu_\\mu(\\tau)\\, \\omega^{\\mu}{}^{ab}(r(\\tau))\\, S_{ab}(\\tau)\n=- \\frac12 \\int\\!\\d\\tau\\, \\Omega{}^{ab}(r(\\tau))\\, S_{ab}(\\tau) \\,.\\label{eq:wsSpinKinematic}\n\\]\nThis is nothing but the spin kinetic term written in equation~\\eqref{eq:fullSimpleAction}.\nIn this way, we see that the worldsheet expression~\\eqref{eq:kerrws} already describes the basic dynamics of spin.\n\n\n\\subsection{Single-Riemann effective operators}\n\nIt is now straightforward to recover the full tower of single-Riemann operators in the Kerr effective action. Returning to the full curved-space case, we may write our action~\\eqref{eq:kerrws} as\n\\[\nS_\\text{EFT} & = \\Re \\!\\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\ne^{i \\lambda \\, a \\cdot \\nabla} i\\, u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\\\ &\n= \\Re \\!\\int\\!\\d\\tau\\, i\\left(\\frac{e^{i a \\cdot \\nabla}-1}{i a \\cdot \\nabla}\\right) u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\\\ &\n= \\sum_{n=0}^\\infty\n\\Re\\!\\int\\!\\d\\tau\\, \\frac{(i a \\cdot \\nabla)^n}{(n+1)!} i\\,u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\,,\n\\]\nwhere we performed the $\\lambda$ integral and expanded the translation operator $e^{i a \\cdot \\nabla}$.\nThe leading contribution is again the spin kinetic term,\nas a short computation demonstrates.\nWe also encounter an infinite series of higher-derivative contributions for $n \\geq 1$. To express them in terms of the Riemann tensor,\nwe recall that it satisfies\n\\[\nR_{ab\\,\\mu\\nu} = e^\\alpha_a e^\\beta_b R_{\\alpha\\beta\\,\\mu\\nu}\n= -\\nabla_\\mu \\omega_{\\nu ab} + \\nabla_\\nu \\omega_{\\mu ab}\n+ \\omega_{\\mu ac}\\,\\omega_{\\nu}{}^c_{~\\,b}\n- \\omega_{\\nu ac}\\,\\omega_{\\mu}{}^c_{~\\,b} \\,.\n\\label{Connection2Riemann}\n\\]\nConsistently omitting the quadratic in $\\omega$ terms from the equation above, as well as the higher-order interaction contributions due to the difference between $p^a$ and $mu^a$,\nwe rewrite a typical effective operator as\n\\begin{align}\n& -\\frac{1}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, u^\\mu p^a a^b\n(i a \\cdot \\nabla)^{n-1} a^\\nu \\nabla_\\nu \\omega^+_{\\mu ab}\n\\big|_{x = r(\\tau)} \\nonumber \\\\ & \\quad\n=-\\frac{1}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, p^a a^b u^\\mu a^\\nu\n(i a \\cdot \\nabla)^{n-1}\n\\big[ R^+_{ab\\,\\mu\\nu} + \\nabla_\\mu \\omega^+_{\\nu ab}\n\\big]_{x = r(\\tau)} + \\ldots \\label{eq:EffOperatorSteps}\\\\ & \\quad\n=-\\frac{m}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, u^a a^b a^\\nu\n(i a \\cdot \\nabla)^{n-1}\n\\bigg[ u^\\mu R^+_{ab\\,\\mu\\nu} + \\frac{D~}{d\\tau} \\omega^+_{\\nu ab}\n\\bigg]_{x = r(\\tau)} + \\ldots \\,.\\nonumber\n\\end{align}\nHere $R^+_{ab\\,\\mu\\nu}=R_{ab\\,\\mu\\nu}+iR^*_{ab\\,\\mu\\nu}$ is defined via the dualisation of the first two indices. Notice that in equation~\\eqref{eq:EffOperatorSteps} we treat \nthe velocity $u$, momentum $p$ and spin $a$ as fields on the worldline, so that they\ncommute with the covariant derivative.\n\nWe may proceed by integrating the $D\/\\d \\tau$ term by parts, after which it acts on factors of velocity and spin.\nThis generates curvature-squared (and higher) operators that we again neglect.\nIn this way, we arrive at the form of the leading interaction Lagrangian\n\\[\nS_\\text{int} =-m\\!\\int\\!\\d\\tau\\,u^a a^b u^\\mu a^\\nu \\Re \\sum_{n=1}^\\infty\n\\frac{(i a \\cdot \\nabla)^{n-1}\\!}{(n+1)!}\n\\bigg[ R_{ab\\,\\mu\\nu} + i R^*_{ab\\,\\mu\\nu}\n\\bigg]_{x = r(\\tau)} + \\ldots \\,.\n\\]\nFinally, separating the even and odd values of $n$ into two distinct sums\n\\[\nS_\\text{int} = m\\!\\int\\!\\d\\tau\n\\bigg[ & \\sum_{n=1}^\\infty \\frac{(-1)^n}{(2n)!} (a \\cdot \\nabla)^{2n-2}\nR_{\\alpha\\beta\\,\\mu\\nu} u^\\alpha a^\\beta u^\\mu a^\\nu \\\\ &\n-  \\sum_{n=1}^\\infty \\frac{(-1)^n}{(2n+1)!}\n(a \\cdot \\nabla)^{2n-1} R^*_{\\alpha\\beta\\,\\mu\\nu} u^\\alpha a^\\beta u^\\mu a^\\nu\n\\bigg]_{x = r(\\tau)} + \\ldots \\,,\\label{eq:LSinteractions}\n\\]\none can verify that this reproduces the leading interactions of a Kerr black hole, as discussed in detail by Levi and Steinhoff~\\cite{Levi:2015msa}. \n\nIt is interesting that the\nworldsheet structure unifies the spin kinetic term with the\nleading interactions of Kerr. The same phenomenon was observed\ndirectly at the level of amplitudes in ref.~\\cite{Chung:2018kqs}.\n\n\n\n\\section{Spinorial equations of motion}\n\\label{sec:spinorEOM}\n\nEquation~\\eqref{eq:rtKerrIntShift} explicitly displays a Newman--Janis shift for the leading interactions of the $\\rootKerr$~solution. Now we take\na first look at the structure of the equations of motion encoding this shift. Since the Newman--Janis shift is chiral, we will find that it is very\nconvenient to describe the dynamics using the method of spinor-helicity, even in a fully classical setting. Our focus here will be to extract\nexpressions for observables from the equations of motion at leading order. Thus we are free to make field redefinitions, dropping total\nderivatives which do not contribute to observables. We will also extend our work to magnetically charged objects, such as spinning dyons\nand the gravitational Kerr--Taub--NUT analogue at the level of equations of motion.\n\nWe may write the leading order action for a $\\rootKerr$~particle with trajectory $r(\\tau)$ and spin $a(\\tau)$ as\n\\[\nS = - \\int \\d \\tau \\left( p \\cdot \\dot r(\\tau) + \\frac12 \\epsilon(p, a, \\Omega) + Q \\Re u^\\mu A_\\mu^+(r + i a) \\right) + \\ldots \\,.\n\\label{eq:rootKerrLeadingAction}\n\\]\nBy varying with respect to the position $r(\\tau)$ it is easy to determine that\n\\[\n\\frac{\\d p^\\mu}{\\d \\tau} &= Q \\Re F^{+\\mu\\nu}(r+ia) u_\\nu + \\ldots\n= \\frac{Q}{m} \\Re F^{+\\mu\\nu}(r+ia) p_\\nu + \\ldots \\,.\n\\label{eq:rtKerrEOMmomentum}\n\\]\nIn the second equality, we replaced the velocity $u = \\dot r$ with the momentum $p\/m$, noting that the difference between the momentum and $mu$ is of \norder $F$. To obtain a similar differential equation for the spin $a^\\mu$, it is helpful to begin by differentiating $a \\cdot p = 0$, finding\n\\[\np_\\mu \\frac{\\d a^\\mu}{\\d \\tau} = p_\\mu \\Re \\frac{Q}{m} F^{+\\mu\\nu}(r+ia) a_\\nu \\,.\n\\]\nBased on this simple result, it is easy to guess that the spin satisfies\n\\[\n\\frac{\\d a^\\mu}{\\d \\tau} = \\frac{Q}{m} \\Re  F^{+\\mu\\nu}(r+ia) a_\\nu + \\ldots \\,,\n\\label{eq:rtKerrEOMspin}\n\\]\nand indeed a more lengthy calculation using the Lagrangian~\\eqref{eq:rootKerrLeadingAction} confirms this guess.\n\nOur expressions~\\eqref{eq:rtKerrEOMmomentum} and~\\eqref{eq:rtKerrEOMspin} for the momentum and spin have the same basic structure,\nand are consistent with the requirements that $p^2$ and $a^2$ are constant while $a \\cdot p = 0$. \nAs discussed in section~\\ref{sec:NJintro}, in the context of scattering amplitudes it has proven to be very convenient to introduce spinor variables describing similar momenta; this logic also applies to spins.\nNotice that there is nothing quantum about using spinor variables for momenta and spin: the momenta of particles in amplitudes need not be\nsmall, and the spin can be arbitrarily large. We are simply taking advantage of the availability of spinorial representations of the Lorentz group.\nA key motivation for introducing spinors in the present context is the chirality structure of eqs.~\\eqref{eq:rtKerrEOMmomentum}\nand~\\eqref{eq:rtKerrEOMspin}, which hint at a more basic description using an intrinsically chiral formalism.\n\nThe little group of a massive momentum is ${SO}(3)$, so to construct the spin vector $a^\\mu$ in terms of spinors we need only\nform a little-group vector representation from little group spinors. The vector representation of ${SO}(3)$ is the symmetric tensor product of two\nspinors, so we will need to symmetrise little group indices. Let $a^{IJ}$ be a constant symmetric two-by-two matrix; then\n\\[\na^\\mu = \\frac12 a^{IJ}\\! \\bra{p_J} \\sigma^\\mu | p_I] \n\\]\nis the spin vector. To understand how these expressions work, it may be helpful to work in a Lorentz frame $p^\\mu = (\\sqrt{p^2}, 0, 0, 0)$. Then\nthe spin is a purely spatial vector, so it is a linear combination of components in the $x$, $y$ and $z$ directions. Thus there is a basis of\nthree possible spins.\nThis is reflected in the three independent components\nof the symmetric two-by-two matrix $a^{IJ}$. The algebra of the spinors\nimmediately guarantees that the spin $a$ and the momentum $p$ are orthogonal.\n\nGiven that we can always reconstruct the momentum and spin from the spinors, all we now need are dynamical equations for the spinors \nthemselves. The leading-order spinorial equations of motion for $\\rootKerr$~are\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= \\frac{Q}{2m} \\maxwell(r(\\tau) + i a(\\tau)) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &=  \\frac{Q}{2m} \\tilde \\maxwell(r(\\tau) - i a(\\tau)) |p_I] \\,.\n\\label{eq:spinorEOMrtKerr}\n\\]\nNotice that the evolution of the spinors is directly determined by the Maxwell spinor of whatever background the particle is moving in. The\nNJ shifts indicated explicitly in equation~\\eqref{eq:spinorEOMrtKerr} are an explicit consequence of the shift~\\eqref{eq:rtKerrIntShift} at the\nlevel of the effective action. It is straightforward to recover the vectorial equations~\\eqref{eq:rtKerrEOMmomentum} \nand~\\eqref{eq:rtKerrEOMspin} from our spinorial equation; for example, for the momentum,\n\\[\n\\frac{dp^\\mu}{d\\tau} &= \\frac{Q}{2m}\\epsilon^{IJ}\\, \\textrm{Re}\\, \\phi(r(\\tau) + ia(\\tau)) \\ket{p_I} |p_J] \\sigma^\\mu\\\\\n&= -\\frac{Q}{2m} \\textrm{Re}\\, F_{\\rho\\sigma}(r(\\tau) + ia(\\tau)) p_\\nu\\, \\textrm{tr}\\left(\\sigma^{\\rho\\sigma} \\sigma^\\mu \\tilde \\sigma^\\nu \\right).\n\\]\nThe trace can be evaluated using standard techniques, which yield the projector of a two-form onto its self-dual part,\n\\begin{equation}\n\\textrm{tr}\\left(\\sigma^\\mu \\tilde \\sigma^\\nu \\sigma^{\\rho\\sigma} \\right) =  \\eta^{\\nu\\rho}\\eta^{\\mu\\sigma} - \\eta^{\\mu\\rho}\\eta^{\\nu\\sigma} + i\\epsilon^{\\nu\\mu\\rho\\sigma}\\,.\\label{eqn:sigmaTrace}\n\\end{equation}\nThe vector algebra now easily leads to~\\eqref{eq:spinorEOMrtKerr}.\n\nTo illustrate the use of spinorial methods, consider scattering two $\\rootKerr$~particles off one another. We will compute both the leading impulse \n$\\Delta p_1$ and the leading angular impulse $\\Delta a_1$ on one of the two particles during the scattering event. The primary goal of this thesis has been to obtain these observables using the methods\nof scattering amplitudes; here, spinorial equations of motion render the computations even simpler. We denote\nthe spinor variables for particle 1 by $\\ket{1, \\tau}$ and $|1, \\tau]$, and \nsimilarly for particle 2; these spinors are explicitly functions of proper time. In a scattering event we denote the initial spinors\nas $\\ket{1} \\equiv \\ket{1, -\\infty}$ (and similarly for $|1]$.) The\nfinal outgoing spinors are then $\\ket{1'} \\equiv \\ket{1, + \\infty}$.\n\nThe impulses on particle 1 are given in terms of a leading order kick of the spinor $\\ket{\\Delta1} \\equiv \\ket{1, +\\infty} - \\ket{1, -\\infty}$ as\\footnote{Notice that we are representing the impulses here as bispinors.}\n\\[\n\\Delta p_1 &= 2 \\epsilon^{IJ} \\Re \\ket{\\Delta 1_{J}} [1_{I}| \\,,\\\\\n\\Delta a_1 &= 2 a_1^{IJ} \\Re \\ket{\\Delta 1_{J}} [1_{I}|  \\,.\n\\label{eq:impulsesFromSpinors}\n\\]\nThus we simply need to compute the kick suffered by the spinor of particle 1 to\ndetermine \\emph{both} impulses, in contrast to other methods available (including using amplitudes.) By direct\nintegration of the spinorial equation~\\eqref{eq:spinorEOMrtKerr} we see that this spinorial kick is\n\\[\n\\ket{\\Delta 1_I} = \\frac{Q_1}{2m_1} \\int_{-\\infty}^\\infty\\! \\d \\tau \\,\\phi(r_1 + i a_1) \\, \\ket {1_I} \\,.\n\\label{eq:deltaSpinorStep}\n\\]\nAt this level of approximation, we may take the trajectory $r_1$ to be a straight line with constant velocity, and take the spin $a_1$ to \nbe constant, under the integral. Notice that we evaluate the Maxwell spinor at the shifted position $r_1 + i a_1$ because of the Newman--Janis\nshift property at the level of interactions.\n\nTo perform the integration we need the Maxwell spinor influencing the motion of particle 1. This is the field of the second of our two particles.\nIt is easy to obtain this field --- indeed, by the standard Newman--Janis shift of the field set up by particle 2, we need only shift the Coulomb\nfield of a point-like charge. The field is\n\\[\n\\phi(x) = 2iQ_2 \\!\\int\\! \\dd^4\\wn k \\, \\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (x+i a_2)}}{\\wn k^2} \\sigma_{\\mu\\nu} \\wn k^\\mu u_2^\\nu \\,.\n\\label{eq:explicitMaxwell}\n\\]\nNote the explicit NJ shift by the spin $a_2$: this is the shift of the background, in contrast to the shift through $a_1$ of \nequation~\\eqref{eq:deltaSpinorStep}. Of course, there is a pleasing symmetry between these shifts. Using the field~\\eqref{eq:explicitMaxwell}\nin our expression~\\eqref{eq:deltaSpinorStep} for the change in the spinors of particle 1, we arrive at an integral expression for the spinor kick,\n\\[\n\\ket{\\Delta 1_I} =  \\frac{i Q_1 Q_2}{2m_1}\\int \\dd^4\\wn k \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} \\wn k^\\mu u_2^\\nu \n\\sigma_{\\mu\\nu} \\ket {1_I} \\,,\n\\]\nwhere $b$ is the impact parameter. \nThis expression contains complete information about both the linear and angular impulses. For example, substituting into \nequation~\\eqref{eq:impulsesFromSpinors} and applying~\\eqref{eqn:sigmaTrace} we find that the angular impulse is\n\\begin{multline}\n\\Delta a_1^\\mu = \\frac{Q_1 Q_2}{m_1} \\Re\\! \\int\\! \\dd^4 \\wn k \\, \\del(\\wn k \\cdot u_1) \\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2}\n\\\\\n\\times\n\\left(\nia_1 \\cdot u_2 \\, \\wn k^\\mu - i\\wn k \\cdot a_1 \\, u_2^\\mu + \\epsilon^\\mu(\\wn k, a_1, u_2) \n\\right) \\,.\n\\end{multline}\n\nSpinorial equations of motion are also available for the leading order interactions of Kerr moving in a gravitational background. They are\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= -\\frac{1}{2} u^\\mu \\omega_\\mu (r + i a) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &= - \\frac{1}{2} u^\\mu \\tilde \\omega_\\mu (r - i a) |p_I] \\,,\n\\label{eq:spinorEOMKerr}\n\\]\nwhere the spin connection is written in terms of spinors:\n\\[\n\\omega_\\mu \\ket{p} = \\omega_\\mu{}^{ab} \\sigma_{ab} \\ket{p} \\,.\n\\]\n\n\nUsing these spinorial equations of motion and a brief calculation in exact analogy with our $\\rootKerr$~discussion above, it is remarkably straightforward\nto recover the 1PM linear and angular impulse due to Kerr\/Kerr scattering in equation~\\eqref{eqn:KerrDeflections}~\\cite{Vines:2017hyw}. \n\nIn fact we can go further and consider the generalisation of Kerr with NUT charge, corresponding in the stationary case to the exact Kerr--Taub--NUT\nsolution. It is known that NUT charge can be introduced by performing the gravitational analogue of electric\/magnetic\nduality \\cite{Talbot:1969bpa}. Working at linearised level, this deforms the linearised spinorial equation of motion to\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= -  \\frac{e^{-i \\theta}}{2} u^\\mu \\omega_\\mu (r + i a) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &= -  \\frac{e^{+i \\theta}}{2} u^\\mu \\tilde \\omega_\\mu (r - i a) |p_I] \\,,\n\\]\nwhere $\\theta$ is a magnetic angle. The particle described by these equations has mass $m \\cos \\theta$ and NUT parameter $m \\sin \\theta$.\nUsing these equations, and defining the rapidity $w$ by $\\cosh w = u_1 \\cdot u_2$, we find that the leading order impulse in a Kerr--Taub--NUT\/Kerr--Taub--NUT scattering event is given by\n\\begin{multline}\n\\Delta p_1^\\mu = - 4\\pi G m_1 m_2 \\Re\\! \\int\\! \\dd^4\\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} e^{i (\\theta_2 -\\theta_1)}\n\\\\\n\\times\n\\left(\ni \\cosh 2w \\,\\wn k^\\mu + 2 \\cosh w \\, \\epsilon^\\mu(\\wn k, u_1, u_2)\n\\right) \\,,\n\\end{multline}\nin agreement with a previous computation performed using scattering amplitudes~\\cite{Emond:2020lwi}. \nIt is also straightforward to compute\nthe angular impulse using these methods; we find that\n\\[\n\\Delta a_1^\\mu = 4\\pi &G m_2 \\Re\\! \\int\\! \\dd^4 \\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} e^{i (\\theta_2 -\\theta_1)}\n\\\\\n\\times&\n\\Big(\n\\cosh 2w \\, \\epsilon^\\mu(\\wn k, u_1, a_1) - 2 \\cosh w \\, u_1^\\mu \\epsilon(\\wn k, u_1, u_2, a_1) \\\\& \\qquad - 2i a_1\\cdot u_2 \\cosh w\\, \\wn k^\\mu + i\\wn k\\cdot a_1 \\left(2 \\cosh w\\, u_2^\\mu - u_1^\\mu\\right)  \n\\Big) \\,.\n\\]\nThese results can be integrated by means of the generalisation of~\\eqref{eqn:I1result},\n\\begin{equation*}\n\\int\\! \\dd^4\\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} \\wn k^\\mu = \\frac{i}{2\\pi \\sqrt{\\gamma^2 - 1}} \\frac{{b}^\\mu + i\\Pi^\\mu{ }_\\nu(a_1 + a_2)^\\nu}{[b + i\\Pi(a_1 + a_2)]^2}\\,,\n\\end{equation*}\nwhere the projector $\\Pi^\\mu{ }_\\nu$ was defined in equation~\\eqref{eqn:projector}. A little algebra is enough to see that, when the magnetic angles are zero, our results precisely match the 1PM impulses for spinning black holes in equation~\\eqref{eqn:KerrDeflections} --- the observables for Kerr--Taub--NUT scattering are simply phase rotations of their Kerr counterparts.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe Newman--Janis shift is often dismissed as a trick, without any underlying geometric justification. The central theme of this chapter is that we\nshould rather view Newman and Janis's work as an important insight. The Kerr solution is simpler than it first seems, and correspondingly\nthe leading interactions of Kerr are simpler than they might otherwise be. It seems appropriate to place the NJ shift at the heart of our\nformalism for describing the dynamics of Kerr black holes, thereby taking maximum advantage of this leading order simplicity.\n\nOur spinorial approach to the classical dynamics of Kerr (and its electromagnetic single-copy, $\\rootKerr\\,$) makes it trivial to include the\nspin (to all orders in $a$) in scattering processes. Computing the evolution of the spinors, rather than the momenta and spin separately,\nreduces the workload in performing these computations, and is even more efficient in some examples than computing with the help of\nscattering amplitudes. However, we only developed these equations at leading order. At higher orders, spinor equations of motion will certainly exist and be worthy of study.\n\nWe found that the effective action for Kerr has the surprising property that it can be formulated in terms of a two-dimensional worldsheet\nintegral instead of the usual one-dimensional worldline effective theory. This remarkable fact provides some kind of geometric basis for\nthe Newman--Janis shift, where it emerges using Stokes's theorem. Our worldsheet actions contain terms integrated over some boundaries,\nand other terms integrated over the ``bulk'' two-dimensional worldsheet. This structure is also familiar from brane world scenarios, but is\nobviously surprising in the context of Kerr black holes. In Minkowski space, this worldsheet is embedded in a complexification of spacetime,\nin a manner somewhat reminiscent of other work on complexified worldlines; see ref.~\\cite{Adamo:2009ru}, for example. However, \nour worldsheet seems to be a bit of a different beast: it is not a complex line, but rather a strip with two boundaries.\n\nThe worldsheet emerged in our work, built up from the physical boundary worldline and geodesics in the direction of spin. This construction\nis very different from the sigma models familiar from string theory. The dynamical variables in our action are the ``near'' worldsheet coordinates,\nthe spin, and body-fixed frame. But perhaps these dynamical variables emerge from a geometric description more reminiscent of the picture for strings.\n\nIt is remains to be seen whether the worldsheet structure persists when higher-order operators, involving two or more powers of the Riemann\ncurvature (or electromagnetic field strength), are included. But we can certainly hope that the surprising simplicity of Kerr persists to higher\norders --- the computation of observables, at finite spin, from both loop-level amplitudes \\cite{Guevara:2018wpp,Chung:2018kqs,Bern:2020buy,Aoude:2020ygw,Kosmopoulos:2021zoq} and post--Minkowskian EFT methods \\cite{Liu:2021zxr} certainly indicates that further progress can be made. The latter reference is particularly inspiring in this regard, due to the fully covariant nature of the results. Meanwhile, precision calculations reliant on the effective action in \\eqref{eq:fullSimpleAction} require including higher-dimension operators in the action  \\cite{Levi:2020kvb,Levi:2020uwu,Levi:2020lfn}. It would be particularly interesting to investigate the symmetry structure of the Kerr worldsheet, with an eye towards placing symmetry constraints on the tower of possible higher-dimension operators.\n\n\n\\chapter{Conclusions}\n\\label{chap:conclusions}\n\nOn-shell scattering amplitudes are the quantum backbone of particle physics. Black holes are archetypes of classical general relativity. The central aim of this thesis has been to show that despite the apparent dissimilarities, the two are intimately connected. Furthermore, we have advocated that utilising purely on-shell data offers a powerful window into black hole physics.\n\nIn the first part of the thesis we gathered the technology needed to compute on-shell observables relevant for gravitational wave astronomy, basing our formalism in the humble study of explicit single particle wavepackets. We introduced our first scattering observable in chapter~\\ref{chap:impulse}: the impulse, or total change in momentum of a scattering point-particle. We constructed explicit expressions for the impulse, valid in any quantum field theory, in terms of on-shell amplitudes. With our explicit wavepackets we were able to rigorously determine the classical regime of this observable in section~\\ref{sec:classicalLimit}, encountering the Goldilocks inequalities, which we showed held the key to calculating the classical limit of scattering amplitudes.\n\nArmed with a full understanding of the classical limit, and in particular a practical knowledge of crucial $\\hbar$ factors, we were able to provide expressions for the classical limit of the impulse observable. Explicit examples at LO and NLO in section~\\ref{sec:examples} led to expressions which agree with classical worldline perturbation theory --- a highly non-trivial constraint. Beyond these orders the impulse does not fully capture the dynamics of a spinless particle, due to the emission of radiation. In chapter~\\ref{chap:radiation} we therefore included the total radiated momentum in our formalism, eventually finding that for inelastic scattering, amplitudes, via the radiation kernel~\\ref{eqn:defOfR}, specify a point-particle's classical worldline current (as first noted in ref.~\\cite{Luna:2017dtq}). We showed the efficacy of the double copy for computing this current, a fact that is crucial for gravitational wave astronomy. We also showed that basing our treatment of radiation and the impulse in quantum mechanics has the enormous advantage of ameliorating the conceptual difficulties inherent in treatments of radiating point-particles in classical field theory, explicitly recovering the LO predictions of the Abraham--Lorentz--Dirac force in electrodynamics from amplitudes.\n\nAlthough we only considered unbound (scattering) events, it is in fact possible to determine the physics of bound states from our observables. This can be done concretely using effective theories \\cite{Cheung:2018wkq}. It should also be possible to connect our observables more directly to bound states using analytic continuation, in a manner similar to the work of K\\\"{a}lin and Porto \\cite{Kalin:2019rwq,Kalin:2019inp}. As in any application of traditional scattering amplitudes, however, time-dependent phenomena are not readily accessible. This reflects the fact that amplitudes are the matrix elements of a time evolution operator from the far past to the far future. For a direct application of our methods to the time-dependent gravitational waveform, we must overcome this limitation. One possible path of future investigation for upgrading the formalism presented here would start from the fact that the observables we have discussed are essentially expectation values. They are therefore most naturally discussed using the time-dependent in-in formalism, which has a well-known Schwinger--Keldysh diagrammatic formulation. Whether the double copy applies in this context remains to be explored.\n\nThe double copy offers an avenue, rooted in scattering amplitudes, to simpler calculations in gravity. Amplitudes methods are also especially potent when applied to the physics of spin, motivating the focus of the second part of the thesis. For a spinning body the impulse and radiation are not enough to uniquely specify the dynamics of a scattering event for two bodies with unaligned spins. For a black hole, uniquely constrained by the no-hair theorem, the full data is gained from knowledge of the angular impulse, or change in the spin vector. Rooting our intuition once again in the physics of single particle states, we identified the Pauli--Lubanski operator as the crucial quantum actor corresponding to the classical spin vector. We used this operator to obtain explicit expressions for the LO angular impulse in terms of amplitudes, finding notably more complex expressions than for the momentum impulse. We did not consider the higher order corrections, but they are very similar to those of the colour impulse in ref.~\\cite{delaCruz:2020bbn} --- these would be necessary to access the results of \\cite{Liu:2021zxr}, for example.\n\nBy carefully studying the classical limit of amplitudes for finite spin, and computing the momentum and angular impulses, we were able to precisely reproduce the leading spin, 1PM scattering data for Kerr black holes. The complete, all-spin classical expressions,~\\eqref{eqn:KerrDeflections}, can be calculated from amplitudes by applying our methods to massive spinor helicity representations in the large spin limit \\cite{Guevara:2019fsj}. These results are one artefact of a beautiful relationship between black holes and amplitudes: minimally coupled graviton amplitudes are the on-shell avatar of the no-hair theorem \\cite{Vaidya:2014kza,Chung:2018kqs}. This provocative idea can provide non-geometric insights into intriguing results in general relativity, such as the Newman--Janis complex map between the Schwarzschild and Kerr solutions, which is naturally explained by the exponentiation of the minimally coupled 3--point amplitude in the large spin limit \\cite{Arkani-Hamed:2019ymq}.\n\nWe extended this on-shell insight into black hole physics in chapter~\\ref{chap:worldsheet}, where we showed that the Newman--Janis shift could be interpreted in terms of a worldsheet effective action. This holds both in gravity, and for the single-copy $\\rootKerr$~solution in electrodynamics. Moreover, at the level of equations of motion we showed that the NJ shift holds also for the leading interactions of the Kerr black hole. These leading interactions were conveniently described using chiral classical equations of motion with the help of the spinor-helicity method familiar from scattering amplitudes. These spinor equations of motion are extremely powerful tools --- they offered remarkably efficient derivations of the on-shell observables calculated from amplitudes earlier in the thesis, encapsulating the full on-shell data for non-aligned black hole scattering in the kick of the holomorphic spinor. It was also a trivial matter to extend the scope of this technology to the magnetically charged Kerr--Taub--NUT black hole, facilitating the first calculation of the angular impulse for this solution. It would be interesting to see if our methods can be generalised to include the full family of parameters of the famous Plebanski--Demianski family \\cite{Debever:1971,Plebanski:1976gy}.\n\nThe full geometry of our worldsheet remains to be explored, in particular its applicability when higher-order curvature terms are accounted for. Such corrections are crucial for the extending the power of the chiral spinor equations. A major motivation for tackling this problem comes from an obstacle towards progress in amplitudes computations of importance for gravitational wave astronomy: a well defined, tree-level expression for the four-point Compton amplitude, valid for any generic spin $s$, is not known. Application of BCFW recursion with arbitrary spin representations leads to spurious poles \\cite{Arkani-Hamed:2017jhn,Aoude:2020onz}; these poles can only be removed by hand ambiguously \\cite{Chung:2018kqs}. As we have seen in chapter~\\ref{chap:impulse}, triangle diagrams play a crucial role in the computation of either the NLO impulse or potential. The Compton amplitude is a key component to the computation of these topologies using generalised unitarity, and thus progress in computing precision post--Minkowksian corrections for particles with spin is impeded by a crucial tree-level input into the calculation.\n\nThe all-spin Compton amplitude should have a well defined, unambiguous classical limit. Physically the amplitude corresponds to absorption and re-emission of a messenger boson, a problem which is tractable in classical field theory. Our worldsheet construction shows that the Newman--Janis shift holds for the leading interactions of Kerr black holes, and thus the results of chapter~\\ref{chap:worldsheet} should offer a convenient way to approach this problem, provided that higher order contributions can be included. We hope that, in this manner, our work will be only the beginning of a programme to exploit the Newman--Janis structure of Kerr black holes to simplify their dynamics.\n\n\\chapter{Lay summary}\n\nBlack holes are some of the most beguiling objects in theoretical physics. Their interiors, shrouded by the cloak of the inescapable event horizon, will always remain one of the\nleast understood environments in modern science. Yet, from the perspective of a (very) safely removed external observer, black holes are remarkably simple objects. So simple in fact that any classical black hole in our leading theory of gravity, general relativity, is exactly specified by just three parameters: its mass, charge and angular momentum. This sparsity of information leads physicists to say that black holes ``\\textit{have no hair}'', in contrast to ``hairy'' bodies, such as stars and PhD students, which unfortunately cannot be exactly identified by three numbers.\n\nBlack holes' ``no hair'' constraints are maybe more reminiscent of the description of fundamental particles, which are catalogued by their mass, charge and quantum spin, than extended bodies with complicated physiognomies. Yet astronomers readily observe influences on the propagation of light, the motion of stars and the structure of galaxies that are precisely predicted by black hole solutions. Moreover, general relativity predicts that black holes can interact and merge with each other. Such interactions take the form of a long courtship, performing a well separated inspiral that steadily, but inescapably, leads to a final unifying merger under immense gravitational forces. These interactions do not happen in isolation from the rest of the universe. They are violent, explosive events which release enormous amounts of energy, exclusively in the form of gravitational waves.\n\nThe frequent experimental detection of these signals from distant black hole mergers marks the beginning of a new era of observational astronomy. Yet despite the enormous total energy released, gravitational wave signals observable on Earth are very, very faint. In addition to advanced instrumentation, extracting these clean whispers from the background hubbub of our messy world requires precise predictions from general relativity, so as to filter out meaningful signals from noise. The difficulty of performing these calculations means that they are rapidly approaching the point of becoming the largest source of errors in experiments; promoting gravitational wave astronomy to a precision science now depends on improving theoretical predictions.\n\nThis has motivated the development of novel ways of looking at gravitational dynamics, and one such route has been to adopt methods from particle physics. Modern particle collider programmes have necessitated the development of an enormous host of theoretical techniques, targeted towards experimental predictions, which have enabled precision measurements to be extracted from swathes of messy data. These techniques focus on calculating \\textit{scattering amplitudes}: probabilities for how particles will interact quantum mechanically. These techniques can be applied to gravity provided that the interactions are weak, and astonishingly theorists have found that they can facilitate precise predictions for the initial inspiral phase of black hole coalescence. This is a relatively new line of research, and one which is growing rapidly.\n\nThis thesis is a small part of this endeavour. It is focussed on obtaining \\textit{on-shell} observables, relevant for black hole physics, from quantum gravity. An on-shell quantity is one that is physical and directly measurable by an experimentalist, such as the change in a body's momentum. Scattering amplitudes are also on-shell, in the sense that they underpin measurable probabilities. We will be interested in direct routes from amplitudes to observables, which seems like an obvious path to choose. However, many of the techniques in a theorist's toolbox instead involve going via an unmeasurable, mathematical midstep which contains all physical data needed to make measurable predictions. One example is the gravitational potential: one cannot measure the potential directly, but it contains everything needed to make concrete predictions of classical dynamics.\n\nMuch of the power of modern scattering amplitudes comes from their reliance on physical, on-shell data. This thesis will show that a direct map to observables opens new insights into the physics of classical black holes. In particular, we can make the apparent similarity between fundamental particles and black holes concrete: certain scattering amplitudes are the on-shell incarnation of the special ``no hair'' constraint for black holes. This relationship is beautiful and provocative. It opens new practical treatments of spinning black holes in particular, and to this end we will use unique complex number properties of these solutions, combined with quantum mechanics, to obtain powerful classical descriptions of how spinning black holes interact with one another.\n\n\\cleardoublepage\n\\phantomsection\n\\chapter{Abstract}\n\n\\noindent On-shell methods are a key component of the modern amplitudes programme. By utilising the power of generalised unitarity cuts, and focusing on gauge invariant quantities, enormous progress has been made in the calculation of amplitudes required for theoretical input into experiments such as the LHC. Recently, a new experimental context has emerged in which scattering amplitudes can be of great utility: gravitational wave astronomy. Indeed, developing new theoretical techniques for tackling the two-body problem in general relativity is essential for future precision measurements. Scattering amplitudes have already contributed new state of the art calculations of post--Minkowskian (PM) corrections to the classical gravitational potential.\n\nThe gravitational potential is an unphysical, gauge dependent quantity. This thesis seeks to apply the advances of modern amplitudes to classical gravitational physics by constructing physical, on-shell observables applicable to black hole scattering, but valid in any quantum field theory. We will derive formulae for the impulse (change in momentum), total radiated momentum, and angular impulse (change in spin vector) from basic principles, directly in terms of scattering amplitudes.\n\nBy undertaking a careful analysis of the classical region of these observables, we derive from explicit wavepackets how to take the classical limit of the associated amplitudes. These methods are then applied to examples in both QED and QCD, through which we obtain new theoretical results; however, the main focus is on black hole physics. We exploit the double copy relationship between gravity and gauge theory to calculate amplitudes in perturbative quantum gravity, from whose classical limits we derive results in the PM approximation of general relativity.\n\nApplying amplitudes to black hole physics offers more than computational power: in this thesis we will show that the observables we have constructed provide particularly clear evidence that massive, spinning particles are the on-shell avatar of the no-hair theorem. Building on these results, we will furthermore show that the classically obscure Newman--Janis shift property of the exact Kerr solution can be interpreted in terms of a worldsheet effective action. At the level of equations of motion, we show that the Newman--Janis shift holds also for the leading interactions of the Kerr black hole. These leading interactions will be conveniently described using chiral classical equations of motion with the help of the spinor helicity method familiar from scattering amplitudes, providing a powerful and purely classical method for computing on-shell black hole observables.\n\\chapter{Acknowledgements}\n\n\\noindent\n\n\\normalsize\n\nMy foremost thanks go to my supervisor, Donal O'Connell. I will be forever grateful to you for helping me learn to navigate the dense jungle of theoretical physics; your capacity for novel ideas has been truly inspirational, and working with you has been a fantastic experience. Thank you for placing your trust in me as a collaborator these last few years, and also for your sage mountaineering advice: don't fall off.\n\nThank you also to my fellow collaborators: Leonardo de la Cruz, Alfredo Guevara, David Kosower, Dhritiman Nandan, Alex Ochirov, Alasdair Ross, Matteo Sergola, and Justin Vines. I am very proud of the science we have done together. I am especially grateful to Justin, for his hospitality at the Albert--Einstein institute, insights into the world of general relativity, and all-round kindness. My thanks also to Andr\\'{e}s Luna for his friendship and warmth; Chris White, for his enthusiastic introduction to the eikonal; and Yu-Tin Huang, Paolo Pichini and Oliver Schlotterer for the successful joint conquest of the great Bavarian mountain Kaiserschmarrn. I'm still full.\n\nI am extremely grateful for the additional hospitality of Daniel Jenkins at Regensburg; Thibault Damour at the IHES; and the organisers of the MIAPP workshop ``Precision Gravity''. These visits, coupled with the QCD meets Gravity conferences, have been highlights of my PhD. Thank you to the wider community for so many exciting scientific discussions in a welcoming environment, and to David, Alessandra Buonanno and Henrik Johansson for offering me the opportunity to continue my tenure; I am sorry that I have not done so.\n\nThe PPT group at Edinburgh has been a great place to work, and one I have sorely missed following the virtual transition. Thank you to all my fellow PhD students and TA's for your support and knowledge, especially Christian Br\\o{}nnum--Hansen, Tomasso Giani, Michael Marshall, Calum Milloy, Izzy Nicholson, Rosalyn Pearson and Saad Nabeebaccus. Teaching in particular has been a great joy throughout; thank you to Lucile Cangemi and Maria Derda for working with me on your MPhys projects, and to JC Denis for encouraging my outreach activities.\n\nResearch is not always a smooth path, but one which is certainly made easier with the support of friends and family: thank you especially to Rosemary Green for her continual love and encouragement. Thank you to Mike Daniels, Laura Glover, Kirsty McIntyre and Sophie Walker for (literally) accommodating my poor decision making and keeping me going, even if you didn't know it! Meanwhile colloquial debates at dinner club tables were more taxing than any calculation, and highlights of my week; I miss them dearly, thank you all. And thank you to everyone I have tied onto a rope with for unwittingly heeding Donal's advice while sharing many fun adventures at the same time. Here's to many more.\n\nFinally, I could not have completed this thesis without the true love of my life, Ruth Green. Thank you for encouraging me to pursue my work in Edinburgh in the first place, for putting up with my geographical superposition from our nest, and for always being there for me.\n\nI acknowledge financial support under STFC studentship ST\/R504737\/1.\n\n\n\n\n\\chapter{Introduction}\n\\label{chap:intro}\n\nThe dawn of gravitational wave astronomy, heralded by the binary black hole and neutron star mergers detected by the LIGO and VIRGO collaborations~\\cite{Abbott:2016blz,Abbott:2016nmj,Abbott:2017oio,Abbott:2017vtc,TheLIGOScientific:2017qsa}, has opened a new observational window on the universe. Future experiments offer the tantalising prospect of unprecedented insights into the physics of black holes, as well as neutron star structure, extreme nuclear matter and general relativity itself. Theorists have a critical role to play in this endeavour: to access such insights, an extensive bank of theoretical waveform templates are required for both event detection and parameter extraction \\cite{Buonanno:2014aza}.\n\nThe vast majority of accessible data in a gravitational wave signal lies in the inspiral regime. This is the phase preceding the dramatic merger, in which the inspiralling pair coalesce and begin to influence each other's motion. The two bodies remain well separated, and one therefore can tackle their dynamics perturbatively: we can begin by treating them as point particles, and then increase precision by calculating corrections at higher orders in a given approximation. For an inspiral with roughly equal mass black holes the most directly applicable perturbative series is the non-relativistic \\textit{post--Newtonian} (PN) expansion, where one expands in powers of the bodies' velocities $v$. Meanwhile the \\textit{post--Minkowskian} (PM) expansion in powers of Newton's constant $G$ is fully relativistic, and thus more naturally suited to scattering interactions; however, it makes crucial contributions to precision inspiral calculations \\cite{Antonelli:2019ytb}. Finally, when one black hole is far heavier than the other a \\textit{self-force} expansion can be taken about the test body limit, expanding in the mass ratio of the black holes but keeping $v$ and $G$ to all orders. \n\nAlthough simple in concept, the inherent non-linearity of general relativity (GR) makes working even with these approximations an extremely difficult task. Yet future prospects in gravitational wave astronomy require perturbative calculations at very high precision~\\cite{Babak:2017tow}. This has spawned interest in new techniques for solving the two-body problem in gravity and generating the required waveforms. Such techniques would complement methods based on the `traditional' Arnowitt--Deser--Misner Hamiltonian formalism~\\cite{Deser:1959zza,Arnowitt:1960es,Arnowitt:1962hi,Schafer:2018kuf}, direct post--Newtonian solutions in harmonic gauge \\cite{Blanchet:2013haa}, long-established effective-one-body (EOB) methods introduced by Buonnano and Damour~\\cite{Buonanno:1998gg,Buonanno:2000ef,Damour:2000we,Damour:2001tu}, numerical-relativity approaches~\\cite{Pretorius:2005gq,Pretorius:2007nq}, and the effective field theory approach pioneered by Goldberger and Rothstein~\\cite{Goldberger:2004jt,Porto:2016pyg,Levi:2018nxp}.\n\nRemarkably, ideas and methods from quantum field theory (QFT) offer a particularly promising avenue of investigation. Here, interactions are encoded by scattering amplitudes. Utilising amplitudes allows a powerful armoury of modern on-shell methods\\footnote{See \\cite{Cheung:2017pzi,Elvang:2015rqa} for an introduction.} to be applied to a problem, drawing on the success of the NLO (next-to-leading order) revolution in particle phenomenology. An appropriate method for extracting observables relevant to the problem at hand is also required. The relevance of a scattering amplitude --- in particular, a loop amplitude --- to the classical potential, for example, is well understood from work on gravity as an effective field theory~\\cite{Iwasaki:1971,Duff:1973zz,Donoghue:1993eb,Donoghue:1994dn,Donoghue:1996mt,Donoghue:2001qc,BjerrumBohr:2002ks,BjerrumBohr:2002kt,Khriplovich:2004cx,Holstein:2004dn,Holstein:2008sx}. There now exists a panoply of techniques for applying modern amplitudes methods to the computation of the classical gravitational potential \\cite{Neill:2013wsa,Bjerrum-Bohr:2013bxa,Bjerrum-Bohr:2014lea,Bjerrum-Bohr:2014zsa,Bjerrum-Bohr:2016hpa,Bjerrum-Bohr:2017dxw,Cachazo:2017jef,Cheung:2018wkq,Caron-Huot:2018ape,Cristofoli:2019neg,Bjerrum-Bohr:2019kec,Cristofoli:2020uzm,Kalin:2020mvi,Cheung:2020gbf}, generating results directly applicable to gravitational wave physics \\cite{Damour:2016gwp,Damour:2017zjx,Bjerrum-Bohr:2018xdl,Bern:2019nnu,Brandhuber:2019qpg,Bern:2019crd,Huber:2019ugz,Cheung:2020gyp,AccettulliHuber:2020oou,Cheung:2020sdj,Kalin:2020fhe,Haddad:2020que,Kalin:2020lmz,Bern:2020uwk,Huber:2020xny,Bern:2021dqo}. By far the most natural relativistic expansion from an amplitudes perspective is the post--Minkowskian expansion (in the coupling constant): indeed, amplitudes methods have achieved the first calculations of the 3PM~\\cite{Bern:2019nnu} and 4PM~\\cite{Bern:2021dqo} potential. Furthermore, it is possible to use analytic continuation to obtain bound state observables directly from the scattering problem \\cite{Kalin:2019rwq,Kalin:2019inp}. \n\nThe gravitational potential is a versatile tool; however it is also coordinate, and thus gauge, dependent. The conservative potential also neglects the radiation emitted from interactions, leading to complications at higher orders. Amplitudes and physical observables, meanwhile, are on-shell and gauge-invariant, and should naturally capture all the physics of the problem. Direct maps between amplitudes and classical physics are well known to hold in certain regimes: for example, the eikonal exponentation of amplitudes in the extreme high energy limit has long been used to derive scattering angles \\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe,Amati:1992zb,Kabat:1992tb,Amati:1993tb,Muzinich:1995uj,DAppollonio:2010krb,Melville:2013qca,Akhoury:2013yua,DAppollonio:2015fly,Ciafaloni:2015vsa,DAppollonio:2015oag,Ciafaloni:2015xsr,Luna:2016idw,Collado:2018isu,KoemansCollado:2019ggb,DiVecchia:2020ymx,DiVecchia:2021ndb}. Furthermore, calculating amplitudes in the high energy regime exposes striking universal features in gravitational scattering \\cite{Bern:2020gjj,Parra-Martinez:2020dzs,DiVecchia:2020ymx}. Meanwhile, a careful analysis of soft limits of amplitudes with massless particles can extract data about both classical radiation \\cite{Laddha:2018rle,Laddha:2018myi,Sahoo:2018lxl,Laddha:2018vbn,Laddha:2019yaj,A:2020lub,Sahoo:2020ryf,Bonocore:2020xuj} and radiation reaction effects \\cite{DiVecchia:2021ndb}. Calculations with radiation can also be accomplished in the eikonal formalism \\cite{Amati:1990xe}, but have proven particularly natural in classical worldline approaches, whereby one applies perturbation theory directly to point-particle worldlines rather than quantum states \\cite{Goldberger:2016iau,Goldberger:2017frp, Goldberger:2017vcg,Goldberger:2017ogt,Chester:2017vcz,Li:2018qap, Shen:2018ebu,Plefka:2018dpa,Plefka:2019hmz,PV:2019uuv,Almeida:2020mrg,Prabhu:2020avf,Mougiakakos:2021ckm}. Using path integrals to develop a worldline QFT enables access to on-shell amplitudes techniques in this context \\cite{Mogull:2020sak,Jakobsen:2021smu}, and this method has been used to calculate the NLO current due to Schwarzschild black hole bremsstrahlung.\n\nWe know how to extract information about classical scattering and radiation from quantum amplitudes in a gauge invariant manner --- but only in specific regimes. It is therefore natural to seek a more generally applicable, on-shell mapping between amplitudes and classical observables: this will form the topic of the first part of this thesis. We will construct general formulae for a variety of on-shell observables, valid in any quantum field theory and for any two-body scattering event. In this context we will also systematically study how to extract the classical limit of an amplitude, developing in the process a precise understanding of how to use quantum amplitudes to calculate observables for classical point-particles. We will show that by studying appropriate observables, expressed directly in terms of amplitudes, we can handle the nuances of the classical relationship between conservative and dissipative physics in a single, systematic approach, avoiding the difficulties surrounding the Abraham--Lorentz--Dirac radiation reaction force in electrodynamics \\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz}. First presented in ref.~\\cite{Kosower:2018adc}, the formalism we will develop has proven particularly useful for calculating on-shell observables for black hole processes involving classical radiation \\cite{Luna:2017dtq,Bautista:2019tdr,Cristofoli:2020hnk,A:2020lub,delaCruz:2020bbn,Mogull:2020sak,Gonzo:2020xza,Herrmann:2021lqe} and spin \\cite{Maybee:2019jus,Guevara:2019fsj,Arkani-Hamed:2019ymq,Moynihan:2019bor,Huang:2019cja,Bern:2020buy,Emond:2020lwi,Monteiro:2020plf}. Wider applications also exist to other aspects of classical physics, such as the Yang--Mills--Wong equations \\cite{delaCruz:2020bbn,Wong:1970fu} and hard thermal loops \\cite{delaCruz:2020cpc}.\n\nEven the most powerful QFT techniques require a precise understanding of how to handle the numerous subtleties involved in taking the classical limit and accurately calculating observables. The reader may therefore wonder whether the philosophy of applying quantum amplitudes to classical physics really offers any fundamental improvement --- after all,  there are many concurrent advances in our understanding of the two-body problem arising from alternative calculational methods. For example, information from the self-force approximation can provide extraordinary simplifications directly at the level of classical calculations \\cite{Bini:2020flp,Bini:2020hmy,Bini:2020nsb,Bini:2020uiq,Bini:2020wpo,Damour:2020tta,Bini:2020rzn}. Aside from the fact that amplitudes methods have achieved state-of-the-art precision in the PM approximation \\cite{Bern:2019nnu,Bern:2021dqo,Herrmann:2021lqe}, such a sweeping judgement would be premature, as we have still yet to encounter two unique facets of the amplitudes programme: the double copy, and the treatment of spin effects.\n\n\\subsection{The double copy}\n\nAn important insight arising from the study of scattering amplitudes is that amplitudes in perturbative quantum gravity are far simpler than one would expect, and in particular are closely connected to the amplitudes of Yang--Mills (YM) theory. This connection is called the double copy, because gravitational amplitudes are obtained as a product of two Yang--Mills quantities. One can implement this double copy in a variety of ways: the original statement, by Kawai, Lewellen and Tye~\\cite{Kawai:1985xq} presents a tree-level gravitational (closed string) amplitude as a sum over terms, each of which is a product of two tree-level colour-ordered Yang--Mills (open string) amplitudes, multiplied by appropriate Mandelstam invariants. More recently, Bern, Carrasco and Johansson~\\cite{Bern:2008qj,Bern:2010ue} demonstrated that the double copy can be understood very simply in terms of a diagrammatic expansion of a scattering amplitude. They noted that any tree-level $m$-point amplitude in Yang--Mills theory could be expressed as a sum over the set of cubic diagrams $\\Gamma$,\n\\begin{equation}\n\\mathcal{A}_{m} = g^{m-2}\\sum_{\\Gamma}\\frac{n_i c_i}{\\Delta_i}\\,,\n\\end{equation}\nwhere $\\Delta_i$ are the propagators, $n_i$ are gauge-dependent kinematic numerators, and $c_i$ are colour factors which are related in overlapping sets of three by Jacobi identities,\n\\begin{equation}\nc_\\alpha \\pm c_\\beta \\pm c_\\gamma = 0\\,.\n\\end{equation}\nThe colour factors are single trace products of $SU(N)$ generators $T^a$, normalised such that $\\textrm{tr}(T^aT^b) = \\delta^{ab}$. Remarkably, BCJ found that gauge freedom makes it possible to always choose numerators satisfying the same Jacobi identities \\cite{Bern:2010yg}. This fundamental property is called \\textit{colour-kinematics duality}, and has been proven to hold for tree-level Yang--Mills theories \\cite{Bern:2010yg}.\n\nWhen colour-kinematics duality holds, the double copy then tells us that\n\\begin{equation}\n\\mathcal{M}_m = \\left(\\frac{\\kappa}{2}\\right)^{m-2}\\sum_{\\Gamma}\\frac{n_i\\tilde{n}_i}{\\Delta_i}\\,\n\\end{equation}\nis the corresponding $m$-point gravity amplitude, obtained by the replacements\n\\begin{equation}\ng\\mapsto\\frac{\\kappa}{2}\\,,\\quad c_i\\mapsto \\tilde{n}_i\\,.\n\\end{equation}\nHere $\\kappa = \\sqrt{32\\pi G}$ is the appropriate gravitational coupling, and $\\tilde{n}_i$ is a distinct second set of numerators satisfying colour-kinematics duality. The choice of numerator determines the resulting gravity theory. To obtain gravity the original numerators are chosen, and thus amplitude numerators for gravity are simply the square of kinematic numerators in Yang--Mills theory, provided that colour-kinematics duality holds. \n\nOne complication is that regardless of the choice of $\\tilde{n}_i$, the result is not a pure theory of gravitons, but instead is a factorisable graviton multiplet. This can easily be seen in pure Yang--Mills theory, where the tensor product $A^\\mu \\otimes A^\\nu \\sim \\phi_\\textrm{d} \\oplus B^{\\mu\\nu} \\oplus h^{\\mu\\nu}$ leads to a scalar dilaton field, antisymmetric Kalb--Ramond axion and traceless, symmetric graviton respectively. In 4 dimensions the axion has only one degree of freedom, so its field strength $H^{\\mu\\nu\\rho} = \\partial^{[\\mu}B^{\\nu\\rho]}$ can be written as\n\\begin{equation}\nH^{\\mu\\nu\\rho}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\partial_\\sigma \\zeta,\\label{eqn:axionscalar}\n\\end{equation}\nwith $\\zeta$ representing the single propagating pseudoscalar degree of freedom. There any many possible ways to deal with the unphysical axion and dilaton modes and isolate the graviton degrees of freedom \\cite{Bern:2019prr} --- we will see some such methods in the course of the thesis. However, the main point here is that Einstein gravity amplitudes can be determined exclusively by gauge theory data.\n\nThis modern formulation of the double copy is particularly exciting as it has a clear generalisation to loop level; one simply includes integrals over loop momentum and appropriate symmetry factors. A wealth of non-trivial evidence  supports this conjecture --- for reviews, see \\cite{Carrasco:2015iwa,Bern:2019prr}. The work of BCJ suggests that gravity may be simpler than it seems, and also more closely connected to Yang--Mills theory than one would guess after inspecting their Lagrangians. Here our simple presentation of colour-kinematics duality was only for tree level, massless gauge theory. However, the double copy can be applied far more generally: it forms bridges between a veritable web of theories, for both massless and massive states \\cite{Johansson:2014zca,Johansson:2015oia,Johansson:2019dnu,Haddad:2020tvs}.\n\nSince perturbation theory is far simpler in Yang--Mills theory than in standard approaches to gravity, the double copy has revolutionary potential for gravitational physics. Indeed, it has proven to be the key tool enabling state-of-the-art calculations of the PM potential from amplitudes \\cite{Bern:2019crd,Bern:2019nnu,Bern:2021dqo}. Furthermore, it has also raised the provocative question of whether exact solutions in general relativity satisfy similar simple relationships to their classical Yang--Mills counterparts, extending the relationship beyond perturbation theory. First explored in \\cite{Monteiro:2014cda}, many exact classical double copy maps are now known to hold between classical solutions of gauge theory and gravity \\cite{Luna:2015paa,Luna:2016hge,Adamo:2017nia,Bahjat-Abbas:2017htu,Carrillo-Gonzalez:2017iyj,Lee:2018gxc,Berman:2018hwd,Carrillo-Gonzalez:2018pjk,Adamo:2018mpq,Luna:2018dpt,CarrilloGonzalez:2019gof,Cho:2019ype,Carrillo-Gonzalez:2019aao,Bah:2019sda,Huang:2019cja,Alawadhi:2019urr,Borsten:2019prq,Kim:2019jwm,Banerjee:2019saj,Bahjat-Abbas:2020cyb,Moynihan:2020gxj,Adamo:2020syc,Alfonsi:2020lub,Luna:2020adi,Keeler:2020rcv,Elor:2020nqe,Alawadhi:2020jrv,Casali:2020vuy,Adamo:2020qru,Easson:2020esh,Chacon:2020fmr,Emond:2020lwi,White:2020sfn,Monteiro:2020plf,Lescano:2021ooe}, even when there is gravitational radiation present~\\cite{Luna:2016due}. \n\nTo emphasise that the classical double copy has not been found simply for esoteric exact solutions, let us briefly consider the original Kerr--Schild map constructed in \\cite{Monteiro:2014cda}. Kerr--Schild spacetimes are a particularly special class of solutions possessing sufficient symmetry that their metrics can be written\n\\begin{equation}\ng_{\\mu\\nu} = \\eta_{\\mu\\nu} + \\varphi k_\\mu k_\\nu\\,,\\label{eqn:KSmetric}\n\\end{equation}\nwhere $\\varphi$ is a scalar function and $k_\\mu$ is null with respect to both the background and full metric, and satisfies the background geodesic equation:\n\\begin{equation}\ng^{\\mu\\nu} k_\\mu k_\\nu = \\eta^{\\mu\\nu} k_\\mu k_\\nu = 0\\,, \\qquad k\\cdot\\partial k_\\mu = 0\\,.\\label{eqn:KSvector}\n\\end{equation}\nThe symmetries of this class of spacetime ensure that the (mixed index placement) Ricci tensor is linearised. It was proposed in \\cite{Monteiro:2014cda} that for such spacetimes there then exists a single copy gauge theory solution,\n\\begin{equation}\nA^a_\\mu = \\varphi\\, c^a k_\\mu\\,.\\label{eqn:singleKScopy}\n\\end{equation}\nwhere $c^a$ is a classical colour charge. This incarnation of the double copy is therefore enacted by replacing copies of the classical colour with the null vector $k_\\mu$, in analogue to the BCJ amplitude replacement rules.\n\nThe crucial importance of the Kerr--Schild double copy is that it encompasses both Schwarzschild and Kerr black holes. Both (exterior) spacetime metrics can be written in the compact Kerr--Schild form, with respective data \\cite{Monteiro:2014cda}\n\\begin{equation}\n\\varphi_\\textrm{Schwz}(r) = \\frac{2GM}{r}\\,, \\quad k^\\mu = \\left(1, \\frac{\\v{x}}{r}\\right)\\\\\n\\end{equation}\nfor Schwarzschild, where $r^2 = \\v{x}^2$; and\n\\begin{equation}\n\\varphi_\\textrm{Kerr}(\\tilde r, \\theta) = \\frac{2GM\\tilde r}{\\tilde r^2 + a^2 \\cos^2\\theta}\\,, \\quad k^\\mu = \\left(1,\\frac{\\tilde r x + ay}{\\tilde r^2 + a^2}, \\frac{\\tilde ry - ax}{\\tilde r^2 + a^2},\\frac{z}{r}\\right)\\label{eqn:blackholesKSforms}\n\\end{equation}\nfor Kerr, where the parameter $a$ is the radius of the Kerr singularity about the $z$ axis. This key parameter is the norm of a pseudovector $a^\\mu$ which fully encodes the spin of the black hole, the \\textit{spin vector}. It is important to note that in the Kerr case $(\\tilde r, \\theta)$ are not the usual polar coordinates, instead satisfying\n\\begin{equation}\n\\frac{x^2 +y^2}{\\tilde r^2 + a^2} + \\frac{z^2}{\\tilde{r}^2} = 1\\label{eqn:KerrKSradial}\n\\end{equation}\nand $z = \\tilde r \\cos\\theta$. The corresponding gauge theory single copies are then given by \\eqn~\\eqref{eqn:singleKScopy}. The Schwarzchild single copy is simply a Coulomb charge. The Kerr single copy meanwhile is a disk of uniform charge rotating about the $z$, axis whose mass distribution exhibits a singularity at $x^2 + y^2 = a^2$ \\cite{Monteiro:2014cda}. We will refer to this unique charged particle by its modern name, $\\rootKerr$ \\cite{Arkani-Hamed:2019ymq}.\n\nThe $\\rootKerr$ solution was first explored by Israel in \\cite{Israel:1970kp}, and will be of great interest for us in the second part of the thesis: its double copy relation to Kerr ensures that the structure and dynamics of Kerr in gravity are precisely mirrored by the behaviour of $\\rootKerr$ in gauge theory, where calculations are often simpler. This is particularly important in the context of the second key area in which applying amplitudes ideas to  black hole interactions can offer a significant computational and conceptual advantage: spin.\n\n\n\\subsection{Spin}\n\nThe astrophysical bodies observed in gravitational wave experiments spin. The spins of the individual bodies in a compact binary coalescence event influence the details of the outgoing gravitational radiation \\cite{Buonanno:2014aza}, and moreover contain information on the poorly-understood formation channels of the binaries \\cite{Mandel:2018hfr}. Measurement of spin is therefore one of the primary physics outputs of gravitational wave observations.\n\nAny stationary axisymmetric extended body has an infinite tower of mass-multipole moments $\\mathcal{I}_\\ell$ and current-multipole moments $\\mathcal{J}_\\ell$, which generally depend intricately on its internal structure and composition. In the point-particle limit it is thus the multipole structure of the body which accurately identifies to an observer whether that object is a neutron star, black hole or other entity. Incorporating spin multipoles into the major theoretical platform for these experiments, the EOB formalism \\cite{Buonanno:1998gg,Buonanno:2000ef}, is well established in the PN approximation \\cite{Damour:2001tu,Damour:2008qf,Barausse:2009aa,Barausse:2009xi,Barausse:2011ys,Damour:2014sva,Bini:2017wfr,Khalil:2020mmr}, and has also been extended to the PM approximation by means of a gauge-invariant spin holonomy \\cite{Bini:2017xzy}. This has been used to compute the dipole (or spin-orbit) contribution to the conservative potential for two spinning bodies through 2PM order \\cite{Bini:2018ywr}. Calculating higher-order PN spin corrections has been a particular strength of the effective field theory treatment of PN dynamics \\cite{Goldberger:2004jt,Porto:2005ac,Porto:2006bt,Porto:2008tb,Levi:2011eq,Levi:2015msa,Levi:2016ofk,Levi:2020kvb,Levi:2020uwu,Levi:2020lfn}, while self-force data has also driven independent progress in this approximation \\cite{Siemonsen:2019dsu,Antonelli:2020aeb,Antonelli:2020ybz}. EFT progress in the handling of spin has also recently been extended to the PM series, yielding the first calculation of finite-size effects beyond leading-order \\cite{Liu:2021zxr}; moreover, these results can be mapped to bound observables by analytic continuation \\cite{Kalin:2019rwq,Kalin:2019inp}.\n\nA common feature of all of these calculations is that they are significantly more complicated than the spinless examples considered previously, and moreover are nearly unanimously restricted to the special case where the spins of the bodies are aligned with each other.\n\nThe black hole case is special. For a Kerr black hole, every multipole is determined by only the mass $m$ and spin vector $a^\\mu$, through the simple relation due to Hansen \\cite{Hansen:1974zz},\n\\begin{equation}\n\\mathcal I_\\ell+i\\mathcal J_\\ell = m\\left(ia\\right)^\\ell \\,.\\label{eqn:multipoles}\n\\end{equation}\nThis distinctive behaviour is a precise reflection of the \\textit{no-hair theorem} \\cite{Israel:1967wq,Israel:1967za,Carter:1971zc}, which ensures that higher multipoles are constrained by the dipole. This simple multipole structure is also reflected in the dynamics of spinning black holes --- for example, remarkable all-spin results are known for black hole scattering at leading order in both the PN and PM approximations \\cite{Vines:2016qwa,Siemonsen:2017yux,Vines:2017hyw}; aligned-spin black hole scattering was also considered at 2PM order for low multipoles in \\cite{Vines:2018gqi}.\n\nMoreover, over the last few years it has become increasingly apparent that an on-shell expression of the no-hair theorem is that black holes correspond to \\textit{minimal coupling} in classical limits of quantum scattering amplitudes for massive spin~$s$ particles and gravitons.  Amplitudes for long-range gravitational scattering of spin 1\/2 and spin 1 particles were found in \\cite{Ross:2007zza,Holstein:2008sx} to give the universal spin-orbit (pole-dipole level) couplings in the post-Newtonian corrections to the gravitational potential.  Further similar work in \\cite{Vaidya:2014kza}, up to spin 2, suggested that the black hole multipoles \\eqref{eqn:multipoles} up to order $\\ell=2s$ are faithfully reproduced from tree-level amplitudes for minimally coupled spin~$s$ particles.\n\nSuch amplitudes for arbitrary spin $s$ were computed in \\cite{Guevara:2017csg}, by adopting the representation of minimal coupling for arbitrary spins presented in \\cite{Arkani-Hamed:2017jhn} using the massive spinor-helicity formalism---see also~\\cite{Conde:2016vxs,Conde:2016izb}. Those amplitudes were shown in \\cite{Guevara:2018wpp,Bautista:2019tdr} to lead in the limit $s\\to\\infty$ to the two-black-hole aligned-spin scattering angle found in \\cite{Vines:2017hyw} at first post--Minkowskian order and to all orders in the spin-multipole expansion, while in \\cite{Chung:2018kqs} they were shown to yield the contributions to the interaction potential (for arbitrary spin orientations) at the leading post--Newtonian orders at each order in spin. Meanwhile in \\cite{Bern:2020buy,Kosmopoulos:2021zoq} amplitudes for arbitrary spin fields were combined with the powerful effective theory matching techniques of \\cite{Cheung:2018wkq} to yield the first dipole-quadrapole coupling calculation at 2PM order. Methods from heavy quark effective theory \\cite{Damgaard:2019lfh,Aoude:2020onz,Haddad:2020tvs} and quantum information \\cite{Aoude:2020mlg} have also proven applicable to spinning black hole scattering, the former leading to the first amplitudes treatment of tidal effects on spinning particles \\cite{Aoude:2020ygw}.\n\nTo replicate the behaviour of Kerr black holes, the massive spin $s$ states in amplitudes must be minimally coupled to the graviton field, by which we mean that the high energy limit is dominated by the corresponding helicity configuration of massless particles \\cite{Arkani-Hamed:2017jhn}. This has been especially emphasised in \\cite{Chung:2018kqs}, where, by matching at tree-level to the classical effective action of Levi and Steinhoff \\cite{Levi:2015msa}, it was shown that the theory which reproduces the infinite-spin limit of minimally coupled graviton amplitudes is an effective field theory (EFT) of spinning black holes. It can be explicitly shown that any deviation from minimal coupling adds further internal structure to the effective theory \\cite{Chung:2019duq,Chung:2019yfs,Chung:2020rrz}, departing the special black hole case. \n\nApplying amplitudes methods to the scattering of any spinning object, black hole or otherwise, we face the familiar problem of requiring an appropriate observable and precise understanding of the classical limit. When the spins of scattering objects are not aligned there no longer exists a well defined scattering plane, and thus the most common observable calculated from classical potentials, the scattering angle, becomes meaningless. We shall therefore apply the methods developed in Part~\\ref{part:observables} to quantum field theories of particles with spin, setting up observables in terms of scattering amplitudes which can fully specify the dynamics of spinning black holes. When the spins are large these methods are known to exactly reproduce established 1PM results \\cite{Guevara:2019fsj,Vines:2017hyw}. \n\nAfter systematically dealing with the classical limit of quantum spinning particles, we will apply insights from amplitudes to the classical dynamics of Kerr and its single copy, $\\rootKerr$. We will utilise the fact that the beautiful relationship between Kerr black holes and minimally coupled amplitudes goes far deeper than simply being a powerful calculational tool. Amplitudes can explain and reveal structures in general relativity that are obscured by geometrical perspectives: for example, the double copy. \n\nAnother key example in the context of spin is the fact, first noted by Newman and Janis in \\cite{Newman:1965tw}, that the Kerr metric can be obtained from Schwarzschild by means of a complex coordinate transformation. This is easy to see when the metrics are in Kerr--Schild form: take the data for Scwharzschild in \\eqn~\\eqref{eqn:blackholesKSforms}. Under the transformation $z \\rightarrow z + ia$,\n\\[\nr^2 \\rightarrow &\\, r^2 + 2iaz - a^2 \\\\ &\\equiv  \\tilde r^2 - \\frac{a^2 z^2}{\\tilde r^2} + 2ia\\tilde{r}\\cos\\theta = (\\tilde r +i a\\cos\\theta)^2\\,,\n\\]\nwhere the Kerr radial coordinate $\\tilde r$ is defined in \\eqn~\\eqref{eqn:KerrKSradial}. Hence under $z \\rightarrow z + ia$ we have that $r \\rightarrow \\tilde r + ia\\cos\\theta$, and moreover,\n\\[\n\\varphi_{\\rm Schwz}(r) \\rightarrow &\\, 2GM\\Re\\left\\{\\frac1{r}\\right\\}\\bigg|_{r\\rightarrow \\tilde r + ia\\cos\\theta}\\\\ &= \\frac{2GM \\tilde r}{(\\tilde r^2 + a^2 \\cos^2\\theta)} \\equiv \\varphi_{\\rm Kerr}(\\tilde r,\\theta)\\,.\n\\]\nIn other words, the Kerr solution looks like a complex translation of the Schwarzschild solution \\cite{Newman:2002mk}. Clearly the same shift holds in the gauge theory single copies. This is but one example of complex maps between spacetimes; further examples were derived by Talbot \\cite{Talbot:1969bpa}, encompassing the Kerr--Newman and Taub--NUT solutions.\n\nA closely related way to understand these properties of classical solutions is to consider their Weyl curvature spinor $\\Psi$. For example, with appropriate coordinates the NJ shift applies exactly to the spinor:\n\\[\n\\Psi^\\text{Kerr}(x) = \\Psi^\\text{Schwarzschild}(x + i a) \\,,\\label{eqn:NJshift}\n\\]\nSimilarly, in the electromagnetic $\\rootKerr$~case~\\cite{Newman:1965tw} it is the Maxwell spinor $\\maxwell$ that undergoes a shift:\n\\[\n\\maxwell^{\\sqrt{\\text{Kerr}}}(x) = \\maxwell^\\text{Coulomb}(x + i a) \\,.\n\\]\nTherefore, $\\rootKerr$~is a kind of complex translation of the Coulomb solution. \n\nAlthough these complex maps are established classically, there is no geometric understanding for \\textit{why} such a complex map holds. However, this is not the case from the perspective of amplitudes --- it was explicitly shown in \\cite{Arkani-Hamed:2019ymq} that the Newman--Janis shift is a simple consequence of the the exponentiation of minimally coupled amplitudes in the large spin limit. The simplicity of minimally coupled amplitudes has since been utilised to explain a wider range of complex mappings between spacetime and gauge theory solutions \\cite{Moynihan:2019bor,Huang:2019cja,Moynihan:2020gxj,Kim:2020cvf}, culminating in a precise network of relationships constructed from the double copy, Newman--Janis shifts and electric-magnetic duality \\cite{Emond:2020lwi}. These investigations have relied on the on-shell observables we will introduce in Part~\\ref{part:observables} \\cite{Kosower:2018adc}.\n\nInspired by the insights offered by amplitudes, we will adopt the complex Newman--Janis shift as a starting point for investigating the classical dynamics of these unique spinning objects. In particular, we will show that interacting effective actions for Kerr and $\\rootKerr$, in the vein of Levi and Steinhoff \\cite{Levi:2015msa}, can be interpreted as actions for a complex worldsheet. We will also apply the power of the massive spinor helicity representations of ref.~\\cite{Arkani-Hamed:2017jhn} to classical dynamics, rapidly deriving on-shell scattering observables for Kerr and $\\rootKerr$ from spinor equations of motion. Although working purely classically, our methodology and philosophy will be entirely drawn from amplitudes-based investigations.\n\n\\section{Summary}\n\nTo summarise, the structure of this thesis is as follows. Part~\\ref{part:observables} is dedicated to the construction of on-shell observables which are well defined in both classical and quantum field theory. We begin in chapter~\\ref{chap:pointParticles} by setting up single particle quantum wavepackets which describe charged scalar point-particles in the classical limit. In chapter~\\ref{chap:impulse} we then turn to descriptions of point-particle scattering by considering our first observable, the impulse, or the total change in the momentum of a scattering particle. We derive general expressions for this observable in terms of amplitudes, and undertake a careful examination of the classical limit, extracting the rules needed to pass from the quantum to the classical regime. In chapter~\\ref{chap:radiation} we introduce the total radiated momentum and demonstrate momentum conservation and the automatic handling of radiation reaction effects in our formalism. We introduce spin in part~\\ref{part:spin}, which is concerned with spinning black holes. In chapter~\\ref{chap:spin} we construct on-shell observables in QFT for spinning particles, reproducing results for Kerr black holes after considering in detail the classical limit of amplitudes with finite spin. We then return to classical dynamics in chapter~\\ref{chap:worldsheet}, using insights from structures in on-shell amplitudes to uncover worldsheet effective actions for $\\rootKerr$ and Kerr particles. We finish by discussing our results in~\\ref{chap:conclusions}. Results in chapters~\\ref{chap:pointParticles}, \\ref{chap:impulse} and~\\ref{chap:radiation} were published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}, chapter~\\ref{chap:spin} is based on \\cite{Maybee:2019jus}, and chapter~\\ref{chap:worldsheet} appeared in \\cite{Guevara:2020xjx}. \n\n\\subsection{Conventions}\n\nIn all the work that follows, our conventions for Fourier transforms are\n\\begin{equation}\nf(x) = \t\\int\\!\\frac{\\d^n q}{(2\\pi)^n}\\, \\tilde{f}(q) e^{-i q\\cdot x}\\,, \\qquad \\tilde{f}(q) = \\int\\! d^4x\\, f(x) e^{i q\\cdot x}\\,.\n\\end{equation}\nWe will consistently work in relativistically natural units where $c=1$, however we will always treat $\\hbar$ as being dimensionful. We work in the mostly minus metric signature $(+,-,-,-)$, where we choose $\\epsilon_{0123} = +1$ for the Levi--Civita tensor. We will occasionally find it convenient to separate a Lorentz vector $x^\\mu$ into its time component $x^0$ and its spatial components $\\v{x}$, so that $x^\\mu = (x^0,x^i) = (x^0, \\v{x})$, where $i=1,2,3$. \n\nFor a given tensor $X$ of higher rank, total symmmetrisation and antisymmetrisation respectively of tensor indices are represented as usual by\n\\begin{equation}\n\\begin{aligned}\nX^{(\\mu_1} \\dots X^{\\mu_n)} &= \\frac1{n!}\\left(X^{\\mu_1} X^{\\mu_2} \\dots X^{\\mu_n} + X^{\\mu_2} X^{\\mu_1} \\dots X^{\\mu_n} + \\cdots\\right)\\\\\nX^{[\\mu_1} \\dots X^{\\mu_n]} &= \\frac1{n!}\\left(X^{\\mu_1} X^{\\mu_2} \\dots X^{\\mu_n} - X^{\\mu_2} X^{\\mu_1} \\dots X^{\\mu_n} + \\cdots\\right).\n\\end{aligned}\n\\end{equation}\nFinally, our definition of the amplitude will consistently differ by a phase factor relative to the standard definition used for the double copy. Here, in either gauge theory or gravity\n\\begin{equation}\ni\\mathcal{A}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) = \\sum \\left(\\text{Feynman diagrams}\\right)\\,,\n\\end{equation}\nwhereas in the convention used in the original work of BCJ~\\cite{Bern:2008qj,Bern:2010ue} the entire left hand side is defined as the amplitude.