diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhqss" "b/data_all_eng_slimpj/shuffled/split2/finalzzhqss" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhqss" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\\label{sec:introduction}\nThe recent development of deep learning has led to dramatic progress in multiple research fields, and this technique has naturally found applications in autonomous vehicles. The use of deep learning to perform perceptive tasks such as image segmentation has been widely researched in the last few years, and highly efficient neural network architectures are now available for such tasks. More recently, several teams have proposed taking deep learning a step further, by training so-called ``end-to-end'' algorithms to directly output vehicle controls from raw sensor data (see, in particular, the seminal work in \\cite{Bojarski2016}).\n\nAlthough end-to-end driving is highly appealing, as it removes the need to design motion planning and control algorithms by hand, handing the safety of the car occupants to a software operating as a black box seems problematic. A possible workaround to this downside is to use ``forensics'' techniques that can, to a certain extent, help understand the behavior of deep neural networks \\cite{Castelvecchi2016}.\n\nWe choose a different approach consisting in breaking down complexity by training simpler, mono-task neural networks to solve specific problems arising in autonomous driving; we argue that the reduced complexity of individual tasks allows much easier testing and validation.\n\nIn this article, we focus on the problem of controlling a car-like vehicle in highly dynamic situations, for instance to perform evasive maneuvers in face of an obstacle. A particular challenge in such scenarios is the important coupling between longitudinal and lateral dynamics when nearing the vehicle's handling limits, which requires highly detailed models to properly take into account \\cite{Gillespie1997}.\nHowever, precisely modeling this coupling involves complex non-linear relations between state variables, and using the resulting model is usually too costly for real-time applications. For this reason, most references in the field of motion planning mainly focus on simpler models, such as point-mass or kinematic bicycle (single track), which are constrained to avoid highly coupled dynamics \\cite{PolackACC2018}.\nSimilarly, research on automotive control usually treats the longitudinal and lateral dynamics separately in order to simplify the problem \\cite{Khodayari2010}.\n\nAlthough these simplifications can yield good results in standard on-road driving situations, they may be problematic for vehicle safety when driving near its handling limits, for instance at high speed or on slippery roads. To handle such situations, some authors have proposed using Model Predictive Control (MPC) with a simplified, coupled dynamic model~\\cite{Falcone2007a} which is limited to extremely short time horizons (a few dozen milliseconds) to allow real-time computation. Other authors have proposed to model the coupling between longitudinal and lateral motions using the concept of ``friction circle'' \\cite{Kritayakirana2012a},\nwhich allows precisely stabilizing a vehicle in circular drifts~\\cite{Goh2016}. However, the transition towards the stabilized drifting phase -- which is critical in the ability, \\emph{e.g.}\\@\\xspace, to perform evasive maneuvers -- remains problematic with this framework.\n\nIn this article, we propose to use deep neural networks to implicitly model highly coupled vehicular dynamics, and perform low-level control in real-time. In order to do so, we train a deep neural network to output low-level controls (wheels torque and steering angle) corresponding to a given initial vehicle state and target trajectory. Compared to classical MPC frameworks which require integrating dynamic equations on-line, this approach allows to perform this task off-line and use only simple mathematical operations on-line, leading to much faster computations.\n\nSeveral authors have already proposed a divide-and-conquer approach by using machine learning on specific sub-tasks instead of performing end-to-end computations, and in particular on the case of motion planning and control. For instance, reference~\\cite{Drews2017} used a Convolutional Neural Network (CNN) to generate a cost function from input images, which is then used inside an MPC framework for high-speed autonomous driving; however, this approach has the same limitations as model predictive control. Other approaches, such as \\cite{Se-YoungOh2000}, used reinforcement learning to output steering controls for a vehicle, but were limited to low-speed applications. Reference~\\cite{Punjani2015} used a Rectified Linear Unit (ReLU) network model to identify the dynamics of a helicopter in order to predict its future accelerations, but this model has not been used for control. \n\nCloser to our work, reference \\cite{Rivals1994} trained neural networks integrating a priori knowledge of the bicycle model for decoupled longitudinal and lateral control of a vehicle; in \\cite{Chen2017}, authors used supervised learning to generate lateral controls for truck and integrated a control barrier function to ensure the safety of the system. Reference \\cite{Cui2017} coupled a standard control and an adaptive neural network to compensate for unknown perturbations in order to perform trajectory tracking for autonomous underwater vehicle. To the best of our knowledge, deep neural networks have not been used in the literature for the coupled control of wheeled vehicles.\n\nThe rest of this article is organized as follows: Section~\\ref{sec:vehicle_model} presents the vehicle model used to generate the training dataset and to simulate the vehicle dynamics on a test track. Section~\\ref{sec:DL_models} introduces two artificial neural networks architectures used to generate the control signals for a given target trajectory, and describes the training procedure used in this article. Section~\\ref{sec:results} compares the performance of these two networks, using simulation on a challenging test track. A comparison to conventional decoupled controllers is also provided. Finally, Section~\\ref{sec:conclusions} concludes this study.\n\n\\section{THE 9~DoF VEHICLE MODEL}\n\\label{sec:vehicle_model}\nIn this section, we present the 9 Degrees of Freedom (9~DoF) vehicle model which is used both to generate the training and testing dataset, and as a simulation model to evaluate the performance of the deep-learning-based controllers.\n\nThe Degrees of Freedom comprise \n3~DoF for the vehicle's motion in a plane ($V_x, V_y, \\dot{\\psi}$), \n2~DoF for the carbody's rotation ($\\dot{\\theta}, \\dot{\\phi}$)\nand 4~DoF for the rotational speed of each wheel ($\\omega_{fl},\\omega_{fr},\\omega_{rl},\\omega_{rr}$). \nThe model takes into account both the coupling of longitudinal and lateral slips and the load transfer between tires. The control inputs of the model are the torques $T_{\\omega_i}$ applied at each wheel $i$ and the steering angle $\\delta$ of the front wheel. The low-level dynamics of the engine and brakes are not considered here. The notations are given in Table \\ref{tab:notations} and illustrated in Figure~\\ref{fig:carSim}.\n\n\\textit{Remark: }the subscript $i=1..4$ refers respectively to the front left ($fl$), front right ($fr$), rear left ($rl$) and rear right ($rr$) wheels.\n\nSeveral assumptions were made for the model:\n\\begin{itemize}\n\t\\item Only the front wheels are steerable.\n\t\\item The roll and pitch rotations happen around the center of gravity.\n\t\\item The aerodynamic force is applied at the height of the center of gravity. Therefore, it does not involve any moment on the vehicle.\n\t\\item The slope and road-bank angle of the road are not taken into account.\n\\end{itemize}\n\n\\begin{table}[h]\n\t\\vspace{+0.08in}\n\t\\caption{Notations}\n\t\\label{tab:notations}\n\t\\begin{tabular}{p{2.4cm} p{12.8cm}}\n\t\t\\hline\n\t\t\\\\\n\t\t$X$, $Y$ & Position of the vehicle in the ground frame\\\\\n\t\t$\\theta$, $\\phi$, $\\psi$ & Roll, pitch and yaw angles of the carbody \\\\\n\t\t$V_x$, $V_y$ & Longitudinal and lateral speed of the vehicle in its inertial frame \\\\\n\t\t$M_T$ & Total mass of the vehicle\\\\\n\t\t$I_x$, $I_y$, $I_z$ & Inertia of the vehicle around its roll, pitch and yaw axis\\\\\n\t\t$I_{r_i}$ & Inertia of the wheel $i$\\\\\n\t\t$T_{\\omega_i}$ & Total torque applied to the wheel $i$\\\\\n\t\t$F_{xp_i}$, $F_{yp_i}$ & Longitudinal and lateral tire forces generated by the road on the wheel $i$ expressed in the tire frame\\\\\n\t\t$F_{x_i}$, $F_{y_i}$ & Longitudinal and lateral tire forces generated by the road on the wheel $i$ expressed in the vehicle frame $(x,y)$\\\\\n\t\t$F_{z_i}$ & Normal reaction forces on wheel $i$\\\\\n\t\t$l_f$, $l_r$ & Distance between the front (resp. rear) axle and the center of gravity\\\\\n\t\t$l_w$ & Half-track of the vehicle\\\\ \n\t\t$h$ & Height of the center of gravity\\\\\n\t\t$r_{eff}$ & Effective radius of the wheel\\\\\n\t\t$\\omega_i$ & Angular velocity of the wheel $i$ \\\\\n\t\t$V_{xp_i}$ & Longitudinal speed of the center of rotation of wheel $i$ expressed in the tire frame\\\\\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[scale=0.4]{6DOF_vehicle_model.png} \n\t\\caption{Vehicle model and notations.