\n\\part{Classical observables from quantum field theory}\t\n\\label{part:observables}\n\n\\chapter{Point-particles}\n\\label{chap:pointParticles}\n\nOn-shell amplitudes in quantum field theory are typically calculated on a basis of plane wave states: the context of the calculation is the physics of states with definite momenta, but indefinite positions. However, to capture the physics of a black hole in quantum mechanics, or indeed any other classical point-particle, this is clearly not sufficient: we need states which are well localised. We also need quantum states that accurately correspond to point-particles when the ``classical limit'' is taken. These requirements motivate the goals of this first chapter. We will precisely specify what we mean by the classical limit, and explicitly construct localised wavepackets which describe single, non-spinning point-particles in this limit. The technology that we develop will provide the foundations for our construction of on-shell observables for interacting black holes in later chapters.\n\nTo ensure full generality we will consider charged particles, studying the classical limits of states in an $SU(N)$ gauge group representation. Such point-particles are described by the Yang--Mills--Wong equations in the classical regime~\\cite{Wong:1970fu}. For gravitational physics one could have in mind an Einstein--Yang--Mills black hole, but there are more interesting perspectives available. YM theory, treated as a classical field theory, shares many of the important physical features of gravity, including non-linearity and a subtle gauge structure. In this respect the YM case has always served as an excellent toy model for gravitational dynamics. But, as we discussed in the previous chapter, our developing understanding of the double copy has taught us that the connection between Yang--Mills theory and gravity is deeper than this; detailed aspects\nof the perturbative dynamics of gravity, including gravitational radiation, can be deduced from Yang--Mills theory and the double copy. Understanding non--trivial gauge, or \\textit{colour}, representation states will thus play a key role in our later calculations of black hole observables. \n\nThis chapter is based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}, in collaboration with Leonardo de la Cruz, David Kosower, Donal O'Connell and Alasdair Ross.\n\n\\section{Restoring $\\hbar$}\n\\label{sec:RestoringHBar}\n\nTo extract the classical limit of a quantum mechanical system describing the physics of point-particles we are of course going to need to be careful in our treatment of Planck's constant, $\\hbar$. A straightforward and pragmatic approach to restoring all factors of $\\hbar$ in an expression is dimensional analysis: we denote the dimensions of mass and length by $[M]$ and $[L]$ respectively.\n\nWe may choose the dimensions of an $n$-point scattering amplitude in four dimensions to be $[M]^{4-n}$ even when $\\hbar \\neq 1$. This is consistent with choosing the dimensions of creation and annihilation operators so that\n\\begin{equation}\n[a_i(p), a^{\\dagger j}(p')] = 2E_p (2\\pi)^3 \\delta^{(3)}(\\v{p} - \\v{p}')\\,\\delta_i{ }^j\\,,\\label{eqn:ladderCommutator}\n\\end{equation}\nHere the indices label the representation $R$ of any Lie group. We define single-particle momentum eigenstates in this representation by\n\\begin{equation}\n|p^i \\rangle = a^{\\dagger i}(p) |0\\rangle\\,.\\label{eqn:singleParticleStateDef}\n\\end{equation}\nSince the vacuum state is taken to be dimensionless, the dimension of $|p^i\\rangle$ is thus $[M]^{-1}$. We further define $n$-particle asymptotic states as tensor products of these normalised single-particle states. In order to avoid an unsightly splatter of factors of $2\\pi$, it is convenient to define\n\\begin{equation}\n\\del^{(n)}(p) \\equiv (2\\pi)^n \\delta^{(n)}(p)\n\\label{eqn:delDefinition}\n\\end{equation}\nfor the $n$-fold Dirac $\\delta$ distribution. With these conventions the state normalisation is\n\\begin{equation}\n\\langle p'_i | p^j \\rangle = 2 E_p \\, \\del^{(3)} (\\v{p}-\\v{p}') \\delta_i{ }^j\\,.\n\\label{eqn:MomentumStateNormalization}\n\\end{equation}\nWe define the amplitudes in four dimensions on this plane wave basis by\n\\begin{multline}\n\\langle p'_1 \\cdots p'_m  | T | p_1 \\cdots p_n \\rangle = \\Ampl(p_1 \\cdots p_n \\rightarrow p'_1 \\cdots p'_m) \\\\ \\times \\del^{(4)}(p_1 + \\cdots p_n - p'_1 - \\cdots - p'_m)\\,.\n\\label{eqn:amplitudeDef}\n\\end{multline}\nThe scattering matrix $S$ and the transition matrix $T$ are both dimensionless, leading to the initially advertised dimensions for amplitudes.\n\nLet us now imagine restoring the $\\hbar$'s in a given amplitude. When $\\hbar = 1$, the amplitude has dimensions of $[M]^{4-n}$.  When $\\hbar \\neq 1$, the dimensions of the momenta and masses in the amplitude are unchanged. Similarly there is no change to the dimensions of polarisation vectors. However, we must remember that the dimensionless coupling in electrodynamics is $e\/\\sqrt{\\hbar}$. Similarly, in gravity a factor of $1\/\\sqrt{\\hbar}$ appears, as the appropriate coupling with dimensions of inverse mass is $\\kappa = \\sqrt{32 \\pi G\/ \\hbar}$. We will see shortly that the situation is a little more intricate in Yang-Mills theory, as the colour factors can carry dimensions of $\\hbar$. However we will establish conventions such that the coupling has the same scaling as the QED\/gravity case.  The algorithm to restore the dimensions of any amplitude in electrodynamics, chromodynamics or gravity is then simple: each factor of a coupling is multiplied by an additional factor of $1\/\\sqrt{\\hbar}$. For example, an $n$-point, $L$-loop amplitude in scalar QED is proportional to $\\hbar^{1-n\/2-L}$. \n\nThis conclusion, though well-known, may be surprising in the present context because it seems na\\\"{i}vely that as $\\hbar \\rightarrow 0$, higher multiplicities and higher loop orders are \\textit{more\\\/} important. However, when restoring powers of $\\hbar$ one must distinguish between the momentum $p^\\mu$ of a particle and its wavenumber, which has dimensions of $[L]^{-1}$. This distinction will be important for us, so we introduce a notation for the wavenumber $\\barp$ associated with a momentum $p$:\n\\begin{align}\n\\wn p \\equiv p \/ \\hbar.\n\\label{eqn:notationWavenumber}\n\\end{align}\nIn the course of restoring powers of $\\hbar$ by dimensional analysis, we will first treat the momenta of all particles as genuine momenta. We will also treat any mass as a mass, rather than the associated (reduced) Compton wavelength $\\ell_c = \\hbar\/m$.\n\nAs we will, the approach to the classical limit --- for observables that make sense classically --- effectively forces the wavenumber scaling upon certain momenta. Examples include the momenta of massless particles, such as photons or gravitons.  In putting the factors of $\\hbar$ back into the couplings, we have therefore not yet made manifest all of the physically relevant factors of $\\hbar$ in an amplitude. This provides one motivation for this part of the thesis: we wish to construct on-shell observables which are both classically and quantum-mechanically sensible. \n\n\\section{Single particle states}\n\\label{sec:stateSetup}\n\nLet us take a generic single particle state expanded on the plane wave basis of \\eqref{eqn:singleParticleStateDef}:\n\\begin{equation}\n|\\psi\\rangle = \\sum_i \\int\\! \\dd^4 p \\, \\delp(p^2 - m^2) \\, \\psi_i(p) \\, |p^i\\rangle\\,,\\label{eqn:InitialState}\n\\end{equation}\nHere $\\dd p$ absorbs a factor of $2 \\pi$; more generally $\\dd^n p$ is defined by\n\\begin{equation}\n\\dd^n p \\equiv \\frac{d^n p}{(2 \\pi)^n}\\,.\n\\label{eqn:ddxDefinition}\n\\end{equation}\nWe restrict the integration to positive-energy solutions of the delta functions of $p^2-m^2$, as indicated by the $(+)$ superscript in $\\delp$, as well as absorbing a factor of $2\\pi$, just as for $\\del(p)$:\n\\begin{equation}\n\\delp(p^2-m^2) \\equiv 2\\pi\\Theta(p^0)\\delta(p^2-m^2)\\,.\n\\label{eqn:delpDefinition}\n\\end{equation}\nWe will find it convenient to further abbreviate the notation for on-shell integrals (over Lorentz-invariant phase space), defining\n\\begin{equation}\n\\df(p) \\equiv \\dd^4 p \\, \\delp(p^2-m^2)\\,.\n\\label{eqn:dfDefinition}\n\\end{equation}\nWe will generally leave the mass implicit, along with the designation of the integration variable as the first summand when the argument is a sum. Note that the right-hand side of~\\eqref{eqn:MomentumStateNormalization} is the appropriately normalised delta function for this measure, \n\\begin{equation}\n\\label{eqn:norm1}\n\\int \\df(p') \\, 2 E_{p'} \\, \\del^{(3)} (\\v{p}-\\v{p}') f(p') = f(p)\\,.\n\\end{equation}\nThus for any function $f(p_1')$, we define\n\\begin{equation}\n\\Del(p-p') \\equiv  2 E_{p'} \\del^{(3)} (\\v{p}-\\v{p}')\\,.\n\\end{equation}\nThe argument on the left-hand side is understood as a function of four-vectors. This leads to a notationally clearer version of \\eqn~\\eqref{eqn:norm1}:\n\\begin{equation}\n\\int \\df(p') \\, \\Del(p - p') f(p') = f(p)\\,,\n\\end{equation}\nand of \\eqn~\\eqref{eqn:MomentumStateNormalization}:\n\\begin{equation}\n\\langle p'_i | p^j \\rangle = \\Del(p-p') \\delta_i{ }^j\\,.\n\\end{equation}\n\nThe full state $|\\psi\\rangle$ is a non-trivial representation, of a Lie group associated with symmetries which constrain the description of our particle. The kinematic data, however, should be independent of these symmetries, and thus a singlet of $R$. The full state is thus a tensor product of momentum and representation states: \n\\begin{align}\n\\ket{\\psi}= \\sum \\ket{\\psi_{\\text{mom}}} \\otimes  \\ket{\\psi_{R}}.\n\\end{align}\nWe will make this explicit by splitting the wavefunctions $\\psi_i(p)$, writing\n\\begin{equation}\n\\sum_i\\psi_i(p)| p^i\\rangle = \\sum_i \\varphi(p) \\chi_i |p^i\\rangle = \\varphi(p) |p\\, \\chi\\rangle\\,.\\label{eqn:wavefunctionSplit}\n\\end{equation}\n\nIn these conventions, \\eqn~\\eqref{eqn:InitialState} becomes\n\\begin{equation}\n| \\psi \\rangle = \\int \\! \\df(p)\\;\n\\varphi(p) | p \\, \\chi \\rangle\\,.\n\\label{eqn:InitialStateSimple}\n\\end{equation}\nUsing this simplified notation, the normalisation condition is\n\\begin{equation}\n\\begin{aligned}\n1 &= \\langle \\psi | \\psi \\rangle \\\\\n&= \\sum_{i,j} \\! \\int \\! \\df(p) \\df(p') \\varphi^*(p_1') \\varphi(p_1) \\chi^{*i} \\chi_j\\, \\Del(p_1 - p_1') \\, \\delta_i{ } ^j\\\\\n&= \\sum_i \\! \\int \\! \\df(p)\\; |\\varphi(p)|^2 |\\chi_i|^2\\,.\n\\end{aligned}\n\\end{equation}\nWe can obtain this normalisation by requiring that both wavefunctions $\\phi(p)$ and $\\chi_i$ be normalised to unity:\n\\begin{equation}\n\\int \\! \\df(p)\\; |\\varphi(p)|^2 = 1\\,, \\qquad \\sum_i \\chi^{i*} \\chi_i = 1\\,.\n\\label{eqn:WavefunctionNormalization}\n\\end{equation}\n\nSince $|\\psi\\rangle$ is expanded on a basis of momentum eigenstates, it is trivial to measure the momentum of the state with the momentum operator $\\mathbb P^\\mu$:\n\\begin{equation}\n\\langle \\psi|\\mathbb{P}^\\mu |\\psi\\rangle = \\int \\! \\df(p)\\; p^\\mu |\\varphi(p)|^2\\label{eqn:momentumExp}\n\\end{equation} \nBut how do we measure the physical charges associated with the representation states $|\\chi\\rangle$?\n\n\\subsection{Review of the theory of colour}\n\\label{sec:setup}\n\nFor a physical particle state, there are two distinct interpretations for the representation $R$: it could be the irreducible representation of the little group for a particle of non-zero spin; or it could be the representation of an internal symmetry group of the theory. In this first part of the thesis we are only interested in scalar particle states. We will therefore restrict to the second option for the time being, returning to little group representations in part~\\ref{part:spin}. \n\nOur ultimate goal is to extract, from QFT, long-range interactions between point particles, mediated by a classical field. We will therefore take $R$ to be any representation of an $SU(N)$ gauge group. The classical dynamics of the corresponding Yang--Mills field $A_\\mu = A_\\mu^a T^a$, coupled to several classical point-like particles, are then described by the Yang--Mills--Wong equations:. \n\\begin{subequations}\n\t\\label{eqn:classicalWong}\n\t\\begin{gather}\n\t\\frac{\\d p_\\alpha^\\mu }{\\d \\tau_\\alpha} = g\\, c^a_\\alpha(\\tau_\\alpha)\\, F^{a\\,\\mu\\nu}\\!(x_\\alpha(\\tau_\\alpha))\\,  v_{\\alpha\\, \\nu}(\\tau_\\alpha)\\,,  \\label{eqn:Wong-momentum} \n\t\\\\\n\t\\frac{\\d c^a_\\alpha}{\\d \\tau_\\alpha}= g f^{abc} v^\\mu_\\alpha(\\tau_\\alpha) A_\\mu^b(x_\\alpha(\\tau_\\alpha))\\,c^c_\\alpha(\\tau_\\alpha)\\,,\n\t\\label{eqn:Wong-color}\n\t\\\\\n\tD^\\mu F_{\\mu\\nu}^a(x) = J^a_\\nu(x) = g \\sum\\limits_{\\alpha= 1}^N \\int\\!\\d\\tau_\\alpha\\,  c^a_\\alpha(\\tau_\\alpha)  v^\\mu(\\tau_\\alpha)\\, \\delta^{(4)}(x-x(\\tau_\\alpha))\\,, \\label{eqn:YangMillsEOM}\n\t\\end{gather}\n\\end{subequations}\nThese equations describe particles, following worldlines $x_\\alpha(\\tau_\\alpha)$ and with velocities $v_\\alpha$, that each carry colour charges $c^a$ which are time-dependent vectors in the adjoint-representation of the gauge group. \n\nLet us review the emergence of of these non-Abelian colour charges from quantum field theory by restricting our attention to scalars $\\pi_\\alpha$ in any representation $R_\\alpha$ of the gauge group, coupled to the Yang--Mills field. The action is\n\\begin{equation}\nS = \\int\\!\\d^4x\\, \\left(\\sum_{\\alpha}\\left[ (D_\\mu \\pi_\\alpha)^\\dagger D^\\mu \\pi_\\alpha - \\frac{m_\\alpha^2}{\\hbar^2} \\pi_\\alpha^\\dagger \\pi_\\alpha\\right] - \\frac14 F^a_{\\mu\\nu} F^{a\\,\\mu\\nu}\\right), \\label{eqn:scalarAction}\n\\end{equation}\nwhere  $D_\\mu = \\partial_\\mu + i g A_\\mu^a T^a_R$. The generator matrices (in a representation $R$) are $T^a_R = (T_R^a)_i{ }^j$, and satisfy the Lie algebra \n$[T_R^a, T_R^b]_i{ }^j = if^{abc} (T_R^c)_i{ }^j$. \n\nLet us consider only a single massive scalar. At the classical level, the colour charge can be obtained from the Noether current $j^a_\\mu$ associated with the global part of the gauge symmetry. The colour charge is explicitly given by\n\\begin{equation}\n\\int\\!\\d^3x\\, j^a_0(t,\\v{x}) = i\\!\\int\\! \\d^3x\\, \\Big(\\pi^\\dagger T^a_R\\, \\partial_0 \\pi - (\\partial_0 \\pi^\\dagger) T^a_R\\, \\pi\\Big)\\,.  \\label{eqn:colourNoetherCharge}\n\\end{equation}\nNotice that a direct application of the Noether procedure has led to a colour charge with dimensions of action, or equivalently, of angular momentum. It is now worth dwelling on dimensional analysis in the context of the Wong equations~\\eqref{eqn:classicalWong}, since they motivate us to make certain choices which may, at first, seem surprising. The Yang--Mills field strength\n\\begin{equation}\nF^a_{\\mu\\nu}  =\\partial_\\mu A^a_\\nu - \\partial_\\nu A^a_\\mu - gf^{abc} A^b_\\mu A^c_\\nu\\,\\label{eqn:fieldStrength}\n\\end{equation}\nis obviously an important actor in these classical equations. Classical equations should contain no factors\\footnote{An equivalent point of view is that any factors of $\\hbar$ appearing in an equation which has classical meaning should be absorbed into parameters of the classical theory.} of $\\hbar$, so we choose to maintain this precise expression for the field strength when $\\hbar \\neq 1$. By inspection it follows that $[g A^{a}_\\mu] = L^{-1}$. We can develop this further; since the action of \\eqn~\\eqref{eqn:scalarAction} has dimensions of angular momentum, the Yang--Mills field strength must have dimensions of $\\sqrt{M\/L^3}$. Thus, from \\eqn~\\eqref{eqn:fieldStrength},\n\\begin{equation}\n[A^a_\\mu] = \\sqrt{\\frac{M}{L}}\\,, \\qquad [g] = \\frac1{\\sqrt{ML}}\\,.\\label{eqn:YMdims}\n\\end{equation}\nThis conclusion about the dimensions of $g$ is in contrast to the situation in electrodynamics, where $[e] = \\sqrt{ML}$. Put another way, in electrodynamics the dimensionless fine structure constant is $e^2 \/ 4\\pi \\hbar$ while in our conventions the Yang--Mills analogue is $\\hbar g^2 \/ 4\\pi\\,$! It is possible to arrange matters such that the YM and EM cases are more similar, but we find the present conventions to be convenient in perturbative calculations.\n\nContinuing with our discussion of dimensions, note that the Yang--Mills version of the Lorentz force, \\eqn~\\eqref{eqn:Wong-momentum}, demonstrates that the quantity $g c^a$ must have the same dimension as the electric charge. This is consistent with our observation above that the colour has dimensions of angular momentum.\n\nAt first our assignment of dimensions of $g$ may seem troubling; the fact that $g$ has dimensions of $1\/\\sqrt{ML}$ implies that the dimensionless coupling at each vertex is $g \\sqrt \\hbar$, so factors of $\\hbar$ associated with the coupling appear with the opposite power to the case of electrodynamics (and gravity). However, because the colour charges are dimensionful the net power of $\\hbar$ turns out to be the same. The classical limit of this aspect of the theory is clarified by the dimensionful nature of the colour --- to see how this works we must quantise. \n\nDimensional analysis demonstrates that the field $\\pi$ has dimensions of $\\sqrt{M\/L}$, so its mode expansion is\n\\begin{equation}\n{\\pi}_i(x) = \\frac1{\\sqrt{\\hbar}} \\int\\!\\df(p)\\, \\left(a_i(p) e^{-ip\\cdot x\/\\hbar} + b_i^\\dagger(p) e^{ip\\cdot x\/\\hbar}\\right).\n\\end{equation}\nThe ladder operators are normalised as in equation~\\eqref{eqn:ladderCommutator}, with the index $i$ again labelling the representation $R$. After quantisation, the colour charge of \\eqn~\\eqref{eqn:colourNoetherCharge} becomes a Hilbert space operator,\n\\begin{equation}\n\\begin{aligned}\n\\C^a &= i\\!\\int\\! \\d^3x\\, \\Big(\\pi^\\dagger T^a_R\\, \\partial_0 \\pi - (\\partial_0 \\pi^\\dagger) T^a_R\\, \\pi\\Big) \\\\\n&= \\hbar\\int\\!\\df(p)\\, \\left( a^\\dagger(p) \\,T^a_R \\, a(p) + b^\\dagger(p)\\, T^a_{\\bar R} \\, b(p)\\right),\\label{eqn:colourOp}\n\\end{aligned}\n\\end{equation}\nwhere we have used that the generators of the conjugate representation $\\bar R$ satisfy $T^a_{\\bar R} = - T^a_R$. The overall $\\hbar$ factor guarantees that the colour has dimensions of angular momentum, as we require. It is important to note that these global colour operators inherit the usual Lie algebra of the generators, modified by factors of $\\hbar$, so that\n\\begin{equation}\n[\\C^a, \\C^b] = i\\hbar f^{abc} \\C^c\\,.\\label{eqn:chargeLieAlgebra}\n\\end{equation}\n\nActing with the colour charge operator of \\eqn~\\eqref{eqn:colourOp} on momentum eigenstates (as defined in \\eqn~\\eqref{eqn:singleParticleStateDef}), we immediately see that\n\\begin{equation}\n\\C^a|p^i\\rangle = \\hbar\\, (T^a_R)_j{ }^i|p^j\\rangle\\,, \\qquad \\langle p_i|\\C^a = \\hbar\\, \\langle p_j|(T^a_R)_i{ }^j\\,.\n\\end{equation}\nThus inner products yield generators scaled by $\\hbar$:\n\\begin{equation}\n\\langle p_i|\\C^a|p^j\\rangle \\equiv (\\newT^a)_i{ }^j = \\hbar\\, (T^a_R)_i{ }^j\\,.\n\\end{equation}\nThe $(C^a)_i{ }^j$ are simply rescalings of the usual generators $T^a_R$ by a factor of $\\hbar$, and thus satisfy the rescaled Lie algebra in \\eqn~\\eqref{eqn:chargeLieAlgebra}; since this rescaling is important for us, it is useful to make the distinction between the two. \n\nWe can now finally act with the colour operator on the single particle state of equation~\\eqref{eqn:InitialStateSimple}:\n\\begin{equation}\n\\C^a|\\psi\\rangle = \\int\\!\\df(p)\\, (\\newT^a)_i{ }^j\\, \\varphi(p) \\chi_j|p^i\\rangle\\,,\n\\end{equation}\nallowing us to define the colour charge of the particle as\n\\begin{equation}\n\\langle \\psi |\\C^a| \\psi \\rangle =  \\chi^{i*} (\\newT^a)_{i}{ }^{j}\\, \\chi_j\\,. \\label{eqn:colourCharge}\n\\end{equation}\n\nAs a final remark on the rescaled generators, let us write out the covariant derivative in the representation $R$. In terms of $\\newT^a$, \nthe $\\hbar$ scaling of interactions is precisely the same as in QED (and in perturbative gravity):\n\\begin{equation}\nD_\\mu = \\partial_\\mu + i \\, g A^a_\\mu T^a = \\partial_\\mu + \\frac{ig}{\\hbar}\\, A^a_\\mu \\newT^a \\,;\\label{eqn:covDerivative}\n\\end{equation}\nfor comparison, the covariant derivative in QED consistent with our discussion in section~\\ref{sec:RestoringHBar} is $\\partial_\\mu + i e A_\\mu\/ \\hbar$. Thus we have arranged that factors of $\\hbar$ appear in the same place in YM theory as in electrodynamics, provided that the colour is measured by $C^a$. This ensures that the basic rules for obtaining the classical limits of amplitudes will be the same; in practical calculations one restores $\\hbar$'s in colour factors and works using $C^a$'s everywhere. However, it is worth emphasising that unlike classical colour charges, the factors $C^a$ do not commute.\n\n\\section{Classical point-particles}\n\\label{sec:PointParticleLimit}\n\nFor the states in equation~\\eqref{eqn:InitialStateSimple} to have a well defined point-particle limit, for any operator $\\mathbb{O}$ they must, at a bare minimum, satisfy the following two constraints in the classical limit \\cite{Yaffe:1981vf}:\n\\[\n\\langle \\psi |\\mathbb{O} | \\psi \\rangle &= \\textrm{finite} \\,, \\\\\n\\langle \\psi |\\mathbb{O} \\, \\mathbb{O}| \\psi \\rangle &= \\langle \\psi |\\mathbb{O} | \\psi \\rangle\\langle \\psi | \\mathbb{O}| \\psi \\rangle + \\textrm{negligible} \\,.\\label{eqn:classicalConstraints}\n\\]\nFurthermore the classical limit is not necessarily injective: distinct quantum states $|\\psi\\rangle$ and $|\\psi'\\rangle$ may yield the same classical limit. Classical physics should of course be independent of the details of quantum states, and therefore we also require that in the limit, the overlap\n\\begin{equation}\n\\langle \\psi' | \\psi \\rangle = \\langle \\psi | \\psi \\rangle + \\textrm{negligible}\\,.\\label{eqn:classicalOverlap}\n\\end{equation}\nSimilarly, the expectation values above should remain unchanged in the limit if taken over distinct but classicaly equivalent states \\cite{Yaffe:1981vf}.\n\nOur goal in this section is to choose suitable momentum and colour wavefunctions, $\\varphi(p)$ and $\\chi$ respectively, which ensure that the observables in equations~\\eqref{eqn:momentumExp} and~\\eqref{eqn:colourCharge} meet these crucial requirements.\n\n\\subsection{Wavepackets}\n\\label{subsec:Wavefunctions}\n\nClassical point particles have well defined positions and momenta. Heuristically, we therefore require well localised quantum states. We will take the momentum space wavefunctions $\\varphi(p)$ to be wavepackets, characterised by a smearing or spread in momenta\\footnote{Evaluating positions and uncertainties therein in\trelativistic field theory is a bit delicate, and we will not consider the question in this thesis.}.   \n\nLet us ground our intuition by first examining nonrelativistic wavefunctions.  An example of a minimum-uncertainty wavefunction in momentum space (ignoring normalisation) for a particle of mass $m$ growing sharper in the $\\hbar\\rightarrow 0$ limit has the form\n\\begin{equation}\n\\exp\\left( -\\frac{\\v{p}\\mskip1mu{}^2}{2 \\hbar m \\lcomp\/ \\lpack^2}\\right)\n= \\exp\\left( -\\frac{\\v{p}\\mskip1mu{}^2}{2m^2 \\lcomp^2\/\\lpack^2}\\right),\n\\label{eqn:NonrelativisticMomentumSpaceWavefunction}\n\\end{equation}\nwhere $\\lcomp$ is the particle's Compton wavelength, and where $\\lpack$ is an additional parameter with dimensions of length. We can obtain the conjugate in position space by Fourier transforming:\n\\begin{equation}\n\\exp\\left( -\\frac{(\\v{x}-\\v{x}_0)^2}{2 \\lpack^2}\\right).\n\\end{equation}\nThe precision with which we know the particle's location is given by $\\lpack$, which we could take as an intrinsic measure of the wavefunction's spread.\n\nThis suggests that in considering relativistic wavefunctions, we should also take the dimensionless parameter controlling the approach to the classical limit in momentum space to be the square of the ratio of the Compton wavelength $\\lcomp$ to the intrinsic spread~$\\lpack$,\n\\begin{equation}\n\\xi \\equiv \\biggl(\\frac{\\lcomp}{\\lpack}\\biggr){\\vphantom{\\frac{\\lcomp}{\\lpack}}}^2\\,.\\label{eqn:defOfXi}\n\\end{equation}\nWe therefore obtain the classical result by studying the behaviour of expectation values as $\\xi\\rightarrow 0$; or alternatively, in the region where\n\\begin{equation}\n\\ell_c \\ll \\ell_w\\,.\\label{eqn:ComptonConstraint1}\n\\end{equation}\nTowards the limit, the wavefunctions must be sharply peaked around the classical value for the momenta, $\\pcl = m \\ucl$, with the classical four-velocity $\\ucl$ normalised to $\\ucl^2 = 1$.  We can express this requirement through the conditions\\footnote{The integration measure for $p$ enforces $\\langle p^2\\rangle = m^2$.}\n\\begin{equation}\n\\begin{aligned}\n\\langle p^\\mu\\rangle &= \\int \\df(p)\\; p^\\mu\\, |\\varphi(p)|^2 = \nm \\uapprox^\\mu f_{p}(\\xi)\\,,\n\\\\ f_{p}(\\xi) &= 1+\\Ord(\\xi^{\\beta'})\\,,\n\\\\ \\uapprox\\cdot \\ucl &= 1+\\Ord(\\xi^{\\beta''})\\,,\n\\\\ \\spread(p)\/m^2 &=\n\\langle \\bigl(p-\\langle p\\rangle\\bigr){}^2\\rangle\/m^2\n\\\\&= \\bigl(\\langle p^2\\rangle-\\langle p\\rangle{}^2\\bigr)\/m^2\n= c_\\Delta \\xi^\\beta\\,,\n\\end{aligned}\n\\label{eqn:expectations}\n\\end{equation}\nwhere $c_\\Delta$ is a constant of order unity, and the $\\beta$'s are simple rational exponents. For the simplest wavefunctions, $\\beta=1$. This spread around the classical value is not necessarily positive, as the difference $p^\\mu-\\langle p^\\mu\\rangle$ may be spacelike, and the expectation of its Lorentz square possibly negative. For that reason, we should resist the usual temptation of taking its square root to obtain a variance.\n\nThese constraints are the specific statement of those in equation~\\eqref{eqn:classicalConstraints} for momentum space wavepackets. What about the vanishing overlap between classically equivalent states,~\\eqref{eqn:classicalOverlap}? To determine a constraint on the wavepackets we need a little more detail of their functional form. Now, because of the on-shell condition $p^2=m^2$ imposed by the phase-space integral over the wavepacket's momenta, the only Lorentz invariant built out of the momentum is constant, and so the wavefunction cannot usefully depend on it. This means the wavefunction must depend on at least one four-vector parameter. The simplest wavefunctions will depend on exactly one four-vector, which we can think of as the (classical) 4-velocity $\\ucl$ of the corresponding particle.  It can depend only on the dimensionless combination $p\\cdot \\ucl\/m$ in addition to the parameter $\\xi$.  The simplest form will be a function of these two in the combination $p\\cdot\\ucl\/(m\\xi)$, so that large deviations from $m \\ucl$ will be suppressed in a classical quantity. The wavefunction will have additional dependence on $\\xi$ in its normalisation.\n\nThe difference between two classically equivalent wavepackets must therefore come down to a characteristic mismatch $q_0$ of their momentum arguments --- without loss of generality, classically equivalent wavepackets are then specified by wavefunctions $\\varphi(p)$ and $\\varphi(p + q_0)$. As one nears the classical limit, both wavefunctions must represent the particle: that is they should be sharply peaked, and in addition their overlap should be $\\Ord(1)$, up to corrections of $\\Ord(\\xi)$. Requiring the overlap to be $\\Ord(1)$ is equivalent to requiring that $\\varphi(p+q_0)$ does not differ much from $\\varphi(p)$, which in turn requires that the derivative at $p$ is small, or that\n\\begin{equation}\n\\frac{q_0\\cdot\\ucl}{m\\xi} \\ll 1\\,.\n\\label{eqn:qConstraint1}\n\\end{equation}\nIf we scale $q$ by $1\/\\hbar$, this constraint takes the following form:\n\\begin{equation}\n\\qb_0\\cdot\\ucl\\,\\lpack \\ll \\sqrt{\\xi}.\n\\label{eqn:qbConstraint1}\n\\end{equation}\nWe have replaced the momentum by a wavenumber. We will see in the next chapter that this constraint is the fundamental constraint forcing the classical scaling advertised in equation~\\eqref{eqn:notationWavenumber} upon certain momenta in scattering amplitudes.\n\nLet us remain in the single particle case and finally examine an explicit example wavefunction satisfying our constraints. We will take a linear exponential,\n\\begin{equation}\n\\varphi(p) = \\Norm m^{-1}\\exp\\biggl[-\\frac{p\\cdot u}{\\hbar\\lcomp\/\\lpack^2}\\biggr]\n= \\Norm m^{-1}\\exp\\biggl[-\\frac{p\\cdot u}{m\\xi}\\biggr]\\,,\n\\label{eqn:LinearExponential}\n\\end{equation}\nwhich shares some features with relativistic wavefunctions discussed in ref.~\\cite{AlHashimi:2009bb}. In spite of the linearity of the exponent in $p$, this function gives rise to the Gaussian of \\eqn~\\eqref{eqn:NonrelativisticMomentumSpaceWavefunction} in the nonrelativistic limit (in the rest frame of $u$).\n\nThe normalisation condition~(\\ref{eqn:WavefunctionNormalization}) requires\n\\begin{equation}\n\\Norm = \\frac{2\\sqrt2\\pi}{\\xi^{1\/2} K_1^{1\/2}(2\/\\xi)}\n\\,,\n\\end{equation}\nwhere $K_1$ is a modified Bessel function of the second kind. For details of this computation and following ones, see appendix~\\ref{app:wavefunctions}.  An immediate corollary is that the overlap\n\\begin{equation}\n\\int \\! \\df(p) \\, \\varphi^*(p + q_0) \\varphi(p) = \\exp\\left[-\\frac{u\\cdot q_0}{m\\xi}\\right] \\equiv \\eta_1(q_0;p)\\,.\\label{eqn:wavefunctionOverlap}\n\\end{equation}\nClearly, for this result to vanish in the limit $\\xi=0$ we must rescale $q_0 = \\hbar \\wn q_0$, which then explicitly recovers the constraint~\\eqref{eqn:qbConstraint1}.\n\nWe can compute the momentum expectation value of the wavepacket straightforwardly, obtaining\n\\begin{equation}\n\\langle p^\\mu\\rangle = m u^\\mu \\frac{K_2(2\/\\xi)}{K_1(2\/\\xi)}\\,.\n\\end{equation}\nAs we approach the classical region, where $\\xi\\rightarrow 0$, the wavefunction indeed becomes sharply peaked, as\n\\begin{equation}\n\\langle p^\\mu \\rangle \\rightarrow m u^\\mu \\left(1 + \\frac34 \\xi\\right) + \\Ord(\\xi^2)\\,.\n\\end{equation}\nMoreover, the spread of the wavepacket\n\\begin{equation}\n\\frac{\\sigma^2(p)}{\\langle p^2\\rangle} = 1 - \\frac{K_2^2(2\/\\xi)}{K_1^2(2\/\\xi)} \\rightarrow -\\frac32 \\xi + \\Ord(\\xi^2)\\,.\n\\end{equation}\n\nFinally, a similar calculation yields\n\\begin{equation}\n\\langle p^\\mu p^\\nu \\rangle = m^2 u^\\mu u^\\nu \\left(1 +  \\frac{2\\xi \\, K_2(2\/\\xi)}{K_1(2\/\\xi)}\\right)  - \\frac{m^2}{2}  \\frac{\\xi\\, K_2(2\/\\xi)}{K_1(2\/\\xi)}\\, \\eta^{\\mu\\nu}\\,,\\label{eqn:doubleMomExp}\n\\end{equation}\nso in the classical region our wavepackets explicitly satisfy\n\\[\n\\langle p^\\mu p^\\nu \\rangle &\\rightarrow m^2 u^\\mu u^\\nu \\left(1 + 2 \\xi\\right)  - \\frac{m^2}{2} \\xi\\, \\eta^{\\mu\\nu}  + \\mathcal{O}(\\xi^2)\\\\ \n&= \\langle p^\\mu \\rangle \\langle p^\\nu \\rangle + \\mathcal{O}(\\xi)\\,.\n\\]\nFrom these results, we see that the conditions in equation~\\eqref{eqn:expectations} are explicitly satisfied, with $c_\\Delta = -3\/2$ and rational exponents $\\beta = \\beta' = \\beta'' = 1$.\n\n\\subsection{Coherent colour states}\n\\label{sec:classicalSingleParticleColour}\n\nWe have seen that the classical point-particle picture emerges from sharply peaked quantum wavepackets. To understand colour, governed by the Yang--Mills--Wong equations in the classical arena, a similar picture should emerge for our quantum colour operator in \\eqn~\\eqref{eqn:colourOp}. We define the classical limit of the colour charge in equation~\\eqref{eqn:colourCharge} to be\n\\begin{equation}\nc^a \\equiv \\langle \\psi |\\C^a| \\psi \\rangle\\,.\n\\end{equation}\nSince the colour operator in \\eqref{eqn:colourOp} explicitly involves a factor of $\\hbar$, another parameter must be large so that the colour expectation $\\langle \\psi |\\C^a | \\psi \\rangle$ is much bigger than $\\hbar$ in the classical region. For states in irreducible representation $R$ the only new dimensionless parameter available is the size of the representation, $n$, and indeed we will see explicitly in the case of $SU(3)$ that we indeed need $n$ large in this limit.\n\nCoherent states are the key to the classical limit very generally~\\cite{Yaffe:1981vf}, and we will choose a coherent state to describe the colour of our particle. The states adopted previously to describe momenta can themselves be  understood as coherent states for a ``first-quantised'' particle --- more specifically they are states for the restricted Poincar\\'e group \\cite{Kaiser:1977ys, TwarequeAli:1988tvp, Kowalski:2018xsw}. By ``coherent\" we mean in the sense of the definition introduced by Perelomov \\cite{perelomov:1972}, which formalises the notion of coherent state for any Lie group and hence can be utilised for both the kinematic and the colour parts.\n\nTo construct explicit colour states we will use the Schwinger boson formalism.  For $SU(2)$, constructing irreducible representations from Schwinger bosons is a standard textbook exercise \\cite{Sakurai:2011zz}. One simply introduces the Schwinger bosons --- that is, creation $a^{\\dagger i}$ and annihilation $a_i$ operators, transforming in the fundamental two-dimensional representation so that $i = 1,2$. The irreducible representations of $SU(2)$ are all symmetrised tensor powers of the fundamental, so the state\n\\[\na^{\\dagger i_1} a^{\\dagger i_2} \\cdots a^{\\dagger i_{2j}} \\ket{0} ,\n\\]\nwhich is automatically symmetric in all its indices, transforms in the spin $j$ representation. \n\nFor groups larger than $SU(2)$, the situation is a little more complicated because the construction of a general irreducible representation requires both symmetrisation and \nantisymmetrisation over appropriate sets of indices. This leads to expressions which are involved already for $SU(3)$ \\cite{Mathur:2000sv,Mathur:2002mx}. We content \nourselves with a  brief discussion of the $SU(3)$ case, which captures all of the interesting features of the general case.\n\nOne can construct all irreducible representations from tensor products only of fundamentals \\cite{Mathur:2010wc,Mathur:2010ey}; however, for our treatment of $SU(3)$ it is helpful to instead make use of the fundamental and antifundamental, and tensor these together to generate representations.  Following \\cite{Mathur:2000sv}, we introduce two sets of ladder operators $a_i$ and $b^i$ , $i=1, 2, 3$, which transform in the $\\mathbf{3}$ and $\\mathbf{3}^*$ respectively.  The colour operator can then be written as\n\\begin{equation}\n\\C^e= \\hbar \\left( a^\\dagger \\frac{\\lambda^e}{2} a -\nb^\\dagger \\frac{\\bar{\\lambda}^e}{2} b  \\right), \\quad e=1, \\dots, 8\\,, \\label{eqn:charge-SU3}\n\\end{equation}\nwhere $\\lambda^e$ are the Gell--Mann matrices and $\\bar\\lambda^e$ are their conjugates. The operators $a$ and $b$ satisfy the usual commutation relations\n\\begin{align}\n[a_i, a^{\\dagger j}]= \\delta_{i}{ }^{j}\\,, \\quad   [b^i, b^{\\dagger}_j]= \\delta^{i}{ }_{j}\\,, \\quad \n[a_i, b^j]= 0\\,, \\quad  [a^{\\dagger i}, b^{\\dagger }_j]= 0\\,.\n\\end{align}\nBy virtue of these commutators, the colour operator \\eqref{eqn:charge-SU3} obeys the commutation relation \\eqref{eqn:chargeLieAlgebra}.\n\nThere are two Casimir operators given by the number operators\\footnote{Here we define $a^{\\dagger}  \\cdot a \\equiv \\sum_{i=1}^3 a^{\\dagger i} a_i$ and\n\t$|\\varsigma|^2\\equiv \\sum_{i=1}^3 |\\varsigma_i|^2$.}\n\\begin{equation}\n\\mathcal{N}_1\\equiv a^\\dagger \\cdot a\\,, \\qquad  \\mathcal{N}_2\\equiv b^\\dagger \\cdot b\\,,\n\\end{equation}\nwith eigenvalues $n_1$ and $n_2$ respectively, so we label irreducible representations by $[n_1, n_2]$. \nNa\\\"ively, the states we are looking for are constructed by acting on the vacuum state as follows:\n\\begin{align}\n\\left(a^{\\dagger i_1} \\cdots  a^{\\dagger i_{n_1}} \\right)\n\\left(b_{j_1}^{\\dagger} \\cdots  b_{j_{n_2}}^{\\dagger} \\right)\n\\ket{0}.\\label{eqn:states-reducible}\n\\end{align}\nHowever, these states are $SU(3)$ reducible and thus cannot be used in our construction of coherent states. We write the irreducible states schematically by acting with a Young projector $\\mathcal{P}$ which appropriately (anti-) symmetrises upper and lower indices, thereby subtracting traces:\n\\begin{align}\n\\ket{\\psi}_{[n_1, n_2]} \\equiv \\mathcal{P} \\left( \\left(a^{\\dagger i_1} \\cdots  a^{\\dagger i_{n_1}} \\right)\n\\left(b^{\\dagger}_{j_1} \\cdots  b_{j_{n_2}}^{\\dagger} \\right)\\ket{0} \\right).\n\\label{eqn:YPstate}\n\\end{align}\nIn general these operations will lead to involved expressions for the states, but we can understand them from their associated Young tableaux (Fig.~\\ref{fig:SU3-YT}). Each double box column represents an operator $b_i^{\\dagger}$ and each single column box represents the operator $a^{\\dagger i}$, and thus for a mixed representation we have $n_2$ double columns and $n_1$ single columns.\n\n\\begin{figure}\n\t\\centering \n\t\\begin{ytableau}\n\t\tj_1 & j_2  & \\dots  &j_{n_2} & i_1 & i_2 & \\cdots & i_{n_1}  \\cr    &  &  &  \n\t\\end{ytableau}\n\t\\caption{Young tableau of $SU(3)$.}\n\t\\label{fig:SU3-YT}\n\\end{figure}\n\nHaving constructed the irreducible states, one can define a coherent state parametrised by two triplets of complex numbers $\\varsigma_i$ and $ \\varrho^i$, $i=1,2, 3$.  These are normalised according to\n\\begin{equation}\n|\\varsigma|^2 = |\\varrho|^2 = 1\\,, \\qquad \\varsigma \\cdot \\varrho=0\\,.\n\\end{equation}\nWe won't require fully general coherent states, but instead their projections onto the $[n_1,n_2]$ representation, which are\n\\begin{equation}\n\\ket{\\varsigma \\,\\varrho}_{[n_1,n_2]}\\equiv  \\frac{1}{\\sqrt{(n_1! n_2!)}} \\left( \\varrho \\cdot b^\\dagger\\right)^{n_2} \\left(\\varsigma \\cdot a^\\dagger \\right)^{n_1} \\ket{0}.\n\\label{eqn:restricted-coherent}\n\\end{equation}\n\nThe square roots ensure that the states are normalised to unity\\footnote{Note that the Young projector in equation~\\eqref{eqn:YPstate} is no longer necessary since the constraint $\\xi \\cdot \\zeta = 0$ removes all the unwanted traces.}. With this normalisation we can write the identity operator as  \n\\begin{equation}\n\\mathbb{I}_{[n_1,n_2]} = \\int \\d \\mu(\\varsigma,\\varrho) \\Big(\\ket{\\varsigma \\,\\varrho}\\bra{\\varsigma \\,\\varrho}\\Big)_{[n_1,n_2]},\\label{eqn:Haar}\n\\end{equation}\nwhere $\\int \\d \\mu(\\varsigma,\\varrho)$ is the $SU(3)$ Haar measure, normalised such that $\\int \\d \\mu(\\varsigma,\\varrho)=1$. Its precise form is irrelevant for our purposes.\n\nWith the states in hand, we can return to the expectation value of the colour operator $\\C^a$ in \\eqn~\\eqref{eqn:colourOp}. The size of the representation, that is $n_1$ and $n_2$, must be large compared to $\\hbar$ in the classical regime so that the final result is finite. To see this let us compute this expectation value explicitly.\nBy definition we have \n\\begin{equation}\n\\langle \\varsigma \\,\\varrho|\\mathbb{C}^e|\\varsigma \\,\\varrho \\rangle_{[n_1,n_2]} =  \\frac{\\hbar}{2} \\left(\\langle \\varsigma \\,\\varrho|a^\\dagger \\lambda^e a|\\varsigma \\,\\varrho \\rangle_{[n_1,n_2]} -\n\\langle \\varsigma \\,\\varrho|b^\\dagger \\bar{\\lambda}^e b|\\varsigma \\,\\varrho\\rangle_{[n_1,n_2]} \\right).\n\\end{equation}\nAfter a little algebra we find that \n\\begin{equation}\n\\langle \\varsigma \\,\\varrho|\\C^e|\\varsigma \\,\\varrho \\rangle = \\frac{\\hbar}{2} \\left( n_1   \\varsigma^{*} \\lambda^e \\varsigma - n_2    \\varrho^*\\bar \\lambda^e \\varrho\\right).\n\\end{equation}\nWe see that a finite charge requires a scaling limit in which we take $n_1$, $n_2$ large  as $\\hbar \\to 0$, keeping the product $\\hbar n_\\alpha$  fixed for at least one value of $\\alpha=1,2$. The classical charge is therefore the finite c-number\n\\begin{equation}\nc^a =  \\langle \\varsigma \\,\\varrho|\\C^a|\\varsigma \\,\\varrho \\rangle_{[n_1, n_2]}  = \\frac{\\hbar}{2} \\left( n_1   \\varsigma^{*}\\lambda^a \\varsigma - n_2 \\varrho^{*} \\bar\\lambda^a \\varrho\\right). \\label{eqn:clas-charge-SU3}\n\\end{equation}\n\nThe other feature we must check is the expectation value of products. Using the result above, a similar calculation for two pairs of charge operators in a large representation leads to\n\\begin{multline}\n\\langle \\varsigma\\,\\varrho|\\C^a\\C^b | \\varsigma\\,\\varrho\\rangle_{[n_1, n_2]}  = \\langle \\varsigma\\,\\varrho|\\C^a| \\varsigma\\,\\varrho\\rangle_{[n_1,n_2]}  \\langle \\varsigma\\,\\varrho|\\C^b | \\varsigma\\,\\varrho\\rangle_{[n_1,n_2]} \\\\ + \\hbar \\left( \\hbar n_1 \\, \\varsigma^*\\lambda^a\\cdot \\lambda ^b \\varsigma - \\hbar n_2\\, \\varrho^* \\bar\\lambda^a\\cdot \\bar\\lambda^b \\varrho \\right).\n\\end{multline}\nThe finite quantity in the classical limit $\\hbar \\to 0, \\, n_\\alpha \\to \\infty$ is the product $\\hbar n_\\alpha$. The term inside the brackets on the second line is itself finite, but comes with a lone $\\hbar$ coefficient, and thus vanishes in the classical limit. Thus,\n\\[\n\\langle \\varsigma\\,\\varrho|\\C^a \\C^b|\\varsigma\\,\\varrho \\rangle _{[n_1,n_2]} = c^a c^b + \\mathcal O(\\hbar)\\,. \\label{eqn:factorisation-charges}\n\\]\nThis is in fact a special case of a more general construction discussed in detail by Yaffe~\\cite{Yaffe:1981vf}. Similar calculations can also be used to demonstrate  that the overlap $\\langle \\chi' | \\chi \\rangle$ is very strongly peaked about $\\chi = \\chi'$, as required by equation~\\eqref{eqn:classicalOverlap}. We have thus constructed explicit colour states which ensure the correct classical behaviour of the colour charges. \n\nFor the remainder of the thesis we will only need to make use of the finiteness and factorisation properties, so we will avoid further use of the explicit form of the representation states. Henceforth we write $\\chi$ for the parameters of a general colour state $\\ket{\\chi}$ with these properties, and $\\d \\mu(\\chi)$ for the Haar measure of the $SU(N)$ colour group.\n\n\\section{Multi-particle wavepackets}\n\nHaving set up appropriate wavepackets for a single particle, we can now consider multiple particles, and thus generalise the generic states adopted in equation~\\eqref{eqn:InitialStateSimple}. We will take two distinguishable scalar particles, associated with distinct quantum fields $\\pi_\\alpha$ with $\\alpha = 1, 2$. The action is therefore as given in \\eqn~\\eqref{eqn:scalarAction}. Both fields $\\pi_\\alpha$ must be in representations $R_\\alpha$ which are large, so that a classical limit is available for the individual colours.\n\nIn anticipation of considering scattering processes in the next section, we will now take our state to be at some initial time in the far past, where we assume that our two particles both have well-defined positions, momenta and colours. In other words, particle $\\alpha$ has a wavepacket $\\varphi_\\alpha(p_\\alpha)$ describing its momentum-space distribution, and a coherent colour wavefunction $\\chi_\\alpha$, as described in the previous section. Then the appropriate generalisation of the multi-particle state is\n\\[\n|\\Psi\\rangle &= \\int\\!\\df(p_1)\\df(p_2)\\, \\varphi_1(p_1) \\varphi_2(p_2)\\, e^{ib\\cdot p_1\/\\hbar}\\, |{p_1}\\, \\chi_1 ; \\, {p_2}\\, \\chi_2 \\rangle \\\\\n&= \\int\\!\\df(p_1)\\df(p_2) \\, \\varphi_1(p_1) \\varphi_2(p_2)\\, e^{ib\\cdot p_1\/\\hbar}\\, \\chi_{1i}\\, \\chi_{2j} |{p_1}^i ; \\, {p_2}^j \\rangle\\,,\\label{eqn:inState}\n\\]\nwhere the displacement operator insertion accounts for the particles' spatial separation.\n\nWe measure observables for multi-particle states by acting with operators which are simply the sum of the individual operators for each of the scalar fields. For example, acting with the colour operator~\\eqref{eqn:colourOp} on the state $|{p_1}\\, \\chi_1 ; \\, {p_2}\\, \\chi_2 \\rangle$ we have\n\\[\n\\C^a |{p_1}\\,\\chi_1 ;&\\, {p_2}\\, \\chi_2 \\rangle = |{p_1}^{i'} \\, {p_2}^{j'} \\rangle \\, \\left( (C^a_{1})_{i'}{}^i \\delta_{j'}{}^j + \\delta_{i'}{}^i (C^a_{2})_{j'}{}^j \\right) \\chi_{1i}\\, \\chi_{2j} \\\\\n&= \\int \\! d\\mu(\\chi'_1) d\\mu(\\chi'_2) \\, \\ket{{p_1} \\, \\chi'_1 ; \\, {p_2}\\, \\chi'_2} \\,\\langle \\chi'_1\\, \\chi'_2| C^a_{1} \\otimes 1 + 1 \\otimes C^a_{2} |\\chi_1\\, \\chi_2 \\rangle\\\\\n&= \\int \\! d\\mu(\\chi'_1) d\\mu(\\chi'_2) \\, \\ket{{p_1} \\, \\chi'_1 ; \\, {p_2}\\, \\chi'_2} \\,\\langle\\chi'_1\\, \\chi'_2| C^a_{1+2}  |\\chi_1\\, \\chi_2\\rangle\\,,\\label{eqn:charge2particleAction}\n\\]\nwhere $C^a_\\alpha$ is the colour in representation $R_\\alpha$ and we have written $C^a_{1+2}$ for the colour operator on the tensor product of representations $R_1$ and $R_2$.\nIn the classical regime, using the property that the overlap between states sets $\\chi'_\\alpha=\\chi_\\alpha $ in the classical limit, it follows that\n\\[\n\\bra{p_1\\,\\chi_1; \\, p_2 \\,\\chi_2} C^a_{1+2} \\ket{p_1\\, \\chi_1 ; \\, p_2\\, \\chi_2 } = c_1^a + c_2^a \\,,\n\\]\nso the colours simply add. A trivial similar result holds for the momenta of the two particles.\n\nSuppose that at some later time the two particles described by our initial state interact --- for example, two black holes scattering elastically. When does a point-particle description  remain appropriate? This will be the crucial topic of the next chapter. We will take the initial separation $b^\\mu$ to be the transverse impact parameter for the scattering of two point-like objects with momenta $p_{1,2}$. (The impact parameter is transverse in the sense that $p_\\alpha \\cdot b = 0$ for $\\alpha = 1, 2$.) At the quantum level, the particles are individually described by the wavefunctions in section~\\ref{sec:PointParticleLimit}. We would expect the point-particle description to be valid when the separation of the two scattering particles is always very large compared to their (reduced) Compton wavelengths, so the point-particle description will be accurate provided that\n\\begin{equation}\n\\sqrt{-b^2} \\gg \\lcomp^{(1,2)}\\,.\n\\end{equation}\nThe impact parameter and the Compton wavelengths are not the only scales we must consider, however --- the spread of the wavepackets, $\\lpack$, is another intrinsic scale. As we will discuss, the quantum-mechanical expectation values of observables are well approximated by the corresponding classical ones when the packet spreads are in the `Goldilocks' zone, $\\lcomp\\ll \\lpack\\ll \\sqrt{-b^2}$. These inequalities will have powerful ramifications on the behaviour of scattering amplitudes in the classical limit. To see this however, we need an on-shell scattering observable.\n\n\\chapter{The impulse}\n\\label{chap:impulse}\n\nAt a gravitational wave observatory we are of course interested in the gravitational radiation emitted by the source of interest. However, gravitational waves also carry information about the potential experienced by, for example, a black hole binary system. This observation motivates our interest in an on-shell observable related to the potential. We choose to explore the \\textit{impulse\\\/} on a particle during a scattering event: at the classical level, this is simply the total change in the momentum of one of the particles --- say particle~1 --- during the collision.\n\nIn this chapter we will begin by examining the change in momentum during a scattering event, without accompanying radiation, extracting the classical values from a fully relativistic quantum-mechanical computation. We examine scattering events in which two widely separated particles are prepared in the state~\\eqref{eqn:inState} at $t \\rightarrow -\\infty$, and then shot at each other with impact parameter $b^\\mu$. We will use this observable as a laboratory to explore certain conceptual and practical issues in approaching the classical limit. Using the explicit wavepackets we have constructed in chapter~\\ref{chap:pointParticles}, we will carefully analyse the small-$\\hbar$ region to understand how scattering amplitudes encode classical physics. We will see that the appropriate treatment is one where point-particles have momenta which are fixed as we take $\\hbar$ to zero, whereas for massless particles and momentum transfers between massive particles, it is the wavenumber which we should treat as fixed in the limit.\n\nOur formalism is quite general, applying in both gauge theory and gravity; for simplicity, we will nonetheless continue to focus on the scattering of two massive, stable quanta of scalar fields described by the Lagrangian in equation~\\eqref{eqn:scalarAction}. We will generalise to higher spin fields in part~\\ref{part:spin}. We will always restrict our attention to scattering processes in which quanta of fields 1 and 2 are both present in the final state. This will happen, for example, if the particles have separately conserved quantum numbers.  We also always assume that no new quanta of fields 1 and 2 can be produced during the collision, for example because the centre-of-mass energy is too small.\n\nIn the first section of this chapter we construct expressions for the impulse in terms of on-shell scattering amplitudes, providing a formal definition of the momentum transfer to a particle in quantum field theory. In \\sect{sec:classicalLimit}, we derive the Goldilocks zone in which the point-particle limit of our wavepackets remains valid, deriving from first principles the behaviour of scattering amplitudes in the classical limit. In \\sect{sec:examples} we apply our formalism explicitly, deriving the NLO impulse in scalar Yang--Mills theory, a result which is analagous to well-established post--Minkowskian results for the scattering of Schwarzschild black holes \\cite{Portilla:1979xx,Portilla:1980uz,Westpfahl:1979gu,Westpfahl:1985tsl}\n\nThis chapter continues to be based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}.\n\n\\section{Impulse in quantum field theory}\n\\label{sec:QFTsetup}\n\nTo define the observable, we place detectors at asymptotically large distances pointing at the collision region. The detectors measure only the momentum of particle 1. We assume that these detectors cover all possible scattering angles. Let $\\mathbb{P}_\\alpha^\\mu$ be the momentum operator for particle~$\\alpha$; the expectation of the first particle's outgoing momentum $\\outp1^\\mu$ is then\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle &= \n{}_\\textrm{out}{\\langle}\\Psi|  \\mathbb{P}_1^\\mu |\\Psi\\rangle_\\textrm{out} \\\\\n&= {}_\\textrm{out}{\\langle}\\Psi|  \\mathbb{P}_1^\\mu U(\\infty,-\\infty)\\,\n|\\Psi\\rangle_\\textrm{in} \\\\\n&= {}_\\textrm{in}{\\langle} \\Psi | \\, U(\\infty, -\\infty)^\\dagger   \\mathbb{P}_1^\\mu U(\\infty, -\\infty) \\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $U(\\infty, -\\infty)$ is the time evolution operator from the far past to the far future.  This evolution operator is just the $S$ matrix, so the expectation value is simply\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle &= \n{}_\\textrm{in}{\\langle} \\Psi | S^\\dagger   \\mathbb{P}_1^\\mu S\\, \n| \\Psi \\rangle_\\textrm{in}\\,.\n\\end{aligned}\n\\end{equation}\nWe can insert a complete set of states and rewrite the expectation value as\n\\begin{equation}\n\\begin{aligned}\n\\langle \\outp1^\\mu \\rangle \n&=\\sum_X \\int \\df(\\finalk_1)\\, \\df(\\finalk_2) \\, d\\mu(\\zeta_1) \\, d\\mu(\\zeta_2)\\; \\finalk_1^\\mu \n\\; \\bigl| \\langle \\finalk_1\\, \\zeta_1; \\finalk_2\\, \\zeta_2; X |S| \\Psi\\rangle\\bigr|^2\\,,\n\\end{aligned}\n\\label{eqn:p1Expectation}\n\\end{equation}\nwhere we can think of the inserted states as the final state of a scattering process. In this equation, $X$ refers to any other particles which may be created. The intermediate state containing $X$ also necessarily contains exactly one particle each corresponding to fields~1 and~2. Their momenta are denoted by $\\finalk_{1,2}$ respectively, while $\\d\\mu(\\zeta_\\alpha)$ is the $SU(N)$ Haar measure for their coherent colour states, as introduced in~\\eqref{eqn:Haar}. The sum over $X$ is a sum over all states, including $X$ empty, and includes phase-space integrals for $X$ non-empty. The expression~(\\ref{eqn:p1Expectation}) already hints at the possibility of evaluating the momentum in terms of on-shell scattering amplitudes.\n\nThe physically interesting quantity is rather the change of momentum of the particle during the scattering, so we define\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle =  \\langle \\Psi |S^\\dagger \\,   \\mathbb{P}^\\mu_1 \\, S |\\Psi\\rangle - \\langle \\Psi | \\,   \\mathbb{P}^\\mu_1 \\,  |\\Psi\\rangle\\,.\n\\end{equation}\nThis impulse is the difference between the expected outgoing and\nthe incoming momenta of particle 1. It is an on--shell observable, defined in both the quantum and the classical theories. Similarly, we can measure the impulse imparted to particle 2. In terms of the momentum operator, $\\mathbb{P}_2^\\mu$, of quantum field 2, this impulse is evidently\n\\begin{equation}\n\\langle \\Delta p_2^\\mu \\rangle =  \\langle \\Psi |S^\\dagger \\,  \\mathbb{P}^\\mu_2 \\, S |\\Psi\\rangle - \\langle \\Psi | \\,   \\mathbb{P}^\\mu_2 \\,  |\\Psi\\rangle.\n\\end{equation}\n\nReturning to the impulse on particle 1, we proceed by writing the scattering matrix in terms of the transition matrix $T$ via $S = 1 + i T$, in order to make contact with the usual scattering amplitudes. The no-scattering (unity) part of the $S$ matrix cancels in the impulse, leaving behind only delta functions that identify the final-state momenta with the initial-state ones in the wavefunction or its conjugate. Using unitarity we obtain the result\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle \n= \\langle \\Psi | \\, i [   \\mathbb{P}_1^\\mu, T ] \\, | \\Psi \\rangle + \\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\label{eqn:defl1}\n\\end{equation}\n\n\\subsection{Impulse in terms of amplitudes}\n\nHaving established a general expression for the impulse, we turn to expressing it in terms of scattering amplitudes. It is convenient to work on the two terms in equation~\\eqref{eqn:defl1} separately. For ease of discussion, we define\n\\begin{equation}\n\\begin{aligned}\n\\ImpA \\equiv \\langle \\Psi | \\, i [ \\mathbb{P}_1^\\mu, T ] \\, | \\Psi \\rangle\\,, \\qquad\n\\ImpB \\equiv \\langle \\Psi | \\, T^\\dagger [ \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,,\n\\end{aligned}\n\\end{equation}\nso that the impulse is $\\langle \\Delta p_1^\\mu \\rangle = \\ImpA + \\ImpB$. Using equation~\\eqref{eqn:inState} to expand the wavepacket in the first term, $\\ImpA$, we find\n\\[\n\\hspace*{-4mm}\\ImpA &= \n\\int \\! \\df(\\initialk_1)\\df(\\initialk_2)\n\\df(\\initialkc_1)\\df(\\initialkc_2)\\;\ne^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar} \\, \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialkc_1) \n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialkc_2) \n\\\\ &\\hspace{40mm}\n\\times i (\\initialkc_1\\!{}^\\mu - \\initialk_1^\\mu) \\, \n\\langle \\initialkc_1\\, \\chi_1'; \\initialkc_2 \\, \\chi_2'| \\,T\\, |\n\\initialk_1 \\, \\chi_1; \\initialk_2\\, \\chi_2 \\rangle\n\\\\&= \\int \\! \\df(\\initialk_1)\\df(\\initialk_2)\n\\df(\\initialkc_1)\\df(\\initialkc_2)\\;\ne^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar} \\, \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialkc_1) \n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialkc_2) \n\\\\ &\\qquad\\qquad\n\\times i \\int \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;(\\finalk_1^\\mu-\\initialk_1^\\mu)\n\\\\ &\\qquad\\qquad\\qquad\\qquad \n\\times \\langle \\initialkc_1 \\, \\chi_1'; \\initialkc_2 \\, \\chi_2'| \\finalk_1 \\, \\zeta_1 ;\\finalk_2\\, \\zeta_2 \\rangle\n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2\\, \\zeta_2 | \\,T\\, |\\initialk_1 \\, \\chi_1; \\initialk_2 \\, \\chi_2\\rangle\n\\,,\n\\label{eqn:defl2}\n\\]\nwhere in the second equality we have re-inserted the final-state momenta $\\finalk_\\alpha$ in\norder to make manifest the phase independence of the result. We label the states in the incoming wavefunction by $\\initialk_{1,2}$, those in the conjugate ones by $\\initialkc_{1,2}$. Let us now introduce the momentum shifts $q_\\alpha = \\initialkc_\\alpha-\\initialk_\\alpha$, and then change variables in the integration from the $p_\\alpha'$ to the $q_\\alpha$. In these variables, the matrix element is\n\\begin{equation}\n\\begin{aligned}\n\\langle p_1'\\, \\chi'_1;\\,p_2'\\, \\chi'_2|T| p_1\\,\\chi_1;\\,p_2\\,\\chi_2\\rangle &= \\langle \\chi'_1\\, \\chi'_2|\\mathcal{A}(p_1,p_2 \\rightarrow p_1',p_2')|\\chi_1\\,\\chi_2\\rangle\\\\ &\\qquad\\qquad\\qquad\\qquad \\times\n\\del^{(4)}(\\initialkc_1+\\initialkc_2-\\initialk_1-\\initialk_2) \n\\\\&\\equiv\n\\langle \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \\initialk_1 + q_1\\,, \\initialk_2 + q_2)\\rangle\n\\del^{(4)}(q_1 + q_2)\\,,\\label{eqn:defOfAmplitude}\n\\end{aligned}\n\\end{equation}\nyielding\n\\begin{equation}\n\\begin{aligned}\n\\ImpA &= \\int \\! \\df(\\initialk_1) \\df(\\initialk_2)\n\\df(q_1+\\initialk_1)\\df(q_2+\\initialk_2)\\;\n\\\\&\\qquad\\times \n\\varphi_1(\\initialk_1) \\varphi_1^*(\\initialk_1 + q_1)\n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialk_2+q_2) \n\\, \\del^{(4)}(q_1 + q_2)\n\\\\&\\qquad\\times \n\\, e^{-i b \\cdot q_1\/\\hbar} \n\\,i q_1^\\mu  \\, \\langle\\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q_1, \\initialk_2 + q_2)\\rangle\\,\n\\,,\n\\end{aligned}\n\\label{eqn:impulseGeneralTerm1a}\n\\end{equation}\nwhere the remaining expectation value is solely over the representation states $\\chi_\\alpha$. Note that we are implicitly using the condition~\\eqref{eqn:classicalOverlap}, in anticipation of the classical limit, for clarity of presentation. Now, recall the shorthand notation introduced earlier for the phase-space measure,\n\\begin{equation}\n\\df(q_1+p_1) = \\dd^4 q_1\\; \\del\\bigl((p_1 + q_1)^2 - m_1^2\\bigr)\n\\Theta(p_1^0 + q_1^0)\\,.\n\\end{equation} \nWe can perform the integral over $q_2$ in \\eqn~\\eqref{eqn:impulseGeneralTerm1a} using the four-fold delta function. Further relabeling $q_1 \\rightarrow q$, we obtain\n\\[\n\\ImpA&= \\int \\! \\df(\\initialk_1)\\df(\\initialk_2) \\dd^4 q  \\; \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\, e^{-i b \\cdot q\/\\hbar}\\\\\n&\\qquad \\times \\Theta(\\initialk_1^0+q^0) \\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_1^*(\\initialk_1 + q)\n\\varphi_2(\\initialk_2) \\varphi_2^*(\\initialk_2-q)\n\\\\& \\qquad\\qquad \\times  \n\\,  i q^\\mu  \\, \\langle\\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\\rangle\\,.\n\\label{eqn:impulseGeneralTerm1}\n\\]\nUnusually for a physical observable, this contribution is linear in the amplitude. We emphasise that the incoming and outgoing momenta of this amplitude do \\textit{not\\\/} correspond to the initial- and final-state momenta of the scattering process, but rather both correspond to the initial-state momenta, as they appear in the wavefunction and in its conjugate.  The momentum $q$ looks like a momentum transfer if we examine the amplitude alone, but for the physical scattering process it represents a difference between the momentum within the wavefunction and that in the conjugate.  Inspired by our discussion in section~\\ref{sec:PointParticleLimit}, we will refer to it as a `momentum mismatch'.  As indicated on the first line of \\eqn~\\eqref{eqn:defl2}, we should think of this term as an interference of a standard amplitude with an interactionless forward scattering. Recalling that in equation~\\eqref{eqn:wavefunctionSplit} we defined ${\\psi_i}_\\alpha(p_\\alpha) = \\varphi_\\alpha(p_\\alpha) {\\chi_i}_\\alpha$, we can write this diagrammatically as\n\\begin{equation}\n\\begin{aligned}\n\\ImpA & = \n\\int \\! \\df(\\initialk_1)\\df(\\initialk_2) \\dd^4 q \\, \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\\\\n& \\qquad \\times  \\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0) \\, e^{-i b \\cdot q\/\\hbar}  \\,  iq^\\mu \\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\n\\vertex (b) ;\n\\vertex [above left=1 and 0.66 of b] (i1) {$\\psi_1(p_1)$};\n\\vertex [above right=1 and 0.33 of b] (o1) {$\\psi_1^*(p_1+q)$};\n\\vertex [below left=1 and 0.66 of b] (i2) {$\\psi_2(p_2)$};\n\\vertex [below right=1 and 0.33 of b] (o2) {$\\psi_2^*(p_2-q)$};\n\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- (o2);\n\\draw[postaction={decorate}] (b) -- (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- (b);\n\\draw[postaction={decorate}] (i2) -- (b);\n\\end{scope}\n\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\filldraw [fill=allOrderBlue] (b) circle [radius=10pt];\n\\end{feynman}\n\\end{tikzpicture} \n\\!\\!\\!\\!.\n\\end{aligned}\n\\end{equation}\n\nTurning to the second term, $\\ImpB$, in the impulse, we again introduce a complete set of states labelled by momenta $\\finalk_1$, $\\finalk_2$ and $X$ so that\n\\begin{equation}\n\\begin{aligned}\n\\ImpB &= \\langle \\Psi | \\, T^\\dagger [ \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\n\\\\&= \\sum_X \\int \\! \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2) \n\\\\& \\qquad\\qquad \\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\rangle \n\\langle \\finalk_1  \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X| [  \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\end{aligned}\n\\end{equation}\nAs above, we can now expand the wavepackets. We again label the momenta in the incoming wavefunction by $\\initialk_{1,2}$, and those in the conjugate ones by $\\initialkc_{1,2}$:\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X \\int \\!\\prod_{\\alpha = 1, 2}  \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \\df(\\initialkc_\\alpha)\n\\; \\varphi_\\alpha(\\initialk_\\alpha) \\varphi^*_\\alpha(\\initialkc_\\alpha) \ne^{i b \\cdot (\\initialk_1 - \\initialkc_1) \/ \\hbar}\n(\\finalk_1^\\mu - \\initialk_1^\\mu) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X) \n\\del^{(4)}(\\initialkc_1+\\initialkc_2 - \\finalk_1 - \\finalk_2 -\\finalk_X)\\\\\n&\\hspace*{15mm}\\times \\langle \\Ampl^*(\\initialkc_1\\,, \\initialkc_2 \\rightarrow \\finalk_1 \\,, \\finalk_2 \\,, \\finalk_X) \n\\Ampl(\\initialk_1\\,,\\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:forcedef2}\n\\end{equation}\nIn this expression we again absorb the representation states $\\chi_\\alpha$ into an expectation value over the amplitudes, while $\\finalk_X$ denotes the total momentum carried by particles in $X$. The second term in the impulse can thus be interpreted as a weighted cut of an amplitude; the lowest order contribution is a weighted two-particle cut of a one-loop amplitude. \n\nIn order to simplify $\\ImpB$, let us again define the momentum shifts $q_\\alpha = \\initialkc_\\alpha-\\initialk_\\alpha$, and change variables in the integration from the $\\initialkc_\\alpha$ to the $q_\\alpha$, so that\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X  \\int \\!\\prod_{\\alpha = 1,2} \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \n\\df(q_\\alpha+\\initialk_\\alpha)\n\\; \\varphi_\\alpha(\\initialk_\\alpha) \\varphi^*_\\alpha(\\initialk_\\alpha+q_\\alpha) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X)\\, \n\\del^{(4)}(q_1+q_2)\\, e^{-i b \\cdot q_1 \/ \\hbar} (\\finalk_1^\\mu - \\initialk_1^\\mu)\\\\\n&\\hspace*{5mm}\\times \\langle\\Ampl^*(\\initialk_1+q_1\\,, \\initialk_2+q_2 \\rightarrow \\finalk_1 \\,,\\finalk_2 \\,, \\finalk_X)\n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned} \\label{eqn:impulseGeneralTerm2a}\n\\end{equation}\nWe can again perform the integral over $q_2$ using the four-fold delta function, and relabel $q_1 \\rightarrow q$ to obtain\n\\begin{equation}\n\\begin{aligned}\n\\ImpB\n&=\n\\sum_X  \\int \\!\\prod_{\\alpha = 1,2} \\df(\\finalk_\\alpha)  \\df(\\initialk_\\alpha) \n\\dd^4 q\\;\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \\\\\n&\\hspace*{5mm}\\times \\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2)\\, \\varphi^*_1(\\initialk_1+q) \\varphi^*_2(\\initialk_2-q) \\\\\n&\\hspace*{5mm}\\times \\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 -\\finalk_X) \\, e^{-i b \\cdot q \/ \\hbar} (\\finalk_1^\\mu - \\initialk_1^\\mu)\n\\\\ &\\hspace*{5mm}\\times \\langle \\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \\finalk_1 \\,,\\finalk_2 \\,, \\finalk_X)\n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2 \\,, \\finalk_X)\\rangle\n\\,.\n\\end{aligned} \n\\label{eqn:impulseGeneralTerm2b}\n\\end{equation}\nThe momentum $q$ is again a momentum mismatch. The momentum transfers $\\xfer_\\alpha\\equiv r_\\alpha-p_\\alpha$ will play an important role in analysing the classical limit, so it is convenient to change variables to them from the final-state momenta $\\finalk_\\alpha$,\n\\begin{equation}\n\\begin{aligned}\n\\ImpB &= \\sum_X  \\int \\!\\prod_{\\alpha = 1,2}  \\df(\\initialk_i) \\dd^4 \\xfer_\\alpha\n\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\qquad\\times\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\, \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2)\\\\\n&\\qquad \\times\\varphi^*_1(\\initialk_1+q) \\varphi^*_2(\\initialk_2-q) e^{-i b \\cdot q \/ \\hbar}\\,\\xfer_1^\\mu \\; \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \n\\\\ &\\qquad\\qquad\\times \n\\langle \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1 \\,,\\initialk_2+\\xfer_2 \\,, \\finalk_X)\n\\\\ &\\qquad\\qquad\\qquad\\qquad\\qquad \\times \n\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2 \\,, \\finalk_X) \\rangle\\,.\n\\end{aligned} \n\\label{eqn:impulseGeneralTerm2}\n\\end{equation}\nDiagrammatically, this second contribution to the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\usetikzlibrary{decorations.markings}\n\\ImpB&= \n\\sum_X {\\int} \\! \\prod_{\\alpha = 1,2}  \\df(\\initialk_\\alpha) \\dd^4 \\xfer_\\alpha\n\\dd^4 q\\; \\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\\,e^{-i b \\cdot q \/ \\hbar}\\,\\xfer_1^\\mu\n\\\\&\\qquad\\times\n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\\\\n& \\hspace*{10mm} \\times \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \\!\\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\begin{scope}\n\\vertex (ip1) ;\n\\vertex [right=3 of ip1] (ip2);\n\\node [] (X) at ($ (ip1)!.5!(ip2) $) {};\n\\begin{scope}[even odd rule]\n\n\\vertex [above left=0.66 and 0.5 of ip1] (q1) {$ \\psi_1(p_1)$};\n\\vertex [above right=0.66 and 0.33 of ip2] (qp1) {$ \\psi^*_1(p_1 + q)$};\n\\vertex [below left=0.66 and 0.5 of ip1] (q2) {$ \\psi_2(p_2)$};\n\\vertex [below right=0.66 and 0.33 of ip2] (qp2) {$ \\psi^*_2(p_2 - q)$};\n\\diagram* {\n\t(ip1) -- [photon]  (ip2)\n};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (q1) -- (ip1);\n\\draw[postaction={decorate}] (q2) -- (ip1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip2) -- (qp1);\n\\draw[postaction={decorate}] (ip2) -- (qp2);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.34 with {\\arrow{Stealth}},\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip1) to [out=90, in=90,looseness=1.2] node[above left] {{$p_1 + w_1$}} (ip2);\n\\draw[postaction={decorate}] (ip1) to [out=270, in=270,looseness=1.2]node[below left] {$p_2 + w_2$} (ip2);\n\\end{scope}\n\n\\node [] (Y) at ($(X) + (0,1.4)$) {};\n\\node [] (Z) at ($(X) - (0,1.4)$) {};\n\\node [] (k) at ($ (X) - (0.65,-0.25) $) {$\\finalk_X$};\n\n\\filldraw [color=white] ($ (ip1)$) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip1) $) circle [radius=8pt];\n\n\\filldraw [color=white] ($ (ip2) $) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip2) $) circle [radius=8pt];\n\n\\end{scope}\n\\end{scope}\n\\filldraw [color=white] ($  (Y) - (3pt, 0) $) rectangle ($ (Z) + (3pt,0) $) ;\n\\draw [dashed] (Y) to (Z);\n\\end{feynman}\n\\end{tikzpicture} .\n\\end{aligned}\n\\end{equation}\n\n\\section{Point-particle scattering}\n\\label{sec:classicalLimit}\n\nThe observable we have discussed  --- the impulse --- is designed to be well-defined in both the quantum and the classical theories. As we approach the classical limit, the quantum expectation values should reduce to the classical impulse, ensuring that we are able to explore the $\\hbar \\rightarrow 0$ limit. Here we explore this limit, and its ramifications on scattering amplitudes, in detail.\n\n\\subsection{The Goldilocks inequalities}\n\nWe have already discussed in section~\\ref{sec:RestoringHBar} how to make explicit the factors of $\\hbar$ in the observables, and in section~\\ref{sec:PointParticleLimit} we selected wavefunctions which have the desired classical point-particle limit, which we established was the region where\n\\begin{equation}\n\\ell_c \\ll \\ell_w\\,.\\label{eqn:comptonConstraint}\n\\end{equation}\nAt this point, we could in principle perform the full quantum calculation, using the specific wavefunctions we chose, and expand in the $\\xi\\!\\rightarrow\\! 0$ limit at the end.  However, having established previously the detailed properties of our wavefunctions $\\varphi_\\alpha$, it is far more efficient to neglect the details and simply use the fact that they allow us to approach the limit as early as possible in calculations.  This will lead us to impose stronger constraints on our choice than the mere existence of a suitable classical limit.\n\nHeuristically, the wavefunctions for the scattered particles must satisfy two separate conditions. As discussed in the single particle case, the details of the wavepacket should not be sensitive to quantum effects. At the same time, now that we aim to describe the scattering of point-particles the spread of the wavefunctions should not be too large, so that the interaction with the other particle cannot peer into the details of the quantum wavepacket.\n\nTo quantify this discussion let us examine $\\ImpA$ in~(\\ref{eqn:impulseGeneralTerm1}) more closely. It has the form of an amplitude integrated over the on-shell phase space for both of the incoming momenta, subject to additional $\\delta$ function constraints --- and then weighted by a phase $e^{-ib\\cdot q\/\\hbar}$ dependent on the momentum mismatch $q$, and finally integrated over all $q$. As one nears the classical limit~\\eqref{eqn:comptonConstraint}, the wavefunction and its conjugate should both represent the particle. The amplitude will vary slowly on the scale of the wavefunction when one is close to the limit. This is therefore precisely the same constraint as we had in the single particle case, and we immediately have\n\\begin{equation}\n\\qb\\cdot\\ucl_\\alpha\\,\\lpack \\ll \\sqrt{\\xi}\\,,\n\\label{eqn:qbConstraint}\n\\end{equation}\nwhere we have scaled $q$ by $1\/\\hbar$, replacing the momentum by a wavenumber.\n\nWe next examine another rapidly varying factor that appears in all our integrands, the delta functions in $q$ arising from the on-shell constraints on the conjugate momenta $\\initialkc_\\alpha$. These delta functions, appearing in equations~(\\ref{eqn:impulseGeneralTerm1} and~\\ref{eqn:impulseGeneralTerm2}), take the form\n\\begin{equation}\n\\del(2p_\\alpha\\cdot q+q^2) = \\frac1{\\hbar m_\\alpha}\\del(2\\qb\\cdot u_\\alpha+\\lcomp \\qb^2)\\,.\n\\label{eqn:universalDeltaFunction}\n\\end{equation}\nThe integration over the initial momenta $\\initialk_\\alpha$ and the initial wavefunctions will smear out these delta functions to sharply peaked functions whose scale is of the same order as the original wavefunctions.  As $\\xi$ gets smaller, this function will turn back into a delta function imposed on the $\\qb$ integration. To see this, let us consider an explicit wavefunction integral similar to $\\ImpA$, but with a simpler integrand:\n\\defT_1{T_1}\n\\begin{equation}\nT_1 = \\int \\df(p_1)\\,\\varphi(p_1)\\varphi^*(p_1+q)\\,\\del(2 p_1\\cdot q+q^2)\\,.\\label{eqn:deltaFunctionIntegral}\n\\end{equation}\nWith $\\varphi$ chosen to be the linear exponential~(\\ref{eqn:LinearExponential}), this integral simplifies to\n\\begin{equation}\nT_1 = \\frac{1}{\\hbar m_1} \\eta_1(\\qb;p_1)\\,\\int \\df(p_1)\\,\n\\del(2 p_1\\cdot\\qb\/m_1+\\hbar\\qb^2\/m_1)\\,|\\varphi(p_1)|^2\\,,\n\\end{equation}\nwhere $\\eta_1(\\qb;p_1)$ is the overlap defined in equation~\\eqref{eqn:wavefunctionOverlap} and we have also replaced $q\\rightarrow\\hbar \\qb$.\n\nThe remaining integrations in $T_1$ are relegated in appendix~\\ref{app:wavefunctions}, but yield\\footnote{The wavenumber transfer is necessarily spacelike.}\n\\begin{multline}\nT_1 = \\frac1{4 \\hbar m_1\\sqrt{(\\qb\\cdot u)^2-\\qb^2}\\,K_1(2\/\\xi)} \n\\\\\\times \\exp\\biggl[-\\frac2{\\xi}\\frac{\\sqrt{(\\qb\\cdot u)^2-\\qb^2}}{\\sqrt{-\\qb^2}}\n\\sqrt{1-\\hbar^2\\qb^2\/(4m_1^2)}\\biggr]\n\\,.\\label{eqn:TIntegral}\n\\end{multline}\nNotice that our result depends on two dimensionless ratios in addition to its dependence on $\\xi$,\n\\begin{equation}\n\\lcomp \\sqrt{-\\qb^2}\n\\qquad\n\\textrm{and}\\qquad\n\\frac{\\qb\\cdot u}{\\sqrt{-\\qb^2}}\\,.\n\\end{equation}\nLet us call $1\/\\sqrt{-\\qb^2}$ a `scattering length' $\\lscatt$.  In terms of this length, our two dimensionless ratios are\n\\begin{equation}\n\\frac{\\lcomp}{\\lscatt}\n\\qquad\n\\textrm{and}\\qquad\n{\\qb\\cdot u}\\,\\lscatt\\,.\\label{eqn:dimensionlessRatios}\n\\end{equation}\n\nAs we approach the $\\hbar,\\xi\\rightarrow 0$ limit, we may expect $T_1$ to be concentrated in a small region in $\\qb$.  Towards the limit, the dependence on the magnitude is just given by the prefactor.  To understand the behaviour in the boost and angular degrees of freedom, we may note that \n\\begin{equation}\n\\frac1{K_1(2\/\\xi)} \\sim \\frac2{\\sqrt{\\pi}\\sqrt{\\xi}} \\exp\\biggl[\\frac2{\\xi}\\biggr]\\,,\\label{eqn:BesselLimit}\n\\end{equation}\nand that $\\hbar\\sqrt{\\xi}$ is of order $\\xi$, so that overall $T_1$ has the form\n\\begin{equation}\n\\frac1{\\xi}\\exp\\biggl[-\\frac{f(\\qb)}{\\xi}\\biggr]\\,.\n\\label{eqn:LimitForm}\n\\end{equation}\nThis will yield a delta function so long as $f(\\qb)$ is positive. To figure out its argument, we recall that $\\qb^2<0$, and parametrise the wavenumber as\n\\begin{equation}\n\\qb^\\mu = \\Eqb\\bigl(\\sinh\\zeta,\\,\\cosh\\zeta\\sin\\theta\\cos\\phi,\n\\,\\cosh\\zeta\\sin\\theta\\sin\\phi,\n\\,\\cosh\\zeta\\cos\\theta\\bigr)\\,,\\label{eqn:qbRapidityParametrisation}\n\\end{equation}\nwith rapidity $\\zeta$ running over $[0,\\infty]$, $\\theta$ over $[0,\\pi]$, and $\\phi$ over $[0,2\\pi]$. Working in the rest frame of $u^\\mu$, the exponent in \\eqn~\\eqref{eqn:TIntegral} (including the term from \\eqn~\\eqref{eqn:BesselLimit}) is\n\\begin{equation}\n-\\frac2{\\xi}\\Bigl(\\cosh\\zeta\\sqrt{1+\\hbar^2 \\Eqb^2\/(4m^2)}-1\\Bigr)\\,,\n\\end{equation}\nso that the delta function will ultimately localise \n\\begin{equation}\n\\cosh\\zeta \\rightarrow \\frac1{\\sqrt{1+\\hbar^2 \\Eqb^2\/(4m^2)}} = \n1-\\frac{\\hbar^2 \\Eqb^2}{8m^2}+\\Ord(\\hbar^4)\n\\end{equation}\nto zero. Thus in terms of the Lorentz--invariant dimensionless ratios in equation~\\eqref{eqn:dimensionlessRatios}, we find that the delta function is\n\\begin{equation}\n\\delta\\!\\left(\\qb\\cdot u\\, \\lscatt + \\frac{\\ell_c^2}{4 \\lscatt^2} \\, \\frac1{\\qb\\cdot u\\, \\lscatt}\\right). \\label{eqn:deltaArgument}\n\\end{equation}\nThe direction-averaging implicit in the integration over $\\initialk_1$ has led to a constraint on two positive quantities built out of the ratios.\n\nRecall that to arrive at this expression we absorbed a factor of $\\hbar\\sim\\sqrt{\\xi}$. Since the argument of the delta function is a polynomial in the dimensionless ratios, both must be independently constrained to be of this order:\n\\begin{subequations}\n\\begin{align}\n{\\qb\\cdot u}\\,\\lscatt &\\lesssim \\sqrt{\\xi}\\,,\\label{eqn:deltaConstraint1}\n\\\\\\frac{\\lcomp}{\\lscatt} &\\lesssim \\sqrt{\\xi}\\,.\\label{eqn:deltaConstraint2}\n\\end{align}\n\\end{subequations}\nIf we had not already scaled out a factor of $\\hbar$ from $q$, these constraints would make it natural to do so. \n\nCombining constraint~\\eqref{eqn:deltaConstraint1} with that in \\eqn~\\eqref{eqn:qbConstraint}, we obtain the constraint $\\lpack\\ll\\lscatt$. Then including constraint~\\eqref{eqn:ComptonConstraint1}, or $\\xi\\ll 1$, we obtain our first version of the `Goldilocks' inequalities,\n\\begin{equation}\n\\lcomp \\ll \\lpack \\ll \\lscatt\\,.\n\\label{eqn:Goldilocks1}\n\\end{equation}\nAs we shall see later in the explicit evaluation of $\\ImpA$, $\\lscatt\\sim \\sqrt{-b^2}$; this follows on dimensional grounds.  This gives us the second version of the `Goldilocks' requirement,\n\\begin{equation}\n\\lcomp\\ll \\lpack\\ll \\sqrt{-b^2}\\,. \n\\label{eqn:Goldilocks2}\n\\end{equation}\n\nNote that the constraint following from~\\eqref{eqn:deltaConstraint2} is weaker, $\\lpack\\lesssim\\lscatt$. Indeed, we should not expect a similar strengthening of this restriction; the sharp peaking of the wavefunctions alone will not force the left-hand side to be much smaller than the right-hand side. This means that we should expect $\\qb\\cdot u$ to be smaller than, but still of order, $\\sqrt{\\xi}\/\\lscatt$.  If we compare the two terms in the argument to the delta function~(\\ref{eqn:universalDeltaFunction}), we see that the second term\n\\begin{equation}\n\\lcomp \\qb^2 \\sim \\frac{\\lcomp}{\\lscatt} \\frac1{\\lscatt} \\ll \\frac{\\sqrt{\\xi}}{\\lscatt}\\,,\\label{eqn:neglectq2}\n\\end{equation}\nso that $\\lcomp \\qb^2 \\ll \\qb\\cdot u_\\alpha$, and the second term should be negligible. In our evaluation of $T_1$, we see that the integral is sharply peaked about the delta function\n\\begin{equation}\n\\delta(\\wn q\\cdot u)\\,.\\label{eqn:universalDeltaFunction2}\n\\end{equation}\nWe are thus free to drop the $\\wn q^2$ correction in the classical limit. There is one important caveat to this simplification, which we will mention below.\n\\begin{figure}[t]\n\t\\center\n\t\\includegraphics[width = 0.75\\textwidth]{Goldilocks}\n\t\\vspace{-3pt}\n\t\\caption{Heuristic depiction of the Goldilocks inequalities.\\label{Goldilocks}}\n\\end{figure}\n\n\\subsection{Taking the limit of observables}\n\nIn computing the classical observable, we cannot simply set $\\xi=0$. Indeed, we don't even want to fully take the $\\xi\\rightarrow 0$ limit. Rather, we want to take the leading term in that limit. This term may in fact be proportional to a power of $\\xi$.  To understand this, we should take note of one additional length scale in the problem, namely the classical radius of the point particle.  In electrodynamics, this is $\\lclass=e^2\/(4\\pi m)$. However,\n\\begin{equation}\n\\lclass = \\frac{\\hbar e^2}{4\\pi\\hbar m} = \\alpha\\lcomp\\,,\n\\end{equation}\nwhere $\\alpha$ is the usual, dimensionless, electromagnetic coupling. Dimensionless ratios of $\\lclass$ to other length scales will be the expansion parameters in classical observables; but as this relation shows, they too will vanish in the $\\xi\\rightarrow 0$ limit. There are really three dimensionless parameters we must consider: $\\xi$; $\\lpack\/\\lscatt$; and $\\lclass\/\\lscatt$.  We want to retain the full dependence on the latter, while considering only effects independent of the first two.\n\nUnder the influence of a perturbatively weak interaction (such as electrodynamics or gravity) below the particle-creation threshold, we expect a wavepacket's shape to be distorted slightly, but not radically changed by the scattering.  We would expect the outgoing particles to be characterised by wavepackets similar to those of the incoming particles. However, using a wavepacket basis of states for the state sums in \\sect{sec:QFTsetup} would be cumbersome, inconvenient, and computationally less efficient than the plane-wave states we used.  We expect the narrow peaking of the wavefunction to impose constraints on the momentum transfers as they appear in higher-order corrections to the impulse $\\ImpB$ in equation~\\eqref{eqn:impulseGeneralTerm2}; but we will need to see this narrowness indirectly, via assessments of the spread as in \\eqn~\\eqref{eqn:expectations}, rather than directly through the presence of wavefunction (or wavefunction mismatch) factors in our observables. We can estimate the spread $\\spread(\\finalk_\\alpha)$ in a final-state momentum $\\finalk_\\alpha$ as follows:\n\\begin{equation}\n\\begin{aligned}\n\\spread(\\finalk_\\alpha)\/m_\\alpha^2 &= \n\\langle\\bigl(\\finalk_\\alpha-\\langle\\finalk_\\alpha\\rangle\\bigr)^2\\rangle\/m_\\alpha^2\n\\\\&= \\bigl(\\langle \\finalk_\\alpha^2\\rangle-\\langle\\finalk_\\alpha\\rangle{}^2\\bigr)\/m_\\alpha^2\n\\\\&= 1-\\bigl(\\langle\\initialk_\\alpha\\rangle+\\expchange\\bigr){}^2\/m_\\alpha^2\n\\\\&= \\spread(p_\\alpha)\/m_\\alpha^2 -\\langle\\Delta p_\\alpha\\rangle\\cdot \n\\bigl(2\\langle\\initialk_\\alpha\\rangle+\\expchange\\bigr)\/m_\\alpha^2\\,.\n\\end{aligned}\n\\end{equation}\nSo long as $\\expchange\/m_\\alpha\\lesssim\\spread(\\initialk_\\alpha)\/m_\\alpha^2$, the second term will not greatly increase the result, and the spread in the final-state momentum will be of the same order as that in the initial-state momentum.   Whether this condition holds depends on the details of the wavefunction. Even if it is violated, so long as $\\expchange\/m_\\alpha \\lesssim c'_\\Delta \\xi^{\\beta'''}$ with $c'_\\Delta$ a constant of $\\Ord(1)$, then the final-state momentum will have a narrow spread towards the limit.  (It would be broader than the initial-state momentum spread, but that does not affect the applicability of our results.)\n\nThe magnitude of $\\expchange$ can be determined perturbatively.  The leading-order value comes from $\\ImpA$, with $\\ImpB$ contributing yet-smaller corrections.  As we shall see, these computations reveal $\\expchange\/m_\\alpha$ to scale like $\\hbar$, or $\\sqrt{\\xi}$, and be numerically much smaller.\n\nThis in turn implies that for perturbative consistency, the `characteristic' values of momentum transfers $w_\\alpha$ inside the definition of $I^\\mu_{(2)}$ must also be very small compared to $m_\\alpha\\sqrt{\\xi}$. This constraint is in fact much weaker than implied by the leading-order value of $\\expchange$. Just as for $q_0$ in \\eqn~\\eqref{eqn:qConstraint1}, we should scale these momentum transfers by $1\/\\hbar$, replacing them by wavenumbers $\\xferb_\\alpha$.  The corresponding scattering lengths $\\tilde\\lscatt^\\alpha = \\sqrt{-w_{\\alpha}^2}$ must again satisfy $\\tilde\\lscatt^\\alpha \\gg \\lpack$. If we now examine the energy-momentum-conserving delta function \nin \\eqn~\\eqref{eqn:impulseGeneralTerm2},\n\\begin{equation}\n\\del^{(4)}(w_1+w_2 + \\finalk_X)\\,,\\label{eqn:radiationScalingDeltaFunction}\n\\end{equation}\nwe see that all other transferred momenta $\\finalk_\\alpha$ must likewise be small compared to $m_\\alpha\\sqrt{\\xi}$: all their energy components must be positive and hence no cancellations are possible inside the delta function. The typical values of these momenta should again by scaled by $1\/\\hbar$ and replaced by wavenumbers. We will see in the next chapter that $I_{(2)}^\\mu$ encodes radiative effects, and the same constraint will force these momenta for emitted radiation to scale as wavenumbers. \n\nWhat about loop integrations?  As we integrate the loop  momentum over all values, it is a matter of taste how we scale it.  If it is the momentum of a (virtual) massless line, however, unitarity considerations suggest that as the natural scaling is to remove a factor of $\\hbar$ in the real contributions to the cut in equation~\\eqref{eqn:impulseGeneralTerm2}, we should likewise do so for virtual lines.  More generally, we should scale those differences of the loop momentum with external legs that correspond to massless particles, and replace them by wavenumbers.  Moreover, unitarity considerations also suggest that we should choose the loop momentum to be that of a massless line in the loop, if there is one.\n\nIn general, we may not be able to approach the $\\hbar\\rightarrow0$ limit of each contribution to an observable separately, because they may contain terms which are singular, having too many inverse powers of $\\hbar$.  We find that such singular terms meet one of two fates: they are multiplied by functions which vanish in the regime of validity of the limit; or they cancel in the sum over all contributions.  We cannot yet offer a general argument that such troublesome terms necessarily disappear in one of these two\nmanners.  We can treat independently contributions whose singular terms ultimately cancel in the sum, so long as we expand each contribution in a Laurent series in $\\hbar$.\n\nFor theories with non-trivial internal symmetries, when identifying singular terms (in both parts of the impulse) it is essential not to forget factors coming from the classical limit of the representation states $\\chi_\\alpha$. Since the scattering particles remain well separated at all times there is no change to the story in chapter~\\ref{chap:pointParticles}. However, it is important to keep in mind that in our conventions colour factors in scattering amplitudes carry dimensions of $\\hbar$, and in particular satisfy the Lie algebra in equation~\\eqref{eqn:chargeLieAlgebra}. There is subsequently an independent Laurent series available when evaluating colour factors, and this can remove terms which appear singular in the kinematic expansion.\n\nFull impulse integrand factors that appear uniformly in all contributions --- that is, factors which appear directly in a final expression after cancellation of terms singular in the $\\hbar\\rightarrow 0$ limit --- can benefit from applying two simplifications to the integrand: setting $p_\\alpha$ to $m_\\alpha\\ucl_\\alpha$, as prescribed by equation~\\eqref{eqn:universalDeltaFunction2}, and truncating at the lowest order in $\\hbar$ or $\\xi$.  For other factors, we must be careful to expand in a Laurent series.  As mentioned above, a consequence of equation~\\eqref{eqn:neglectq2} is that inside the on-shell delta functions $\\del(2p_\\alpha\\cdot \\qb\\pm \\hbar \\qb^2)$ we can neglect the $\\hbar \\qb^2$ term; this is true so long as the factors multiplying these delta functions are not singular in $\\hbar$. If they are indeed nonsingular (after summing over terms), we can safely neglect the second term inside such delta functions, and replace them by $\\del(2p_\\alpha\\cdot \\qb)$.  A similar argument allows us to neglect the $\\hbar \\qb^0$ term inside the positive-energy theta functions; the $\\qb$ integration then becomes independent of them.  Similar arguments, and caveats, apply to the squared momentum-transfer terms $\\hbar \\xferb_\\alpha^2$ appearing inside on-shell delta functions in higher-order contributions, along with the energy components $\\xferb_\\alpha^0$ appearing inside positive-energy theta functions.  They can be neglected so long as the accompanying factors are not singular in $\\hbar$.  If accompanying factors \\textit{are} singular as $\\hbar\\rightarrow 0$, then we may need to retain such formally suppressed $\\hbar \\qb^2$ or $\\hbar \\xferb_\\alpha^2$ terms inside delta functions.\nWe will see an example of this in the calculation of the NLO contributions to the impulse in section~\\ref{sec:examples}.\n\n\\subsubsection{Summary}\n\nFor ease of future reference, let us collect the rules we have derived for calculating classical scattering observables from quantum field theory. We have all together established that, in the classical limit, we must apply the following constraints when evaluating amplitudes in explicit calculations:\n\\begin{itemize}\n\t\\item The momentum mismatch $q = p'_1 - p_1$ scales as a wavenumber, $q = \\hbar \\wn q$.\n\t\\item The momentum transfers $w_\\alpha$ in $I_{(2)}^\\mu$ scale as wavenumbers.\n\t\\item Massless loop momenta scale as wavenumbers.\n\t\\item The $\\hbar \\wn q^2$ factors in on--shell delta functions can be dropped, but only when there are no terms singular in $\\hbar$.\n\t\\item Any amplitude colour factors are evaluated using the commutation relation~\\eqref{eqn:chargeLieAlgebra}.\n\\end{itemize}\nWe derived these rules using the impulse, but they hold for any on-shell observable constructed in the manner of section~\\ref{sec:QFTsetup}. Furthermore, it will be convenient to introduce a notation to allow us to manipulate integrands under the eventual approach to the $\\hbar\\rightarrow0$ limit; we will use large angle brackets for the purpose,\n\\begin{multline}\n\\Lexp f(p_1,p_2,\\cdots) \\Rexp = \\int\\! \\df(p_1) \\df(p_2) |\\varphi(p_1)|^2 |\\varphi(p_2)|^2\\\\ \\times \\langle \\chi_1\\,\\chi_2|f(p_1,p_2,\\cdots)|\\chi_1\\,\\chi_2\\rangle\\,,\n\\label{eqn:angleBrackets}\n\\end{multline}\nwhere the integration over both $\\initialk_1$ and $\\initialk_2$ is implicit.  Within the angle brackets, we have approximated $\\varphi_\\alpha(p\\pm q)\\simeq \\varphi_\\alpha(p)$ and $\\chi'_\\alpha \\simeq  \\chi_\\alpha$. Then, relying on our detailed study of the momentum and colour wavefunctions in sections~\\ref{sec:PointParticleLimit} and~\\ref{sec:classicalLimit}, to evaluate the integrals and representation expectation values implicit in the large angle brackets we can simply set $p_\\alpha\\simeq m_\\alpha \\ucl_\\alpha$, and replace quantum colour charges $C_\\alpha^a$ with their (commuting) classical counterparts $c^a_\\alpha$.\n\n\\subsection{The classical impulse}\n\\label{sec:classicalImpulse}\n\nWe have written the impulse in terms of two terms, $\\langle \\Delta p_1^\\mu \\rangle = \\ImpA + \\ImpB$, and expanded these in terms of wavefunctions in equations~\\eqref{eqn:impulseGeneralTerm1} and ~\\eqref{eqn:impulseGeneralTerm2}. We will now discuss the classical limit of these terms in detail, applying the rules gathered above.\n\nWe begin with the first and simplest term in the impulse, $\\ImpA$, given in \\eqn~\\eqref{eqn:impulseGeneralTerm1}, and here recast in the notation of \\eqn~\\eqref{eqn:angleBrackets} in preparation:\n\\begin{multline}\n\\ImpAcl =  \\Lexp i\\!\\int \\!\\dd^4 q  \\; \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0)\\\\\n \\times  e^{-i b \\cdot q\/\\hbar} \n\\, q^\\mu  \\, \\Ampl(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\\,\\Rexp\\,.\n\\label{eqn:impulseGeneralTerm1recast}\n\\end{multline}\nRescale $q \\rightarrow \\hbar\\qb$; drop the $q^2$ inside the on-shell delta functions;\nand also remove the overall factor of $\\tilde g^2$ and accompanying $\\hbar$'s from the amplitude, to obtain the leading-order (LO) contribution to the classical impulse,\n\\begin{multline}\n\\DeltaPlo \\equiv \\ImpAclsup{(0)} =  \\frac{i\\tilde g^2}{4} \\Lexp \\hbar^2\\! \\int \\!\\dd^4 \\qb  \\; \n\\del(\\qb\\cdot p_1) \\del(\\qb\\cdot p_2) \n\\\\\\times  \ne^{-i b \\cdot \\qb} \n\\, \\qb^\\mu  \\, \\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow \np_1 + \\hbar\\qb, p_2 - \\hbar\\qb)\\,\\Rexp\\,.\n\\label{eqn:impulseGeneralTerm1classicalLO}\n\\end{multline}\nWe denote by $\\AmplB^{(L)}$ the reduced $L$-loop amplitude, that is the $L$-loop amplitude with a factor of the (generic) coupling $\\tilde g\/\\sqrt{\\hbar}$ removed for every interaction: in the gauge theory case, this removes a factor of $g\/\\sqrt{\\hbar}$, while in the gravitational case, we would remove a factor of $\\kappa\/\\sqrt{\\hbar}$.  In general, this rescaled fixed-order amplitude depends only on $\\hbar$-free ratios of couplings; in pure electrodynamics or gravitational theory, it is independent of couplings.  In pure electrodynamics, it depends on the charges of the scattering particles. While it is free of the powers of $\\hbar$ discussed in section~\\ref{sec:RestoringHBar}, it will in general still scale with an overall power of $\\hbar$ thanks to dependence on momentum mismatches or transfers. As we shall see in the next section, additional inverse powers of $\\hbar$ emerging from $\\AmplB$ will cancel the $\\hbar^2$ prefactor and yield a nonvanishing result.\n\nAs a reminder, while this contribution to a physical observable is linear in an amplitude, it arises from an expression involving wavefunctions multiplied by their conjugates.  This is reflected in the fact that both the `incoming' and `outgoing' momenta in the amplitude here are in fact initial-state momenta. Any phase which could be introduced by hand in the initial state would thus cancel out of the observable.\n\nThe LO classical impulse is special in that only the first term~(\\ref{eqn:impulseGeneralTerm1}) contributes. In general however, it is only the sum of the two terms in \\eqn~\\eqref{eqn:defl1} that has a well-defined classical limit.  We may write this sum as\n\\begin{multline}\n\\Delta p_1^\\mu = \n\\Lexp i\\hbar^{-2}\\!\n\\int \\!\\dd^4 q  \\; \\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2)\\\\ \\times \n\\Theta(\\initialk_1^0+q^0)\\Theta(\\initialk_2^0-q^0) \\; e^{-i b \\cdot q\/\\hbar} \\; \\impKer \\Rexp \\,,\n\\label{eqn:partialClassicalLimitNLO}\n\\end{multline}\nwhere the \\textit{impulse kernel\\\/} $\\impKer$ is defined as\n\\begin{equation}\n\\begin{aligned}\n\\impKer \\equiv&\\,  \\hbar^2 q^\\mu \\, \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + q, \\initialk_2 - q)\n\\\\& -i \\hbar^2 \\sum_X \\int \\!\\prod_{\\alpha = 1,2}  \\dd^4 \\xfer_\\alpha\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\hphantom{-} \\times \\xfer_1^\\mu\\, \\del^{(4)}(\\xfer_1+\\xfer_2+\\finalk_X) \\Ampl(\\initialk_1 \\initialk_2 \\rightarrow \\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2 \\,, \\finalk_X)\n\\\\ &\\hspace*{25mm}\\times \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \\initialk_1+\\xfer_1 \\,,\\initialk_2+\\xfer_2 \\,, \\finalk_X)\\,.\n\\end{aligned}\n\\label{eqn:FullImpulse}\n\\end{equation}\nThe prefactor in \\eqn~\\eqref{eqn:partialClassicalLimitNLO} and the normalization of $\\impKer$ are chosen so that the latter is $\\Ord(\\hbar^0)$ in the classical limit. At leading order, \nthe only contribution comes from the tree-level four-point amplitude in the first term, and after passing to the classical limit, we recover \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} as expected. At next-to-leading order (NLO), both terms contribute. The contribution from\nthe first term is from the one-loop amplitude, while that from the second term has $X=\\emptyset$, so that both the amplitude and conjugate inside the integral are tree level four-point amplitudes.\n\nFocus on the NLO contributions, and pass to the classical limit. As discussed in section~\\ref{subsec:Wavefunctions} we may neglect the $q^2$ terms in the delta functions present in \\eqn~\\eqref{eqn:partialClassicalLimitNLO} so long as any singular terms in the impulse\nkernel cancel. We then rescale $q \\rightarrow \\hbar\\qb$; and remove an overall factor of $\\tilde g^4$ and accompanying $\\hbar$''s from the amplitudes. In addition, we may rescale $\\xfer\\rightarrow \\hbar\\xferb$. However, since singular terms may be present in the individual summands of the impulse kernel --- in general, they will cancel against singular terms emerging from the loop integration in the first term in \\eqn~\\eqref{eqn:FullImpulse} ---  \nwe are not entitled to drop the $w^2$ inside the on-shell delta functions. We obtain\n\\begin{equation}\n\\DeltaPnlo= \\frac{i\\tilde g^4}{4}\\Lexp \\int \\!\\dd^4 \\qb  \\; \\del(\\initialk_1 \\cdot \\qb ) \n\\del(\\initialk_2 \\cdot \\qb) \n\\; e^{-i b \\cdot \\qb} \\;  \\impKerCl \\Rexp \\,,\n\\label{eqn:classicalLimitNLO}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl &= \\hbar \\qb^\\mu \\, \\AmplB^{(1)}(\\initialk_1 \\initialk_2 \\rightarrow \n\\initialk_1 + \\hbar\\qb  , \\initialk_2 -  \\hbar\\qb)\n\\\\&\\hphantom{=} \n-i \\hbar^3 \\int \\! \\dd^4 \\xferb \\; \n\\del(2p_1\\cdot \\xferb+ \\hbar\\xferb^2)\\del(2p_2\\cdot \\xferb- \\hbar\\xferb^2)  \\; \\xferb^\\mu \\;\n\\\\&\\hspace*{15mm}\\times \n\\AmplB^{(0)}(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+ \\hbar \\xferb\\,, \\initialk_2- \\hbar\\xferb)\n\\\\&\\hspace*{15mm}\\times \n\\AmplB^{(0)*}(\\initialk_1+ \\hbar\\qb\\,, \\initialk_2-  \\hbar\\qb \\rightarrow \n\\initialk_1+\\hbar\\xferb  \\,,\\initialk_2- \\hbar\\xferb) \\,.\n\\label{eqn:impKerClDef}\n\\end{aligned}\n\\end{equation}\nOnce again, we will see in the next section that additional inverse powers of $\\hbar$ will arise from the amplitudes, and will yield a finite and nonvanishing answer\nin the classical limit.\n\n\\section{Examples}\n\\label{sec:examples}\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\n\nTo build confidence in the formalism we have developed, let us use it to conduct explicit calculations of the classical impulse. We will work in the context of scalar Yang--Mills theory, as defined by the Lagrangian in equation~\\eqref{eqn:scalarAction}, using the double copy where our interest is in perturbative gravity.\n\nBefore we begin to study the impulse at leading and next-to-leading order, note that it is frequently convenient to write amplitudes in Yang--Mills theory in colour-ordered form; for example, see~\\cite{Ochirov:2019mtf} for an application to amplitudes with multiple different external particles. The full amplitude $\\mathcal{A}$ is decomposed onto a basis of colour factors times partial amplitudes $A$. The colour factors are associated with some set of Feynman topologies. Once a basis of independent colour structures is chosen, the corresponding partial amplitudes must be gauge invariant. Thus,\n\\[\n\\mathcal{A}(p_1,p_2 \\rightarrow p_1',p_2') = \\sum_D \\mathcal{C}(D)\\, A_D(p_1,p_2 \\rightarrow p_1',p_2')\\,,\\label{eqn:colourStripping}\n\\]\nwhere $\\mathcal{C}(D)$ is the colour factor of diagram $D$ and $A_D$ is the associated partial amplitude. Expectation values of the representation states $\\chi_\\alpha$ can now be taken as being purely over the colour structures.\n\n\\subsection{Leading-order impulse}\n\\label{sec:LOimpulse}\n\\newcommand{\\tree}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .125 and .275 of v1] (o1);\n\t\\vertex [below right = .125 and .275 of v2] (o2);\n\t\\vertex [above left = .125 and .275 = of v1] (i1);\n\t\\vertex [below left = .125 and .275 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\subsubsection{Gauge theory}\n\nWe begin by computing in YM theory the impulse, $\\DeltaPlo$, on particle 1 at leading order. At this order, only $\\ImpA$ contributes, as expressed in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. To evaluate the impulse, we must first compute the $2\\rightarrow 2$ tree-level scattering amplitude. The reduced amplitude $\\AmplB^{(0)}$ is\n\\begin{equation}\ni\\AmplB^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = \\!\\!\\!\\!\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\\begin{feynman}\n\\vertex (v1);\n\\vertex [below = 0.97 of v1] (v2);\n\\vertex [above left=0.5 and 0.66 of v1] (i1) {$p_1$};\n\\vertex [above right=0.5 and 0.33 of v1] (o1) {$p_1+\\hbar \\qb$};\n\\vertex [below left=0.5 and 0.66 of v2] (i2) {$p_2$};\n\\vertex [below right=0.5 and 0.33 of v2] (o2) {$p_2-\\hbar \\qb$};\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o2);\n\\diagram*{(v1) -- [gluon] (v2)};\n\\end{feynman}\t\n\\end{tikzpicture}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!= i \\newT_1\\cdot\\newT_2 \\frac{4 p_1 \\cdot p_2 + \\hbar^2 \\qb^2}\n{\\hbar^2 \\qb^2}\\,.\n\\label{eqn:ReducedAmplitude1}\n\\end{equation}\nClearly, the colour decomposition of the amplitude is trivial:\n\\begin{equation}\\label{eqn:treeamp}\n\\bar{A}_{\\scalebox{0.5}{\\tree}} =  \\frac{4 p_1\\cdot p_2 +\\hbar \\barq^2}{\\hbar^2 \\barq^2}\\,, \\qquad \\mathcal{C}\\!\\left(\\tree\\right) = \\newT_1\\cdot\\newT_2\\,.\n\\end{equation}\nWe can neglect the second term in the numerator, which is subleading in the classical limit.\n\nSubstituting this expression into \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}, we obtain\n\\begin{equation}\n\\DeltaPlo = i g^2 \\Lexp \\int \\!\\dd^4 \\qb  \\; \\del(\\qb\\cdot p_1) \\del(\\qb\\cdot p_2)\\, e^{-i b \\cdot \\qb} \\newT_1\\cdot\\newT_2 \\frac{p_1 \\cdot p_2}{\\qb^2}\\, \\qb^\\mu\\,\\Rexp\\,.\n\\label{eqn:impulseClassicalLOa}\n\\end{equation}\nAs promised, the leading-order expression is independent of $\\hbar$. Evaluating the $p_{1,2}$ integrals, in the process applying the simplifications explained in section~\\ref{sec:classicalLimit}, namely replacing $p_\\alpha\\rightarrow m_\\alpha\\ucl_\\alpha$, we find that\n\\begin{equation}\n\\DeltaPlo = i g^2 c_1\\cdot c_2 \\int \\!\\dd^4 \\qb  \\; \\del(\\qb\\cdot \\ucl_1) \\del(\\qb\\cdot \\ucl_2) \ne^{-i b \\cdot \\qb} \\frac{\\ucl_1 \\cdot \\ucl_2}{\\qb^2} \\, \\qb^\\mu\\,.\n\\label{eqn:impulseClassicalLO}\n\\end{equation}\nNote that evaluating the double angle brackets has also replaced quantum colour factors with classical colour charges. Replacing the classical colour with electric charges $Q_\\alpha$ yields the result for QED; this expression then has intriguing similarities to quantities that arise in the high-energy\nlimit of two-body scattering~\\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe,Amati:1992zb,Kabat:1992tb,Amati:1993tb,Muzinich:1995uj,DAppollonio:2010krb,Melville:2013qca,Akhoury:2013yua,DAppollonio:2015fly,Ciafaloni:2015vsa,DAppollonio:2015oag,Ciafaloni:2015xsr,Luna:2016idw,Collado:2018isu}. The eikonal approximation used there is known to\nexponentiate, and it would be interesting to explore this connection further. \n\nNote that it is natural that the Yang--Mills LO impulse is a simple colour dressing of its QED counterpart, since at leading order the gluons do not self interact.\n\nIt is straightforward to perform the integral over $\\qb$ in \\eqn~\\eqref{eqn:impulseClassicalLO} to obtain an explicit expression for the leading order impulse. To do so, we work in the rest frame of particle 1, so that $\\ucl_1 = (1, 0, 0, 0)$. Without loss of generality we can orientate the spatial coordinates in this frame so that particle 2 is moving along the $z$ axis, with proper velocity $\\ucl_2 = (\\gamma, 0, 0, \\gamma \\beta)$. We have introduced the standard Lorentz gamma factor $\\gamma = \\ucl_1 \\cdot \\ucl_2$ and the velocity parameter $\\beta$ satisfying $\\gamma^2 ( 1- \\beta^2 ) = 1$. In terms of these variables, the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPlo &= i g^2 c_1\\cdot c_2 \\int \\!\\dd^4 \\qb  \\;\n\\del(\\qb^0) \\del(\\gamma \\qb^0 - \\gamma \\beta \\qb^3) \\;\ne^{-i b \\cdot \\qb} \\frac{\\gamma}{\\qb^2}\n\\, \\qb^\\mu \\\\\n&= -i \\frac{g^2 c_1 \\cdot c_2 }{4\\pi^2 |\\beta|}\\int \\! d^2 \\qb  \\;\ne^{i \\v{b} \\cdot \\v{\\qb}_\\perp} \\frac{1}{\\v \\qb_\\perp^2}\n\\, \\qb^\\mu \\, ,\n\\end{aligned}\n\\end{equation}\nwhere $\\qb^0 = \\qb^3 = 0$ and the non-vanishing components of $\\qb^\\mu$ in the $xy$ plane of our corrdinate system are $\\v \\qb_\\perp$. It remains to perform the two dimensional integral over $\\v \\qb_\\perp$, which is easily done using polar coordinates. Let the magnitude of $\\v \\qb_\\perp$ be $\\chi$ and orient the $x$ and $y$ axes so that $\\v b \\cdot \\v \\qb_\\perp = | \\v b| \\chi \\cos \\theta$. Then the non-vanishing components of $\\qb^\\mu$ are $\\qb^\\mu = (0, \\chi \\cos \\theta, \\chi \\sin \\theta, 0)$ and the impulse is\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPlo &= -i \\frac{g^2 c_1\\cdot c_2 }{4\\pi^2 |\\beta|}\\int_0^\\infty d \\chi \\; \\chi \\int_{-\\pi}^\\pi d \\theta   \\;\ne^{i | \\v b| \\chi \\cos \\theta} \\frac{1}{\\chi^2}\n\\, (0, \\chi \\cos \\theta, \\chi \\sin \\theta, 0) \\\\\n&= -i \\frac{g^2 c_1\\cdot c_2 }{4\\pi^2 |\\beta|}\\int_0^\\infty d \\chi \\; \\int_{-\\pi}^\\pi d \\theta   \\;\ne^{i | \\v b| \\chi \\cos \\theta} \n\\, (0, \\cos \\theta, \\sin \\theta, 0) \\\\\n&= \\frac{g^2 c_1\\cdot c_2 }{2\\pi |\\beta|}\\int_0^\\infty d \\chi \\; \nJ_1 ( |\\v b| \\chi) \\; \\hat{\\v b} \\\\ \n&= \\frac{g^2 c_1\\cdot c_2 }{2\\pi |\\beta|} \\; \\frac{\\hat{\\v b}}{| \\v b|} \\, ,\\label{eqn:LOimpulseIntegral}\n\\end{aligned}\n\\end{equation}\nwhere $\\hat {\\v b}$ is the spatial unit vector in the direction of the impact parameter. To restore manifest Lorentz invariance, note that\n\\begin{equation}\n\\frac{1}{| \\beta|} = \\frac{\\gamma}{\\sqrt{\\gamma^2 - 1}}\\,, \n\\quad \\frac{\\hat{\\v b}}{|\\v b|} = - \\frac{b^\\mu}{b^2}\\,.\n\\end{equation}\n(Recall that $b^\\mu$ is spacelike, so $-b^2>0$.) With this input, we may write the impulse as\n\\begin{equation}\n\\DeltaPlo  \n= -\\frac{g^2 c_1\\cdot c_2}{2\\pi} \\frac{\\gamma}{\\sqrt{\\gamma^2 - 1}} \\frac{b^\\mu}{b^2}\\,.\n\\end{equation}\n\nStripping away the colour and adopting the QED coupling $e$, in the non-relativistic limit this should match a familiar formula: the expansion of the Rutherford scattering angle $\\theta(b)$ as a function of the impact parameter. To keep things simple, we consider Rutherford scattering of a light particle (for example, an electron) off a heavy particle (a nucleus). Taking particle 1 to be the moving light particle, particle 2 is very heavy and we work in its rest frame. Expanding the textbook Rutherford result to order $e^2$, we find\n\\begin{equation}\n\\theta(b) = 2 \\tan^{-1} \\frac{e^2}{4 \\pi m v^2 b} \\simeq \\frac{e^2}{2 \\pi m v^2 b},\n\\end{equation}\nwhere $v$ is the non-relativistic velocity of the particle. To recover this simple result from equation~\\eqref{eqn:impulseClassicalLO}, recall that in the non-relativistic limit $\\gamma \\simeq 1 + v^2\/2$. The scattering angle, at this order, is simply $\\Delta v\/v$.  We will make use of this frame in later sections as well.\n\nWe note in passing that the second term in the numerator \\eqn~\\eqref{eqn:ReducedAmplitude1} is a quantum correction.  It will ultimately be suppressed by $\\lcomp^2\/b^2$, and in addition would contribute only a contact interaction, as it leads to a $\\delta^{(2)}(b)$ term in the impulse.\n\n\\subsubsection{Gravity}\n\nRather than compute gravity amplitudes using the Feynman rules associated with the Einstein--Hilbert action, we can easily just apply the double copy where we have knowledge of their gauge theory counterparts. The generalisation of the traditional BCJ gauge theory replacement rules \\cite{Bern:2008qj,Bern:2010ue} to massive matter states was developed by Johansson and Ochirov \\cite{Johansson:2014zca}.  In our context the colour-kinematics replacement is simple: the amplitude only has a $t$-channel diagram, making the Jacobi identity trivial. Thus by replacing the colour factor with the desired numerator we are guaranteed to land on a gravity amplitude, provided we replace $g\\rightarrow\\frac{\\kappa}{2}$, where $\\kappa = \\sqrt{32\\pi G}$ is the coupling in the Einstein--Hilbert Lagrangian.\n\nA minor point before double-copying is to further rescale\\footnote{We choose this normalisation as it simplifies the colour replacements in the double copy.} the (dimensionful) colour factors as $\\tilde{\\newT}^a$ = $\\sqrt{2}\\newT^a$, such that\n\\begin{equation}\n\\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = \\frac{g^2}{\\hbar^3}\\frac{2p_1 \\cdot p_2 + \\mathcal{O}(\\hbar)}{\\wn q^2} \\tilde{\\newT}_1 \\cdot \\tilde{\\newT}_2\\,.\\label{eqn:scalarYMamp}\n\\end{equation}\nThen replacing the colour factor with the (rescaled) scalar numerator from equation~\\eqref{eqn:treeamp}, we immediately obtain the gravity tree amplitude\\footnote{The overall sign is consistent with the replacements in \\cite{Bern:2008qj,Bern:2010ue} for our amplitudes' conventions.}\n\\begin{equation}\n\\mathcal{M}^{(0)}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = -\\frac{4}{\\hbar^3}\\left(\\frac{\\kappa}{2}\\right)\\frac{(p_1 \\cdot p_2)^2 + \\mathcal{O}(\\hbar)}{\\wn q^2} \\,.\n\\end{equation}\nThis is not quite an amplitude in Einstein gravity: the interactions suffer from dilaton pollution, as can immediately be seen by examining the amplitude's factorisation channels:\n\\[\n\\lim\\limits_{\\wn q^2 \\rightarrow 0} \\left(\\wn q^2 \\hbar^3 \\mathcal{M}^{(0)}\\right) &= -4\\left(\\frac{\\kappa}{2}\\right)^2\\, p_1^\\mu p_1^{\\tilde{\\mu}} \\left(\\mathcal{P}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} + \\mathcal{D}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\right) p_2^\\nu p_2^{\\tilde{\\nu}}\\,,\n\\]\nwhere\n\\begin{equation}\n\\mathcal{P}^{(D)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} = \\eta_{\\mu(\\nu}\\eta_{\\tilde{\\nu})\\tilde{\\mu}} - \\frac{1}{D-2}\\eta_{\\mu\\tilde{\\mu}}\\eta_{\\nu\\tilde{\\nu}} \\qquad \\text{and} \\qquad\n\\mathcal{D}^{(D)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}} = \\frac{1}{D-2}\\eta_{\\mu\\tilde{\\mu}}\\eta_{\\nu\\tilde{\\nu}}\\label{eqn:gravityProjectors}\n\\end{equation}\nare the $D$-dimensional de-Donder gauge graviton and dilaton projectors respectively. The pure Einstein gravity amplitude can now just be read off as the part of the amplitude contracted with the graviton projector. We find that\n\\begin{equation}\n\\mathcal{M}^{(0)}_{\\rm GR}(p_1, p_2 \\rightarrow p_1+\\hbar\\qb\\,, p_2-\\hbar\\qb) = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\hbar^3\\,\\wn q^2} \\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right).\n\\end{equation}\nFollowing the same steps as those before~\\eqref{eqn:impulseClassicalLO}, we find that the LO impulse for massive scalar point-particles in general relativity (such as Schwarzschild black holes) is\n\\begin{equation}\n\\Delta p_1^{\\mu,(0)} = -2i m_1 m_2 \\left(\\frac{\\kappa}{2}\\right)^2\\! \\int \\!\\dd^4 \\qb  \\; \\del(2\\qb\\cdot \\ucl_1) \\del(2\\qb\\cdot \\ucl_2) \ne^{-i b \\cdot \\qb} \\frac{(2\\gamma^2 - 1)}{\\qb^2} \\, \\qb^\\mu\\,.\n\\end{equation}\nIntegrating as in equation~\\eqref{eqn:LOimpulseIntegral} yields the well known 1PM result~\\cite{Westpfahl:1979gu,Portilla:1980uz}\n\\begin{equation}\n\\Delta p_1^{\\mu,(0)} = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}} (2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2}\\,.\n\\end{equation}\n\n\\subsection{Next-to-leading order impulse}\n\\label{sec:nloQimpulse}\n\nAt the next order in perturbation theory, a well-defined classical impulse is only obtained by combining all terms in the impulse $\\langle \\Delta p_1^\\mu \\rangle$ of order $\\tilde g^4$. As we discussed in section~\\ref{sec:classicalImpulse}, both $\\ImpA$ and $\\ImpB$ contribute. We found in \\eqn~\\eqref{eqn:classicalLimitNLO} that the impulse is a simple integral over an impulse kernel $\\impKerCl$, defined in \\eqn~\\eqref{eqn:impKerClDef}, which has a well-defined classical limit. \n\nThe determination of the impulse kernel at this order requires us to compute the four-point one-loop amplitude along with a cut amplitude; that is, an integral over a term quadratic in the tree amplitude. We will compute the NLO impulse in scalar Yang--Mills theory. As the one-loop amplitude in gauge theory is simple, we compute using on-shell renormalised perturbation theory in Feynman gauge.\n\n\\subsubsection{Purely Quantum Contributions}\n\\label{sec:PurelyQuantum}\n\nThe contributions to the impulse in the quantum theory can be divided into three classes, according to the prefactor in the charges they carry. For simplicity of counting, let us momentarily restrict to Abelian gauge theory, with charges $Q_\\alpha$. There are then three classes of diagrams: $\\Gamma_1$, those proportional to $Q_1^3 Q_2$; $\\Gamma_2$, those to $Q_1^2 Q_2^2$; and $\\Gamma_3$, those to $Q_1 Q_2^3$. The first class can be further subdivided into $\\Gamma_{1a}$, terms which would be proportional to $Q_1 (Q_1^2+n_s Q_3^2) Q_2$ were we to add $n_s$ species of a third scalar with charge $Q_3$, and into $\\Gamma_{1b}$, terms which would retain the simple $Q_1^3 Q_2$ prefactor.  Likewise, the last class can be further subdivided into $\\Gamma_{3a}$, terms which would be proportional to $Q_1 (Q_2^2+n_s Q_3^2) Q_2$, and into $\\Gamma_{3b}$, those whose prefactor would remain simply $Q_1 Q_2^3$.\n\nClasses $\\Gamma_{1a}$ and $\\Gamma_{3a}$ consist of gauge boson self-energy corrections along with renormalisation counterterms.  They appear only in the 1-loop corrections to the four-point amplitude, in the first term in the impulse kernel $\\impKerCl$. As one may intuitively expect, they give no contribution in the classical limit. Consider, for example, the self-energy terms, focussing on internal scalars of mass $m$ and charge $Q_i$. We define the self-energy via\n\\begin{equation}\n\\hspace{-10pt}Q_i^2 \\Pi(q^2) \\left( q^2 \\eta^{\\mu\\nu} - q^\\mu q^\\nu \\right) \\equiv \\scalebox{0.9}{\\feynmandiagram [inline = (a.base), horizontal=a to b, horizontal=c to d] { a -- [photon, momentum'=\\(q\\)] b -- [fermion, half left] c -- [fermion, half left] b -- [draw = none] c -- [photon] d};\n\\,  + \\!\\!\\!\\! \\feynmandiagram[inline = (a.base), horizontal=a to b]{a -- [photon, momentum'=\\(q\\)] c -- [out=45, in=135, loop, min distance=2cm]c -- [photon] b};\n\\!\\!\\!\\! + \\, \\feynmandiagram[inline = (a.base), layered layout, horizontal=a to b] { a -- [photon, momentum'=\\(q\\)] b [crossed dot] -- [photon] c};} \\,,\n\\label{eqn:SelfEnergyContributions}\n\\end{equation}\nwhere we have made the projector required by gauge invariance manifest, but have not included factors of the coupling. We have extracted the charges $Q_i$ for later convenience. The contribution of the photon self-energy to the reduced 4-point amplitude is\n\\begin{equation}\n\\AmplB_\\Pi = {Q_1 Q_2 Q_i^2} \\frac{(2p_1 + \\hbar \\qb) \\cdot (2p_2 -\\hbar \\qb)}\n{\\hbar^2 \\qb^2} \\Pi(\\hbar^2 \\qb^2)\\,.\n\\end{equation}\nThe counterterm is adjusted to impose the renormalisation condition that $\\Pi(0) = 0$, \nrequired in order to match the identification of the gauge coupling with its classical counterpart. As a power series in the dimensionless ratio $q^2 \/ m^2 = \\hbar^2 \\qb^2 \/ m^2$, which is of order $\\lcomp^2 \/ b^2$,\n\\begin{equation}\n\\Pi(q^2) = \\hbar^2 \\Pi'(0) \\frac{\\qb^2}{m^2} \n+ \\mathcal{O}\\biggl(\\frac{\\lcomp^4}{b^4} \\biggr)\\,.\n\\end{equation}\nThe renormalisation condition is essential in eliminating possible contributions of $\\Ord(\\hbar^0)$. One way to see that $\\AmplB_\\Pi$ is a purely quantum correction is to follow the powers of $\\hbar$. As $\\Pi(q^2)$ is of order $\\hbar^2$, $\\AmplB_\\Pi$ is of order $\\hbar^0$. This gives a contribution of $\\Ord(\\hbar)$ to the impulse kernel~(\\ref{eqn:impKerClDef}), which in turn gives a contribution of $\\Ord(\\hbar)$ to the impulse, as can be seen in \\eqn~\\eqref{eqn:classicalLimitNLO}.\n\nAlternatively, one can consider the contribution of these graphs to $\\Delta p \/ p$. Counting each factor of $\\qb$ as of order $b$, and using $\\Pi(q^2) \\sim \\lcomp^2 \/ b^2$, it is easy to see that these self-energy graphs yield a contribution to $\\Delta p \/ p$ of order $\\alpha^2 \\hbar^3 \/ (mb)^3 \\sim (\\lclass^2 \/ b^2) \\,( \\lcomp \/ b)$.\n\nThe renormalisation of the vertex is similarly a purely quantum effect. Since the classes $\\Gamma_{1b}$ and $\\Gamma_{3b}$ consisted of vertex corrections, wavefunction renormalisation, and their counterterms, they too give no contribution in the classical limit.\n\nThese conclusions continue to hold in the non--Abelian theory, with charges promoted to colour factors $C_\\alpha$. The different colour structures present in each class of diagram introduces a further splitting of topologies, but one that does not disrupt our identification of quantum effects.\n\n\\subsubsection{Classical colour basis}\n\\label{sec:colourDecomp}\n\\newcommand{\\boxy}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.25 of v1] (v2);\n\t\\vertex [above = 0.25 of v1] (v3);\n\t\\vertex [right = 0.25 of v3] (v4);\n\t\\vertex [above left = 0.15 and 0.15 of v3] (o1);\n\t\\vertex [below left = 0.15 and 0.15 of v1] (i1);\n\t\\vertex [above right = 0.15 and 0.15 of v4] (o2);\n\t\\vertex [below right = 0.15 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (v3);\n\t\\draw (v3) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v4);\n\t\\draw (v4) -- (o2);\n\t\\draw (v1) -- (v2);\n\t\\draw (v3) -- (v4);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\crossbox}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.3 of v1] (v2);\n\t\\vertex [above = 0.3 of v1] (v3);\n\t\\vertex [right = 0.3 of v3] (v4);\n\t\\vertex [above left = 0.125 and 0.125 of v3] (o1);\n\t\\vertex [below left = 0.125 and 0.125 of v1] (i1);\n\t\\vertex [above right = 0.125 and 0.125 of v4] (o2);\n\t\\vertex [below right = 0.125 and 0.125 of v2] (i2);\n\t\\vertex [above right = 0.1 and 0.1 of v1] (g1);\n\t\\vertex [below left = 0.1 and 0.1 of v4] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (v2);\n\t\\draw (v3) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v3) -- (v4);\n\t\\draw (v4) -- (o2);\n\t\\draw (v4) -- (g2);\n\t\\draw (g1) -- (v1);\n\t\\draw (v2) -- (v3);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\newcommand{\\triR}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left = 0.25 and 0.17of v1] (v2);\n\t\\vertex [above right = 0.25 and 0.17 of v1] (v3);\n\t\\vertex [below right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [below left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [above right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [above left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v2) -- (v1);\n\t\\draw (v3) -- (v1);\n\t\\end{feynman}\n\t\\end{tikzpicture}}\n\\newcommand{\\triL}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below left = 0.25 and 0.17of v1] (v2);\n\t\\vertex [below right = 0.25 and 0.17 of v1] (v3);\n\t\\vertex [above right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [above left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [below right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [below left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v2) -- (v1);\n\t\\draw (v3) -- (v1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\nonAbL}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.15 of v1] (g1);\n\t\\vertex [below left = 0.2 and 0.175 of g1] (v2);\n\t\\vertex [below right = 0.2 and 0.175 of g1] (v3);\n\t\\vertex [above right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [above left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [below right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [below left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v1) -- (g1);\n\t\\draw (v2) -- (g1);\n\t\\draw (v3) -- (g1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\nonAbR}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above = 0.15 of v1] (g1);\n\t\\vertex [above left = 0.2 and 0.175 of g1] (v2);\n\t\\vertex [above right = 0.2 and 0.175 of g1] (v3);\n\t\\vertex [below right = 0.2 and 0.275 of v1] (o1);\n\t\\vertex [below left = 0.2 and 0.275 of v1] (i1);\n\t\\vertex [above right = 0.1 and 0.15 of v3] (o2);\n\t\\vertex [above left = 0.1 and 0.15 of v2] (i2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (v3);\n\t\\draw (v3) -- (o2);\n\t\\draw (v1) -- (g1);\n\t\\draw (v2) -- (g1);\n\t\\draw (v3) -- (g1);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\nThis leaves us with contributions of class $\\Gamma_2$; these appear in both terms in the impulse kernel. These contributions to the 1-loop amplitude in the first term take the form\n\\begin{equation}\n\\begin{aligned}\ni \\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') &= \\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}] \n\\begin{feynman}\n\\vertex (b) ;\n\\vertex [above left=1 and 0.66 of b] (i1) {$p_1$};\n\\vertex [above right=1 and 0.33 of b] (o1) {$p_1+q$};\n\\vertex [below left=1 and 0.66 of b] (i2) {$p_2$};\n\\vertex [below right=1 and 0.33 of b] (o2) {$p_2-q$};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- (o2);\n\\draw[postaction={decorate}] (b) -- (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- (b);\n\\draw[postaction={decorate}] (i2) -- (b);\n\\end{scope}\t\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\draw [pattern=north west lines, pattern color=patternBlue] (b) circle [radius=10pt];\n\\filldraw [fill=white] (b) circle [radius=6pt];\n\\end{feynman}\n\\end{tikzpicture} \n\\\\\n&\\hspace*{-30mm}=\n\\scalebox{1.1}{\n\t\\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.9 of v1] (v2);\n\t\\vertex [above = 0.97 of v1] (v3);\n\t\\vertex [right = 0.9 of v3] (v4);\n\t\\vertex [above left = 0.5 and 0.5 of v3] (i2);\n\t\\vertex [below left = 0.5 and 0.5 of v1] (i1);\n\t\\vertex [above right = 0.5 and 0.5 of v4] (o2);\n\t\\vertex [below right = 0.5 and 0.5 of v2] (o1);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (v4);\n\t\\draw [postaction={decorate}] (v4) -- (o2);\n\t\\diagram*{(v3) -- [gluon] (v1)};\n\t\\diagram*{(v2) -- [gluon] (v4)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [right = 0.9 of v1] (v2);\n\t\\vertex [above = 0.97 of v1] (v3);\n\t\\vertex [right = 0.9 of v3] (v4);\n\t\\vertex [above left = 0.5 and 0.5 of v3] (i2);\n\t\\vertex [below left = 0.5 and 0.5 of v1] (i1);\n\t\\vertex [above right = 0.5 and 0.5 of v4] (o2);\n\t\\vertex [below right = 0.5 and 0.5 of v2] (o1);\n\t\\vertex [above right = 0.45 and 0.485 of v1] (g1);\n\t\\vertex [below left = 0.4 and 0.4 of v4] (g2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (v4);\n\t\\draw [postaction={decorate}] (v4) -- (o2);\n\t\\diagram*{(v4) -- [gluon] (v1)};\n\n\t\\filldraw [color=white] (g1) circle [radius=6.4pt];\n\t\\diagram*{(v2) -- [gluon] (v3)};\n\t\\end{feynman}\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left = 0.94 and 0.45 of v1] (v2);\n\t\\vertex [above right = 0.94 and 0.45 of v1] (v3);\n\t\\vertex [below right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [below left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [above right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [above left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(v1) -- [gluon] (v2)};\n\t\\diagram*{(v1) -- [gluon] (v3)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below left = 0.94 and 0.45 of v1] (v2);\n\t\\vertex [below right = 0.94 and 0.45 of v1] (v3);\n\t\\vertex [above right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [above left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [below right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [below left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(v2) -- [gluon] (v1)};\n\t\\diagram*{(v3) -- [gluon] (v1)};\n\t\\end{feynman}\t\n\t\\end{tikzpicture} } \\\\ &\\hspace{-10mm}+\\scalebox{1.1}{\\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above = 0.55 of v1] (g1);\n\t\\vertex [above left = 0.45 and 0.45 of g1] (v2);\n\t\\vertex [above right = 0.45 and 0.45 of g1] (v3);\n\t\\vertex [below right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [below left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [above right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [above left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(g1) -- [gluon] (v1)};\n\t\\diagram*{(v2) -- [gluon] (g1)};\n\t\\diagram*{(g1) -- [gluon] (v3)};\n\n\t\\end{feynman}\t\n\t\\end{tikzpicture} +\t\\begin{tikzpicture}[baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.55 of v1] (g1);\n\t\\vertex [below left = 0.45 and 0.45 of g1] (v2);\n\t\\vertex [below right = 0.45 and 0.45 of g1] (v3);\n\t\\vertex [above right = 0.5 and 0.95 of v1] (o1);\n\t\\vertex [above left = 0.5 and 0.95 of v1] (i1);\n\t\\vertex [below right = 0.4 and 0.5 of v3] (o2);\n\t\\vertex [below left = 0.4 and 0.5 of v2] (i2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (v3);\n\t\\draw [postaction={decorate}] (v3) -- (o2);\n\t\\diagram*{(g1) -- [gluon] (v1)};\n\t\\diagram*{(g1) -- [gluon] (v2)};\n\t\\diagram*{(v3) -- [gluon] (g1)};\n\n\t\\end{feynman}\t\n\t\\end{tikzpicture} + \\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\t\\begin{feynman}\n\t\\vertex (v1) ;\n\t\\vertex [above left= 0.6 and 1 of v1] (i1);\n\t\\vertex [above right= 0.6 and 1 of v1] (o1);\n\t\\vertex [below = 0.7 of v1] (v2);\n\t\\vertex [below left= 0.6 and 1 of v2] (i2);\n\t\\vertex [below right= 0.6 and 1 of v2] (o2);\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v2);\n\t\\draw [postaction={decorate}] (v2) -- (o2);\n\t\\diagram*{(v1) -- [gluon, half left] (v2)};\n\t\\diagram*{(v2) -- [gluon, half left] (v1)};\n\t\\end{feynman}\n\t\\end{tikzpicture}}.\n\\end{aligned}\n\\end{equation}\nIn each contribution, we count powers of $\\hbar$ following the rules in section~\\ref{subsec:Wavefunctions}, replacing $\\ell\\rightarrow\\hbar\\ellb$ and $q\\rightarrow\\hbar \\qb$.  In the final double-seagull contribution, we will get four powers from the loop measure, and four inverse powers from the two photon propagators.  Overall, we will not get enough inverse powers to compensate the power in front of the integral in \\eqn~\\eqref{eqn:impKerClDef}, and thus the seagull will die in the classical limit. \n\nWe will refer to the remaining topologies as the box $B$, cross box $C$, triangles $T_{\\alpha\\beta}$, and non-Abelian diagrams $Y_{\\alpha\\beta}$, respectively. Applying the colour decomposition of equation~\\eqref{eqn:colourStripping}, the 1-loop amplitude contributing classically to the linear part of the impulse is\n\\begin{multline}\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') = \\mathcal{C}\\!\\left(\\boxy \\right) B + \\mathcal{C}\\!\\left(\\crossbox \\right) C + \\mathcal{C}\\!\\left(\\triR \\right) T_{12} \\\\ + \\mathcal{C}\\!\\left(\\triL \\right) T_{21} + \\mathcal{C}\\!\\left(\\nonAbR \\right) Y_{12} + \\mathcal{C}\\!\\left(\\nonAbL \\right) Y_{21}\\,.\n\\end{multline}\nA first task is to choose a basis of independent colour structures. The complete set of colour factors can easily be calculated:\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{C}\\!\\left(\\boxy \\right) = \\newT_1^a \\newT_2^a \\newT_1^b \\newT_2^b\\,, \\qquad\n\\mathcal{C}\\!\\left(\\crossbox \\right) = \\newT_1^a \\newT_2^b \\newT_1^b \\newT^a_2\\,,\\\\\n\\mathcal{C}\\!\\left(\\nonAbL \\right) = \\hbar\\, \\newT_1^a f^{abc} \\newT_2^b \\newT_2^c\\,, \\qquad \n\\mathcal{C}\\!\\left(\\nonAbR \\right) = \\hbar\\, \\newT_1^a \\newT_1^b f^{abc} \\newT_2^c\\,,\\\\\n\\mathcal{C}\\!\\left(\\triL \\right) = \\frac12\\, \\mathcal{C}\\!\\left(\\boxy \\right) + \\frac12\\, \\mathcal{C}\\!\\left(\\crossbox \\right) = \\mathcal{C}\\!\\left(\\triR \\right).\n\\end{gathered}\n\\end{equation}\nAt first sight, we appear to have a basis of four independent colour factors: the box, cross box and the two non-Abelian triangles. However, it is very simple to see that the latter are in fact both proportional to the tree colour factor of \\eqn~\\eqref{eqn:treeamp}; for example, \n\\[\n\\mathcal{C}\\!\\left(\\nonAbL \\right) = \\frac{\\hbar}{2}\\, \\newT_1^a f^{abc} [\\newT_2^b, \\newT_2^c] &= \\frac{i\\hbar^2}{2} f^{abc} f^{bcd} \\newT_1^a \\newT_2^d\\\\\n& = \\frac{i\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right),\n\\]\nwhere we have used \\eqn~\\eqref{eqn:chargeLieAlgebra}. Moreover, similar manipulations demonstrate that the cross-box colour factor is not in fact linearly independent:\n\\[\n\\mathcal{C}\\!\\left(\\crossbox \\right) &= \\newT_1^a \\newT_1^b \\left( \\newT_2^a \\newT_2^b - i\\hbar f^{abc} \\newT_2^c\\right)\\\\\n&= (\\newT_1 \\cdot \\newT_2) (\\newT_1 \\cdot \\newT_2 ) - \\frac{i\\hbar}{2} [\\newT_1^a, \\newT_2^b] f^{abc} \\newT_2^c\\\\\n& = \\mathcal{C}\\!\\left(\\boxy \\right) + \\frac{\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right).\n\\]\nThus at 1-loop the classically significant part of the amplitude has a basis of two colour structures: the box and tree. Hence the decomposition of the 1-loop amplitude into partial amplitudes and colour structures is\n\\begin{multline}\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') =  \\mathcal{C}\\!\\left(\\boxy \\right) \\bigg[B + C + T_{12} + T_{21}\\bigg] \\\\ + \\frac{\\hbar^2}{2}\\, \\mathcal{C}\\!\\left(\\tree \\right) \\bigg[C + \\frac{ T_{12}}{2} + \\frac{T_{21}}{2} + iY_{12} + iY_{21}\\bigg]\\,.\\label{eqn:1loopDecomposition}\n\\end{multline}\nThis expression for the amplitude is particularly useful when taking the classical limit. The second term is proportional to two powers of $\\hbar$, while the only possible singularity in $\\hbar$ at one loop order is a factor $1\/\\hbar$ in the evaluation of the kinematic parts of the diagrams. Thus, it is clear that the second line of the expression must be a quantum \ncorrection, and can be dropped in calculating the classical impulse. Perhaps surprisingly, these terms include the sole contribution from the non-Abelian triangles $Y_{\\alpha\\beta}$, and thus we will not need to calculate these diagrams. We learn that classically, the 1-loop scalar YM amplitude has a basis of only one colour factor:\n\\[\n\\AmplB^{(1)}(p_1,p_2 \\rightarrow p_1', p_2') &=  \\mathcal{C}\\!\\left(\\boxy \\right) \\bigg[B + C + T_{12} + T_{21}\\bigg] + \\mathcal{O}(\\hbar)\\,.\\label{eqn:OneLoopImpulse}\n\\]\nMoreover, the impulse depends on precisely the same topologies as in QED \\cite{Kosower:2018adc}.\n\n\\subsubsection{Triangles}\n\\label{sec:Triangles}\n\nLet us first examine the two (colour stripped) triangle diagrams in \\eqn~\\eqref{eqn:OneLoopImpulse}.  They are related by swapping particles~1 and~2. The first diagram is\n\\begin{equation}\ni T_{12} = \n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (i2) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (o1) {$p_2$};\n\\vertex [below=2.5 of i2] (o2) {$p_2 - q$};\n\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of i2] (v2);\n\\vertex [above right=1.1 and 1.25 of o1] (v3);\n\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (i2);\n\\draw [postaction={decorate}] (o1) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (o2);\n\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\ell\\)] (v1);\n\t(v2) -- [gluon] (v3);\n};\n\\end{feynman}\n\\end{tikzpicture}\n= -2 \\!\\int \\!\\dd^D \\ell\\, \\frac{(2p_1 + \\ell) \\cdot (2 p_1 + q + \\ell)}\n{\\ell^2 (\\ell - q)^2 (2p_1 \\cdot \\ell + \\ell^2 + i \\epsilon)}\\,.\n\\end{equation}\nIn this integral, we use a dimensional regulator in a standard way ($D=4-2\\varepsilon$)\nin order to regulate potential divergences. We have retained an explicit $i \\epsilon$ in the massive scalar propagator, because it will play an important role below.\n\nTo extract the classical contribution of this integral to the amplitude, we recall from section~\\ref{subsec:Wavefunctions} that we should set $q =  \\hbar\\qb$ and $\\ell =  \\hbar\\ellb$, and therefore that the components of $q$ and $\\ell$ are all small compared \nto $m$. Consequently, the triangle simplifies to\n\\begin{equation}\nT_{12} = \\frac{4 i m_1^2}{\\hbar} \\!\\int\\! \\dd^4 \\bar \\ell \\, \\frac{1}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2 (p_1 \\cdot \\bar \\ell + i \\epsilon)}\\,.\n\\label{eqn:triangleIntermediate1}\n\\end{equation}\nHere, we have taken the limit $D\\rightarrow 4$, as the integral is now free of divergences.\nNotice that we have exposed one additional inverse power of $\\hbar$. Comparing to the definition of $\\impKerCl$ in \\eqn~\\eqref{eqn:impKerClDef}, we see that this inverse power of $\\hbar$ will cancel against the explicit factor of $\\hbar$ in $\\ImpAclsup{(1)}$, signalling a classical contribution to the impulse.\n\nAt this point we employ a simple trick which simplifies the loop integral appearing in \\eqn~\\eqref{eqn:triangleIntermediate1}, and which will be of great help in simplifying the more complicated box topologies below. The on-shell condition for the outgoing particle 1 requires that $p_1 \\cdot \\qb = - \\hbar \\qb^2\/2$, so replace $\\ellb \\rightarrow \\ellb' = \\qb - \\ellb$ in $T_{12}$:\n\\begin{equation}\n\\begin{aligned}\nT_{12} &= -\\frac{4 i m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell'\\, \\frac{1}{\\bar \\ell'^2 (\\bar \\ell' - \\bar q)^2 (p_1 \\cdot \\bar \\ell' + \\hbar \\qb^2 - i \\epsilon)} \\\\\n&= -\\frac{4 i m_1^2}{\\hbar} \\!\\int \\!\\dd^4 \\bar \\ell' \\,\\frac{1}{\\bar \\ell'^2 (\\bar \\ell' - \\bar q)^2 (p_1 \\cdot \\bar \\ell' - i \\epsilon)} + \\mathcal{O}(\\hbar^0)\\,,\n\\end{aligned}\n\\end{equation}\nBecause of the linear power of $\\hbar$ appearing in \\eqn~\\eqref{eqn:impKerClDef}, the second term\nis in fact a quantum correction. We therefore neglect it, and write\n\\begin{equation}\nT_{12}= -\\frac{4 i m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell\\, \\frac{1}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2 (p_1 \\cdot \\bar \\ell - i \\epsilon)} \\, ,\n\\end{equation}\nwhere we have dropped the prime on the loop momentum: $\\ell' \\rightarrow \\ell$. Comparing with our previous expression, \\eqn~\\eqref{eqn:triangleIntermediate1}, for the triangle, the net result of these replacements has simply been to introduce an overall sign while, crucially, also switching the sign of the $i \\epsilon$ term. Symmetrising over the two expressions for $T_{12}$, we learn that\n\\begin{equation}\nT_{12}= \\frac{2 m_1^2}{\\hbar}\\! \\int\\! \\dd^4 \\bar \\ell\\, \\frac{\\del(p_1 \\cdot \\bar \\ell)}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2} \\,,\n\\end{equation}\nusing the identity\n\\begin{equation}\n\\frac{1}{x-i \\epsilon} - \\frac{1}{x+i \\epsilon} = i \\del(x)\\,.\n\\label{eqn:deltaPoles}\n\\end{equation}\n\nThe second triangle contributing to the amplitude, $T_{21}$, can be obtained from $T_{12}$ simply by interchanging the labels 1 and 2:\n\\begin{equation}\nT_{21} = \\frac{2 m_2^2}{\\hbar}\\! \\int \\! \\dd^4 \\bar \\ell \\,\n\\frac{\\del(p_2 \\cdot \\bar \\ell)}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2}\\,.\n\\end{equation}\nThese triangles contribute to the impulse kernel via\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl \\big|_\\mathrm{triangle} &= \\hbar\\qb^\\mu\\, \\mathcal{C}\\!\\left(\\boxy \\right) (T_{12} + T_{21}) \n\\\\&= 2\\left(\\newT_1\\cdot \\newT_2 \\right)^2 \\bar q^\\mu\\! \\int \\! \\frac{\\dd^4 \\bar \\ell}{\\bar \\ell^2 (\\bar \\ell - \\bar q)^2} \\left(m_1^2\\del(p_1 \\cdot \\bar \\ell) + m_2^2\\del(p_2 \\cdot \\bar \\ell) \\right).\n\\end{aligned}\n\\end{equation}\nRecall that we must integrate over the wavefunctions in order to obtain the classical impulse from the impulse kernel. As we have discussed in section~\\ref{subsec:Wavefunctions}, because the inverse power of $\\hbar$ here is cancelled by the linear power present explicitly in \\eqn~\\eqref{eqn:classicalLimitNLO}, we may evaluate the wavefunction integrals by replacing the $p_\\alpha$ by their classical values $m_\\alpha \\ucl_\\alpha$. The result for the contribution to the kernel is\n\\begin{equation}\n\\impKerTerm1 \\equiv\n{2 (c_1\\cdot c_2)^2} \\qb^\\mu \\!\\int \\! \\dd^4 \\ellb\\;\n\\frac{1}{\\ellb^2 (\\ellb - \\qb)^2} \n\\biggl(m_1{\\del(\\ucl_1 \\cdot \\ellb)} \n+ m_2{\\del(\\ucl_2 \\cdot \\ellb)} \\biggr)\\,.\n\\label{eqn:TriangleContribution}\n\\end{equation}\nOne must still integrate this expression over $\\qb$, as in \\eqn~\\eqref{eqn:classicalLimitNLO}, to\nobtain the contribution to the impulse.\n\n\\subsubsection{Boxes}\n\\label{sec:Boxes}\n\nThe one-loop amplitude also includes boxes and crossed boxes, and the NLO contribution to the impulse includes as well a term quadratic in the tree amplitude which we can think of as the cut of a one-loop box. Because of the power of $\\hbar$ in front of the first term in \\eqn~\\eqref{eqn:impKerClDef}, we need to extract the contributions of all of these quantities at order $1\/\\hbar$. However, as we will see, each individual diagram also contains singular terms of order $1\/\\hbar^2$.  We might fear that these terms pose an obstruction to the \nvery existence of a classical limit of the observable in which we are interested. As we will see, this fear is misplaced, as these singular terms cancel completely, leaving a well-defined classical result. It is straightforward to evaluate the individual contributions, but making the cancellation explicit requires some care. We begin with the colour-stripped box:\n\\begin{equation}\n\\begin{aligned}\n\\hspace*{-7mm}i B &= \\hspace*{-2mm}\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - q$};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\ell\\)] (v1);\n\t(v2) -- [gluon] (v4);\n};\n\\end{feynman}\n\\end{tikzpicture} \n\\hspace*{-9mm}\n= \\int \\! \\dd^D \\ell \\;\n\\frac{(2 p_1 \\tp \\ell) \\td (2p_2 \\tm \\ell)\\,(2 p_1 \\tp q\\tp \\ell) \\td (2 p_2\\tm q\\tm \\ell)}\n{\\ell^2 (\\ell \\tm q)^2 (2 p_1 \\cdot \\ell \\tp \\ell^2 \\tp i \\epsilon)\n\t(-2p_2 \\cdot \\ell \\tp \\ell^2 \\tp i \\epsilon)}\n=\\hspace*{-8mm}\n\\\\[-2mm]\n\\\\&\\hspace*{-5mm} \\frac{1}{\\hbar^{2+2\\varepsilon}} \\!\\!\\int \\! \\dd^D \\ellb\\;\n\\frac{\\bigl[4 p_1\\td p_2\\tm 2\\hbar(p_1\\tm p_2)\\td\\ellb\\tm \\hbar^2\\ellb^2\\bigr]\n\t\\bigl[4 p_1\\td p_2\\tm 2\\hbar(p_1\\tm p_2)\\td(\\ellb\\tp \\qb)\\tm \\hbar^2(\\ellb\\tp \\qb)^2\\bigr]}\n{\\ellb^2 (\\ellb - \\qb)^2 (2 p_1 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)\n\t(-2p_2 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)}\\,,\\hspace*{-8mm}\n\\end{aligned}\n\\end{equation}\nwhere as usual, we have set $q = \\hbar \\qb$, $\\ell = \\hbar \\ellb$. We get four powers of $\\hbar$ from changing variables in the measure, but six inverse powers from the propagators\\footnote{We omit fractional powers of $\\hbar$ in this counting as they will disappear when we take $D \\rightarrow 4$.}. We thus encounter an apparently singular $1\/\\hbar^2$ leading behaviour.  We must extract both this singular, $\\Ord(1\/\\hbar^2)$, term \nas well as the terms contributing in the classical limit, which here are $\\Ord(1\/\\hbar)$.\nConsequently, we must take care to remember that the on-shell delta functions enforce $\\qb \\cdot p_1 = - \\hbar \\qb^2 \/ 2$ and $\\qb \\cdot p_2 = \\hbar \\qb^2 \/ 2$. \n\nPerforming a Laurent expansion in $\\hbar$, truncating after order $1\/\\hbar$, and separating different orders in $\\hbar$, we find that the box's leading terms are given by\n\\begin{equation}\n\\begin{aligned}\nB &= B_{-1}+B_0\\,,\n\\\\ B_{-1} &= \\frac{4 i}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\int \\frac{\\dd^D \\ellb}{\\ellb^2 (\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb - i \\epsilon)} \\,,\n\\\\ B_{0} &= -\\frac{2i }{\\hbar^{1+2\\varepsilon}} p_1 \\cdot p_2 \n\\int \\frac{\\dd^D \\ellb}{\\ellb^2 (\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb - i \\epsilon)}\n\\\\& \\hspace*{30mm}\\times\n\\biggl[ 2{(p_1 - p_2)\\cdot \\ellb}\n+ \\frac{ (p_1 \\cdot p_2) \\ellb^2}{(p_1 \\cdot \\ellb + i \\epsilon)} \n- \\frac{(p_1 \\cdot p_2) \\ellb^2}{(p_2 \\cdot \\ellb - i \\epsilon)}\\biggl]\\,.\n\\end{aligned}\n\\label{eqn:BoxExpansion}\n\\end{equation}\nNote that pulling out a sign from one of the denominators has given the appearance of\nflipping the sign of one of the denominator $i\\epsilon$ terms.  We must also bear in mind that the integral in $B_{-1}$ is itself \\textit{not\\\/} $\\hbar$-independent, so that we will later need to expand it as well.\n\nSimilarly, the crossed box is\n\\begin{align}\ni C &= \n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + q$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - q$};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- node [above] {$\\scriptstyle{p_1 + \\ell}$} (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw [postaction={decorate}] (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\diagram*{(v3) -- [gluon] (v2);};\n\\filldraw [color=white] ($ (v3) !.5! (v2) $) circle [radius = 4.3pt];\n\\diagram*{\t(v1) -- [gluon] (v4);};\n\\end{feynman}\n\\end{tikzpicture} \n\\\\\n&=  \\int \\! \\dd^D \\ell\\, \\frac{(2 p_1 + \\ell) \\cdot (2p_2 - 2q + \\ell)(2 p_1 +q+ \\ell) \\cdot (2 p_2 -q + \\ell)}{\\ell^2 (\\ell - q)^2 (2 p_1 \\cdot \\ell + \\ell^2 + i \\epsilon)(2p_2 \\cdot (\\ell-q) + (\\ell-q)^2 + i \\epsilon)}\\nonumber\n\\\\\n&= \\frac{1}{\\hbar^{2+2\\varepsilon}} \\!\\! \\int \\! \\dd^D \\ellb\\, \n\\frac{(2 p_1 + \\hbar\\ellb) \\cdot (2p_2 - 2\\hbar\\qb + \\hbar\\ellb)\\,\n\t(2 p_1 +\\hbar\\qb+ \\hbar\\ellb) \\cdot (2 p_2 -\\hbar\\qb + \\hbar\\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2 (2 p_1 \\cdot \\ellb + \\hbar\\ellb^2 + i \\epsilon)\n\t(2p_2 \\cdot (\\ellb-\\qb) + \\hbar(\\ellb-\\qb)^2 + i \\epsilon)}\\,.\\nonumber\n\\end{align}\nUsing the on-shell conditions to simplify $p_\\alpha\\cdot \\qb$ terms in the denominator\nand numerator, and once again expanding in powers of $\\hbar$, truncating after order $1\/\\hbar$, and separating different orders in $\\hbar$,  we find\n\\begin{equation}\n\\begin{aligned}\nC &= C_{-1}+C_0\\,,\n\\\\ C_{-1} &= -\\frac{4i}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int \\!\\frac{\\dd^D \\ellb}{\\ellb^2(\\ellb - \\qb)^2} \n\\frac{1}\n{(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb + i \\epsilon)} \n\\\\ C_{0} &= -\\frac{2i}{\\hbar^{1+2\\varepsilon}} p_1 \\cdot p_2\n\\!\\int \\!\\frac{\\dd^D \\ellb}{\\ellb^2(\\ellb - \\qb)^2\n\t(p_1 \\cdot \\ellb + i \\epsilon)(p_2 \\cdot \\ellb + i \\epsilon)} \n\\\\& \\qquad \\times\n\\biggl[2  (p_1 + p_2) \\cdot \\ellb\n- \\frac{(p_1 \\cdot p_2) \\ellb^2}{(p_1 \\cdot \\bar \\ell + i \\epsilon)} \n- \\frac{(p_1 \\cdot p_2) [(\\ellb - \\qb)^2 - \\qb^2]}\n{(p_2 \\cdot \\ellb + i \\epsilon)}\\biggr]\\,.\\hspace*{-20mm}\n\\end{aligned}\n\\label{eqn:CrossedBoxExpansion}\n\\end{equation}\nComparing the expressions for the $\\Ord(1\/\\hbar^2)$ terms in the box and the crossed box, \n$B_{-1}$ and $C_{-1}$ respectively, we see that there is only a partial cancellation of \nthe singular, $\\mathcal{O}(1\/\\hbar^2)$, term in the reduced amplitude $\\AmplB^{(1)}$. The impulse kernel, \\eqn~\\eqref{eqn:impKerClDef}, does contain another term, which is quadratic in the tree-level reduced amplitude  $\\AmplB^{(0)}$.  We will see below that taking this additional contribution into account leads to a complete cancellation of the singular term; \nbut the classical limit does not exist for each of these terms separately.\n\n\\subsubsection{Cut Box}\n\\label{sec:CutBoxes}\n\nIn order to see the cancellation of the singular term we must incorporate the term in the impulse kernel which is quadratic in tree amplitudes. As with the previous loop diagrams, let us begin by splitting the colour and kinematic information as in equation~\\eqref{eqn:colourStripping}. Then the quadratic term in~\\eqref{eqn:impKerClDef} can be written as\n\\begin{equation}\n\\impKerCl \\big|_\\textrm{non-lin} = \\mathcal{C}\\!\\left({\\scalebox{1}{\\tree}} \\right)^\\dagger \\mathcal{C}\\!\\left({\\scalebox{1}{\\tree}}\\right) \\cutbox^\\mu\\,,\\label{eqn:cutBoxColDecomp}\n\\end{equation}\nwhere the kinematic data $\\cutbox^\\mu$ can be viewed as proportional to the cut of the one-loop box, weighted by the loop momentum $\\hbar \\xferb^\\mu$:\n\\begin{equation}\n\\cutbox^\\mu = -i\\hbar^2\\int \\! \\dd^4 \\xferb \\, \\xferb^\\mu \\, \\del(2 p_1 \\cdot \\xferb + \\hbar \\xferb^2) \\del(2p_2 \\cdot \\xferb - \\hbar \\xferb^2) \\times\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}] \n\\begin{feynman}\n\\vertex (i1) {$p_1$};\n\\vertex [right=2.5 of i1] (o1) {$p_1 + \\hbar \\qb$};\n\\vertex [below=2.5 of i1] (i2) {$p_2$};\n\\vertex [below=2.5 of o1] (o2) {$p_2 - \\hbar\\qb$};\n\\node [] (cutTop) at ($ (i1)!.5!(o1) $) {};\n\\node [] (cutBottom) at ($ (i2)!.5!(o2) $) {};\n\\vertex [below right=1.1 of i1] (v1);\n\\vertex [below left=1.1 of o1] (v2);\n\\vertex [above right=1.1 of i2] (v3);\n\\vertex [above left=1.1 of o2] (v4);\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw (v1) -- (v2);\n\\draw [postaction={decorate}] (v2) -- (o1);\n\\draw [postaction={decorate}] (i2) -- (v3);\n\\draw (v3) -- (v4);\n\\draw [postaction={decorate}] (v4) -- (o2);\n\\filldraw [color=white] ($  (cutTop) - (3pt, 0) $) rectangle ($ (cutBottom) + (3pt,0) $) ;\n\\draw [dashed] (cutTop) -- (cutBottom);\n\\diagram*{\n\t(v3) -- [gluon, momentum=\\(\\hbar\\xferb\\)] (v1);\n\t(v2) -- [gluon] (v4);\n};\n\\end{feynman}\n\\end{tikzpicture}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!;.\n\\end{equation}\nNote that an additional factor of $\\hbar$ in the second term of \\eqn~\\eqref{eqn:impKerClDef} will be multiplied into \\eqn~\\eqref{CombiningBoxes} below, as it parallels the factor in the first term of \\eqn~\\eqref{eqn:impKerClDef}. Evaluating the Feynman diagrams, we obtain\n\\begin{multline}\n\\cutbox^\\mu = -i\\frac{1}{\\hbar^2}\\! \\int \\! \\dd^4 \\xferb \\, \n\\del(2 p_1 \\cdot \\xferb + \\hbar \\xferb^2) \\del(2p_2 \\cdot \\xferb - \\hbar \\xferb^2) \\,\n\\frac{\\xferb^\\mu}{\\xferb^2 (\\xferb - \\qb)^2} \\\\\n\\quad \\times (2 p_1 + \\hbar\\xferb ) \\cdot (2p_2 - \\xferb \\hbar)\\,\n(2 p_1 + \\hbar\\qb + \\hbar\\xferb ) \\cdot (2 p_2 - \\hbar\\qb  -  \\hbar\\xferb)\\, .\\label{eqn:cutBoxFull}\n\\end{multline}\nAs in the previous subsection, expand in $\\hbar$, and truncate after order $1\/\\hbar$,\nso that\n\\begin{align}\n\\cutbox^\\mu &= \\cutbox_{-1}^\\mu + \\cutbox_{0}^\\mu\\,,\\nonumber\n\\\\ \\cutbox_{-1}^\\mu &=  -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\frac{\\dd^4 \\ellb \\; \\ellb^\\mu}{\\ellb^2 (\\ellb - \\qb)^2}\n\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb) \\,, \\label{eqn:CutBoxExpansion}\n\\\\ \\cutbox_{0}^\\mu &=  -\\frac{2i}{\\hbar} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\frac{\\dd^4 \\ellb \\; \\ellb^\\mu}{\\ellb^2 (\\ellb - \\qb)^2}\\,\n{\\ellb^2} \\Big(\\del'(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)\n- \\del(p_1 \\cdot \\ellb) \\del'(p_2 \\cdot \\ellb) \\Big)\\,.\\nonumber\n\\end{align}\nWe have relabelled $\\xferb\\rightarrow\\ellb$ in order to line up terms more transparently with corresponding ones in the box and crossed box contributions.\n\nFinally, it is easy to see that the cut box colour factor in~\\eqref{eqn:cutBoxColDecomp} is simply\n\\begin{equation}\n\\mathcal{C}\\!\\left(\\tree \\right)^\\dagger \\mathcal{C}\\!\\left(\\tree \\right) = (\\newT_2\\cdot \\newT_1) (\\newT_1 \\cdot \\newT_2) = \\mathcal{C}\\!\\left(\\boxy \\right)\\,.\n\\end{equation}\nThus there is only one relevant colour structure in the NLO momentum impulse, that of the box. This will be important in the following.\n\n\\subsubsection{Combining Contributions}\n\\label{sec:CombiningTerms}\n\nWe are now in a position to assemble the elements computed in the three previous subsections in order to obtain the NLO contributions to the impulse kernel $\\impKerCl$, and thence the NLO contributions to the impulse using \\eqn~\\eqref{eqn:classicalLimitNLO}. Let us begin by examining the singular terms. We must combine the terms from the box, crossed box, and cut box. We can simplify the cut-box contribution $\\cutbox_{-1}^\\mu$ by exploiting the linear change of variable $\\ellb' = \\qb - \\ellb$:\n\\begin{align}\n\\cutbox_{-1}^\\mu \n&= -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \\nonumber\n\\!\\int\\! \\dd^4 \\ellb' \\;\\frac{ (\\qb^\\mu - \\ellb'^\\mu)}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\del(p_1 \\cdot \\ellb'-p_1\\cdot\\qb\n\\del(p_2 \\cdot \\ellb'-p_2\\cdot \\qb)\n\\\\&= -\\frac{4i}{\\hbar^2} (p_1 \\cdot p_2)^2  \n\\!\\int\\! \\dd^4 \\ellb' \\;\\frac{ (\\qb^\\mu - \\ellb'^\\mu)}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\del(p_1 \\cdot \\ellb'+\\hbar\\qb^2\/2)\n\\del(p_2 \\cdot \\ellb'-\\hbar\\qb^2\/2)\\nonumber\n\\\\&= -\\frac{2i}{\\hbar^2} (p_1 \\cdot p_2)^2 \\qb^\\mu\n\\! \\int\\! \\dd^4 \\ellb \\;\\frac{\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2} +\\Ord(1\/\\hbar)\\,,\\label{eqn:CutSingular}\n\\end{align}\nwhere we have used the on-shell conditions to replace $p_1\\cdot\\qb\\rightarrow -\\hbar\\qb^2\/2$\nand $p_2\\cdot\\qb\\rightarrow \\hbar\\qb^2\/2$, and where the last line arises from averaging over the two equivalent expressions for $\\cutbox_{-1}^\\mu$.\n\n\nWe may similarly simplify the singular terms from the box and cross box. Indeed, using the identity~\\eqref{eqn:deltaPoles} followed by the linear change of variable, we have\n\\begin{equation}\n\\begin{aligned}\nB_{-1} + C_{-1} &= \n-\\frac{4 }{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int \\! \\frac{\\dd^D \\ellb}{\\ellb^2 (\\bar \\ell - \\bar q)^2} \n\\frac{1}{(p_1 \\cdot \\ellb + i \\epsilon)} \\del(p_2 \\cdot \\ellb)\n\\\\&= \\frac{4 Q_1^2 Q_2^2}{\\hbar^{2+2\\varepsilon}} (p_1 \\cdot p_2)^2\n\\!\\int\\! \\frac{\\dd^D \\ellb'}{\\ellb'^2 (\\ellb' - \\qb)^2}\n\\frac{\\del(p_2 \\cdot \\ellb'-\\hbar\\qb^2\/2) }{(p_1 \\cdot \\ellb'+\\hbar\\qb^2\/2 - i \\epsilon)} \n\\\\&= \\frac{2i}{\\hbar^2} (p_1 \\cdot p_2)^2\n\\!\\int\\! \\frac{\\dd^4 \\ellb\\;\\del(p_1 \\cdot \\ellb)\\del(p_2 \\cdot \\ellb) }\n{\\ellb^2 (\\ellb - \\qb)^2} + \\Ord(1\/\\hbar)\\,,\n\\end{aligned}\n\\label{eqn:BoxSingular}\n\\end{equation}\nwhere we have averaged over equivalent forms, and then used \\eqn~\\eqref{eqn:deltaPoles} a second time in obtaining the last line. At the very end, we took $D\\rightarrow 4$.\n\nCombining \\eqns{eqn:CutSingular}{eqn:BoxSingular}, we find that the potentially singular contributions to the impulse kernel in the classical limit are\n\\begin{equation}\n\\begin{aligned}\n\\impKerCl &\\big|_\\textrm{singular} =\n\\hbar\\qb^\\mu\\, \\mathcal{C}\\!\\left(\\boxy \\right) (B_{-1}+C_{-1}) +\\hbar\\, \\mathcal{C}\\!\\left(\\boxy \\right) \\cutbox^\\mu_{-1} \n\\\\&\\hspace*{-6mm} = \\frac{2i}{\\hbar} (p_1 \\cdot p_2)^2\\qb^\\mu \\left(\\newT_1\\cdot \\newT_2 \\right)^2  \\Bigg[\n\\int \\frac{\\dd^4 \\ellb\\;\\del(p_1 \\cdot \\ellb)\\del(p_2 \\cdot \\ellb) }\n{\\ellb^2 (\\ellb - \\qb)^2}\n\\\\ &\\hspace{50mm} - \\!\\int \\dd^4 \\ellb \\;\\frac{\\del(p_1 \\cdot \\ellb) \\del(p_2 \\cdot \\ellb)}\n{\\ellb^2 (\\ellb - \\qb)^2}\\Bigg]  +\\Ord(\\hbar^0)\n\\\\&\\hspace*{-6mm}=\\Ord(\\hbar^0)\\,.\n\\end{aligned}\n\\label{CombiningBoxes}\n\\end{equation}\nSince all terms have common colour factors the dangerous terms cancel, leaving only well-defined contributions.\n\n\\defZ{Z}\nIt remains to extract the $\\Ord(1\/\\hbar)$ terms from the box, crossed box, and cut box contributions, and to combine them with the triangles~(\\ref{eqn:TriangleContribution}), which are of this order.  In addition to $B_0$ from \\eqn~\\eqref{eqn:BoxExpansion}, $C_0$ from \\eqn~\\eqref{eqn:CrossedBoxExpansion}, and $\\cutbox^\\mu_0$ from \\eqn~\\eqref{eqn:CutBoxExpansion}, we must also include the $\\Ord(1\/\\hbar)$ terms left implicit in \\eqns{eqn:CutSingular}{eqn:BoxSingular}.  In the former contributions, we can now set $p_\\alpha\\cdot \\qb = 0$, as the $\\hbar$ terms in the on-shell delta functions would give rise to contributions of $\\Ord(\\hbar^0)$ to the impulse kernel, which in turn will give contributions of $\\Ord(\\hbar)$ to the impulse.  In combining all these terms, we make use of summing over an expression and the expression after the linear change of variables;\nthe identity~(\\ref{eqn:deltaPoles}); and the identity\n\\begin{equation}\n\\del'(x) = \\frac{i}{(x-i\\epsilon)^2} - \\frac{i}{(x+i\\epsilon)^2}\\,.\n\\end{equation}\nOne finds that\n\\begin{equation}\n\\begin{aligned}\n&\\hbar\\qb^\\mu (B_0 + C_0) +\\bigl[\\hbar\\qb^\\mu (B_{-1} + C_{-1})\\bigr]\\big|_{\\Ord(\\hbar^0)}\n= Z^\\mu\n\\\\& + 2  (p_1 \\cdot p_2)^2 \\qb^\\mu\n\\!\\int \\!\\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\biggl(\\del(p_2 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(p_1 \\td \\ellb \\tp i \\epsilon )^2} \n+ \\del(p_1 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(p_2 \\td \\ellb \\tm i \\epsilon )^2}\\biggr) \n\\,,\n\\\\ &\\hbar\\cutbox^\\mu_0 +\\bigl[\\hbar\\cutbox^\\mu_{-1}\\bigr]\\big|_{\\Ord(\\hbar^0)}\n= - Z^\\mu \n\\\\& -\\!2 i (p_1 \\cdot p_2)^2 \n\\! \\int\\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\ellb^\\mu \\, \\ellb \\td (\\ellb \\tm \\qb)\\,\n\\bigl( \\del^\\prime(p_1 \\td \\ellb) \\del(p_2 \\td \\ellb) - \\del^\\prime(p_2 \\td \\ellb)\n\\del(p_1 \\td \\ellb)\\bigr) \\,,\n\\end{aligned}\n\\end{equation}\nwhere we have now taken $D\\rightarrow4$, and where the quantity $Z^\\mu$ is\n\\begin{multline}\nZ^\\mu = i (p_1 \\cdot p_2)^2 \\qb^\\mu \n\\!