}\n\t\\label{fig:carSim}\n\\end{figure}\n\n\\subsection{Vehicle dynamics}\n\\label{ssec:CarModel}\n\nEquations~(\\ref{eq:veh_dyn_1}-\\ref{eq:veh_dyn_5}) give the expression of the vehicle dynamics:\n\\begin{subequations}\n\t\\label{eq:vehicle_dynamics}\n\t\\begin{eqnarray}\n\t\\label{eq:veh_dyn_1}\n\tM_T \\dot{V}_x & = & M_T\\dot{\\psi} V_y + \\sum_{i=1}^4 F_{x_i} - F_{aero}\\\\\n\t\\label{eq:veh_dyn_2}\n\tM_T\\dot{V}_y & = & - M_T\\dot{\\psi} V_x + \\sum_{i=1}^4 F_{y_i}\\\\\n\t\\label{eq:veh_dyn_3}\n\tI_x\\ddot{\\theta} & = & l_w (F_{z_1}+F_{z_3}-F_{z_2}-F_{z_4}) + h \\sum_{i=1}^4 F_{y_i}\\\\\n\t\\label{eq:veh_dyn_4}\n\tI_y\\ddot{\\phi} & = & l_r (F_{z_3} + F_{z_4}) -l_f (F_{z_1} +F_{z_2}) - h \\sum_{i=1}^4 F_{x_i}\\\\\n\t\\label{eq:veh_dyn_5}\n\tI_z\\ddot{\\psi} & = & l_f (F_{y_1} +F_{y_2}) - l_r (F_{y_3} + F_{y_4}) \\\\ \\nonumber\n\t& + & l_w (F_{x_2}+F_{x_4}-F_{x_1}-F_{x_3})\n\t\\end{eqnarray}\t\n\n\t$F_{x_i}$ and $F_{y_i}$ denote respectively the longitudinal and the lateral tire forces expressed in the vehicle frame; ${F_{aero}=\\frac{1}{2}\\rho_{air}C_x S V_x^2}$ denote the aerodynamic drag forces with $\\rho_{air}$ the mass density of air, $C_x$ the aerodynamic drag coefficient and $S$ the frontal area of the vehicle; $F_{z_i}$ denote the damped mass\/spring forces depending on the suspension travel $\\zeta_i$ due to the roll $\\theta$ and pitch $\\phi$ angles according to Equation~(\\ref{eq:veh_dyn_Fz}). The parameters $k_s$ and $d_s$ are respectively the stiffness and the damping coefficients of the suspensions.\n\t\\begin{eqnarray}\n\t\t\\label{eq:veh_dyn_Fz}\n\t\tF_{z_i} = - k_s \\zeta_i(\\theta, \\phi) - d_s \\dot{\\zeta_i}(\\theta, \\phi)\n\t\\end{eqnarray}\n\t\n\tThe position $(X,Y)$ of the vehicle in the ground frame can then be derived using Equations~(\\ref{eq:veh_dyn_X}) and (\\ref{eq:veh_dyn_Y}).\n\t\\begin{eqnarray}\n\t\\label{eq:veh_dyn_X}\n\t\\dot{X} & = & V_x \\cos\\psi - V_y \\sin\\psi\\\\\n\t\\label{eq:veh_dyn_Y}\n\t\\dot{Y} & = & V_x \\sin\\psi + V_y \\cos\\psi\t\n\t\\end{eqnarray}\n\\end{subequations}\n\t\n\t\\subsection{Wheel dynamics}\n\t\\label{ssec:WheelModel}\n\tThe dynamics of each wheel $i=1..4$ expressed in the pneumatic frame is given by Equation~(\\ref{eq:wheel_dyn}):\n\t\\begin{eqnarray} \n\t\t\\label{eq:wheel_dyn}\n\t\tI_r \\dot{\\omega}_i & = & T_{{\\omega}_i}-r_{eff} F_{xp_i}\n\t\\end{eqnarray}\n\n\\subsection{Tire dynamics}\n\\label{ssec:TireModel}\nThe longitudinal force $F_{xp_i}$ and the lateral force $F_{yp_i}$ applied by the road on each tire $i$ and expressed in the pneumatic frame are functions of the longitudinal slip ratio $\\tau_{x_i}$, the side-slip angle $\\alpha_i$, the normal reaction force $F_{z_i}$ and the road friction coefficient $\\mu$: \n\\begin{subequations}\n\t\\begin{eqnarray}\n\tF_{xp_i} & = & f_x(\\tau_{x_i}, \\alpha_i, F_{z_i}, \\mu)\\\\\n\tF_{yp_i} & = & f_y(\\alpha_i, \\tau_{x_i}, F_{z_i}, \\mu)\n\t\\end{eqnarray}\n\\end{subequations}\n\nThe longitudinal slip ratio of the wheel $i$ is defined as following:\n\\begin{eqnarray}\n\t\\label{eq:slip_ratio}\n\t\\;\\tau_{x_i} = \\left\\{\n\t\\begin{array}{ll}\n\t\\frac{r_{eff} \\omega_i - V_{xpi}}{r_{eff}|\\omega_i|} & $if $ r_{eff}\\omega_i \\geq V_{xp_i} $ (Traction phase)$\\\\\n\t\\frac{r_{eff} \\omega_i - V_{xpi}}{|V_{xpi}|} & $if $ r_{eff}\\omega_i < V_{xpi} $ (Braking phase)$ \n\t\\end{array}\n\t\\right.\n\\end{eqnarray}\n\nThe lateral slip-angle $\\alpha_i$ of tire $i$ is the angle between the direction given by the orientation of the wheel and the direction of the velocity of the wheel (see Figure~\\ref{fig:carSim}):\n\\begin{eqnarray}\n\\small\n\\alpha_f = \\delta - \\atan \\left(\\frac{V_y + l_f \\dot{\\psi}}{V_x \\pm l_w \\dot{\\psi}}\\right) ; \\; \\alpha_r = - \\atan \\left (\\frac{V_y - l_r \\dot{\\psi}}{V_x \\pm l_w \\dot{\\psi}}\\right)\n\\end{eqnarray}\n\nIn order to model the functions $f_x$ and $f_y$, we used the combined slip tire model presented by Pacejka in \\cite{Pacejka2002} (cf. Equations (4.E1) to (4.E67)) which takes into account the interaction between the longitudinal and lateral slips on the force generation. Therefore, the friction circle due to the laws of friction (see Equation (\\ref{eq:friction_circle})) is respected. Finally, the impact of load transfer between tires is also taken into account through $F_z$. \n\\begin{eqnarray}\n\\label{eq:friction_circle}\n||\\vec{F}_{xp}+\\vec{F}_{yp}|| \\leq \\mu ||\\vec{F}_z||\n\\end{eqnarray} \n\nLastly, the relationships between the tire forces expressed in the vehicle frame $F_x$ and $F_y$ and the ones expressed in the pneumatic frame $F_{xp}$ and $F_{yp}$ are given in Equation~(\\ref{eq:frame_F_change}):\n\\begin{subequations}\n\t\\label{eq:frame_F_change}\n\t\\begin{eqnarray}\n\n\t\\small{F_{x_i}} & = & \\small{(F_{xp_i}\\cos\\delta_i-F_{yp_i}\\sin\\delta_i)\\cos\\phi-F_{z_i}\\sin\\phi}\\\\\n\t\\small{F_{y_i}} & = & \\small{(F_{xp_i}\\cos\\delta_i-F_{yp_i}\\sin\\delta_i)\\sin\\theta\\sin\\phi}\\\\ \\nonumber\n\t& + & \\small{(F_{yp_i}\\cos\\delta_i+F_{xp_i}\\sin\\delta_i)\\cos\\theta + F_{z_i}\\sin\\theta\\cos\\phi}\t\t\n\t\\end{eqnarray}\n\\end{subequations}\n\nMore details on vehicle dynamics can be found in \\cite{Gillespie1997} and \\cite{Rajamani2012}.\n\n\\section{DEEP LEARNING MODELS}\n\\label{sec:DL_models}\n\nWe propose two different artificial neural network architectures to learn the inverse dynamics of a vehicle, in particular the coupled longitudinal and lateral dynamics. \nAn artificial neural network is a network of simple functions called neurons. Each neuron computes an internal state (activation) depending on the input it receives and a set of trainable parameters, and returns an output depending on the input and the activation. Most neural networks are organized into groups of units called layers and arranged in a tree-like structure, where the output of a layer is used as input for the following one. The training of the neural network consists in finding the set of parameters (weights and biases) minimizing the error (or \\emph{loss}) between predicted and actual values on a training dataset. In this paper, this training dataset is computed using the 9 DoF vehicle model presented in Section~\\ref{sec:vehicle_model}.\n\n\\subsection{Dataset}\n\\label{ssec:dataset}\n \nThe dataset generated by the 9DoF vehicle model has a total of 43241 instances: it is divided into a train set of 28539 instances and a test set of 14702 instances. The following procedure was used to generate each instance: \n\nFirst, a control $u$ to apply is generated randomly, as well as an initial state $\\xi^{(0)}$ of the vehicle.\nMore precisely, the vehicle is chosen to be either in an acceleration phase or in a deceleration phase with equiprobability. \nIn the first case, the torques at the front wheels $T_{\\omega_1}$ and $T_{\\omega_2}$ are set equal to each other and drawn from a uniform distribution between $0$Nm and $750$Nm, while the torques at the rear wheels $T_{\\omega_3}$ and $T_{\\omega_4}$ are set equal to zero (the vehicle is assumed to be a front-wheel drive one). In the second case, the torques of each wheel are set equal to each other and drawn from a uniform distribution between $-1250$Nm and $0$Nm. In both cases, the steering angle $\\delta$ is drawn from a uniform distribution between $-0.5$ and $+0.5$rad.\nThe initial state $\\xi^{(0)}$ is composed of the initial position $(X^{(0)},Y^{(0)})$ of the vehicle in the ground frame, the longitudinal and lateral velocities $V_x^{(0)}$ and $V_y^{(0)}$, the roll, pitch and yaw angles and their derivatives, and the rotational speed $\\omega_i^{(0)}$ of the each wheels. The initial longitudinal speed $V_x^{(0)}$ is drawn from a uniform distribution between $5$ and $40$m.s$^{-1}$; the initial lateral speed $V_y^{(0)}$ is drawn from a uniform distribution whose parameters depend of $V_x^{(0)}$; the rotational speed $\\omega_i^{(0)}$ is chosen such that the longitudinal slip ratio is zero. All the other initial states are set to zero. \n\nSecondly, the 9~DoF vehicle model is run for $3$s, starting from the initial state $\\xi^{(0)}$ and keeping the control $u$ constant during the whole simulation. \n\nThe resulting trajectories are downsampled to 301 timesteps, corresponding to a sampling time of $10$ms.\n\nConsequently, each instance of the dataset consists in: an initial state $\\xi^{(0)}$ of the vehicle, a control $(T_{\\omega_1}, T_{\\omega_2}, T_{\\omega_3}, T_{\\omega_4}, \\delta)$ kept constant over time, and the associated trajectory obtained $(X^{(0)}, Y^{(0)}), \\ldots, (X^{(300)}, Y^{(300)})$.\nThe dataset generation method is summarized in Algorithm \\ref{algo:dataset_gen_algo}.\n\n\\begin{algorithm}\n\\caption{Dataset Generation}\n\\begin{algorithmic}[1]\n\\Function{generate instance}{}\n \\State $\\mathtt{is\\_accelerating} \\sim {\\mathcal {B}}(0, 1)$ \\Comment{Coin flipping}\n \n \\If{$\\mathtt{is\\_accelerating} = 1$}\n \\State $u_1 \\sim {\\mathcal {U}}(0, 750)$ \\Comment{uniform; in N.m}\n \\State $\\delta \\sim {\\mathcal {U}}(-0.5, +0.5)$ \\Comment{uniform; in rad}\n \\State $u \\gets [u_1, u_1, 0, 0, \\delta]$\n \\ElsIf{$\\mathtt{is\\_accelerating} = 0$}\n \\State $u_1 \\sim {\\mathcal {U}}(-1250, 0)$ \\Comment{uniform; in N.m}\n \\State $\\delta \\sim {\\mathcal {U}}(-0.5, +0.5)$ \\Comment{uniform; in rad}\n \\State $u \\gets [u_1, u_1, u_1, u_1, \\delta]$\n \\EndIf\n\n \\State $V_x^{(0)} \\sim {\\mathcal {U}}(5, 40)$ \\Comment{uniform; in m.s$^{-1}$}\n \\State $V_y^{(0)} \\sim {\\mathcal {U}}(a, b)$ \\Comment{uniform; in m.s$^{-1}$}\n \\State where $a=\\max\\left(-1, - \\frac{V_x^{(0)}}{3}\\right)$\n \\State and $b=\\min \\left(+1, + \\frac{V_x^{(0)}}{3}\\right)$\n\n\t \\State $\\mathtt{trajectory} \\gets 9DoF(\\xi^{(0)}, u, T_{sim}=3s) \\big|_{(X,Y)}$\n \\State \\textbf{save} $(\\xi^{(0)}, u,\\ \\mathtt{trajectory})$\n\\EndFunction\n\\Function{generate dataset}{$n=43241$}\n\\For{$i\\gets 1 \\ldots n$}\n\t\\textproc{generate instance()}\n\\EndFor\n\\EndFunction\n\\end{algorithmic}\n\\label{algo:dataset_gen_algo}\n\\end{algorithm}\n\n\\subsection{Model 1: Multi-Layer Perceptron}\n\\label{ssec:Model_MLP}\n\nA Multi-Layer Perceptron (MLP), or multi-layer feedforward neural network, is a neural network $f$ whose equations are:\n\\begin{subequations}\n\\begin{eqnarray}\n\\vecbold{h}^{(0)} & = & \\vecbold x \\\\\n\\vecbold{h}^{(k)} & = &\\sigma^{(k)}( \\vecbold{W}^{(k)\\top} \\vecbold{h}^{(k-1)} + \\vecbold{b}^{(k)}), \\; \\text{for } {k=1..L}\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\vecbold{x}$ denotes the input vector, $\\vecbold{h}^{(k)}$ the output of layer $k \\in \\llbracket 1, L \\rrbracket$, $L \\in \\mathbb N^{*}$ the number of layers of the MLP and $\\sigma^{(k)}$ denotes the $k$-th activation function. $\\vecbold{h}^{(L)} = f(\\vecbold{x})$ denotes the output vector of the neural network.\n\nThe MLP, presented in Figure~\\ref{fig:mlp-full}, is used to predict the constant control $(T_{\\omega_1}, T_{\\omega_2}, T_{\\omega_3}, T_{\\omega_4}, \\delta)$ to apply given an initial state $\\xi^{(0)}$ and a desired trajectory $(X^{(0)}, Y^{(0)}), \\ldots, (X^{(300)}, Y^{(300)})$. It is trained on the dataset presented in subection \\ref{ssec:dataset}.\nIt comprises $L=5$ layers, respectively containing 32, 32, 128, 32 and 128 neurons. All the activations functions of the network are rectified linear units (ReLU): ${\\sigma(x) = \\max(0, x)}$. The loss function used, as well as weights initialization or regularization are discussed in the section~\\ref{ssec:Training}, as they are common for the two neural networks proposed. We performed a grid search to choose the sizes of the layers among $3^5=243$ possibilities by allowing each layer to have a size of either 32, 64, or 128 neurons, training the corresponding MLP for 200 epochs and evaluating its performance on the test dataset.\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[scale=0.6]{mlp.pdf} \n\t\\caption{Multi-Layer Perceptron}\n\t\\label{fig:mlp-full}\n\\end{figure}\n\n\\subsection{Model 2: Convolutional Neural Network}\n\\label{ssec:Model_CNN}\n\nConvolutional Neural Networks (CNN) are neural networks that use convolution in place of general matrix multiplication in at least one of their layers. A traditional CNN model almost always involves a sequence of convolution and pooling layers.\nCNNs have a proven history of being successful for processing data that has a known grid-like topology. For instance, numerous authors make use of CNNs for classification \\cite{dong2015vehicle}, or semantic segmentation \\cite{badrinarayanan2017segnet} purposes. \n\nWe propose to use convolutions to pre-process the vehicle trajectory before feeding it to the MLP, as illustrated in Figure~\\ref{fig:cnn-full}. Trajectories are time-series data, which can be thought of as a 1D grid taking samples at regular time intervals, and thus are very good inputs to process with a CNN.\nWe decided to process the X and Y coordinates separately.\nFor each channel $\\vecbold{x}$ (either $X$ or $Y$), we construct the following CNN module, which is depicted in Figure~\\ref{fig:cnn-module}:\n\\begin{subequations}\n\t\\begin{eqnarray}\n\t\\vecbold{h}^{(0)} & = & \\vecbold{x} \\\\\n\t\\vecbold{h}^{(k)} & = & \\sigma^{(k)}( \\pi^{(k)} ( \\vecbold{W}^{(k)} * \\vecbold{h}^{(k-1)} + \\vecbold{b}^{(k)} ) ), \\; \\text{for } {k=1..L'} \\quad \\; \\;\n\t\\end{eqnarray}\n\\end{subequations}\nwhere $\\vecbold{h}^{(L')}$ is the output of the CNN module, $L' \\in \\mathbb N^{*}$ the number of layers, $\\sigma^{(k)}$ the $k$-th activation function and $\\pi^{(k)}$ the $k$-th pooling function. \n\nThe parameters of the CNN module are $L'=3$, with a convolution kernel size of 3 for all convolutions. The activation functions are all ReLU and the pooling functions are all average-pooling of size 2. \nThe first two convolutions have 4 feature maps while the last convolution has only 1 feature map. \n\nAs the longitudinal and lateral dynamics are quite different, distinct sets of weights are used for the $X$ and $Y$ convolutions. After processing the X and Y 1D-trajectories by their dedicated CNN module, their output are concatenated.\nThis new output is then fed to the former MLP whose characteristics remain the same except from the dimension of its input.\nThe whole model shown in Figure~\\ref{fig:cnn-full} is designated as the ``CNN model'' in the rest of this work. \n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[scale=0.5]{cnn-im1.pdf} \n\t\\caption{Convolutional Neural Network}\n\t\\label{fig:cnn-full}\n\\end{figure}\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[scale=0.6]{cnn-im2.pdf} \n\t\\caption{CNN Module}\n\t\\label{fig:cnn-module}\n\\end{figure}\n\n\\subsection{Training procedure}\n\\label{ssec:Training}\n\nThe training procedure is the same for the two neural networks:\n\n\t\\subsubsection{Weights Initialization \\& Batching}\n\t\\label{sssec:weights_initialization}\n\t\n\tEach training batch is composed of 32 instances of the dataset.\n\tThe Xavier initialization \\cite{Glorot2010} (also known as GLOROT uniform initialization) is used to set the initial random weights for all the weights of our model.\n\n \\subsubsection{Loss function, Regularization \\& Optimizer}\n \\label{sssec:loss_and_regu_and_optim}\n \n The objective of the training is to reduce the mean square error (MSE) between the controls predicted $u^{pred}$ by the neural network and the ones $u^{real}$ that were really applied to obtain the given trajectory.\n The neural network is trained in order to minimize the loss function $L$ defined by Equation~(\\ref{eq:loss_function}) on the train dataset, before evaluation on the test dataset.\n\t\\begin{eqnarray}\n\t\t\\label{eq:loss_function}\n\t\tL = \\gamma L_{\\delta} + (1 - \\gamma) L_{T} + L_{reg}\n\t\\end{eqnarray}\n\twhere \n\t\\begin{subequations}\n\t\\begin{eqnarray}\n\t\t\\label{eq:loss_function1}\n\t\tL_{\\delta}(\\delta^{real}, \\delta^{pred}) & = & \\frac{1}{0.5} MSE(\\delta^{real}, \\delta^{pred})\\\\\n\t\t\\label{eq:loss_function2}\n\t\tL_{T}(T_{\\omega_i}^{real}, T_{\\omega_i}^{pred}) & = & \\frac{1}{4 \\times 2000} \\sum_{i=1}^4 MSE(T_{\\omega_i}^{real}, T_{\\omega_i}^{pred}) \\qquad\\\\\n\t\t\\label{eq:loss_function3}\n\t\tL_{reg}(W) & = & \\gamma_{reg} ||W||_2^2\n\t\\end{eqnarray}\n\t\\end{subequations}\n\t\n\tThe scaling factors $1\/0.5$ and $1\/(4\\times2000)$ were chosen in order to normalize the steering and the torques. The parameter $\\gamma=0.99$ was chosen in order to prioritize the lateral dynamics over the longitudinal one. Equation~(\\ref{eq:loss_function3}) is an L2 regularization of our model, where $W$ is the vector containing all the weights of the network. We set $\\gamma_{reg} = 10^{-5}$.\n\t\n\tTo train our model, we used the Adam optimization algorithm \\cite{Kingma2014}. It calculates an exponential moving average of the gradient and the squared gradient. For the decay rates of the moving averages, we used the parameters $\\beta_1 = 0.9$, $\\beta_2 = 0.999$. The values of other parameters were $\\alpha = 10^{-3}$ for the learning rate, and $\\epsilon = 10^{-8}$ to avoid singular values.\n\n\\section{RESULTS}\n\\label{sec:results}\nIn order to compare their ability to learn the vehicle dynamics, the two different artificial neural networks are used as ``controllers\"\\footnote{Properly speaking, they are not real controllers as they to not learn how to reject disturbances and modeling errors.}. The reference track, presented in Figure~\\ref{fig:track_circuit}, comprises both long straight lines and narrow curves. The reference speed is set to $V_{ref}=10$m\/s on the whole track. \n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[]{track.pdf} \n\t\\caption{Top view of the test track; numbers $1$ to $7$ refers to different road sections delimited by dashed lines in order to facilitate the matching with Figures~\\ref{fig:steering} to \\ref{fig:lateral_error}.