\\int \\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2} \n\\;(2 \\ellb \\cdot \\qb - \\ellb^2 )\n\\bigl( \\del^\\prime(p_1 \\td \\ellb) \\del(p_2 \\td \\ellb) \\\\ - \\del^\\prime(p_2 \\td \\ellb)\n\\del(p_1 \\td \\ellb)\\bigr) \\, .\n\\end{multline}\n\nFinally, we integrate over the external wavefunctions. The possible singularity in $\\hbar$ has cancelled, so as discussed in section~\\ref{subsec:Wavefunctions}, we perform the integrals by replacing the momenta $p_\\alpha$ with their classical values $m_\\alpha \\ucl_\\alpha$, and replace the quantum colour factors with classical colour charges. The box-derived contribution is therefore\n\\begin{multline}\n\\impKerTerm2 \\equiv\n2 (c_1\\cdot c_2)^2 \\gamma^2 \\qb^\\mu\n\\! \\int \\!\\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\biggl(m_2\\del(\\ucl_2 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(\\ucl_1 \\td \\ellb \\tp i \\epsilon )^2} \n\\\\ + m_1\\del(\\ucl_1 \\td \\ellb) \n\\frac{\\ellb \\td (\\ellb \\tm\\qb) }{(\\ucl_2 \\td \\ellb \\tm i \\epsilon )^2}\\biggr) \\,,\n\\end{multline}\nwhile that from the cut box is\n\\begin{multline}\n\\impKerTerm3 \\equiv\n-2 i (c_1\\cdot c_2)^2 \\, \\gamma^2 \n\\! \\int\\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb \\tm \\qb)^2}  \n\\ellb^\\mu \\, \\ellb \\td (\\ellb \\tm \\qb)\\\\\n\\times\\Big( m_2\\del^\\prime(\\ucl_1 \\td \\ellb)  \\del(\\ucl_2 \\td \\ellb) - m_1\\del^\\prime(\\ucl_2 \\td \\ellb) \\del(\\ucl_1 \\td \\ellb)\\Big)\\,.\n\\end{multline}\nIn both contributions we have dropped the $Z^\\mu$ term which cancels\nbetween the two. The full impulse kernel is given by the sum $\\impKerTerm1+\\impKerTerm2+\\impKerTerm3$, and the impulse by\n\\begin{equation}\n\\begin{aligned}\n\\DeltaPnlo &= \\frac{i g^4}{4}\\hbar\\! \\int \\! \\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \ne^{-i \\qb\\cdot b} \n\\left( \\impKerTerm1 + \\impKerTerm2 + \\impKerTerm3 \\right)\n\\\\&= \\frac{ig^4}{2} (c_1\\cdot c_2)^2\\!\n\\int \\! \\frac{\\dd^4 \\ellb}{\\ellb^2 (\\ellb - \\qb)^2}\\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) e^{-i \\qb\\cdot b}\n\\\\&\\hphantom{=}\\times\\biggl[\n\\qb^\\mu \\biggl( \\frac{\\del(\\ucl_1 \\cdot \\ellb)}{m_2}\n+ \\frac{\\del(\\ucl_2 \\cdot \\ellb)}{m_1} \\biggr)\n\\\\&\\hphantom{=} \\hphantom{\\times\\biggl[}\n+\\gamma^2\\qb^\\mu \\biggl(\\frac{\\del(\\ucl_2 \\td \\ellb)}{m_1}\n\\frac{\\ellb \\td (\\ellb -\\qb) }{(\\ucl_1 \\td \\ellb + i \\epsilon )^2} \n+ \\frac{\\del(\\ucl_1 \\td \\ellb)}{m_2}\n\\frac{\\ellb \\td (\\ellb -\\qb) }{(\\ucl_2 \\td \\ellb - i \\epsilon )^2}\\biggr) \n\\\\&\\hphantom{=} \\hphantom{\\times\\biggl[} -\ni \\gamma^2\\ellb^\\mu \\, \\ellb \\td (\\ellb - \\qb)\\,\n\\biggl( \\frac{\\del^\\prime(\\ucl_1 \\td \\ellb) \\del(\\ucl_2 \\td \\ellb)}{m_1}\n- \\frac{\\del^\\prime(\\ucl_2 \\td \\ellb) \\del(\\ucl_1 \\td \\ellb)}{m_2}\\biggr)\\biggr]\\,.\n\\end{aligned}\n\\label{eqn:NLOImpulse}\n\\end{equation}\nIt was shown in \\cite{delaCruz:2020bbn} that this result is precisely reproduced by applying worldline perturbation theory to iteratively solve the Yang--Mills--Wong equations in equation~\\eqref{eqn:classicalWong}. Moreover, our the final result for the impulse in non-Abelian gauge theory is in fact identical to QED \\cite{Kosower:2018adc} (in which context this calculation was first performed), but with the charge to colour replacement $Q_1 Q_2 \\rightarrow c_1 \\cdot c_2$. This is a little peculiar, as it is natural to expect the non-linearity of the Yang--Mills field to enter at this order (and it does so in the quantum theory). The origin of the result is the colour basis decomposition in equation~\\eqref{eqn:1loopDecomposition}, and in particular the fact that the non-Abelian triangle diagrams only contribute to the $\\hbar^2$ suppressed second colour structure.\n\nWith the relevant Yang--Mills amplitude at hand, one may of course wonder about the prospect of double copying to obtain the NLO impulse in gravity. The construction of colour-kinematics dual numerators at loop level following our methods is highly non-trivial; however, recent progress with massive particles may now make this problem tractable \\cite{Carrasco:2015iwa}. It is also interesting to compare our methods to those of Shen~\\cite{Shen:2018ebu}, who implemented the double copy at NLO wholly within the classical worldline formalism following ground-breaking work of Goldberger and Ridgway~\\cite{Goldberger:2016iau}. Shen found it necessary to include vanishing terms involving structure constants in his work. Similarly, in our context, some colour factors are paired with kinematic numerators proportional to $\\hbar$. It would be interesting to use the tools developed in these chapters to explore the double copy construction of Shen~\\cite{Shen:2018ebu} from the perspective of amplitudes.\n\nThe agreement of~\\eqref{eqn:NLOImpulse} with worldline perturbation theory offers a strong check on our formalism, and is of greater importance than the evaluation of the remaining integrals, which also arise in the classical theory. Their evaluation is surprisingly intricate; however, we can gain some interesting insights into the physics just from considering momentum conservation.\n\n\\subsubsection{On-Shell Cross Check}\n\nAs we have seen, careful inclusion of boxes, crossed boxes as well as cut boxes are necessary to determine the impulse in the classical regime. This may seem to be at odds with other work on the classical limit of amplitudes, which often emphasises the particular importance of triangle diagrams to the classical potential at next to leading order. However, in the context of the potential, the partial cancellation between boxes and crossed boxes is well-understood~\\cite{Donoghue:1996mt}, and it is because of this fact that triangle diagrams are particularly important. The residual phase is known to exponentiate so that it does not effect classical physics. Meanwhile, the relevance of the subtraction of iterated (cut) diagrams has long been a topic of discussion~\\cite{Sucher:1994qe,BjerrumBohr:2002ks,Neill:2013wsa}.\n\nNevertheless, in the case of the impulse it may seem that the various boxes play a more significant role, as they certainly contribute to the classical result for the impulse. In fact, it is easy to see that these terms must be included to recover a physically sensible result. The key observation is that the final momentum, $\\finalk_1^\\mu$, of the outgoing particle after a classical scattering process must be on shell, $\\finalk_1^2 = m_1^2$.\n\nWe may express the final momentum in terms of the initial momentum and the impulse, so that\n\\begin{equation}\n\\finalk_1^\\mu = p_1^\\mu + \\Delta p_1^\\mu\\,.\n\\end{equation}\nThe on-shell condition is then\n\\begin{equation}\n(\\Delta p_1)^2 + 2 p_1 \\cdot \\Delta p_1 = 0\\,.\n\\end{equation}\nAt order $g^2$, this requirement is satisfied trivially. At this order $(\\Delta p_1)^2$ is negligible, while\n\\begin{equation}\np_1 \\cdot \\Delta p_1 = i m_1 g^2 c_1\\cdot c_2 \\!\\int \\! \\dd^4 \\qb \\,\n\\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \n\\, e^{-i \\qb \\cdot b} \\, \\qb \\cdot \\ucl_1 \\frac{\\ucl_1 \\cdot \\ucl_2 }{\\qb^2} = 0\\,,\n\\end{equation}\nusing our result for the LO impulse in \\eqn~\\eqref{eqn:impulseClassicalLO}.\n\nThe situation is less trivial at order $g^4$, as neither $p_1 \\cdot \\Delta p_1$ nor $(\\Delta p_1)^2$ vanish. In fact, at this order we may use \\eqn~\\eqref{eqn:impulseClassicalLO} once again to find that\n\\begin{multline}\n(\\Delta p_1)^2 = - g^4  (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n\\times \\int \\! \\dd^4 \\qb \\,\\dd^4 \\qb' \\, \\del(\\qb \\cdot \\ucl_1) \n\\del(\\qb \\cdot \\ucl_2) \\del(\\qb' \\cdot \\ucl_1) \\del(\\qb' \\cdot \\ucl_2) \n\\, e^{-i (\\qb + \\qb') \\cdot b} \\,  \\frac{\\qb \\cdot \\qb'}{\\qb^2 \\, \\qb'^2}\\,.\n\\label{eqn:DeltaPsquared}\n\\end{multline}\nMeanwhile, to evaluate $p_1 \\cdot \\Delta p_1$ we must turn to our NLO result for the impulse, \\eqn~\\eqref{eqn:NLOImpulse}. Thanks to the delta functions present in the impulse, we \nfind a simple expression:\n\\begin{multline}\n2 p_1 \\cdot \\Delta p_1 = g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2\\! \\int \\! \\dd^4 \\qb \\, \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \\, e^{-i \\qb\\cdot b} \\\\\n\\times \\int \\! \\dd^4 \\ellb\\; \\ellb \\cdot \\ucl_1 \\, \\del'(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2) \\, \\frac{\\ellb \\cdot (\\ellb - \\qb)}{\\ellb^2 (\\ellb - \\qb)^2}\\,.\n\\label{eqn:pDotDeltaP}\n\\end{multline}\nTo simplify this expression, it may be helpful to imagine working in the restframe of the timelike vector $u_1$. Then, the $\\ellb$ integral involves the distribution $\\ellb_0 \\, \\del'(\\ellb_0)$, while $\\qb_0 = 0$. Thus the $\\ellb_0$ integral has the form\n\\begin{equation}\n\\int \\! \\dd \\ellb_0 \\, \\ellb_0 \\, \\del'(\\ellb_0) \\, f(\\ellb_0{}^2) = -\\!\\int \\! \\dd \\ellb_0 \\, \\del(\\ellb_0) \\, f(\\ellb_0{}^2)\\,.\n\\end{equation}\nUsing this observation, we may simplify equation~\\eqref{eqn:pDotDeltaP} to find\n\\begin{equation}\n\\begin{aligned}\n\\vspace{-2mm}2 p_1 \\cdot \\Delta p_1 &= -g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n& \\,\\,\\, \\times \n\\int \\! \\dd^4 \\qb\\, \\dd^4 \\ellb \\; \\del(\\qb \\cdot \\ucl_1) \\del(\\qb \\cdot \\ucl_2) \n\\del(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2)  \ne^{-i \\qb\\cdot b}  \\frac{\\ellb \\cdot (\\ellb - \\qb)}{\\ellb^2 (\\ellb - \\qb)^2}  \\\\\n&=  g^4 (c_1\\cdot c_2)^2 \\, (\\ucl_1 \\cdot \\ucl_2)^2 \\\\\n& \\,\\,\\,\\times\n\\int \\! \\dd^4 \\ellb \\, \\dd^4 \\qb' \\;\n\\del(\\ellb \\cdot \\ucl_1) \\del(\\ellb \\cdot \\ucl_2) \\del(\\qb' \\cdot \\ucl_1) \n\\del(\\qb' \\cdot \\ucl_2) \\, e^{-i (\\ellb+\\qb')\\cdot b} \n\\frac{\\ellb \\cdot \\qb'}{\\ellb^2\\, \\qb'^2}\\,,\n\\end{aligned}\n\\end{equation}\nwhere in the last line we set $\\qb' = \\qb - \\ellb$. This expression is equal but opposite to \\eqn~\\eqref{eqn:DeltaPsquared}, and so the final momentum is on-shell as it must be.\n\nIt is worth remarking that the part of the NLO impulse that is relevant in this cancellation arises solely from the cut boxes. One can therefore view this phenomenon as an analogue of the removal of iterations of the tree in the potential.\n\n\\section{Beyond next-to-leading-order}\n\\label{sec:NNLO}\n\nWe have worked in this chapter under the premise of studying conservative scattering. Yet the LO and NLO impulse are only conservative in the sense that momentum is simply exchanged from particle 1 to particle 2 at these orders. However, beyond these lowest orders in perturbation theory physics does not clearly distinguish between conservative and dissipative behaviour: we will see shortly that at NNLO momentum can be radiated away, and moreover back-reacts on the impulse. To complete our on-shell formalism we must therefore incorporate radiation --- the interplay between the impulse and the radiated momentum forms the subject of our next chapter.\n\\chapter{Radiation: emission and reaction}\n\\label{chap:radiation}\n\n\\section{Introduction}\n\nGravitational wave astronomy relies on extracting measurable data from radiation. In this chapter we will therefore apply the methods developed for the impulse to construct a second on-shell and quantum-mechanical observable, the total emitted radiation.\n\nThese two observables are not independent. Indeed, the relation between them goes to the heart of one of the difficulties in traditional approaches to classical field theory with point sources. In two-particle scattering in classical electrodynamics, for example, momentum is transferred from one particle to the other via the electromagnetic field, as described by the Lorentz force. But the energy-momentum lost by point-particles to radiation is not accounted for by the Lorentz force. Conservation of momentum is restored by taking into account an additional force, the Abraham--Lorentz--Dirac (ALD) force~\\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz,LandauLifshitz}; see e.g. refs.~\\cite{Higuchi:2002qc,Galley:2006gs,Galley:2010es,Birnholtz:2013nta,Birnholtz:2014fwa,Birnholtz:2014gna} for more recent treatments. Inclusion of this radiation reaction force is not without cost: rather, it leads to the celebrated issues of runaway solutions or causality violations in the classical electrodynamics of point sources.\n\nUsing quantum mechanics to describe charged-particle scattering in should cure these ills. Indeed, we will see explicitly that a quantum-mechanical description will conserve energy and momentum in particle scattering automatically. First, in section~\\ref{sec:radiatedmomentum} we will set up expressions for the total radiated momentum in quantum field theory, and show that when combined with the impulse of the previous chapter, momentum is automatically conserved to all orders in perturbation theory. We will apply our previous investigation of the classical limit in section~\\ref{sec:classicalradiation}, introducing the radiation kernel and discussing how it relates to objects familiar from classical field theory. In section~\\ref{sec:LOradiation} we explicitly compute the radiation kernel at leading order in gauge and gravitational theories, and using its form in QED explicitly show that our impulse formalism from chapter~\\ref{chap:impulse} reproduces the predictions of the classical Abraham--Lorentz--Dirac force. We discuss the results of this and the previous chapter in section~\\ref{sec:KMOCdiscussion}.\n\nThis chapter continues to be based on work published in refs.~\\cite{Kosower:2018adc,delaCruz:2020bbn}.\n\n\\section{The momentum radiated during a collision}\n\\label{sec:radiatedmomentum}\n\nA familiar classical observable is the energy radiated by an accelerating particle, for example during a scattering process. More generally we can compute the four-momentum radiated. In quantum mechanics there is no precise prediction for the energy or the momentum radiated by localised particles; we obtain a continuous spectrum if we measure a large number of events. However we can compute the expectation value of the four-momentum radiated during a scattering process. This is a well-defined observable, and as we will see it is on-shell in the sense that it can be expressed in terms of on-shell amplitudes.\n\nTo define the observable, let us again surround the collision with detectors which measure outgoing radiation of some type. We will call the radiated quanta `messengers'. Let $\\mathbb{K}^\\mu$ be the momentum operator for whatever field is radiated; then the expectation of the radiated momentum is\n\\begin{equation}\n\\begin{aligned}\n\\langle k^\\mu \\rangle = {}_\\textrm{out}{\\langle} \\Psi |  \\mathbb{K}^\\mu S \\, | \\Psi \\rangle_\\textrm{in} = {}_\\textrm{in}{\\langle} \\Psi | \\, S^\\dagger   \\mathbb{K}^\\mu S\\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $|\\Psi\\rangle_\\textrm{in}$ is again taken as the wavepacket in equation~\\eqref{eqn:inState}. Once again we can anticipate that the radiation will be expressed in terms of amplitudes. Rewriting $S = 1 + i T$, the expectation value becomes\n\\begin{align}\n\\Rad^\\mu \\equiv \\langle k^\\mu \\rangle &= {}_\\textrm{in}{\\langle} \\Psi | \\, S^\\dagger   \\mathbb{K}^\\mu S \\, | \\Psi \\rangle_\\textrm{in}\n= {}_\\textrm{in}\\langle \\Psi | \\, T^\\dagger   \\mathbb{K}^\\mu T \\, | \\Psi \\rangle_\\textrm{in}\\,,\n\\end{align}\nbecause $ \\mathbb{K}^\\mu |\\Psi \\rangle_\\textrm{in} = 0$ since there are no quanta of radiation in the incoming state. \n\nWe can insert a complete set of states $|X ; k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2\\rangle$ containing at least one radiated messenger of momentum $k$, and write the expectation value of the radiated momentum as follows:\n\\begin{equation}\n\\begin{aligned}\n\\hspace{-2mm}\\Rad^\\mu = \\sum_X \\int\\! \\df(k) \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\nk_X^\\mu \\bigl|\\langle k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\,| \\, T \\,\n| \\Psi \\rangle\\bigr|^2\\,.\n\\end{aligned}\n\\label{eqn:radiationTform}\n\\end{equation}\nIn this expression, $X$ can again be empty, and $k_X^\\mu$ is the sum of the explicit messenger momentum $k^\\mu$ and the momenta of any messengers in the state $X$. Notice that we are including explicit integrals for particles 1 and 2, consistent with our assumption that the number of these particles is conserved during the process. The state $| k \\rangle$ describes a radiated messenger; the phase space integral over $k$ implicitly includes a sum over its helicity.\n\nExpanding the initial state, we find that the expectation value of the radiated momentum is given by\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu &= \\sum_X \\int\\! \\df(k) \\df(\\finalk_1) \\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\,\nk_X^\\mu \\\\ \n& \\times\\bigg| \\int\\! \\df(\\initialk_1)\\df(\\initialk_2)  e^{i b \\cdot \\initialk_1\/\\hbar} \\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \\del^{(4)}(\\initialk_1 + \\initialk_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X) \\\\\n& \\hspace{40mm} \\times \\langle \\zeta_1 \\, \\zeta_2 \\, X|\\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X) |\\chi_1\\, \\chi_2 \\rangle \\bigg|^2\n\\,,\n\\label{eqn:ExpectedMomentum}\n\\end{aligned}\n\\end{equation}\nwhere we have accounted for any representation states in $X$, with the appropriate Haar measure implicity contained in the external sum. We can again introduce momentum transfers, $q_\\alpha=\\initialkc_\\alpha-\\initialk_\\alpha$, and trade the integrals over $\\initialkc_\\alpha$ for integrals over the $q_\\alpha$.  One of the four-fold $\\delta$ functions will again become $\\del^{(4)}(q_1+q_2)$, and we can use it to perform the $q_2$ integrations.  We again relabel $q_1\\rightarrow q$. The integration leaves behind a pair of on-shell $\\delta$ functions and positive-energy $\\Theta$ functions, just as in \\eqns{eqn:impulseGeneralTerm1}{eqn:impulseGeneralTerm2}:\n\\begin{equation}\n\\begin{aligned}\n\\hspace{-8pt}\\Rad^\\mu =&\\, \\sum_X \\int\\! \\df(k) \\prod_{\\alpha=1,2}\\df(\\finalk_\\alpha) \\df(\\initialk_\\alpha) \\dd^4 q\\;\n\\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \n\\varphi_1^*(\\initialk_1+q) \\varphi_2^*(\\initialk_2-q) \\,\n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1{}^0+q^0)\\Theta(\\initialk_2{}^0-q^0)\n\\\\&\\times \nk_X^\\mu \\, e^{-i b \\cdot q\/\\hbar} \n\\,\\del^{(4)}(\\initialk_1 + \\initialk_2 \n- \\finalk_1 - \\finalk_2 - k - \\finalk_X)%\n\\\\&\\times \n\\langle \\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X)\n \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \\finalk_1\\,, \\finalk_2\\,, k\\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2b}\n\\end{equation}\nRepresentation states have been absorbed into an expectation value as in equation~\\eqref{eqn:defOfAmplitude}. We emphasise that this is an all-orders expression: the amplitude $\\Ampl(\\initialk_1,\\squeeze \\initialk_2 \\squeeze\\rightarrow \\squeeze \\finalk_1,\\squeeze \\finalk_2,\\squeeze k,\\squeeze \\finalk_X)$ includes all loop corrections, though of course it can be expanded in perturbation theory. The corresponding real-emission contributions are present in the sum over states $X$.   If we truncate the amplitude at a fixed order in perturbation theory, we should similarly truncate the sum over states. Given that the expectation value is expressed in terms of an on-shell amplitude, it is also appropriate to regard this observable as a fully on-shell quantity.\n\nIt can be useful to represent the observables diagrammatically. Two equivalent expressions for the radiated momentum are helpful:\n\\begin{multline}\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{positioning}\n\\Rad^\\mu =\n\\sum_X  \\mathlarger{\\int}\\! \\df(k)\\df(\\finalk_1)\\df(\\finalk_2)\\;\nk_X^\\mu \n\\\\ \\times\\left| \\mathlarger{\\int}\\! \\df(\\initialk_1) \\df(\\initialk_2)\\; \ne^{i b \\cdot \\initialk_1\/\\hbar} \\, \n\\del^{(4)}\\!\\left(\\sum p\\right) \\hspace*{-13mm}\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([xshift=-5cm,yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\vertex (b) ;\n\\vertex [above left=of b] (i1) {$\\psi_1(\\initialk_1)$};\n\\vertex [above right=of b] (o1) {$\\finalk_1$};\n\\vertex [above right =0.2 and 1.4 of b] (k) {$k$};\n\\vertex [below right =0.2 and 1.4 of b] (X) {$\\finalk_X$};\n\\vertex [below left=1 and 1 of b] (i2) {${\\psi_2(\\initialk_2)}$};\n\\vertex [below right=1 and 1 of b] (o2) {$\\finalk_2$};\n\\diagram* {(b) -- [photon, photonRed] (k)};\n\\diagram*{(b) -- [boson] (X)};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (b) -- node [right=4pt] {}(o2);\n\\draw[postaction={decorate}] (b) -- node [left=4pt] {} (o1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.42 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (i1) -- node [left=4pt] {} (b);\n\\draw[postaction={decorate}] (i2) --node [right=4pt] {} (b);\n\\end{scope}\t\n\\filldraw [color=white] (b) circle [radius=10pt];\n\\filldraw [fill=allOrderBlue] (b) circle [radius=10pt];\t\n\\end{feynman}\n\\end{tikzpicture}\n\\right|^2,\n\\label{eqn:RadiationPerfectSquare}\n\\end{multline}\nwhich is a direct pictorial interpretation of equation~\\eqref{eqn:ExpectedMomentum}, and\n\\begin{equation}\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{positioning}\n\\begin{aligned}\n\\Rad^\\mu &=\n\\sum_X \\mathlarger{\\int}\\! \\df(k) \\prod_{\\alpha = 1, 2} \\df(\\finalk_\\alpha)\n \\df(\\initialk_\\alpha) \\df(\\initialkc_\\alpha)\\; k_X^\\mu  \\, e^{i b \\cdot (\\initialk_1 - \\initialkc_1)\/\\hbar}   \\\\\n& \\times \\del^{(4)}(\\initialk_1 + \\initialk_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X)\\, \\del^{(4)}(\\initialkc_1 + \\initialkc_2 - \\finalk_1 - \\finalk_2 - k - \\finalk_X) \\\\\n& \\hspace{50mm}\\times\n\\begin{tikzpicture}[scale=1.0, \nbaseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax]\n\tcurrent bounding box.center)},\n] \n\\begin{feynman}\n\\begin{scope}\n\\vertex (ip1) ;\n\\vertex [right=2 of ip1] (ip2);\n\\node [] (X) at ($ (ip1)!.5!(ip2) $) {};\n\\begin{scope}[even odd rule]\n\\begin{pgfinterruptboundingbox}\n\\path[invclip] ($  (X) - (4pt, 30pt) $) rectangle ($ (X) + (4pt,30pt) $) ;\n\\end{pgfinterruptboundingbox} \n\n\\vertex [above left=0.66 and 0.33 of ip1] (q1) {$ \\psi_1(\\initialk_1)$};\n\\vertex [above right=0.66 and 0.33 of ip2] (qp1) {$ \\psi^*_1(\\initialkc_1)$};\n\\vertex [below left=0.66 and 0.33 of ip1] (q2) {$ \\psi_2(\\initialk_2)$};\n\\vertex [below right=0.66 and 0.33 of ip2] (qp2) {$ \\psi^*_2(\\initialkc_2)$};\n\n\\diagram* {(ip1) -- [photon, out=30, in=150, photonRed]  (ip2)};\n\\diagram*{(ip1) -- [photon, out=330, in=210]  (ip2)};\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.4 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (q1) -- (ip1);\n\\draw[postaction={decorate}] (q2) -- (ip1);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.7 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip2) -- (qp1);\n\\draw[postaction={decorate}] (ip2) -- (qp2);\n\\end{scope}\n\\begin{scope}[decoration={\n\tmarkings,\n\tmark=at position 0.38 with {\\arrow{Stealth}},\n\tmark=at position 0.74 with {\\arrow{Stealth}}}] \n\\draw[postaction={decorate}] (ip1) to [out=90, in=90,looseness=1.7] node[above left] {{$ \\finalk_1$}} (ip2);\n\\draw[postaction={decorate}] (ip1) to [out=270, in=270,looseness=1.7]node[below left] {${\\finalk_2}$} (ip2);\n\\end{scope}\n\n\\node [] (Y) at ($(X) + (0,1.5)$) {};\n\\node [] (Z) at ($(X) - (0,1.5)$) {};\n\\node [] (k) at ($ (X) - (0.35,-0.55) $) {$k$};\n\\node [] (x) at ($ (X) - (0.35,0.55) $) {$\\finalk_X$};\n\n\\filldraw [color=white] ($ (ip1)$) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip1) $) circle [radius=8pt];\n\n\\filldraw [color=white] ($ (ip2) $) circle [radius=8pt];\n\\filldraw  [fill=allOrderBlue] ($ (ip2) $) circle [radius=8pt];\n\n\\end{scope} \n\\end{scope}\n\\draw [dashed] (Y) to (Z);\n\\end{feynman}\n\\end{tikzpicture},\n\\end{aligned}\n\\end{equation}\nwhich demonstrates that we can think of the expectation value as the weighted cut of a loop amplitude. As $X$ can be empty, the lowest-order contribution arises from the weighted cut of a two-loop amplitude.\n\n\\subsection{Conservation of momentum}\n\\label{sect:allOrderConservation}\n\nThe expectation of the radiated momentum is not independent of the impulse. In fact the relation between these quantities is physically rich. In the classical electrodynamics of point~particles, for example, the impulse is due to a total time integral of the usual Lorentz force,~\\eqref{eqn:Wong-momentum}. However, when the particles emit radiation the point-particle approximation leads to well-known issues. This is a celebrated problem in classical field theory.  Problems arise because of the singular nature of the point-particle source. In particular, the electromagnetic field at the position of a point charge is infinite, so to make sense of the Lorentz force acting on the particle the traditional route is to subtract the particle's own field from the full electromagnetic field in the force law. The result is a well-defined force, but conservation of momentum is lost.\n\nConservation of momentum is restored by including another force, the Abraham--Lorentz--Dirac (ALD) force~\\cite{Lorentz,Abraham:1903,Abraham:1904a,Abraham:1904b,Dirac:1938nz}, acting on the particles. This gives rise to an impulse on particle 1 in addition to the impulse due to the Lorentz force. The Lorentz force exchanges momentum between particles 1 and 2, while the radiation reaction impulse,\n\\begin{equation}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^2 Q_1^2}{6\\pi m_1}\\int_{-\\infty}^\\infty\\! d\\tau \\left(\\frac{d^2p_1^\\mu}{d\\tau^2} + \\frac{p_1^\\mu}{m_1^2}\\frac{dp_1}{d\\tau}\\cdot\\frac{dp_1}{d\\tau}\\right),\n\\label{eqn:ALDclass}\n\\end{equation}\naccounts for the irreversible loss of momentum due to radiation. Of course, the ALD force is a notably subtle issue in the classical theory.\n\nIn the quantum theory of electrodynamics there can be no question of violating conservation of momentum, so the quantum observables we have defined must already include all the effects which would classically be attributed to both the Lorentz and ALD forces. This must also hold for the counterparts of these forces in any other theory. In particular, it must be the case that our definitions respect conservation of momentum; it is easy to demonstrate this formally to all orders using our definitions. Later, in section~\\ref{sec:ALD}, we will indicate how the radiation reaction is included in the impulse more explicitly.\n\nOur scattering processes involve two incoming particles. Consider, then,\n\\begin{equation}\n\\begin{aligned}\n\\langle \\Delta p_1^\\mu \\rangle + \\langle \\Delta p_2^\\mu \\rangle &= \n\\langle \\Psi | i [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle \n+ \\langle \\Psi | T^\\dagger [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle \\\\\n&= \\bigl\\langle \\Psi \\big| i \\bigl[ \\textstyle{\\sum_\\alpha}   \\mathbb{P}_\\alpha^\\mu, T \\bigr] \n\\big| \\Psi \\bigr\\rangle \n+ \\langle \\Psi | T^\\dagger [  \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle\\,,\n\\end{aligned}\n\\end{equation}\nwhere the sum $\\sum   \\mathbb{P}_\\alpha^\\mu$ is now over all momentum operators in the theory, not just those for the two initial particles. The second equality above holds because $  \\mathbb{P}_\\alpha^\\mu | \\Psi \\rangle = 0$ for $\\alpha \\neq 1,2$; only quanta of fields 1 and 2 are present in the incoming state. Next, we use the fact that the total momentum is time independent, or in other words\n\\begin{equation}\n\\Bigl[ \\sum  \\mathbb{P}_\\alpha^\\mu, T \\Bigr] = 0\\,,\n\\end{equation}\nwhere the sum extends over all fields. Consequently,\n\\begin{equation}\n\\langle \\Psi | i [   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu, T ] | \\Psi \\rangle =  \n\\bigl\\langle \\Psi \\big| i \\bigl[ \\textstyle{\\sum_\\alpha}   \\mathbb{P}_\\alpha^\\mu, T \\bigr] \\big| \n\\Psi \\bigr\\rangle = 0\\,.\n\\label{eqn:commutatorVanishes}\n\\end{equation}\nThus the first term $\\langle \\Psi | i [   \\mathbb{P}_1^\\mu, T ] | \\Psi \\rangle$ in the impulse~(\\ref{eqn:defl1}) describes only the exchange of momentum between particles~1 and~2; in this sense it is associated with the classical Lorentz force (which shares this property) rather than with the classical ALD force (which does not). The second term in the impulse, on the other hand, includes radiation. To make the situation as clear as possible, let us restrict attention to the case where the only other momentum operator is $  \\mathbb{K}^\\mu$, the momentum operator for the messenger field. Then we know that $[   \\mathbb{P}_1^\\mu +   \\mathbb{P}_2^\\mu +   \\mathbb{K}^\\mu, T] = 0$, and conservation of momentum at the level of expectation values is easy to demonstrate:\n\\begin{equation}\n\\langle \\Delta p_1^\\mu \\rangle + \\langle \\Delta p_2^\\mu \\rangle = \n- \\langle \\Psi | T^\\dagger [   \\mathbb{K}^\\mu, T ] | \\Psi \\rangle = \n- \\langle \\Psi | T^\\dagger    \\mathbb{K}^\\mu T  | \\Psi \\rangle = \n- \\langle k^\\mu \\rangle = - \\Rad^\\mu\\,,\n\\end{equation}\nonce again using the fact that there are no messengers in the incoming state.\n\nIn the classical theory, radiation reaction is a subleading effect, entering for two-body scattering at order $e^6$ in perturbation theory in electrodynamics. This is also the case in the quantum theory. To see why, we again expand the operator product in the second term of \\eqn~\\eqref{eqn:defl1} using a complete set of states:\n\\begin{multline}\n\\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle = \\sum_X \\int \\! \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\n\\\\ \\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X \\rangle \n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X | [  \\mathbb{P}_1^\\mu, T] \\, |\\Psi \\rangle\\,.\n\\end{multline}\nThe sum over $X$ is over all states, including an implicit integral over their momenta and a sum over any other quantum numbers. The inserted-state momenta of particles 1 and~2 (necessarily present) are labeled by $\\finalk_\\alpha$, and the corresponding integrations over these momenta by $\\df(\\finalk_\\alpha)$.  These will ultimately become integrations over the final-state momenta in the scattering. To make the loss of momentum due to radiation explicit at this level, we note that\n\\begin{multline}\n\\langle \\Psi | \\, T^\\dagger [   \\mathbb{P}_1^\\mu +    \\mathbb{P}_2^\\mu, T] \\, |\\Psi \\rangle \n= -\\sum_X \\int \\! \\df(\\finalk_1)\\df(\\finalk_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\;\n\\\\\\times\\langle \\Psi | \\, T^\\dagger | \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X\\rangle  \n\\langle \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X | \\,   \\mathbb{P}_X^\\mu T \\,  |\\Psi \\rangle\\,,\n\\end{multline}\nwhere $ \\mathbb{P}_X$ is the sum over momentum operators of all quantum fields other than the scalars~1 and 2. The sum over all states $X$ will contain, for example, terms where the state $X$ includes messengers of momentum $k^\\mu$ along with other massless particles. We can further restrict attention to the contributions of the messenger's momentum to $\\mathbb{P}_X^\\mu$. This contribution produces a net change of momentum of particle 1 given by\n\\begin{multline}\n-\\sum_X \\int \\!  \\df(k) \\df(\\finalk_1)\\df(\\finalk_2)\\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\; k^\\mu \\, \n\\\\\\times\\langle \\Psi | \\, T^\\dagger | k ; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X\\rangle\n\\langle k; \\finalk_1 \\, \\zeta_1; \\finalk_2 \\, \\zeta_2; X| \\, T \\,  |\\Psi \\rangle \n= - \\langle k^\\mu \\rangle\\,,\n\\end{multline} \nwith the help of equation~\\eqref{eqn:radiationTform}. Thus we explicitly see the net loss of momentum due to radiating messengers. In any theory this quantity is suppressed by factors of the coupling $\\tilde g$ because of the additional state. The lowest order case corresponds to $X = \\emptyset$; as there are two quanta in $|\\psi \\rangle$, we must compute the modulus squared of a five-point tree amplitude. The term is proportional to $\\tilde g^6$, where $\\tilde g$ is the coupling of an elementary three-point amplitude; as far as the impulse is concerned, it is a next-to-next-to-leading order (NNLO) effect. Other particles in the state $X$, and other contributions to its momentum, describe higher-order effects.\n\n\\section{Classical radiation}\n\\label{sec:classicalradiation}\n\\defK^\\mu{K^\\mu}\n\nFollowing our intensive study of the classical limit of the impulse in the previous chapter, the avenue leading to the classical limit of $R^\\mu$ is clear: provided we work with the wavefunctions of chapter~\\ref{chap:pointParticles} in the the Goldilocks zone $\\ell_c \\ll \\ell_w \\ll \\lscatt$, we can simply adopt the rules of section~\\ref{sec:classicalLimit}. In particular the radiated momentum $k$ will scale as a wavenumber in the classical region. This is enforced by the energy-momentum-conserving delta function in \\eqn~\\eqref{eqn:ExpectedMomentum2b}, rewritten in terms of momentum transfers $w_\\alpha = r_\\alpha - p_\\alpha$:\n\\begin{equation}\n\\del^{(4)}(w_1+w_2 + k + \\finalk_X)\\,.\n\\end{equation}\nThe arguments given after equation~\\eqref{eqn:radiationScalingDeltaFunction} then ensure that the typical values of all momenta in the argument should again by scaled by $1\/\\hbar$ and replaced by wavenumbers.\n\nWith no new work required on the formalities of the classical limit, let us turn to explicit expressions for the classical radiated momentum in terms of amplitudes. Recall that our expressions for the total emitted radiation in section~\\ref{sec:radiatedmomentum} depended on $q$, which represents a momentum mismatch rather than a momentum transfer. However, we expect the momentum transfers to play an important role in the classical limit, and so it is convenient to change variables from the $r_\\alpha$ to make use of them:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu &= \\sum_X \\int\\! \\df(k) \\prod_{\\alpha=1,2} \\df(\\initialk_\\alpha) \\dd^4\\xfer_\\alpha\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\Theta(\\initialk_1{}^0+q^0)\\Theta(\\initialk_2{}^0-q^0)\\,\\varphi_1(\\initialk_1) \\varphi_2(\\initialk_2) \n\\\\&\\qquad\\times \\varphi_1^*(\\initialk_1+q) \\varphi_2^*(\\initialk_2-q) \\, k_X^\\mu \\, e^{-i b \\cdot q\/\\hbar}  \\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X)\n\\\\&\\qquad\\qquad\\times \n\\langle\\Ampl^*(\\initialk_1+q\\,, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\qquad\\times \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\rangle\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2}\n\\end{equation}\nWe can now recast this expression in the notation of \\eqn~\\eqref{eqn:angleBrackets}:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class &= \\sum_X\\, \\Lexp \\int\\! \\df(k) \\prod_{\\alpha=1,2} \\dd^4\\xfer_\\alpha\\,\\dd^4 q\\;\n\\del(2p_\\alpha\\cdot \\xfer_\\alpha+\\xfer_\\alpha^2)\\Theta(p_\\alpha^0+\\xfer_\\alpha^0)\\,  k_X^\\mu \n\\\\&\\times \n\\del(2\\initialk_1 \\cdot q + q^2) \\del(2 \\initialk_2 \\cdot q - q^2) \n\\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X) \\Theta(\\initialk_1{}^0+q^0)%\n\\\\& \\times \\Theta(\\initialk_2{}^0-q^0)\\, e^{-i b \\cdot q\/\\hbar} \\, \\Ampl^*(\\initialk_1+q, \\initialk_2-q \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\qquad\\times \n\\Ampl(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\\,\\Rexp\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2recast}\n\\end{equation}\nWe will determine the classical limit of this expression using precisely the same logic as in the preceding chapter. Let us again focus on the leading contribution, with $X=\\emptyset$.  Once again, rescale $q \\rightarrow \\hbar\\qb$, and drop the $q^2$ inside the on-shell delta functions. Here, remove an overall factor of $\\tilde g^6$ and accompanying $\\hbar$'s from the amplitude and its conjugate. In addition, rescale the momentum transfers $\\xfer\\rightarrow \\hbar\\xferb$ and the radiation momenta, $k\\rightarrow\\hbar\\wn k$. At leading order there is no sum, so there will be no hidden cancellations, and we may drop the $\\xfer_\\alpha^2$ inside the on-shell delta functions to obtain\n\\def\\bar k{\\bar k}\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class &= \n\\tilde g^6 \\Lexp \\hbar^4\\! \\int\\! \\df(\\bar k) \\prod_{\\alpha=1,2} \\dd^4\\xferb_\\alpha\\dd^4 \\qb\\, \\del(2\\xferb_\\alpha\\cdot p_\\alpha)\n\\del(2\\qb\\cdot p_1) \\del(2\\qb\\cdot p_2) \\, e^{-i b \\cdot \\qb}\n\\\\& \\qquad \\times \\bar k^\\mu \\, \\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb, \\initialk_2-\\hbar \\qb \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2\\,, \\hbar\\bar k)\n\\\\& \\qquad\\times \n\\AmplB^{(0)}(\\initialk_1, \\initialk_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2\\,, \\hbar\\bar k)\\,\\del^{(4)}(\\xferb_1+\\xferb_2+ \\bar k)\\,\\Rexp\n\\,.\n\\end{aligned}\n\\label{eqn:ExpectedMomentum2classicalLO}\n\\end{equation}\nWe will make use of this expression below to verify that momentum is conserved as expected.\n\nOne disadvantage of this expression for the leading order radiated momentum is that it is no longer in a form of an \nintegral over a perfect square, such as shown in \\eqn~\\eqref{eqn:RadiationPerfectSquare}. Nevertheless we can recast \\eqn~\\eqref{eqn:ExpectedMomentum2recast} in such a form.\nTo do so, perform a change of variable, including in the (momentum space) wavefunctions. To begin, it is helpful to write \\eqn~\\eqref{eqn:ExpectedMomentum2recast} as\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class =&\\, \\sum_X \\prod_{\\alpha=1,2}  \\int \\! \\df(\\initialk_\\alpha)\\,  |\\varphi_\\alpha(\\initialk_\\alpha)|^2 \\int \\! \\df(k) \\df( \\xfer_\\alpha+\\initialk_\\alpha) \\df(q_\\alpha+\\initialk_\\alpha) \\; \n\\\\& \\times \\del^{(4)}(\\xfer_1+\\xfer_2+ k+ \\finalk_X) \\del^{(4)}(q_1 + q_2) \\, e^{-i b \\cdot q_1\/\\hbar} \\, k_X^\\mu \\,   %\n\\\\&\\qquad\\times \n\\langle \\Ampl^*(\\initialk_1+q_1\\,, \\initialk_2 + q_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\times \\Ampl(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\xfer_1\\,, \\initialk_2+\\xfer_2\\,, k\\,, \\finalk_X)\n\\rangle\\,\n\\,.\n\\end{aligned}\n\\end{equation}\n\\def\\tilde\\initialk{\\tilde\\initialk}\n\\def\\tilde\\xfer{\\tilde\\xfer}\n\\def\\tilde q{\\tilde q}\n\\noindent We will now re-order the integration and perform a change of variables. Let us define $\\tilde\\initialk_\\alpha=\\initialk_\\alpha - \\tilde\\xfer_\\alpha$, $\\tilde q_\\alpha = q_\\alpha + \\tilde\\xfer_\\alpha$, and $\\tilde\\xfer_\\alpha = - \\xfer_\\alpha$, changing variables from $\\initialk_\\alpha$ to $\\tilde\\initialk_\\alpha$, from $q_\\alpha$ to $\\tilde q_\\alpha$, and from $\\xfer_\\alpha$ to $\\tilde\\xfer_\\alpha$:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class =&\\, \\sum_X \\prod_{\\alpha=1,2}  \\int \\! \\df(\\tilde\\initialk_\\alpha)  \\df(k) \\df(\\tilde\\xfer_\\alpha+\\tilde\\initialk_\\alpha) \\df (\\tilde q_\\alpha+\\tilde\\initialk_\\alpha) |\\varphi_\\alpha(\\tilde\\initialk_\\alpha+\\tilde\\xfer_\\alpha)|^2 \\; \n\\\\& \\times \\del^{(4)}(\\tilde \\xfer_1+ \\tilde \\xfer_2- k- \\finalk_X) \\del^{(4)}(\\tilde q_1 + \\tilde q_2 - k - \\finalk_X)\\, e^{-i b \\cdot (\\tilde q_1 - \\tilde\\xfer_1)\/\\hbar} \\, k_X^\\mu\n\\\\&\\qquad\\times \n\\langle\\Ampl^*(\\tilde\\initialk_1+ \\tilde q_1\\,, \\tilde\\initialk_2 + \\tilde q_2 \\rightarrow \n\\tilde\\initialk_1\\,, \\tilde\\initialk_2\\,, k\\,, \\finalk_X)\n\\\\&\\qquad\\qquad\\times \\Ampl(\\tilde\\initialk_1 + \\tilde\\xfer_1\\,, \\tilde\\initialk_2 + \\tilde\\xfer_2\\rightarrow \n\\tilde\\initialk_1\\,, \\tilde\\initialk_2\\,, k\\,, \\finalk_X)\n\\rangle\\,\n\\,.\n\\end{aligned}\n\\end{equation}\nAs the $\\tilde\\xfer_\\alpha$ implicitly carry a factor of $\\hbar$, just as argued in \\sect{subsec:Wavefunctions} for the momentum mismatch $q$, we may neglect the shift in the wavefunctions.  Dropping the tildes, and associating the $\\xfer_\\alpha$ integrals with $\\Ampl$ and the $q_\\alpha$ integrals with $\\Ampl^*$, our expression is revealed as an integral over a perfect square,\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class\n& = \\sum_X \\prod_{\\alpha=1,2} \\Lexp \\int \\! \\df(k) \\, k_X^\\mu\n\\biggl | \\int \\! \\df(\\xfer_\\alpha + \\initialk_\\alpha)  \\; \n\\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X)\\\\\n& \\hspace*{25mm} \\times   e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X) \\biggr|^2 \\Rexp\n\\,.\n\\label{eqn:radiatedMomentumClassicalAllOrder}\n\\end{aligned}\n\\end{equation}\nThe perfect-square structure allows us to define a \\textit{radiation kernel\\\/},\n\\begin{equation}\n\\begin{aligned}\n\\RadKer(k, \\finalk_X)\n&\\equiv \\hbar^{3\/2} \\prod_{\\alpha = 1, 2}  \\int \\! \\df( \\initialk_\\alpha +  \\xfer_\\alpha)  \\; \n\\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X) \\\\\n& \\qquad \\qquad \\times  e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X), \\\\\n= & \\hbar^{3\/2}\\prod_{\\alpha = 1, 2} \\int \\! \\dd^4 \\xfer_\\alpha\\; \n\\del(2 p_\\alpha \\cdot \\xfer_\\alpha + \\xfer_\\alpha^2)\\, \\del^{(4)}( \\xfer_1+  \\xfer_2- k- \\finalk_X) \\\\\n& \\quad\\times \\Theta(\\initialk_\\alpha^0+\\xfer_\\alpha^0)\\, e^{i b \\cdot  \\xfer_1\/\\hbar}  \\,   \n\\Ampl( \\initialk_1 +  \\xfer_1,  \\initialk_2 +  \\xfer_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, k\\,, \\finalk_X)\\,,\n\\label{eqn:defOfR}\n\\end{aligned}\n\\end{equation}\nso that\n\\begin{equation}\n\\begin{aligned}\n\\Rad^\\mu_\\class &=  \\sum_X \\hbar^{-3}\\Lexp \\int \\! \\df(k) \\, k_X^\\mu\n\\left |\\RadKer(k, \\finalk_X) \\right|^2 \\Rexp\n\\,.\n\\label{eqn:radiatedMomentumClassical}\n\\end{aligned}\n\\end{equation}\nThe prefactor along with the normalization of $\\RadKer$ are again chosen so that the classical limit of the radiation kernel will be of $\\Ord(\\hbar^0)$. Let us now focus once more on the leading contribution, with $X=\\emptyset$.  As usual, rescale $\\xfer \\rightarrow \\hbar\\xferb$, and remove an overall factor of $\\tilde g^6$ and accompanying $\\hbar$'s from the amplitude and its conjugate. Then the LO radiation kernel is\n\\begin{equation}\n\\begin{aligned}\n\\RadKerCl(\\wn k) \n& \\equiv \\hbar^2 \\prod_{\\alpha = 1, 2} \\int \\! \\dd^4 \\xferb_\\alpha \\, \\del(2p_\\alpha \\cdot \\xferb_\\alpha + \\hbar\\xferb_\\alpha^2) \\,\n\\del^{(4)}( \\xferb_1+  \\xferb_2- \\wn k)\ne^{i b \\cdot  \\xferb_1}  \n\\\\& \\hspace{70pt} \\times  \\AmplB^{(0)}( \\initialk_1 +  \\hbar \\xferb_1,  \\initialk_2 + \\hbar \\xferb_2\\rightarrow \n\\initialk_1\\,,  \\initialk_2\\,, \\hbar \\wn k)\\,,\n\\label{eqn:defOfRLO}\n\\end{aligned}\n\\end{equation}\nensuring that the leading-order momentum radiated is simply\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu, (0)}_\\class &= \\tilde g^6 \\Lexp \\int \\! \\df(\\wn k) \\, \\wn k^\\mu \\left | \\RadKerCl(\\wn k) \\right|^2 \\Rexp\\,.\n\\label{eqn:radiatedMomentumClassicalLO}\n\\end{aligned}\n\\end{equation}\n\n\\subsubsection{Conservation of momentum}\n\\label{sect:classicalConservation}\n\nConservation of momentum certainly holds to all orders, as we saw in \\sect{sect:allOrderConservation}.  However, it is worth making sure that we have not spoiled this critical physical property in our previous discussion, or indeed in our discussion of the classical impulse in \\sect{sec:classicalImpulse}. One might worry, for example, that there is a subtlety with the order of limits.\n\nThere is no issue at LO and NLO for the impulse, because\n\\begin{equation}\n\\DeltaPlo + \\DeltaPloTwo = 0 ,\\quad \\DeltaPnlo + \\DeltaPnloTwo = 0.\n\\end{equation}\nThese follow straightforwardly from the definitions of the observables, \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} and \\eqn~\\eqref{eqn:classicalLimitNLO}. The essential point is that the amplitudes entering into these orders in the impulse conserve momentum for two particles. At LO, for example, using \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} the impulse on particle 2 can be written as\n\\begin{multline}\n\\DeltaPloTwo=  \\frac{i\\tilde g^2}{4} \\Lexp \\hbar^2\\! \\int \\!\\dd^4 \\qb_1  \\dd^4 \\qb_2 \\; \n\\del(\\qb_1\\cdot p_1) \\del(\\qb_1\\cdot p_2)  \\del^{(4)}(\\qb_1 + \\qb_2)\n\\\\\\times  \ne^{-i b \\cdot \\qb_1} \n\\, \\qb_2^\\mu  \\, \\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow \np_1 + \\hbar\\qb_1, p_2 + \\hbar\\qb_2)\\,\\Rexp.\n\\end{multline}\nIn this equation, conservation of momentum at the level of the four point amplitude $\\AmplB^{(0)}(p_1,\\,p_2 \\rightarrow p_1 + \\hbar\\qb_1, p_2 + \\hbar\\qb_2)$ is expressed by the presence of the four-fold delta function $\\del^{(4)}(\\qb_1 + \\qb_2)$. Using this delta function, we may replace $\\qb_2^\\mu$ with $- \\qb_1^\\mu$ and then integrate over $\\qb_2$, once again using the delta function. The result is manifestly $-\\DeltaPlo$,  \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. A similar calculation goes through at NLO.\n\nIn this sense, the scattering is conservative at LO and at NLO. At NNLO, however, we must take radiative effects into account.  This backreaction is entirely described by the quadratic part of the impulse, $\\ImpB$. As indicated in \\eqn~\\eqref{eqn:commutatorVanishes}, $\\ImpA$ is always conservative. From our perspective here, this is because it involves only four-point amplitudes.  Thus to understand conservation of momentum we need to investigate $\\ImpB$. The lowest order case in which a five point amplitude can enter $\\ImpB$ is at NNLO. Let us restrict attention to this lowest order case, taking the additional state $X$ to be a messenger.\n\nFor $\\ImpB$ the lowest order term inolving one messenger is, in the classical regime,\n\\begin{equation}\n\\hspace*{-3mm}\\begin{aligned}\n\\ImpBclsup{(\\textrm{rad})} =&\\, \\tilde g^6\n\\Lexp \\hbar^{4}\\!\\int \\! d\\Phi(\\wn k)  \\prod_{\\alpha = 1,2} \\dd^4\\xferb_\\alpha\\, \n\\dd^4 \\qb_1 \\dd^4 \\qb_2 \\;\\del(2 \\xferb_\\alpha\\cdot p_\\alpha + \\xferb_\\alpha^2)\n\\\\&\\times  \\del(2 \\qb_1\\cdot p_1) \\del(2 \\qb_2\\cdot p_2)\\,\ne^{-i b \\cdot \\qb_1}\\,\\xferb_1^\\mu\\,\n\\del^{(4)}(\\xferb_1+\\xferb_2 + \\bar k)\\, \\del^{(4)}(\\qb_1+\\qb_2)\n\\\\&\\quad\\times \\AmplB^{(0)}(\\initialk_1\\,, \\initialk_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2, \\hbar\\bar k)\n\\\\&\\qquad\\times \n\\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb_1\\,, \\initialk_2 + \\hbar \\qb_2 \\rightarrow \n\\initialk_1+\\hbar\\xferb_1\\,, \\initialk_2+\\hbar\\xferb_2, \\hbar\\bar k)\n\\,\\Rexp\\,.\n\\end{aligned} \n\\label{eqn:nnloImpulse}\n\\end{equation}\nTo see that this balances the radiated momentum, we use \\eqn~\\eqref{eqn:ExpectedMomentum2classicalLO}. The structure of the expressions are almost identical; conservation of momentum holds because the factor $\\bar k^\\mu$ in \\eqn~\\eqref{eqn:ExpectedMomentum2classicalLO} is balanced by $\\xferb_1^\\mu$ in \\eqn~\\eqref{eqn:nnloImpulse} and $\\xferb_2^\\mu$ in the equivalent expression for particle 2.\n\nThus conservation of momentum continues to hold in our expressions once we have passed to the classical limit, at least through NNLO. At this order there is non-zero momentum\nradiated, so momentum conservation is non-trivial from the classical point of view. We will see by explicit calculation in QED that our classical impulse correctly incorporates the impulse from the ALD force in addition to the Lorentz force.\n\n\\subsection{Perspectives from classical field theory}\n\\defx{x}\n\nBefore jumping into examples, it is useful to reflect on the total radiated momentum, expressed in terms of amplitudes, by digressing into classical field theory. To do so we must classically describe the distribution and flux of energy and momentum in the radiation field itself. Although our final conclusions also hold in YM theory and gravity, let us work in electrodynamics for simplicity. Here the relevant stress-energy tensor is\n\\begin{equation}\nT^{\\mu\\nu}(x) = F^{\\mu\\alpha}(x) F_\\alpha{}^\\nu(x) + \\frac 14 \\eta^{\\mu\\nu} F^{\\alpha\\beta}(x) F_{\\alpha\\beta}(x) \\,.\\label{eqn:EMfieldStrength}\n\\end{equation}\nIn particular, the (four-)momentum flux through a three dimensional surface $\\partial \\Omega$ with surface element $\\d\\Sigma_\\nu$ is\n\\begin{equation}\nK^\\mu = \\int_{\\partial \\Omega}\\!\\! \\d \\Sigma_\\nu T^{\\mu\\nu}(x)\\,.\n\\end{equation}\nWe are interested in the total momentum radiated as two particles scatter. At each time $t$, we therefore surround the two particles with a large sphere. The instantaneous flux of momentum is measured by integrating over the surface area of the sphere; the total momentum radiated is then the integral of this instantaneous flux over all times. It is straightforward to determine the momentum radiated by direct integration over these spheres using textbook methods --- see appendix D of \\cite{Kosower:2018adc}.\n\nA simpler but more indirect method is the following. We wish to use the Gauss theorem to write\n\\begin{equation}\nK^\\mu = \\int_{\\partial \\Omega}\\!\\! \\d \\Sigma_\\nu T^{\\mu\\nu}(x) =  \\int \\! \\d^4x \\, \\partial_\\nu T^{\\mu\\nu}(x)\\,.\n\\end{equation}\nHowever, the spheres surrounding our particle are not the boundary of all spacetime: they do not include the timelike future and past boundaries. To remedy this, we use a trick due to Dirac~\\cite{Dirac:1938nz}. \n\nThe radiation we have in mind is causal, so we solve the Maxwell equation with retarded boundary conditions. We denote these fields by $F^{\\mu\\nu}_\\textrm{ret}(x)$.\nWe could equivalently solve the Maxwell equation using the advanced Green's function. If we wish to determine precisely the same fields $F^{\\mu\\nu}_\\textrm{ret}(x)$ but using the advanced Green's function, we must add a homogeneous solution of the Maxwell equation. Fitting the boundary conditions in this way requires subtracting the incoming radiation field $F^{\\mu\\nu}_\\textrm{in}(x)$ which is present in the advanced solution (but not in the retarded solution) and adding the outgoing radiation field (which is present in the retarded solution, but not the advanced solution.) In other words,\n\\begin{equation}\nF^{\\mu\\nu}_\\textrm{ret}(x) - F^{\\mu\\nu}_\\textrm{adv}(x) = - F^{\\mu\\nu}_\\textrm{in}(x) + F^{\\mu\\nu}_\\textrm{out}(x)\\,.\n\\end{equation}\nNow, the radiated momentum $K^\\mu$ in which we are interested is described by $F^{\\mu\\nu}_\\textrm{out}(x)$. The field $F^{\\mu\\nu}_\\textrm{in}(x)$ transports the same total amount of momentum in from infinity, ie it transports momentum $-K^\\mu$ out. Therefore the difference between the momenta transported out to infinity by the retarded and by the advanced fields is simply $2 K^\\mu$. This is useful, because the contributions of the point-particle sources cancel in this difference.\n\nThe relationship between the momentum transported by the retarded and advanced field is reflected at the level of the Green's functions themselves. \nThe difference in the Green's function takes an instructive form:%\n\\begin{equation}\n\\begin{aligned}\n\\tilde G_\\textrm{ret}(\\bar k) - \\tilde G_\\textrm{adv}(\\bar k) &= \n\\frac{(-1)}{(\\bar k^0 + i \\epsilon)^2 - \\v{\\bar k}^2} \n- \\frac{(-1)}{(\\bar k^0 - i \\epsilon)^2 - \\v{\\bar k}^2} \n\\\\&=  i     \\left( \\Theta(\\bar k^0) - \\Theta(-\\bar k^0) \\right) \\del(\\bar k^2)\\,.\n\\end{aligned}\n\\end{equation}\nIn this equation, $\\v{\\bar k}$ denotes the spatial components of wavenumber four-vector $\\bar k$. This difference is a homogeneous solution of the wave equation since it is supported \non $\\bar k^2 = 0$. The two terms correspond to positive and negative angular frequencies. As we will see, the relative sign ensures that the momenta transported to infinity add.\n\nWith this in mind, we return to the problem of computing the momentum radiated and write\n\\begin{equation}\n2 K^\\mu = \\int_{\\partial \\Omega} \\!\\!\\d \\Sigma_\\nu \\Big(T^{\\mu\\nu}_\\textrm{ret}(x) -T^{\\mu\\nu}_\\textrm{adv}(x) \\Big)\\,.\n\\end{equation}\nIn this difference, the contribution of the sources at timelike infinity cancel, so we may regard the surface $\\partial \\Omega$ as the boundary of spacetime. Therefore,\n\\begin{equation}\n2K^\\mu = \\int \\! d^4 x \\, \\partial_\\nu \\!\\left(T^{\\mu\\nu}_\\textrm{ret}(x) -T^{\\mu\\nu}_\\textrm{adv}(x) \\right) =- \\int \\! d^4 x \\left( F^{\\mu\\nu}_\\textrm{ret}(x) - F^{\\mu\\nu}_\\textrm{adv}(x)\\right) J_\\nu(x)\\,,\n\\end{equation}\nwhere the last equality follows from the equations of motion. We now pass to momentum space, noting that\n\\begin{equation}\nF^{\\mu\\nu}(x) = -i\\! \\int \\! \\dd^4 \\bar k \\left( \\bar k^\\mu \\tilde A^\\nu(\\bar k) - \\bar k^\\nu \\tilde A^\\mu(\\bar k) \\right) e^{-i \\bar k \\cdot x}\\,.\n\\end{equation}\nUsing conservation of momentum, the radiated momentum becomes\n\\begin{equation}\n\\begin{aligned}\n2K^\\mu \n&= i\\! \\int \\! \\dd^4 \\bar k \\; \\bar k^\\mu \\left( \\tilde A^\\nu_\\textrm{ret}(\\bar k) - \\tilde A^\\nu_\\textrm{adv}(\\bar k) \\right) \\tilde J_\\nu^*(\\bar k), \\\\\n&= -\\int \\! \\dd^4 \\bar k \\; \\bar k^\\mu \\left(\\Theta(\\bar k^0) - \\Theta(-\\bar k^0)\\right) \\del(\\bar k^2) \\tilde J^\\nu(\\bar k) \\tilde J_\\nu^*(\\bar k)\\,.\n\\label{eqn:momentumMixed}\n\\end{aligned}\n\\end{equation}\nThe two different $\\Theta$ functions arise from the outgoing and incoming radiation fields. Setting $k'^\\mu = - k^\\mu$ in the second term, and then dropping the prime, it is easy to see that the two terms add as anticipated. We arrive at a simple general result for the momentum radiated:\n\\begin{equation}\n\\begin{aligned}\nK^\\mu &= -\\int \\! \\dd^4 \\wn k \\,  \\Theta(\\wn k^0)\\del(\\wn k^2) \\, \\wn k^\\mu \\, \\tilde J^\\nu(\\wn k) \\tilde J_\\nu^*(\\wn k) \\\\\n&= -\\int \\! \\df( \\wn k) \\, \\bar k^\\mu \\, \\tilde J^\\nu(\\bar k) \\tilde J_\\nu^*(\\bar k) \\,.\n\\label{eqn:classicalMomentumRadiated}\n\\end{aligned}\n\\end{equation}\nIt is now worth pausing to compare this general classical formula for the radiated momentum to the expression we derived previously in \\eqn~\\eqref{eqn:radiatedMomentumClassical}. Evidently the radiation kernel we defined in \\eqn~\\eqref{eqn:defOfR} is related to the classical current $\\tilde J^\\mu(\\bar k)$. This fact was anticipated in ref.~\\cite{Luna:2017dtq}. Indeed, if we introduce a basis of polarisation vectors $\\varepsilon^{h}_\\mu(\\bar k)$ associated with the wavevector $\\bar k$ with helicity $h$, we may write the classical momentum radiated as\n\\begin{equation}\nK^\\mu = \\sum_h \\int \\! \\df(\\bar k) \\, \\bar k^\\mu \\, \n\\left| \\varepsilon^{h} \\cdot \\tilde J(\\bar k) \\right|^2\\,,\n\\label{eqn:classicalMomentumRadiated1}\n\\end{equation}\nwhere here we have written the sum over helicities explicitly. Similar expressions hold in classical YM theory and gravity \\cite{Goldberger:2016iau}.\n\n\\section{Examples}\n\\label{sec:LOradiation}\n\\def\\varepsilon{\\varepsilon}\n\nAt leading-order the amplitude appearing in the radiation kernel in equation~\\eqref{eqn:defOfRLO} is a five-point, tree amplitude (figure~\\ref{fig:5points}) that can be readily computed. In Yang--Mills theory,\n\\newcommand{\\treeLa}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [above right = .05 and .125 of v1] (g1);\n\t\\vertex [below right = 0.125 and 0.125 of g1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (g1);\n\t\\draw (g1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeLb}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [above left = .05 and .125 of v1] (g1);\n\t\\vertex [below left = 0.125 and 0.125 of g1] (g2);\n\t\\draw (i1) -- (g1);\n\t\\draw (g1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeLc}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [below right = 0.15 and 0.25 of v1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (v1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\newcommand{\\treeYM}{\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [below = 0.3 of v1] (v2);\n\t\\vertex [above right = .1 and .25 of v1] (o1);\n\t\\vertex [above left = .1 and .25 of v1] (i1);\n\t\\vertex [below right = .1 and .25 of v2] (o2);\n\t\\vertex [below left = .1 and .25 of v2] (i2);\n\t\\vertex [below = .15 of v1] (g1);\n\t\\vertex [right = 0.275 of g1] (g2);\n\t\\draw (i1) -- (v1);\n\t\\draw (v1) -- (o1);\n\t\\draw (i2) -- (v2);\n\t\\draw (v2) -- (o2);\n\t\\draw (g1) -- (g2);\n\t\\draw (v1) -- (v2);\n\t\\end{feynman}\t\n\t\\end{tikzpicture}}\n\\[\n\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) &=\n\\sum_D \\mathcal{C}^a(D) \\bar{A}^{(0)}_D(p_1 + w_1, p_2 + w_2\\rightarrow p_1, p_2; k, h) \\\\\n&= \\Big[\\mathcal{C}^a\\!\\left(\\treeLa \\right)\\!A_{\\scalebox{0.5}{\\treeLa}} \n+ \\mathcal{C}^a\\!\\left(\\treeLb \\right)\\! A_{\\scalebox{0.5}{\\treeLb}} \n\\\\ & \\qquad\\qquad + \\mathcal{C}^a\\!\\left(\\treeLc \\right)\\! A_{\\scalebox{0.5}{\\treeLc}}\n+ (1\\leftrightarrow 2) \\Big]\n- i\\,\\mathcal{C}^a\\!\\left(\\treeYM \\right)\\! A_{\\scalebox{0.5}{\\treeYM}}\\,.\n\\]\nExplicitly, the colour factors are given by\n\\begin{equation}\n\\begin{gathered}\n\\mathcal{C}^a\\!\\left(\\treeLa \\right) = (\\newT_1^a \\cdot \\newT_1^b) \\newT_2^b\\,, \\qquad\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) = (\\newT_1^b \\cdot \\newT_1^a) \\newT_2^b\\,,\\\\\n\\mathcal{C}^a\\!\\left(\\treeLc \\right) = \\frac12\\mathcal{C}^a\\!\\left(\\treeLa \\right) + \\frac12\\mathcal{C}^a\\!\\left(\\treeLb \\right), \\qquad\n\\mathcal{C}^a\\!\\left(\\treeYM \\right) = i\\hbar f^{abc} \\newT_1^b \\newT_2^c\\,,\\label{eqn:radColourFactors}\n\\end{gathered}\n\\end{equation}\nwith the replacement $1\\leftrightarrow2$ for diagrams with gluon emission from particle 2. Just as in the 4-point case at  1-loop, this is an overcomplete set for specifying a basis, because \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) = (\\newT_1^a\\cdot \\newT_1^b) \\newT_2^b +  i\\hbar f^{bac} \\newT_1^c \\newT_2^b = \\mathcal{C}^a\\!\\left(\\treeLa \\right) +  \\mathcal{C}^a\\!\\left(\\treeYM \\right).\\label{eqn:JacobiSetUp}\n\\end{aligned}\n\\end{equation}\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tikzpicture}[decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\t\\begin{feynman}\n\t\\vertex (v1);\n\t\\vertex [above left=1 and 0.66 of v1] (i1) {$\\initialk_1+\\xfer_1$};\n\t\\vertex [above right=1 and 0.8 of v1] (o1) {$\\initialk_1$};\n\t\\vertex [right=1.2 of v1] (k) {$k$};\n\t\\vertex [below left=1 and 0.66 of v1] (i2) {$\\initialk_2+\\xfer_2$};\n\t\\vertex [below right=1 and 0.8 of v1] (o2) {$\\initialk_2$};\n\t\\draw [postaction={decorate}] (i1) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o1);\n\t\\draw [postaction={decorate}] (i2) -- (v1);\n\t\\draw [postaction={decorate}] (v1) -- (o2);\n\t\\diagram*{(v1) -- [photon] (k)};\n\t\\filldraw [color=white] (v1) circle [radius=10pt];\n\t\\draw [pattern=north west lines, pattern color=patternBlue] (v1) circle [radius=10pt];\n\t\\end{feynman}\t\n\t\\end{tikzpicture} \n\t\\caption[The amplitude appearing in the leading-order radiation kernel.]{The amplitude $\\Ampl^{(0)}(\\initialk_1+\\xfer_1\\,,\\initialk_2+\\xfer_2\\rightarrow\n\t\t\\initialk_1\\,,\\initialk_2\\,,k)$ appearing in the radiation kernel at leading order.}\n\t\\label{fig:5points}\n\\end{figure}\nHence the full basis of colour factors is only 3 dimensional, and the colour decomposition of the 5-point tree is\n\\begin{multline}\n\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) = \\mathcal{C}^a\\!\\left(\\treeLa \\right)\\Big(A_{\\scalebox{0.5}{\\treeLa}} + A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}}\\Big) \\\\ \n+ \\frac12\\mathcal{C}^a\\!\\left(\\treeYM \\right)\\Big(-iA_{\\scalebox{0.5}{\\treeYM}} + 2 A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}}\\Big) + (1\\leftrightarrow2)\\,.\n\\end{multline}\nGiven that the second structure is $\\mathcal{O}(\\hbar)$, it would appear that we could again neglect the second term as a quantum correction. However, this intuition is not quite correct, as calculating the associated partial amplitude shows:\n\\begin{multline}\n-iA_{\\scalebox{0.5}{\\treeYM}} + 2  A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}} = -\\frac{4\\,\\varepsilon^{h}_\\mu(\\wn k)}{\\hbar^2} \\bigg[\\frac{2p_1\\cdot p_2}{{\\wn w_2^2\\, p_1\\cdot\\wn k}}\\, \\frac{p_1^\\mu}{\\hbar} + \\frac{1}{\\hbar\\, \\wn w_1^2 \\wn w_2^2}\\Big(2 p_2\\cdot\\wn k\\, p_1^\\mu \\\\- p_1\\cdot p_2 \\,(\\wn w_1^\\mu - \\wn w_2^\\mu) - 2p_1\\cdot\\wn k\\, p_2^\\mu\\Big) + \\mathcal{O}(\\hbar^0) \\bigg]\\,,\\label{eqn:radSingular}\n\\end{multline}\nwhere we have used $p_1\\cdot\\wn w_2 = p_1\\cdot\\wn k + \\hbar\\wn w_1^2\/2$ on the support of the on-shell delta functions in the radiation kernel~\\eqref{eqn:defOfRLO}. The partial amplitude appears to be singular, as there is an extra power of $\\hbar$ downstairs. However, this will cancel against the extra power in the colour structure, yielding a classical contribution. Meanwhile in the other partial amplitude the potentially singular terms cancel trivially, and the contribution is classical:\n\\begin{multline}\nA_{\\scalebox{0.5}{\\treeLa}} + A_{\\scalebox{0.5}{\\treeLb}} + A_{\\scalebox{0.5}{\\treeLc}} = \\frac{2}{\\hbar^2} \\frac{\\varepsilon^{h}_\\mu(\\wn k)}{\\wn w_2^2\\, p_1\\cdot\\wn k}\\bigg[ 2p_1\\cdot p_2\\,\\wn w_2^\\mu +  \\frac{p_1\\cdot p_2}{p_1\\cdot\\wn k} p_1^\\mu(\\wn w_1^2 - \\wn w_2^2) \\\\ - 2p_1\\cdot\\wn k\\, p_2^\\mu + 2p_2\\cdot \\wn k\\, p_1^\\mu + \\mathcal{O}(\\hbar)\\bigg]\\,.\n\\end{multline}\nSumming all colour factors and partial amplitudes, the classical part of the 5-point amplitude is\n\\begin{align}\n&\\bar{\\mathcal{A}}^{(0)}(\\wn k^a) = \\sum_D \\mathcal{C}^a(D) \\bar{A}^{(0)}_D(p_1 + w_1, p_2 + w_2\\rightarrow p_1, p_2; k, h) \\nonumber\\\\\n&= - \\frac{4\\varepsilon_\\mu^h(\\wn k)}{\\hbar^2} \\bigg\\{ \\frac{\\newT_1^a (\\newT_1\\cdot \\newT_2)}{\\wn w_2^2 \\, \\wn k \\cdot p_1} \\bigg[-(p_1\\cdot p_2)\\left(\\wn w_2^\\mu - \\frac{\\wn k\\cdot\\wn w_2}{\\wn k\\cdot p_1} p_1^\\mu\\right) + \\wn k\\cdot p_1 \\, p_2^\\mu  - \\wn k\\cdot p_2\\, p_1^\\mu\\bigg] \\nonumber \\\\ &\\qquad + \\frac{if^{abc}\\,\\newT_1^b \\newT_2^c}{\\wn w_1^2 \\wn w_2^2}\\bigg[2\\wn k\\cdot p_2\\, p_1^\\mu\n - p_1\\cdot p_2\\, \\wn w_1^\\mu + p_1\\cdot p_2 \\frac{\\wn w_1^2}{\\wn k\\cdot p_1}p_1^\\mu\\bigg] + (1\\leftrightarrow 2)\\bigg\\}\\,,\n\\end{align}\nwhere we have used that $\\wn w_1^2 - \\wn w_2^2 = -2\\wn k\\cdot \\wn w_2$ since the outgoing radiation is on-shell. Finally, we can substitute into the radiation kernel in \\eqn~\\eqref{eqn:defOfRLO} and take the classical limit. Averaging over the wavepackets sets $p_\\alpha = m_\\alpha u_\\alpha$ and replaces quantum colour charges with their classical counterparts, yielding\n\\[\n&\\mathcal{R}^{a,(0)}_\\text{YM}(\\wn k) = -\\int\\!\\dd^4\\,\\wn w_1 \\dd^4 \\wn w_2 \\, \\del^{(4)}(\\wn k - \\wn w_1 - \\wn w_2) \\del(u_1\\cdot\\wn w_1) \\del(u_2\\cdot\\wn w_2)\\, e^{ib\\cdot\\wn w_1} \\varepsilon_\\mu^{h}\\\\ \n&\\times \\bigg\\{ \\frac{c_1\\cdot c_2}{m_1} \\frac{c_1^a}{\\wn w_2^2 \\, \\wn k \\cdot u_1} \\left[-(u_1\\cdot u_2)\\left(\\wn w_2^\\mu - \\frac{\\wn k\\cdot\\wn w_2}{\\wn k\\cdot u_1} u_1^\\mu\\right) + \\wn k\\cdot u_1 \\, u_2^\\mu - \\wn k\\cdot u_2\\, u_1^\\mu\\right]\\\\\n& \\qquad + \\frac{if^{abc}\\,c_1^b c_2^c}{\\wn w_1^2 \\wn w_2^2}\\left[2\\wn k\\cdot u_2\\, u_1^\\mu - u_1\\cdot u_2\\, \\wn w_1^\\mu + u_1\\cdot u_2 \\frac{\\wn w_1^2}{\\wn k\\cdot u_1}\\,u_1^\\mu\\right] + (1\\leftrightarrow 2)\\bigg\\}\\,.\\label{eqn:LOradKernel}\n\\]\nOur result is equal to the leading order current $\\tilde{J}^{\\mu,(0)}_a$ obtained in \n\\cite{Goldberger:2016iau} by iteratively solving the Wong equations in \\eqn~\\eqref{eqn:Wong-momentum} and \\eqn~\\eqref{eqn:Wong-color} for timelike particle worldlines.\n\n\\subsection{Inelastic black hole scattering}\n\\label{sec:inelasticBHscatter}\n\nLet us turn to an independent application of our LO Yang--Mills radiation kernel~\\eqref{eqn:LOradKernel}. By returning to the colour-kinematics structure of the underlying amplitude, we can readily use the double copy to calculate results for gravitational wave emission from black hole scattering.\n\nTo apply the double copy we need the overcomplete set of colour factors in equation~\\eqref{eqn:radColourFactors}. This is because the object of fundamental interest is now the Jacobi identity that follows from~\\eqref{eqn:JacobiSetUp},\n\\begin{equation}\n\\mathcal{C}^a\\!\\left(\\treeLb \\right) - \\mathcal{C}^a\\!\\left(\\treeLa \\right) = \\mathcal{C}^a\\!\\left(\\treeYM \\right),\n\\end{equation}\nwith an identical identity holding upon exchanging particles 1 and 2. Unlike our example in section~\\ref{sec:LOimpulse}, this is a non-trivial relation, and we must manipulate the numerators of each topology into a colour-kinematics dual form. This can be readily achieved by splitting the topologies with four-point seagull vertices, and adding their kinematic information to diagrams with the same colour structure. It is simple to verify that, in the classical limit, a basis of colour-kinematics dual numerators for this amplitude is\n\\begin{align}\n\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeLb}} &= 4(p_1\\cdot p_2)\\, p_1\\cdot\\varepsilon^{h}_k + 2\\hbar\\Big(p_1\\cdot \\wn k\\, (p_1 + p_2)^\\mu + p_1\\cdot p_2\\, (\\wn w_1 - \\wn w_2)^\\mu\\Big)\\cdot\\varepsilon^h_k + \\mathcal{O}(\\hbar^2)\\nonumber\\\\\n\\hspace{-7mm}\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeLa}} &= 4(p_1\\cdot p_2)\\, p_1\\cdot\\varepsilon^{h}_k + 2\\hbar\\Big(p_1\\cdot \\wn k \\, (p_1 - p_2)^\\mu + 2 p_2\\cdot\\wn k\\, p_1^\\mu\\Big)\\cdot\\varepsilon^h_k + \\mathcal{O}(\\hbar^2)\\label{eqn:vecnums}\\\\\n\\sqrt{2}\\, n_{\\scalebox{0.5}{\\treeYM}} &= 2\\hbar\\Big(2 p_1\\cdot \\wn k\\, p_2^\\mu - 2p_2\\cdot\\wn k\\, p_1^\\mu + p_1\\cdot p_2\\, (\\wn w_1 - \\wn w_2)^\\mu\\Big)\\cdot\\varepsilon^h_k +\\mathcal{O}(\\hbar^2)\\,, \\nonumber\n\\end{align}\nwhere $\\varepsilon^h_k \\equiv \\varepsilon^h_\\mu(\\wn k)$. It is crucial that we keep $\\mathcal{O}(\\hbar)$ terms, as we know from equation~\\eqref{eqn:radSingular} that when the YM amplitude is not written on a minimal basis of colour factors there are terms which are apparently singular in the classical limit. The factors of $\\sqrt{2}$ are to account for the proper normalisation of colour factors involved in the double copy --- see discussion around equation~\\eqref{eqn:scalarYMamp}.\n\nWith a set of colour-kinematics dual numerators at hand, we can now double copy by replacing colour factors with these numerators, leading to\n\\begin{multline}\n\\hbar^{\\frac{7}{2}}  {\\mathcal{M}}^{(0)}_\\textrm{JNW}(\\wn k) = \\left(\\frac{\\kappa}{2}\\right)^3 \\bigg[ \\frac{1}{\\hbar\\,\\wn w_2^2} \\bigg(\\frac{n^\\mu_{\\scalebox{0.5}{\\treeLa}} n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{n^\\mu_{\\scalebox{0.5}{\\treeLb}}n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar \\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2}n^\\mu_{\\scalebox{0.5}{\\treeYM}} n^\\nu_{\\scalebox{0.5}{\\treeYM}} \\bigg] e_{\\mu\\nu}^h(\\wn k)\\,,\\label{eqn:JNWamplitude}\n\\end{multline}\nwhere we have used the outer product of the polarisation vectors (of momentum $\\wn k$) from the numerators,\n\\begin{equation}\ne_{\\mu\\nu}^{h} = \\frac12\\left(\\varepsilon^{h}_\\mu \\varepsilon^{h}_\\nu + \\varepsilon^{h}_\\nu \\varepsilon^{h}_\\mu - P_{\\mu\\nu}\\right) + \\frac12\\left(\\varepsilon^{h}_\\mu \\varepsilon^{h}_\\nu - \\varepsilon^{h}_\\nu \\varepsilon^{h}_\\mu\\right) + \\frac12 P_{\\mu\\nu}\\,.\\label{eqn:polarisationOuterProd}\n\\end{equation}\nHere $P_{\\mu\\nu} = \\eta_{\\mu\\nu} - \\left(\\wn k_\\mu \\wn r_\\nu + \\wn k_\\nu \\wn r_\\mu\\right)\/(\\wn k\\cdot \\wn r)$ is a transverse projector with reference momentum $\\wn r$. Since the amplitude is symmetric in its numerators, we can immediately restrict attention to the initial symmetric and traceless piece --- the polarisation tensor for a graviton. It now simply remains to Laurent expand in $\\hbar$ to retrieve the parts of the amplitude which contribute to the classical radiation kernel for gravitational radiation. \n\nRather than doing so at this point, it is more pertinent to note that our result is not yet an amplitude in Einstein gravity. As in our 4-point discussion in the previous chapter, this result is polluted by dilaton interactions. In particular, the corresponding current is for the scattering of JNW naked singularities in Einstein--dilaton gravity \\cite{Goldberger:2016iau}. A convenient way to remove the dilaton states is to use a scalar ghost \\cite{Johansson:2014zca}, as shown explicitly for the classical limit of this amplitude in ref.~\\cite{Luna:2017dtq}. We introduce a new massles, adjoint-representation scalar $\\chi$, minimally coupled to the YM gauge field, and use the double copy of its amplitude to remove the pollution. The ghost couples to the scalar fields in our action~\\eqref{eqn:scalarAction} via the interaction term\n\\begin{equation}\n\\mathcal{L}_{\\chi \\textrm{int}} = -2g \\sum_{\\alpha=1,2} \\Phi_\\alpha^\\dagger \\chi \\Phi_\\alpha\\,.\\label{eqn:adjointScalar}\n\\end{equation}\nOn the same 5-point kinematics as our previous YM amplitude, the equivalent numerators for the topologies in~\\eqref{eqn:vecnums} with interactions mediated by the new massless scalar are\n\\[\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeLb}} &= -4 (p_1 - \\hbar\\wn w_2) \\cdot\\varepsilon^h_k \\\\\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeLa}} &= -4 p_1 \\cdot\\varepsilon^h_k \\\\\n\\sqrt{2}\\, \\tilde n_{\\scalebox{0.5}{\\treeYM}} &= -2\\hbar (\\wn w_1 - \\wn w_2)\\cdot\\varepsilon^h_k\\,. \\label{eqn:scalarnums}\n\\]\nSince $\\chi$ is in the adjoint representation, the appropriate colour factors are again those in equation~\\eqref{eqn:radColourFactors} (there is now no seagull topology). The numerators hence trivially satisfy colour-kinematics duality. We can therefore double copy to yield the ghost amplitude,\n\\begin{multline}\n\\hbar^{\\frac72} {\\mathcal{M}}^{(0)}_\\textrm{ghost}(\\wn k) = \\left(\\frac{\\kappa}{2}\\right)^3 \\bigg[\\frac1{\\hbar\\,\\wn w_2^2}\\bigg(\\frac{\\tilde n^\\mu_{\\scalebox{0.5}{\\treeLa}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{\\tilde n^\\mu_{\\scalebox{0.5}{\\treeLb}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar\\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2} \\tilde n^\\mu_{\\scalebox{0.5}{\\treeYM}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeYM}}\\bigg] e^h_{\\mu\\nu}(\\wn k)\\,,\\label{eqn:ghostAmp}\n\\end{multline}\nThis is an amplitude for a ghost in the sense that we can now write \\cite{Luna:2017dtq}\n\\begin{equation}\n\\mathcal{M}^{(0)}_\\textrm{Schwz}(\\wn k) = \\mathcal{M}^{(0)}_\\textrm{JNW} - \\frac1{D-2}\\mathcal{M}^{(0)}_\\textrm{ghost}\\,,\n\\end{equation}\nwhere the appearance of the spacetime dimension comes from matching the ghost to the dilaton propagator. Expanding the numerators, in $D=4$ one finds that\n\\begin{multline}\n\\hbar^2 \\bar{\\mathcal{M}}^{(0)}_\\textrm{Schwz}(\\wn k) =   \\bigg[\\frac{{P}_{12}^\\mu {P}_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} + \\frac{p_1\\cdot p_2}{2\\wn w_1^2 \\wn w_2^2}\\left({Q}^\\mu_{12} {P}_{12}^\\nu + {Q}_{12}^\\nu {P}_{12}^\\mu\\right) \\\\ + \\frac14\\left((p_1\\cdot p_2)^2 - \\frac12\\right)\\left(\\frac{{Q}^\\mu_{12} {Q}^\\nu_{12}}{\\wn w_1^2 \\wn w_2^2} - \\frac{{P}^\\mu_{12} {P}^\\nu_{12}}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2}\\right)\\bigg]e_{\\mu\\nu}^{h} + \\mathcal{O}(\\hbar)\\,,\\label{eqn:ampSchwarzschild}\n\\end{multline}\nwhere\n\\begin{subequations}\n\t\\begin{gather}\n\tP_{12}^\\mu = (\\wn k\\cdot p_1)\\, p_2^\\mu - (\\wn k\\cdot p_2)\\, p_1^\\mu\\,,\\label{eqn:gaugeinvariant1}\\\\\n\tQ_{12}^\\mu = (\\wn w_1 - \\wn w_2)^\\mu - \\frac{\\wn w_1^2}{\\wn k\\cdot p_1} p_1^\\mu + \\frac{\\wn w_2^2}{\\wn k\\cdot p_2} p_2^\\mu\\,,\\label{eqn:gaugeinvariant2}\n\t\\end{gather}\\label{eqn:gaugeInvariants}\n\\end{subequations}\nare two gauge invariant functions of the kinematics. Substituting the amplitude into the LO radiation kernel~\\eqref{eqn:defOfRLO} yields the LO current for the scattering of two Schwarzschild black holes. Note that whereas we took the classical limit before double copying, this result was first obtained in ref.~\\cite{Luna:2017dtq} by only taking the classical limit (via a large mass expansion) once the gravity amplitudes were at hand.\n\n\\subsubsection{Reissner--Nordstr\\\"{o}m black holes}\n\nIn our previous calculation, the adjoint massless scalar introduced in equation~\\eqref{eqn:adjointScalar} merely acted as a useful computational trick to remove internal dilaton pollution. However, if we consider other black hole species it can be promoted to a far more fundamental role. For example, let us consider gravitational radiation emitted by the scattering of two Reissner--Nordstr\\\"{o}m black holes. RN black holes are solutions to Einstein--Maxwell theory rather than vacuum general relativity, and thus have non-zero electric (or magnetic) charges $Q_\\alpha$. At leading order their gravitational interactions are the same as for Schwarzschild black holes, but the total current due to gravitational radiation will be different, precisely because of terms sourced from electromagnetic interactions, mediated by a massless vector field.\n\nThis is in contrast to our previous example, where we ``squared'' the vector numerators to obtain tensor interactions, in the sense that $A^\\mu \\otimes A^\\mu \\sim H_{\\mu\\nu}$. To obtain electromagnetic interactions in the gravity amplitude guaranteed by the double copy we need to use numerators in a vector and scalar representation respectively, such that we have $A^\\mu \\otimes \\phi \\sim \\tilde{A}^\\mu$. This is exactly what the sets of numerators in equations~\\eqref{eqn:vecnums} and~\\eqref{eqn:scalarnums} provide.\n\nThe double copy does not require that one square numerators, merely that colour data is replaced with numerators satisfying the same Jacobi identities. Thus the double copy construction\n\\begin{multline}\n\\hbar^{\\frac72}  {\\mathcal{M}}^{(0)}(\\wn k) = \\frac{\\kappa}{2} e^2 Q_1 Q_2\\bigg[ \\frac{1}{\\hbar\\,\\wn w_2^2} \\bigg(\\frac{n^\\mu_{\\scalebox{0.5}{\\treeLa}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeLa}}}{2p_1\\cdot \\wn k} - \\frac{n^\\mu_{\\scalebox{0.5}{\\treeLb}}\\tilde n^\\nu_{\\scalebox{0.5}{\\treeLb}}}{2p_1\\cdot \\wn k + \\hbar \\wn w_1^2 - \\hbar \\wn w_2^2}\\bigg) + (1\\leftrightarrow 2)\\\\  + \\frac{1}{\\hbar^2 \\wn w_1^2 \\wn w_2^2}n^\\mu_{\\scalebox{0.5}{\\treeYM}} \\tilde n^\\nu_{\\scalebox{0.5}{\\treeYM}} \\bigg] e_{\\mu\\nu}^h\\label{eqn:RNamp}\n\\end{multline}\nis guaranteed to be a well-defined gravity amplitude. Note that we have altered the coupling replacement in the double copy appropriately.  \n\nA consequence of using kinematic numerators from alternative sets is that the amplitude is asymmetric in its Lorentz indices; specifically, for the scalar diagrams there are no seagull vertex terms, while the scalar boson triple vertex term is manifestly different to the pure vector case. Since the graviton polarisation tensor is symmetric and traceless, the graviton amplitude is obtained by symmetrising over the Lorentz indices in~\\eqref{eqn:RNamp}. Substituting the result into equation~\\eqref{eqn:defOfRLO} yields the LO gravitational radiation kernel due to electromagnetic interactions,\n\\begin{multline}\n\\mathcal{R}_{\\rm RN,grav}^{(0)} = \\frac{e^2\\kappa}{4} Q_1 Q_2 \\!