}\n\t\\label{fig:track_circuit}\n\\end{figure}\n\n\\subsection{Generating the control commands}\n\\label{ssec:results_tracking}\n\nIn order to compute the control commands to be applied to the vehicle, the artificial neural network needs to know the trajectory the vehicle has to follow in the next $3$s, as in the train dataset. One problem that arises is that it has only learned to follow trajectories starting from its actual position such as in Figure~\\ref{fig:ex_trajdataset}. However, in practice, the vehicle is almost never exactly on the reference path. Therefore, a path section starting from the actual position of the vehicle and bringing it back to the reference path is generated: for that purpose, cubic Bezier curves were chosen as illustrated in Figure~\\ref{fig:ex_trajbezier}. \nThus, at each iteration, \n(i) a Bezier curve with length $3$s is computed to link the actual position of the vehicle to the reference trajectory; \n(ii) a query comprising the previously computed Bezier curve is sent to the artificial neural network;\n(iii) the artificial neural network returns the torques at each wheel and the front steering angle to apply\nuntil the next control commands are obtained. The computation sequence is run every $300$ms, even though the query takes less than $2$ms. \n\n\\subsection{Comparison of the models}\n\\label{ssec:results_comparison}\n\nThe results obtained for the MLP and the CNN models are displayed respectively in blue and in red in Figures~\\ref{fig:steering} to \\ref{fig:lateral_error}. The resulting videos, obtained using the software PreScan \\cite{Prescanurl}, are available online\\footnote{https:\/\/www.youtube.com\/watch?v=yyWy1uavlXs}.\nClearly, it appears that the results obtained using a CNN are better than a MLP.\nFirst, we observe that the control commands are smoother in curves with the CNN. There are steep steering (see Figure~\\ref{fig:steering}) and front torques (see Figure~\\ref{fig:Tfront}) variations for the MLP around $s=360$m \nin road sections n$^\\circ4$ and around $s=480$m in road sections n$^\\circ6$. In the latter case, the steering angle reaches its saturation value $+0.5$rad and the wheel torques change suddently from $1000$Nm to $-1000$Nm and vice-versa, which is impossible in practice. On the contrary, the control signals of the CNN model remains always smooth and within a reasonable range of values. Secondly, both the longitudinal and lateral errors are smaller for the CNN than the MLP as shown respectively in Table~\\ref{tab:perf_longi} and \\ref{tab:perf_lateral}.\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Comparison of the longitudinal performances of the MLP and CNN controllers (in m\/s).}\n\t\\label{tab:perf_longi}\n\t\\begin{tabular}{|l|cccc|}\n\t\t\\hline\n\t\tmodel & RMS & average & std. dev. & max\\\\\n\t\t\\hline\n\t\tMLP & 0.76 & -0.29 & 0.70 & -4.94\\\\\n\t\tCNN & 0.60 & -0.39 & 0.46 & -2.33\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Comparison of the lateral performances of the MLP and CNN controllers (in m).}\n\t\\label{tab:perf_lateral}\n\t\\begin{tabular}{|l|cccc|}\n\t\t\\hline\n\t\tmodel & RMS & average & std. dev. & max\\\\\n\t\t\\hline\n\t\tMLP & 0.61 & 0.003 & 0.61 & 3.26\\\\\n\t\tCNN & 0.43 & 0.014 & 0.43 & 1.7\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nHowever, unlike classic controllers, stability cannot be ensured for these ``controllers\" as they are black boxes. In particular, for the CNN, we observe a lateral static error in straight lines. This static error is caused in fact by the Bezier curves which do not converge fast enough to the reference track on straight lines as only the first $300$ms are really followed by the CNN model (see Figure~\\ref{fig:CNN_Bezier_static_error}). Moreover, Figure~\\ref{fig:steering} shows that the steering angle applied during straight lines is the same for MLP and CNN.\n\n\\subsection{Coupling between longitudinal and lateral dynamics}\n\\label{ssec:results_coupling}\n\nThe speed limit a kinematic bicycle model can reach in a curve of radius $R$ is given by Equation~(\\ref{eq:V_lim6}) where $\\mu=1$ is the road friction coefficient and $g$ the gravity constant \\cite{PolackACC2018}. This corresponds to $9.9$m\/s ($R=20$m) in road section n$^\\circ$2 and $7.0$m\/s ($R=10$m) in road section n$^\\circ$6. \nAs the reference speed is set to $10$m\/s throughout the track, conventional decoupled longitudinal and lateral controllers (based on a kinematic bicycle model) will not perform well in road section n$^\\circ$6.\n\n\\begin{eqnarray}\n\\label{eq:V_lim6}\nV_{kbm_{lim}} & = & \\sqrt{0.5 \\mu g R}\n\\end{eqnarray}\n\nOn the contrary, both models (especially the CNN) are able to pass this road section, showing the ability of artificial neural networks to handle coupled longitudinal and lateral dynamics. More precisely, we observe in Figure~\\ref{fig:V_speed} that the speed is reduced in section n$^\\circ6$ because the artificial neural networks deliberately brake (see Figure~\\ref{fig:Tfront} and \\ref{fig:Trear}), even though the speed of the vehicle is below the reference speed. This is due to the loss function used during training and given by Equation~(\\ref{eq:loss_function}) that penalizes more steering angle errors than torque errors. Hence, the models prioritize the lateral over the longitudinal dynamics.\n \nTherefore, such ``controllers\" are particularly interesting for highly dynamic maneuvers such as emergency situations or aggressive driving where the longitudinal and lateral dynamics are strongly coupled. However, they should be used sparingly as they are only black boxes, or should at least be supervised by model-based systems. Moreover, for normal driving situations, conventional decoupled longitudinal and lateral controller should be preferred.\n\n\\newpage\n\n\\begin{minipage}[c]{\\textwidth}\n\t\\centering\n\t\\includegraphics[]{delta_plot_VF.pdf} \n\t\\captionof{figure}{Comparison of the steering command computed by the\n\t\tdifferent controllers. The numbers 1 to 7 correspond to the different road sections presented in Figure~\\ref{fig:track_circuit}.}\n\t\\label{fig:steering}\n\t\n\t\\includegraphics[]{Tfront_plot_VF.pdf} \n\t\\captionof{figure}{Comparison of the torque applied at the front wheels computed by the different controllers.}\n\t\\label{fig:Tfront}\n\t\n\t\\includegraphics[]{Trear_plot_VF.pdf} \n\t\\captionof{figure}{Comparison of the torque applied at the rear wheels computed by the different controllers.}\n\t\\label{fig:Trear}\n\t\n\t\\includegraphics[]{V_speed_VF.pdf} \n\t\\captionof{figure}{Comparison of the total speed obtained with the different controllers.}\n\t\\label{fig:V_speed}\n\t\n\t\\includegraphics[]{lateral_error_VF.pdf} \n\t\\captionof{figure}{Comparison of the absolute value of the lateral error obtained with the different controllers.\n\t}\n\t\\label{fig:lateral_error}\n\\end{minipage}\n\n\\clearpage\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[]{train_example.pdf} \n\t\\caption{Example of a training dataset instance: in red, the reference trajectory, in blue the one obtained from the control predicted by the CNN model.}\n\t\\label{fig:ex_trajdataset}\n\\end{figure}\n\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[]{Bezier1.pdf} \n\t\\caption{Example of a Bezier curve (in red) joining the actual position of the vehicle (the red circle) to the reference trajectory (in green). The actual trajectory followed by the vehicle is shown in blue.}\n\t\\label{fig:ex_trajbezier}\n\\end{figure}\n\n\\begin{figure}[h!\n\t\\centering\n\t\\includegraphics[]{Bezier2.pdf} \n\t\\caption{Example of a Bezier curve on a straight line section of the reference trajectory. The lateral error is not corrected since the convergence of the Bezier curve to the reference trajectory is too slow.}\n\t\\label{fig:CNN_Bezier_static_error}\n\\end{figure}\n\n\\subsection{Comparison with decoupled controllers}\n\\label{ssec:results_decoupled_ctrl_compare}\nFinally, the ``controllers\" obtained with the MLP and CNN models are compared with commonly used decoupled controllers: the lateral controller is either a pure-pursuit (PP) \\cite{Coulter1992} or a Stanley \\cite{Thrun2006} controller while in both cases, the longitudinal controller is ensured by a Proportional-Integral (PI) controller with gains $K_P = 600$ and $K_I = 10$. The gain for the front lateral error is $0.75$ for the Stanley controller. The preview distance of the pure-pursuit controller is defined as a function of the total speed $V_g$ at the center of gravity: $L_P = l_f + T_A V_g$ where $T_A = 1.5$s is the anticipation time. The results of the PP and the Stanley controllers are shown respectively in green and grey in Figures~\\ref{fig:steering} to \\ref{fig:lateral_error}. Clearly, a decrease of performance can be observed when using these decoupled controllers in the challenging part of the track. In particular, the lateral error becomes huge in both cases during the sharp turn of road section n$^\\circ6$ while the CNN was able to perform reasonnably well.\n\n\\section{CONCLUSIONS}\n\\label{sec:conclusions}\n\nThis work presented some preliminary results on deep learning applied to trajectory tracking for autonomous vehicles. Two different approaches, namely a MLP and a CNN, were trained on a high-fidelity vehicle dynamics model in order to compute simultaneously the torque to apply on each wheel and the front steering angle from a given reference trajectory. It turns out that the CNN model provides better results, both in terms of accuracy and smoothness of the control commands. Moreover, compared to most of the existing controllers, it is able to handle situations with strongly coupled longitudinal and lateral dynamics in a very short time. However, the controller obtained is a black-box and should not be used in standalone.\n\nThe results proved the ability of deep learning algorithms to learn the vehicle dynamics characteristics. This opens a wide range of new possible applications of such techniques, for example for generating dynamically feasible trajectories. Future work will focus on (i) replacing the complex dynamics models by a learned off-line model in Model Predictive Control for motion planning, (ii) using Generative Adversarial Networks (GAN) to generate safe trajectories where the learned dynamics is used as constraint, and (iii) performing real-world experiments with our approach on a real car. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn the context of non-commutative probability spaces\nthere are only very few possibilities for universal notions of independence. If we require that this notion is commutative (i.e., $x$ independent from $y$ is the same as $y$ independent from $x$) and that\nconstants are independent from everything then there are only two such concepts, namely the classical independence and the free independence. On\nthe level of algebras, equipped with a state, this means that there are only two universal kind of product constructions, namely the tensor product and the\nreduced free product. We refer the reader to \\cite{Sp-universal,Mur,BGSch} for more details on this.\n\nSo if we have a collection of variables which are independent (in this univeral sense) then there are only two possibilities; they are either all classically independent or they are all freely \nindependent. On the other hand, we can gain some more flexibility if we\ndo not ask for the same kind of independence between all of them. This \nraises the question about mixtures of the two forms of independences.\nOf course, one can create quite easily such situations by starting with two sets\nof variables $X$ and $Y$ which are free; then split each of them into \ntwo subsets $X=X_1\\cup X_2$ and $Y=Y_1\\cup Y_2$, such that $X_1$ and\n$X_2$ are classically independent and $Y_1$ and $Y_2$ are freely independent. One can continue in this fashion and get so a collection of variables where some pairs of them are free and other pairs are classically independent. However, this is restricted to situations where we can group our\nvariables in sets with specific kind of independence among them. We are interested in a generalization of this, by trying to prescribe arbitrarily free or classical independence for any pair. An example for this would be to ask for five\nvariables $x_1,x_2,x_3,x_4,x_5$ such that\n\\begin{itemize}\n\\item\n$x_1$ and $x_2$ are free, \n\\item\n$x_2$ and $x_3$ are free, \n\\item\n$x_3$ and $x_4$ are\nfree, \n\\item\n$x_4$ and $x_5$ are free, \n\\item\n$x_5$ and $x_1$ are free,\n\\item\nbut all other pairs are independent. \n\\end{itemize}\n(Here and in the following we will always mean ``classically independent'' when we say ``independent''.)\n\n\nSuch a situation cannot be generated\nby the above dividing into groups, and it is not clear apriori whether such\na requirement can be satisfied in any meanigful way. In \\cite{Mlo}\nM\\l otkowski showed that this can, indeed, be achieved for any prescription of the mixture of free and classical independence. For this he introduced the general notion of $\\Lambda$-freeness. It seems that his work did not get the attention it deserves and we hope that our work will stimulate new interest in this concept. To his original results we will add here a description of the combinatorial structure of $\\Lambda$-freeness, featuring in particular a formula for mixed moments in terms of free cumulants. This will be taken up\nin the subsequent paper \\cite{SW} and will lead to new forms of quantum groups, with partial commutation relations.\n\nOn the level of groups or semi-groups the prescription of commutation relations for some fixed pairs of generators is of course not new;\nin the group case this goes, among others, under the names of ``right angled Artin groups'' (see \\cite{Cha}), ``free partially commutative groups'' or ``trace groups'', in the case of semi-groups one talks about ``Cartier-Foata monoids'' (see \\cite{foata1969problemes}) or ``trace monoids''.\nActually, there is also the notion of a corresponding mixed product of groups, which is usually called the ``graph product of groups'' and was introduced by Green in \\cite{thesis}.\nIn a sense $\\Lambda$-freeness reveals the notion of ``independence'' for the group algebras of such graph products of groups with respect to their canonical trace. We will make this connection precise in Proposition\n\\ref{prop:on-groups}.\n\nOur interest in $\\Lambda$-freeness\narouse out of discussions on similar constructions of the second author, on mixtures between monotone and boolean \\cite{J-bm} and boolean and free independences \\cite{J-bf}. \nMuch motivation is also taken from recent work on bi-freeness \\cite{Voi-bifree,MaNi-bifree,CNS-bifree}. Bifreeness does not fit in the frame presented here, but there are some similarities, in particular, concerning the underlying combinatorics.\n\n\n\n\\section{The setting}\nThe notion of $\\Lambda$-freeness is defined in terms of a matrix\nwhich specifies the choice which pairs should be free and which should be independent. \nM\\l otkowski denoted this matrix by $\\Lambda$; we prefer here to call it $\\varepsilon$, and hence we will also speak of $\\varepsilon$-freeness or, alternatingly, $\\varepsilon$-independence.\n\nSo let $I$ be an index set (finite or infinite). \nFor any given collection of algebras $\\mathcal{A}_i$,\nfor all $i\\in I$, we want to embed the $\\mathcal{A}_i$ in a bigger algebra $\\mathcal{A}$,\nsuch that for each pair of algebras we have that they are either free or independent. In order to specify this choice we will use a symmetric matrix\n$\\varepsilon=(\\varepsilon_{ij})_{i,j\\in I}$ with non-diagonal entries either 0 or 1. This $\\varepsilon$ should specify our \nmixture according to: \n\\begin{itemize}\n\\item\n$\\mathcal{A}_i$ and $\\mathcal{A}_j$ are free if $\\varepsilon_{ij}=0$, and\n\\item\n$\\mathcal{A}_i$ and $\\mathcal{A}_j$ are independent if $\\varepsilon_{ij}=1$ (which includes in\nparticular, that $\\mathcal{A}_i$ and $\\mathcal{A}_j$ commute\n\\end{itemize}\nIt will be convenient to set $\\varepsilon_{ii}=0$ for all $i\\in I$.\n\nIn the following such a matrix $\\varepsilon$ will be fixed.\nOf course, we can identify such a matrix with the adjacency matrix of a simple (i.e., no loops, no multiple edges) graph; then the edges of the graph give us the independence relations between the involved algebras, which correspond to the vertices of the graph.\n\nFor the basic notions and facts about non-commutative probability spaces, non-crossing partitions or free cumulants we refer to \\cite{NS}.\n\n\\section{The definition of $\\varepsilon$-independence}\n\n\\begin{notation}\nLet us use the following notation. Given some subalgebras \n$\\mathcal{A}_i$ ($i\\in I$) and an index-tuple $\\text{\\bf i}=(i(1),\\dots,i(n))\\in I^n$ we write\n$(a_1,\\dots,a_n)\\in \\mathcal{A}_\\text{\\bf i}$ for: $a_k\\in \\mathcal{A}_{i(k)}$ for $k=1,\\dots,n$.\n\\end{notation}\n\n\\begin{definition}\n1) By $I_n^\\varepsilon$ we denote those $n$-tuples of indices from $I$ for which neigbours are different modulo our $\\varepsilon$-relations; more precisely, $\\text{\\bf i}=(i(1),\\dots,i(n))\\in I_n^\\varepsilon$ if and only if: if we have $i(k)=i(l)$ for $1\\leq k20 R_{\\rm in} $ that would be \ninferred from the centrally peaked CO fundamental emission \nprofiles of most T Tauri stars (Najita et al.\\ 2003). \nThus, the outer radius of the V836 Tau CO emission is in the range \n$0.3-0.5$\\,AU. \n\n\nWe can provide constraints on the mean conditions in the emitting gas\nby modeling the CO spectrum with a radially constant\nexcitation temperature and column density.\nThe similar shape and strength of the $v$=1--0 \nemission lines over a wide range in $J$ reveals that the emission \nlines are optically thick, i.e., that the gas column density is \n$> 0.001\\rm \\,g\\,cm^{-2}$ if turbulent line broadening is negligible. \nIn addition, the absence of detectable $v$=1--0 $^{13}$CO emission lines \nlimits the column density to $< 0.03\\rm \\,g\\,cm^{-2}$ \nfor a CO abundance of $3\\times 10^{-4}$ relative to hydrogen and \nan interstellar $^{13}$CO\/$^{12}$CO ratio of 90. \n\n\nThe weakness of the $v$=2--1 and $v$=3--2 lines in the spectrum \nrequires either a low average excitation temperature or that the vibrational \nlevels above $v$=1 depart significantly from thermal equilibrium. \nFor LTE level populations, \nthe relative strengths of the $v$=1--0 lines and the limit on the\nstrength of the $v$=2--1 transitions constrain \nthe mean excitation temperature to the range 700--1100\\,K.