\\int\\! \\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(p_1\\cdot\\wn w_1) \\del(p_2\\cdot\\wn w_2) \\del^{(4)}(\\wn w_1 + \\wn w_2 - \\wn k) e^{ib\\cdot\\wn w_1} \\\\\\times \\bigg[\\frac{{Q}_{12}^\\mu {P}_{12}^\\nu + {Q}_{12}^\\nu {P}_{12}^\\mu}{\\wn w_1^2 \\wn w_2^2} + (p_1\\cdot p_2) \\bigg(\\frac{{Q}_{12}^\\mu {Q}_{12}^\\nu}{\\wn w_1^2 \\wn w_2^2} - \\frac{{P}_{12}^\\mu {P}_{12}^\\nu}{(\\wn k\\cdot p_1)^2 (\\wn k\\cdot p_2)^2} \\bigg)\\bigg] e_{(\\mu\\nu)}^{h}\\,.\\label{eqn:RNradKernel}\n\\end{multline}\nWe verify this result in appendix~\\ref{app:worldlines}, by calculating the corresponding classical energy-momentum tensor from perturbative solutions to the field equations of Einstein--Maxwell theory.\n\nTo obtain an amplitude for graviton emission we symmetrised the result from the double copy. This rather conveniently restricted to a graviton amplitude. However, because we used double copy numerators from different theories there is also a non-zero contribution from the antisymmetric part of the polarisation tensor, $e_{[\\mu\\nu]}^{h}$. For gravity, this corresponds to the axion mode $B_{\\mu\\nu}$, and thus $\\mathcal{M}^{[\\mu\\nu]}$ should correspond to axion emission. It is a simple matter to show that\n\\begin{equation}\n\\hbar^{\\frac52} \\mathcal{M}^{(0)}_{\\rm RN, axion} = -\\frac{e^2\\kappa}{\\wn w_1^2 \\wn w_2^2} Q_1 Q_2 \\left({Q}^{\\nu}_{12} {P}^\\mu_{12} - {Q}^\\mu_{12} {P}^\\nu_{12}\\right) e_{[\\mu\\nu]}^{h} + \\mathcal{O}(\\hbar)\\,.\\label{eqn:axion}\n\\end{equation}\nAs well as being antisymmetric, for an on-shell state with momentum $\\wn k^\\mu$ the axion polarisation tensor must satisfy $e^h_{[\\mu\\nu]}(\\wn k) \\wn k^\\mu = 0$ . Thus it has the explicit form $e^h_{[\\mu\\nu]}(\\wn k) = \\epsilon_{\\mu\\nu\\rho\\sigma} \\wn k^\\rho \\xi^\\sigma\/(\\wn k \\cdot \\xi)$, where $\\xi^\\sigma$ is an unspecified reference vector. We will find it convenient to take $\\xi^\\sigma = p_1^\\sigma + p_2^\\sigma$. Expanding the amplitude and then including this choice we have\n\\[\n\\hbar^{\\frac72} \\bar{\\mathcal{M}}^{(0)}_{\\rm RN,axion} &= -\\frac{2e^2\\kappa}{\\wn w_1^2 \\wn w_2^2} Q_1 Q_2 \\Big[\\left((\\wn k\\cdot p_1)\\, p_2^\\mu  - (\\wn k\\cdot p_2)\\, p_1^\\mu\\right)(\\wn w_1 - \\wn w_2)^\\nu \\\\ & \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad - \\left(\\wn w_1^2 - \\wn w_2^2\\right) p_2^\\mu p_1^\\nu \\Big] \\epsilon_{\\mu\\nu\\rho\\sigma} \\frac{\\wn k^\\rho \\xi^\\sigma}{\\wn k \\cdot \\xi}\\\\\n&= \\frac{4 e^2\\kappa}{\\wn w_1^2 \\wn w_2^2}Q_1 Q_2\\, \\epsilon_{\\mu\\nu\\rho\\sigma} p_1^\\mu p_2^\\nu \\wn w_1^\\rho \\wn w_2^\\sigma\\,,\n\\]\nleading to a LO axion radiation kernel\n\\begin{multline}\n\\mathcal{R}_{\\rm RN,axion}^{(0)} = e^2\\kappa Q_1 Q_2 \\! \\int\\! \\dd^4\\wn w_1 \\dd^4\\wn w_2\\, \\del(p_1\\cdot\\wn w_1) \\del(p_2\\cdot\\wn w_2) \\del^{(4)}(\\wn w_1 + \\wn w_2 - \\wn k) \\\\ \\times \\frac{e^{ib\\cdot\\wn w_1}}{\\wn w_1^2 \\wn w_2^2} \\epsilon_{\\mu\\nu\\rho\\sigma} u_1^\\mu u_2^\\nu \\wn w_1^\\rho \\wn w_2^\\sigma\\,.\n\\end{multline}\nThe electromagnetic scattering is only able to radiate axions due to their coupling with the vector bosons. It is not possible to couple axions to a scalar particle in the absence of spin, as to do so breaks diffeomorphism invariance and axion gauge symmetry \\cite{Goldberger:2017ogt}. The amplitude of \\eqn~(\\ref{eqn:axion}) is therefore purely due to the antisymmetrisation of the triple gauge vertex; given that we have an isolated single vertex responsible, we can verify that the radiation is indeed axionic by using Einstein--Maxwell axion-dilaton coupled gravity. The action is\n\\begin{equation}\n\\hspace{-3mm} S=\\frac1{2\\kappa}\\int\\! \\d^4x \\,{\\sqrt{g}}\\left[R - \\frac{1}{2}(\\partial_\\mu\\phi_\\textrm{d})^2 - \\frac{1}{2}e^{4\\phi_\\textrm{d}}(\\partial_\\mu\\zeta)^2 -  e^{-2\\phi_\\textrm{d}}F^2\\! - \\zeta F^{\\mu\\nu}{F}^*_{\\mu\\nu}\\right].\n\\end{equation}\nHere ${F}^*_{\\mu\\nu}$ is the dual electromagnetic field strength, $\\phi_\\textrm{d}$ is the dilaton field, $\\zeta$ is the axion pseudoscalar defined in \\eqn~\\eqref{eqn:axionscalar}, and $g=-\\det(g_{\\mu\\nu})$. Treating the axion-photon interaction in the final term perturbatively and integrating by parts gives\n\\begin{equation}\n\\mathcal{L}_{\\rm int} \\sim A_\\lambda\\partial_{[\\alpha}A_{\\beta]}H^{\\lambda\\alpha\\beta}\\,.\n\\end{equation}\nAllocating momenta to the axion-photon vertex in the same way as in the scattering amplitudes, this interaction term corresponds to the Feynman rule\n\\hspace{-0.5cm}\n\\begin{tabular}[h!]{cccc}\n\t\\begin{minipage}[c]{0.03\\textwidth}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{0.18\\textwidth}\n\t\t\\centering\n\t\t\\scalebox{0.8}{\n\t\t\\begin{tikzpicture}[thick, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={zigzag}]\n\t\t\\tikzset{zigzag\/.style={decorate, decoration=zigzag}}\n\t\t\\begin{feynman}\n\t\t\\vertex (v1);\n\t\t\\vertex [above = 1.44 of v1] (v2) {$\\mu \\nu$};\n\t\t\\vertex [below right = 1 and 1.5 of v1] (v3);\n\t\t\\vertex [below left = 1 and 1.5 of v1] (v4);\n\t\t\\vertex [left = 0.06 of v1] (a1) ;\n\t\t\\vertex [right = 0.06 of v1] (a2) ;\n\t\t\\vertex [above = 1.44 of a1] (a3) ;\n\t\t\\vertex [above = 1.44 of a2] (a4) ;\n\t\t\\vertex [right = 0.2 of v1] (a5) ;\n\t\t\\vertex [above = 1.44 of a5] (a6) ;\n\t\t\\vertex [below right = 1 and 1.5 of v1] (v33) {$\\sigma$};\n\t\t\\vertex [below left = 1 and 1.5 of v1] (v44) {$\\rho$};\n\t\t\\draw[zigzag] (a1) -- (a3);\n\t\t\\draw[zigzag] (a2) -- (a4);\n\t\t\\diagram*{(a5) -- [white, scalar, momentum' = {[arrow style=black,thick]\\(k\\)}] (a6)};\n\t\t\\diagram*{(v3) -- [photon, momentum = \\(q_2\\)] (v1)};\n\t\t\\diagram*{(v4) -- [photon, momentum = \\(q_1\\)] (v1)};\n\t\t\\filldraw [color=black] (v1) circle [radius=2pt];\n\t\t\\end{feynman}\n\t\t\\end{tikzpicture}}\n\t\\end{minipage}&\n\t\\hspace{-0.7cm}\n\t\\begin{minipage}[c]{0.8\\textwidth}\n\t\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t=-2i\\kappa\\,\\big\\{(q_2-q_1)_\\mu  k_{[\\rho} \\eta_{\\sigma]\\nu} -& (q_2-q_1)_\\nu k_{[\\rho}\\eta_{\\sigma]\\mu} \\\\ &+ \\eta_{\\mu[\\rho}\\eta_{\\sigma]\\nu} \\left[q_1\\cdot k - q_2\\cdot k\\right]\\big\\}.\n\t\t\\end{aligned}\n\t\t\\end{equation}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{0.03\\textwidth}\n\t\\end{minipage}\n\t\\vspace{6pt}\n\\end{tabular}\n\\noindent Constructing a five-point amplitude for massive two external scalars with this axion emission vertex and taking the classical limit then precisely reproduces the results of \\eqn~\\eqref{eqn:axion}, verifying that the antisymmetric double copy artefact is indeed axionic.\n\n\\subsection{Momentum conservation and radiation reaction}\n\\label{sec:ALD}\n\nLet us return to gauge theory, and in particular the YM radiation kernel in equation~\\eqref{eqn:LOradKernel}. An immediate corollary of this result is the LO radiation in classical electrodynamics; replacing colour with electric charges and ignoring the structure constant terms yields\n\\begin{equation}\n\\begin{aligned}\n\\RadKerCl_\\text{EM}(\\bar k)&=\\int \\! \\dd^4 \\xferb_1 \\dd^4 \\xferb_2 \\;\n\\del(\\ucl_1\\cdot\\xferb_1) \\del(\\ucl_2\\cdot\\xferb_2) \n\\del^{(4)}(\\bar k - \\xferb_1 - \\xferb_2) \\, e^{i\\xferb_1 \\cdot b} \n\\\\& \\hphantom{\\rightarrow}\\times\\biggl\\{\n\\frac{1}{m_1} \\frac{Q_1^2Q_2^{\\vphantom{2}}}{\\xferb_2^2} \n\\biggl[-\\ucl_2\\cdot\\varepsilon^h_k + \\frac{(\\ucl_1\\cdot \\ucl_2)(\\xferb_2\\cdot\\varepsilon^h_k)}{\\ucl_1\\cdot\\bar k} \n+ \\frac{(\\ucl_2\\cdot\\bar k)(\\ucl_1\\cdot\\varepsilon^h_k)}{\\ucl_1\\cdot\\bar k} \n\\\\&\\hspace*{40mm} \n- \\frac{(\\bar k\\cdot\\xferb_2)(\\ucl_1\\cdot \\ucl_2)(\\ucl_1\\cdot\\varepsilon^h_k)}{(\\ucl_1\\cdot\\bar k)^2}\\bigg] \n+ (1 \\leftrightarrow 2)\\biggr\\} \\,.\n\\label{eqn:Rcalculation}\n\\end{aligned}\n\\end{equation}\nIt is a simple calculation to see that the LO current which solves the Maxwell field equation has precisely the same expression, up to an overall sign~\\cite{Kosower:2018adc}.\n\nNow, we have already seen that conservation of momentum holds exactly (in \\sect{sect:allOrderConservation}) and in our classical expressions (in \\sect{sect:classicalConservation}). Let us ensure that there is no subtlety in these discussions by explicit calculation.\n\nTo do so, we calculate the part of the NNLO impulse $\\ImpBclsup{(\\textrm{rad})}$ which encodes radiation reaction, defined in \\eqn~\\eqref{eqn:nnloImpulse}. The two amplitudes appearing in equation~\\eqref{eqn:nnloImpulse} are in common with the amplitudes relevant for the radiated momentum, equation~\\eqref{eqn:defOfRLO}, though they are evaluated at slightly different kinematics. It will be convenient to change the sign of $\\xferb_\\alpha$ here; with that change, the amplitudes are\n\\begin{multline}\n\\AmplB^{(0)}(\\initialk_1, \\initialk_2 \\rightarrow \\initialk_1-\\hbar\\xferb_1\\,, \\initialk_2-\\hbar\\xferb_2 \\, , \\bar k) = \\frac{4Q_1^2 Q_2^{\\vphantom{2}}}{\\hbar^{2}\\, \\xferb_2^2} \n\\bigg[-p_2\\td\\varepsilon^h_k + \\frac{(p_1\\td p_2)(\\xferb_2\\td\\varepsilon^h_k)}{p_1\\cdot\\bar{k}} \n\\\\+ \\frac{(p_2\\cdot\\bar k)(p_1\\td\\varepsilon^h_k)}{p_1\\td\\bar k} \n- \\frac{(\\bar k\\td\\xferb_2)(p_1\\td p_2)(p_1\\td\\varepsilon^h_k)}{(p_1\\cdot\\bar k)^2}\\bigg] + ( 1 \\leftrightarrow 2)\n\\end{multline}\nand\n\\begin{multline}\n\\AmplB^{(0)*}(\\initialk_1+\\hbar \\qb_1\\,, \\initialk_2 + \\hbar \\qb_2 \\rightarrow \n\\initialk_1-\\hbar\\xferb_1\\,, \\initialk_2-\\hbar\\xferb_2 \\,, \\bar k)\n= \\frac{4Q_1^2 Q_2^{\\vphantom{2}}}{\\hbar^{2}\\, \\xferb_2'^2} \n\\bigg[\\tm p_2\\td\\varepsilon^{h*}_k\n\\\\\\tp \\frac{(p_1\\td p_2)(\\xferb'_2\\td\\varepsilon^{h*}_k)}{p_1\\cdot\\bar k} \n\\tp \\frac{(p_2\\td\\bar k)(p_1\\td\\varepsilon_k^{h*})}{p_1\\cdot\\bar k} \n\\tm \\frac{(\\bar k\\td\\xferb'_2)(p_1\\td p_2)(p_1\\td\\varepsilon^{h*}_k)}{(p_1\\cdot\\bar k)^2}\\bigg] \n + ( 1 \\leftrightarrow 2)\\,,\n\\end{multline}\nwhere we find it convenient to define $\\xferb_\\alpha' = \\qb_\\alpha + \\xferb_\\alpha$ (after the change of sign). \n\nWe can now write the impulse contribution as\n\\begin{multline}\n\\ImpBclsup{(\\textrm{rad})} = -e^6 \\Lexp \\int \\! d\\Phi(\\bar k) \n\\prod_{\\alpha = 1,2} \\int \\dd^4\\xferb_\\alpha\\, \\dd^4 \\xferb'_\\alpha \\; \\xferb_1^\\mu \\; \n\\\\ \\times \\mathcal{X}^*(\\xferb'_1, \\xferb'_2, \\bar k) \\mathcal{X}(\\xferb_1, \\xferb_2, \\bar k)\n\\Rexp\\,  ,\n\\label{eqn:impulseNNLO}\n\\end{multline}\nwhere \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{X}(\\xferb_1, \\xferb_2, \\bar k) &= {4}  \\, \\del(2 \\xferb_1\\cdot p_1)\n\\del(2 \\xferb_2\\cdot p_2) \\del^{(4)}(\\bar k - \\xferb_1 - \\xferb_2) \n\\, e^{i b \\cdot  \\xferb_1}\n\\\\ \n& \\quad \\times \n\\biggl\\{Q_1^2 Q_2^{\\vphantom{2}} \\frac{\\varepsilon_{\\mu}^h(\\wn k) }{\\xferb_2^2}\n\\bigg[-p_2^\\mu + \\frac{p_1\\cdot p_2 \\, \\xferb_2^\\mu}{p_1\\cdot\\bar k} \n+ \\frac{p_2\\cdot\\bar k \\, p_1^\\mu}{p_1\\cdot\\bar k} \n\\\\&\\hspace*{35mm}\n- \\frac{(\\bar k\\cdot\\xferb_2)(p_1\\cdot p_2) \\, p_1^\\mu}{(p_1\\cdot\\bar k)^2}\\bigg] \n+ (1 \\leftrightarrow 2)\\biggr\\}\\,.\n\\end{aligned}\n\\label{eqn:X1}  \n\\end{equation}\nThis expression is directly comparable to those for radiated momentum: \\eqn~\\eqref{eqn:impulseNNLO}, and the equivalent impulse contribution to particle 2, balance the radiated momentum \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO} using $\\xferb_1^\\mu + \\xferb_2^\\mu = \\bar k^\\mu$, provided that the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation}, is related to integrals over $\\mathcal{X}$. Indeed this relationship holds: the integrations present in the radiation kernel are supplied by the $\\xferb_\\alpha$ and $\\xferb'_\\alpha$ integrals in \\eqn~\\eqref{eqn:impulseNNLO}; these integrations disentangle in the sum of impulses on particles 1 and 2 when we impose $\\xferb_1^\\mu + \\xferb_2^\\mu = \\bar k^\\mu$, and then form the square of the radiation kernel.\n\nIt is interesting to compare this radiated momentum with the situation in traditional formulations of classical physics, where one must include the ALD radiation reaction force \nby hand in order to enforce momentum conservation. Because the situation is simplest when only one particle is dynamical, let us take the mass $m_2$ to be very large compared to \n$m_1$ in the remainder of this section, and work in particle 2's rest frame.  In this frame, it does not radiate, and the only radiation reaction is on particle 1 --- the radiated momentum is precisely balanced by  the impulse on particle 1 due to the ALD force. We can therefore continue our discussion with reference to our expression for radiated momentum, \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO}, and the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation}. In this situation we may also simplify the kernels by dropping the $(1 \\leftrightarrow 2)$ instruction: these terms will be dressed by an inverse power of $m_2$, and so are subdominant when $m_2 \\gg m_1$. \n\nWe will soon compute the impulse due to the ALD force directly from its classical expression in \\eqn~\\eqref{eqn:ALDclass}. But in preparation for that comparison there is one step which we must take. Classical expressions for the force---which involve only the particle's momentum and its derivatives---do not involve any photon phase space. So we must perform the integration over $\\df(\\wn k)$ which is present in \\eqn~\\eqref{eqn:radiatedMomentumClassicalLO}. \n\nTo organise the calculation, we integrate over the $\\qb_1$ variables in the radiation kernel, \\eqn~\\eqref{eqn:Rcalculation} using the four-fold delta function, so that we may write the radiated momentum as\n\\begin{align}\n\\hspace*{-3mm}\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{m_1^2}\\! \\int\\!\\dd^4\\qb\\, \\dd^4\\qb'\\;\ne^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb')) \n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2} \\phInt\\,,\\label{eqn:rrmidstage}\n\\end{align}\nwhere we renamed the remaining variables, $\\xferb_2\\rightarrow \\qb$ and $\\xferb'_2\\rightarrow \\qb'$, in order to match the notation used later. After some algebra we find\n\\begin{multline}\n\\phInt = \\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu \\,\n\\left[ 1 + \\frac{(\\ucl_1\\cdot \\ucl_2)^2(\\qb\\cdot \\qb')}{\\wn E^2} \n+ \\frac{(\\ucl_2\\cdot\\bar k)^2}{\\wn E^2}  \\right. \\\\ \n\\left. - \\frac{(\\ucl_1\\cdot \\ucl_2)(\\ucl_2\\cdot\\bar k)\\,\\bar k\\cdot(\\qb+\\qb')}{\\wn E^3} + \\frac{(\\ucl_1\\cdot \\ucl_2)^2(\\bar k\\cdot\\qb)(\\bar k\\cdot\\qb')}{\\wn E^4}  \\right].\n\\label{eqn:phaseSpaceIntegral}\n\\end{multline}\nThe quantity $\\wn E$ is defined to be $\\wn E = \\ucl_1 \\cdot \\wn k$; in view of the delta function, the integral is constrained so that $\\wn E = \\ucl_1 \\cdot \\qb$. This quantity is the wavenumber of the photon in the rest frame of particle 1, and is fixed from the point of view of the phase space integration. As a result, the integrals are simple: there are two delta functions (one explicit, one in the phase space measure) which can be used to perform the $\\bar k^0$ integration and to fix the magnitude of the spatial wavevector. The remaining integrals are over angles. The relevant results were calculated in appendix~C of ref.~\\cite{Kosower:2018adc}, and are\n\\begin{equation}\n\\begin{gathered}\n\\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu = \\frac{\\wn E^2}{2\\pi} u_1^\\mu \\Theta(\\wn E)\\,,\\\\\n\\int \\! \\df (\\bar k) \\, \\del(\\ucl_1\\cdot \\bar k - \\wn E) \\, \\bar k^\\mu \\bar k^\\nu \\bar k^\\rho = \\frac{\\wn E^4}{\\pi}\\left(u_1^\\mu u_1^\\nu u_1^\\rho - \\frac12 u_1^{(\\mu} \\eta^{\\nu\\rho)}\\right) \\Theta(\\wn E)\\,.\n\\end{gathered}\n\\end{equation}\nThe radiated momentum then takes a remarkably simple form after the phase space integration:\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{3\\pi m_1^2} &\\int\\!\\dd^4\\qb \\,\n\\dd^4\\qb'\\; e^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb'))\n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2}\n\\\\ &\\times\\Theta(\\ucl_1\\cdot\\qb)\\,\\left[(\\ucl_1\\cdot\\qb)^2 \n+ \\qb\\cdot\\qb'(\\ucl_1\\cdot \\ucl_2)^2\\right] \\ucl_1^\\mu \\,.\n\\label{eqn:radTheta}\n\\end{aligned}\n\\end{equation}\n\nThe $\\Theta$ function is a remnant of the photon phase space volume, so it will be convenient to remove it. The delta functions in the integrand in \\eqn~\\eqref{eqn:radTheta} constrain the components of the vectors $\\qb$ and $\\qb'$ which lie in the two dimensional space spanned by $u_1$ and $u_2$. Let us call the components of $q$ and $q'$ in this plane to be $q_\\parallel$ and $q'_\\parallel$. Then the delta functions set $q_\\parallel = q'_\\parallel$. As a result, the integrand (ignoring the $\\Theta$ function) is symmetric in $q_\\parallel \\rightarrow - q_\\parallel$. Consequently we may symmetrise to find\n\\begin{equation}\n\\begin{aligned}\n\\Rad^{\\mu,(0)}_\\class = -\\frac{e^6 Q_1^4 Q_2^2}{6\\pi m_1^2} &\\int\\!\\dd^4\\qb\\, \\dd^4\\qb'\\;\ne^{-ib\\cdot(\\qb - \\qb')} \\del(\\ucl_1\\cdot(\\qb - \\qb')) \n\\frac{\\del(\\ucl_2\\cdot\\qb)}{\\qb^2} \\frac{\\del(\\ucl_2\\cdot\\qb')}{\\qb'^2} \n\\\\ &\\hspace*{10mm}\\times\\left[(\\ucl_1\\cdot\\qb)^2 \n+ \\qb\\cdot\\qb'(\\ucl_1\\cdot \\ucl_2)^2\\right]\\ucl_1^\\mu \\,.\n\\label{eqn:rrResult}\n\\end{aligned}\n\\end{equation}\n\nIt is now remarkably simple to see that this expression is equal but opposite to the impulse obtained from the classical ALD force in \\eqn~\\eqref{eqn:ALDclass}. Working in perturbation theory, the lowest order contribution to $dp_1 \/ d\\tau$ is of order $e^2$, due to the (colour-stripped) LO Lorentz force~\\eqref{eqn:Wong-momentum}. We can determine this explicitly using the methods of appendix~\\ref{app:worldlines}: with particle 2 kept static, one finds\n\\begin{equation}\n\\frac{d p^{\\mu,(0)}_1}{d \\tau_1} =i e^2 Q_1 Q_2 \\int \\!\\dd^4 \\qb \\, \\del(\\qb \\cdot u_2) \\, e^{- i \\qb \\cdot (b + u_1 \\tau_1)} \\,  \\frac{\\qb^\\mu \\, u_1 \\cdot u_2 - u_2^\\mu \\, \\qb \\cdot u_1}{\\qb^2}\\,.\n\\label{eqn:LOforce}\n\\end{equation}\nTherefore $\\Delta {p^\\mu_1}_{\\rm ALD}$ is at least of order $e^4$. However, this potential contribution to the ALD impulse vanishes. To see this, observe that the acceleration due to the LO Lorentz force gives rise to an ALD impulse of\n\\begin{equation}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^4 Q_1^3 Q_2}{6\\pi m_1}\\int\\!\\dd^4\\qb\\, \\del(\\qb\\cdot u_1) \\del(\\qb\\cdot u_2) \\, e^{-i\\wn{q}\\cdot b} \\, \\qb\\cdot u_1 \\, \\big(\\cdots\\big) = 0\\,.\n\\end{equation}\nAn alternative point of view on the same result is to perform the time integral in equation~\\eqref{eqn:ALDclass}, noting that the second term in the ALD force is higher order. The impulse is then proportional to $f^\\mu(+\\infty) - f^\\mu(-\\infty)$, the difference in the asymptotic Lorentz forces on particle 1. But at asymptotically large times the two particles are infinitely far away, so the Lorentz forces must vanish. Since this second argument does not rely on perturbation theory we may ignore the first term in the ALD force law.\n\nThus, the first non-vanishing impulse due to radiation reaction is of order $e^6$. Since we only need the leading order Lorentz force to evaluate the ALD impulse, we can anticipate that the result will be very simple. Indeed, integrating the ALD force, we find that the impulse on particle 1 due to radiation reaction is\n\\begin{multline}\n\\Delta {p^\\mu_1}_{\\rm ALD} = \\frac{e^6 Q_1^4 Q_2^2}{6\\pi m_1^2} u_1^\\mu \\!\\int\\! \\dd^4\\qb\\,\\dd^4\\qb'\\, \\del(\\qb\\cdot u_2) \\del(\\qb'\\cdot u_2)  \\del(u_1\\cdot(\\qb-\\qb')) \\, e^{-ib\\cdot(\\qb-\\qb')} \\\\ \\times\\frac{1}{\\qb^2 \\qb'^2} \\left[(\\qb\\cdot u_1)^2 + \\qb\\cdot\\qb'(u_1\\cdot u_2)^2 \\right].\n\\label{eqn:classicalRadiationImpulse}\n\\end{multline}\nThis is precisely the expression~\\eqref{eqn:rrResult} we found using our quantum mechanical approach.\n\n\n\n\\section{Discussion}\n\\label{sec:KMOCdiscussion}\n\nIn order to apply on-shell scattering amplitudes to the calculation of classically observable quantities for black holes, one needs a definition of the observables in the quantum theory.  One also needs a path and clear set of rules for taking the classical limit of the quantum observables.  In this first part of the thesis we have constructed one such path. Our underlying motivation is to understand the dynamics of classical general relativity through the double copy. In particular, we are interested in the relativistic two-body problem which is so central to the physics of the compact binary coalescence events observed by LIGO and Virgo. Consequently, we focused on observables in two-body events.\n\nWe have shown how to construct two observables relevant to this problem: the momentum transfer or impulse~(\\ref{eqn:defl1}) on a particle; and the momentum emitted as radiation~(\\ref{eqn:radiationTform}) during the scattering of two charged but spinless point particles.  We have shown how to restore $\\hbar$'s and classify momenta in \\sect{sec:RestoringHBar}; in \\sect{sec:PointParticleLimit}, how to choose suitable wavefunctions for localised single particle states; and established in section~\\ref{sec:classicalLimit} the conditions under which the classical limit is simple for point-particle scattering. With these formalities at hand we were able to further provide simplified leading and next-to-leading-order expressions in terms of on-shell scattering amplitudes for the impulse in \\eqns{eqn:impulseGeneralTerm1classicalLO}{eqn:classicalLimitNLO}, and for the radiated momentum in \\eqn~\\eqref{eqn:radiatedMomentumClassical}.  These expressions apply directly to both gauge theory and gravity. In sections~\\ref{sec:examples} and \\ref{sec:LOradiation}, we used explicit expressions for amplitudes in QED, Yang--Mills theory and perturbative Einstein gravity to obtain classical results. We have been careful throughout to ensure that our methods correctly incorporate conservation of momentum, without the need to introduce an analogue of the Abraham--Lorentz--Dirac radiation reaction.\n\nOther momentum observables should be readily accessible by similar derivations: for example the total radiated angular momentum is of particular current interest \\cite{Damour:2020tta}, and is moreover accessible from worldline QFT \\cite{Jakobsen:2021smu}; it would be very interesting to understand how this observable fits into our formalism. Higher-order corrections, to the extent they are unambiguously defined in the classical theory, require the harder work of computing two- and higher-loop amplitudes, but the formalism of these chapters will continue to apply.\n\nOur setup has features in common with two related, but somewhat separate, areas of current interest. One area is the study of the potential between two massive bodies. The second is the study of particle scattering in the eikonal. Diagrammatically, the study of the potential is evidently closely related to the impulse of chapter~\\ref{chap:impulse}. To some extent this is by design: we wished to construct an on-shell observable related to the potential. But we have also been able to construct an additional observable, the radiated momentum, which is related to the gravitational flux.\n\nIt is interesting that classical physics emerges in the study of the high-energy limit of quantum scattering~\\cite{Amati:1987wq,tHooft:1987vrq,Muzinich:1987in,Amati:1987uf,Amati:1990xe}, see also refs.~\\cite{Damour:2016gwp,Damour:2017zjx}. \nIndeed the classical centre-of-momentum scattering angle can be obtained from the eikonal function (see, for example ref.~\\cite{DAppollonio:2010krb}). This latter function must therefore be related as well to the impulse, even though we have not taken any high-energy limit. Indeed, the impulse and the scattering angle are equivalent at LO and NLO, \nbecause no momentum is radiated at these orders. Therefore the scattering angle completely determines the change in momentum of the particles (and vice versa). The connection to the eikonal function should be interesting to explore.\n\nAt NNLO, on the other hand, the equivalence between the angle and the impulse fails. This is because of radiation: knowledge of the angle tells you where the particles went,\nbut not how fast. In this respect the impulse is more informative than the angle. Eikonal methods are still applicable in the radiative case~\\cite{Amati:1990xe},\nso they should reproduce the high-energy limit of the expectation value of the radiated momentum. Meanwhile at low energies, methods based on soft theorems could provide a bridge between the impulse and the radiated momentum~\\cite{Laddha:2018rle,Laddha:2018myi,Sahoo:2018lxl}. Indeed, a first step in these directions was recently made in \\cite{A:2020lub}. Radiation reaction physics can also be treated in this regime \\cite{DiVecchia:2021ndb}, and we look forward to future progress in understanding how these references overlap with our formalism.\n\nThe NLO scattering angle is, in fact, somewhat simpler than the impulse: see ref.~\\cite{Luna:2016idw} for example. Thanks to the exponentiation at play in the eikonal limit, it is the triangle diagram which is responsible for the NLO correction. But the impulse contains additional contributions, as we discussed in \\sect{sec:nloQimpulse}. Perhaps this is because the impulse must satisfy an on-shell constraint, unlike the angle.\n\nThe focus of our study of inelastic scattering has been the radiation kernel introduced in~\\eqref{eqn:defOfR}. Equivalent to a classical current, this object has proven to be especially versatile in the application of amplitudes methods to gravitational radiation. It has played a direct role in studies of the Braginsky--Thorne memory effect \\cite{Bautista:2019tdr} and gravitational shock waves \\cite{Cristofoli:2020hnk}; derivations of the connections between amplitudes, soft limits and classical soft theorems \\cite{A:2020lub}; and calculations of Newman--Penrose spinors for long-range radiation in split-signature spacetimes \\cite{Monteiro:2020plf}. The kernel is also closely related to progress in worldline QFT \\cite{Mogull:2020sak,Jakobsen:2021smu}. However, the reader may object that we have not in fact calculated the total emitted radiation. This was recently achieved in ref.~\\cite{Herrmann:2021lqe}, using integral evaluation techniques honed in $\\mathcal{N}=8$ supergravity \\cite{Parra-Martinez:2020dzs}. To calculate the full observable the authors of \\cite{Herrmann:2021lqe} took a similar approach to our treatment of the non-linear impulse contributions at 1-loop, treating the product of radiation kernels as a cut of a 2-loop amplitude. Their application of the formalism presented here has led to state-of-the-art results for post--Minkowskian bremsstrahlung, already partially recovered from PM effective theory \\cite{Mougiakakos:2021ckm}. \n\nIn these two chapters we restricted attention to spinless scattering. In this context, for colourless particles such as astrophysical black holes the impulse (or equivalently, the angle) is the only physical observable at LO and NLO, and completely determines the interaction Hamiltonian between the two particles~\\cite{Damour:2016gwp,Damour:2017zjx}. The situation is richer in the case of arbitrarily aligned spins --- then the change in spins of the particles is an observable which is not determined by the scattering angle. Fully specifying the dynamics of black holes therefore requires including spin in our formalism: this is the topic of the second part of the thesis.\n\\part{Spinning black holes}\n\\label{part:spin}\n\n\\chapter{Observables for spinning particles}\n\\label{chap:spin}\n\n\\section{Introduction}\n\nTo begin this part of the thesis we will continue in the vein of previous chapters and use quantum field theory, now for particles with non-zero spin, to calculate observables for spinning point-particles. Our focus will be the leading-order scattering of black holes, however the formalism is applicable more widely \\cite{Maybee:2019jus}. In chapter~\\ref{chap:intro} we discussed at some length how scattering amplitudes have been applied to the dynamics of spinning Kerr black holes, and more fundamentally how minimally coupled amplitudes behave as the on-shell avatar of the no-hair theorem. Here we remove the restriction to the aligned-spin configuration in the final results of \\cite{Guevara:2018wpp,Bautista:2019tdr}, and the restriction to the non-relativistic limit in the final results of \\cite{Chung:2018kqs}.  We use on-shell amplitudes to directly compute relativistic classical observables for generic spinning-particle scattering, reproducing such results for black holes obtained by classical methods in \\cite{Vines:2017hyw}, thereby providing more complete evidence for the correspondence between minimal coupling to gravity and classical black holes.\n\nWe will accomplish this by relaxing the restriction to scalars in previous chapters. In addition to the momentum impulse $\\Delta p^\\mu$, there is now another relevant on-shell observable, the change $\\Delta s^\\mu$ in the spin (pseudo-)vector $s^\\mu$, which we will call the \\textit{angular impulse}.  We introduce this quantity in \\sect{sec:GRspin}, where we also review classical results from \\cite{Vines:2017hyw} for binary black hole scattering at 1PM order. \nIn \\sect{sec:QFTspin} we consider the quantum analogue of the spin vector, the Pauli--Lubanski operator; manipulations of this operator allow us to write expressions for the angular impulse akin to those for the impulse of chapter~\\ref{chap:impulse}. Obtaining the classical limit requires some care, which we discuss before constructing example gravity amplitudes in \\sect{sec:amplitudes} from the double copy. \n\nRather than working at this stage with massive spinor representations valid for any quantum spin $s$, in the vein of \\cite{Arkani-Hamed:2017jhn}, we will ground our intuition in explicit field representations of the Poincar\\'{e} group --- we will work with familiar spin 1\/2 Dirac fermions and massive, spin 1 bosons. Explicit representations can indeed be chosen for any generic spin $s$ field; the applications of these representations to black hole physics was studied in ref.~\\cite{Bern:2020buy}. However, for higher spins the details become extremely involved. In \\sect{sec:KerrCalcs} we show that substituting familiar low spin examples into our general formalism exactly reproduces the leading terms of all-multipole order expressions for the impulse and angular impulse of spinning black holes \\cite{Vines:2017hyw}. Finally, we discuss how our results further entwine Kerr black holes and scattering amplitudes in \\sect{sec:angImpDiscussion}.\n\nThis chapter is based on work conducted in collaboration with Donal O'Connell and Justin Vines, published in \\cite{Maybee:2019jus}.\n\n\\section{Spin and scattering observables in classical gravity}\\label{sec:GRspin}\n\nBefore setting up our formalism for computing the angular impulse, let us briefly review aspects of this observable in relativistic classical physics. \n\n\\subsection{Linear and angular momenta in asymptotic Minkowski space}\n\nTo describe the incoming and outgoing states for a weak scattering process in asymptotically flat spacetime we can use special relativistic physics, working as in Minkowski spacetime. There, any isolated body has a constant linear momentum vector $p^\\mu$ and an antisymmetric tensor field $J^{\\mu\\nu}(x)$ giving its total angular momentum about the point $x$, with the $x$-dependence determined by $J^{\\mu\\nu}(x')=J^{\\mu\\nu}(x)+2p^{[\\mu}(x'-x)^{\\nu]}$, or equivalently $\\nabla_\\lambda J^{\\mu\\nu}=2p^{[\\mu}\\delta^{\\nu]}{}_\\lambda$.\n\nRelativistically, centre of mass (CoM) position and intrinsic and orbital angular momenta are frame-dependent concepts, but a natural inertial frame is provided by the direction of the momentum $p^\\mu$, giving the proper rest frame.  We define the body's proper CoM worldline to be the set of points $r$ such that $J^{\\mu\\nu}(r)p_\\nu=0$, i.e.\\ the proper rest-frame mass-dipole vector about $r$ vanishes, and we can then write\n\\begin{equation}\\label{eqn:Jmunu}\nJ^{\\mu\\nu}(x)=2p^{[\\mu}(x-r)^{\\nu]}+S^{\\mu\\nu},\n\\end{equation}\nwhere $r$ can be any point on the proper CoM worldline, and where $S^{\\mu\\nu}=J^{\\mu\\nu}(r)$ is the intrinsic spin tensor, satisfying\n\\begin{equation}\nS^{\\mu\\nu} p_\\nu=0.\\label{eqn:SSC}\n\\end{equation}\nEquation \\eqref{eqn:SSC} is often called the ``covariant'' or Tulczyjew--Dixon spin supplementary condition (SSC) \\cite{Fokker:1929,Tulczyjew:1959} in its (direct) generalization to curved spacetime in the context of the Mathisson--Papapetrou--Dixon equations \\cite{Mathisson:1937zz,Mathisson:2010,Papapetrou:1951pa,Dixon1979,Dixon:2015vxa} for the motion of spinning extended test bodies.\nGiven the condition \\eqref{eqn:SSC}, the complete information of the spin tensor $S^{\\mu\\nu}$ is encoded in the momentum $p^\\mu$ and the spin pseudovector \\cite{Weinberg:1972kfs},\n\\begin{equation}\ns_\\mu = \\frac{1}{2m}\\epsilon_{\\mu\\nu\\rho\\sigma} p^\\nu S^{\\rho\\sigma} = \\frac{1}{2m}\\epsilon_{\\mu\\nu\\rho\\sigma} p^\\nu J^{\\rho\\sigma}(x),\\label{eqn:GRspinVec}\n\\end{equation}\nwhere $\\epsilon_{0123} = +1$ and $p^2=m^2$.  Note that $s\\cdot p=0$; $s^\\mu$ is a spatial vector in the proper rest frame.\nGiven \\eqref{eqn:SSC}, the inversion of the first equality of \\eqref{eqn:GRspinVec} is\n\\begin{equation}\nS_{\\mu\\nu} =\\frac{1}{m} \\epsilon_{\\mu\\nu\\lambda\\tau} p^\\lambda s^\\tau.\\label{eqn:SSCintrinsicSpin}\n\\end{equation}\nThe total angular momentum tensor $J^{\\mu\\nu}(x)$ can be reconstructed from $p^\\mu$, $s^\\mu$, and a point $r$ on the proper CoM worldline, via \\eqref{eqn:SSCintrinsicSpin} and \\eqref{eqn:Jmunu}.\n\n\n\n\n\\subsection{Scattering of spinning black holes in linearised gravity}\n\nFollowing the no-hair property emphasised by equation~\\eqref{eqn:multipoles} of chapter~\\ref{chap:intro}, the full tower of gravitational multipole moments of a spinning black hole, and thus also its (linearised) gravitational field, are uniquely determined by its monopole $p^\\mu$ and dipole $J^{\\mu\\nu}$.  This is reflected in the scattering of two spinning black holes, in that the net changes in the holes' linear and angular momenta depend only on their incoming linear and angular momenta.  It has been argued in \\cite{Vines:2017hyw} that the following results concerning two-spinning-black-hole scattering, in the  1PM approximation to GR, follow from the linearised Einstein equation and a minimal effective action description of spinning black hole motion, the form of which is uniquely fixed at 1PM order by general covariance and appropriate matching to the Kerr solution.\n\n\n\n\nConsider two black holes with incoming momenta $p_1^\\mu=m_1 u_1^\\mu$ and $p_2^\\mu=m_2 u_2^\\mu$, defining the 4-velocities $u^\\mu=p^\\mu\/m$ with $u^2=1$, and incoming spin vectors $s_1^\\mu=m_1 a_1^\\mu$ and $s_2^\\mu=m_2 a_2^\\mu$, defining the rescaled spins $a^\\mu=s^\\mu\/m$ (with units of length, whose magnitudes measure the radii of the ring singularities).  Say the holes' zeroth-order incoming proper CoM worldlines are orthogonally separated at closest approach by a vectorial impact parameter $b^\\mu$, pointing from 2 to 1, with $b\\cdot u_1 =b\\cdot u_2=0$.  Then, according to the analysis of \\cite{Vines:2017hyw}, the net changes in the momentum and spin vectors of black hole 1 are given by\n\\begin{alignat}{3}\n\\begin{aligned}\n\\Delta p_1^{\\mu} &= \\textrm{Re}\\{\\mathcal Z^\\mu\\}+O(G^2),\n\\\\\n\\Delta s_1^{\\mu} &= - u_1^\\mu a_1^\\nu\\, \\textrm{Re}\\{\\mathcal Z_\\nu\\} - \\epsilon^{\\mu\\nu\\alpha\\beta} u_{1\\alpha} a_{1\\beta}\\, \\textrm{Im}\\{\\mathcal Z_\\nu\\}+O(G^2),\n\\end{aligned}\\label{eqn:KerrDeflections}\n\\end{alignat}\nwhere\n\\begin{equation}\n\\mathcal Z_\\mu = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}}\\Big[(2\\gamma^2 - 1)\\eta_{\\mu\\nu} - 2i\\gamma \\epsilon_{\\mu\\nu\\alpha\\beta} u_1^\\alpha u_2^\\beta\\Big]\\frac{ b^\\nu + i\\Pi^\\nu{ }_\\rho (a_1+a_2)^\\rho}{[b + i\\Pi(a_1+a_2)]^2}\\,,\n\\end{equation}\nwith $\\gamma = u_1\\cdot u_2$ the relative Lorentz factor, and with\n\\begin{equation}\n\\begin{aligned}\n\\Pi^\\mu{ }_\\nu &= \\epsilon^{\\mu\\rho\\alpha\\beta} \\epsilon_{\\nu\\rho\\gamma\\delta} \\frac{{u_1}_\\alpha {u_2}_\\beta u_1^\\gamma u_2^\\delta}{\\gamma^2 - 1}\\\\ &= \\delta^\\mu{ }_\\nu +\\frac1{\\gamma^2 - 1}\\bigg(u_1^\\mu({u_1}_\\nu - \\gamma {u_2}_\\nu) + u_2^\\mu({u_2}_\\nu - \\gamma {u_1}_\\nu)\\bigg) \\label{eqn:projector}\n\\end{aligned}\n\\end{equation}\nthe projector into the plane orthogonal to both incoming velocities.\nThe analogous results for black hole 2 are given by interchanging the identities $1\\leftrightarrow 2$.\n\nIf we take black hole 2 to have zero spin, $a_2^\\mu\\to0$, and if we expand to quadratic order in the spin of black hole 1, corresponding to the quadrupole level in 1's multipole expansion, then we obtain the results shown in \\eqref{eqn:JustinImpResult} and \\eqref{eqn:JustinSpinResult} below.  In the remainder of this chapter, developing necessary tools along the way, we show how those results can be obtained from classical limits of scattering amplitudes. In particular, we will consider one-graviton exchange between a massive scalar particle and a massive spin $s$ particle, with minimal coupling to gravity, with $s=1\/2$ to yield the dipole level, and with $s=1$ to yield the quadrupole level.\n\n\\section{Spin and scattering observables in quantum field theory}\n\\label{sec:QFTspin}\n\nWe have already established general formulae in quantum field theory for the impulse and radiated momentum; as the angular impulse is also on-shell similar methods should be applicable. A first task is to understand what quantum mechanical quantity corresponds to the classical spin pseudovector of equation~\\eqref{eqn:GRspinVec}. This spin vector is a quantity associated with a single classical body, and we therefore momentarily return to discussing single-particle states. \n\nParticle states of spin $s$ are irreducible representations of the little group. For massive particles in 4 dimensions the little group is isomorphic to $SU(2)$, and thus we can adopt the simplest coherent states considered in section~\\ref{sec:classicalSingleParticleColour}. The size of the representation is now determined by the spin quantum number $s$ associated with the states. For fractional spins the normalisation in \\eqn~\\eqref{eqn:ladderCommutator} of course generalises to an anticommutation relation.\n\nFor spinning particles we will thus adopt the wavepackets in equation~\\eqref{eqn:InitialStateSimple}, with the important distinction that the representation states now refer to the little group, not a gauge group. To make this distinction clear we will denote the little group states by $|\\xi\\rangle$ rather than $|\\chi\\rangle$ (not to be confused with the parameter~\\eqref{eqn:defOfXi}). The momentum space wavefunctions $\\varphi(p)$ remain entirely unchanged.\n\n\\subsection{The Pauli--Lubanski spin pseudovector}\n\nWhat operator in quantum field theory is related to the classical spin pseudovector of equation~\\eqref{eqn:GRspinVec}? We propose that the correct quantum-mechanical\ninterpretation is that the spin is nothing but the expectation value of the \\textit{Pauli--Lubanski} operator,\n\\begin{equation}\n\\W_\\mu = \\frac{1}{2}\\epsilon_{\\mu\\nu\\rho\\sigma} \\P^\\nu \\J^{\\rho\\sigma}\\,,\\label{eqn:PLvec}\n\\end{equation}\nwhere $\\P^\\mu$ and $\\mathbb{J}^{\\rho\\sigma}$ are translation and Lorentz generators respectively. In particular,\nour claim is that the expectation value\n\\begin{equation}\n\\langle s^\\mu \\rangle = \\frac1{m} \\langle \\mathbb{W}^\\mu\\rangle = \\frac1{2m} \\epsilon^{\\mu\\nu\\rho\\sigma} \\langle \\mathbb{P}_\\nu \\mathbb{J}_{\\rho\\sigma}\\rangle\n\\end{equation}\nof the Pauli--Lubanski operator on a single particle state~\\eqref{eqn:InitialStateSimple} is the quantum-mechanical generalisation of the classical spin pseudovector. Indeed, a simple comparison of equations~\\eqref{eqn:GRspinVec} and~\\eqref{eqn:PLvec} indicates a connection between the two quantities. We will provide abundant evidence for this link in the remainder of this chapter --- it is shown in greater detail in appendix~B of ref.~\\cite{Maybee:2019jus}.\n\nThe Pauli--Lubanski operator is a basic quantity in the classification of free particle states, although it receives less attention in introductory accounts of quantum field theory than it should. With the help of the Lorentz algebra\n\\[\n[\\J^{\\mu\\nu}, \\P^\\rho] &= i \\hbar (\\eta^{\\mu\\rho} \\P^\\nu - \\eta^{\\nu\\rho} \\P^\\mu) \\,, \\\\\n[\\J^{\\mu\\nu}, \\J^{\\rho\\sigma}] &= i \\hbar (\\eta^{\\nu\\rho} \\J^{\\mu\\sigma} - \\eta^{\\mu\\rho} \\J^{\\nu\\sigma} - \\eta^{\\nu\\sigma} \\J^{\\mu\\rho} + \\eta^{\\mu\\rho} \\J^{\\mu\\sigma}) \\, ,\n\\]\nit is easy to establish the important fact that the Pauli--Lubanski operator commutes with the momentum:\n\\[\n[\\P^\\mu, \\W^\\nu] = 0\\,.\\label{eqn:PWcommute}\n\\] \nFurthermore, as $\\W^\\mu$ is a vector operator it satisfies\n\\[\n[\\J^{\\mu\\nu}, \\W^\\rho] = i\\hbar (\\eta^{\\mu\\rho} \\W^\\nu - \\eta^{\\nu\\rho} \\W^\\mu) \\,.\n\\]\nIt then follows that the commutation relations of $\\W$ with itself are\n\\[\n[\\W^\\mu, \\W^\\nu] = i\\hbar \\epsilon^{\\mu\\nu\\rho\\sigma} \\W_\\rho \\P_\\sigma\\,.\n\\]\nOn single particle states this last commutation relation takes a particularly instructive form. Working in the rest frame of our massive particle state, evidently $W^0 = 0$. The remaining generators satisfy\\footnote{We normalise $\\epsilon^{123} = +1$, as usual.}\n\\[\n[ \\W^i, \\W^j] = i \\hbar m \\,\\epsilon^{ijk} \\W^k \\,,\n\\]\nso that the Pauli--Lubanski operators are nothing but the generators of the little group. Not only is this the basis for their importance, but also we will find that these commutation relations are directly useful in our computation of the change in a particle's spin during scattering.\n\nBecause the $\\mathbb{W}^\\mu$ commutes with the momentum, we have\n\\begin{equation}\n\\langle p'\\, j| \\W^\\mu |p\\, i \\rangle \\propto \\del_\\Phi(p-p')\\,.\n\\end{equation}\nWe define the matrix elements of $\\W$ on the states of a given momentum to be\n\\[\n\\langle p'\\, j| \\W^\\mu |p\\, i \\rangle \\equiv m \\s^\\mu_{ij}(p)\\, \\del_\\Phi(p-p') \\,,\\label{eqn:PLinnerProd}\n\\]\nso that the expectation value of the spin vector over a single particle wavepacket is\n\\[\n\\langle \\s^\\mu \\rangle = \\sum_{i,j} \\int d\\Phi(p) \\, | \\varphi(p) |^2 \\, \\xi^*_i \\s^\\mu_{ij} \\xi_j\\,.\n\\]\nThe matrix $\\s^\\mu_{ij}(p)$, sometimes called the spin polarisation vector, will be important below. These matrices inherit the commutation relations of the Pauli--Lubanski vector, so that in particular\n\\[\n[\\s^\\mu(p), \\s^\\nu(p) ] = \\frac{i\\hbar}{m} \\, \\epsilon^{\\mu\\nu\\rho\\sigma} \\s_\\rho(p) p_\\sigma \\,.\n\\]\n\nSpecialising now to a particle in a given representation, we may derive well known \\cite{Ross:2007zza,Holstein:2008sx,Bjerrum-Bohr:2013bxa,Guevara:2017csg} explicit expressions for the spin polarisation $s^\\mu_{ij}(p)$ by starting with the Noether current associated with angular momentum. Such derivations for the simple spin 1\/2 and 1 cases were given in appendix B of~\\cite{Maybee:2019jus} --- for a Dirac spin $1\/2$ particle, the spin polarisation is\n\\[\ns^\\mu_{ab}(p) = \\frac{\\hbar}{4m} \\bar{u}_a(p) \\gamma^\\mu \\gamma^5 u_b(p)\\,.\\label{eqn:spinorSpinVec}\n\\]\nMeanwhile, for massive vector bosons we have\n\\begin{equation}\ns_{ij}^\\mu(p) = -\\frac{i\\hbar}{m} \\epsilon^{\\mu\\nu\\rho\\sigma} p_\\nu \\varepsilon{^*_i}_\\rho(p) {\\varepsilon_j}_\\sigma(p)\\,. \\label{eqn:vectorSpinVec}\n\\end{equation}\nWe have these normalised quantities to be consistent with the algebraic properties of the Pauli--Lubanski operator.\n\n\\subsection{The change in spin during scattering}\n\nNow that we have a quantum-mechanical understanding of the spin vector, we move on to discuss the dynamics of the spin vector in a scattering process. Following the set-up of chapter~\\ref{chap:impulse} we consider the scattering of two stable, massive particles which are quanta of different fields, and are separated by an impact parameter $b^\\mu$. We will explicitly consider scattering processes mediated by vector bosons and gravitons. The relevant incoming two-particle state is therefore that in equation~\\eqref{eqn:inState}, but with little group states $\\xi_\\alpha$.\n\nThe initial spin vector of particle 1 is\n\\[\n\\langle s_1^\\mu \\rangle = \\frac1{m_1} \\langle \\Psi |\\W^\\mu_1 |\\Psi \\rangle\\,,\n\\]\nwhere $\\W^\\mu_1$ is the Pauli--Lubanski operator of the field corresponding to particle 1. Since the $S$ matrix is the time evolution operator from the far past to the far future, the final spin vector of particle 1 is\n\\[\n\\langle s_1'^\\mu \\rangle = \\frac1{m_1} \\langle \\Psi | S^\\dagger \\W^\\mu_1 S| \\Psi \\rangle\\,.\n\\]\nWe define the angular impulse on particle 1 as the difference between these quantities:\n\\begin{equation}\n\\langle \\Delta s_1^\\mu \\rangle = \\frac1{m_1}\\langle\\Psi|S^\\dagger \\mathbb{W}_1^\\mu S|\\Psi\\rangle - \\frac1{m_1}\\langle\\Psi|\\mathbb{W}_1^\\mu|\\Psi\\rangle\\,.\\label{eqn:defOfAngImp}\n\\end{equation}\nWriting $S = 1 + iT$ and making use of the optical theorem yields\n\\begin{equation}\n\\langle \\Delta s_1^\\mu\\rangle = \\frac{i}{m_1}\\langle\\Psi|[\\mathbb{W}_1^\\mu,{T}]|\\Psi\\rangle + \\frac{1}{m_1}\\langle\\Psi|{T}^\\dagger[\\mathbb{W}_1^\\mu,{T}]|\\Psi\\rangle\\,.\\label{eqn:spinShift}\n\\end{equation}\nJust as with equation~\\eqref{eqn:defl1}, it is clear that the second of these terms will lead to twice as many powers of the coupling constant for a given interaction. Therefore only the first term is able to contribute at leading order. We will be exclusively considering tree level scattering $\\mathcal{A}^{(0)}$, so the first term is the sole focus of our attention\\footnote{The expansion of the second term is very similar to that of the colour impulse in ref.~\\cite{delaCruz:2020bbn}.}.\n\nOur goal now is to express the leading-order angular impulse in terms of amplitudes. To that end we substitute the incoming state in equation~\\eqref{eqn:inState} into the first term of \\eqn~\\eqref{eqn:spinShift}, and the leading-order angular impulse is given by\n\\begin{multline}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = \\frac{i}{m_1}\\prod_{\\alpha=1,2} \\int\\! \\df(p_\\alpha') \\df(p_\\alpha) \\,\\varphi_\\alpha^*(p_\\alpha') \\varphi_\\alpha(p_\\alpha) e^{ib\\cdot(p_1-p'_1)\/\\hbar} \\\\ \\times \\left\\langle p_1'\\,\\xi'_1 ; p_2'\\, \\xi'_2\\left|\\W_1^\\mu\\, {T} -{T}\\, \\W_1^\\mu \\right|p_1 \\, \\xi_1; p_2\\, \\xi_2 \\right\\rangle.\n\\end{multline}\nScattering amplitudes can now be explicitly introduced by inserting a complete set of states between the spin and interaction operators, as in equation~\\eqref{eqn:p1Expectation}. In their first appearance this yields\n\\begin{multline}\n\\int\\! \\df(r_1) \\df(r_2) \\d\\mu(\\zeta_1) \\d\\mu(\\zeta_2)\\langle p'_1\\, \\xi'_1; p'_2\\, \\xi'_2|\\W_1^\\mu|r_1\\, \\zeta_1; r_2, \\zeta_2\\rangle \\langle r_1\\, \\zeta_1; r_2\\, \\zeta_2|T|p_1\\, \\zeta_1 ; p_2\\, \\zeta_2\\rangle \\\\\n =  m_1 \\langle \\xi'_1\\, \\xi'_2 |{\\s}^\\mu_1(p'_1)\\mathcal{A}(p_1, p_2 \\rightarrow p'_1, p'_2)|\\xi_1\\,\\xi_2\\rangle\\,  \\del^{(4)}(p'_1 + p'_2 - p_1 - p_2)\\,,\n\\end{multline}\nwhere, along with the definition of the scattering amplitude, we have used the definition of the spin polarisation vector~\\eqref{eqn:spinorSpinVec}. The result for the other ordering of $T$ and $\\W^\\mu_1$ is very similar. \n\nAn essential point is that under the little group state inner product above, the spin polarisation vector and amplitude do not commute: they are both matrices in the little group representation, and we have simply suppressed the explicit indices. This novel feature of the angular impulse will become extremely important. Substituting into the full expression for $\\langle\\Delta s_1^{\\mu,(0)}\\rangle$ and integrating over the delta functions, we find that the observable is\n\\begin{equation}\n\\begin{aligned}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = i \\prod_{\\alpha = 1, 2} &\\int \\!\\df(p_\\alpha') \\df(p_\\alpha) \\,\\varphi_\\alpha^*(p_\\alpha') \\varphi_\\alpha (p_\\alpha) \\del^{(4)}(p_1' + p_2' - p_1 - p_2)  \\\\ & \\times  e^{ib\\cdot(p_1-p'_1)\/\\hbar} \\langle\\xi'_1\\, \\xi'_2| \\s_{1}^\\mu(p_1') \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p'_1, p'_2) \\\\ &\\hspace{30mm} - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p'_1, p'_2) \\s_{1}^{\\mu}(p_1)|\\xi_1\\, \\xi_2\\rangle\\,.\n\\end{aligned}\n\\end{equation}\nWe now eliminate the delta function by introducing the familiar momentum mismatches $q_\\alpha = p'_\\alpha - p_\\alpha$ and performing an integral. The leading-order angular impulse becomes\n\\begin{equation}\n\\begin{aligned}\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle = i & \\int\\! \\df(p_1) \\df(p_2)\\, \\dd^4q\\,\\del(2p_1\\cdot q + q^2) \\del(2p_2\\cdot q - q^2) \\Theta(p_1^0 + q^0) \\\\ \\times& \\Theta(p_2^0 - q^0) \\varphi_1^*(p_1 + q) \\varphi_2^*(p_2 - q) \\varphi_1(p_1) \\varphi_2(p_2) e^{-ib\\cdot q\/\\hbar} \\\\\\times& \\langle \\xi'_1\\, \\xi'_2|\\s_{1}^{\\mu}(p_1 + q) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s^{\\mu}(p_1)|\\xi_1\\, \\xi_2\\rangle\\,.\\label{eqn:exactAngImp}\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Passing to the classical limit}\n\\label{sec:classicalLim}\n\nThe previous expression is an exact, quantum formula for the leading-order change in the spin vector during conservative two-body scattering. As a well-defined observable, we can extract the classical limit of the angular impulse by following the formalism introduced in part~\\ref{part:observables}.\n\nRecall that the basic idea is simple: the momentum space wavefunctions must localise the particles, without leading to a large uncertainty in their momenta. They therefore have a finite but small width $\\Delta x = \\ell_w$ in position space, and $\\Delta p = \\hbar\/\\ell_w$ in momentum space. This narrow width restricts the range of the integral over $q$ in equation~\\eqref{eqn:exactAngImp} so that $q \\lesssim \\hbar \/\\ell_w$. We therefore introduce the wavenumber $\\qb = q\/\\hbar$. We further found that our explicit choice of wavefunctions $\\varphi_\\alpha$ were very sharply peaked in momentum space around the value $\\langle p_\\alpha^\\mu \\rangle = m_\\alpha u_\\alpha^\\mu$, where $u_\\alpha^\\mu$ is a classical proper velocity. We neglect the small shift $q = \\hbar \\qb$ in the wavefunctions present in equation~\\eqref{eqn:exactAngImp}, and also the term $q^2$ compared to the dominant $2 p \\cdot q$ in the delta functions, arriving at\n\\[\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle =&\\, i \\!  \\int \\! \\df(p_1) \\df(p_2)\\, \\dd^4q\\,\\del(2p_1\\cdot q ) \\del(2p_2\\cdot q) |\\varphi_1(p_1)|^2 |\\varphi_2(p_2)|^2 \\\\&\\times e^{-ib\\cdot q\/\\hbar}\\, \\bigg\\langle\\s_{1}^{\\mu}(p_1 + q) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s_{1}^{\\mu}(p_1)\\bigg\\rangle   \\,.\n\\]\nNotice we have also dropped the distinction between the little group states, simply writing an expectation value as in~\\eqref{eqn:amplitudeDef}. This is permissible since we established coherent states suitable for the classical limit of $SU(2)$ states in section~\\ref{sec:classicalSingleParticleColour}. Adopting the notation for the large angle brackets from \\eqn~\\eqref{eqn:angleBrackets}, the angular impulse takes the form\n\\[\n\\langle\\Delta s^{\\mu,(0)}_1\\rangle =  &\\, \\Lexp i\\! \\int \\!\\dd^4 q\\, \\del (2p_1 \\cdot q) \\del (2p_2\\cdot q) e^{-ib\\cdot q\/\\hbar}  \\\\ & \\times\\bigg(\\s^{\\mu}(p_1 +  \\hbar\\qb) \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\\\ &\\qquad\\qquad - \\mathcal{A}^{(0)}(p_1, p_2 \\rightarrow p_1 + q, p_2 - q) \\s_{1}^{\\mu}(p_1)\\bigg)\\Rexp\\,,\n\\label{eqn:intermediate}\n\\]\n\nAn important $\\hbar$ shift remaining is that of the spin polarisation vector $\\s_{1}^{\\mu}(p_1 + \\hbar\\wn q)$. This object is a Lorentz boost of $\\s_{1}^{\\mu}(p_1)$. In the classical limit $q$ is small, so the Lorentz boost $\\Lambda^{\\mu}{ }_\\nu p_1^\\nu = p_1^\\mu + \\hbar\\wn q^\\mu$ is infinitesimal. In the vector representation an infinitesimal Lorentz transformation is $\\Lambda^\\mu\\,_\\nu=\\delta^\\mu_\\nu + w^\\mu{ }_\\nu$, so for our boosted momenta $w^\\mu{ }_\\nu p_1^\\nu = \\hbar\\wn q^\\mu$. The appropriate generator is\n\\begin{equation}\nw^{\\mu\\nu} = -\\frac{\\hbar}{m_1^2}\\left(p_1^\\mu \\wn q^\\nu - \\wn q^\\mu p_1^\\nu\\right)\\,.\\label{eqn:LorentzParameters}\n\\end{equation} \nThis result is valid for particles of any spin as it is purely kinematic, and therefore can be universally applied in our general formula for the angular impulse. In particular, since $w_{\\mu\\nu}$ is explicitly $\\mathcal{O}(\\hbar)$ the spin polarisation vector transforms as\n\\begin{equation}\n\\s_{1\\,ij}^{\\mu}(p_1 + \\hbar\\wn q) = \\s_{1\\,ij}^{\\mu}(p_1) - \\frac\\hbar{m^2} p^\\mu \\qb \\cdot s_{ij}(p_1)\\,.\n\\label{eqn:infinitesimalBoost}\n\\end{equation}\nThe angular impulse becomes\n\\begin{multline}\n\\langle \\Delta s_1^{\\mu,(0)} \\rangle \\rightarrow \\Delta s_1^{\\mu,(0)}\\label{eqn:limAngImp}\n= \\Lexp i\\! \\int\\!\\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q)e^{-ib\\cdot\\wn q} \\\\ \\times \\bigg(-\\hbar^3 \\frac{p_1^\\mu}{m_1^2} \\qb \\cdot \\s_{1} (p_1) \\mathcal{A}^{(0)}(\\wn q) + \\hbar^2 \\big[\\s^\\mu_1(p_1), \\mathcal{A}^{(0)}(\\wn q)\\big] \\bigg)\\Rexp\\,.\n\\end{multline}\nThe little group states have manifested themselves in the appearance of a commutator. The formula appears to be of a non-uniform order in $\\hbar$, but fortunately this is not really the case: any terms in the amplitude with diagonal indices will trivially vanish under the commutator; alternatively, any term with a commutator will introduce a factor of $\\hbar$ through the algebra of the Pauli--Lubanski vectors.\nTherefore all terms have the same weight, $\\hbar^3$, independently of factors appearing in the amplitude. \nThe analogous formula for the leading order, classical momentum impulse was given in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}.\nWe will make use of both the momentum and angular impulse formulae below.\n\nThere is a caveat regarding the uncertainty principle in the context of our spinning particles. In the following examples we restrict to low spins: spin 1\/2 and spin 1. Consequently the expectation of the spin vector $\\langle s^\\mu \\rangle$ is of order $\\hbar$; indeed $\\langle s^2 \\rangle = s(s+1) \\hbar^2$. This requires us to face the quantum-mechanical distinction between $\\langle s^\\mu s^\\nu \\rangle$ and $\\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$. Because of the uncertainty principle, the uncertainty $\\sigma_1^2$ associated with the operator $s^1$, for example, is of order $\\hbar$, and therefore the difference between $\\langle s_1^2 \\rangle$ and $\\langle s_1 \\rangle^2$ is of order $\\hbar^2$. Thus the difference $\\langle s^\\mu s^\\nu \\rangle - \\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$ is of order $\\langle s^\\mu s^\\nu \\rangle$. We are therefore not entitled to replace $\\langle s^\\mu s^\\nu \\rangle$ by $\\langle s^\\mu \\rangle \\langle s^\\nu \\rangle$ for any arbitrary states, and will make the distinction between these quantities below. As we showed explicitly (for $SU(3)$ states) in chapter~\\ref{chap:pointParticles}, this limitation can be overcome by studying very large spin representations. To elucidate the details of taking the classical limit of amplitudes with spin we will primarily work with the explicit spin 1\/2 and spin 1 fields in this chapter; however, we will comment on the all--spin generalisation of our results, studied in \\cite{Guevara:2019fsj}. The large spin limit will then play a crucial role in the next chapter.\n\nThe procedure for passing from amplitudes to a concrete expectation value is as follows. Once one has computed the amplitude, and evaluated any commutators, explicit powers of $\\hbar$ must cancel. We then evaluate the integrals over the on-shell phase space of the incoming particles simply by evaluating the momenta $p_\\alpha$ as $p_\\alpha = m_\\alpha u_\\alpha$. An expectation value over the spin wave functions $\\xi_\\alpha$ remains; these are always of the form $\\langle s^{\\mu_1} \\cdots s^{\\mu_n}\\rangle$ for various values of $s$. Only when the spin $s$ is large can we factorise this expectation value.\n\n\\section{Classical limits of amplitudes with finite spin}\n\\label{sec:amplitudes}\n\nWe have constructed a general formula for calculating the leading classical contribution to the angular impulse from scattering amplitudes. In the limit these amplitudes are Laurent expanded in $\\hbar$, with only one term in the expansion providing a non-zero contribution. How this expansion works in the case for charged scalar amplitudes was established in part~\\ref{part:observables}, but now we need to consider examples of amplitudes for particles with spin. The identification of the spin polarisation vector defined in \\eqn~\\eqref{eqn:PLinnerProd} will be crucial to this limit.\n\nWe will again look at the two lowest spin cases, considering tree level scattering of a spin $1\/2$ or spin $1$ particle off a scalar in Yang--Mills theory and gravity. Tree level Yang--Mills amplitudes will now be denoted by $\\mathcal{A}_{s_1-0}$, and those for Einstein gravity as $\\mathcal{M}_{s_1-0}$. To ensure good UV behaviour of our amplitudes, we adopt minimally coupled interactions between the massive states and gauge fields. This has the effect of restricting the classical value of the gyromagnetic ratio to $g_L =2$, for all values of $s$ \\cite{Chung:2018kqs,Ferrara:1992yc}.\n\n\\subsection{Gauge theory amplitudes}\n\nWe will continue to consider Yang--Mills theory minimally coupled to matter in some representation of the gauge group. The common Lagrangian is that in equation~\\eqref{eqn:scalarAction}. Our calculations will be in the vein of section~\\ref{sec:LOimpulse}, and we will always have the same $t$-channel colour factor. The amplitude for scalar-scalar, $\\mathcal{A}_{0-0}$, scattering is of course that in equation~\\eqref{eqn:treeamp}.\n\n\\subsubsection*{Spinor-scalar}\n\nWe can include massive Dirac spinors $\\psi$ in the Yang--Mills amplitudes by using a Lagrangian $\\mathcal{L} = \\mathcal{L}_0 + \\mathcal{L}_{\\textrm{Dirac}}$, where the Dirac Lagrangian \n\\begin{equation}\n\\mathcal{L}_\\textrm{Dirac} = \\bar{\\psi}\\left(i\\slashed{D} - m\\right)\\psi\\label{eqn:DiracL}\n\\end{equation}\nincludes a minimal coupling to the gauge field, and $\\mathcal{L}_0$ is the scalar Langrangian in equation~\\eqref{eqn:scalarAction}. The tree level amplitude for spinor-scalar scattering is then\n\\begin{equation}\ni\\mathcal{A}^{ab}_{1\/2-0} = \\frac{ig^2}{2 \\hbar q^2}\\bar{u}^a(p_1+q)\\gamma^\\mu u^{b}(p_1)(2p_2 - q)_\\mu\\, \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,,\n\\end{equation}\nwhere we have normalised the (dimensionful) colour factors consistent with the double copy. We are interested in the pieces of this amplitude that survive to the classical limit. To extract them we must set the momentum transfer as $q = \\hbar\\wn q$ and expand the amplitude in powers of $\\hbar$. \n\nThe subtlety here is the on-shell Dirac spinor product. In the limit, when $q$ is small, we can follow the logic of \\eqn~\\eqref{eqn:infinitesimalBoost} and interpret $\\bar{u}^a(p_1 + \\hbar\\wn q) \\sim \\bar{u}^a(p_1) + \\Delta\\bar{u}^a(p_1)$ as being infinitesimally Lorentz boosted, see also~\\cite{Lorce:2017isp}. One expects amplitudes for spin 1\/2 particles to only be able to probe up to linear order in spin (i.e. the dipole of a spinning body) \\cite{Vaidya:2014kza,Guevara:2017csg,Guevara:2018wpp}, so in deriving the infinitesimal form of the Lorentz transformation we expand to just one power in the spin. The infinitesimal parameters $w_{\\mu\\nu}$ are exactly those determined in \\eqn~\\eqref{eqn:LorentzParameters}, so in all the leading terms of the spinor product are\n\\begin{equation}\n\\bar{u}^a(p_1 + \\hbar\\wn q) \\gamma_\\mu u^b(p_1) = 2{p_1}_\\mu \\delta^{ab} + \\frac{\\hbar}{4m^2}\\bar{u}^a(p_1) p{_1}^\\rho \\wn q^\\sigma [\\gamma_\\rho,\\gamma_\\sigma] \\gamma_\\mu u^b(p_1) + \\mathcal{O}(\\hbar^2)\\,.\\label{eqn:spinorshift}\n\\end{equation}\nEvaluating the product of gamma matrices via the identity\n\\begin{equation}\n[\\gamma_\\mu, \\gamma_\\nu] \\gamma_\\rho = 2\\eta_{\\nu\\rho}\\gamma_\\mu - 2\\eta_{\\mu\\rho}\\gamma_\\nu -  2i\\epsilon_{\\mu\\nu\\rho\\sigma} \\gamma^\\sigma \\gamma^5\\,,\\label{eqn:3gammaCommutator}\n\\end{equation}\nwhere $\\epsilon_{0123} = +1$ and $\\gamma^5 = i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3$, the spinor product is just\n\\begin{multline}\n\\bar{u}^a(p_1 + \\hbar\\wn q) \\gamma_\\mu u^b(p_1) = 2{p_1}_\\mu\\delta^{ab}  + \\frac{\\hbar}{2m_1^2} \\bar{u}^a(p_1) p_1^\\rho \\wn q^\\sigma \\left(\\gamma_\\sigma\\eta_{\\mu\\rho} - \\gamma_\\rho\\eta_{\\mu\\sigma}\\right) u^b(p_1) \\\\ - \\frac{i\\hbar}{2m_1^2} \\bar{u}^a(p_1) p{_1}^\\rho \\wn q^\\sigma \\epsilon_{\\rho\\sigma\\mu\\delta} \\gamma^\\delta\\gamma^5 u^b(p_1) + \\mathcal{O}(\\hbar^2)\\,.\n\\end{multline}\nComparing with our result from \\eqn~\\eqref{eqn:spinorSpinVec}, the third term clearly hides an expression for the spin 1\/2 polarisation vector. Making this replacement and substituting the spinor product into the amplitude yields, for on-shell kinematics, only two terms at an order lower than $\\mathcal{O}(\\hbar^2)$:\n\\begin{equation}\n\\hbar^3\\mathcal{A}^{ab}_{1\/2-0} = \\frac{2g^2}{\\wn q^2} \\left((p_1\\cdot p_2)\\delta^{ab} - \\frac{i}{m_1}  \\epsilon( p{_1}, \\wn q, p_2, s^{ab}_1) + \\mathcal{O}(\\hbar^2)\\right) \\tilde{\\newT}_1\\cdot\\tilde{\\newT}_2\\,,\\label{eqn:spinorYMamp}\n\\end{equation}\nwhere here and below we adopt the short-hand notation ${s_1}^{\\mu}_{ab} = {s_1}^{\\mu}_{ab}(p_1)$ and\n\\[\n\\epsilon(a,b,c,d) = \\epsilon_{\\mu\\nu\\rho\\sigma} a^\\mu b^\\nu c^{\\rho} d^{\\sigma} \\,,\n\\qquad \\epsilon_\\mu(a,b,c) = \\epsilon_{\\mu\\nu\\rho\\sigma} a^\\nu b^\\rho c^\\sigma\\,.\n\\]\nUpon substitution into the impulse in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO} or angular impulse in \\eqn~\\eqref{eqn:limAngImp} the apparently singular denominator in the $\\hbar\\rightarrow 0$ limit is cancelled. It is only these quantities, not the amplitudes, that are classically well defined and observable.\n\n\\subsubsection*{Vector-scalar}\n\nNow consider scattering a massive vector rather than spinor. The minimally coupled gauge interaction can be obtained by applying the Higgs mechanism to the Yang--Mills Lagrangian\\footnote{Regardless of minimal coupling, for vector states with masses generated in this way the classical value of $g_L=2$ \\cite{Chung:2018kqs,Ferrara:1992yc}.}, which when added to the scalar Lagrangian $\\mathcal{L}_0$ yields the tree-level amplitude\n\\begin{multline}\ni\\mathcal{A}^{ij}_{1-0} = -\\frac{ig^2}{2\\hbar q^2} \\varepsilon{_i^*}^\\mu(p_1+q) \\varepsilon^\\nu_j(p_1) \\left(\\eta_{\\mu\\nu}(2p_1+q)_\\lambda - \\eta_{\\nu\\lambda}(p_1-q)_\\mu \\right.\\\\\\left. - \\eta_{\\lambda\\mu}(2q+p_1)_\\nu \\right)(2p_2 - q)^\\lambda\\, \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,.\n\\end{multline}\nTo obtain the classically significant pieces of this amplitude we must once more expand the product of on-shell tensors, in this case the polarisation vectors. In the classical limit we can again consider the outgoing polarisation vector as being infinitesimally boosted, so $\\varepsilon{_i^*}^\\mu(p_1 + \\hbar\\wn q) \\sim \\varepsilon{_i^*}^\\mu(p_1) + \\Delta\\varepsilon{_i^*}^\\mu(p_1)$. \n\nHowever, from spin 1 particles we expect to be able to probe $\\mathcal{O}(s^2)$, or quadrupole, terms \\cite{Vaidya:2014kza,Guevara:2017csg,Guevara:2018wpp}. Therefore it is salient to expand the Lorentz boost to two orders in the Lorentz parameters $w_{\\mu\\nu}$, so under infinitesimal transformations we take\n\\begin{equation}\n\\varepsilon_i^\\mu(p) \\mapsto \\Lambda^\\mu{ }_\\nu\\, \\varepsilon_i^\\nu(p) \\simeq \\left(\\delta^\\mu{ }_\\nu - \\frac{i}{2} w_{\\rho\\sigma} (\\Sigma^{\\rho\\sigma})^\\mu{ }_\\nu - \\frac18 \\left((w_{\\rho\\sigma} \\Sigma^{\\rho\\sigma})^2\\right)^\\mu{ }_\\nu \\right) \\!\\varepsilon_i^\\nu(p)\\,,\n\\end{equation}\nwhere $(\\Sigma^{\\rho\\sigma})^\\mu{ }_\\nu = i\\left(\\eta^{\\rho\\mu} \\delta^\\sigma{ }_\\nu - \\eta^{\\sigma\\mu} \\delta^\\rho{}_\\nu\\right)$. Since the kinematics are again identical to those used to derive \\eqn~\\eqref{eqn:LorentzParameters}, we get\n\\begin{equation}\n\\varepsilon{_i^*}^\\mu(p_1 + \\hbar\\wn q)\\, \\varepsilon^\\nu_j(p_1) = \\varepsilon{_i^*}^\\mu \\varepsilon^\\nu_j - \\frac{\\hbar}{m_1^2} (\\wn q \\cdot \\varepsilon_i^*) p_1^\\mu \\varepsilon^\\nu_j - \\frac{\\hbar^2}{2 m_1^2} (\\wn q\\cdot\\varepsilon_i^*) \\wn q^\\mu \\varepsilon^\\nu_j + \\mathcal{O}(\\hbar^3)\\,,\\label{eqn:vectorS}\n\\end{equation}\nwhere now $\\varepsilon_i$ will always be a function of $p_1$, so in the classical limit $\\varepsilon_i^*\\cdot p_1 = \\varepsilon_i\\cdot p_1=0$. Using this expression in the full amplitude, the numerator becomes\n\\begin{multline}\nn_{ij} = 2(p_1\\cdot p_2)(\\varepsilon_i^*\\cdot\\varepsilon_j) - 2\\hbar (p_2\\cdot\\varepsilon_i^*)(\\wn q\\cdot\\varepsilon_j) + 2\\hbar (p_2\\cdot\\varepsilon_j)(\\wn q\\cdot\\varepsilon_i^*) \\\\ + \\frac{1}{m_1^2}\\hbar^2 (p_1\\cdot p_2)(\\wn q\\cdot\\varepsilon_i^*)(\\wn q\\cdot\\varepsilon_j) + \\frac{\\hbar^2}{2}\\wn q^2 (\\varepsilon_i^*\\cdot \\varepsilon_j) + \\mathcal{O}(\\hbar^3)\\,.\n\\end{multline}\nHow the spin vector enters this expression is not immediately obvious, and relies on Levi--Civita tensor identities. At $\\mathcal{O}(\\hbar)$, $\\epsilon^{\\delta\\rho\\sigma\\nu} \\epsilon_{\\delta\\alpha\\beta\\gamma} = -3!\\, \\delta^{[\\rho}{ }_\\alpha \\delta^{\\sigma}{ }_\\beta \\delta^{\\nu]}{ }_\\gamma$ leads to\n\\begin{align}\n\\hbar (p_2\\cdot\\varepsilon{_i^*})(\\wn q \\cdot \\varepsilon_j) - \\hbar (p_2 \\cdot \\varepsilon_j)(\\wn q \\cdot \\varepsilon_i^*) = \\frac{\\hbar}{m_1^2}&p_1^\\rho \\wn q^\\sigma p_2^\\lambda \\epsilon_{\\delta\\rho\\sigma\\lambda}\\epsilon^{\\delta\\alpha\\beta\\gamma}{\\varepsilon_i^*}_\\alpha {\\varepsilon_j}_\\beta {p_1}_\\gamma \\nonumber \\\\ &\\equiv  -\\frac{i}{m_1} \\epsilon(p_1, \\wn q, p_2, {s_1}_{ij})\\,,\n\\end{align}\nwhere again we are able to identify the spin 1 polarisation vector calculated in \\eqn~\\eqref{eqn:vectorSpinVec} and introduce it into the amplitude. There is also a spin vector squared contribution entering at $\\mathcal{O}(\\hbar^2)$; observing this is reliant on applying the identity $\\epsilon^{\\mu\\nu\\rho\\sigma} \\epsilon_{\\alpha\\beta\\gamma\\delta} = -4!\\, \\delta^{[\\mu}{ }_\\alpha \\delta^{\\nu}{ }_\\beta \\delta^{\\rho}{ }_\\gamma \\delta^{\\sigma]}{ }_\\delta$ and the expression in \\eqn~\\eqref{eqn:vectorSpinVec} to calculate\n\\begin{equation}\n\\sum_k \\left(\\wn q \\cdot s_1^{ik}\\right) (\\wn q \\cdot s_1^{kj}) = -\\hbar^2 (\\wn q \\cdot \\varepsilon_i^*) (\\wn q\\cdot \\varepsilon_j) - \\hbar^2 \\wn q^2 \\delta_{ij} + \\mathcal{O}(\\hbar^3)\\,.\n\\end{equation}\nThis particular relationship is dependent on the sum over helicities $\\sum_h {\\varepsilon^*_h}^\\mu \\varepsilon^\\nu_h = -\\eta^{\\mu\\nu} + \\frac{p_1^\\mu p_1^\\nu}{m_1^2}$ for massive vector bosons, an additional consequence of which is that $\\varepsilon_i^*\\cdot \\varepsilon_j = -\\delta_{ij}$. Incorporating these rewritings of the numerator in terms of spin vectors, the full amplitude is\n\\begin{multline}\n\\hbar^3\\mathcal{A}^{ij}_{1-0} = \\frac{2g^2}{\\wn q^2}\\left((p_1\\cdot p_2)\\delta^{ij} - \\frac{i}{m_1} \\epsilon(p_1, \\wn q, p_2, s_1^{ij}) + \\frac{1}{2 m_1^2}(p_1\\cdot p_2)(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) \\right.\\\\\\left. + \\frac{\\hbar^2 \\wn q^2}{4m_1^2}\\left(2p_1\\cdot p_2 + m_1^2\\right) + \\mathcal{O}(\\hbar^3)\\right) \\tilde{\\newT}_1\\cdot \\tilde{\\newT}_2\\,.\n\\end{multline}\nThe internal sum over spin indices in the $\\mathcal{O}(s^2)$ term will now always be left implicit. In classical observables we can also drop the remaining $\\mathcal{O}(\\hbar^2)$ term, as this just corresponds to a quantum correction from contact interactions.\n\n\\subsection{Gravity amplitudes}\n\nRather than re-calculate these amplitudes in perturbative gravity, let us apply the double copy\\footnote{One can easily verify that direct calculations with graviton vertex rules given in \\cite{Holstein:2008sx} reproduce our results.}. Note that for massive states with spin this ability is reliant on our gauge theory choice of $g_L=2$, as was noted in \\cite{Goldberger:2017ogt}. Only with this choice is the gravitational theory consistent with the low energy spectrum of string theory \\cite{Goldberger:2017ogt,Chung:2018kqs}, of which the double copy is an intrinsic feature.\n\nFor amplitudes in the LO impulse the Jacobi identity is trivial, as we saw in section~\\ref{sec:LOimpulse}. We can therefore simply replace colour factors with the desired numerator. In particular, if we replace the colour factor in the previous spin $s$--spin 0 Yang--Mills amplitudes with the scalar numerator from \\eqn~\\eqref{eqn:scalarYMamp} we will obtain a spin $s$--spin 0 gravity amplitude, as the composition of little group irreps is simply $(\\mathbf{2s + 1})\\otimes\\mathbf{1}=\\mathbf{2s + 1}$. Using the scalar numerator ensures that the spin index structure passes to the gravity theory unchanged. Thus we can immediately obtain that the classically significant part of the spin $1\/2$--spin 0 gravity amplitude is\n\\begin{equation}\n\\hbar^3 \\mathcal{M}^{ab} = -\\left(\\frac{\\kappa}{2}\\right)^2\\frac{4}{\\wn q^2} \\bigg[(p_1\\cdot p_2)^2\\delta^{ab} - \\frac{i}{m_1}(p_1\\cdot p_2)\\, \\epsilon(p_1, \\wn q, p_2, s_1^{ab}) + \\mathcal{O}(\\hbar^2)\\bigg]\\,,\\label{eqn:spinorScalarGravAmp}\n\\end{equation}\nwhile that for spin 1--spin 0 scattering is\n\\begin{multline}\n\\hbar^3\\mathcal{M}^{ij} = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\wn q^2} \\left[(p_1\\cdot p_2)^2\\delta^{ij} - \\frac{i}{m_1}(p_1\\cdot p_2)\\, \\epsilon(p_1, \\wn q, p_2, s_1^{ij}) \\right.\\\\\\left. + \\frac{1}{2 m_1^2}(p_1\\cdot p_2)^2(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) + \\mathcal{O}(\\hbar^2)\\right]. \\label{eqn:vectorScalarGravAmp}\n\\end{multline}\nNotice that the $\\mathcal{O}(s)$ parts of these amplitudes are exactly equal, up to the different spin indices. This is a manifestation of gravitational universality: the gravitational coupling to the spin dipole should be independent of the spin of the field, precisely as we observe.\n\nWe have deliberately not labelled these as Einstein gravity amplitudes, because the gravitational modes in our amplitudes contain both gravitons and scalar dilatons. To see this, let us re-examine the factorisation channels in the $t$ channel cut, but now for the vector amplitude:\n\\[\n\\lim\\limits_{\\wn q^2 \\rightarrow 0} \\left(\\wn q^2 \\hbar^3 \\mathcal{M}^{ij}\\right) &= -4\\left(\\frac{\\kappa}{2}\\right)^2\\, \\bigg(p_1^\\mu p_1^{\\tilde{\\mu}} \\delta^{ij} - \\frac{i}{m_1}p_1^\\mu \\epsilon^{\\tilde{\\mu}\\rho\\sigma\\delta} {p_1}_\\rho \\wn q_\\sigma {s_1}_\\delta^{ij} \\\\ & \\qquad\\qquad\\qquad + \\frac{1}{2m_1^2} (\\wn q\\cdot s_1^{ik})(\\wn q\\cdot s_1^{kj}) p_1^\\mu p_1^{\\tilde{\\mu}}\\bigg) \\mathcal{P}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\, p_2^\\nu p_2^{\\tilde{\\nu}} \\\\ & - 4\\left(\\frac{\\kappa}{2}\\right)^2\\left(p_1^\\mu p_1^{\\tilde{\\mu}} \\delta^{ij} + \\frac{(\\wn q\\cdot s_1^{ik})(\\wn q\\cdot s_1^{kj})}{2 m_1^2} p_1^\\mu p_1^{\\tilde{\\mu}}\\right) \\mathcal{D}^{(4)}_{\\mu\\tilde{\\mu}\\nu\\tilde{\\nu}}\\, p_2^\\nu p_2^{\\tilde{\\nu}}\\,,\n\\]\nwhere we have utilised the de-Donder gauge graviton and dilaton projectors from equation~\\eqref{eqn:gravityProjectors}. As for the scalar case, the pure Einstein gravity amplitude for classical spin 1--spin 0 scattering can just be read off as the part of the amplitude contracted with the graviton projector. We find that\n\\begin{multline}\n\\hbar^3\\mathcal{M}^{ij}_{1-0} = -\\left(\\frac{\\kappa}{2}\\right)^2 \\frac{4}{\\wn q^2} \\left[\\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right)\\delta^{ij} - \\frac{i}{m_1}(p_1\\cdot p_2)\\,\\epsilon(p_1, \\wn q, p_2, s_1^{ij}) \\right.\\\\\\left. + \\frac{1}{2 m_1^2}\\left((p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right)(\\wn q\\cdot s_1^{ik}) (\\wn q \\cdot s_1^{kj}) + \\mathcal{O}(\\hbar^2)\\right]. \\label{eqn:vectorGravAmp}\n\\end{multline}\nThe spinor--scalar Einstein gravity amplitude receives the same correction to the initial, scalar component of the amplitude. \n\nNote that dilaton modes are coupling to the scalar monopole and $\\mathcal{O}(s^2)$ quadrapole terms in the gravity amplitudes, but not to the $\\mathcal{O}(s)$ dipole component. We also do not find axion modes, as observed in applications of the classical double copy to spinning particles \\cite{Li:2018qap,Goldberger:2017ogt}, because axions are unable to couple to the massive external scalar.\n\n\\section{Black hole scattering observables from amplitudes}\n\\label{sec:KerrCalcs}\n\nWe are now armed with a set of classical tree-level amplitudes and formulae for calculating the momentum impulse $\\Delta p_1^\\mu$ and angular impulse $\\Delta s_1^\\mu$ from them. We also already have a clear target where the analogous classical results are known: the results for 1PM scattering of spinning black holes found in \\cite{Vines:2017hyw}. \n\nGiven our amplitudes only reach the quadrupole level, we can only probe lower order terms in the expansion of \\eqn~\\eqref{eqn:KerrDeflections}. Expanding in the rescaled spin $a_1^\\mu$, and setting $a_2^\\mu\\to0$, the momentum impulse is\n\\begin{multline}\n\\Delta p_1^{\\mu} = \\frac{2 G m_1m_2}{\\sqrt{\\gamma^2 - 1}} \\left\\{(2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2} + \\frac{2\\gamma}{b^4} \\Big( 2{b}^\\mu {b}^\\nu-b^2\\Pi^{\\mu\\nu}\\Big)\\, \\epsilon_{\\nu\\rho}(u_1, u_2)\\,  a_1^\\rho\\right.\n\\\\\n\\left. - \\frac{2\\gamma^2 - 1}{{b}^6} \\Big(4b^\\mu b^\\nu b^\\rho-3b^2 b^{(\\mu}\\Pi^{\\nu\\rho)}\\Big)a_{1\\nu}a_{1\\rho} + \\mathcal{O}(a^3) \\right\\}+\\mathcal O(G^2)\\,, \\label{eqn:JustinImpResult}\n\\end{multline}\nwhere $\\Pi^\\mu{}_\\nu$ is the projector into the plane orthogonal to $u_1^\\mu$ and $u_2^\\mu$ from \\eqref{eqn:projector}.\nMeanwhile the angular impulse to the same order is\n\\begin{multline}\n\\Delta s_1^{\\mu} =-u_1^\\mu a_{1\\nu}\\Delta p_1^\\nu -\\frac{2G m_1m_2}{\\sqrt{\\gamma^2-1}}\\left\\{\n2\\gamma \\epsilon^{\\mu\\nu\\rho\\sigma}  u_{1\\rho}\\, \\epsilon_{\\sigma} (u_1,u_2, b) \\frac{a_{1\\nu} }{b^2}\\right.\n\\\\\n\\left. - \\frac{2\\gamma^2 - 1}{b^4} \\epsilon^{\\mu\\nu\\kappa\\lambda} u_{1\\kappa}  \\Big( 2{b}_\\nu {b}_\\rho-b^2\\Pi_{\\nu\\rho} \\Big)a_{1\\lambda} a_1^\\rho \n+ \\mathcal{O}(a^3) \\right\\}+\\mathcal O(G^2)\\,. \\label{eqn:JustinSpinResult}\n\\end{multline}\nIn this section we demonstrate that both of these results can be recovered by using the classical pieces of our Einstein gravity amplitudes.\n\n\\subsection{Momentum impulse}\n\nTo calculate the momentum impulse we substitute $\\mathcal{M}_{1-0}$ into the general expression in \\eqn~\\eqref{eqn:impulseGeneralTerm1classicalLO}. Following the prescription in \\sect{sec:classicalLimit}, the only effect of the momentum integrals in the expectation value is to set $p_\\alpha \\rightarrow m_\\alpha u_\\alpha$ in the classical limit. This then reduces the double angle bracket to the single expectation value over the spin states:\n\\begin{equation}\n\\begin{aligned}\n\\Delta p_1^{\\mu,(0)} &= -i m_1 m_2 \\left(\\frac{\\kappa}{2}\\right)^2 \\!