\nSmaller excitation temperatures \nrequire larger emitting areas to produce the required line flux, as \nwell as larger inclinations in order to produce the emission over \nthe required range of velocities. As a result, at temperatures \n$\\lesssim 700$\\,K, the requirement on the line flux drives the \nemitting radii to values large enough that the required velocities \ncannot be obtained at any inclination. For a radially constant \nexcitation temperature \n$\\gtrsim 1200$\\,K, the $v$=2-1 lines are too strong \nrelative to the $v$=1--0 lines. \nIn addition, given the constraints on the radial range (and therefore \nthe emitting area) of the emission, the strength of\nthe $v$=1--0 lines are overpredicted at such high temperatures. \n\n\nWe can obtain a better fit to the average line profiles by allowing the\ntemperature to vary as a function of radius.\nConsistent with the above considerations, \na gas temperature that varies slowly with \nradius ($T = 1200\\,{\\rm K} (r\/R_{\\rm in})^{-0.30}$) and \na line-of-sight disk column density of \n$\\Sigma = 0.003\\rm \\,g\\,cm^{-2}$ \nthat is radially constant between \n$R_{\\rm in}= 13 R_\\odot$ and $R_{\\rm out} = 5 R_{\\rm in}$, \ngives a reasonable fit to \nthe spectra assuming LTE level populations (Fig.~6).\nIn addition to providing a reasonable fit to \nthe relative line strengths of the $v$=1--0 lines, \nthe model also reasonably fits the average line profiles of the \n$v$=1--0 lines (Figs.~4 and 5; heavy dashed line).\n\nWe can also fit the spectra with a model that uses a steep \ntemperature gradient rather than a specified outer radius \nto limit the radial extent of the emission. \nWith a temperature profile \n$T=1400\\,{\\rm K}(r\/R_{\\rm in})^{-0.6}$ \nand a radially-constant line-of-sight column density \n$\\Sigma=0.0037 \\rm \\,g\\,cm^{-2}$ and $R_{\\rm in}=16.2R_\\odot$, \nthe 1--0 emission decreases sharply beyond $7-8R_{\\rm in}$ \nbecause the Planck function contributes little at $4.7\\hbox{$\\mu$m}$ \nat the low temperatures achieved at these radii \n($\\lesssim 400$\\,K). \nThe high-$J$ P-branch lines are also optically thin at \nthese radii.\nThe model provides a reasonable fit to the relative strengths \nof the $v$=1--0 lines. The average line profiles are reasonably \nwell fit (Figs.\\ 4 and 5; heavy dash-dot line), \nalthough the central dip in the $v$=1--0 R lines \nis shallower than observed. This is because the low-$J$ R lines \nremain optically thick (and continue to produce emission) to larger \nradii than the high-$J$ P lines. In contrast, the outer \nradius to the emission used in the first model, produces \nsimilar line profiles for the low-$J$ R and high-$J$ P lines. \n\nTo summarize, the observed spectra can be explained \nwith an abrupt truncation of the CO emission \nbeyond an outer radius of $7-8R_{\\rm in}$. \nA steep temperature gradient where the temperature drops \nto $\\lesssim 400$\\,K at $7-8 R_{\\rm in}$ is an alternative explanation. \nNote that there is a slight inconsistency in the model \nparameters used in the above fits. The adopted inclination \nformally implies a distance of 165\\,pc rather than the \nnominal Taurus distance of 140\\,pc that we have assumed \nin the fits. If V836 Tau is located at a larger distance \nthan 140\\,pc, the CO emission can be fit with similar model \nparameters to those used above, with the modification that \n$M_* = 0.88M_{\\odot}$ and the emission arises from radii \n1.18 times larger than assumed above. \nThe higher mass is within the mass range allowed by the \nuncertainty in the stellar spectral type. \n\n\n\n\\section{Discussion}\n\n\\subsection{Truncated Excitation or Truncated Disk?} \n\nThe CO emission from V836 Tau shows similarly double-peaked \nline profiles for lines spanning a large range in excitation \ntemperature. The line profile shapes indicate that the \nCO emission is truncated beyond $\\sim 0.4$\\,AU. \nThe similarity in the relative strengths of the 1--0 transitions \nshow that the 1--0 transitions are optically thick over the \nemitting region. \nAs described in the previous section, \nthese properties could be produced by a {\\it physical} truncation \nof the gaseous disk beyond a radius of $\\sim 0.4$\\,AU or \na truncation of the CO emission (but not the disk gas \ncolumn density) at the same radius. \n\nAlthough it is difficult to determine which of these \ninterpretations is correct based on the available information, \nsome possibilities can be ruled out. For example, \nthe truncation of the CO emission is unlikely to result from \nthe thermal dissociation of CO since the excitation \ntemperature of the gas is much less than the thermal \ndissocation temperature of CO ($\\sim 4000$\\,K at the \ndensities of inner disks). \n\nWe might also consider the possible explanations that \nhave been put forward for the origin of the \ndouble-peaked line profiles \nobserved in the $v$=2--0 CO overtone bandhead emission \nfrom T Tauri stars and Herbig Ae stars at $2.3\\hbox{$\\mu$m}$\n(e.g., Carr et al.\\ 1993; Najita et al.\\ 1996, 2000). \nSuch emission is detected only from sources with \nhigh accretion rates, probably a consequence of the \ntemperatures and high column densities that are needed to produce the \novertone emission.\nSince these systems have disks that are believed to be \nradially continuous, the double-peaked lines are unlikely \nto arise from a radially truncated disk. \nWe have previously described two possible explanations for \nthe double-peaked lines that make up the CO $v$=2--0 bandhead: \n(1) the outer radius is a dust sublimation front, or \n(2) the transition from atomic H to H$_2$ at decreasing temperature \ndepopulates the higher CO vibrational levels. \n\nIn the first scenario, a dust sublimation front renders \nthe line emitting layer optically thick in the continuum \nat temperatures below $\\sim 1500$\\,K, \neliminating the contrast of the CO $v$=2--0 emission above the \ncontinuum at these radii (e.g., Carr 1989). \nThis explanation is difficult to apply to the CO \nfundamental lines, because \ntypical dust temperatures at the inferred \nouter radius of $\\sim 0.4$\\,AU for the CO fundamental emission \nare much below the dust sublimation temperature \n(e.g., D'Alessio et al.\\ 1999) and measured dust sublimation \nradii are much smaller \nthan the inferred outer radius of the CO fundamental emission \n(Eisner et al.\\ 2005; Muzerolle et al.\\ 2003). \nTherefore, the outer \nradius observed for the CO fundamental emission is \nunlikely to result from dust sublimation. \n\nAs an alternative scenario, \nwe previously speculated that \nthe transition from atomic to molecular hydrogen at decreasing \ndisk temperature leads to the depopulation of the higher \nvibrational levels of CO due to the lower collisional \ncross-section of molecular hydrogen with CO compared to \nthat of atomic hydrogen with CO (Najita et al.\\ 1996).\nIn chemical equilibrium, disks would \ntransition from a mixture of CO and atomic hydrogen at small \ndisk radii ($T > 2000$\\,K) to a mixture of CO and molecular \nhydrogen at large disk radii ($T < 2000$\\,K). \nIn this situation, we estimated that radiative trapping would \nbe able to maintain the $v \\ge 2$ vibrational populations down \nto a temperature of $\\sim 1500$\\,K, below which significant \ndepopulation would occur.\nUsing a non-LTE model of this kind, we were able to reproduce \nthe line intensities and shapes of CO overtone emission lines \ncovering a wide range of excitation conditions ($v$=2--0 to $v$=5--3) \nin the spectra of two Herbig Ae stars. \n\nThe same effect is unlikely to apply in detail to the \ntruncation of the CO fundamental emission seen in V836 Tau \n($R_{\\rm out} \\sim 0.4$\\,AU). \nWhile chemical equilibrium may be relevant at the large \ncolumn densities needed to produce the overtone emission, \ndisks are expected to have significant vertical structure \nand to depart significantly from chemical equilibrium \nat the smaller column densities needed to produce the \nfundamental emission. \nThermal-chemical models of T Tauri disks irradiated by stellar \nX-rays (Glassgold et al.\\ 2004; Meijerink et al.\\ 2008) \nindicate that overlying the large column densities \nwhere the overtone lines form is a warm surface layer \nthat is expected to be \nconducive to the production of CO fundamental emission \nover a large range in radii (Glassgold et al.\\ 2004). \n\nThe X-ray irradiated disk models, which have currently studied \nthe structure of disks over the region 0.25--2\\,AU, find that throughout \nthis range of radii disks possess a warm ($\\sim 1000$\\,K) \nsurface layer ($\\gtrsim 10^{21}\\rm \\,cm^{-2}$) of CO mixed with \natomic hydrogen (Glassgold et al.