\\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot\\wn q) e^{-ib\\cdot\\wn q} \\frac{\\wn q^\\mu}{\\wn q^2} \\\\&\\times \\left\\langle \\frac12(2\\gamma^2 - 1) - \\frac{i \\gamma}{m_1} \\epsilon(u_1, \\wn q, u_2, s_1) + \\frac{2\\gamma^2 - 1}{4m_1^2} (\\wn q\\cdot s_1) (\\wn q \\cdot s_1) \\right\\rangle\\\\\n&\\equiv -4i m_1 m_2 \\pi G \\bigg((2\\gamma^2 - 1) I^\\mu  -  2i\\gamma u_1^\\rho u_2^\\nu \\epsilon_{\\rho\\sigma\\nu\\delta}\\, \\big\\langle a_1^\\delta \\big\\rangle I^{\\mu\\sigma} \\\\ &\\hspace{60mm} + \\frac{2\\gamma^2 - 1}{2} \\big\\langle {a_1}_\\nu {a_1}_\\rho \\big\\rangle I^{\\mu\\nu\\rho} \\bigg)\\,,\n\\end{aligned}\n\\end{equation}\nwhere we have rescaled $a^\\mu = s^\\mu\/m$ and defined three integrals of the general form\n\\begin{equation}\nI^{\\mu_1\\cdots \\mu_n} = \\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot \\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\wn q^2} \\wn q^{\\mu_1} \\cdots \\wn q^{\\mu_n}\\,.\\label{eqn:defOfI}\n\\end{equation}\n\nThe lowest rank integral of this type was evaluated in chapter~\\ref{chap:impulse}, with the result\n\\begin{equation}\nI^\\mu = \\frac{i}{2\\pi \\sqrt{\\gamma^2 - 1}} \\frac{{b}^\\mu}{b^2}\\,.\\label{eqn:I1result}\n\\end{equation}\nTo evaluate the higher rank examples, note that the results must lie in the plane orthogonal to the four velocities. This plane is spanned by the impact parameter $b^\\mu$, and the projector $\\Pi^\\mu{ }_\\nu$ defined in \\eqn~\\eqref{eqn:projector}. Thus, for example,\n\\begin{equation}\nI^{\\mu\\nu} = \\alpha_2 b^\\mu b^\\nu + \\beta_2 \\Pi^{\\mu\\nu}\\,.\n\\end{equation}\nGiven that we are working away from the threshold value $b = 0$, the left hand side is traceless and $\\beta_2 = - \\alpha_2\\, b^2 \/2$. Then contracting both sides with $b_\\nu$, one finds\n\\begin{equation}\n\\alpha_2 b^2\\, b^\\mu = 2\\int\\!\\dd^4\\wn q\\, \\del(u_1\\cdot\\wn q) \\del(u_2\\cdot \\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\wn q^2} \\wn q^{\\mu} (b\\cdot\\wn q) = \\frac{1}{\\pi \\sqrt{\\gamma^2 - 1}} \\frac{b^\\mu}{b^2}\\,,\n\\end{equation} \nwhere we have used the result of \\eqn~\\eqref{eqn:I1result}. Thus the coefficient $\\alpha_2$ is uniquely specified, and we find\n\\begin{equation}\nI^{\\mu\\nu} = \\frac{1}{\\pi b^4 \\sqrt{\\gamma^2 - 1}} \\left(b^\\mu b^\\nu - \\frac12 b^2 \\Pi^{\\mu\\nu} \\right)\\label{eqn:I2result}.\n\\end{equation}\nFollowing an identical procedure for $I^{\\mu\\nu\\rho}$, we can then readily determine that\n\\begin{equation}\nI^{\\mu\\nu\\rho} = -\\frac{4i}{\\pi b^6 \\sqrt{\\gamma^2 - 1} } \\left(b^\\mu b^\\nu b^\\rho - \\frac34 b^2 b^{(\\mu} \\Pi^{\\nu\\rho)} \\right).\\label{eqn:I3result}\n\\end{equation}\n\nSubstituting the integral results into the expression for the leading order classical impulse, and expanding the projectors from \\eqn~\\eqref{eqn:projector}, then leads to\n\\begin{multline}\n\\Delta p_1^{\\mu,(0)} = \\frac{2G m_1 m_2}{\\sqrt{\\gamma^2 - 1}}\\left((2\\gamma^2 - 1) \\frac{{b}^\\mu}{b^2} + \\frac{2\\gamma}{b^4} ( 2 b^\\mu {b}^\\alpha-b^2\\Pi^{\\mu\\alpha}) \\epsilon_{\\alpha\\rho}(u_1, u_2) \\big\\langle a_1^\\rho \\big\\rangle  \\right.\n\\\\\n\\left.- \\frac{2\\gamma^2 - 1}{{b}^6} (4b^\\mu b^\\nu b^\\rho-3b^2 b^{(\\mu}\\Pi^{\\nu\\rho)})\\langle a_{1\\nu}a_{1\\rho}\\rangle\\right).\\label{eqn:QuantumImpRes}\n\\end{multline}\nComparing with \\eqn~\\eqref{eqn:JustinImpResult} we observe an exact match, up to the appearance of spin state expectation values, between our result and the $\\mathcal{O}(a^2)$ expansion of the result for spinning black holes from \\cite{Vines:2017hyw}. \n\n\\subsection{Angular impulse}\n\nOur expression, equation~\\eqref{eqn:limAngImp}, for the classical leading-order angular impulse naturally has two parts: one term has a commutator while the other term does not. For clarity we will handle these two parts separately, beginning with the term without a commutator, which we will call the direct term.\n\n\\subsubsection*{The direct term}\n\nSubstituting our $\\mathcal{O}(s^2)$ Einstein gravity amplitude, equation~\\eqref{eqn:vectorGravAmp}, into the direct part of the general angular impulse formula, we find\n\\begin{align}\n&\\Delta s_1^{\\mu,(0)}\\big|_{\\textrm{direct}} \\nonumber\n\\equiv\n\\Lexp i\\! \\int\\!\\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q)e^{-ib\\cdot\\wn q} \\bigg(-\\hbar^3 \\frac{p_1^\\mu}{m_1^2} \\qb \\cdot \\s_{1} (p_1) \\mathcal{M}_{1-0} \\bigg) \\Rexp \\\\\n&\\;= \\Lexp \\frac{i \\kappa^2}{m_1^2} \\int\\! \\dd^4\\wn q\\,\\del(2p_1\\cdot\\wn q) \\del(2p_2\\cdot\\wn q) \\frac{e^{-ib\\cdot\\wn q}}{\\qb^2}\\, p_1^\\mu \\wn q \\cdot \\s_1(p_1) \\\\ &\\qquad\\times\\bigg(\\bigg((p_1\\cdot p_2)^2  - \\frac{1}{2}m_1^2m_2^2\\bigg)  - \\frac{i}{m_1}(p_1\\cdot p_2) \\, \\epsilon(p_1, \\wn q, p_2,s_1) \\bigg)+ \\mathcal{O}(s^3) \\Rexp\\,.\\nonumber\n\\end{align}\nAs with the momentum impulse, we can reduce the double angle brackets to single, spin state, angle brackets by replacing $p_\\alpha \\rightarrow m_\\alpha u_\\alpha$, so that\n\\begin{equation}\n\\hspace{-3pt}\\Delta s^{\\mu,(0)}_1\\big|_{\\textrm{direct}}\\!\\! = 4\\pi G m_2\\, u_1^\\mu\\!  \\left(\\! i  \\left(2\\gamma^2 - 1\\right)\\! \\langle s_1^\\nu \\rangle I_\\nu + \\frac{2}{m_1} \\gamma \n\\, u_1^\\alpha u_2^\\gamma \\epsilon_{\\alpha\\beta\\gamma\\delta} \\langle s_{1\\nu}  s_1^{\\delta} \\rangle \\, I^{\\nu\\beta} \\!\\right),\n\\end{equation}\nwhere the integrals are again defined by \\eqn~\\eqref{eqn:defOfI}. We can now just substitute our previous evaluations of these integrals, equations~\\eqref{eqn:I1result} and~\\eqref{eqn:I2result}, to learn that\n\\begin{multline}\n\\Delta a_1^{\\mu,(0)}\\big|_{\\textrm{direct}} = -\\frac{2G m_2}{\\sqrt{\\gamma^2 -1}} u_1^\\mu \\left((2\\gamma^2 - 1)\\frac{{b}_\\nu}{b^2} \\spinExp{a_1^\\nu} \\right. \\\\ \\left. + \\frac{2\\gamma}{b^4} \\left( 2{b}^\\nu {b}^\\alpha-{b^2}\\Pi^{\\nu\\alpha}\\right) \\epsilon_{\\alpha\\rho} (u_1, u_2) \\spinExp{{a_1}_\\nu a_1^\\rho} \\right).\\label{eqn:linPiece}\n\\end{multline}\n\n\\subsubsection*{The commutator term}\n\nNow we turn to the commutator piece of \\eqn~\\eqref{eqn:limAngImp}. The scalar part of our Einstein gravity amplitude, equation~\\eqref{eqn:vectorGravAmp}, has diagonal spin indices, so its commutator vanishes. We encounter two non-vanishing commutators:\n\\[\n[\\s_1^\\mu, \\s_1^\\nu]&=\\frac{i \\hbar}{m_1} \\, \\epsilon^{\\mu\\nu}(s_{1} , p_{1}) \\,,\\\\\n[\\s_1^\\mu, \\qb \\cdot \\s_1 \\, \\qb\\cdot \\s_1] &= \\frac{2 i \\hbar}{m_1} \\, \\qb \\cdot s_1 \\, \\epsilon^{\\mu} (\\qb, s_{1}, p_1) + \\mathcal{O}(\\hbar^2) \\,,\n\\]\nomitting a term which is higher order. Using these expressions in the commutator term, the result is\n\\[\n&\\Delta s_1^{\\mu, (0)}|_\\textrm{com} = i\\, \\Lexp \\int\\! \\dd^4 \\qb \\, \\del(2p_1 \\cdot \\qb) \\del(2p_2 \\cdot \\qb) e^{-ib \\cdot \\qb} \\, \\hbar^2 [s^\\mu(p_1), \\mathcal{M}_{1-0}] \\Rexp \\\\\n&= i \\kappa^2 \\Lexp \\int\\! \\dd^4 \\qb \\, \\del(2p_1 \\cdot \\qb) \\del(2p_2 \\cdot \\qb) \\frac{e^{-ib \\cdot \\qb}}{\\qb^2} \\bigg( (p_1 \\cdot p_2) \\, \\epsilon(p_1, \\qb, p_2)_\\sigma\\epsilon^{\\mu\\nu\\rho\\sigma} s_{1 \\nu} \\frac{p_{1 \\rho}}{m_1^2}  \\\\\n& \\qquad\\qquad\\qquad\\qquad- \\frac{i}{m_1^3} \\left( (p_1\\cdot p_2)^2 - \\frac12 m_1^2 m_2^2\\right) \\qb \\cdot \\s_1 \\, \\epsilon^{\\mu\\nu}(\\qb, p_1) s_{1\\nu} \\bigg) \\Rexp\\,.\n\\]\nAs is familiar by now, we evaluate the integrals over the momentum-space wavefunctions by setting $p_\\alpha = m_\\alpha u_\\alpha$, but expectation values over the spin-space wavefunctions remain. The result can be organised in terms of the integrals $I^\\alpha$ and $I^{\\alpha\\beta}$ defined in equation~\\eqref{eqn:defOfI}:\n\\begin{multline}\n\\Delta s_1^{\\mu, (0)}|_\\textrm{com} = 2\\pi i \\, G m_2 \\bigg( 4\\gamma \\epsilon^{\\mu\\nu\\rho\\sigma} \\langle s_{1\\,\\nu} \\rangle u_{1\\,\\rho} \\epsilon_{\\sigma\\alpha} (u_1, u_2) I^\\alpha \n\\\\ - \\frac{2 i}{m_1} (2\\gamma^2 - 1) \\epsilon^{\\mu\\nu\\rho\\sigma} u_{1\\, \\rho} \\langle s_{1\\, \\sigma} {s_{1}}^\\alpha \\rangle I_{\\alpha\\nu}\n\\bigg) \\, .\n\\end{multline}\nFinally, we perform the integrals using equations~\\eqref{eqn:I1result} and~\\eqref{eqn:I2result}, rescale the spin vector to $a_1^\\mu$ and combine the result with the direct contribution in \\eqn~\\eqref{eqn:linPiece}, to find that the angular impulse at $\\mathcal{O}(a^2)$ is\n\\[\n\\hspace{-4pt}\\Delta s_1^{\\mu,(0)}\n= -\\frac{2Gm_1 m_2}{\\sqrt{\\gamma^2 -1}}& \\bigg\\{(2\\gamma^2 - 1) u_1^\\mu \\frac{{b}_\\nu}{b^2} \\spinExp{a_1^\\nu} + \\frac{2\\gamma}{b^2} \\epsilon^{\\mu\\nu\\rho\\sigma} \\spinExp{a_{1\\,\\nu}} u_{1\\,\\rho} \\epsilon_{\\sigma}(u_1, u_2, b) \\\\ &- \\frac{2\\gamma}{b^2} u_1^\\mu \\left(\\eta^{\\nu\\alpha} - \\frac{2{b}^\\nu {b}^\\alpha}{b^2}\\right) \\epsilon_{\\alpha\\rho} (u_1, u_2) \\spinExp{{a_1}_\\nu a_1^\\rho} \\\\ & + \\frac{(2\\gamma^2 - 1)}{b^2} \\epsilon^{\\mu\\nu\\rho\\sigma} {u_1}_\\rho \\spinExp{{a_1}_\\sigma {a_1}_\\lambda} \\left(\\Pi^\\lambda{}_\\nu - \\frac{2{b}_\\nu {b}^\\lambda}{b^2}\\right)  \n \\bigg\\}\\,.\n\\label{eqn:QuantumAngImpRes}\n\\]\nThis final result agrees with the classical result of equation~\\eqref{eqn:JustinSpinResult}, modulo the remaining spin expectation values.\n\n\n\\section{Discussion}\n\\label{sec:angImpDiscussion}\n\nStarting from a quantum field theory for massive spinning particles with arbitrary long-range interactions (mediated e.g.\\ by gauge bosons or gravitons), we have followed a careful analysis of the classical limit $(\\hbar\\to0)$ for long-range scattering of spatially localised wavepackets.   We have thereby arrived at fully relativistic expressions for the angular impulse, the net change in the intrinsic angular momentum of the massive particles, due to an elastic two-body scattering process.  This, our central result of the chapter, expressed in terms of on-shell scattering amplitudes, is given explicitly at leading order in the coupling by  \\eqref{eqn:limAngImp}.  Our general formalism places no restrictions on the order in coupling, and the expression \\eqref{eqn:spinShift} for the angular impulse, like its analogues for the momentum and colour impulse found in earlier chapters, should hold at all orders. Since the publication of this formalism in ref.~\\cite{Maybee:2019jus}, general, fully classical formulae have been proposed in \\cite{Bern:2020buy,Kosmopoulos:2021zoq} for the angular impulse, expressed in terms of commutators of a modified eikonal phase. It would be very interesting to establish a firm relationship between this proposal and our results. Our 1PM calculations of the spin-squared parts of the fully covariant momentum and angular impulses have also been confirmed using post--Minkowskian EFT methods, and extended to next-to-leading-order in the coupling \\cite{Liu:2021zxr}, equivalent to a 1-loop computation.\n\nIn this chapter we applied our general results to the examples of a massive spin 1\/2 or spin 1 particle (particle 1) exchanging gravitons with a massive spin 0 particle (particle 2), imposing minimal coupling.  The results for the linear and angular impulses for particle 1, $\\Delta p_1^\\mu$ and $\\Delta s_1^\\mu$, due to its scattering with the scalar particle 2, are given by \\eqref{eqn:QuantumImpRes} and \\eqref{eqn:QuantumAngImpRes}.  These expressions are valid to linear order in the gravitational constant $G$, or to 1PM order, having arisen from the tree level on-shell amplitude for the two-body scattering process.  By momentum conservation (in absence of radiative effects at this order), $\\Delta p_2^\\mu=-\\Delta p_1^\\mu$, and the scalar particle has no intrinsic angular momentum, $s_2^\\mu=\\Delta s_2^\\mu=0$.  The spin 1\/2 case provides the terms through linear order in the rescaled spin $a_1^\\mu=s_1^\\mu\/m_1$, and the spin 1 case yields the same terms through linear order plus terms quadratic in $a_1^\\mu$.\n\nOur final results \\eqref{eqn:QuantumImpRes} and \\eqref{eqn:QuantumAngImpRes} from the quantum analysis are seen to be in precise agreement with the results \\eqref{eqn:JustinImpResult} and \\eqref{eqn:JustinSpinResult} from \\cite{Vines:2017hyw} for the classical scattering of a spinning black hole with a non-spinning black hole, through quadratic order in the spin --- except for the appearance of spin-state expectation values $\\langle a_1^\\mu\\rangle$ and $\\langle a_1^\\mu a_1^\\nu\\rangle$ in the quantum results replacing $a_1^\\mu$ and $a_1^\\mu a_1^\\nu$ in the classical result.  For any quantum states of a finite-spin particle, these expectation values cannot satisfy the  appropriate properties of their classical counterparts, e.g., $\\langle a^\\mu a^\\nu\\rangle \\ne \\langle a^\\mu\\rangle \\langle a^\\nu\\rangle$.  Furthermore, we know from section~\\ref{sec:classicalSingleParticleColour} that the intrinsic angular momentum of a quantum spin-$s$ particle scales like $\\langle s^\\mu\\rangle=m\\langle a^\\mu\\rangle\\sim s\\hbar$, and we would thus actually expect any spin effects to vanish in a classical limit where we take $\\hbar\\to0$ at fixed spin quantum number $s$.\n\nFor a fully consistent classical limit yielding non-zero contributions from the intrinsic spin we of course would need to take the spin $s\\to\\infty$ as $\\hbar\\to0$, so as to keep $\\langle s^\\mu\\rangle\\sim s\\hbar$ finite. However, the expansions in spin operators of the minimally coupled amplitudes and impulses, expressed in the forms we have derived here, are found to be universal, in the sense that going to higher spin quantum numbers $s$ continues to reproduce the same expressions at lower orders in the spin operators.  We have seen this explicitly here for the linear-in-spin level, up to spin 1.\nThis pattern was confirmed to hold for minimally coupled gravity amplitudes for arbitrary spin $s$ in ref.~\\cite{Guevara:2019fsj}. The authors showed that applying the formalism we have presented here to amplitudes in the limit $s\\to\\infty$ fully reproduces the full Kerr 1PM observables listed in \\eqref{eqn:KerrDeflections}, up to spin state expectation values. Using the coherent states in section~\\ref{sec:classicalSingleParticleColour}, one can indeed then take the limit where $\\langle a^\\mu a^\\nu\\rangle = \\langle a^\\mu\\rangle \\langle a^\\nu\\rangle$ and so forth. The precise forms of $1\/s$ corrections to the higher-multipole couplings were discussed in \\cite{Chung:2019duq}.\n\nOur formalism provides a direct link between gauge-invariant quantities, on-shell amplitudes and classical asymptotic scattering observables, with generic incoming and outgoing states, for relativistic spinning particles.  It is tailored to be combined with powerful modern techniques for computing relevant amplitudes, such as unitarity methods and the double copy.  Already, with our examples at the spin 1\/2 and spin 1 levels, we have explicitly seen that it produces new evidence (for generic spin orientations, and without taking the non-relativistic limit) for the beautiful correspondence between classical spinning black holes and massive spinning quantum particles minimally coupled to gravity, as advertised in chapter~\\ref{chap:intro}. Let us now turn to a striking application of this on-shell relationship.\n\\chapter{A worldsheet for Kerr}\n\\label{chap:worldsheet}\n\n\\section{NJ shifts from amplitude exponentiation}\n\\label{sec:NJintro}\n\nThe Newman--Janis (NJ) shift in equation~\\eqref{eqn:NJshift} is a remarkable exact property of the Kerr solution, relating it to the simpler non-spinning Schwarzschild solution via means of a complex translation.\nA partial understanding of this phenomenon is available in the context of minimally-coupled scattering amplitudes. \nRather than consider field representations of a definite spin as in the last chapter, here it is more instructive to follow Arkani--Hamed, Huang and Huang~\\cite{Arkani-Hamed:2017jhn}, and consider massive little group representations of arbitrary spin. Specifically, we will introduce spinors $\\ket{p_I}$ and $|p_I]$ with $SU(2)$ little group indices $I=1,2$, such that that the momentum is written\\footnote{We adopt the conventions of ref.~\\cite{Chung:2018kqs}.}\n\\[\np^\\mu = \\frac12 \\epsilon^{IJ} \\bra{p_J} \\sigma^\\mu | p_I] = \\frac12 \\bra{p^I} \\sigma^\\mu | p_I] \\,.\n\\]\nWe raise and lower the little group indices $I, J, \\ldots$ with two-dimensional Levi--Civita tensors, as usual. The $\\sigma^\\mu$ matrices\nare a basis of the Clifford algebra, and we use the common choice\n\\[\n\\sigma^\\mu = (1, \\sigma_x, \\sigma_y, \\sigma_z) \\,.\n\\]\nMinimally coupled three-point amplitudes take a particularly simple form when written in these spinor helicity representations. In particular, the amplitude for a spin $s$ particle of mass $m$ and charge $Q$ absorbing a photon with positive-helicity polarisation vector $\\varepsilon_k^+$ is simply given by \\cite{Chung:2018kqs,Arkani-Hamed:2019ymq}\n\\begin{equation}\n\\begin{tikzpicture}[scale=1.0, baseline={([yshift=-\\the\\dimexpr\\fontdimen22\\textfont2\\relax] current bounding box.center)}, decoration={markings,mark=at position 0.6 with {\\arrow{Stealth}}}]\n\\begin{feynman}\n\\vertex (v1);\n\\vertex [below left = 0.66 and 1 of v1] (i1);\n\\vertex [below left = 0.5 and 0.9 of v1] (i11) {$1^s$};\n\\vertex [below right= 0.66 and 1 of v1] (o1);\n\\vertex [below right = 0.5 and 0.9 of v1] (o11) {$2^s$};\n\\vertex [above = 1 of v1] (v2) ;\n\\vertex [above = 1 of v1] (v22) {$ {}_{+}$};\n\\draw [postaction={decorate}] (i1) -- (v1);\n\\draw [postaction={decorate}] (v1) -- (o1);\n\\diagram*{(v2) -- [photon,momentum=\\(k\\)] (v1)};\n\\filldraw [color=black] (v1) circle [radius=2pt];\n\\end{feynman}\t\n\\end{tikzpicture} = - \\frac{Q}{\\sqrt{\\hbar}}\\, \\varepsilon^{+}_k\\cdot(p_1 + p_2)\\,\\frac{\\langle p_1^I \\, p_2^J\\rangle^{\\odot 2s}}{m^{2s}}\\,,\n\\end{equation}\nwhere the exponent includes a prescription to symmetrise over the little group indices. We can deduce the classical limit of this generic amplitude in the same manner as in section~\\ref{sec:amplitudes} for specific field representations. The photon momentum $k$ behaves as a wavenumber in the limit, so the outgoing momentum $p_2$ can be viewed as an infinitesimal Lorentz boost, with generators~\\eqref{eqn:LorentzParameters}. In terms of spinors,\n\\begin{equation}\n|p_2^I\\rangle = |p_1^I\\rangle - \\frac{\\hbar}{2m_1} \\wn k \\cdot \\sigma \\,|p_1^I]\\,.\n\\end{equation}\nAs in our earlier examples we expand the amplitude in the spin polarisation vector, which in these variables is in general given by \\cite{Arkani-Hamed:2019ymq,Chung:2018kqs}\n\\begin{equation}\ns^\\mu_{IJ}(p) = \\frac{1}{m} s\\hbar \\langle p_I| \\sigma^\\mu|p_J]\\,.\n\\end{equation}\nFrom our single particle wavefunctions in chapter~\\ref{chap:pointParticles}, we know that as $\\hbar\\rightarrow0$ the spin $s$ must tend to infinity (since it is the size of the little group representation), such that the combination $s \\hbar$ appearing in state expectation values is finite. Thus, in the classical limit, the three-point amplitude is\n\\begin{equation}\n\\mathcal{A}_{3,+} = - \\frac{2Q}{\\sqrt{\\hbar}} (p_1\\cdot\\varepsilon_k^+) \\lim_{s\\rightarrow\\infty} \\left(\\mathbb{I} - \\frac{\\bar k \\cdot a}{2s}\\right)^{2s} = - \\frac{2Q}{\\sqrt{\\hbar}} (p_1\\cdot\\varepsilon_k^+) e^{-\\bar k \\cdot a}\\,,\n\\end{equation}\nwhere recall that $a^\\mu = s^\\mu\/m$. The denominator $\\hbar$ factor is a remnant of the correct normalisation for amplitude coupling constants from section~\\ref{sec:RestoringHBar} --- such factors will be unimportant in this chapter and thus uniformly neglected. \n\nSince $-2Q(p_1\\cdot\\varepsilon_k^+) = \\mathcal{A}_{3,+}^\\textrm{Coulomb}$ is the usual QED amplitude for a scalar of charge $Q$ and momentum $p$ absorbing a positive-helicity photon, we can conclude that\n\\[\n\\mathcal{A}_{3,+}^{\\sqrt{\\text{Kerr}}} =\ne^{-\\wn k \\cdot a} \\mathcal{A}_{3,+}^{\\text{Coulomb}} \\,.\n\\label{eq:rootKerrAmp}\n\\]\nSimilarly, the gravitational three-point amplitude for a massive particle is\n\\[\n\\mathcal{M}_{3,+}^{{\\text{Kerr}}} =\ne^{-\\wn k \\cdot a} \\mathcal{M}_{3,+}^{\\text{Schwarzschild}} \\,,\n\\]\nin terms of the ``Schwarzschild'' amplitude for a scalar particle\ninteracting with a positive-helicity graviton of momentum $k$.\nA straightforward way to establish\nthe connection of these amplitudes to spinning black holes\n\\cite{Guevara:2018wpp,Chung:2018kqs,Guevara:2019fsj}\nis to then use them to compute the impulse on a scalar probe at leading-order in the Kerr background \\cite{Arkani-Hamed:2019ymq}.\nThe calculation can be performed using the formalism of this thesis on the one hand, in particular equation~\\eqref{eqn:impulseGeneralTerm1classicalLO}; or\nusing classical equations of motion on the other.\nA direct comparison of the two approaches makes it evident\n\\cite{Guevara:2019fsj,Arkani-Hamed:2019ymq}\nthat the NJ shift of the background\nis captured by the exponential factors $e^{\\pm k\\cdot a}$.\n\nThis connection between the NJ shift and scattering amplitudes\nsuggests that the NJ shift should extend beyond the exact Kerr solution\nto the \\emph{interactions} of spinning black holes.\nIndeed, it is straightforward to scatter two Kerr particles\n(by which we mean massive particles with classical spin lengths\n$a_1$ and $a_2$)\noff one another using amplitudes.\nThe purpose of this chapter is to investigate\nthe classical interpretation of this fact.\nTo do so, we turn to the classical effective theory describing the worldline interactions of a Kerr particle \\cite{Porto:2005ac,Porto:2006bt,Porto:2008tb,Steinhoff:2015ksa,Levi:2015msa}.\nWe will see that the NJ property endows this worldline action\nwith a remarkable two-dimensional worldsheet structure.\nThe Newman--Janis story emerges via Stokes's\ntheorem on this worldsheet with boundary, and indeed persists\nfor at least the leading interactions.\nWe will see that novel equations of\nmotion, making use of the spinor-helicity formalism in a purely classical context, allow us to make the shift manifest in the leading interactions.\n\nOur effective action is constructed only from the information in the three-point amplitudes. At higher orders, information from four-point\nand higher amplitudes (or similar sources) is necessary to fully specify the effective action. Therefore our action is in principle supplemented\nby an infinite tower of higher-order operators. We may hope, however, that the worldsheet structure may itself constrain the allowed higher-dimension operators. \n\nAs further applications of our methods, we will use a generalisation of the Newman--Janis shift \\cite{Talbot:1969bpa} to introduce magnetic charges (in electrodynamics) and NUT parameters (in gravity) for the particles described by our equations of motion.\nAs an example, we compute the leading impulse on a probe particle with mass,\nspin and NUT charge moving in a Kerr--Taub--NUT background.\nThe charged generalisation of the NJ complex map can similarly be connected to the behaviour of three-point amplitudes in the classical limit \\cite{Moynihan:2019bor,Huang:2019cja,Chung:2019yfs,Moynihan:2020gxj,Emond:2020lwi,Kim:2020cvf}, and we will reproduce results recently derived from this perspective~\\cite{Emond:2020lwi},\nfurthermore calculating the leading angular impulse for the first time.\n\nThe material in this chapter is organised as follows. We begin our discussion in the context of electrodynamics, constructing the effective action for a $\\rootKerr$~\nprobe in an arbitrary electromagnetic background. In this case it is rather easy to understand how the worldsheet emerges. We discuss key\nproperties of the worldsheet, including the origin of the Newman--Janis shift, in this context. It turns out to be useful to perform the matching\nin a spacetime with ``split'' signature $(+,+,-,-)$, largely because the three-point amplitude does not exist on-shell in Minkowski space.\nThe structure of the worldsheet is particularly simple\nin split-signature spacetimes.\nIn section~\\ref{sec:gr} we turn to the gravitational case, showing that the worldsheet naturally describes the dynamics of a spinning Kerr particle.\nWe discuss equations of motion in section~\\ref{sec:spinorEOM},\nfocussing on the leading-order interactions which are not sensitive to\nterms in the effective action which we have not constrained. In this section, we will see how useful the methods of spinor-helicity are for capturing the\nchiral dynamics associated with the NJ shift, as well as magnetic charges.\n\nThis chapter is based on material first published in \\cite{Guevara:2020xjx}, in collaboration with Alfredo Guevara, Alexander Ochirov, Donal O'Connell and Justin Vines.\n\n\\section{From amplitude to action}\n\\label{sec:rootKerrEFT}\n\nWe begin by concentrating on the slightly simpler example of\nthe $\\rootKerr$~particle in electromagnetism.\nWe wish to construct an effective action\nfor a massive, charged particle with spin angular momentum $S^{\\mu\\nu}$.\nBuilding on the work of Porto, Rothstein, Levi and Steinhoff~\\cite{Porto:2005ac,Porto:2006bt,Porto:2008tb,Levi:2014gsa,Levi:2015msa},\nwe write the worldline action as\n\\[\nS = \\int\\!\\d\\tau \\bigg\\{ {-m}\\sqrt{u^2} - \\frac12 S_{\\mu\\nu} \\Omega^{\\mu\\nu} - Q A \\cdot u \\bigg\\} +  S_\\text{EFT} \\,,\n\\label{eq:fullSimpleAction}\n\\]\nwhere $u^\\mu$ and $\\Omega^{\\mu\\nu}$ are the linear\nand angular velocities,\\footnote{We will be fixing $\\tau$\nto be the proper time,\nso the velocity $u^\\mu = \\d r^\\mu\/\\d\\tau$ will satisfy $u^2=1  $.\nThe angular velocity can be defined through a body-fixed frame $e^a_\\mu(\\tau)$ on the worldline as\n\\[\n\\Omega^{\\mu\\nu}(\\tau) = e^\\mu_a(\\tau) \\frac{\\d~}{\\d\\tau} e^{a\\nu}(\\tau) \\,.\n\\label{eq:angMom}\n\\]\nThe tetrad allows us to pass from body-fixed frame indices $a, b, \\ldots$\nto Lorentz indices $\\mu,\\nu, \\ldots $, as usual.\nMore details on spinning particles in effective theory\ncan be found in recent reviews~\\cite{Porto:2016pyg,Levi:2018nxp}.}\nand $S_\\text{EFT}$ contains additional operators\ncoupling the spinning particle to the electromagnetic field. The momentum is defined as the canonical conjugate of the velocity,\n\\begin{equation}\np_\\mu = -\\frac{\\partial L}{\\partial u^\\mu} = m u_\\mu + {\\cal O}(A)\\,.\n\\end{equation}\nWe will continue to assume the spin tensor to be transverse\naccording to the Tulczyjew covariant spin supplementary condition in equation~\\eqref{eqn:SSC}.\nWe can therefore relate the spin angular momentum to the spin pseudovector $a^\\mu$ by\n\\[\na^{\\mu} = \\frac1{2p^2} \\epsilon^{\\mu\\nu\\rho\\sigma} p_\\nu S_{\\rho\\sigma} \\equiv \\frac{1}{2p^2} \\epsilon^\\mu(p,S)\\qquad \\Leftrightarrow \\qquad\nS_{\\mu\\nu} = \\epsilon_{\\mu\\nu}(p, a) \\,.\n\\label{eq:spinEquivs}\n\\]\nThe effective action~\\eqref{eq:fullSimpleAction} can be written independently of the choice of SSC, at the expense of introducing an additional term from minimal coupling \\cite{Yee:1993ya,Porto:2008tb,Steinhoff:2015ksa}. This has played an important role in recent work pushing the gravitational effective action beyond linear-in-curvature terms \\cite{Levi:2020kvb,Levi:2020uwu,Levi:2020lfn},\nbut for our present purposes a fixed SSC will suffice.\nNote that any differences in the choice of the spin tensor $S_{\\mu\\nu}$\nare projected out from the pseudovector $a^\\mu$ by definition,\nand it is the latter that will be central to our discussion.\n\nWe will only consider the effective operators in $S_\\text{EFT}$\nthat involve one power of the electromagnetic field $A_\\mu$,\nwhich can be fixed by the three-point amplitudes.\nSince these amplitudes are parity-even, \nthe possible single-photon operators are\n\\begin{multline}\nS_\\text{EFT} = Q\\sum_{n=1}^\\infty \\int\\!\\d\\tau \\, u^\\mu a^\\nu\n\\Big[ B_n  (a\\cdot \\partial)^{2n-2} F_{\\mu\\nu}^*(x)  \\\\ + C_n (a \\cdot \\partial)^{2n-1} F_{\\mu\\nu}(x) \\Big]_{x=r(\\tau)} \\,.\n\\label{eq:introEFT}\n\\end{multline}\nNotice that an odd number of spin pseudovectors is accompanied\nby the dual field strength\n\\[\nF^*_{\\mu\\nu} = \\frac12 \\epsilon_{\\mu\\nu\\rho\\sigma} F^{\\rho \\sigma} \\,,\n\\]\nwhile the plain field strength goes together with an even power of $a$.\nBy dimensional analysis,\nthe unknown constant coefficients $B_n$ and $C_n$ are dimensionless. \n\n\\subsection{Worldsheet from source}\n\nTo determine the unknown coefficients,\nwe choose to match our effective action\nto a quantity that can be derived directly from the three-point\n$\\rootKerr$~amplitude~\\eqref{eq:rootKerrAmp}.\nA convenient choice is the classical Maxwell spinor given by the amplitude for an incoming photon,\nwhich is~\\cite{Monteiro:2020plf}\n\\[\n\\phi(x) = -\\frac{\\sqrt{2}}{m} \\Re \\int\\!\\d\\Phi(\\wn k) \\, \\del(\\wn k \\cdot u) \\, \\ket{\\wn k} \\bra{\\wn k} \\, e^{- i \\wn k \\cdot x } \\mathcal{A}_{3,+} \\,.\n\\]\nIn this expression the integration is over on-shell massless phase space, cast in the notation of equation~\\eqref{eqn:dfDefinition}.\nThis Maxwell spinor is defined in (2,2) signature.\nIndeed, in Minkowski space the only solution of the zero-energy condition $\\wn k \\cdot u$ for\na massless, on-shell momentum\nis $\\wn k^\\mu=0$, so the three-point amplitude cannot exist on-shell for non-trivial kinematics.\nHowever, there is no such issue in (2,2) signature, which motivates analytically continuing from Minkowski space. (The spinor $\\ket{\\wn k}$ is constructed from the on-shell null momentum $\\wn k$ as usual in spinor-helicity.)\n\nIn fact, the Newman--Janis shift makes it extremely natural for us to analytically continue to split signature even in the classical sense, without any consideration of three-point amplitudes.\nThe Maxwell spinor for a static $\\rootKerr$~particle is explicitly\n\\[\n\\phi^{\\sqrt{\\text{Kerr}}}(x) = -\\frac{Q}{4\\pi} \\frac{1}{(x^2 + y^2 + (z + i a)^2)^{3\/2}} (x, y, z + ia) \\cdot \\boldsymbol{\\sigma} \\,.\n\\label{eq:phiKerrMink}\n\\]\nIn preparation for the analytic continuation $z= -iz'$,\nwe may choose to order the Pauli matrices\nas $\\boldsymbol{\\sigma}=(\\sigma_z, \\sigma_x, \\sigma_y)$.\nThen the spinor structure in equation~\\eqref{eq:phiKerrMink} becomes real,\nwhile the radial fall-off factor in the Maxwell spinor simplifies to\n\\[\n\\frac{1}{(x^2 + y^2 - (z - a)^2)^{3\/2}} \\,, \\nonumber\n\\]\nwhere we have dropped the prime sign of $z$.\nIn short, we have a real Maxwell spinor in (2,2) signature,\nand the spin $a$ is now a real translation in the timelike $z$ direction.\n\nWe now analytically continue the action~\\eqref{eq:introEFT}\nby choosing the spin direction to become timelike.\nIn doing so, we also continue the component of the EM field in \nthe spin direction, consistent with a covariant derivative\n$\\partial + i Q A$.\nIn split signature, it is convenient to rewrite the effective action\nansatz in terms of self- and anti-self-dual field strengths,\nwhich we define as\n\\[\nF^\\pm_{\\mu\\nu}(x) = F_{\\mu\\nu}(x) \\pm F^*_{\\mu\\nu}(x) \\,.\n\\]\nOur action then depends on a new set of unknown Wilson coefficients\n$\\tilde{B}_n$ and $\\tilde{C}_n$:\n\\[\nS_\\text{EFT} = Q \\sum_{n=0}^\\infty \\int\\!\\d\\tau \\, u^\\mu a^\\nu\n\\big[ \\tilde{B}_n (a \\cdot \\partial)^n F_{\\mu\\nu}^+(x)\n+ \\tilde{C}_n (a \\cdot \\partial)^n F_{\\mu\\nu}^-(x)\n\\big]_{x=r(\\tau)} \\,.\\label{eq:2,2actionUnmatched}\n\\]\nTo determine these coefficients we can match to the three-point amplitude by computing the Maxwell spinor for the radiation field sourced by the $\\rootKerr$~particle, which we assume to have constant spin $a^\\mu$ and constant proper velocity $u^\\mu$. In (2,2) signature the exponential factor in equation~\\eqref{eq:rootKerrAmp} also picks up a factor $-i$, so we match our action to\n\\[\n\\phi(x) = -\\frac{\\sqrt{2}}{m} \\Re\\! \\int\\! \\d\\Phi(\\wn k) \\, \\del(\\wn k \\cdot u) \\, \\ket{\\wn k} \\bra{\\wn k} \\, e^{- i \\wn k \\cdot x } \\mathcal{A}_{3,+}^\\text{Coulomb} e^{i \\wn k\\cdot a} \\,.\\label{eq:rKthreePointSpinor}\n\\]\nOur matching calculation hinges upon the field strength sourced by the particle, which is determined by the $\\rootKerr$~worldline current $\\tilde j^{\\mu}(\\wn k)$. In solving the Maxwell equation we impose retarded boundary conditions precisely as in~\\cite{Monteiro:2020plf}, placing our observation point $x$ in the future with respect to one time coordinate $t^0$, but choosing the proper velocity $u$ to point along the orthogonal time direction. It is useful to make use of the result\n\\[\n\\frac{1}{\\wn k^2_\\text{ret}} = -i \\sign \\wn k^0 \\del(\\wn k^2) + \\frac{1}{\\wn k^2_\\text{adv}} \\,,\n\\]\nwhere the $\\text{ret}$ and $\\text{adv}$ subscripts indicate retarded and advanced Green's functions respectively. Since the advanced Green's\nfunction has support for $t^0 < 0$, we may simply replace\n\\[\n\\frac{1}{\\wn k^2_\\text{ret}} = -i \\sign(\\wn k^0) \\del(\\wn k^2)  \\,.\\label{eq:2,2retardedProp}\n\\]\nThe field strength sourced by the current in split signature is therefore\n\\[\nF^{\\mu\\nu}(x)  &= 2\\!\\int\\!\\dd^4\\wn k\\, \\sign(\\wn k^0) \\del(\\wn k^2) \\, \\wn k^{[\\mu} \\tilde j^{\\nu]}\\,e^{-i\\wn k\\cdot x}\\\\\n&= 2\\!\\int\\!\\dd^4\\wn k\\, \\left(\\Theta(\\wn k^0) - \\Theta(-\\wn k^0)\\right) \\del(\\wn k^2) \\,  \\wn k^{[\\mu} \\tilde j^{\\nu]}\\,e^{-\\wn ik\\cdot x}\\,,\n\\]\nNotice that the appropriate integral measure is now precisely the invariant phase-space measure~\\eqref{eqn:dfDefinition}; substituting the worldline current for our $\\rootKerr$~effective action in equation~\\eqref{eq:2,2actionUnmatched}, evaluated on a leading-order trajectory, we thus have\n\\begin{multline}\nF^{\\mu\\nu}(x) = 4Q \\Re\\! \\int\\!\\d\\Phi(\\wn k)\\,\\del(\\wn k\\cdot u) \\bigg\\{\\wn k^{[\\mu} u^{\\nu]}\\Big[1 + ia\\cdot \\wn k \\sum_{n=0}^\\infty \\left(\\tilde B_n (ia\\cdot \\wn k)^{n} + \\tilde C_n (ia\\cdot \\wn k)^{n}\\right)\\Big]\\\\\n+ i\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\sum_{n=0}^\\infty\\left(\\tilde B_n(ia\\cdot \\wn k)^{n} - \\tilde C_n (ia\\cdot \\wn k)^{n}\\right)\\bigg\\} \\, e^{-i\\wn k\\cdot x}\\,.\\label{eq:unmatchedField}\n\\end{multline}\n\nTo match to the three-point $\\rootKerr$~amplitude, we need to compute the Maxwell spinor $\\phi$ and its conjugate, $\\tilde \\phi$. To do so, we introduce a basis of positive and negative helicity polarisation vectors $\\varepsilon_k^{\\pm}$. On the support of the delta function in~\\eqref{eq:unmatchedField}, manipulations using the spinorial form of the polarisation vectors given in \\cite{Monteiro:2020plf} then lead to\n\\[\n\\wn k^{[\\mu} u^{\\nu]} \\sigma_{\\mu\\nu} &= +\\frac{\\sqrt{2}}{2}\n \\varepsilon_k^+\\cdot u\\, |\\wn k\\rangle \\langle \\wn k|\\\\\n\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} a\\cdot \\wn k\\ \\varepsilon_k^+\\cdot u \\,|\\wn k\\rangle \\langle \\wn k|\\,.\n\\]\nThe latter equality relies upon the identity $\\wn k^{[\\mu} \\epsilon^{\\nu\\rho\\sigma\\lambda]} = 0$, and the fact that $\\sigma_{\\mu\\nu}$ is self-dual in this signature. With these expressions in hand, it is easy to see that the Maxwell spinor has a common spinorial basis, and takes the simple form\n\\begin{multline}\n\\phi(x) = 2\\sqrt{2} Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k\\rangle \\langle \\wn k| \\,\\varepsilon_k^+\\cdot u  \\\\ \\times \\left(1 + \\sum_{n=0}^\\infty 2 \\tilde C_n (ia\\cdot\\wn k)^{n+1}\\right).\\label{eq:unmatchedMaxwell}\n\\end{multline}\nRecall from equation~\\eqref{eq:2,2actionUnmatched} that the Wilson coefficients $\\tilde B_n$ and $\\tilde C_n$ were identified with self- and anti-self-dual field strengths, respectively. Since a positive-helicity wave is associated with an anti-self-dual field strength, it is no surprise that the Maxwell spinor should depend only on this part of the $\\rootKerr$~effective action. Fixing the $\\tilde B_n$ coefficients requires the dual spinor, which is given by\n\\begin{multline}\n\\tilde \\phi(x) = -2\\sqrt{2} Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k]\\, [\\wn k|\\, \\varepsilon_k^-\\cdot u \\\\ \\times \\left(1 + \\sum_{n=0}^\\infty 2 \\tilde B_n (ia\\cdot\\wn k)^{n+1}\\right)\\,.\n\\label{eq:unmatchedMaxwellDual}\n\\end{multline}\nHere we have used that\n\\[\n\\wn k^{[\\mu} u^{\\nu]} \\tilde\\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} \\varepsilon_k^-\\cdot u\\, |\\wn k]\\,[\\wn k|\\\\\n\\wn k^{[\\mu} \\epsilon^{\\nu]}(\\wn k,u,a) \\tilde\\sigma_{\\mu\\nu} &= -\\frac{\\sqrt{2}}{2} a\\cdot\\wn k\\ \\varepsilon_k^-\\cdot u \\,|\\wn k]\\,[\\wn k|\\,,\n\\]\nrecalling that $\\tilde \\sigma$ is anti-self-dual in split signature spacetimes.\n\nIt now only remains to match to the Maxwell spinors for the three-point~amplitude, as given in equation~\\eqref{eq:rKthreePointSpinor}. The scalar Coulomb amplitudes for photon absorption are just $\\mathcal{A}_\\pm = -2mQ\\, u\\cdot\\varepsilon_k^\\pm$, so for the $\\rootKerr$~three-point amplitude\n\\[\n\\phi(x) &= + 2\\sqrt{2}  Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\, |\\wn k\\rangle \\langle\\wn k| \\, \\varepsilon_k^+\\cdot u \\, e^{i\\wn k\\cdot a}\\,,\\\\\n\\tilde\\phi(x) &= -2\\sqrt{2}  Q \\Re\\! \\int\\! \\d\\Phi(\\wn k)\\, \\del(\\wn k\\cdot u) e^{-i\\wn k\\cdot x}\\,  |\\wn k]\\, [\\wn k|\\, \\varepsilon_k^-\\cdot u\\,e^{-i \\wn k\\cdot a}\\,.\n\\]\nExpanding the exponentials and matching to eqs.~\\eqref{eq:unmatchedMaxwell} and \\eqref{eq:unmatchedMaxwellDual} identifies\n\\begin{equation}\n\\tilde B_n = \\frac{(-1)^{n+1}}{2(n+1)!}\\,, \\qquad \\quad\n\\tilde C_n = \\frac{1}{2(n+1)!}\\,,\n\\end{equation}\nwhich upon substitution into equation~\\eqref{eq:2,2actionUnmatched} finally yields the $\\rootKerr$~effective action in split-signature:\n\\[\nS_\\text{EFT}  &= Q \\sum_{n=0}^\\infty \\int\\!\\d\\tau\\, u^\\mu a^\\nu\n\\bigg[ {-\\frac{(-a\\cdot \\partial)^n}{2(n+1)!}}  F_{\\mu\\nu}^+(x)\n+ \\frac{(a\\cdot \\partial)^n}{2(n+1)!} F_{\\mu\\nu}^-(x)\n\\bigg]_{x=r(\\tau)} \\\\ \n=& -\\frac{Q}{2} \\int\\!\\d\\tau\\, u^\\mu a^\\nu\n\\bigg[ \\left(\\frac{e^{-a \\cdot \\partial} - 1}{-a \\cdot \\partial}\\right) F_{\\mu\\nu}^+(x)\n- \\left(\\frac{e^{ a \\cdot \\partial} - 1}{a \\cdot \\partial}\\right) F_{\\mu\\nu}^-(x)\n\\bigg]_{x=r(\\tau)} \\,.\\label{eq:2,2actionMatchedCoefficients}\n\\]\n\nSo far, the Newman--Janis structure is hinted at by the translation\noperators $e^{\\pm a \\cdot \\partial}$ appearing in the effective action.\nWe can make this structure more manifest by writing the effective action equivalently as\n\\[\nS_\\text{EFT} & = -\\frac{Q}{2}\n\\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\, u^\\mu a^\\nu\n\\big[ e^{-\\lambda \n(a \\cdot \\partial)} F_{\\mu\\nu}^+(x)\n- e^{\\lambda (a \\cdot \\partial)} F_{\\mu\\nu}^-(x)\n\\big]_{x=r(\\tau)} \\\\ &\n= -\\frac{Q}{2} \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\, u^\\mu a^\\nu\n\\big[ F^+_{\\mu\\nu}(r - \\lambda a) - F^-_{\\mu\\nu}(r + \\lambda a) \\big] \\,.\n\\label{eq:seftStep}\n\\]\nOur effective action is now an integral over a two-dimensional region\n--- a worldsheet, rather than a worldline.\n\nTo see that this worldsheet is indeed connected to the Newman--Janis shift,\nlet us recover this shift for the Maxwell spinor.\nFirst, we can read off the worldsheet current~$J^\\mu$ from the action\n$S_\\text{EFT}-Q\\!\\int\\!\\d\\tau A_\\mu u^\\mu = -\\!\\int\\!\\d^4x A_\\mu J^\\mu$.\nThen, the gauge field $A_\\mu$ set up at a point~$x$ by this source\nmay be written as an integral of a Green's function $G(x-y)$ over the worldsheet:\n\\begin{align}\nA^\\mu(x)  = \\int\\!\\d^4y\\,G(x-y) J^\\mu(y) \\hphantom{+ \\frac{1}{2}\\!\\int_0^1\\!\\d\\lambda}&\\\\ \n=  Q\\!\\int\\!\\d\\tau\n\\bigg\\{ u^\\mu G(x-r)\n+ \\frac{1}{2}\\!\\int_0^1\\!\\d\\lambda \\Big(\n&\\big[ u^\\mu (a \\cdot \\partial)\n+ \\epsilon^{\\mu}(u, a, \\partial)\n\\big] G(x-r+\\lambda a) \\nonumber \\\\ \n- &\\big[ u^\\mu (a \\cdot \\partial)\n- \\epsilon^{\\mu} (u, a, \\partial)\n\\big] G(x-r-\\lambda a) \\Big)\\!\n\\bigg\\} \\,. \\nonumber\n\\end{align}\nThe field strength follows by differentiation, after which contraction with $\\sigma$ matrices yields the Maxwell spinor,\n\\[\n\\phi(x) = 2Q\\!\\int\\!\\d\\tau\\, \\sigma_{\\mu\\nu}\nu^{\\nu} \\partial^{\\mu}\\! \\left[ G(x-r) - \\int_0^1\\!\\d\\lambda\n\\,(a \\cdot \\partial) G(x-r-\\lambda a) \\right] ,\n\\label{eq:NJbyintegration}\n\\]\nwhere the first term comes from the non-spinning part of the action~\\eqref{eq:fullSimpleAction}.\nNow the $a\\cdot\\partial$ operator acting on the Green's function\ncan be understood as a derivative with respect to $\\lambda$.\nThis produces a $\\lambda$ integral of a total derivative,\nwhich reduces to the boundary terms.\nCancelling the first term in equation~\\eqref{eq:NJbyintegration}\nagainst the boundary contribution at $\\lambda = 0$, we find simply that\n\\[\n\\phi(x) = 2Q\\!\\int\\!\\d\\tau \\, \\sigma_{\\mu\\nu} u^\\nu \\partial^\\mu  G(x-r-a)  \\,.\n\\]\nThe Maxwell spinor depends only on the anti-self-dual part of the effective action, shifted by the spin length. The real translation in $(2,2)$ signature\nis a result of this real worldsheet structure. We will shortly see that this structure persists for interactions.\n\n\n\n\\subsection{Worldsheet for interactions}\n\nLet us analytically continue\nthe action~\\eqref{eq:seftStep} back to Minkowski space:\n\\[\nS_\\text{EFT} & = \\frac{Q}{2} \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\nu^\\mu a^\\nu \\big[ i F^+_{\\mu\\nu}(r + i \\lambda a)\n- i F^-_{\\mu\\nu}(r - i \\lambda a) \\big] \\\\ &\n= Q \\Re \\int_\\Sigma \\! \\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(r + i \\lambda a) \\, u^\\mu a^\\nu \\,.\n\\]\nHere the self-dual and anti-self-dual field strengths are\n\\[\nF^\\pm_{\\mu\\nu} = F_{\\mu\\nu} \\pm i F^*_{\\mu\\nu}\n= \\pm \\frac{i}{2} \\epsilon_{\\mu\\nu\\rho\\sigma} F^{\\pm\\,\\rho\\sigma} \\,,\n\\label{eq:rKwsaction}\n\\]\nand $\\Sigma$ is the worldsheet, with $\\tau$ running over $(-\\infty,\\infty)$, and $\\lambda$ over $[0,1]$.\n\nNow that we are back in Minkowski space,\nlet us turn to the Newman--Janis structure of interactions.\nSuppose that our spinning particle is moving\nunder the influence of an external electromagnetic field, generated by distant sources. The total interaction Lagrangian contains the worldsheet\nterm~\\eqref{eq:rKwsaction} as well as the usual worldline minimal coupling:\n\\[\nS_\\text{int} = -Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!\\!\\d\\tau A_\\mu(r) u^\\mu\n+ Q \\Re\\! \\int_\\Sigma\\!\\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(r + i \\lambda a) \\, u^\\mu a^\\nu + \\ldots \\,,\n\\label{eq:sint}\n\\]\nwhere $\\partial \\Sigma_\\text{n}$ is the ``near'' boundary of the worldsheet, at $\\lambda = 0$, as shown in figure~\\ref{rootKerrIntegration}. We will similarly refer to the boundary at $\\lambda = 1$ as the ``far'' boundary. The near boundary is the physical location of the object, while the far boundary is a timelike line embedded in the complexification of Minkowski space. We have also indicated the presence of unknown additional operators (involving at least two powers of the field strength) in the action by the ellipsis in equation~\\eqref{eq:sint}.\n\nIt is convenient to introduce a complex coordinate $z = r + i \\lambda a$ on the worldsheet. In terms of this coordinate, we may write the two-form\n\\[\nF^+(z) =\n\\frac{1}{2} F^+_{\\mu\\nu}(z) \\d z^\\mu \\wedge \\d z^\\nu = i F^+_{\\mu\\nu}(z) (u^\\mu + i \\lambda \\dot a^\\mu) a^\\nu \\, \\d \\tau \\wedge \\d \\lambda \\,,\n\\]\nwhere $\\dot a^\\mu = \\d a^\\mu \/ \\d \\tau$. Since in the absence of interactions the spin is constant, $\\dot a$ must be of order $F$.\nTherefore, we may rewrite our interaction action as\n\\[\nS_\\text{int} = -Q \\int_{\\partial \\Sigma_\\text{n}} \\!\\!\\! A_\\mu(r) \\d r^\\mu\n+ \\frac{Q}{2} \\Re\\! \\int_\\Sigma F^+_{\\mu\\nu}(z) \\, \\d z^\\mu \\wedge \\d z^\\nu + \\ldots \\,.\n\\label{eq:sintNicer}\n\\]\nIn doing so, we have redefined the higher-order operators indicated by the ellipsis.\n\n\\begin{figure}[t]\n\t\\center\n\t\\includegraphics[width = 0.5\\textwidth]{contour.pdf}\n\t\\vspace{-3pt}\n\t\\caption[Geometry of the Kerr worldsheet effective action.]{Geometry of the effective action: boundary $\\partial \\Sigma_\\text{n}$ of the  complex worldsheet (translucent plane) is  fixed to the particle worldline in real space (solid plane).\\label{rootKerrIntegration}}\n\\end{figure}\n\nWhen the electromagnetic fields appearing in the action~\\eqref{eq:sintNicer} are generated by external sources,\nboth $F$ and $F^*$ are closed two-forms,\nso we may introduce potentials $A$ and $A^*$\nsuch that $F = \\d A$ and $F^* = \\d A^*$.\nThe dual gauge potential $A^*$ is related to $A$ by duality,\nbut this relationship need not concern us here:\nwe only require that both potentials exist in the vicinity\nof the $\\rootKerr$~particle.\nHence we may also write $F^+ = \\d A + i\\, \\d A^* = \\d A^+$.\nThen the action~\\eqref{eq:sintNicer} becomes\n\\[\nS_\\text{int} &\n=-Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!A\n+ Q \\Re\\!\\int_\\Sigma\\!\\d A^+ + \\ldots \\\\ &\n=-Q \\int_{\\partial \\Sigma_\\text{n}}\\!\\!A\n+ Q \\Re\\!\\int_{\\partial \\Sigma_\\text{n}}\\!\\!A^+\n- Q \\Re\\!\\int_{\\partial \\Sigma_\\text{f}}\\!\\!A^+ + \\ldots\\,,\n\\]\nwhere the boundary consists of two disconnected lines (the far and near boundaries).\nThe orientation of the integration contour was set by $F^+$,\nas depicted in figure~\\ref{rootKerrIntegration}.\n\nNow, notice that on the near boundary $z = r(\\tau)$ is real.\nHence $\\Re A^+ = A$,\nso we are only left with the far-boundary contribution in the action:\n\\[\nS_\\text{int} = -Q \\Re\\!\\int_{\\partial \\Sigma_\\text{f}}\\!\\!A^+ + \\ldots\n=-Q \\Re\\!\\int\\!\\d \\tau \\, u^\\mu A_\\mu^+(r + i a) + \\ldots \\,.\n\\label{eq:rtKerrIntShift}\n\\]\nThus we explicitly see that the interactions\nof a $\\rootKerr$~particle can be described with a Newman--Janis shift.\nWe will exploit this fact explicitly in section~\\ref{sec:spinorEOM}.\nBefore we do, we turn to gravitational interactions.\n\n\n\n\\section{Spin and gravitational interactions}\n\\label{sec:gr}\n\nAs a step towards a worldsheet action for a probe Kerr in a non-trivial background, it is helpful to understand how to make the electromagnetic\neffective action~\\eqref{eq:rKwsaction} generally covariant. In a curved spacetime, we cannot simply add a vector $\\lambda a$ to a point $r$.\nTo see what to do, let us reintroduce translation operators as in equation~\\eqref{eq:seftStep}. The worldsheet EFT term in Minkowski space is\n\\[\nS_\\text{EFT} &= Q \\Re\\! \\int_\\Sigma \\! \\d\\tau \\d\\lambda\\, \ni\\, e^{i \\lambda  \\, a \\cdot \\partial} F^+_{\\mu\\nu}(x)\\, u^\\mu a^\\nu \\Big|_{x=r(\\tau)}  \\\\\n&= Q \\Re \\int_\\Sigma \\! \\d\\tau \\d\\lambda \\, i\n\\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\partial)^n  F^+_{\\mu\\nu}(x) \\, u^\\mu a^\\nu\\Big|_{x=r(\\tau)} \\,.\n\\]\nNow, it is clear that a minimal way to make this term generally covariant is to replace the partial derivatives $\\partial$ with covariant \nderivatives $\\nabla$, so in curved space we have\n\\[\nS_\\text{EFT} &= Q \\Re \\!\\int_\\Sigma \\! \\d\\tau \\d\\lambda \\,\n\\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\nabla)^n F^+_{\\mu\\nu}(x) \\, i u^\\mu a^\\nu  \\Big|_{x=r(\\tau)} \\,.\n\\label{eq:rKeftCurvedStep}\n\\]\n\nIt is therefore natural for us to consider a covariant translation operator\n\\[\ne^{i\\lambda\\,a \\cdot \\nabla} \\equiv \\sum_{n=0}^\\infty \\frac{1}{n!} (i \\lambda \\, a \\cdot \\nabla)^n \\,.\n\\label{eq:explicitTranslation}\n\\]\nThis operator generates translations along geodesics in the direction $a$.\nTo see\nwhy, note that the perturbative expansion of such a geodesic beginning at a point\n$x_0$ in the direction $a$ with parameter $\\ell$ is\n\\[\nx^\\mu(\\ell) = x_0^\\mu + \\ell \\, a^\\mu - \\frac{\\ell^2}{2} \\Gamma^\\mu_{\\nu\\rho}(x_0) a^\\nu a^\\rho + \\ldots \\,.\n\\]\nNow consider the perturbative expansion of a scalar function $f(x)$ along such a\ngeodesic. We have\n\\begin{align}\nf(x(\\ell)) &= f(x_0) + \\ell \\, a^\\mu \\partial_\\mu f(x_0) + \\frac{\\ell^2}2 a^\\mu a^\\nu \n\\left(\n\\partial_\\mu \\partial_\\nu f(x_0) - \\Gamma^\\alpha_{\\mu\\nu}(x_0) \\partial_\\alpha f(x_0)\n\\right) \n+ \\ldots \\nonumber\\\\\n&=\nf(x_0) + \\ell (a \\cdot \\nabla) f(x_0) + \\frac{\\ell^2}2 (a \\cdot \\nabla) (a \\cdot \\nabla) f(x_0)\n+ \\ldots \\\\\n&= e^{\\ell \\, a\\cdot \\nabla} f(x_0) \\,.\\nonumber\n\\end{align}\n\nA traditional point of view on equation~\\eqref{eq:rKeftCurvedStep} is that the\noperators only act on the two-form $F^+_{\\mu\\nu}$. However, we can alternatively\nthink of the operator acting on a scalar function $F^+_{\\mu\\nu} u^\\mu a^\\nu$, \nprovided we extend the definitions of the velocity $u$ and the spin $a$\nso that they become fields on the domain of the translation operator. We can simply\ndo this by parallel-transporting $u(r(\\tau))$ \nand $a(r(\\tau))$ along the geodesic beginning at $r(\\tau)$ in the direction $a(\\tau)$\n(using the Levi--Civita connection). We denote these geodesics by $z(\\tau, \\lambda)$; \nexplicitly,\n\\[\nz^\\mu(\\tau, \\lambda) = r^\\mu(\\tau) + i \\lambda a^\\mu(\\tau) + \\frac{\\lambda^2}{2} \\Gamma^\\mu_{\\nu\\rho}(r(\\tau)) a^\\nu a^\\rho + \\ldots \\,.\n\\]\n(Notice that the translation operator~\\eqref{eq:explicitTranslation} has parameter\n$i \\lambda$.)\nThe parallel-transported vectors, with initial conditions $a(z(\\tau, 0)) = a(\\tau)$ \nand $u(z(\\tau, 0)) = u(\\tau)$,\nhave the similar perturbative expansions\n\\[\nu^\\mu(z(\\tau, \\lambda)) &= u^\\mu(\\tau) - i \\lambda \\Gamma^\\mu_{\\nu\\rho} (r(\\tau)) \na^\\nu(\\tau) u^\\rho(\\tau) + \\ldots \\,,\\\\\na^\\mu(z(\\tau, \\lambda)) &= a^\\mu(\\tau) - i \\lambda \\Gamma^\\mu_{\\nu\\rho} (r(\\tau)) \na^\\nu(\\tau) a^\\rho(\\tau) + \\ldots \\,.\n\\]\nWe now view the translation operator in equation~\\eqref{eq:rKeftCurvedStep}\nas acting on the scalar quantity $F^+_{\\mu\\nu} u^\\mu a^\\nu$:\n\\[\ne^{i \\lambda a \\cdot \\nabla} F_{\\mu\\nu}(r(\\tau)) a^\\mu(\\tau) u^\\nu(\\tau) \n= F_{\\mu\\nu}(z(\\tau,\\lambda)) a^\\mu(z(\\tau,\\lambda)) u^\\nu(z(\\tau, \\lambda)) \\,.\n\\]\nExpanding perturbatively to first order in $\\lambda$, we have\n\\begin{align}\ne^{i \\lambda a \\cdot \\nabla} F_{\\mu\\nu}(r(\\tau)) a^\\mu(\\tau) u^\\nu(\\tau)\n&= \\left(F_{\\mu\\nu}(r(\\tau)) + i \\lambda a^\\rho \\left(\n\\partial_\\rho F_{\\mu\\nu} - \\Gamma^\\alpha_{\\mu\\rho} F_{\\alpha\\nu} \n-\\Gamma^\\alpha_{\\nu\\rho} F_{\\mu \\alpha} \n\\right)\\right) a^\\mu u^\\nu \\nonumber \\\\\n&= \\left( F_{\\mu\\nu}(r(\\tau)) + i \\lambda a^\\rho \\nabla_\\rho F_{\\mu\\nu}\\right)a^\\mu u^\\nu \\,.\n\\end{align}\nThe final expression is precisely the same as the picture in which \nthe derivatives act only on the field strength: these are equivalent points of\nview.\n\nThe worldsheet arises from interpreting the translation operators as genuine\ntranslations. In curved space, the operators replace the straight-line sum \n$r+i a \\lambda$ appearing in our action~\\eqref{eq:rKwsaction} with the \nnatural generalisation --- a geodesic in the direction $a$.\\footnote{In general, \nthese geodesics may become singular. We assume that such singularities do\nnot arise. If they were to arise, there would also be a divergence in the\ninterpretation of the EFT as an infinite sum of operators.}\nWe can express the curved-space effective action as\n\\[\nS_\\text{EFT} = Q \\Re \\!\\int_\\Sigma \\! \\d\\tau \\d \\lambda \\, iF^+_{\\mu\\nu}(z) \\, u^\\mu(z) a^\\nu(z) \\Big|_{z=z(\\tau,\\lambda)} \\,.\n\\label{eq:rkCovActionVelocity}\n\\]\nThe surface $\\Sigma$ is built up from the worldline of the particle, augmented by the geodesics in the direction $a$ for each $\\tau$.\n\nNote that, since we neglect higher-order interactions,\nwe may replace the velocity vector field $u(\\tau,\\lambda)$ \nin the action~\\eqref{eq:rkCovActionVelocity}\nwith the similarly defined momentum field $p(\\tau, \\lambda)$.\nIndeed, at $\\lambda = 0$ the difference adds another order in the gauge field,\nand this persists for $\\lambda \\neq 0$ after parallel translation along the geodesics.\nTherefore, up to $F^2$ operators that we are neglecting,\nthe $\\rootKerr$~action may be written as\n\\[\nS_\\text{EFT} = \\frac{Q}{m} \\Re\\!\\int_\\Sigma\\!\\d\\tau \\d\\lambda \\, iF^+_{\\mu\\nu}(z) \\, p^\\mu(z) a^\\nu(z) \\Big|_{z=z(\\tau,\\lambda)} \\,.\n\\label{eq:rkCovActionMomentum}\n\\]\n\nWe are now ready for the fully gravitational Kerr worldsheet action,\nwhich is naturally motivated as a classical double copy of this covariantised worldsheet action.\nRecalling that we should double-copy from non-Abelian gauge theory rather than electrodynamics, we promote the field strength to the Yang--Mills case:\n\\[\nQ F^+_{\\mu\\nu}(z) ~\\rightarrow~ c^A(z) F^{A+}_{\\mu\\nu}(z) \\,,\n\\]\nwhere $c^A(z(\\tau,\\lambda))$ is a vector in the colour space (generated by parallel transport from the classical colour vector of a particle,\nas described by the Yang--Mills--Wong equations~\\eqref{eqn:classicalWong}).\nThe double copy replaces colour by kinematics, so we anticipate a replacement of the form $c^A \\rightarrow u^\\mu$.\nMoreover, to replace $F^{A}_{\\mu\\nu}$ we need an object with three indices,\nantisymmetric in two of them,\nfor which the spin connection\n\\[\n\\omega_\\mu{}^{ab} = e^b_\\nu\\, \\nabla_\\mu e^{a\\nu} =\ne^b_\\sigma \\big( \\partial_\\mu e^{a\\sigma} + \\Gamma^\\sigma_{\\mu\\nu} e^{a\\nu} \\big)\\label{eq:spinConnection}\n\\]\nis the natural candidate.\nSince it is defined via a derivative of the (body-fixed) spacetime tetrad~$e_a^\\mu$, which is\na dimensionless quantity, on dimensional grounds the replacement should be\nof the form $F^{A}_{\\mu\\nu} \\to m \\omega_\\mu{}^{ab}$.\nIndeed, we find that the correct worldsheet action for Kerr is\n\\[\nS_\\text{EFT} = \\Re\\! \\int_\\Sigma \\d\\tau \\d \\lambda \\, i\\,u_\\mu(z) \\, \\omega^{+\\mu}{}_{ab}(z) \\, p^a(z) a^b(z) \\Big|_{z=z(\\tau,\\lambda)} \\,,\n\\label{eq:kerrws}\n\\]\nwhere $\\omega^+$ is a self-dual part of the spin connection,\ndefined explicitly by\n\\[\n\\omega^{+\\mu}{}_{ab}(x) = \\omega^{\\mu}{}_{ab}(x) + i\\,\\omega^{*\\mu}{}_{ab}(x) \\,, \\qquad\n\\omega^{*\\mu}{}_{ab}(x) = \\frac12 \\epsilon_{abcd}\\,\\omega^{\\mu\\,cd}(x) \\,.\n\\label{eq:omegaDefs}\n\\]\nIn writing these equations, we have extended the body-fixed frame $e_a = e_a^\\mu \\partial_\\mu$ of vectors to every point of the complex worldsheet. We do so by parallel transport.\nAs usual, the frame indices $a, b, \\cdots$ take values from 0 to 3,\nand $\\epsilon_{abcd}$ is the flat-space Levi--Civita tensor,\nwith $\\epsilon_{0123} = + 1$.\n\n\n\n\\subsection{Flat-space limit}\n\nWe will shortly prove that the worldsheet term~\\eqref{eq:kerrws} reproduces all single-curvature terms in the known effective action for a Kerr black hole in an arbitrary background~\\cite{Porto:2006bt,Porto:2008tb,Levi:2015msa}. But first we wish to show that the term is non-trivial even in\nflat space, and is in fact the standard kinetic term for a spinning particle in Minkowski space~\\cite{Porto:2005ac} in that context.\n\nIn flat space and Cartesian coordinates, the worldsheet effective term~\\eqref{eq:kerrws} is\n\\[\nS_\\text{EFT} = \\Re\\! \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\ni\\,u_\\mu(\\tau)\\,\\omega^{+\\mu}{}_{ab}(r + i \\lambda a)\\, p^a(\\tau) a^b(\\tau) \\,, \n\\label{eq:recoverSpinKineticStep}\n\\]\nsince the parallel transport of the vectors $u, p$ and $a$ is now trivial, and the geodesics reduce to straight lines. In flat space, \nthe frame $e^a_\\mu(\\tau, \\lambda)$ is also independent of $\\lambda$, since it is\ngenerated by parallel transport. \nThus, the spin connection is $\\lambda$-independent and the $\\lambda$ integral in equation~\\eqref{eq:recoverSpinKineticStep} becomes trivial.\n\nGiven the $\\lambda$ independence of the spin connection, we may write\n\\[\nS_\\text{EFT} & = \\Re\\! \\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,i\\,\nu_\\mu(\\tau)\\,\\omega^{+\\mu}{}_{ab}(r(\\tau))\\, p^a(\\tau) a^b(\\tau) \\\\ &\n=-\\!\\int\\!\\d\\tau\\,\nu_\\mu(\\tau)\\,\\omega^{*\\mu}{}_{ab}(r(\\tau))\\, p^a(\\tau) a^b(\\tau) \\,.\n\\]\nRecalling the definitions of the dual spin connection~$\\omega^*$ and the spin pseudovector~$a$, eqs.~\\eqref{eq:omegaDefs} and \\eqref{eq:spinEquivs}, we equivalently have\n\\[\nS_\\text{EFT} = - \\frac12 \\int\\!\\d\\tau\\,\nu_\\mu(\\tau)\\, \\omega^{\\mu}{}^{ab}(r(\\tau))\\, S_{ab}(\\tau)\n=- \\frac12 \\int\\!\\d\\tau\\, \\Omega{}^{ab}(r(\\tau))\\, S_{ab}(\\tau) \\,.\\label{eq:wsSpinKinematic}\n\\]\nThis is nothing but the spin kinetic term written in equation~\\eqref{eq:fullSimpleAction}.\nIn this way, we see that the worldsheet expression~\\eqref{eq:kerrws} already describes the basic dynamics of spin.\n\n\n\\subsection{Single-Riemann effective operators}\n\nIt is now straightforward to recover the full tower of single-Riemann operators in the Kerr effective action. Returning to the full curved-space case, we may write our action~\\eqref{eq:kerrws} as\n\\[\nS_\\text{EFT} & = \\Re \\!\\int\\!\\d\\tau\\!\\int_0^1\\!\\d\\lambda\\,\ne^{i \\lambda \\, a \\cdot \\nabla} i\\, u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\\\ &\n= \\Re \\!\\int\\!\\d\\tau\\, i\\left(\\frac{e^{i a \\cdot \\nabla}-1}{i a \\cdot \\nabla}\\right) u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\\\ &\n= \\sum_{n=0}^\\infty\n\\Re\\!\\int\\!\\d\\tau\\, \\frac{(i a \\cdot \\nabla)^n}{(n+1)!} i\\,u_\\mu(\\tau) \\, \\omega^{+\\mu}{}_{ab}(r(\\tau)) \\, p^a(\\tau) a^b(\\tau) \\,,\n\\]\nwhere we performed the $\\lambda$ integral and expanded the translation operator $e^{i a \\cdot \\nabla}$.\nThe leading contribution is again the spin kinetic term,\nas a short computation demonstrates.\nWe also encounter an infinite series of higher-derivative contributions for $n \\geq 1$. To express them in terms of the Riemann tensor,\nwe recall that it satisfies\n\\[\nR_{ab\\,\\mu\\nu} = e^\\alpha_a e^\\beta_b R_{\\alpha\\beta\\,\\mu\\nu}\n= -\\nabla_\\mu \\omega_{\\nu ab} + \\nabla_\\nu \\omega_{\\mu ab}\n+ \\omega_{\\mu ac}\\,\\omega_{\\nu}{}^c_{~\\,b}\n- \\omega_{\\nu ac}\\,\\omega_{\\mu}{}^c_{~\\,b} \\,.\n\\label{Connection2Riemann}\n\\]\nConsistently omitting the quadratic in $\\omega$ terms from the equation above, as well as the higher-order interaction contributions due to the difference between $p^a$ and $mu^a$,\nwe rewrite a typical effective operator as\n\\begin{align}\n& -\\frac{1}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, u^\\mu p^a a^b\n(i a \\cdot \\nabla)^{n-1} a^\\nu \\nabla_\\nu \\omega^+_{\\mu ab}\n\\big|_{x = r(\\tau)} \\nonumber \\\\ & \\quad\n=-\\frac{1}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, p^a a^b u^\\mu a^\\nu\n(i a \\cdot \\nabla)^{n-1}\n\\big[ R^+_{ab\\,\\mu\\nu} + \\nabla_\\mu \\omega^+_{\\nu ab}\n\\big]_{x = r(\\tau)} + \\ldots \\label{eq:EffOperatorSteps}\\\\ & \\quad\n=-\\frac{m}{(n+1)!} \\Re\\!\\int\\!\\d\\tau\\, u^a a^b a^\\nu\n(i a \\cdot \\nabla)^{n-1}\n\\bigg[ u^\\mu R^+_{ab\\,\\mu\\nu} + \\frac{D~}{d\\tau} \\omega^+_{\\nu ab}\n\\bigg]_{x = r(\\tau)} + \\ldots \\,.\\nonumber\n\\end{align}\nHere $R^+_{ab\\,\\mu\\nu}=R_{ab\\,\\mu\\nu}+iR^*_{ab\\,\\mu\\nu}$ is defined via the dualisation of the first two indices. Notice that in equation~\\eqref{eq:EffOperatorSteps} we treat \nthe velocity $u$, momentum $p$ and spin $a$ as fields on the worldline, so that they\ncommute with the covariant derivative.\n\nWe may proceed by integrating the $D\/\\d \\tau$ term by parts, after which it acts on factors of velocity and spin.\nThis generates curvature-squared (and higher) operators that we again neglect.\nIn this way, we arrive at the form of the leading interaction Lagrangian\n\\[\nS_\\text{int} =-m\\!\\int\\!\\d\\tau\\,u^a a^b u^\\mu a^\\nu \\Re \\sum_{n=1}^\\infty\n\\frac{(i a \\cdot \\nabla)^{n-1}\\!}{(n+1)!}\n\\bigg[ R_{ab\\,\\mu\\nu} + i R^*_{ab\\,\\mu\\nu}\n\\bigg]_{x = r(\\tau)} + \\ldots \\,.\n\\]\nFinally, separating the even and odd values of $n$ into two distinct sums\n\\[\nS_\\text{int} = m\\!\\int\\!\\d\\tau\n\\bigg[ & \\sum_{n=1}^\\infty \\frac{(-1)^n}{(2n)!} (a \\cdot \\nabla)^{2n-2}\nR_{\\alpha\\beta\\,\\mu\\nu} u^\\alpha a^\\beta u^\\mu a^\\nu \\\\ &\n-  \\sum_{n=1}^\\infty \\frac{(-1)^n}{(2n+1)!}\n(a \\cdot \\nabla)^{2n-1} R^*_{\\alpha\\beta\\,\\mu\\nu} u^\\alpha a^\\beta u^\\mu a^\\nu\n\\bigg]_{x = r(\\tau)} + \\ldots \\,,\\label{eq:LSinteractions}\n\\]\none can verify that this reproduces the leading interactions of a Kerr black hole, as discussed in detail by Levi and Steinhoff~\\cite{Levi:2015msa}. \n\nIt is interesting that the\nworldsheet structure unifies the spin kinetic term with the\nleading interactions of Kerr. The same phenomenon was observed\ndirectly at the level of amplitudes in ref.~\\cite{Chung:2018kqs}.\n\n\n\n\\section{Spinorial equations of motion}\n\\label{sec:spinorEOM}\n\nEquation~\\eqref{eq:rtKerrIntShift} explicitly displays a Newman--Janis shift for the leading interactions of the $\\rootKerr$~solution. Now we take\na first look at the structure of the equations of motion encoding this shift. Since the Newman--Janis shift is chiral, we will find that it is very\nconvenient to describe the dynamics using the method of spinor-helicity, even in a fully classical setting. Our focus here will be to extract\nexpressions for observables from the equations of motion at leading order. Thus we are free to make field redefinitions, dropping total\nderivatives which do not contribute to observables. We will also extend our work to magnetically charged objects, such as spinning dyons\nand the gravitational Kerr--Taub--NUT analogue at the level of equations of motion.\n\nWe may write the leading order action for a $\\rootKerr$~particle with trajectory $r(\\tau)$ and spin $a(\\tau)$ as\n\\[\nS = - \\int \\d \\tau \\left( p \\cdot \\dot r(\\tau) + \\frac12 \\epsilon(p, a, \\Omega) + Q \\Re u^\\mu A_\\mu^+(r + i a) \\right) + \\ldots \\,.\n\\label{eq:rootKerrLeadingAction}\n\\]\nBy varying with respect to the position $r(\\tau)$ it is easy to determine that\n\\[\n\\frac{\\d p^\\mu}{\\d \\tau} &= Q \\Re F^{+\\mu\\nu}(r+ia) u_\\nu + \\ldots\n= \\frac{Q}{m} \\Re F^{+\\mu\\nu}(r+ia) p_\\nu + \\ldots \\,.\n\\label{eq:rtKerrEOMmomentum}\n\\]\nIn the second equality, we replaced the velocity $u = \\dot r$ with the momentum $p\/m$, noting that the difference between the momentum and $mu$ is of \norder $F$. To obtain a similar differential equation for the spin $a^\\mu$, it is helpful to begin by differentiating $a \\cdot p = 0$, finding\n\\[\np_\\mu \\frac{\\d a^\\mu}{\\d \\tau} = p_\\mu \\Re \\frac{Q}{m} F^{+\\mu\\nu}(r+ia) a_\\nu \\,.\n\\]\nBased on this simple result, it is easy to guess that the spin satisfies\n\\[\n\\frac{\\d a^\\mu}{\\d \\tau} = \\frac{Q}{m} \\Re  F^{+\\mu\\nu}(r+ia) a_\\nu + \\ldots \\,,\n\\label{eq:rtKerrEOMspin}\n\\]\nand indeed a more lengthy calculation using the Lagrangian~\\eqref{eq:rootKerrLeadingAction} confirms this guess.\n\nOur expressions~\\eqref{eq:rtKerrEOMmomentum} and~\\eqref{eq:rtKerrEOMspin} for the momentum and spin have the same basic structure,\nand are consistent with the requirements that $p^2$ and $a^2$ are constant while $a \\cdot p = 0$. \nAs discussed in section~\\ref{sec:NJintro}, in the context of scattering amplitudes it has proven to be very convenient to introduce spinor variables describing similar momenta; this logic also applies to spins.\nNotice that there is nothing quantum about using spinor variables for momenta and spin: the momenta of particles in amplitudes need not be\nsmall, and the spin can be arbitrarily large. We are simply taking advantage of the availability of spinorial representations of the Lorentz group.\nA key motivation for introducing spinors in the present context is the chirality structure of eqs.~\\eqref{eq:rtKerrEOMmomentum}\nand~\\eqref{eq:rtKerrEOMspin}, which hint at a more basic description using an intrinsically chiral formalism.\n\nThe little group of a massive momentum is ${SO}(3)$, so to construct the spin vector $a^\\mu$ in terms of spinors we need only\nform a little-group vector representation from little group spinors. The vector representation of ${SO}(3)$ is the symmetric tensor product of two\nspinors, so we will need to symmetrise little group indices. Let $a^{IJ}$ be a constant symmetric two-by-two matrix; then\n\\[\na^\\mu = \\frac12 a^{IJ}\\! \\bra{p_J} \\sigma^\\mu | p_I] \n\\]\nis the spin vector. To understand how these expressions work, it may be helpful to work in a Lorentz frame $p^\\mu = (\\sqrt{p^2}, 0, 0, 0)$. Then\nthe spin is a purely spatial vector, so it is a linear combination of components in the $x$, $y$ and $z$ directions. Thus there is a basis of\nthree possible spins.\nThis is reflected in the three independent components\nof the symmetric two-by-two matrix $a^{IJ}$. The algebra of the spinors\nimmediately guarantees that the spin $a$ and the momentum $p$ are orthogonal.\n\nGiven that we can always reconstruct the momentum and spin from the spinors, all we now need are dynamical equations for the spinors \nthemselves. The leading-order spinorial equations of motion for $\\rootKerr$~are\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= \\frac{Q}{2m} \\maxwell(r(\\tau) + i a(\\tau)) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &=  \\frac{Q}{2m} \\tilde \\maxwell(r(\\tau) - i a(\\tau)) |p_I] \\,.\n\\label{eq:spinorEOMrtKerr}\n\\]\nNotice that the evolution of the spinors is directly determined by the Maxwell spinor of whatever background the particle is moving in. The\nNJ shifts indicated explicitly in equation~\\eqref{eq:spinorEOMrtKerr} are an explicit consequence of the shift~\\eqref{eq:rtKerrIntShift} at the\nlevel of the effective action. It is straightforward to recover the vectorial equations~\\eqref{eq:rtKerrEOMmomentum} \nand~\\eqref{eq:rtKerrEOMspin} from our spinorial equation; for example, for the momentum,\n\\[\n\\frac{dp^\\mu}{d\\tau} &= \\frac{Q}{2m}\\epsilon^{IJ}\\, \\textrm{Re}\\, \\phi(r(\\tau) + ia(\\tau)) \\ket{p_I} |p_J] \\sigma^\\mu\\\\\n&= -\\frac{Q}{2m} \\textrm{Re}\\, F_{\\rho\\sigma}(r(\\tau) + ia(\\tau)) p_\\nu\\, \\textrm{tr}\\left(\\sigma^{\\rho\\sigma} \\sigma^\\mu \\tilde \\sigma^\\nu \\right).\n\\]\nThe trace can be evaluated using standard techniques, which yield the projector of a two-form onto its self-dual part,\n\\begin{equation}\n\\textrm{tr}\\left(\\sigma^\\mu \\tilde \\sigma^\\nu \\sigma^{\\rho\\sigma} \\right) =  \\eta^{\\nu\\rho}\\eta^{\\mu\\sigma} - \\eta^{\\mu\\rho}\\eta^{\\nu\\sigma} + i\\epsilon^{\\nu\\mu\\rho\\sigma}\\,.\\label{eqn:sigmaTrace}\n\\end{equation}\nThe vector algebra now easily leads to~\\eqref{eq:spinorEOMrtKerr}.\n\nTo illustrate the use of spinorial methods, consider scattering two $\\rootKerr$~particles off one another. We will compute both the leading impulse \n$\\Delta p_1$ and the leading angular impulse $\\Delta a_1$ on one of the two particles during the scattering event. The primary goal of this thesis has been to obtain these observables using the methods\nof scattering amplitudes; here, spinorial equations of motion render the computations even simpler. We denote\nthe spinor variables for particle 1 by $\\ket{1, \\tau}$ and $|1, \\tau]$, and \nsimilarly for particle 2; these spinors are explicitly functions of proper time. In a scattering event we denote the initial spinors\nas $\\ket{1} \\equiv \\ket{1, -\\infty}$ (and similarly for $|1]$.) The\nfinal outgoing spinors are then $\\ket{1'} \\equiv \\ket{1, + \\infty}$.\n\nThe impulses on particle 1 are given in terms of a leading order kick of the spinor $\\ket{\\Delta1} \\equiv \\ket{1, +\\infty} - \\ket{1, -\\infty}$ as\\footnote{Notice that we are representing the impulses here as bispinors.}\n\\[\n\\Delta p_1 &= 2 \\epsilon^{IJ} \\Re \\ket{\\Delta 1_{J}} [1_{I}| \\,,\\\\\n\\Delta a_1 &= 2 a_1^{IJ} \\Re \\ket{\\Delta 1_{J}} [1_{I}|  \\,.\n\\label{eq:impulsesFromSpinors}\n\\]\nThus we simply need to compute the kick suffered by the spinor of particle 1 to\ndetermine \\emph{both} impulses, in contrast to other methods available (including using amplitudes.) By direct\nintegration of the spinorial equation~\\eqref{eq:spinorEOMrtKerr} we see that this spinorial kick is\n\\[\n\\ket{\\Delta 1_I} = \\frac{Q_1}{2m_1} \\int_{-\\infty}^\\infty\\! \\d \\tau \\,\\phi(r_1 + i a_1) \\, \\ket {1_I} \\,.\n\\label{eq:deltaSpinorStep}\n\\]\nAt this level of approximation, we may take the trajectory $r_1$ to be a straight line with constant velocity, and take the spin $a_1$ to \nbe constant, under the integral. Notice that we evaluate the Maxwell spinor at the shifted position $r_1 + i a_1$ because of the Newman--Janis\nshift property at the level of interactions.\n\nTo perform the integration we need the Maxwell spinor influencing the motion of particle 1. This is the field of the second of our two particles.\nIt is easy to obtain this field --- indeed, by the standard Newman--Janis shift of the field set up by particle 2, we need only shift the Coulomb\nfield of a point-like charge. The field is\n\\[\n\\phi(x) = 2iQ_2 \\!\\int\\! \\dd^4\\wn k \\, \\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (x+i a_2)}}{\\wn k^2} \\sigma_{\\mu\\nu} \\wn k^\\mu u_2^\\nu \\,.\n\\label{eq:explicitMaxwell}\n\\]\nNote the explicit NJ shift by the spin $a_2$: this is the shift of the background, in contrast to the shift through $a_1$ of \nequation~\\eqref{eq:deltaSpinorStep}. Of course, there is a pleasing symmetry between these shifts. Using the field~\\eqref{eq:explicitMaxwell}\nin our expression~\\eqref{eq:deltaSpinorStep} for the change in the spinors of particle 1, we arrive at an integral expression for the spinor kick,\n\\[\n\\ket{\\Delta 1_I} =  \\frac{i Q_1 Q_2}{2m_1}\\int \\dd^4\\wn k \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} \\wn k^\\mu u_2^\\nu \n\\sigma_{\\mu\\nu} \\ket {1_I} \\,,\n\\]\nwhere $b$ is the impact parameter. \nThis expression contains complete information about both the linear and angular impulses. For example, substituting into \nequation~\\eqref{eq:impulsesFromSpinors} and applying~\\eqref{eqn:sigmaTrace} we find that the angular impulse is\n\\begin{multline}\n\\Delta a_1^\\mu = \\frac{Q_1 Q_2}{m_1} \\Re\\! \\int\\! \\dd^4 \\wn k \\, \\del(\\wn k \\cdot u_1) \\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2}\n\\\\\n\\times\n\\left(\nia_1 \\cdot u_2 \\, \\wn k^\\mu - i\\wn k \\cdot a_1 \\, u_2^\\mu + \\epsilon^\\mu(\\wn k, a_1, u_2) \n\\right) \\,.\n\\end{multline}\n\nSpinorial equations of motion are also available for the leading order interactions of Kerr moving in a gravitational background. They are\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= -\\frac{1}{2} u^\\mu \\omega_\\mu (r + i a) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &= - \\frac{1}{2} u^\\mu \\tilde \\omega_\\mu (r - i a) |p_I] \\,,\n\\label{eq:spinorEOMKerr}\n\\]\nwhere the spin connection is written in terms of spinors:\n\\[\n\\omega_\\mu \\ket{p} = \\omega_\\mu{}^{ab} \\sigma_{ab} \\ket{p} \\,.\n\\]\n\n\nUsing these spinorial equations of motion and a brief calculation in exact analogy with our $\\rootKerr$~discussion above, it is remarkably straightforward\nto recover the 1PM linear and angular impulse due to Kerr\/Kerr scattering in equation~\\eqref{eqn:KerrDeflections}~\\cite{Vines:2017hyw}. \n\nIn fact we can go further and consider the generalisation of Kerr with NUT charge, corresponding in the stationary case to the exact Kerr--Taub--NUT\nsolution. It is known that NUT charge can be introduced by performing the gravitational analogue of electric\/magnetic\nduality \\cite{Talbot:1969bpa}. Working at linearised level, this deforms the linearised spinorial equation of motion to\n\\[\n\\frac{\\d}{\\d \\tau} \\ket{p_I} &= -  \\frac{e^{-i \\theta}}{2} u^\\mu \\omega_\\mu (r + i a) \\ket{p_I}\\,, \\\\\n\\frac{\\d}{\\d \\tau} |p_I] &= -  \\frac{e^{+i \\theta}}{2} u^\\mu \\tilde \\omega_\\mu (r - i a) |p_I] \\,,\n\\]\nwhere $\\theta$ is a magnetic angle. The particle described by these equations has mass $m \\cos \\theta$ and NUT parameter $m \\sin \\theta$.\nUsing these equations, and defining the rapidity $w$ by $\\cosh w = u_1 \\cdot u_2$, we find that the leading order impulse in a Kerr--Taub--NUT\/Kerr--Taub--NUT scattering event is given by\n\\begin{multline}\n\\Delta p_1^\\mu = - 4\\pi G m_1 m_2 \\Re\\! \\int\\! \\dd^4\\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} e^{i (\\theta_2 -\\theta_1)}\n\\\\\n\\times\n\\left(\ni \\cosh 2w \\,\\wn k^\\mu + 2 \\cosh w \\, \\epsilon^\\mu(\\wn k, u_1, u_2)\n\\right) \\,,\n\\end{multline}\nin agreement with a previous computation performed using scattering amplitudes~\\cite{Emond:2020lwi}. \nIt is also straightforward to compute\nthe angular impulse using these methods; we find that\n\\[\n\\Delta a_1^\\mu = 4\\pi &G m_2 \\Re\\! \\int\\! \\dd^4 \\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i\\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} e^{i (\\theta_2 -\\theta_1)}\n\\\\\n\\times&\n\\Big(\n\\cosh 2w \\, \\epsilon^\\mu(\\wn k, u_1, a_1) - 2 \\cosh w \\, u_1^\\mu \\epsilon(\\wn k, u_1, u_2, a_1) \\\\& \\qquad - 2i a_1\\cdot u_2 \\cosh w\\, \\wn k^\\mu + i\\wn k\\cdot a_1 \\left(2 \\cosh w\\, u_2^\\mu - u_1^\\mu\\right)  \n\\Big) \\,.\n\\]\nThese results can be integrated by means of the generalisation of~\\eqref{eqn:I1result},\n\\begin{equation*}\n\\int\\! \\dd^4\\wn k  \\, \\del(\\wn k \\cdot u_1)\\del(\\wn k \\cdot u_2) \\frac{e^{-i \\wn k \\cdot (b+i a_1+i a_2)}}{\\wn k^2} \\wn k^\\mu = \\frac{i}{2\\pi \\sqrt{\\gamma^2 - 1}} \\frac{{b}^\\mu + i\\Pi^\\mu{ }_\\nu(a_1 + a_2)^\\nu}{[b + i\\Pi(a_1 + a_2)]^2}\\,,\n\\end{equation*}\nwhere the projector $\\Pi^\\mu{ }_\\nu$ was defined in equation~\\eqref{eqn:projector}. A little algebra is enough to see that, when the magnetic angles are zero, our results precisely match the 1PM impulses for spinning black holes in equation~\\eqref{eqn:KerrDeflections} --- the observables for Kerr--Taub--NUT scattering are simply phase rotations of their Kerr counterparts.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe Newman--Janis shift is often dismissed as a trick, without any underlying geometric justification. The central theme of this chapter is that we\nshould rather view Newman and Janis's work as an important insight. The Kerr solution is simpler than it first seems, and correspondingly\nthe leading interactions of Kerr are simpler than they might otherwise be. It seems appropriate to place the NJ shift at the heart of our\nformalism for describing the dynamics of Kerr black holes, thereby taking maximum advantage of this leading order simplicity.\n\nOur spinorial approach to the classical dynamics of Kerr (and its electromagnetic single-copy, $\\rootKerr\\,$) makes it trivial to include the\nspin (to all orders in $a$) in scattering processes. Computing the evolution of the spinors, rather than the momenta and spin separately,\nreduces the workload in performing these computations, and is even more efficient in some examples than computing with the help of\nscattering amplitudes. However, we only developed these equations at leading order. At higher orders, spinor equations of motion will certainly exist and be worthy of study.\n\nWe found that the effective action for Kerr has the surprising property that it can be formulated in terms of a two-dimensional worldsheet\nintegral instead of the usual one-dimensional worldline effective theory. This remarkable fact provides some kind of geometric basis for\nthe Newman--Janis shift, where it emerges using Stokes's theorem. Our worldsheet actions contain terms integrated over some boundaries,\nand other terms integrated over the ``bulk'' two-dimensional worldsheet. This structure is also familiar from brane world scenarios, but is\nobviously surprising in the context of Kerr black holes. In Minkowski space, this worldsheet is embedded in a complexification of spacetime,\nin a manner somewhat reminiscent of other work on complexified worldlines; see ref.~\\cite{Adamo:2009ru}, for example. However, \nour worldsheet seems to be a bit of a different beast: it is not a complex line, but rather a strip with two boundaries.\n\nThe worldsheet emerged in our work, built up from the physical boundary worldline and geodesics in the direction of spin. This construction\nis very different from the sigma models familiar from string theory. The dynamical variables in our action are the ``near'' worldsheet coordinates,\nthe spin, and body-fixed frame. But perhaps these dynamical variables emerge from a geometric description more reminiscent of the picture for strings.\n\nIt is remains to be seen whether the worldsheet structure persists when higher-order operators, involving two or more powers of the Riemann\ncurvature (or electromagnetic field strength), are included. But we can certainly hope that the surprising simplicity of Kerr persists to higher\norders --- the computation of observables, at finite spin, from both loop-level amplitudes \\cite{Guevara:2018wpp,Chung:2018kqs,Bern:2020buy,Aoude:2020ygw,Kosmopoulos:2021zoq} and post--Minkowskian EFT methods \\cite{Liu:2021zxr} certainly indicates that further progress can be made. The latter reference is particularly inspiring in this regard, due to the fully covariant nature of the results. Meanwhile, precision calculations reliant on the effective action in \\eqref{eq:fullSimpleAction} require including higher-dimension operators in the action  \\cite{Levi:2020kvb,Levi:2020uwu,Levi:2020lfn}. It would be particularly interesting to investigate the symmetry structure of the Kerr worldsheet, with an eye towards placing symmetry constraints on the tower of possible higher-dimension operators.\n\n\n\\chapter{Conclusions}\n\\label{chap:conclusions}\n\nOn-shell scattering amplitudes are the quantum backbone of particle physics. Black holes are archetypes of classical general relativity. The central aim of this thesis has been to show that despite the apparent dissimilarities, the two are intimately connected. Furthermore, we have advocated that utilising purely on-shell data offers a powerful window into black hole physics.\n\nIn the first part of the thesis we gathered the technology needed to compute on-shell observables relevant for gravitational wave astronomy, basing our formalism in the humble study of explicit single particle wavepackets. We introduced our first scattering observable in chapter~\\ref{chap:impulse}: the impulse, or total change in momentum of a scattering point-particle. We constructed explicit expressions for the impulse, valid in any quantum field theory, in terms of on-shell amplitudes. With our explicit wavepackets we were able to rigorously determine the classical regime of this observable in section~\\ref{sec:classicalLimit}, encountering the Goldilocks inequalities, which we showed held the key to calculating the classical limit of scattering amplitudes.\n\nArmed with a full understanding of the classical limit, and in particular a practical knowledge of crucial $\\hbar$ factors, we were able to provide expressions for the classical limit of the impulse observable. Explicit examples at LO and NLO in section~\\ref{sec:examples} led to expressions which agree with classical worldline perturbation theory --- a highly non-trivial constraint. Beyond these orders the impulse does not fully capture the dynamics of a spinless particle, due to the emission of radiation. In chapter~\\ref{chap:radiation} we therefore included the total radiated momentum in our formalism, eventually finding that for inelastic scattering, amplitudes, via the radiation kernel~\\ref{eqn:defOfR}, specify a point-particle's classical worldline current (as first noted in ref.~\\cite{Luna:2017dtq}). We showed the efficacy of the double copy for computing this current, a fact that is crucial for gravitational wave astronomy. We also showed that basing our treatment of radiation and the impulse in quantum mechanics has the enormous advantage of ameliorating the conceptual difficulties inherent in treatments of radiating point-particles in classical field theory, explicitly recovering the LO predictions of the Abraham--Lorentz--Dirac force in electrodynamics from amplitudes.\n\nAlthough we only considered unbound (scattering) events, it is in fact possible to determine the physics of bound states from our observables. This can be done concretely using effective theories \\cite{Cheung:2018wkq}. It should also be possible to connect our observables more directly to bound states using analytic continuation, in a manner similar to the work of K\\\"{a}lin and Porto \\cite{Kalin:2019rwq,Kalin:2019inp}. As in any application of traditional scattering amplitudes, however, time-dependent phenomena are not readily accessible. This reflects the fact that amplitudes are the matrix elements of a time evolution operator from the far past to the far future. For a direct application of our methods to the time-dependent gravitational waveform, we must overcome this limitation. One possible path of future investigation for upgrading the formalism presented here would start from the fact that the observables we have discussed are essentially expectation values. They are therefore most naturally discussed using the time-dependent in-in formalism, which has a well-known Schwinger--Keldysh diagrammatic formulation. Whether the double copy applies in this context remains to be explored.\n\nThe double copy offers an avenue, rooted in scattering amplitudes, to simpler calculations in gravity. Amplitudes methods are also especially potent when applied to the physics of spin, motivating the focus of the second part of the thesis. For a spinning body the impulse and radiation are not enough to uniquely specify the dynamics of a scattering event for two bodies with unaligned spins. For a black hole, uniquely constrained by the no-hair theorem, the full data is gained from knowledge of the angular impulse, or change in the spin vector. Rooting our intuition once again in the physics of single particle states, we identified the Pauli--Lubanski operator as the crucial quantum actor corresponding to the classical spin vector. We used this operator to obtain explicit expressions for the LO angular impulse in terms of amplitudes, finding notably more complex expressions than for the momentum impulse. We did not consider the higher order corrections, but they are very similar to those of the colour impulse in ref.~\\cite{delaCruz:2020bbn} --- these would be necessary to access the results of \\cite{Liu:2021zxr}, for example.\n\nBy carefully studying the classical limit of amplitudes for finite spin, and computing the momentum and angular impulses, we were able to precisely reproduce the leading spin, 1PM scattering data for Kerr black holes. The complete, all-spin classical expressions,~\\eqref{eqn:KerrDeflections}, can be calculated from amplitudes by applying our methods to massive spinor helicity representations in the large spin limit \\cite{Guevara:2019fsj}. These results are one artefact of a beautiful relationship between black holes and amplitudes: minimally coupled graviton amplitudes are the on-shell avatar of the no-hair theorem \\cite{Vaidya:2014kza,Chung:2018kqs}. This provocative idea can provide non-geometric insights into intriguing results in general relativity, such as the Newman--Janis complex map between the Schwarzschild and Kerr solutions, which is naturally explained by the exponentiation of the minimally coupled 3--point amplitude in the large spin limit \\cite{Arkani-Hamed:2019ymq}.\n\nWe extended this on-shell insight into black hole physics in chapter~\\ref{chap:worldsheet}, where we showed that the Newman--Janis shift could be interpreted in terms of a worldsheet effective action. This holds both in gravity, and for the single-copy $\\rootKerr$~solution in electrodynamics. Moreover, at the level of equations of motion we showed that the NJ shift holds also for the leading interactions of the Kerr black hole. These leading interactions were conveniently described using chiral classical equations of motion with the help of the spinor-helicity method familiar from scattering amplitudes. These spinor equations of motion are extremely powerful tools --- they offered remarkably efficient derivations of the on-shell observables calculated from amplitudes earlier in the thesis, encapsulating the full on-shell data for non-aligned black hole scattering in the kick of the holomorphic spinor. It was also a trivial matter to extend the scope of this technology to the magnetically charged Kerr--Taub--NUT black hole, facilitating the first calculation of the angular impulse for this solution. It would be interesting to see if our methods can be generalised to include the full family of parameters of the famous Plebanski--Demianski family \\cite{Debever:1971,Plebanski:1976gy}.\n\nThe full geometry of our worldsheet remains to be explored, in particular its applicability when higher-order curvature terms are accounted for. Such corrections are crucial for the extending the power of the chiral spinor equations. A major motivation for tackling this problem comes from an obstacle towards progress in amplitudes computations of importance for gravitational wave astronomy: a well defined, tree-level expression for the four-point Compton amplitude, valid for any generic spin $s$, is not known. Application of BCFW recursion with arbitrary spin representations leads to spurious poles \\cite{Arkani-Hamed:2017jhn,Aoude:2020onz}; these poles can only be removed by hand ambiguously \\cite{Chung:2018kqs}. As we have seen in chapter~\\ref{chap:impulse}, triangle diagrams play a crucial role in the computation of either the NLO impulse or potential. The Compton amplitude is a key component to the computation of these topologies using generalised unitarity, and thus progress in computing precision post--Minkowksian corrections for particles with spin is impeded by a crucial tree-level input into the calculation.\n\nThe all-spin Compton amplitude should have a well defined, unambiguous classical limit. Physically the amplitude corresponds to absorption and re-emission of a messenger boson, a problem which is tractable in classical field theory. Our worldsheet construction shows that the Newman--Janis shift holds for the leading interactions of Kerr black holes, and thus the results of chapter~\\ref{chap:worldsheet} should offer a convenient way to approach this problem, provided that higher order contributions can be included. We hope that, in this manner, our work will be only the beginning of a programme to exploit the Newman--Janis structure of Kerr black holes to simplify their dynamics.\n\\chapter{Lay summary}\n\nBlack holes are some of the most beguiling objects in theoretical physics. Their interiors, shrouded by the cloak of the inescapable event horizon, will always remain one of the\nleast understood environments in modern science. Yet, from the perspective of a (very) safely removed external observer, black holes are remarkably simple objects. So simple in fact that any classical black hole in our leading theory of gravity, general relativity, is exactly specified by just three parameters: its mass, charge and angular momentum. This sparsity of information leads physicists to say that black holes ``\\textit{have no hair}'', in contrast to ``hairy'' bodies, such as stars and PhD students, which unfortunately cannot be exactly identified by three numbers.\n\nBlack holes' ``no hair'' constraints are maybe more reminiscent of the description of fundamental particles, which are catalogued by their mass, charge and quantum spin, than extended bodies with complicated physiognomies. Yet astronomers readily observe influences on the propagation of light, the motion of stars and the structure of galaxies that are precisely predicted by black hole solutions. Moreover, general relativity predicts that black holes can interact and merge with each other. Such interactions take the form of a long courtship, performing a well separated inspiral that steadily, but inescapably, leads to a final unifying merger under immense gravitational forces. These interactions do not happen in isolation from the rest of the universe. They are violent, explosive events which release enormous amounts of energy, exclusively in the form of gravitational waves.\n\nThe frequent experimental detection of these signals from distant black hole mergers marks the beginning of a new era of observational astronomy. Yet despite the enormous total energy released, gravitational wave signals observable on Earth are very, very faint. In addition to advanced instrumentation, extracting these clean whispers from the background hubbub of our messy world requires precise predictions from general relativity, so as to filter out meaningful signals from noise. The difficulty of performing these calculations means that they are rapidly approaching the point of becoming the largest source of errors in experiments; promoting gravitational wave astronomy to a precision science now depends on improving theoretical predictions.\n\nThis has motivated the development of novel ways of looking at gravitational dynamics, and one such route has been to adopt methods from particle physics. Modern particle collider programmes have necessitated the development of an enormous host of theoretical techniques, targeted towards experimental predictions, which have enabled precision measurements to be extracted from swathes of messy data. These techniques focus on calculating \\textit{scattering amplitudes}: probabilities for how particles will interact quantum mechanically. These techniques can be applied to gravity provided that the interactions are weak, and astonishingly theorists have found that they can facilitate precise predictions for the initial inspiral phase of black hole coalescence. This is a relatively new line of research, and one which is growing rapidly.\n\nThis thesis is a small part of this endeavour. It is focussed on obtaining \\textit{on-shell} observables, relevant for black hole physics, from quantum gravity. An on-shell quantity is one that is physical and directly measurable by an experimentalist, such as the change in a body's momentum. Scattering amplitudes are also on-shell, in the sense that they underpin measurable probabilities. We will be interested in direct routes from amplitudes to observables, which seems like an obvious path to choose. However, many of the techniques in a theorist's toolbox instead involve going via an unmeasurable, mathematical midstep which contains all physical data needed to make measurable predictions. One example is the gravitational potential: one cannot measure the potential directly, but it contains everything needed to make concrete predictions of classical dynamics.\n\nMuch of the power of modern scattering amplitudes comes from their reliance on physical, on-shell data. This thesis will show that a direct map to observables opens new insights into the physics of classical black holes. In particular, we can make the apparent similarity between fundamental particles and black holes concrete: certain scattering amplitudes are the on-shell incarnation of the special ``no hair'' constraint for black holes. This relationship is beautiful and provocative. It opens new practical treatments of spinning black holes in particular, and to this end we will use unique complex number properties of these solutions, combined with quantum mechanics, to obtain powerful classical descriptions of how spinning black holes interact with one another.\n\n\\cleardoublepage\n\\phantomsection\n\\chapter{Abstract}\n\n\\noindent On-shell methods are a key component of the modern amplitudes programme. By utilising the power of generalised unitarity cuts, and focusing on gauge invariant quantities, enormous progress has been made in the calculation of amplitudes required for theoretical input into experiments such as the LHC. Recently, a new experimental context has emerged in which scattering amplitudes can be of great utility: gravitational wave astronomy. Indeed, developing new theoretical techniques for tackling the two-body problem in general relativity is essential for future precision measurements. Scattering amplitudes have already contributed new state of the art calculations of post--Minkowskian (PM) corrections to the classical gravitational potential.\n\nThe gravitational potential is an unphysical, gauge dependent quantity. This thesis seeks to apply the advances of modern amplitudes to classical gravitational physics by constructing physical, on-shell observables applicable to black hole scattering, but valid in any quantum field theory. We will derive formulae for the impulse (change in momentum), total radiated momentum, and angular impulse (change in spin vector) from basic principles, directly in terms of scattering amplitudes.\n\nBy undertaking a careful analysis of the classical region of these observables, we derive from explicit wavepackets how to take the classical limit of the associated amplitudes. These methods are then applied to examples in both QED and QCD, through which we obtain new theoretical results; however, the main focus is on black hole physics. We exploit the double copy relationship between gravity and gauge theory to calculate amplitudes in perturbative quantum gravity, from whose classical limits we derive results in the PM approximation of general relativity.\n\nApplying amplitudes to black hole physics offers more than computational power: in this thesis we will show that the observables we have constructed provide particularly clear evidence that massive, spinning particles are the on-shell avatar of the no-hair theorem. Building on these results, we will furthermore show that the classically obscure Newman--Janis shift property of the exact Kerr solution can be interpreted in terms of a worldsheet effective action. At the level of equations of motion, we show that the Newman--Janis shift holds also for the leading interactions of the Kerr black hole. These leading interactions will be conveniently described using chiral classical equations of motion with the help of the spinor helicity method familiar from scattering amplitudes, providing a powerful and purely classical method for computing on-shell black hole observables.\n\\chapter{Acknowledgements}\n\n\\noindent\n\n\\normalsize\n\nMy foremost thanks go to my supervisor, Donal O'Connell. I will be forever grateful to you for helping me learn to navigate the dense jungle of theoretical physics; your capacity for novel ideas has been truly inspirational, and working with you has been a fantastic experience. Thank you for placing your trust in me as a collaborator these last few years, and also for your sage mountaineering advice: don't fall off.\n\nThank you also to my fellow collaborators: Leonardo de la Cruz, Alfredo Guevara, David Kosower, Dhritiman Nandan, Alex Ochirov, Alasdair Ross, Matteo Sergola, and Justin Vines. I am very proud of the science we have done together. I am especially grateful to Justin, for his hospitality at the Albert--Einstein institute, insights into the world of general relativity, and all-round kindness. My thanks also to Andr\\'{e}s Luna for his friendship and warmth; Chris White, for his enthusiastic introduction to the eikonal; and Yu-Tin Huang, Paolo Pichini and Oliver Schlotterer for the successful joint conquest of the great Bavarian mountain Kaiserschmarrn. I'm still full.\n\nI am extremely grateful for the additional hospitality of Daniel Jenkins at Regensburg; Thibault Damour at the IHES; and the organisers of the MIAPP workshop ``Precision Gravity''. These visits, coupled with the QCD meets Gravity conferences, have been highlights of my PhD. Thank you to the wider community for so many exciting scientific discussions in a welcoming environment, and to David, Alessandra Buonanno and Henrik Johansson for offering me the opportunity to continue my tenure; I am sorry that I have not done so.\n\nThe PPT group at Edinburgh has been a great place to work, and one I have sorely missed following the virtual transition. Thank you to all my fellow PhD students and TA's for your support and knowledge, especially Christian Br\\o{}nnum--Hansen, Tomasso Giani, Michael Marshall, Calum Milloy, Izzy Nicholson, Rosalyn Pearson and Saad Nabeebaccus. Teaching in particular has been a great joy throughout; thank you to Lucile Cangemi and Maria Derda for working with me on your MPhys projects, and to JC Denis for encouraging my outreach activities.\n\nResearch is not always a smooth path, but one which is certainly made easier with the support of friends and family: thank you especially to Rosemary Green for her continual love and encouragement. Thank you to Mike Daniels, Laura Glover, Kirsty McIntyre and Sophie Walker for (literally) accommodating my poor decision making and keeping me going, even if you didn't know it! Meanwhile colloquial debates at dinner club tables were more taxing than any calculation, and highlights of my week; I miss them dearly, thank you all. And thank you to everyone I have tied onto a rope with for unwittingly heeding Donal's advice while sharing many fun adventures at the same time. Here's to many more.\n\nFinally, I could not have completed this thesis without the true love of my life, Ruth Green. Thank you for encouraging me to pursue my work in Edinburgh in the first place, for putting up with my geographical superposition from our nest, and for always being there for me.\n\nI acknowledge financial support under STFC studentship ST\/R504737\/1.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nIn any economy the distribution of wealth $P(m)$ among individuals follows a pattern for large values \nof wealth $m$, to be specific, it decays \nas $P(m) \\sim  m^{-(1+\\nu)}$ for large $m$ where $\\nu$ is called the Pareto exponent \\cite{Pareto}. Pareto exponent usually varies between $1$\nand $3$ \\cite{Mandelbrot:1960, EIWD, EWD05, ESTP, SCCC, Yakovenko:RMP, datapap,DraguYakov0}. \nA number of models have been proposed\nto reproduce the observed features of an economy \\cite{marjitIspolatov, Dragulescu:2000,\nChakraborti:2000,Chatterjee:rev,Chatterjee:2010}. One important objective of several such econophysical models \nis to reproduce the Pareto tail \nin the wealth\/income distribution. Some of the models were inspired by the kinetic theory of \ngases which derives the average macroscopic behaviour from the microscopic interactions among molecules. \nIn these models traders\/agents are treated as molecules of gas. A typical trading process between two such \ntraders\/agents maintaining local conservation of wealth \ncan be compared to an interaction between two gas molecules maintaining local energy conservation in gas. \nThese models follow a microcanonical description, i.e., the total wealth \nis a conserved quantity. \nSeveral such models are studied \\cite{EIWD, EWD05, Toscani2018} where debt is allowed for a trader\/agent. However, in our case  \nwe consider no agent can end up with a negative wealth, i.e., debt is not allowed in a trading. \n\nThus, if there are two agents $i$ and $j$ who before taking part in the trading had wealth \n$m_i(t)$ and $m_j(t)$ respectively at time $t$, will have wealths according to the following relations \nat the next time step $t+1$:\\\\\n\\begin{align}\\label{mimj}\n\\begin{split}\nm_i(t+1) = m_i(t) + \\Delta m; \\\\\nm_j(t+1) = m_j(t) - \\Delta m.\\\\\n\\end{split}\n\\end{align}\n\nThere are several other models of the wealth distribution which do not consider the kinetic theory concept.  \nIn \\cite{Bouchaud-Mezard}, a very simple model of economy was discussed, where the time evolution\nwas described by an equation involving exchange between individuals and\nrandom speculative trading in such a way that under an arbitrary change of monetary units \nthe fundamental symmetry of the economy\nis obeyed.\nA mean-field limit of this\nequation was investigated there and the distribution of wealth came out to be of the Pareto type.\nAnother model is the Lotka-Volterra \nmodel which is again a kind of mean field model where wealth of an agent at a particular time \ndepends on her\/his wealth in the previous step as well as the average \nwealth of all agents \\cite{Solomon,Malcai}. Apart from these, there are other \nmodels which depend on stochastic processes \\cite{Garl,Sornette}. The main problem in the last two type \nmodels is that here wealth \nexchange between agents is not allowed \nand therefore cannot be realized as a real trading process. Although in \\cite{Bouchaud-Mezard}, wealth exchange is \nconsidered, according to the authors, it is again not a fully realistic one, as mean field concept is used.\nIn some models, \ninstead of considering binary collision-like trading, just as in case of a rarefied classical gas, simultaneous\nmultiple interactions are taken into account to model a socio-economic phenomena \nin a multi-agent system \\cite{Toscani}. \\\\\n\nIn the gas-like models, the wealth exchange between agents follow the same rule as energy exchange between two gas \nmolecules in kinetic theory; that is why they are called \\textit{kinetic wealth exchange models}. \nBachelier in his $1900$ PhD. thesis developed a 'theory  of speculation' \\cite{Bachelier1900}, \nwhere he suggested a practical connection between stochastic theory \nand financial analysis.\nThe idea that velocity distribution for gas molecules and income distribution for agents \ncan be compared was first addressed in \\cite{SahaSrivastava}, \nalthough no specific reason behind this was addressed. The \nfirst simplest conservative model of this kind was proposed by  \nDragulescu and Yakovenko (DY model) \\cite{Dragulescu:2000}. In that model, $N$ agents \nrandomly exchange wealth pairwise keeping the total wealth $M$ constant. \nIt is shown that the steady-state ($t \\rightarrow \\infty$) wealth there follows a Boltzmann-Gibbs distribution:\n$P(m)=(1\/T)\\exp(-m\/T)$; $T=M\/N$ ~\\cite{Chatterjee:rev}.\\\\\n\nA modification to this model considering the fact that agents save a definite fraction of their wealth $\\lambda$ before \ntaking part in any trading, termed as \\textit{saving propensity}, was addressed first by Chakraborti \nand Chakrabarti~\\cite{Chakraborti:2000} \n(CC model).    \nThis results in a wealth \ndistribution close to Gamma distributions~\\cite{Patriarca:2004,Repetowicz:2005} and is seen\nto fit well to empirical data for low and middle wealth regime of an economy \\cite{datapap}.\nLater, a  model was proposed by Chatterjee et. al. \\cite{Chatterjee:2004} \n(CCM model) where distributed saving propensities were assumed for individuals. \nThe importance of the model is that it led to a wealth distribution \nwith a Pareto-tail. Apart from wealth distribution, people often study network like features in these models\n\\cite{tummi2, Gabaix, manna1,goswami1}, a few of which address preferential interaction between agents. In \\cite{goswami1}\nit was considered that two agents will interact with more probability if their wealths are ``close'' or if\nthey have interacted before.\\\\\n\nIn a real economy, however, this preferential interaction often\ndepend on some other factors. Restriction in interaction may arise in some situations as we have recently \nseen in the Pandemic situation. This type of restricted interaction is studied in \\cite{Toscani2}. \nAlso during the economic crisis in Argentina during $2000-2009$ another restricted interaction was studied in \\cite{Ferrero}. \nThere may be restriction in interaction for other reasons too. \nIt is known that \na poverty line exists in any economy \\cite{gov_report, brady}. In various cases, the poverty line is estimated \nnear $40\\%-60\\%$ of the median of income\\cite{brady}. People below the poverty line often get\nsubsidy from the Government. Also, it is a general notion that a person feels insecure if her\/his \nwealth falls below a specific level, may be termed as a \\textit{threshold line}. \nIn this work, a \\textit{threshold line} is introduced in an otherwise CCM like model, which is below \nthe defined poverty line in \nan economy. The wealths of $N$ agents \nare chosen from the uniform distribution and the total wealth is taken as $M$. \nThe agents whose wealth are assigned below the threshold line are Below Threshold Line (BTL) agents. The subsidy \nis given to the BTL agents in such a way that it can just promote \nthe agents above threshold line. \nOnce an agent is marked a BTL one, he\/she remains eligible for subsidy always.  \nHowever, those who are \nabove the threshold line at the beginning are not getting any subsidy \neven if their wealth fall below  the threshold \nline after a certain number of interactions. \nAlso an interaction will not occur at all if both the interacting agents are having wealth below the threshold line\nbecause of human psychology of insecurity. \\\\\n\nIn some earlier works to study the dynamics of the transactions, a walk was conceived for the\nagents in an abstract $1$-D Gain-Loss space (GLS) \\cite{chattsen, goswami2}. \nThe corresponding walk was compared to a biased random walk. \nIn this work, we compare the walk of a tagged agent $k$ to a lazy random walk for different values of \nsaving propensity $\\lambda_k$. The difference from earlier work is that, here a \ntagged agent is a BTL one and except from moving Right\/Left, the walker \nmay stay put to its position in the GLS if the interaction does not occur at all. It is seen that average distance \ntravelled in the GLS, i.e.,\n$\\langle x \\rangle = 0$ for some $\\lambda_k=\\lambda_{k}^*$. The value of $\\lambda_{k}^*$ is slightly different from what \nwe found in \\cite{goswami2}. The wealth distribution and several other features \nof the lazy walker are studied in this context. The objective of this study is to check whether there is any difference \nin the agent's upliftment in the wealth space\nif the subsidy is given repeatedly to a BTL agent or only once.\\\\\n\n\\section{Model Description}\nWe consider CCM model with $N=256$ agents. The total money $M$ is distributed randomly among the agents. \nThe key feature of CCM model is that here the saving propensities of agents are chosen from a uniform distribution. \nThe wealth exchange between two traders $i$ and $j$ can be represented as:\\\\\n\\begin{align}\\label{mimj_CCM}\n\\begin{split}\nm_i(t+1)=\\lambda_i m_i(t) + \\epsilon_{ij} \\left[(1-\\lambda_i)m_i(t) + (1-\\lambda_j)m_j(t)\\right],\\\\\nm_j(t+1)=\\lambda_j m_j(t) + (1-\\epsilon_{ij}) \\left[(1-\\lambda_i)m_i(t) + (1-\\lambda_j)m_j(t)\\right];\n\\end{split}\n\\end{align}\nHere $\\lambda_i, \\lambda_j$ are the saving propensities of agents $i,j$ respectively and $\\epsilon_{ij}$ is a random \nfraction related to stochastic nature of a trading process.\nIn addition to this regular interaction, a threshold line $m_{L}$ is proposed here. \nWe assumed the poverty line near $40\\%$ of the average wealth of the economy as indicated in \\cite{brady} \nand the threshold line is chosen below that. \nAt the time of wealth assignment if \nan agent is found to be below the line she\/he will be marked as BTL and a \nsubsidy is assigned. \nAs the wealths of agents are assigned from a uniform distribution the median is same as the average wealth.\nThis means a certain fraction of the people will get subsidy. \nHowever, during course of interaction, an agent who is not a BTL one, falls below the threshold \nline she\/he is not eligible for subsidy. Two types of models are studied here as follows:\n\n\\begin{itemize}\n \\item Model A : For this, the BTL agents are stamped as ``BTL'' at the time of wealth assignment and \n the subsidy equal to $m_L$ is given to the BTL agents at the beginning of each configuration. \n \\item Model B : In this, again, some are stamped ``BTL'' at the beginning but \nthe subsidy equal to $m_L$ is given to them at each Monte Carlo (MC) step,\nwhere one MC step consists of $\\frac{N(N-1)}{2}$ interactions. This means if a BTL agent goes above the threshold line \nafter one or few MC steps, she\/he is still eligible for subsidy (exactly as in any caste based system).\n\\end{itemize}\n \nIn both the cases, the subsidy given promotes the BTL agent above the threshold line $m_L$, if the agent is below that. \nAt every interaction, the wealths of the interacting agents are checked. An \ninteraction is prohibited only if \nboth of the interacting agents fall below the threshold line. In subsequent interactions, however, \nthere is always a chance that such an agent is promoted above the line.\nThe stationary state is obtained after a typical relaxation time and the distribution of wealth and the walk \nin the GLS are studied. It is to be noted that the subsidy here is given from the tax payed by the \npeople above the threshold line. In this way, the economy is \nclosed and total wealth remains conserved. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[trim=5cm 0cm 0cm 0cm, width=6cm, angle=-90,clip]{mnydist256_overall.eps}\n\\caption{Overall wealth distribution $P(m)$ for model A (Left) and model B (right) for $m_L=0.1$ (violet), $0.4$ (green). The Pareto exponent \n$\\nu$ is found to be close to $1$.}\n\\label{mnydist_overall}\n\\end{figure}\n\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[trim=5cm 0cm 0cm 0cm, width=6cm, angle=-90]{mnydist256_0.1.eps}\n\\caption{Left: Wealth distribution for model A for $m_L=0.1$ for $\\lambda_k=0.0$(violet),$0.2$(green),$0.4$(blue),$0.6$(yellow),\n$0.9$(red), \nRight: Wealth distribution for model B for same $m_L=0.1$ and same $\\lambda_k$ s. Both \nare for a tagged BTL agent. The plots indicate higher probability close to $m_L$.}\n\\label{mnydist}\n\\end{figure}\n\n\\section{Agent Dynamics}\nAlthough the actual form of wealth distribution \n$P(m)$ depends on the form of saving propensity distribution, there is one thing common for all. \nThe Pareto tail is present whatever be the form of the saving propensity distribution; the only difference is in the value of \nthe exponent. Here, the saving propensities of all the agents are \nchosen from a uniform distribution which is the simplest one. \nAlthough in this work, we are actually interested in the dynamics of a tagged agent, \nthe behaviour of overall distribution of wealth $P(m)$ is also of great importance. \nThe overall wealth distribution shows the Pareto exponent to be roughly $1$ as in case of conventional CCM model \n\\cite{Chatterjee:rev} except from the fact that near $m_L$ there is a sharp change in the profile. This behaviour \ncan be understood by studying individual agent's wealth distribution which will be addressed in the next section.   \nThe wealth distribution $P(m)$ is shown in Fig. \\ref{mnydist_overall} \nfor model A (Left) and model B (right) for two different $m_L$. \n\n\n\nWe perform numerical simulation for a system of $N$ agents\nand look for the dynamics of a tagged agent who is a BTL one with a predefined saving propensity.\nAs we have stated earlier we use two models, namely A and B, for assigning subsidy. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[trim=5cm 0cm 0cm 0cm, width=6 cm, angle=-90,clip]{mnydist256_compare.eps}\n\\caption{Left: Wealth Distribution of the BTL tagged agent for both models A and B for $\\lambda_k=0.0$. \nRight: Same for $\\lambda_k=0.9$. The data are shown \nfor $m_L=0.1$(red), $0.4$ (black). The plots indicate that for model B the wealth distribution \nextends upto a larger $m$ as we are repeating the wealth assignment. The effect is more prominent for larger values \nof $m_L$.}\n\\label{mnydist_compare}\n\\end{figure}\n\n\\subsection{Wealth Distribution}\nThe distribution of wealth $P(m_k|\\lambda_k)$ for different $\\lambda_k$ of the tagged BTL agent are shown in \nFig. \\ref{mnydist} for both the models A and B for $m_{L}=0.1$. For both the cases the nature of wealth \ndistribution is similar to what observed in \\cite{chattsen} and earlier \\cite{Chatterjee:rev, Chatterjee:2004} \nwith the exception that \nas the tagged agent is a BTL one and she\/he is assigned a wealth equal to $m_L$, \nthere is a higher probability near $m_L$ than what observed earlier. For small values of $\\lambda_k$ the agent has a \nsmall amount of wealth compared to the average wealth of the economy and for higher $\\lambda_k$ the wealth possessed by \nthe agent is comparable to the average wealth. It is to be noted that as our agent is a BTL one, the\naverage wealth she\/he possessed is smaller compared to that predicted by the usual CCM model.  \nAs we increase $m_L$ to higher values, higher $m$ is more probable. \nThe models A and B show similar distribution for all $\\lambda_k$ if $m_L$ \nis low but their nature is different for higher $m_L$. It is seen that \nfor higher threshold line the distribution for model B shifts to higher $m$ compared to model A. This can \nbe understood easily. For model B, we are assigning the subsidy to BTL agents at every MC steps without checking \nwhether they are below the threshold line or not. Therefore, higher value of wealth is more probable.  \nThis can also be realized from another aspect of the CCM model. When we set a higher $m_L$, that means \na larger number of agents is below that line compared to the case of a smaller $m_L$. That means we are \nmoving closer to the usual CCM picture where we choose an agent irrespective of the initial wealth possessed by her\/him. \nThese are shown in Fig. \n\\ref{mnydist_compare} for $m_L=0.1$ and $m_L=0.4$ for $\\lambda_k=0.0, 0.9$.\n\n\n\\begin{figure}[!h]\n\\includegraphics[width=9 cm, angle=-90]{slopesAB_0.1.eps}\n\\caption{Variation of $\\langle x \\rangle$ against $t$ for $\\lambda_k = 0.0$ (violet),$0.4$ (yellow) and $0.9$ (blue) for the \nBTL tagged agent walker for both models A (solid lines) and B (dotted lines) for $N = 256$ for $m_L=0.1$. \nInset shows the plot of slope $a(\\lambda_k)$ as a function of $\\lambda_k$ obtained from the slopes \nof $\\langle x \\rangle$ versus $t$ plot for the walker for both the models A and B.}\n\\label{slopeAB}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[trim=5cm 0cm 0cm 0cm,width=6cm, angle=-90,clip]{slopevariation_mL.eps}\n\\caption{Variation of $a(\\lambda_k)$ against $m_L$ for $\\lambda_k = 0.0$ (violet),$0.4$ (yellow) and $0.9$ (blue) for the \nBTL tagged agent walker for both models A (Left) and B (Right) for $N = 256$.}\n\\label{slopevariation_mL}\n\\end{figure}\n\n\n\\section{Walk in the GLS: Comparison to a Lazy Walker}\nTo investigate the dynamics of this model in more detail at the microscopic level,\none may conceive a walk for the agents in the GLS.\nIt is well known that the usual CCM walk can be compared to a biased \nrandom walk (BRW) whose forward bias decreases as we increase $\\lambda$ \nfrom zero. The walk has no bias at a particular $\\lambda_k=0.469$ and then it decreases further and becomes negative \non increasing $\\lambda_k$ \\cite{chattsen,goswami2}. The steps in those \nstudies were taken as Right\/Left according to whether it is Gain\/Loss. \nIn this study we are going to take a similar approach\nfor this CCM walk with a modification. Except for Gain and Loss, there is a third possibility.\nWhen an agent \n gains she\/he moves a step towards Right and if she\/he incurs a\nloss moves a step towards Left. Apart from these two, the third possibility demands \nthat the BTL tagged agent may not interact \nwith another one if any one of them or both possesses a wealth less than $m_L$ and \ntherefore the corresponding walker may stay put at its position in the GLS. \nThe walks are correlated as when two\nagents interact, if any one takes a right step, the other has to move towards left. Also if one is stay put, \nthe other should stay put too.\\\\\n\n\\subsection{Measurement of bias}\n\nFor a lazy walker we know that it can have steps $1,0,-1$. It is obvious therefore that the CCM walk in this study can be\ncompared to a lazy walk. If the agent gains then the corresponding walker moves one step to \nthe right, if loses, the walker moves towards left. If the interaction is missed due to either one of them \nor both falling below the threshold line, the walker remains in its \nposition in the GLS. Just like earlier works, here also \nthe amount of gain\/loss is not important. \\\\\n\nConsider a biased lazy walker with probability of going towards right $p_R$, towards left $p_L$ and probability to stay put \n$p_0$. Obviously $p_R+p_L+p_0=1$. The average distance traveled by such a walker is linear in $t$. Precisely, the average \ndistance traversed can be written as\n$\\langle x \\rangle = a(\\lambda_k)t$. Here $a(\\lambda_k)=[2p_R-(1-p_0)]$ is the slope of the line, \na measure for the amount of drift. \nAs in  any ballistic diffusion, here we have $(\\langle x^2 \\rangle - \\langle x\\rangle^2) \\sim t^2$ for all $\\lambda_k$ except  \nwhen $\\lambda_k \\rightarrow \\lambda_{k}^*$. For $\\lambda_k \\rightarrow \\lambda_{k}^*$, we have observed that \n$(\\langle x^2 \\rangle - \\langle x\\rangle^2) \\sim t$. \nIn Fig. \\ref{slopeAB}, the variation \nof $\\langle x \\rangle$ against $t$ is shown for $\\lambda_k = 0.0,0.4$ and $0.9$ for the \nBTL tagged agent walker for both models A and B. Here we have taken $N = 256$ and $m_L=0.1$. \nInset shows the plot of drift $a(\\lambda_k)$ as a function of $\\lambda_k$. The drifts are obtained from the slopes  \nof $\\langle x \\rangle$ versus $t$ plot. \nAt $\\lambda_{k}^*$, for both the models we have found that $p_R=p_L$ and therefore drift $a(\\lambda_{k}^*)=0$. \nThe precise value of $\\lambda_{k}^*$ is found to be \n$0.471$ for model A and $0.443$ for model B. \nFor any $\\lambda_k$ \nthe slope is more positive for low $m_L$ and less positive for larger $m_L$. However, close to $\\lambda_{k}^*$, the effect is \nalmost negligible.\nThis is shown in Fig. \n\\ref{slopevariation_mL}. This can be interpreted in the following way. The BTL agents' subsidy come \nfrom other agents' taxes, i.e., at the cost of others. As we increase $m_L$, number of BTL agents \nincrease and the subsidy amount coming from the taxes of others increase. Therefore possibility of having an interaction \ndecreases and the tendency to gain also decreases. As our definition for model B demands giving subsidy \nto the agent at every MC step, and that has to come from the others above the threshold line, therefore,  \nthe effect is more pronounced for model B compared to A. \nHowever, the amount of wealth possessed by a BTL agent increases as we \nincrease $m_L$ for any specific $\\lambda_k$. \\\\\n\nWe check the exact number of right, left and zero steps for the modified CCM walk and try to find out how \nthe probabilities $p_R, p_L, p_0$ change with $\\lambda_k$ for a specific $m_L$. As we know that \n$a(\\lambda_k)=2p_R-(1-p_0)$, from Fig. \\ref{slopeAB} it is clear that,\n$p_R, p_L$ and $p_0$ are functions of $\\lambda_k$. \nThe specific probabilities for \ntwo $m_L$ values will be found \nin the Table \\ref{p0pRpL}. For both the models the variation of $p_0(\\lambda_k)$ against $\\lambda_k$ \nare shown in Fig. \\ref{slopeverific} \nfor two different $m_L$. \nIt is seen that the nature of variation matches well with the form \n$p_0(\\lambda_k)\\sim a_0\\exp(-b_0 x^2)$ for both the models A and B, \nwhere $a_0$ and $b_0$ \nare two parameters, for low values of $\\lambda_k$. The plots show discrepancy for high $\\lambda_k$ values \nfrom the predicted behaviour. \nNow if we simulate a lazy \nwalker with those $p_0, p_R$ and $p_L$ values that should show a similar $\\langle x \\rangle$ versus $t$ behaviour.  \nIn the inset of Fig.\n\\ref{slopeverific}, the $\\langle x \\rangle$ versus $t$ graph is compared with the same of a \nlazy walker for a few specific $\\lambda_k$ and therefore $p_0, p_R$ and $p_L$ values values.\\\\\n\n\n\\begin{table}[!h]\n\\begin{center}\n\\caption{$p_0, p_R, p_L$ values for specified $m_L$ and $\\lambda_k$ values for Model A and B \nfor the modified CCM walker of BTL agent.\n\\begin{tabular}{|c|c|c|c|c|c|c|c|\n\\hline\n$m_L$  &  $\\lambda_k$   &   \\multicolumn{3}{c|}{Model A}   & \\multicolumn{3}{c|}{Model B} \\\\\\cline{3-8}\n   &  & $p_0$ &$p_R$ &$p_L$ & $p_0$ &$p_R$ &$p_L$ \\\\\n\\hline\n0.1 & 0.0 & 0.285 & 0.384 & 0.331 & 0.282 & 0.386 & 0.332 \\\\\n    & 0.4 & 0.111 & 0.449 & 0.439 & 0.113 & 0.446 & 0.441 \\\\\n    & 0.9  & 0.0 & 0.482 & 0.518 & 0.0 & 0.480 & 0.520 \\\\\n\\hline\n0.05 & 0.0 & 0.078 & 0.492 & 0.430 & 0.110 & 0.474 & 0.416 \\\\\n     & 0.4 & 0.013 & 0.496 & 0.490 & 0.020 & 0.492 & 0.488 \\\\\n     & 0.9 & 0.0   & 0.481 & 0.519 & 0.0 & 0.480 & 0.520 \\\\\n\\hline\n\\end{tabular}\n\\label{p0pRpL}\n\\end{center}\n\n\\end{table\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=9cm,angle=-90]{slopeverification_p0_lambda.eps}\n\\caption{Variation of $p_0(\\lambda_k)$ against $\\lambda_k$ for models A (Left) and B (Right) for $N = 256$ for $m_L=0.05$ (yellow), $0.1$ (violet). \nBlack dotted line shows the nature of $a_0\\exp(-b_0 x^2)$. \nInset shows the comparison of slope obtained from $\\langle x \\rangle$ versus $t$ data (solid line) \nand using the parameters in lazy walk (dotted line).}\n\\label{slopeverific}\n\\end{figure}\n\n\n\n\\subsection{Distribution of Path Lengths in the GLS}\nWe have seen in the previous section, how the probabilities $p_R, p_L, p_0$ changes with $\\lambda_k$. To have a \ndetailed understanding about the probabilities we are now going to study how the quantities vary with walk length $X$.\nA path length $X$ here signifies the length traversed at a stretch without changing direction. For the Right\/Left \ndirection, this means the agent will gain\/lose for $X$ steps continuously and after that it will either make a loss\/gain \nor stay put. Here we study three such quantities $W_R(X), W_L(X)$ and $W_0(X)$ where the suffix indicates whether \nit is a gain or loss or no interaction. The distribution of path lengths in the GLS is an interesting quantity to study and \nwas studied earlier in \\cite{goswami2} where there were only Right\/Left movements. \nHere, it is clear that:\n\\begin{equation}\nW_i(X)\\propto p_{i}^{X}(1-p_i)^2;  ~~~~~~i=0,R,L. \\\\\n\\end{equation}\n\nWe now wish to extract the values of $p_R$ for a specific $\\lambda_k$ and $m_L$ from the distribution of path lengths \nat a stretch. \nFor high $\\lambda_k$ values, e.g., $\\lambda_k=0.8,0.9$, the probability \n$p_0$ is extremely small, and therefore the walk is very \nsimilar to a biased random walk. In that case, it is easy to extract some $p_R^{eff}(X, \\lambda_k, m_L)$ as then \nwe can approximately write\n\\begin{equation}\n \\frac{W_R(X)}{W_L(X)}=(\\frac{p_R}{1-p_R})^{X-2}.\n \\label{compLR}\n\\end{equation}\n$\\frac{W_R(X)}{W_L(X)}$ is calculated numerically for a $\\lambda_k$ and $m_L$.\nThe $p_R^{eff}$ is shown as a function of $X$ in Fig. \\ref{walklength_pR} for a few $m_L$ values for both model A and B. \nThe $p_R$ value has some variation over $X$ and not constant as predicted in Table \\ref{p0pRpL}. \n\\begin{figure}[!h]\n\\centering \n\\includegraphics[trim=5cm 0cm 0cm 0cm,width=6cm, angle=-90,clip]{walklength_pR.eps}\n\\caption{Variation of $p_R^{eff}$ as a function of $X$ for models A ($\\lambda_k=0.8, m_L=0.1$ (violet) and $\\lambda_k=0.9, m_L=0.05$ (blue) shown) \nand B ($\\lambda_k=0.8, m_L=0.05$ and $\\lambda_k=0.9, m_L=0.1$ shown) for $N = 256$ obtained by distribution of walk lengths at a \nstretch. Black dotted lines show the corresponding $p_R$ values obtained from slopes.  \n}\n\\label{walklength_pR}\n\\end{figure}\n\n\nFor smaller $\\lambda_k$ however we cannot use Eq. \\ref{compLR}.\nAs for a lazy walker there are three parameters involved, we can use the obtained value of $p_0$ to check whether \nwe are getting the same $p_R$ value as in Table \\ref{p0pRpL} from this path length distribution data or not. \nFor this we use the following: \n\\begin{equation}\n \\frac{W_R(X)}{W_0(X)}=\\frac{p_{R}^{X}(1-p_R)^2}{p_{0}^{X}(1-p_0)^2}.\n\\end{equation}\nFor the BTL tagged agent's walk in GLS, we calculate $\\frac{W_R(X)}{W_0(X)}$ numerically for specific values of \n$\\lambda_k$ and $m_L$. \nThe obtained $p_R^{eff}(X, \\lambda_k, m_L)$ \nvalues do not match \nwell except for the low values of $X$. \n\nFrom the above two aspects, therefore, we can say that the walker is not behaving like an usual biased lazy walker.\nThe variation of $W_0, W_R$ and $W_L$ as a function of $X$ are shown in Fig. \\ref{WX_relall} a, b for A and B. \nAs it can be seen the individual path distributions vary approximately as an exponential.\nIt is to be noted here that \nall the variations are shown such that $\\sum_X W_i(X)=1$ where $i=0, R, L$. Also the relative variation of path distributions\n$W_0, W_R$ and $W_L$ as a function of $X$ are shown in Fig. \\ref{WX_relall} c, d, \nconsidering $W_0(X)+W_R(X)+W_L(X)=1$ for all \n$X$. It is clear from Fig. \\ref{WX_relall} c, d that for large $X$,  \n$W_R$ and $W_L$ decreases and finally comes to zero. That means long paths at a stretch \nfor Gain\/Loss are less probable. However, \nlong $W_0$ paths are possible.\n\n\\begin{figure}[!h]\n\\centering \n\\includegraphics[width=9cm, angle=-90]{WX_rel.eps}\n\\caption{Top Left : Variation of $W_0(X)$ against $X$ for models A and B for $\\lambda_k=0.0, 0.4$ and $m_L=0.1$ (violet and blue for model A and green and yellow for model B). Top Right :  \nVariation of $W_R(X)$ (violet and blue) and $W_{L}(X)$ (green and yellow) against $X$ for model A for $\\lambda_k=0.0, 0.4$ and $m_L=0.1$. (Behaviour of the quantities \nfor model B is similar as those of A and therefore it is not shown.)\nBottom Left : Relative variation of $W_0(X)$ (violet), $W_R(X)$ (red), $W_L(X)$ (green) against $X$ for model A for $\\lambda_k=0.0, 0.4$ and $m_L=0.1$.\nBottom Right : Relative variation of $W_0(X)$ (violet), $W_R(X)$ (red), $W_L(X)$ (green) against $X$ for model B for \n$\\lambda_k=0.0, 0.4$ and $m_L=0.1$.}\n\\label{WX_relall}\n\\end{figure}\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[trim=5cm 0cm 0cm 0cm,width=6cm, angle=-90,clip]{dirrevAB.eps}\n\\caption{Left : Direction Reversal Probability $f_d$ against $\\lambda_k$ for model A for $m_L=0.1$ (violet) , $m_L=0.05$ (yellow). Solid lines indicate $f_d$ from distribution of free path and dashed lines indicate those from \nthe simulated lazywalk. Right : Same for \nModel B}\n\\label{dirrev}\n\\end{figure}\n\n\nAnother interesting quantity to check here is the direction reversal probability. For lazy walker, the direction \nreversal probability is $2(p_0+p_R-p_0^2-p_R^2-p_0p_R)$. For our walk we consider the following quantity:\n\\begin{equation}\n \\langle X \\rangle=\\sum_X (XW_R(X)+XW_L(X)+XW_0(X)).\n\\label{avgX}\n\\end{equation}\nHere $\\langle X \\rangle$ is the average distance traveled in any particular direction (Gain, Loss or no movement) \nat a stretch. Obviously, the direction reversal probability is given by \n$f_{d}=\\frac{1}{\\langle X \\rangle}$. \n\nThe results are shown for some specific $\\lambda_k$ and $m_L$ for A and B. It is seen that for our walker, probability \nis very high for all $\\lambda_k$ values which saturates to a value close to $0.5$ for high $\\lambda_k$. This is expected as \nwhen $\\lambda_k$ is high, number of missed interactions become negligible. Therefore, the  \ndynamics is controlled only through $p_R$ and the situation becomes similar as in case of \nRight\/Left movement. These are shown in Fig. \\ref{dirrev}. We also \nsimulated a lazy walk using the parameters $p_0, p_R, p_L$ and calculated $f_d$ for that. \nFor the lazy walker although the nature of variation is similar and tends to $0.5$ for high $\\lambda_k$ \nas in case of our walker,\nthe exact probabilities do not match. \n\n\n\n\\section{Correlation}\nWe have seen that the steps of the walker have \nthree possible values. Therefore, we need to analyze the time series for such a walk. \nLet the step taken at a time $t$ be written as  $s(t)=0, \\pm 1$. The corresponding time correlation \nfunction can be written as:\n$C(t)=\\langle s(t_0) s(t_0+t)\\rangle - s_0^2$\nwhere $s_0^2=\\langle s(t_0)\\rangle \\langle s(t_0+t)\\rangle$. This can be written as we know that here \n$\\langle s(t_0)\\rangle$ is independent of $t$ and therefore $\\langle s(t_0)\\rangle=\\langle s(t_0+t)\\rangle=s_0$. \nJust as in CCM walk, here $\\langle s_0 \\rangle \\neq 0$ unless near $\\lambda_{k}^*$ for both the model A and B. The correlations for model \nA and B are shown in Fig. \\ref{corr}. It is seen that there is a strong correlation near $t=1$ which gradually decreases \nwith increasing $\\lambda_k$.The short time correlations in both models A and B are negative. \nThis  is consistent with the fact that direction\nreversal probability is greater than $1\/2$.  However, for one particular $\\lambda_k$ the correlation ultimately \nsaturates to a value $C_{sat}$ when $t\\rightarrow \\infty$. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[height=.8\\textwidth, angle=-90]{corrAB.eps}\n\\caption{Left : The correlation of steps $C(t)$ averaging over all possible initial\ntimes $t_0$ for the walker for model A for $m_L=0.1$ and $\\lambda_k=0.0$ (red), $0.4$ (yellow), $0.9$ (blue). Right : Same for Model B.}\n\\label{corr}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[trim=5cm 0cm 0cm 0cm,width=6cm, angle=-90,clip]{C1Csat.eps}\n\\caption{Left : The saturation value of correlation of steps $C_{sat}$ against $\\lambda_k$ for $m_L=0.1$ for A (violet) and B (yellow). \nMiddle : The strongest correlation $C(1)$ as a function of $\\lambda_k$ for $m_L=0.1$ for the models A (red) and B (black). \nRight : $C(1)$ against $m_L$ for A (green) and B (yellow) for $\\lambda_k=0.0$ (solid line) and $\\lambda_k=0.4$ (dashed line).}\n\\label{c1csat}\n\\end{figure}\n\nThis kind of feature was earlier noticed in \\cite{goswami2}. \nThe saturation value $C_{sat}$ is estimated by averaging near the end over a few hundred values of $t$. $C_{sat}$ as a function \nof $\\lambda_k$ \nis shown for $m_L=0.1$ for both models A and B in Fig. \\ref{c1csat} a. It is seen that \n $C_{sat}$ reaches a minimum close to $\\lambda_{k}^*$ which is $0.471$ for model A and $0.443$ for model B. \n The minimum value of \n $C_{sat}$ as observed for a $\\lambda_k \\approx \\lambda_{k}^*$ is $\\sim O(10^{-5})$. Also as we observe \nthe strongest correlation $C(1)$ is changing with $\\lambda_k$, we plot the same in Fig. \\ref{c1csat} b. \nIt is observed that as we \nincrease threshold line $m_L$, the correlation over one step $C(1)$ becomes weaker and weaker for lower $\\lambda_k$ values. \nHowever for high $\\lambda_k$ for all $m_L$ values, $C(1)$ is almost a constant. \nThis feature is similar in model A and B  but for higher values of $m_L$ the strength of maximum correlation, i.e., $C(1)$\nis weaker for a particular $m_L$ in model $B$. This is shown in Fig. \\ref{c1csat} c. \n\n\\section{Reason Behind High Direction Reversal}\nIt is already seen that the probability of direction reversal in the GLS is very high. This signature is also clear from \nthe strong correlation for small $t$. In this part, we are going to analyze why this direction reversal is preferred \nby an agent in this walk. \\\\\n\n\nFor our \nconvenience, we choose the DY model which is the simplest among all and the form of wealth distribution is well known. \nFor this we mimic a situation when the agent ends up with gain and then again interacts. The probability that \nthe agent will incur a loss in the next step requires that she\/he has to interact with another with low value of wealth. We \nconsider our agent ends up with wealth $m'$ in the first step. \nHowever, there are two possibilities in this case. If the second agent's wealth is between $0$ and $m_L$ then \nthere will be no interaction at all and if that is higher than $m_L$ and less than $m'$, our agent may lose some. \nThe conditional probability that the agent will have a loss after a gain is\n\\begin{equation}\n W'(LG)=\\frac{\\int_{m_1}^{\\infty} P(m_1)dm_1 \\int_{m_L}^{m_1} P(m_2) dm_2}{\\int_{m_1}^{\\infty} P(m_1)dm_1}\n\\end{equation}\nand the probability that the interaction will not occur is\n\\begin{equation}\n W'(0G)=\\frac{\\int_{m_1}^{\\infty} P(m_1)dm_1 \\int_{0}^{m_L} P(m_2) dm_2}{\\int_{m_1}^{\\infty} P(m_1)dm_1}\n\\end{equation}\nBut the probability that the previous step was a gain is $p_R$. Considering the wealth distribution of the form \n$P(m)\\sim \\exp(-m)$, the probability that it will either lose or stay put is \n$W(iG)=p_R(\\lambda_k)\\times(1-\\frac{1}{2}\\exp(-m_1))$ where $i=L\/0$.\\\\\n\nSimilarly the conditional probability of having a Loss in the first step and then either a Gain or a no interaction is \n\\begin{equation}\n W'(jL)=\\frac{1}{2}[1+\\exp(-m_1)]+1-\\exp(-m_L)\n\\end{equation}\nwhere $j=G\/0$.\nTherefore $W(jL)=p_L(\\lambda_k)W'(jL)=p_L(\\lambda_k)[\\frac{1}{2}[1+\\exp(-m_1)]+1-\\exp(-m_L)]$.\\\\\n\nProceeding in a similar manner we can show that \n\\begin{equation}\n W(i0)=p_0(\\lambda_k)\n\\end{equation}\nwhere $i=G\/L$. \\\\\n\nThus, the probability of direction reversal is \n\\begin{equation}\n f_d(\\lambda_k)=p_0(\\lambda_k)+p_R(\\lambda_k)(1-\\frac{1}{2}\\exp(-m_1))+p_L(\\lambda_k)[\\frac{1}{2}[1+\\exp(-m_1)]+1-\\exp(-m_L)].\n\\end{equation}\nIn any case we can show that \n\\begin{equation}\n\\label{fd}\n f_d(\\lambda_k) > p_0(\\lambda_k)+\\frac{1}{2}p_R(\\lambda_k)+p_L(\\lambda_k)[\\frac{3}{2}-\\exp(-m_L)]\n\\end{equation}\nUsing the obtained values of $p_0, p_R$ and $p_L$ for different $\\lambda_k$ and $m_L$ we check \nthe lower bound of the direction reversal probabilities from the  Eq. \\ref{fd}. $f_d$ for a few $\\lambda_k$ and \n$m_L$ values obtained using Eq. \\ref{fd} and from the model are compared in Table \\ref{table2}.\n\n\n\\begin{table}[!h]\n\\begin{center}\n\\caption{Comparison of direction reversal probabilities $f_d$ obtained from Eq. \\ref{fd} and from the model A}\n\\begin{tabular}{|c|c|c|c|c|\n\\hline\n$m_L$  &  $\\lambda_k$ &   \\multicolumn{3}{c|}{$f_d$} \\\\\\cline{3-5}\n   &  & From Eq. \\ref{fd} & From Lazywalk & From model \\\\\n\\hline\n  0.1 & 0.0 & 0.672  & 0.661 & 0.580\\\\\n    & 0.4 & 0.598 & 0.593 & 0.593 \\\\\n    & 0.9  & 0.548 & 0.499 & 0.508 \\\\\n\\hline\n0.05 & 0.0 & 0.560 & 0.567 & 0.600 \\\\\n     & 0.4 & 0.530 & 0.513 & 0.552 \\\\\n     & 0.9 & 0.525 & 0.499 & 0.508\\\\\n\\hline\n\\end{tabular}\n\\label{table2}\n\\end{center}\n\\end{table\n\nAs we can see from Table \\ref{table2}, there is good agreement of the lower bound of the inequality with the lazy walker \nexcept when $\\lambda_k$ is very high. Also the agreement is not that good for the data obtained from our model. This \nindicates that the walk of our agent is not exactly a lazy walk.\nAs $\\lambda_k$ becomes very high the DY model approximation is no longer valid and therefore there is discrepancy from \nthe direction reversal probability obtained from Eq. \\ref{fd}.\n\n\\section{Summary and Conclusion}\nIn this work, the nature of transactions\nmade in CCM model with some modification is studied. We used the idea of threshold line below which an agent is \nidentified as a BTL one at the time of assigning wealth. \nThese agents are always eligible for subsidy. This is similar to giving opportunity to \nsome backward class people in any caste-based system.\nThe threshold line is important in another way. \nIt dictates whether an interaction will occur or not. Any agent, either BTL or not, having \nwealth below that line at any point of interaction is considered insecure as in \nany real situation. If either one or both the interacting agents' wealth is below the \nthreshold line, the interaction will not occur. We also considered\nan equivalent picture of a $1$D walk in an abstract space for Gains and Losses. Here amount of Gain\/Loss is not important; \nwe have just used the information whether it is a Gain or Loss. As a tagged agent may have Gain, Loss or no interaction, \nthe corresponding walker is a lazy walker which in addition to \nLeft\/Right movement, may stay put at a position. The high direction reversal probability indicates \nthat there is a high tendency of individuals to make\na gain or to stay put immediately after a loss and vice versa. This kind of effect was studied for usual CCM model before. \nHere also we found this to be compatible with\nhuman psychology. From the high direction reversal probability it is clear that after a 'no' interaction the agent will \ntry to interact immediately at the next step. Also a person may take part in an interaction which may lead either to a loss \nor to no interaction, when she\/he had a\ngain in the previous step. After suffering a loss, in a same manner, a person \nwill either try to have a gain or may stay put due to \ninsecurity. This\neffect is maximum for zero saving propensity and decreases with increased saving. The data obtained \nfrom correlation for one step \nalso indicates that there will be high tendency of gain after loss and vice versa. (of course, there may be stay \nputs in between). \\\\\n\nThe subsidy is given here in two ways, firstly at the time of assigning the wealth to the agents initially (Model A), \nand secondly, \nat each MC step (Model B). \nIt is seen that if subsidy is given repeatedly (i.e., at each MC step), \nthe agent moves with a less positive drift in the GLS for Model B compared \nto Model A for any particular \n$\\lambda_k$ (Fig. \\ref{slopeAB}). This can be understood, as, giving repeated subsidy to a BTL agent will affect the wealth of others \nand as the walks are actually correlated, it affects the tagged BTL walk in turn. The amount of wealth possessed by such an agent \nis greater in Model B compared to Model A. The parameters $p_0, p_R$ and $p_L$ are \nobtained from the walk of the tagged BTL agent and with those a lazy walk is simulated. It is seen that the BTL agent's  \nCCM-like walk is not exactly similar to a biased lazy walk. The walk has no bias for some saving propensity $\\lambda_{k}^*$ which is \ndifferent for models A and B. \\\\\n\nThe distribution of path traveled at a stretch $X$ is studied for Right\/Left movements and staying put. \nThe quantities are $W_R(X), W_L(X)$ and $W_0(X)$ where the suffix indicates whether \nit is a gain or loss or no interaction.\nUsing that we calculated the direction reversal probability $f_d$ and the same is calculated analytically for a DY model. \nUsing the lazy walker \nparameters we compared the analytical direction reversal probability with those obtained from the walk. The analytical \napproach indicates that $f_d(\\lambda_k)$ must be greater than \n$p_0(\\lambda_k)+\\frac{1}{2}p_R(\\lambda_k)+p_L(\\lambda_k)[\\frac{3}{2}-\\exp(-m_L)]$. \nHowever, the analytical treatment was done using the \nDY model wealth distribution and therefore is not in good agreement to our case, when saving propensity is high. \nFrom different calculations shown here it is clear that for Model A, $\\lambda_k=\\lambda_{k}^*\\simeq 0.471$ \nand for model B it is close to $0.443$. Although for Model A it is close to \nthe $\\lambda_{k}^*$ obtained for usual CCM model, i.e., $0.469$, for Model B the result is different.\n\nFinally, our study shows a modified CCM model with the concept of a Threshold line.   \nThe threshold line in addition to identification of BTL agents, puts restriction to interactions. \nWe have seen that putting restriction changes the distribution which is dependent on the threshold line $m_L$ as \nshown in Fig. \\ref{mnydist_compare}.\nThe dynamics of a tagged BTL agent is compared to a Lazy walker in GLS. \nThe high direction reversal probability for such a walker indicates that\none can afford to have a loss or may stay put he\/she has gained begore. Also after suffering a loss, it may \nstay put or may try to make a gain.\nThis is completely compatible with \nhuman psychology. \nAlso, it is seen that the value of $\\lambda_{k}^{*}$ is \nindependent of $m_L$ for both the models A and B as shown in Fig. \\ref{slopevariation_mL}. \nThe effect of giving subsidy once over giving repeated subsidy \nto such agents \nis checked from the point of view of the walk in GLS. The \nwalk is seen to have more positive drift when the subsidy is given once.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\\label{intro}\n\nHow galaxies form and evolve is one of the primary outstanding problems\nin extragalactic astronomy. The initial conditions \nled to the collapse of dark matter halos which clustered hierarchically \ninto progressively larger \nstructures. In the halo interiors, gas formed rotating disks \nwhich underwent star formation (SF) to produce\nstellar disks (Cole 2000; Steinmetz \\& Navarro 2002).  The subsequent\ngrowth of galaxies is thought to have proceeded through a combination\nof major mergers,\n(e.g., Toomre 1977; Barnes 1988; Khochfar \\& Silk 2006, 2009), \nminor mergers (e.g., Oser et al. 2012, Hilz et al. 2013), cold-mode gas\naccretion (Birnboim \\& Dekel 2003; Kere{\\v s} et al. 2005, 2009; \nDekel \\& Birnboim 2006; Brooks et al. 2009; Ceverino et al. 2010; \nDekel et al. 2009a, b), and secular processes (Kormendy \\& Kennicutt 2004).\n\nIn early simulations focusing on gas-poor mergers, the major merger of two \nspiral galaxies with mass ratio $M_1\/M_2 \\geq 1\/4$ would inevitably destroy the \npre-existing stellar disks by violent relaxation, producing a remnant bulge or \nelliptical having a puffed-up distribution of stars with a low ratio of \nordered-to-random motion ($V\/\\sigma$) and a steep de Vaucouleurs  $r^{1\/4}$ \nsurface brightness profile\\footnote{A de  Vaucouleurs  $r^{1\/4}$ profile\ncorresponds to a S\\'ersic (1968) profile with index $n=4$.} (Toomre 1977).\nImproved simulations (Robertson  et al. 2006; Naab et al. 2006;  \nGovernato et al. 2007; Hopkins et al. 2009a, b) significantly revised this \npicture. In unequal-mass major mergers, violent relaxation of stellar disks is \nnot complete. Furthermore, for major mergers where the progenitors have \nmoderate-to-high gas fractions, \ngas-dissipative processes build disks on small and large scales \n(Hernquist \\& Mihos 1995; Robertson et al. 2006; Hopkins et al. 2009a, b; \nKormendy et al. 2009). The overall single S\\'ersic index $n$ of such \nremnants are typically $2\\lesssim n \\lesssim 4$ (Naab et al. 2006; \nNaab \\& Trujillo 2006; Hopkins et al. 2009a). \nThe subsequent accretion of gas from the halo, cold streams, and minor mergers \ncan further build large-scale stellar disks, whose size depends on the specific \nangular momentum of the accreted gas (Steinmetz \\& Navarro 2000; \nBirnboim \\& Dekel 2003; Kere{\\v s} et al. 2005, 2009; Dekel \\& Birnboim 2006; \nRobertson et al. 2006; Dekel et al. 2009a, b; Brooks et al. 2009; \nHopkins et al. 2009b; Ceverino et al. 2010). Additionally, \nBournaud, Elmegreen, \\& Elmegreen (2007) and Elmegreen et al. (2009) discuss\nbulge formation via the merging of clumps forming within very gas-rich, \nturbulent disk in high-redshift galaxies. These bulges can have a range of \nS\\'ersic indices, ranging from $n<2$ to $n=4$.\n\nAs far as the structure of galaxies is concerned, we are still actively \nstudying and debating the epoch and formation pathway for the main stellar \ncomponents of galaxies, namely flattened, dynamically cold, disk-dominated \ncomponents (including outer disks, circumnuclear disks, and pseudobulges) \nversus puffy, dynamically hot spheroidal or triaxial bulges\/ellipticals. \nGetting a census of dynamically hot bulges\/ellipticals and dynamically cold, \nflattened disk-dominated components on large and small scales in galaxies \nprovides a powerful way of evaluating the importance of violent bulge-building \nprocesses, such as violent relaxation, versus gas-dissipative disk-building\nprocesses.\n\nWe adopt throughout this paper the widely used definition of a bulge as the \nexcess light above an outer disk in an S0 or spiral galaxy (e.g., \nLaurikainen et al. 2007, 2009, 2010; Fisher \\& Drory 2008; Gadotti 2009; \nWeinzirl et al. 2009). The central bulge falls in three main categories called \nclassical bulges, disky pseudobulges (Kormendy 1993; \nKormendy \\& Kennicutt 2004; Jogee, Scoville, \\& Kenney 2005; Athanassoula 2005; \nKormendy \\& Fisher 2005; Fisher \\& Drory 2008), and boxy pseudobulges \n(Combes \\& Sanders 1981; Combes et al. 1990; \nPfenniger \\& Norman 1990;  Bureau \\& Athanassoula 2005; Athanassoula 2005; \nMartinez-Valpuesta et al. 2006). Some bulges are composite mixtures \nof the first two classes (Kormendy \\& Barentine 2010; \nBarentine \\& Kormendy 2012). For remainder of the paper we refer to classical \nbulges simply as ``bulges'' when the context is unambiguous. \n\nNumerous observational efforts have been undertaken to derive\nsuch a census among galaxies in the field environment. Photometric\nstudies (e.g., Kormendy 1993; Graham 2001; Balcells et al. 2003, 2007b; \nLaurikainen et al. 2007; Graham \\& Worley 2008;  Fisher \\& Drory 2008;\nWeinzirl et al. 2009; Gadotti 2009; Kormendy et al. 2010)\nhave dissected field galaxies into  outer stellar disks\nand different types of central bulges (classical, disky\/boxy pseudobulges) \nassociated with different S\\'ersic index,\nand compiled the stellar bulge-to-total light or mass ratio ($B\/T$)\nof spirals and S0s. It is found that low-$B\/T$ and bulgeless galaxies are \ncommon in the field  at low redshifts, both among low-mass\nor late-type galaxies (B{\\\"o}ker et al. 2002; Kautsch et al. 2006;\nBarazza et al. 2007, 2008) and among high-mass spirals or early-type spirals\n(Kormendy 1993; Balcells et al. 2003, 2007b; Laurikainen et al. 2007; Graham \\& Worley 2008; Weinzirl et al. 2009; Gadotti 2009; Kormendy et al. 2010).\nBalcells et al. (2003) highlighted the paucity of $r^{1\/4}$ profiles in the \nbulges of early-type disk galaxies. Working on a bigger sample,\nWeinzirl et al. (2009) report that the majority ($66.4 \\pm 4.4\\%$)\nof massive ($M_\\star \\geq 10^{10}$~$M_\\odot$) field spirals have\nlow $B\/T$ ($\\leq0.2$) and bulges with low S\\'ersic index ($n \\leq 2$).\n\nThese empirical results can be used to test models of the\nassembly history of field galaxies. For instance,\nWeinzirl et al. (2009) find that the results reported above \nare consistent with hierarchical semi-analytic models of galaxy evolution from\nKhochfar \\& Silk (2006) and Hopkins et al. (2009a), which predict that\nmost ($\\sim 85\\%$) massive field spirals have had no major merger since $z=2$.\nWhile this work reduces the tension between theory and observations for field \ngalaxies, one should note that hydrodynamical models still face challenges in \nproducing purely bulgeless massive galaxies in  different environments. \n\nIt is important to extend such studies from the field environment to rich \nclusters. Hierarchical models predict differences in galaxy merger history as \na function of galaxy mass, environment, and redshift (Cole et al. 2000; \nKhochfar \\& Burkert 2001). Furthermore, cluster-specific physical processes, \nsuch as  ram-pressure stripping (Gunn \\& Gott 1972; Fujita \\& Nagashima 1999), \ngalaxy harassment (Barnes \\& Hernquist 1991; Moore et al. 1996, 1998, 1999;\nHashimoto et al. 1998; Gnedin 2003), and strangulation (Larson et al.\n1980), can alter SF history and galaxy stellar components (disks, bulges, bars).\n\nEfforts to establish accurate demographics of galaxy components in clusters \nare ongoing. In the nearby Virgo cluster, Kormendy et al. (2009) find more \nthan 2\/3 of the stellar mass is in classical bulges\/ellipticals, including the \nstellar mass contribution from M87\\footnote{M87 is considered as a giant ellipticals by \nsome authors and as a cD by others. The detection of intra-cluster light around \nM87 (Mihos et al. 2005, 2009) strongly supports the view that it is a cD \ngalaxy. In this paper (e.g.,\nTable~\\ref{tabmdr}) we consider M87 as a cD when making comparisons (e.g., \nSection~\\ref{census}) to Virgo.}. Furthermore, there is clear \nevidence for ongoing environmental effects in Virgo; \nsee Kormendy \\& Bender (2012) for a comprehensive review. \n\nYet Virgo is not very rich compared with more typical clusters \n(Heiderman et al. 2009). The Coma cluster at $z=0.024$ ($D=100$~Mpc) \nhas a central number density 10,000~Mpc$^{-3}$ (The \\& White 1986) and is the\ndensest cluster in the local universe. However, ground-based data do not \nprovide high enough resolution \n($1^{\\prime\\prime} -2^{\\prime\\prime} =500-1000$~pc) for accurate structural \ndecomposition, an obstacle to earlier work. \n\nIn this paper we make use of data from the $Hubble$ $Space$ $Telescope$\n($\\textit{HST}$) Treasury Survey (Carter et al. 2008) of Coma which provides \nhigh-resolution (50~pc) imaging from the Advanced Camera for Surveys (ACS). \nOur goal is to derive the demographics of galaxy components, in particular \nclassical bulges\/ellipticals and flattened disk-dominated components \n(including both large-scale disks and disky pseudobulges), in the Coma cluster, \nand to compare the results with lower-density environments and to theoretical \nmodels, to constrain the assembly history of galaxies.\n\nIn Section~\\ref{data0} we present our mass-complete sample of cluster galaxies\nwith stellar mass $M_\\star \\geq 10^9$~$M_\\odot$. In  Section~\\ref{method} \nwe describe our structural decomposition strategy.\nSection~\\ref{approach} describes our working assumption in this paper of \nusing S\\'ersic index as a proxy for tracing the disk-dominated structures \nand classical bulges\/ellipticals. Section~\\ref{special} outlines our procedure \nfor structural decomposition, and we refer the reader to Appendix~\\ref{app} for \na more detailed description.\nSection~\\ref{scheme} overviews the scheme we use to assign morphological\ntypes to galaxies. In Section~\\ref{mdr}, we quantitatively assign galaxy types \nbased on the structural decompositions. We also make a census \n(Section~\\ref{census}) of structures built by dissipation versus violent \nstellar processes, explore how stellar mass is distributed in different galaxy \ncomponents (Section~\\ref{discussmass}), and consider galaxy scaling relations \n(Section~\\ref{discussScaling}). In Section~\\ref{discuss3}, we evaluate and \ndiscuss the effect of cluster environmental processes. In Section~\\ref{theory} \nwe compare our empirical results with theoretical models, \nafter first identifying Coma-like environments in the simulations. Readers not\ninterested in the complete details about the theoretical model can\nskip to Sections~\\ref{globalprop} and \\ref{datavmodel}.\nWe summarize our results in Section~\\ref{summary}.\n\nWe adopt a flat $\\Lambda$CDM \ncosmology with $\\Omega_\\Lambda = 0.7$ and $H_0 = 73$~km~s$^{-1}$~Mpc$^{-1}$.\nWe use AB magnitudes throughout the paper, except where indicated otherwise.\n\n\n\n\n\\section{Data and Sample Selection}\\label{data0}\nThis study is based on the data products from the $\\textit{HST}$\/ACS Coma \nCluster Treasury Survey (Carter et al. 2008), which provides ACS Wide Field \nCamera images for 25 pointings  spanning 274 arcmin$^2$ in the F475W and F814W \nfilters.  The total ACS exposure time per pointing is typically 2677 seconds in \nF475W and 1400 seconds in F814W. Most (19\/25) pointings are located within \n0.5~Mpc from the central cD galaxy NGC~4874, and the other 6\/25 pointings are \nbetween 0.9 and 1.75~Mpc southwest of the cluster center.  \nThe FWHM of the ACS point-spread function (PSF) is $\\sim 0\\farcs1$ (Hoyos et al. 2011), corresponding to $\\sim 50$~pc at the 100~Mpc distance of the Coma \ncluster (Carter et al. 2008). Note the 19 pointings cover only \n$19.7\\%$ by area of the projected central 0.5~Mpc of Coma. This limited \nspatial coverage of ACS in the projected central 0.5~Mpc of Coma may introduce \na possible bias in the sample due to cosmic variance. We quantify this effect \nin Appendix~\\ref{cosmicvar} and discuss the implications throughout the paper.\n\nHammer et al. (2010) discuss the images and \\texttt{SExtractor} source \ncatalogs for Data Release 2.1 (DR2.1). The F814W $5\\sigma$ limit for point \nsources is 26.8 mag (Hammer et al. 2010), and we estimate the $5\\sigma$ F814W \nsurface brightness limit for extended sources within a $0\\farcs7$ diameter \naperture to be 25.6 mag\/arcsec$^2$. Several of the ACS images in DR2.1 suffer \nfrom bias offsets on the \ninter-chip and\/or inter-quadrant scale that cause difficulty in removing the sky\nbackground. We use the updated ACS images reprocessed to reduce the impact of \nthis issue. The DR2.1 images are used where this issue is not present.\n\n\\subsection{Selection of Bright Cluster Members}\\label{datas1}\n\nWe select our sample based on the eyeball catalog of N. Trentham et al. (in\npreparation), with updates from Marinova et al. (2012). This catalog provides\nvisually determined morphologies and cluster membership status for galaxies\nwith an apparent magnitude F814W $\\leq 24$ mag. \nMorphology classifications in this catalog come from a combination of \nRC3 (de Vaucouleurs et al. 1991) and visual inspection. In Section~\\ref{mdr}\nwe assign Hubble types based only on our own multi-component decompositions.\n\nCluster membership is ranked from 0 to 4 following the method of Trentham \\& \nTully (2002). Membership class 0 means the galaxy is a spectroscopically \nconfirmed cluster member.  The subset of spectroscopically confirmed cluster \nmembers was identified based on published redshifts (Colless \\& Dunn 1996; \nAdelman-McCarthy et al. 2008; Mobasher et al. 2001; Chiboucas et al. 2010) and\nis approximately complete in surface brightness at the galaxy half-light radius \n($\\mu_\\mathrm{e, F814W}$) to $\\sim 22.5$ mag\/arcsec$^2$ (den Brok et al. 2011). \nThe remaining galaxies without spectroscopic confirmation are assigned a rating \nof 1 (very probable cluster member), 2 (likely cluster member), 3 (plausible \ncluster member), or  4 (likely background object) based on a visual estimation \nthat also considers surface brightness and morphology.\n\nFrom this catalog, we define a sample S1 of 446 cluster members having F814W \n$\\leq 24$ mag and membership rating 0-3 located within the projected central \n0.5~Mpc of Coma, which is the projected radius probed by the central ACS \npointings. To S1 we add the second central cD galaxy NGC~4889, which is not \nobserved by the ACS data. The majority (179) of S1 galaxies have member class \n0, and 30, 131, and 106 have member \nclass 1, 2, and 3, respectively. \n\n\\subsection{Calculation of Stellar Masses}\\label{datas2}\n\nStellar masses are a thorny issue. Uncertainties in the mass-to-light \nratios of stellar populations ($M\/L$) arise from a poorly known initial mass \nfunction (IMF) as well as degeneracies between age and \nmetallicity. We calculate stellar masses based on the $HST$ F475W and \nF814W-band photometry. First, we convert the $HST$ (AB) photometry to the \nCousins-Johnson (Vega) system using\n\\begin{equation}\n\\rm{ I=F814W -0.38}\n\\end{equation}\nfrom the WFPC2 Photometry Cookbook and \n\\begin{equation}\n\\rm{ B-I=1.287\\left(F475W - F814W\\right) + 0.538 }\n\\end{equation}\nfrom Price et al. (2009).\n\nNext, we calculate $I$-band $M\/L$ from the calibrations of Into \\& Portinari \n(2013) for a Kroupa et al. (1993) initial mass function (IMF) with\n\\begin{equation}\n\\rm{ I_{lum}=10^{\\left( -0.4\\left( I-35-4.08\\right) \\right)} }\n\\end{equation}\nand\n\\begin{equation}\n\\rm{ M_\\star=I_{lum} \\times 10^{ \\left( 0.641\\left(B-I\\right) -0.997 \\right) }},\n\\end{equation}\nwhere $I$ corresponds to the apparent \\texttt{MAG\\_AUTO} \\texttt{SExtractor} \nmagnitude\\footnote{For galaxies COMAi125935.698p275733.36 $=$ NGC~4874 and COMAi125931.103p275718.12, \n\\texttt{SExtractor} vastly underestimates the total F814W luminosity, and the\ncalculation is instead made with the total luminosity derived from structural\ndecomposition (Section~\\ref{special}).}, 35 is the distance modulus to Coma, and 4.08 is the solar absolute \nmagnitude in $I$-band.\n\nWe use the above method to calculate stellar masses for all galaxies in S1\nexcept NGC~4889, which does not have ACS data. \nFor NGC~4889, we use $gr$ Petrosian magnitudes from SDSS DR10 \n(Ahn et al. 2013). Stellar masses are determined using the \nrelations of Bell et al. (2003) and assuming a Kroupa IMF, namely\n\\begin{equation}\n\\rm{ g_{lum}=10^{\\left( -0.4\\left( g-35-5.10\\right) \\right)} }\n\\end{equation}\nand\n\\begin{equation}\n\\rm{ M_\\star=g_{lum} \\times 10^{ \\left( -0.499+1.519\\left( g-r\\right) -0.1 \\right) }},\n\\end{equation}\nwhere $g$ and $r$ are apparent SDSS magnitudes, 35 is the distance modulus to Coma, and\n5.10 is the solar absolute magnitude in $g$-band\\footnote{The Kroupa IMF offset term reported as -0.15 in \nBell et al. 2003 was calculated assuming unrealistic conditions (Bell, E., private \ncommunication). The correct value is -0.1 and is used in Borch et al. (2006).}.\n\n\nIt is hard to derive the stellar mass of cD galaxies for several reasons.\nThe stellar $M\/L$ ratio of cDs is believed to be high ($M_{\\rm dyn}\/L_B>100$; \nSchneider 2006),  but is very uncertain as most of the light\nof a cD is in an outer envelope made of intra-cluster light \nand galaxy debris. Another problem is that even if one knew the correct \nstellar $M\/L$ ratio, it is likely that the available photometry from ACS and\nSDSS is missing light from the extended low surface brightness envelope.\nGiven all these factors, it is likely that the above equations, which are \ntypically used to convert  color to $M_\\star$ for normal representative \ngalaxies, are underestimating the $M\/L$ ratios and stellar masses of the cDs, \nso that the adopted stellar masses for the cDs \n($M_\\star \\sim6-8\\times 10^{11}$~$M_\\odot$) are lower limits. Due to the \nuncertain stellar masses of the cDs, we present many of our results without \nthem, and we take care to consider them separately from the less massive galaxy \npopulation of E, S0, and spiral galaxies.\n\n\\subsection{Selection of Final Sample of Massive Galaxies}\\label{datas3}\n\nThe left panels of Figure~\\ref{lummasstype} show the distributions of F814W \nmagnitudes (upper panel) and stellar masses (lower panels) for sample S1, \nwhile in the right panels of the same figure the correlations of stellar \nmasses with F814W magnitudes (upper panel) and $g-r$ colors (lower panel) \nare shown.\n\nIn this paper, we focus on massive ($M_\\star \\geq 10^9$~$M_\\odot$) galaxies. \nOur rationale is that we are specifically interested in \nunderstanding the evolution of the most massive cluster galaxies through \ncomparisons with model clusters (Section~\\ref{theory}) which show mass \nincompleteness at galaxy stellar masses $M_\\star < 10^9$~$M_\\odot$.\nWe found for sample S1 that imposing the mass cut \\\n$M_\\star \\geq 10^9$~$M_\\odot$ \neffectively removes most galaxies identified in the Trentham et al. catalog as \ndwarf\/irregular and very low surface brightness galaxies. With this cut,\nwe are left with 75 galaxies that consist primarily of E, S0, and spiral \ngalaxies, two cDs, and only six dwarfs. Three out of 75 galaxies are \nsignificantly cutoff the ACS detector, and we ignore these sources.  \nOf the remaining 72 galaxies, 69\/72 have spectroscopic redshifts. The 3\/72\ngalaxies without spectroscopic redshifts appear too red to be in Coma\n(Figure~\\ref{lummasstype}d), and the estimated SDSS DR10 photometric redshifts \nare much larger than the redshift of Coma (0.024). We also neglect these three \nsources as they are unlikely to be Coma members. Our final working sample S2 \nconsists of the 69 galaxies inside the projected central 0.5~Mpc with stellar \nmass $M_\\star \\geq 10^9$~$M_\\odot$ and spectroscopic redshifts.\nTable~\\ref{tab:cross_ids} cross references our sample with other datasets.\n\n\\section{Method and Analysis}\\label{method}\n\n\\subsection{Using S\\'ersic Index as a Proxy For Tracing Disk-Dominated Structures \nand Classical Bulges\/Ellipticals}\\label{approach}\nAs outlined in Section~\\ref{intro}, galaxy bulges and stellar disks hold \ninformation on galaxy assembly history. The overall goal in this work is to \nseparate galaxy components into groups of classical bulges\/ellipticals versus \ndisk-dominated structures.\n\nIt is common practice (e.g., Laurikainen et al. 2007; Gadotti 2009; \nWeinzirl et al. 2009) to characterize galaxy structures\n(bulges, disks, and bars) with generalized ellipses whose radial light\ndistributions are described by the  S\\'ersic (1968) profile:\n\\begin{equation}\n\\rm{ I(r)=I_e \\ exp \\left( -b_n \\left[ \\left( r\\over{r_\\mathrm{e}} \\right)  ^{1\/n} -1 \\right]  \\right) },\n\\end{equation}\nwhere $I_e$ is the surface brightness at the effective radius $r_\\mathrm{e}$ \nand $b_n$ \\footnote{The precise values of $b_n$\nare given from the roots of the equation $\\Gamma(2n) - 2\\gamma(2n,b_n)=0$, \nwhere $\\Gamma$ is the gamma function and $\\gamma$ is the incomplete gamma \nfunction.} is a constant that depends on S\\'ersic index $n$.  \n\nIn this paper, we adopt the working assumption that in intermediate and \nhigh-mass ($M_\\star \\geq 10^9$ $M_\\odot$) galaxies, a low S\\'ersic index $n$ \nbelow a threshold value $n_{\\rm disk\\_max}$ corresponds to a dynamically cold \ndisk-dominated structure. Note we specify ``disk-dominated'' rather than \n``pure disk'' as we refer to barred disks and thick disks.   While this \nassumption is not necessarily waterproof, it is based on multiple lines of \ncompelling evidence that are outlined below. \n\n\\begin{enumerate}\n\\item\nFreeman (1970) showed that many large-scale disks of S0 and spiral galaxies are \ncharacterized by an exponential light  profile (S\\'ersic index $n=1$) over 4-6 \ndisk scalelengths.  Since then, it has become standard practice in studies of \ngalaxy structure to model the outer disk of S0s and spirals with an exponential \nprofile (e.g., Kormendy 1977; Boroson 1981; Kent 1985; de Jong 1996; \nBaggett et al. 1998; Byun \\& Freeman 1995; Allen et al. 2006; Laurikainen 2007; \nGadotti 2009; Weinzirl et al. 2009). \n\n\\item\nOn smaller scales, flattened, rotationally supported inner disks with high \n$V\/\\sigma$ (i.e., disky pseudobulges) have been associated with low S\\'ersic \nindex $n\\lesssim2$ (Kormendy 1993; Kormendy \\& Kennicutt 2004; Jogee, \nScoville, \\& Kenney 2005; Athanassoula 2005; Kormendy \\& Fisher 2005; \nFisher \\& Drory 2008; Fabricius et al. 2012). This suggests $n_{\\rm disk\\_max}$ \nshould be close to 2. \n\nFabricius et al. (2012) explore the major-axis kinematics of 45 S0-Scd galaxies \nwith high-resolution spectroscopy. They demonstrate a systematic agreement \nbetween the shape of the velocity dispersion profile and the bulge type as \nindicated by the S\\'ersic index. Low S\\'ersic index bulges have both increased \nrotational support (higher $<V^2>\/<\\sigma^2>$ values) and on average lower \ncentral velocity dispersions. Classical bulges (disky pseudobulges)\nshow have centrally peaked (flat) velocity dispersion profiles whether\nidentified visually or by a high S\\'ersic index.\n\n\\item\nAt high ($z\\sim2$) redshift, where it is not yet possible to fully resolve \ngalaxy substructures, it has become conventional to use the global S\\'ersic \nindex $n\\lesssim2$ in massive galaxies to separate disk-dominated versus \nbulge-dominated galaxies (e.g., Ravindranath et al. 2004; \nvan der Wel et al. 2011; Weinzirl et al. 2011). Weinzirl et al. (2011) further \nexplore the distributions of ellipticities ($1-b\/a$) for the massive $z\\sim2$ \ngalaxies with low ($n\\leq2$) and high ($n>2$) global S\\'ersic index. They find \ngalaxies with low global S\\'ersic index $n\\leq2$ have a distribution of \nprojected ellipticities more similar to massive $z\\sim0$ spirals than to \nmassive $z\\sim0$ ellipticals.\n\n\\end{enumerate}\n\nThe above does not allow for low-$n$, dynamically hot structures. \nA low-$n$ dynamically hot structure would be considered in our study as\na pure photometric disk, a low-$n$ bulge, or an unbarred S0 galaxy. The \nerror due to misunderstood objects in the first two groups is expected to be \nsmall or nonexistent.  There is only one pure photometric disk in the sample\n(Section~\\ref{mdr}) and low-$n$ bulges ($N=20$) only make up 2.2\\% of galaxy \nstellar mass (excluding the cDs, Section~\\ref{census}).\nFurthermore, Figure~15 of Fabricius et al. (2012) shows that no low-$n$ bulge \nturns out to be dynamically hot. \n\nThere are 20 unbarred S0 galaxies in our sample, and these \naccount for 18.5\\% of the galaxy stellar mass (excluding the cD galaxies). \nAbout 75\\% of these objects have stellar mass and luminosity consistent with \ndwarf spheroidal galaxies (Kormendy et al. 2009).  Even if some of these \nsystems are actually dwarf spheroidals, they may not be dynamically hot as\nsome studies (e.g., Kormendy et al. 2009, Kormendy \\& Bender 2012) claim that \nmany dwarfs are actually disk systems closely related to \ndIrr, which have been stripped of gas via supernova feedback or environmental \neffects. The remaining 25\\% would be misclassified elliptical\ngalaxies as they are too bright and massive to be dwarfs.  Note,\nhowever, that Figure 33 of Kormendy et al. (2009) shows that elliptical\ngalaxies with $M_V < -18$ and S\\'ersic $n<2$ are very rare. In the\nworse-case scenario that all of our unbarred S0 galaxies are dynamically hot \nstructures, our measurement of the dynamically hot stellar mass in \nSection~\\ref{census} would be too low by $\\sim 30\\%$.\n\nThe second natural  related working assumption in our paper is\n that in intermediate and high-mass \n($M_\\star \\geq 10^9$ $M_\\odot$) galaxies, components with S\\'ersic \n$n> n_{\\rm disk\\_max}$ \nare classical bulge\/elliptical components (defined in Section~\\ref{intro}).\nSuch bulges\/ellipticals are formed by the redistribution of stars during major \nand minor galaxy collisions. $N$-body simulations\nshow that minor mergers consistently raise the bulge S\\'ersic index\n(Aguerri et al. 2001; Eliche-Moral et al. 2006; Naab \\& Trujillo 2006).\nThe effect of successive minor mergers is cumulative (Aguerri et al. 2001; \nBournaud, Jog, \\& Combes 2007; Naab et al. 2009; Hilz et al. 2012).\n\nWe empirically set $n_{\\rm disk\\_max}$ to 1.66 based on looking at the \nS\\'ersic $n$ of outer disks in those Coma galaxies that are barred, and by \ndefinition, must harbor outer \ndisk since bars are disk features. Appendix~\\ref{2cp} and Appendix~\\ref{nuperr} \ndiscuss the empirical details behind this choice.\n\n\\subsection{Overview of Our Structural Decomposition Procedure}\\label{special}\n\nFor our mass-complete sample of 69 intermediate-to-high mass\n($M_\\star \\geq 10^9$~$M_\\odot$) galaxies, we use deep, high-resolution \n($0\\farcs1$ or 50~pc), F814W-band images of Coma from $\\textit{HST}$\/ACS, \nwhich allow for accurate structural decomposition.\nWe fit galaxies with one, two, or three S\\'ersic profiles, \nplus a nuclear point source, when needed  (see Appendix~\\ref{app} \nfor details).  We use GALFIT (Peng et al. 2002). \nIn a model with one or more S\\'ersic profiles, there is expected to be coupling \nbetween the free parameters, particularly $r_\\mathrm{e}$ and $n$, although most \nprevious studies have generally ignored this effect. Weinzirl et al. (2009) \nexplores the issue of parameter coupling for barred and unbarred spiral \ngalaxies.\n\nWe take some precautions to ensure accurate decompositions:\n\n\\begin{enumerate}\n\\item  \nWe fit all structures with a generalized S\\'ersic profile where\nthe S\\'ersic index is a free parameter (Section~\\ref{scheme}).\nThis limits the number of a priori assumptions on the physical \nnature or shape of galaxy structures.\n\n\\item  \nIn clusters, the featureless (i.e., no spiral arms delineated by young stars, \nrings of SF, or gas\/dust lanes) outer disks of gas-poor S0s are not readily\ndistinguished from the equally featureless outer stellar components of \nclassical ellipticals. We do this in essence by applying $n_{\\rm disk\\_max}$\nto the S\\'ersic index $n$ of the outer galaxy structure.\n\n\\item Not requiring outer disks to have an exponential $n=1$ profile \naccommodates non-exponential disk structures (e.g., disks with down-bending \ntruncations or up-bending anti-truncations Freeman 1970;\nvan der Kruit 1979; van der Kruit \\& Searle 1981a, 1981b;\nde Grijs et al. 2001; Pohlen et al. 2002; Matthews \\& Gallagher 1997;\nErwin et al. 2005; Pohlen \\& Trujillo 2006; Maltby et al. 2012) that are \nrotationally supported. \n\n\\item\nStellar bars, ovals\/lenses, and nuclear point sources are modeled when needed, \nwhich is critical for obtaining a reliable characterization of the bulge \n(e.g., Balcells et al. 2003; Laurikainen et al. 2005, 2007; \nWeinzirl et al. 2009). \n\\end{enumerate}\n\nOur structural decomposition scheme and decision sequence are described\nin detail in Appendix~\\ref{decomp}, illustrated in Figures\n\\ref{fitflow} and  \\ref{fitflow2}, and briefly outlined below:\n\n\\begin{itemize}\n\\item\n{\\it Stage 1 (Single S\\'ersic fit with nuclear point source if needed):}\nThe single S\\'ersic model is adopted if either the galaxy does not show any \ncoherent structures (e.g., inner\/outer disks, bars, bulges, rings, or spiral \narms) indicating the need for additional S\\'ersic components, or, \nalternatively, if the galaxy has a core - a light profile that deviates \ndownward from the inward extrapolation of the S\\'ersic profile (see \nAppendix~\\ref{core}). Such galaxies are interpreted as photometric ellipticals \nif the single S\\'ersic index is above a threshold value $n_{\\rm disk\\_max}$ \nassociated with disks (Section~\\ref{approach}, Appendix~\\ref{2cp}, and \nAppendix~\\ref{nuperr}); otherwise they are considered photometric disks. Three \ngalaxies show convincing evidence for being cores, and these are luminous \nobjects with high single S\\'ersic $n > n_{\\rm disk\\_max}$ (see \nAppendix~\\ref{2cp}, Table~\\ref{tabcore}, Appendix~\\ref{core}). The results of \nStage 1 are listed in Table~\\ref{1cptab}. See Appendix~\\ref{1cp} for \nadditional details on the single S\\'ersic fits. \n\n\\item\n{\\it Stage 2 (Double S\\'ersic model with nuclear point source if needed):}\nGalaxies showing coherent structure in the Stage 1 residuals are subjected to \na two-component S\\'ersic + S\\'ersic fit, with nuclear point source if needed \n(see Figure~\\ref{fitflow2}). This two-component model is intended to model the \ninner (C1) and outer (C2) galaxy structures. \n\nThere are two possible outcomes.  a) If the outer component C2 is an outer disk \nbased on having S\\'ersic index $n\\leq n_{\\rm disk\\_max}$, then the galaxy is \nconsidered a spiral or S0 with an outer disk having a photometric bulge and, \nin some cases, a large-scale bar. b) If the outer component C2 does not meet \nour definition of an outer disk, then the galaxy is considered a photometric \nelliptical having an outer component C2 with $n>n_{\\rm disk\\_max}$ and an inner\ncomponent C1 of any $n$. See Appendix~\\ref{2cp} for details.\n\n\\item\n{\\it Stage 3 (Triple S\\'ersic model with nuclear point source if needed):}\nCase (a) in Stage 2 identifies spiral and S0 galaxies with an outer disk. \nThese galaxies are further processed as follows: a) If there is evidence for a \nlarge-scale bar (see Appendix~\\ref{2cp}), then a triple S\\'ersic profile is \nfitted in Stage 3 for the photometric bulge, disk, and bar.  b) Otherwise, the \ngalaxy is considered as unbarred and the double S\\'ersic fit for a photometric \nbulge and disk is adopted.  In both cases (a) and (b), it is important to note \nthat the photometric bulge is allowed to have any S\\'ersic index $n$, thus \nallowing for structures with $n\\leq n_{\\rm disk\\_max}$ and structures with \n$n>n_{\\rm disk\\_max}$.\n\\end{itemize}\n\n\\subsection{Overview of  Our Galaxy Classification Scheme}\\label{scheme}\n\nThe decomposition scheme discussed above and in Figures \\ref{fitflow} and \n\\ref{fitflow2} leads naturally to the galaxy classification system outlined in \nFigure~\\ref{galaxyclass}, where there are five main galaxy types, G1 to G5.  \nSystems best fitted by single S\\'ersic models (plus a nuclear point source if \npresent) represent galaxies of type G1 and G2. Systems best fitted by two or \nthree S\\'ersic profiles (plus a nuclear point source if present) represent \ngalaxies of type G3 to G5.\n\n\\begin{enumerate}\n\\item\nG1: Photometric disk with $n\\leq n_{\\rm disk\\_max}$ (plus a nuclear point \nsource if present).\n\n\\item\nG2: Photometric elliptical with $n > n_{\\rm disk\\_max}$ (plus a nuclear point \nsource if present).\n\n\\item\nG3: Unbarred S0 or spiral having an outer disk with $n\\leq n_{\\rm disk\\_max}$ \nand an inner photometric bulge of any $n$ (plus a nuclear point source if \npresent).\n\n\\item\nG4: Barred S0 or spiral having an outer disk with $n\\leq n_{\\rm disk\\_max}$, \na bar, and an inner photometric bulge of any $n$ (plus a nuclear point \nsource if present).\n\n\\item\nG5: Photometric elliptical having an outer component with \n$n > n_{\\rm disk\\_max}$ and an inner component of any $n$.\n\\end{enumerate}\n\nThis galaxy classification scheme has multiple advantages.  Firstly, it \nallows us to identify low-$n$ disk-dominated structures within galaxies, both \non large scales and in the \ncentral regions, in the form of outer disks with $n\\leq n_{\\rm disk\\_max}$ \nin spirals and S0s, photometric bulges with $n\\leq n_{\\rm disk\\_max}$ in \nspirals and S0s (representing disky pseudobulges), \nand inner disks within ellipticals represented by a component C1 having \n$n\\leq n_{\\rm disk\\_max}$. Furthermore, it allows a census of galaxy \ncomponents with $n > n_{\\rm disk\\_max}$ more akin to \nclassical bulges\/ellipticals. Our scheme does not allow for\nlow-$n$ dynamically hot components. As discussed in \nSection~\\ref{approach}, this is not a problem because in our sample \nsuch structures are not expected to be present in large numbers.\n\nTable~\\ref{tabdecomp} lists \nthe distribution of best-fit models for the sample of galaxies with stellar\n mass $M_\\star \\geq 10^9$~$M_\\odot$, and the breakdown of galaxies into classes \nG1 to G5. Table~\\ref{multicptab} lists the structural parameters from the best\nsingle or multi-component model.  In summary, we fit 6, 38, and 25 galaxies \nwith 1, 2, and 3 S\\'ersic profiles, respectively. Our best-fit models have \nreduced $\\chi^2$ of order one. In terms of galaxy types G1 \nto G5, we assign 1, 5, 24, 25, and 14 objects to classes  G1, G2, G3, G4, and \nG5, respectively. The number of Stage 3 fits implies the bar fraction among \ngalaxies with an extended outer disk is $50.0 \\pm 7.1\\%$, and this is\nconsistent with the bar fraction in Coma derived by Marinova et al. (2012). \n\n\\section{Empirical Results on Galaxy Structure}\\label{results}\n\n\\subsection{Galaxy Types and Morphology-Density Relation in the Center of Coma}\\label{mdr}\n\nWe next map classes G1 to G5 to more familiar Hubble types, namely cD, \nphotometric E, S0, and spiral. The Hubble types assigned here depend \nonly on the morphology classes (G1 to G5) associated with structural \ndecomposition; they are independent of the morphological types from the \nTrentham et al. (in prep.) catalog discussed in Section~\\ref{data0}. The \nresults are shown in Table~\\ref{tabdecomp}, and this process is explained in \ndetail below. \n\nThe one object in class G1 (photometric disk) has a single S\\'ersic \nindex~$n \\leq n_{\\rm disk\\_max}$ and a nuclear point source. This object has \nno visible spiral arms, so it is an S0. Objects assigned to class G2 \n(photometric ellipticals) have single S\\'ersic index $n > n_{\\rm disk\\_max}$ \nand include two known central cD galaxies, NGC~4874 and NGC~4889. We label \nthese two sources separately as cD galaxies because they contain a \ndisproportionately large fraction of the stellar mass.\nClasses G3 (unbarred S0, spiral) and G4 (barred S0, spiral) represent S0 or\nspiral disk galaxies with a possible large-scale bar. We label the six \ngalaxies in either class G3 or G4 showing spiral arms in the data or\nresidual images as spirals, while the remaining sources are labeled S0.\nClass G5 objects are identified as photometric ellipticals \nhaving an outer component with $n > n_{\\rm disk\\_max}$ and an inner \ncomponent of any $n$. \n\nConsidering the Hubble types assigned above, we find evidence of a \nstrong absence of spiral galaxies. In the projected central 0.5~Mpc of the \nComa cluster, there are 2 cDs  (NGC~4874 and NGC~4889), spirals are rare, and \nthe morphology breakdown of (E+S0):spirals is (25.3\\%+65.7\\%):9.0\\% by numbers \nand (32.0\\%+62.2\\%):5.8\\% by stellar mass. Note that our ratio of \nE-to-S0 galaxies is lower than found elsewhere for Coma (e.g., \nGavazzi et al. 2003) and for other clusters (e.g., Dressler 1980; \nFasano et al. 2000; Poggianti et al. 2009), where it is $\\sim1-2$. This is \ndriven by the effect of cosmic variance on our sample \n(Appendix~\\ref{cosmicvar}). Also, the total stellar mass cited here does not \ninclude the cDs as their stellar mass is quite uncertain (see \nSection~\\ref{datas2}).\n\nIn contrast to the central parts of Coma, lower-density environments are \ntypically dominated by spirals. This is quantitatively illustrated by  \nTable~\\ref{tabmdr}, \nwhich compares the results in Coma with the lower-density Virgo cluster and the \nfield. We note that Virgo has significantly lower projected galaxy number \ndensities and halo mass (Binggeli et al. 1987) than the center of Coma.\nMcDonald et al. (2009) study a sample of 286  Virgo cluster member \ngalaxies that is complete down to $B_T=16$ (Vega mag). At stellar mass \n$M_\\star \\geq 10^9$~$M_\\odot$, if M87 is counted as a giant elliptical,  the \n(E+S0):spirals breakdown is (34.1\\%+31.6\\%):34.8\\%  by numbers and  \n(59.2\\%+19.3\\%):21.4\\% by stellar mass. There is evidence  \n(Mihos et al. 2005, 2009; Kormendy et al. 2009) that M87 has a cD halo, and \nafter excluding M87, the (E+S0):spirals breakdown  changes slightly to \n(33.5\\%+31.6\\%):34.8\\%  by numbers and (57.2\\%+20.3\\%):22.5\\% by stellar mass. \nIn the field, the  (E+S0):spiral morphology breakdown is  $\\sim$~20\\%:80\\% by \nnumber for bright galaxies (Dressler 1980).\n\n\\subsection{What Fraction of Total Galactic Stellar Mass is in Disk-Dominated Structures \nVersus Classical Bulges\/Ellipticals?}\\label{census}\nHere and in Section~\\ref{discussmass}, we discuss the stellar mass breakdown \namong galaxy components within each galaxy type. Our results are summarized \nin Tables~\\ref{masstable2} and \\ref{masstable}. \n\nRecall that in Section~\\ref{datas2}, the total stellar masses were \ncomputed through applying calibrations of $M\/L$ to the $HST$ F475W and F814W \nphotometry. To calculate the stellar mass in galaxy substructures we assume a \nconstant $M\/L$ ratio and simply multiply the F814W light ratio of each \ncomponent by the total galaxy stellar mass. \nA more rigorous approach is to also perform the decompositions\nin the F475W band and to fold the colors of galaxy substructures into\nthe calculation. In Appendix~\\ref{colorgrad}, we\nconsider the effect of galaxy color gradients for a subset of galaxies;\nthe effect of the color gradients on the stellar mass fractions is small \n($\\sim5\\%$) and does not impact our conclusions.\n\nTable~\\ref{masstable2} summarizes our attempt at providing a census of \nthe stellar mass among disk-dominated components and classical \nbulges\/ellipticals, in the projected central 0.5~Mpc\nof Coma, excluding the two cDs. We highlight the main results below.\n\n\\begin{enumerate}\n\n\\item{\\it Stellar mass in low-$n$ flattened disk-dominated structures ($43\\%$):}\\\\\nThe total stellar mass in small and large-scale disk-dominated components is $\\sim36.0\\%$. Bars are disk-dominated components in\nthe sense that they are flattened non-axisymmetric components.  \nBar proportions typically range from 2.5:1 to 5:1 in their equatorial \nplane (Binney \\& Tremaine 1987). \nThe stellar mass percentage in bars is 6.8\\%. \nThus, the total fraction mass in disk-dominated components is $43\\%$.\\\\\n\n\\item{\\it Stellar mass in high-$n$ classical bulges\/ellipticals ($57\\%$):}\\\\\nThe remaining stellar mass is in components with $n> n_{\\rm disk\\_max}$.\nThese components include the outer components of photometric ellipticals,\nthe central components with $n> n_{\\rm disk\\_max}$ in photometric\nellipticals, and the bulges of S0s and spirals with $n> n_{\\rm disk\\_max}$.\nThe percent stellar mass in these systems is 57\\%.  \\\\\n\n\\item{\\it Environmental dependence of disk-dominated structures :}\\\\\nFinally, we discuss how $f_{\\rm disk\\_dominated}$, the fraction of galactic stellar \nmass in disk-dominated structures, varies with environment.\nFor the\nlower density field-like environments studied by Weinzirl et al.\n(2009), this fraction  $f_{\\rm disk\\_dominated}$  is $\\sim89.6\\%$  \nfor galaxies with $M_\\star  \\geq 10^{10}$~$M_\\odot$. Applying the same\nmass cut in Coma, the fraction  $f_{\\rm disk\\_dominated}$ \nis $\\sim40.1\\%$, which is lower than in the field\nas expected.\n\nDue to the effect of cosmic variance on our sample \n(Appendix~\\ref{cosmicvar}), our measurement of disk-dominated stellar mass \nis larger by an estimated factor of 1.27, compared to what would be \nobtained from an unbiased sample. This is estimated by weighting the \nfraction of hot and cold stellar mass in elliptical, S0, and spiral \ngalaxies (Table~\\ref{masstable}) with the morphology-density\ndistribution from GOLD Mine for the projected central 0.5~Mpc of Coma.\n\nWe also note here the results for the Virgo cluster, in which Kormendy\net al. (2009) find that in galaxies with \n$M_\\star \\gtrsim 5\\times 10^9$~$M_\\odot$,\nmore than 2\/3 of the stellar mass is in classical\nbulges\/ellipticals, implying that $f_{\\rm disk\\_dominated}$\nis less than 1\/3.  It may seem surprising that our value\nof  $f_{\\rm disk\\_dominated}$ in Coma is higher than the value of 1\/3 for Virgo.\nHowever, we believe this apparent discrepancy is due to the fact that\nthe Virgo study includes the giant elliptical galaxy M87, which is\nmarginally classified as a cD (Kormendy et al. 2009), while our study excludes\nthe two cDs in the central part of Coma.  If we include these 2 cDs\nand adopt a conservative lower limit for their stellar mass,\nthen the fraction $f_{\\rm disk\\_dominated}$  of stellar mass in the low-$n$\ncomponent would be less than 27\\%,  since the cDs add their mass\nto high-$n$ stellar components (see Appendix~\\ref{scD}).\n\n\\end{enumerate}\n\n\\subsection{What Fraction of Stellar Mass within S0, E, Spirals is in \nDisk-Dominated Structures versus Classical Bulges\/Ellipticals?}\\label{discussmass}\n\nWe now discuss how the stellar mass is distributed among E, S0, and spiral \nHubble types in the projected central 0.5~Mpc of Coma. As above, fractional \nstellar masses are reported without including the cD galaxies. \n\n\\begin{enumerate}\n\\item {\\it Mass distribution among high-$n$ classical bulges\/ellipticals versus low-$n$ disky pseudobulges in Coma S0s and spirals:}\\\\\nBulges account for $\\sim30.5\\%$ of the stellar mass across E, S0, and spiral \ngalaxies. The ratio $R$ of stellar mass in high-$n$ ($n\\gtrsim1.7$) classical\nbulges to low-$n$  ($n\\lesssim1.7$) disky pseudobulges is $28.3\\%\/2.2\\%$ or \n12.9.\\\\\n\n\\item {\\it Mass distributions among bulges in Coma S0s versus S0s in lower-density environments:}\\\\\nWe next compare the bulges of Coma S0s versus S0s in lower-density\nenvironments (LDEs). The results are summarized in Table~\\ref{tabS0}.\nWe base this comparison on the results of Laurikainen et al. (2010), who\nderive structural parameters from $2D$ multi-component decompositions of \n117 S0s in LDEs that include a mix of field and Virgo environments. \nFor S0s in these LDEs with $M_\\star \\geq 7.5\\times 10^9$ $M_\\odot$,\nthe ratio $R$ of stellar mass  in high-$n$ ($n\\gtrsim1.7$) \nclassical bulges to low-$n$  ($n\\lesssim1.7$) disky pseudobulges is\n30.6\\%\/4.7\\% or 6.5, while it is\n41.7\\%\/2.4\\% or 17.4 in the projected central 0.5~Mpc of Coma.\nNote the difference in mass stored in high-$n$ and low-$n$ bulges is not \ndue to a greater frequency of high-$n$ bulges, which\nis similar at this mass range.\\\\\n\n\\item {\\it Mass distribution in outer and inner components of photometric ellipticals in Coma:}\\\\\nBy definition in Section~\\ref{mdr}, photometric ellipticals have no outer disk.\nThe outer components of these ellipticals have S\\'ersic $n$ from 1.72 to 6.95, \nwith a median value of 2.1. The total fractional stellar mass of the outer \nstructures in ellipticals relative to our sample (minus the cDs) is \n$\\sim25.9\\%$. Photometric ellipticals may contain an inner component of any \nS\\'ersic $n$, and we find a range in $n$ of 0.31 to 5.88 in S\\'ersic index, \nwith a median of 1.0. Inner components with $n\\leq n_{\\rm disk\\_max}$ represent \ncompact inner disks analogous to the disky pseudobulges in S0s and spirals; \nmost of these inner components (9\/14 or $64.3 \\pm 12.8\\%$) qualify as inner \ndisks.\n\\end{enumerate}\n\n\\subsection{Scaling Relations for Outer Disks and Bulges}\\label{discussScaling}\nHere, we explore scaling relations for the bulges and outer disks \nin the projected central 0.5~Mpc of the Coma cluster. We assess \nhow these structures compare with outer disks and bulges in LDEs, such as \nfield, groups, and even low-density clusters similar to the Virgo cluster, \nwhere environmental processes and merger histories are likely to be different. \n\nFor this comparison, we use the results of Gadotti (2009), who studies face-on\n($b\/a\\geq0.9$) galaxies from the SDSS Data Release~2 in a volume limited sample \nat $0.02\\leq z \\leq 0.07$. He derives galaxy structure from 2D decompositions\nof multi-band $gri$ images that account for bulge, disk, and bar components. \nThe Coma sample S0s\/spirals have stellar mass \n$10^9 \\leq M_\\star \\leq 6\\times 10^{10}$~$M_\\odot$, and for this comparison we \nconsider only galaxies with stellar mass \n$5\\times 10^9 \\leq M_\\star \\leq 6\\times 10^{10}$~$M_\\odot$. We proceed with the\ncaveat that the sample from Gadotti (2009) is incomplete in mass for\n$M_\\star \\lesssim 5\\times10^{10}$~$M_\\odot$.\n\nFigure~\\ref{scaling16} compares properties of large-scale disks (size, \nluminosity) with galaxy $M_\\star$. Figure~\\ref{scaling16}a explores the \n\\textit{projected} half-light radius in the $i$-band ($r_\\mathrm{e}$) of outer \ndisks along the major axis at a given galaxy $M_\\star$ in Coma versus LDEs. It \nshows that at a given galaxy $M_\\star$, the average disk $r_\\mathrm{e}$ is \nsmaller in the projected central 0.5~Mpc of Coma compared with LDEs by \n$\\sim30-82\\%$. \nWhile the scatter in disk $r_\\mathrm{e}$ is large, the separation \nbetween the two mean values in each mass bin is larger than the sum of the \nerrors. The suggestion that outer disks in Coma are more compact is\nconsistent with the results of previous analyses of disk \nstructure in Coma (Guti{\\'e}rrez et al. 2004; Aguerri et al. 2004).\nFigure~\\ref{scaling16}b makes a similar comparison for the outer disk\nluminosity between Coma and LDEs. We use here the ACS F814W photometry for\nComa and the SDSS $i$-band photometry from Gadotti (2009). At a given stellar\nmass, the average outer disk luminosities are fainter by $\\sim40-70\\%$,\nexcluding the lowest mass bin.\n\nWe next consider the effect of $M\/L$ to test if the difference in outer disk \nluminosity could imply a a difference in outer disk mass.\nFor Coma, we show the {\\it galaxy-wide} $(M\/L)_i$ ratio \nestimated, while for the Gadotti (2009) sample we show $i$-band $M\/L$ ratios in the outer disks, $(M\/L)_{d,i}$.\nFigure~\\ref{scaling16}c compares the resulting \n$(M\/L)$ values against galaxy $M_\\star$. The average $(M\/L)_i$ in Coma is larger than \nthe average $(M\/L)_{d,i}$ in LDEs by a factor of $\\sim1.3-2$ at a \ngiven galaxy $M_\\star$, excluding \nthe lowest mass bin. This difference in $(M\/L)_i$ accounts for $\\sim48-80\\%$ \nof the average offset in disk luminosity. This suggests {\\it some} of the difference in \nouter disk luminosity might be driven by a real difference in outer disk mass.\nCappellari (2013), in comparison, concludes that spirals in Coma transformed\ninto fast rotating early-type galaxies while decreasing in\n\\textit{global} half-light radius with little mass variation.\n\nFigure~\\ref{scaling17} examines how bulge size ($r_\\mathrm{e}$), bulge \nluminosity, bulge S\\'ersic index,  and bulge-to-disk {\\it light ratio} ($B\/D$) \nscale with galaxy \n$M_\\star$. Figures~\\ref{scaling17}a-c show that bulge size, bulge luminosity, \nand bulge S\\'ersic index as a function of galaxy $M_\\star$ are not\nsystematically offset in Coma versus LDEs. \nFigure~\\ref{scaling17}d shows there is a great scatter in $B\/D$ versus \ngalaxy $M_\\star$. \n\nFigure~\\ref{scaling18}a shows $B\/D$ versus bulge S\\'ersic index. At a \ngiven bulge S\\'ersic index, galaxies in Coma show a systematically higher \naverage $B\/D$ ratio than galaxies in LDEs. A linear regression fit reveals \na clear offset in $B\/D$ for a given bulge index. Figure~\\ref{scaling18}b \nindicates that at a given bulge S\\'ersic index the bulge luminosities in Coma \nand LDEs are very consistent. Figure~\\ref{scaling18}c, on the other hand, shows \na clear offset in disk luminosity ($\\sim0.6$ mag),\nindicating that differences in $B\/D$ are due, at least in part, to outer disk \nsize\/luminosity.\n\nFrom this investigation, we have learned of a reduction in the average \nsizes and luminosities in the outer disks of Coma galaxies that may translate \ninto a lower mean outer disk stellar mass. This may be explained in part by \ncluster environmental effects. We consider this point further in \nSection~\\ref{discuss3}.\n\n\\subsection{Environmental Processes in Coma}\\label{discuss3}\n\nMany studies provide evidence for the action of environmental processes in Coma.\nThe predominantly intermediate or old stellar populations in the center\nof the cluster (e.g., Poggianti et al. 2001; Trager et al. 2008; Edwards \\& Fadda 2011) \nare indirect evidence for the action of starvation.\nFurthermore, the properties of Coma S0s display radial cluster trends that \nfavor formation processes that are environment-mediated \n(Rawle et al. 2013, Head et al. 2014). Several examples of \nram-pressure stripping have been directly observed in Coma (Yagi et al. 2007, \n2010; Yoshida et al. 2008; Smith et al. 2010; Fossati et al. 2012). \nThere is also much evidence for the violent effects of tidal forces.\nThe presence of a diffuse intra-cluster medium around Coma central galaxies \nNGC~4874 and NGC~4889 has long been discussed (Kormendy \\& Bahcall 1974; \nMelnick et al. 1977; Thuan \\& Kormendy 1977; Bernstein et al. 1995; \nAdami et al. 2005; Arnaboldi 2011). At the cluster center, the intra-cluster \nlight represents up to $20\\%$ of the cluster galaxy luminosity (Adami et al. \n2005).  This central intra-cluster light is not uniform given the presence of \nplumes and tidal tails (Gregg \\& West 1998; Adami et al. 2005), and debris \nfields are also found further outside the cluster \ncenter (Gregg \\& West 1998; Trentham \\& Mobasher 1998).\n\nBelow, we comment on how our results add to this picture.\n\n\\begin{enumerate}\n\\item {\\it  Reduced Growth and Truncations of Outer Disks in  Coma S0s\/spirals:}\\\\\nIn Section~\\ref{discussScaling}, we found that at a given galaxy stellar mass, \nthe average half-light radius ($r_\\mathrm{e}$) of the outer disk in S0s\/spirals is $\\sim30-82\\%$ smaller, and the average disk $i$-band luminosity is \n$\\sim40-70\\%$ fainter in Coma than in  lower-density environments \n(Figure~\\ref{scaling16}).\nThese observations may be explained in part by cluster environmental effects\n(e,g., strangulation, ram-pressure stripping, tidal stripping) that\nsuppress the growth of large-scale disks.\nHot gas stripping (strangulation) can \nplausibly suppress disk growth by limiting the \namount of gas that can cool and become part of the outer disk. Tidal stripping \nvia galaxy harassment is predicted (e.g., Moore et al. 1999) to be particularly\nefficient at removing mass from extended disks. Ram-pressure stripping is\nmost effective at removing HI gas in the outskirts of a large scale-disk.\nThe evidence (Yagi et al. 2007, 2010;\nYoshida et al. 2008; Fossati et al. 2012) suggests ram-pressure \nstripping happens quickly, and if so it should be\neffective at preventing the growth of large-scale disks after the host galaxy\nenters the cluster.\\\\\n\n\\item {\\it Low S\\'ersic index in S0\/spiral outer disks:}\\\\\nFigure~\\ref{diskn} demonstrates the majority \nof outer disks have low S\\'ersic index ($66.0\\pm8.2\\%$ with $n<1$ and \n$18.0\\pm12.8\\%$ with $n<0.5$). \nThis effect is not artificially driven by bars because\nthe low $n<1$ disks  include barred and unbarred galaxies\nto similar proportions, and additionally, the disks are fitted\nseparately from the bars in our work.\nSimilar examples have been found\nin Virgo. Kormendy \\& Bender (2012) find several examples of Gaussian \n($n\\sim0.5$) disks among both barred and unbarred galaxies, which commonly\noccur in barred galaxies (e.g., Kormendy \\& Kennicutt 2004). \nGaussian-like disks among\nunbarred galaxies are much more surprising (Kormendy \\& Bender 2012).\nFigure~\\ref{diskn} shows that the large fraction of $n<1$ outer disks in Coma \nis not driven by barred galaxies alone.\nIt is not easy to compare the fraction of low $n<1$ disks in\nComa versus LDEs because most work to date in LDEs\n(e.g., Allen et al. 2006; Laurikainen 2007, 2010; Weinzirl et al. 2009)  fit the outer disk with a \nfixed $n=1$ exponential profile. \n\nEnvironmental processes could be creating the Gaussian-like disks. \nKormendy \\& Bender (2012) have suggested this and invoked dynamical heating. We \ncould be seeing a stronger and\/or different manifestation in Coma.\nRam-pressure stripping and tidal stripping can plausibly reduce the \nS\\'ersic $n$ by cutting off the outskirts of the outer stellar\/gaseous \ndisk. \n\\\\\n\n\\item {\\it Bulge-to-disk ratio ($B\/D$):}\\\\\nThe mean bulge S\\'ersic index rises with mean $B\/D$ light ratio\nin both the central part of Coma and LDEs, consistent with the idea that\nthe development of high $B\/D$ ratio in galaxies is usually associated\nwith processes, such as major mergers, which naturally results in a\nhigh $n$. Such a correlation was also found previously in field spirals\n(e.g., Andredakis et al. 1995; Weinzirl et al. 2009).\n\nWe also find that at a given bulge index, the  $B\/D$ light\nratio is higher for Coma. This environmental effect appears to be \ndue, at least in part, to \nthe fact that at a given bulge $n$, the bulge luminosity is similar in\nComa and LDEs, but the outer disks have lower luminosity by a factor\nof a few in Coma  (Figure~\\ref{scaling18}).  \nThis reduced disk growth is likely due to cluster environmental effects\nsuppressing the growth of large-scale outer disks.\nThis conclusion for Coma nicely parallel studies of \nram-pressure stripping \n(Cayette et al. 1990, 1994; Kenney et al. 2004, 2008; Chung et al. 2007, 2009)\nand dynamical heating (Kormendy \\& Bender 2012) in the less extreme Virgo cluster.\n\n\\end{enumerate}\n\n\\section{Comparison of Empirical Results With Theoretical Predictions}\\label{theory}\n\\subsection{Overview of the Models}\\label{theoryoverview}\nIn this section, we compare our empirical results for Coma with simulations of \nclusters. \nThe simulated clusters are derived from a semi-analytical model (SAM) based on \nNeistein \\& Weinmann (2010). The SAM is able to produce reasonable matches \n(Wang, Weinmann, \\& Neistein 2012) to the galaxy stellar mass function \ndetermined by (Li \\& White 2009) for massive \n$M_\\star \\gtrsim 5\\times 10^8$~$M_\\odot$ \ngalaxies at low redshift ($0.001<z<0.5$) over all environments \n(including Virgo and Coma) probed in the northern hemisphere component of SDSS\nData Release 7. A brief summary of the SAM formalism is given below. Interested readers should see Neistein \\& Weinmann (2010) and Wang, Weinmann, \\& \nNeistein (2012) for additional details.\n\nThe SAM uses merger trees extracted from the Millennium $N$-body simulation \n(Springel et al. 2005). \nGalaxies are modeled as vectors of stellar mass, cold gas, and hot gas.\nBaryonic physics are handled with semi-analytic prescriptions. \nIn between merger events, the efficiencies of quiescent evolutionary processes, such as cold and hot gas accretion, gas \ncooling, star formation, and supernovae feedback, are modeled as functions \nof halo mass and redshift only. The star formation rate \nis proportional to the amount of cold gas, and the star formation\nefficiency is a function of halo mass and redshift.\nIn the model, the baryonic mass (i.e., the sum of stellar and cold gas mass)\nis used to define major ($M_1\/M_2 \\geq 1\/4$) and minor ($1\/10 < M_1\/M_2 < 1\/4$) mergers. As we will\ndiscuss in Section~\\ref{massratio}, the results are highly sensitive to whether\nthe stellar mass ratio or  baryonic mass ratio are used.\n\nImmediately after a major merger, the remnant's stellar $B\/T$ ratio is always \none. This is because the model assumes any existing stellar disks are \ndestroyed, and all stars undergo violent relaxation to form a bulge\/elliptical.\nAfter a major merger, an extended stellar disk is rebuilt via gas cooling, \ncausing $B\/T$ to fall. Any further major mergers will reset $B\/T$ to one. \nDuring a minor merger, the stellar component of the satellite\nof baryonic mass $M_1$ is added to the bulge.  \n\nDuring a major\/minor merger, some fraction of cold gas is converted to stars\nin a short induced starburst $\\sim10$~Myr in duration. The amount of \nmerger-induced star formation depends explicitly on the cold gas mass. \nStars formed in \n{\\it major} merger-induced starbursts are considered part of the bulge\n(see Section~\\ref{datavmodel}). \nThis is a reasonable assumption given that a) the\nbursts of star formation are much shorter than the overall duration of the\nmergers and b) all existing stars {\\it from both progenitors} are violently\nrelaxed during final coalescence. \nIt seems less likely the starburst stars induced in minor mergers should be\nviolently relaxed since minor mergers are not very efficient at violently\nrelaxing stars in the host galaxy. We consider this issue further in\nSection~\\ref{datavmodel}.\nTherefore, in the model used in this paper, the bulge stellar mass traces \nthe mass assembled via major and minor mergers. Galaxies without bulges have \nhad no resolvable merger history. The model does not build bulges through the \ncoalescence of clumps condensing in violent disk instabilities (Bournaud, \nElmegreen, \\& Elmegreen 2007; Elmegreen et al. 2009).\n\nGalaxy clusters impose additional environmental effects that complicate\nmodeling with SAMs. The SAM used here accounts for stripping of hot gas \n(i.e., strangulation; Larson et al. 1980) by assuming hot gas is stripped \nexponentially with a timescale of 4 Gyr. Other processes like ram-pressure,\nstripping\/disruption of stellar mass (Moore et al. 1996, 1998, 1999; \nGnedin 2003), dynamical friction heating by satellite (El-Zant et al. 2004), \nand gravitational heating by infalling substructures (Khochfar \\& Ostriker \n2008) are neglected.\nIt is not clear how much the inclusion of ram-pressure stripping in the SAM\nwould affect our results. While hydrodynamical simulations clearly demonstrate\nthe strong influence of ram-pressure stripping on gas mass, galaxy morphology, \nand star formation (e.g., Quilis et al. 2000; Tonnesen \\& Bryan 2008, 2009, \n2010), some SAMs (e.g., Okamoto \\& Nagashima 2003; Lanzoni et al. 2005) \nsuggest that accounting for ram-pressure stripping has only a small affect. \nTidal stripping creates a \npopulation of intra-cluster stars that can contribute between $10-40\\%$ of the\noptical light in rich clusters (e.g., Bernstein et al. 1995; \nFeldmeier et al. 2004; Zibetti et al. 2005). \nThe inclusion of tidal stripping in SAMs is important for addressing\na wide range of systematic effects (e.g., Bullock et al. 2001; \nWeinmann et al. 2006; Henriques et al. 2008, 2010), but tidal stripping is not\npresent in this SAM.\n\n\\subsection{The Mass Function and Cumulative Number Density in Coma}\\label{comavagc}\n\nIn order to compare galaxies in the simulations with those in the center of \nComa, we first need to identify model clusters that best represent Coma. \nWe do this based on the global properties of Coma, namely the halo mass and \nsize, galaxy stellar mass function, and radial profile of cumulative projected \ngalaxy number density. As the ACS coverage of Coma encompasses a fraction \n(19.7\\%) of the projected central 0.5~Mpc, we calculate these properties in \nComa with Data Release 7 (DR7) of the NYU Value-Added Galaxy Catalog \n(NYU-VAGC, Blanton et al. 2005), which  provides full spatial coverage of Coma. \nNYU-VAGC DR7 is based on SDSS DR7 data (Abazajian et al. 2009) and provides \ncatalogs generated from an independent, and improved, reduction of the public \ndata (Padmanabhan et al. 2008).\n\nWe select Coma cluster member galaxies from NYU-VAGC \nassuming Coma cluster galaxies have radial velocity in the range \nv$_{\\rm min}$ = 4620 km\/s to v$_{\\rm max}$ = 10,000 km\/s, which is the range \nin radial velocity among spectroscopically confirmed members in the ACS \nsurvey. We also adopt the Coma virial radius and virial mass to be \n2.9$h^{-1}_{70}$~Mpc and $1.4\\times10^{15}h^{-1}_{70}$~$M_\\odot$, respectively,\nmeasured by Lokas \\& Mamon (2003) with a 30\\% accuracy, where\n$h^{-1}_{70}=H_0\/70$.\nFor our adopted $H_0$ of 73, we scale these numbers by $(73\/70)^{-1}$, so that\nthe virial radius and virial mass are \n2.8$h^{-1}_{73}$~Mpc and $1.3\\times10^{15}h^{-1}_{73}$~$M_\\odot$, respectively.\nWe select galaxies with the following criteria:\n\\begin{enumerate}\n\\item Radial velocity in range 4620 to 10,000 km\/s.\n\\item Projected radius, ${\\rm{R_p}}$ from the cluster center (i.e., NGC~4874) \nless than the virial radius. \n\\item Brightness exceeding the SDSS spectroscopic completeness limit of \n$r=17.7$ mag, or $M_r\\leq -17.3$ mag at the 100~Mpc distance of Coma. This \ncorresponds to a stellar mass of $1.3\\times 10^9$~$M_\\odot$ assuming a $g-r$ \ncolor of 0.67, which is the average among Coma galaxies in the NYU-VAGC\nselected in this manner.\n\\end{enumerate}\n\nPanel (b) of Figure~\\ref{virp} shows the resulting projected galaxy\ndensity profile for this set of Coma galaxies.  \n\nWe next calculate the global galaxy stellar mass function within the virial \nradius. Figure~\\ref{virp}c shows the result. This mass function includes\nnormal massive galaxies  (E, S0, spiral) as well as the two cDs  \n(NGC~4874 and NGC~4889). As described in Section~\\ref{datas2},\nwe derive the stellar mass by applying Equations 5 and 6 to SDSS $gr$ \nphotometry.\n\nUsing the cD galaxy stellar masses as lower limits\nat the high mass end of the galaxy stellar mass function in Figure~\\ref{virp}c, \nwe measure a slope \n$\\alpha = -1.16$ and characteristic mass $M^* = 1.25 \\times 10^{11}~M_\\odot$ \nfor the global galaxy stellar mass function of Coma inside the cluster virial \nradius.\n\n\\subsection{Global Properties of Model Clusters Versus Coma}\\label{globalprop}\nNext, we compare the above global properties of the Coma cluster with the \nsimulated clusters  in the theoretical model in order to identify the model \nclusters that best represent Coma. We consider all 160 Friend-of-Friend (FOF, \nDavis et al. 1985) groups in the Millennium simulation having a halo mass in \nthe range $5\\times10^{14} - 10^{16}$~$M_\\odot$. We refer to the most massive \nhalo, and its gravitationally bound subhaloes, in each FOF group as a \n`cluster'. \n\nTo find potential matching clusters, we identify massive \n($M_\\star \\geq 10^9$~$M_\\odot$) member galaxies in each \ncluster in a way that is consistent with the selection of Coma member galaxies \nin Section~\\ref{comavagc}: \n\n\\begin{enumerate}\n\\item  Radial velocity matching the range in line-of-sight velocities in the \nxy, xz, yz projections of the cluster.\n\\item Projected radius, ${\\rm{R_p}}$, from the cluster center less than the \ncluster virial radius.\n\\item Luminosity brighter than the SDSS spectroscopic completeness limit of \n$M_r\\leq -17.3$ mag at the 100~Mpc distance of Coma. \n\\end{enumerate}\n\nTo gauge how well the simulated clusters compare with Coma in terms of\nglobal properties, we examine the match in cumulative number density, mass \nfunction, and halo parameters (virial mass and radius). \n\nIn Figure~\\ref{virp},  we gauge how the global properties of Coma compare\nwith those of  all 160 cluster simulations.\nFigure~\\ref{virp}a shows the combinations of virial radius and\nhalo masses of the simulated clusters. The Coma halo parameters (virial mass \nand radius) adopted in Section~\\ref{comavagc} are well matched to the largest \nand most massive model clusters.\n\nFigure~\\ref{virp}b shows the radial profile of\ncumulative galaxy  number density.\nThe central galaxy number densities in the simulated clusters span three orders\nof magnitude from $\\sim10^4$ to $\\sim4\\times 10^5$~Mpc$^{-3}$,\noverlapping with the high central density in Coma \n($\\sim3\\times10^4$~Mpc$^{-3}$). The thick dotted line denotes the \ncluster model with the best-matching halo \nparameters from Figure~\\ref{virp}a. This halo model does a \ngood job at matching the galaxy number density profile of Coma at projected\nradius $R_p>0.7$~Mpc, but not at smaller projected radii. In comparison, the\nmodel with the best matching cumulative number density profile, shown as\nthe open circle in Figure~\\ref{virp}a, is smaller by $\\sim60\\%$ in halo mass \nthan Coma. The next nine best matches to cumulative number density also\ndiffer in halo mass by $\\sim30\\%$ or more from the halo mass in Coma, which\nis estimated to be accurate to within 30\\% (Section~\\ref{comavagc}).\n\nFigure~\\ref{virp}c compares the galaxy stellar mass\nfunction  between Coma and the simulated model clusters.\nAll the model clusters produce too many extremely massive\n($M_\\star \\gtrsim 5\\times 10^{11}$~$M_\\odot$) galaxies.\nThese very high-mass galaxies are not devoid of ongoing star \nformation like ellipticals in Coma are (Section~\\ref{discuss3}). Rather,\nthese galaxies have present day SFR of $\\sim10$~$M_\\odot$~yr$^{-1}$.\nFurthermore, the cluster mass functions show slopes that are marginally too \nsteep ($\\alpha \\sim -1.5$ versus $\\alpha = -1.16$) on the low-mass end \n(Section~\\ref{comavagc}).\n\nWe note that when this SAM model was compared with SDSS observations of \ngalaxies averaged over all environments at low redshift (Wang, Weinmann, \\& \nNeistein 2012), the model galaxy stellar mass function shows a similar,  but \nless extreme, discrepancy with the galaxy stellar mass function of Li \\& White \n(2009)  in terms of producing too many of the most massive galaxies. \nFigure~\\ref{virp}c includes the galaxy stellar mass function from Li \\& White \n(2009) as a dashed line for comparison.\n\nIn Figure~\\ref{bestclusters} we make the comparison with three sets of \nmodel clusters (a total of 30 model clusters) containing the 10 best \nmatches to Coma in terms of the cumulative galaxy number density, galaxy \nstellar mass function, and halo parameters.\nMatching to one criterion (e.g., cumulative number density) does not ensure a good match to the other two criteria.\n\nWe are left with the sobering conclusion that the simulations cannot \nproduce a model cluster simultaneously matching multiple global properties \nof Coma, our local benchmark for one of the richest nearby galaxy clusters.\nThe large discrepancy in the  galaxy stellar mass function between the model \nand Coma could be due to a number of factors. The model currently does not \ninclude tidal stripping\/disruption of stars and ram-pressure stripping\n(Section~\\ref{theoryoverview}),\nwhich would reduce the stellar mass of galaxies on all mass scales.\nThe importance of ram pressure stripping is further discussed in \nSection~\\ref{coldgasmass}, where we find that the cold gas fraction in the \nmodel galaxies is much higher in Coma galaxies. \n\n\\subsection{Strong Dependence of Results on Mass Ratio Used to Define Mergers}\\label{massratio}\n\nMerger history and galaxy $B\/T$ are highly dependent on the mass used \n(stellar mass, baryonic mass, halo mass) to define merger mass ratio $M_1\/M_2$.\nFor a single representative cluster model, Figure~\\ref{majordef} highlights \nthe key differences that arise when $M_1\/M_2$ is defined as the ratio of \nstellar mass (Def 1, left column) versus cold gas plus stars (Def 2, right \ncolumn).  This representative cluster was selected because it is the best \nmatching cluster to the cumulative galaxy number density distribution in Coma \n(Figure~\\ref{bestclusters}). The first row of Figure~\\ref{majordef} shows the \ncumulative percentage of galaxies with a major merger since redshift $z$. \nIn the second row of Figure~\\ref{majordef}, the histograms show the percentage \nof galaxies with a last major merger at redshift $z$. \nThe third row shows the percentage of galaxies with a given $B\/T$ value, sorted\nby galaxies with and without a major merger. \nFinally, the last row of Figure~\\ref{majordef} gives the distribution of \npresent-day $B\/T$ versus redshift of the last major merger. \n\nIn the following sections, we consider a model where the merger mass ratio \n$M_1\/M_2$ depends on stellar mass plus cold gas, as this ratio is understood to \nbe the most appropriate definition (Hopkins et al. 2009b). \nTraditionally, observers have tended to  use stellar mass ratios in identifying\nmergers (e.g., Lin et al. 2004; Bell et al. 2006; Jogee et al. 2009;\nRobaina et al. 2010) as stellar masses are readily measured for a large number\nof galaxies. However, with the advent of ALMA, it will be increasingly possible\nto incorporate the cold gas mass for a large number of galaxies.\n\n\\subsection{Cold Gas Mass in Coma Galaxies Versus Model Galaxies}\\label{coldgasmass}\n\nIn the SAM used here,  the cold gas fraction  $f_{\\rm gas}$ (defined as the \nratio of cold gas to the baryonic mass made of cold gas, hot gas, and stars)  \nand  the ratio ($M_{\\rm cold\\_gas}\/M_\\star$) of cold gas to stellar mass are \nboth overly high.  The issue of high cold gas fraction in this model was \nhighlighted and discussed in Wang, Weinmann, \\& Neistein (2012).\nHere, we quantify how far off the model values are compared with what is \nexpected for a rich cluster like Coma.\n\nFigure~\\ref{boselli} illustrates the degree to which the ratio \n($M_{\\rm cold\\_gas}\/M_\\star$) is overestimated by comparing with data from \nBoselli et al. (1997), who measure atomic ($M_{HI}$) and molecular gas \n($M_{H_2}$) masses for Coma cluster member galaxies and non-cluster galaxies.  \nThe top panel shows that the average ratio of cold gas to stellar mass \n($M_{\\rm cold\\_gas}\/M_\\star$) \nranges from $\\sim1-12$ for a representative model cluster. The bottom panel\nshows that the ratio of $M_{HI+H_2}\/M_\\star$ for Coma cluster galaxies from \nBoselli et al. (1997) \nis usually $<0.1$; non-cluster galaxies are more gas rich, but the ratio of \n$M_{HI+H_2}\/M_\\star$ is still $\\ll 1$.  At \n$10^{10} \\lesssim M_\\star \\lesssim 10^{11}$~$M_\\odot$, the model predicts a \ncold gas to stellar mass ratio that is a factor $\\sim25-87$ times \nhigher than the median in Coma cluster galaxies. \n\n\\subsection{Data Versus Model Predictions for Stellar Mass in Dynamically Hot and Cold\nComponents}\\label{datavmodel}\n\nWe next proceed to compare the observed versus model predictions for the \ndistribution of mass in dynamically hot and cold stellar components.\nThe following comparisons are made in the projected central 0.5~Mpc of Coma \nand the model clusters.\n\nWe first start by describing how the model builds bulges and ellipticals.\nIn the model, the total bulge stellar mass  $M_{\\rm \\star,Bulge,model}$ \nconsists of stellar mass accreted  in major and minor mergers, plus\nstellar mass from SF induced in both types of mergers.\n\nNext, we discuss how to compare the model with the data. \nFor our sample of  Coma galaxies (excluding the 2 cD systems)\n with $M_\\star \\geq 10^9$~$M_\\odot$, we compute the ratio R1$_{\\rm data}$\nas the the stellar mass in all components with $n > n_{\\rm disk\\_max}$\nto the sum of galaxy stellar mass.\nThe reasons for not including the cD systems were discussed in \nSection~\\ref{datas2}.\nFrom Section~\\ref{census},  R1$_{\\rm data}$ is 57\\%.\n\nWe next compare this ratio to the corresponding quantity\nin the model. The comparison is not entirely straightforward as the model\ndoes not give a S\\'ersic index. We therefore have to associate components\nin the model to the corresponding high \n$n>n_{\\rm disk\\_max}$ classical bulges\/ellipticals in the data.\nThe most natural step is to assume that the stellar mass built\nduring major mergers is redistributed into such high-$n$ components.\nWe call the result R2$_{\\rm model}$. We find that for \n$M_\\star \\geq 10^9$~$M_\\odot$, R2$_{\\rm model}$ has a wide dispersion: \n$\\sim35-79\\%$ for the 30 model clusters shown in Figure~\\ref{bestclusters}, \nwith a median value of $\\sim66\\%$. The representative cluster discussed in \nSection~\\ref{massratio} and Figures~\\ref{majordef}-\\ref{boselli} \nhas a value of $\\sim72\\%$.\n\nGuidance on the S\\'ersic index of structures formed during minor mergers can be \ngleaned from Hopkins et al. (2009b). In the \ngeneral case of an unequal mass merger, the coalescence of the smaller \nprogenitor (mass $M_1$) with the center of the primary will destroy (i.e., \nviolently relax) the smaller galaxy and also potentially violently relax an \nadditional mass $\\leq M_1$ in the primary.  The stars that are \nviolently relaxed in the minor merger become part of the bulge in the primary \ngalaxy. Thus, we define R3$_{\\rm model}$ to be R2$_{\\rm model}$ plus the \nstellar accretion from minor mergers. For $M_\\star \\geq 10^9$~$M_\\odot$, \nR3$_{\\rm model}$ is only slightly higher than\nR2$_{\\rm model}$ by a few percent. R3$_{\\rm model}$ ranges from $\\sim35-82\\%$,\nwith a median value of $\\sim71\\%$,\nand the representative cluster (Section~\\ref{massratio}, \nFigures~\\ref{majordef}-\\ref{boselli}) has a value of $\\sim71\\%$.\n\nThe comparison of R1$_{\\rm data}$ with R2$_{\\rm model}$  and  R3$_{\\rm model}$ \nis a global comparison of the total stellar mass fraction within high-$n$ \ncomponents {\\it summed over all} the galaxies with \n$M_\\star \\geq 10^9$~$M_\\odot$. Next, we push the data versus model comparison \none step further by doing it in bins of stellar mass, as shown in \nFigure~\\ref{mtcompare}.\n\nThe top panel of Figure~\\ref{mtcompare} plots the mean ratio of stellar mass\nfraction in dynamically hot components ($f_{\\rm \\star,hot}$) as a function\nof total galaxy stellar mass, for data versus model.\nFor each stellar mass bin shown in Figure~\\ref{mtcompare},\nthe value of $f_{\\rm \\star,hot}$ is calculated for {\\it each galaxy}\nas $M_{\\rm \\star,hot}\/M_{\\star}$.\nIn the data, $M_{\\rm \\star,hot}$ is taken as the stellar mass of any\nhigh $n>n_{\\rm disk\\_max}$ component in the galaxy.\nThe model shown here is the best cluster model matched by cumulative galaxy\nnumber density (see Figure~\\ref{bestclusters}, column 1).\nFor this model, two lines are shown: the solid line takes $M_{\\rm \\star,hot}$\nas the stellar mass accreted and formed during major mergers, \nwhile the dotted line also adds in the stellar mass accreted during minor\nmergers. \n\nIn the top panel of Figure~\\ref{mtcompare} there is \nsignificant disagreement between the fractions of $f_{\\rm \\star,hot}$\nfor the Coma data and the model. As shown by the second dotted \nmodel curve in Figure~\\ref{mtcompare}, adding in the stellar mass accreted in \nminor mergers to the model only changes the fraction by a few percent.\nThe values of $f_{\\rm \\star,hot}$ are chiefly representative\nof the contributions from major mergers.\n\nThe bottom panel of Figure~\\ref{mtcompare} plots the analogous mean ratio of \nstellar mass fraction in dynamically cold components \n($f_{\\rm \\star,cold} =M_{\\rm \\star,cold}\/M_{\\star}$)\nas a function of total galaxy stellar mass. In the model, the two lines show \ntwo  different expressions for $M_{\\rm \\star,cold}$.\nFor the solid  line, we  take $M_{\\rm \\star,cold}$ to be the\nmass of the outer disk $M_{\\rm \\star,Outer\\_disk}$, which \nrepresents the difference between the bulge mass ($M_{ \\rm \\star,Bulge,model}$) \nand the total stellar mass.\nOne problem with this approach is that it ignores small-scale  nuclear disks\nformed in the bulge region. We tackle this problem by defining a second\ndotted model line \nthat accounts for stars formed via induced SF during minor mergers.\nIt is clear in the bottom panel of Figure~\\ref{mtcompare} that the model\noverpredicts the mass in disks as a function of galaxy stellar mass. \nNote the contribution to $f_{\\rm \\star,cold}$ from minor merger induced SF is \n$\\lesssim17\\%$ in a stellar mass given bin.\n\nThe main conclusion from Figure~\\ref{mtcompare} is that the best-matching\ncluster model is underpredicting the mean fraction $f_{\\rm \\star,hot}$ of \nstellar mass locked in hot components over a wide range in galaxy stellar mass\n($10^9 \\leq M\\star \\lesssim 8\\times10^{10}$~$M_\\odot$).\nSimilarly this model overpredicts the mean value for $f_{\\rm \\star,cold}$.\nThe effect of cosmic variance on our sample (Section~\\ref{census} and \nAppendix~\\ref{cosmicvar}) means\nour measured $f_{\\rm \\star,hot}$ is lower than the true value \nby an estimated factor of 1.16.\nTherefore, the underprediction of $f_{\\rm \\star,hot}$ in the\nmodel is worse than what we are citing.\nWhile the discussion in this section focused only on a single model cluster,\nthe results and conclusions would be similar if we had analyzed alternate\nsimulated clusters, such as those matched to the cluster galaxy stellar mass\nfunction (see Figure~\\ref{bestclusters}, column 2) or halo parameters\n(see Figure~\\ref{bestclusters}, column 3). \n\nThere could be several explanations as to why the models are underproducing the \nfraction of dynamically hot stellar mass ($f_{\\rm \\star,hot}$) and \noverproducing the fraction of dynamically cold stellar mass \n($f_{\\rm \\star,cold}$). One possibility \nis that the absence of key cluster processes (especially ram-pressure stripping \nand tidal stripping) in the models is leading to the overproduction of the \nmodel galaxy's cold gas reservoir (Section~\\ref{coldgasmass}), compared to a \nreal cluster galaxy, whose outer gas would be removed.  This means that in the \nmodels, SF in gas that would otherwise be removed from the\ngalaxy builds additional dynamically cold stellar mass following\nthe last major merger.  Another possibility is that the models  ignore\nthe production of bulges via the merging of star forming clumps\n(Bournaud, Elmegreen, \\& Elmegreen 2007; Elmegreen et al. 2009).\nIt is still debated whether this mode can efficiently produce\nclassical bulges, but if it does, then its non-inclusion in the models\ncould lead to the underprediction of $f_{\\rm \\star,hot}$.\n\nIn summary, our comparison of empirical results to theoretical\npredictions underscores the need to include in SAMs\nenvironmental processes, such as ram-pressure\nstripping and tidal stripping, which affect the cold gas content of galaxies,\nas well as more comprehensive models of bulge assembly.\nIt is clear that galaxy evolution is a\nfunction of \\textit{both} stellar mass and environment.\n\n\\section{Summary \\& Conclusions}\\label{summary}\n\nWe present a study of the Coma cluster in which we constrain galaxy assembly\nhistory in the projected central 0.5~Mpc  by performing multi-component \nstructural decomposition on  a mass-complete sample of 69 galaxies with \nstellar mass $M_\\star \\geq 10^9$~$M_\\odot$. Some strengths of this study \ninclude the use of superb high-resolution ($0\\farcs1$), F814W images from the \n$\\textit{HST}$\/ACS Treasury Survey of the Coma cluster, and the adoption of a \nmulti-component decomposition strategy where no a priori assumptions are made \nabout the  S\\'ersic index of bulges, bars  or disks. We use structural \ndecomposition to identify the two fundamental kinds of galaxy structure -- \ndynamically cold, disk-dominated components and dynamically hot classical\nbulges\/ellipticals -- by adopting the working assumption that the S\\'ersic \nindex $n$ is a reasonable proxy for tracing different structural components.\nWe define disk-dominated structures as components with a low S\\'ersic index\n$n$ below an empirically determined threshold value $n_{\\rm disk\\_max}\\sim1.7$\n(Section~\\ref{approach}). Galaxies with an outer disk are called spirals or\nS0s. We explore the effect of environment by performing a census of \ndisk-dominated structures versus classical bulges\/ellipticals in Coma.\nWe also compare our empirical results on galaxies in the\ncenter of the Coma cluster  with theoretical predictions \nfrom a semi-analytical model. Our main results are summarized below.\n\n\\begin{enumerate}\n\n\n\n\n\\item {\\it Breakdown of stellar mass in Coma between\nlow-$n$ disk-dominated structures and high-$n$ classical\nbulges\/ellipticals:}\\\\\nWe make the first attempt (Section~\\ref{census} and \nTables~\\ref{masstable2}--\\ref{masstable}) at exploring the distribution of \nstellar mass in Coma in terms of dynamically hot versus dynamically cold \nstellar components. After excluding the 2 cDs because of their uncertain \nstellar masses, we find that in the projected central 0.5 Mpc of the\nComa cluster, galaxies with stellar mass $M_\\star \\geq 10^9$~$M_\\odot$ have\n57\\% of their  cumulative stellar mass  locked up in\nhigh-$n$  ($n \\gtrsim 1.7$)  classical bulges\/ellipticals\nwhile the remaining  43\\% is in the form of\nlow-$n$ ($n\\lesssim1.7$) disk-dominated structures (outer disks,\ninner disks, disky pseudobulges, and bars).\nAccounting for the effect of cosmic variance and color gradients in \ncalculating these stellar mass fractions would not significantly change\nthis census (Appendices~\\ref{cosmicvar}--\\ref{colorgrad}).\\\\\n\n\n\\item {\\it Impact of environment on morphology-density relation:}\\\\\nUsing our structural decomposition to assign galaxies the Hubble types E, S0, \nor spiral, we find evidence of a  strong morphology-density relation.\nIn the projected central 0.5~Mpc of the Coma cluster, spirals are rare, and \nthe morphology breakdown of (E+S0):spirals is (91.0\\%):9.0\\% by numbers and \n(94.2\\%):5.8\\% by stellar mass (Section~\\ref{mdr} and Table~\\ref{tabmdr}).\\\\\n\n\n\\item {\\it Impact of environment on outer disks:}\\\\\nIn the central parts of Coma, the properties of large scale disks are\nlikely indicative of environmental processes that suppress disk growth or \ntruncate disks (Section~\\ref{discuss3}). In particular, \nat a given galaxy stellar mass, outer disks are smaller\nby $\\sim30-82\\%$ and fainter in the $i$-band by $\\sim40-70\\%$\n(Figure~\\ref{scaling16}).\nThe suggestion that outer disks in Coma are more compact is\nconsistent with the results of previous analyses of disk\nstructure in Coma (Guti{\\'e}rrez et al. 2004; Aguerri et al. 2004).\\\\\n\n\\item {\\it Impact of environment on bulges:}\\\\\nThe ratio $R$ of stellar mass in high-$n$ ($n\\gtrsim1.7$) classical bulges to \nlow-$n$ ($n\\lesssim1.7$) disky pseudobulges is 17.3 in Coma. We measure $R$ to\nbe a factor of $\\sim2.2-2.7$ higher in Coma compared with various samples from\nLDEs (Sections~\\ref{census}--\\ref{discussmass}, \nTables~\\ref{masstable2}--\\ref{masstable}).\nWe also find that at a given bulge  S\\'ersic index $n$, the bulge-to-total\nratio $B\/D$, and the $i$-band light ratio are offset to higher values in Coma\ncompared with LDEs. This effect appears to be due, at least in part, to the \nabove-mentioned lower disk luminosity in Coma.\\\\\n\n\n\n\\item {\\it Comparison of data to theoretical predictions:}\\\\\nWe  compare our empirical results on galaxies in the\ncenter of the Coma cluster  with theoretical predictions based on\ncombining the Millennium cosmological simulations of dark matter\n (Springel et al. 2005) with baryonic physics from a semi-analytical model\n(Neistein \\& Weinmann 2010; Wang, Weinmann, \\& Neistein 2012).\n\nIt is striking that no model cluster can simultaneously match the global \nproperties\n(halo mass\/size,  cumulative galaxy number density, galaxy stellar mass \nfunction) of Coma (Figures~\\ref{virp} and \\ref{bestclusters}), and the \ncold gas to stellar mass ratios in the model clusters are at least 25 times \nhigher than is measured in Coma.\n\nAs suggested by Hopkins et al. (2009b), we find galaxy merger history is \nhighly dependent on how the merger mass ratio $M_1\/M_2$ is defined.\nSpecifically, there is a factor of $\\sim5$ difference\nin merger rate when the merger mass ratio is based on the baryonic mass versus \nthe stellar mass (Figure~\\ref{majordef}).\nTraditionally, observers have tended to use stellar mass ratios in identifying\nmergers, but  with the advent of ALMA, it will be increasingly possible  and\nimportant to incorporate the cold gas mass.\n\nFor representative ``best-match'' simulated clusters, we compare the empirical \nand theoretically predicted fraction $f_{\\rm \\star,hot}$ and \n$f_{\\rm \\star,cold}$ of stellar mass locked, respectively, in high-$n$, \ndynamically hot versus low-$n$, dynamically cold stellar components.\nOver a wide range of galaxy stellar mass \n($10^9 \\leq M\\star \\lesssim 8\\times10^{10}$),\nthe model underpredicts the mean fraction $f_{\\rm \\star,hot}$ of stellar\nmass locked in hot components by a factor of $\\gtrsim1.5$. Similarly, the model\noverpredicts the mean value for  $f_{\\rm \\star,cold}$ (Section~\\ref{datavmodel}\nand Figure~\\ref{mtcompare}). \n\nWe suggest this disagreement might be \ndue to two main factors. Firstly, key cluster processes (especially \nram-pressure stripping and tidal stripping), which impact the cold gas content \nand disk-dominated components of galaxies, are absent. Secondly, the models \nignore the production of bulges via the merging of star forming clumps\n(Bournaud, Elmegreen, \\& Elmegreen 2007; Elmegreen et al. 2009).\nThese results underscore the need to implement  in theoretical models\nenvironmental processes, such as ram-pressure stripping and tidal stripping,\nas well as more comprehensive models of bulge assembly. It is clear that\ngalaxy evolution is not a solely a function of stellar mass, but it also depends\non environment.\n\n\n\\end{enumerate}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}