\\ 2004; Meijerink et al.\\ 2008). \nThus, the CO fundamental transitions could plausibly \nbe excited over radii within and beyond 1\\,AU. \nThis expectation is in agreement with observations of \nCO fundamental emission from T Tauri stars. \nEmpirically, we find that \nalmost all accreting T Tauri stars show CO fundamental emission \nand that the majority of CO fundamental emission profiles \nare centrally peaked. This indicates that the emission \narises from a wide range of disk radii \n($R_{\\rm out}\/R_{\\rm in} > 20$, $R_{\\rm out} \\simeq 1-2$\\,AU), \nwithout a sharp truncation in excitation \nas a function of velocity (Najita et al.\\ 2003).\n\n\n\nHowever, it is possible that these general trends, obtained \nfor typical T Tauri stars, do not apply to V836 Tau. \nWhereas typical T Tauri stars have strong near-infrared \nexcesses and stellar accretion rates $\\sim 10^{-8}M_{\\odot}\\,{\\rm yr}^{-1}$, \nV836 Tau has a weak near-infrared excess and a low \nstellar accretion rate $\\sim 10^{-9}M_{\\odot}\\,{\\rm yr}^{-1}$ \n(Hartigan et al.\\ 1995 scaled to Gullbring et al.\\ 1998; \nHerczeg et al.\\ 2006). \nThe weak near-infrared excess might indicate either \na significant settling of grains out of the disk atmosphere \nor a lack of dust at small disk radii. \n\nIn this situation, one might imagine that a lack of small \ngrains could reduce the heating of the gaseous atmosphere,\nif gas-grain collisions dominate the heating of the gaseous \ndisk. In a cooler atmosphere, vibrational CO emission would \nbe more difficult to produce, perhaps contributing, thereby, \nto the truncation of the CO emission. \nThis does not seem likely in the context of recent X-ray \nirradiated disk atmosphere models, in which the \ngas is heated directly by X-rays or accretion-related processes \nand the grains function primarily as a coolant for the gas \nthrough collisions. \nIn such a model, we might expect grain growth and settling \nto produce warmer gaseous atmospheres, rather than a radial \ntruncation of the CO emission. \n\nAlternatively, one might imagine that CO might be less abundant \nin a grain-poor disk atmosphere if the CO is synthesized from \nH$_2$ that forms on grains. However, in the X-ray irradiated \ndisk atmosphere models, the grains are typically warmer than \n100\\,K within 1\\,AU, and have therefore been assumed to play \nno significant role in the synthesis of CO and other molecules \n(Glassgold et al.\\ 2004). \nAs a result, grain growth and settling is not expected to \nsignificantly reduce the strength of the CO emission from the \ndisk atmosphere. \n\nAnother possibility is that the low accretion rate of V836 Tau \ncompared to the average T Tauri accretion rate \n($\\sim 10^{-8}M_{\\odot}\\,{\\rm yr}^{-1}$; Hartmann et al.\\ 1998)\nmight play a role in truncating the CO emission. \nSince accretion-related processes may heat the gaseous \natmosphere (Glassgold et al.\\ 2004), \nthe gaseous atmosphere may experience reduced heating \nat the lower accretion rate of V836 Tau. \nWhile the available heating is clearly able to produce \ndetectable $v$=1--0 CO fundamental emission from V836 Tau, \ncould reduced heating radially truncate the emission, perhaps \nvia a non-LTE effect? \n\nA simple estimate indicates that the vibrational populations\ncould be in non-LTE.\nThe critical density $n_{\\rm cr}$ for the $v$=1 level of CO, obtained\nfrom the Einstein A-values for the $v$=1--0 transitions (e.g.,\nGoorvitch \\& Chackerian 1994) \nand the $v$=1--0 collision rates for CO with atomic\nhydrogen (Glass \\& Kironde 1983), varies as $T^{-1\/2}$ and equals\n$\\sim 5 \\times 10^{12}\\rm \\,cm^{-3}$ at a temperature of 1000\\,K\n(see discussion in Najita et al.\\ 1996).\nOur modeling of the CO emission from V836 Tau shows \nthat the $v$=1--0 CO emission arises from a gas column density \n$\\sim 0.01 \\rm \\,g\\,cm^{-2}$ at disk radii $\\lesssim 0.4$ AU. In the \nD'Alessio et al.\\ (1999) disk model, the density in the disk \natmosphere over this range of column density is \n$n_{\\rm H} \\sim 10^{11}-10^{12} \\rm \\,cm^{-3}$ at radii 0.1--0.4 AU. \nTherefore, a rough estimate is $n_{\\rm H}\/n_{\\rm cr} \\sim 0.1$ \nin the CO emitting region. \n\nOur modeling of the CO emission further shows that the \n$v$=1--0 CO lines are optically thick, with $\\tau \\sim 10$. \nSince the escape probability for Gaussian lines depends \nasymptotically on the line optical depth as $\\sim \\tau^{-1}$ \n(Mihalas 1978), the line optical depth approximately compensates \nfor the low value of $n_{\\rm H}\/n_{\\rm cr}$ so that \nthe $v=1$ level could be in LTE. \nThe situation for the higher vibrational levels is possibly \nsimilar, with the critical densities for these levels being \ncomparable to the critical density for $v$=1.\nThis because the A-values of the higher vibrational levels are\nlarger (proportional to $v$), but the collision rates may also\nbe larger because of the contribution from $\\Delta v > 1$ collisions \n(A. Glassgold, personal communication).\nThus, the CO vibrational level populations could plausibly \nbe in LTE, consistent with the assumption made in \\S 3.3. \n\nHowever, both lower temperatures and lower densities in the CO \nemitting region will tend to drive departures from LTE. Reduced \naccretion heating can produce both lower temperatures and lower \ndensities. As the accretion heating is reduced, the vertical temperature\ninversion that puts the CO into emission will become increasing\nlimited to the lower column density surface region that can be heated\nby external irradiation (e.g., by stellar X-rays). This surface\nregion will be characterized by lower densities.\nDetailed calculations of the \nthermal, chemical, and density structure of disk atmospheres \nare needed to address this issue quantitatively.\n\nBecause such calculations are currently lacking, we might \ntake instead a more empirical approach and compare the \nCO fundamental line profiles of V836 Tau with \nthose of other T Tauri stars with low stellar \naccretion rates and\/or low near-infrared excesses. \nAs an example of such a comparison, Figure 7 (top panel) \nshows the CO fundamental emission from LkCa\\,15, \na T Tauri star with a stellar accretion rate \nsimilar to that of V836 Tau \n(Hartmann et al.\\ 1998; see also Najita, Strom, \\& \nMuzerolle 2007b).\nThe spectrum shows the data reported in \nNajita et al.\\ (2003), but re-reduced using the approach \ndescribed in \\S 2. \nThe $v$=1--0 CO emission from LkCa\\,15 is centered at the \nstellar velocity (short vertical lines, bottom panel), \nbut compared to V836 Tau, \nit shows a centrally peaked profile. \n\nAlthough the signal-to-noise ratio of the spectrum is \nlimited, weak stellar photospheric features appear to \nbe present. \nIn the bottom panel of Figure 7, we show the CO emission \nfrom LkCa\\,15 after correction for a stellar photospheric \ncontribution, following the approach described in \\S 3.1. \nThe stellar photospheric model assumes \nan Allard stellar atmosphere model with \na gravity of $\\log g = 4.0$, \nan effective temperature of 4400\\,K to match the \nK5 spectral type of LkCa\\,15 (Herbig \\& Bell 1988), \na stellar rotational velocity $v \\sin i =12.5\\,{\\rm km}\\,{\\rm s}^{-1}$ \n(Hartmann, Soderblom, \\& Stauffer 1987), \nand an observed (topocentric) radial velocity \nof $-39.8 \\,{\\rm km}\\,{\\rm s}^{-1}$, which is appropriate for \nthe measured stellar radial velocity \n(Hartmann et al.\\ 1987) and the observation date. \nA veiling of 2.5 times the stellar continuum roughly \nreproduces the strength of the stellar photospheric \nfeatures near the 1--0 P30 line. \nThis level of veiling is also consistent with the veiling at \n$5\\hbox{$\\mu$m}$ implied by the SED (e.g., Furlan \net al. 2006). \nSubtracting the veiled stellar photospheric component \nproduces a disk CO emission profile that is even more \ncentrally peaked (Fig.\\ 7, bottom panel). \n\nIn addition to its low stellar accretion rate, \nLkCa\\,15 is also similar to V836 Tau in that it has the \ncharacteristics of a transition object \nwith a weak near-infrared continuum and an \noptically thick outer disk (e.g., Bergin et al.\\ 2004; \nEspaillat et al.\\ 2007). \nIts properties therefore probe empirically how both the reduction \nin small grains and reduced accretion-related heating \nmight affect the CO fundamental emission from the disk. \nThe LkCa\\,15 spectrum shows that in at least some \ncases, these effects do not radially truncate the CO \nfundamental emission within 1 AU. \n\n\nTo summarize, the double-peaked line profile of the CO fundamental \nemission from V836 Tau may indicate that the gaseous disk extends \nfrom close to the star ($\\sim 0.05$\\,AU) out to a physical \ntruncation radius ($\\sim 0.4$\\,AU). \nIf the gaseous disk in V836 Tau is instead continuous beyond \nthis radius, the double-peaked CO \nprofile would indicate a sudden truncation of the \nCO emission beyond a radius of 0.4\\,AU. \nAs discussed in \\S 4.2, an abrupt decrement in excitation \nis not expected empirically, since \nthe majority of CO emission profiles observed to date \n(both typical classical T Tauri stars and transition objects \nlike LkCa\\,15)\nshow centrally peaked CO profiles with no comparable \ndecrement in excitation with radius.\n\nHowever, a truncated emission profile may result from either \nan (anomalously) steep temperature gradient or departures from \nLTE that become important in low accretion rate systems.\nThese two possible explanations can be explored and \npotentially distinguished with a higher signal-to-noise spectrum \nthat measures the strengths of the $v$=2--1 lines. \nA theoretical study of the possibility of non-LTE CO level populations \nwould also be welcome in sorting out whether an excitation effect \nis a possible explanation for the outer radius of the emission.\nAdditional observations of low accretion rate sources \nwould be useful in exploring this issue empirically. \n\nMost definitive of all would be observations of spectral line \ntransitions that robustly probe the disk atmosphere of V836 Tau \nat excitation \ntemperatures $< 400$\\,K. These observations would complement \nthe insensitivity of the CO fundamental transitions to low \ntemperature gas. \nIf little emission is detected with these diagnostics at disk \nradii $>0.4$\\,AU, that would strongly suggest that the gaseous \ninner disk in V836 Tau is physically truncated at $\\sim 0.4$\\,AU.\nSome of the mid-infrared molecular emission diagnostics recently \nreported in the spectrum of AA Tau (C$_2$H$_2$, HCN, H$_2$O, OH; \nCarr \\& Najita 2008) may prove useful in this regard.\n\n\n\\subsection{Nature of V836 Tau}\n\nThe possibility that the disk of V836 Tau is physically \ntruncated beyond 0.4\\,AU may bear on our \nunderstanding of the nature of the system. \nAs noted in \\S 1, V836 Tau has been previously classified as \na transition object (Strom et al.\\ 1989), \na system with an optically thin inner disk (within $R_{\\rm hole}$) \nand an optically thick outer disk (beyond $R_{\\rm hole}$). \nThe SEDs of these systems have been variously explained as \na consequence of grain growth and planetesimal formation \n(e.g., Strom et al.\\ 1989; Dullemond \\& Dominik 2005), \ngiant planet formation \n(e.g., Skrutskie et al.\\ 1990; Marsh \\& Mahoney 1992), \nor photoevaporation \n(e.g., Clarke et al.\\ 2001; Alexander et al.\\ 2006). \n\nWhile all of these processes can produce optically thin regions \nin the disk (inner holes or gaps), they make different \npredictions for stellar accretion rates and disk masses, \nas well as the radial distribution of the gaseous disk. \nAs a result, measuring the radial distribution of disk gas \nand comparing the stellar accretion rates and disk masses \nof transition objects with those of accreting T Tauri stars of \ncomparable age can potentially sort among the possible \nexplanations for a transitional SED. For example, \na recent study using the latter approach showed \nthat transition objects in Taurus (including V836 Tau) \nhave stellar accretion rates that are on average $\\sim 10$ \ntimes lower than those of non-transitional \nT Tauri stars with comparable disk masses (Najita et al.\\ 2007b). \nSuch a reduced stellar accretion rate \nis predicted for disks that have formed Jovian mass planets\n(e.g., Lubow \\& D'Angelo 2006), \nsuggesting that giant planet formation may play a role in \nexplaining the origin of at least some transition objects. \n\nStudies of the radial distribution of the gaseous disk in the \nsystem provide an additional way to distinguish the nature \nof individual transition objects. \nFor example, although grain growth can render the inner disk \noptically thin (within $R_{\\rm hole}$), the gas in the same region of the \ndisk is not expected to be altered significantly; gas would \ntherefore fill the region within $R_{\\rm hole}$. \nIf a giant planet has formed with a mass sufficient to open a gap, \nboth the gas and dust would be expected to be cleared dynamically \nfrom the vicinity of the orbit of the planet, \ncreating an inner disk \n(within $R_{\\rm inner} < R_{\\rm hole}$) that is fed by accretion streams from an \nouter disk (beyond $R_{\\rm hole}$; Lubow, Seibert, \\& Artymowicz 1999; \nBryden et al.\\ 1999; Kley 1999; D'Angelo et al.\\ 2003; \nLubow \\& D'Angelo 2006). \nAccretion streams are not expected in the case of a massive \ngiant planet ($\\sim 10 M_J$; e.g., Lubow et al.\\ 1999), \nand no significant gas or dust is expected anywhere within \n$R_{\\rm hole}$ in this case. \nA lack of gas or dust within $R_{\\rm hole}$ is also expected \nin the photoevaporation case; such systems are further expected \nto have a very low disk mass \n($\\sim 0.001 M_{\\odot}$; e.g., Alexander \\& Armitage 2007). \n\nOur results for V836 Tau are intriguing in this context. \nCompared to the SEDs of well-studied transition objects \nsuch as GM Aur and DM Tau, where a strong infrared excess \nappears only beyond $\\sim 10\\hbox{$\\mu$m}$ indicating an optically \nthin region 3--20\\,AU in size (Calvet et al.\\ 2005), \nthe SED of V836 Tau \nshows a significant infrared excess at a shorter wavelength \n$\\sim 5-10\\hbox{$\\mu$m}$. \nTherefore, if the V836 Tau system has an optically thin inner \nregion, it is comparatively small and plausibly within the \nrange of disk radii probed by CO \nfundamental emission (within $\\lesssim 2$\\,AU; Najita et al.\\ 2003). \nThe SED of V836 Tau can be fit with a \nsimple model of a flared optically thick disk with an \ninner hole $\\sim 1$\\,AU in radius (see Appendix). \nIn comparison, the SED of the transition object LkCa\\,15 \ncan be interpreted as indicating an optically thin region \nwithin $\\sim 3$\\,AU (Bergin et al.\\ 2004), \nor a radial gap extending from 5--46\\,AU (Espaillat et al.\\ 2007). \nA planet orbiting at such large distances ($\\sim 3$\\,AU or $\\sim 40$\\,AU)\nmay create a gap in the disk, but not \nradially truncate the gaseous disk significantly within 1\\,AU. \nIn contrast, systems like V836 Tau, in which the SED may indicate \nan optically thin region at much smaller radii ($< 1$\\,AU), \nare the ones for which the CO fundamental emission would \nin principle be capable of diagnosing a physically truncated \ninner disk if an orbiting companion is present.\nThe radially truncated CO emission that we observe may \nsupport this picture. \n\nAs an alternative interpretation of the available data, \nthe short wavelength SED of V836 Tau \n($\\lambda < 10\\hbox{$\\mu$m}$) \ncan also be fit with a simple model of an inclined ($i=60$), \ngeometrically flat disk (see Appendix) that possibly \nresults from significant grain growth and settling.\nIn such a situation, the gaseous disk would be radially continuous \nand we might expect to observe a CO profile like those of other \nnon-transition T Tauri stars. Such a profile, typically \ncentrally peaked, is not observed. \n\nIn the photoevaporation\nand massive giant planet \nscenarios for the origin of a transitional SED, \nlittle gas is expected to be present within $R_{\\rm hole}$, \nin contrast to the observed situation where a gaseous disk \nis present at 0.05--0.4\\,AU and ongoing (possibly intermittent) \nstellar accretion is observed. The relatively high \ndisk mass of V836 Tau ($\\sim 0.01M_{\\odot}$; Andrews \\& Williams 2005), \ncompared to the much smaller masses at which photoevaporation \nis expected to be able to create an inner hole \n($\\sim 0.001M_{\\odot}$), further argues against the photoevaporation \nscenario. The possibility of a massive planet is also \nrestricted by current limits on the stellar radial velocity \nof V836 Tau, which constrain the mass of a companion within \n$0.4 - 1$\\,AU to $<5-10 M_J$ (L. Prato, personal communication). \n\nThese arguments are schematic in that they rely on theoretical\npredictions that have not been verified observationally. For\nexample, the sizes of gaps that will be induced by an orbiting\ncompanion of a given mass, and the extent to which stellar accretion\nwill be reduced, are poorly known from an observational point of\nview. Stellar companions have been found to establish large inner\nholes and to terminate stellar accretion in some transition objects\n(CoKu Tau\/4---Ireland \\& Kraus 2008; D'Alessio et al.\\ 2005) and\nnot in others (CS Cha---Guenther et al.\\ 2007; Espaillat et al.\\\n2007). The situation for lower mass companions is essentially\nunexplored. As theoretical predictions are tested, it will\nbe useful to examine in detail the range of companion masses that\nare consistent with the properties of V836 Tau.\n\n\n\\section{Summary and Future Directions}\n\nV836 Tau has been classified as a transition object (Strom et al.\\ 1989), \na system that may be on the verge of dissipating its disk, \npossibly as a consequence of planetesimal formation or giant \nplanet formation. These processes can reduce the continuum \nopacity in certain regions of the disk, producing an optically \nthin inner hole or a low column density gap. \nIf V836 Tau has such an optically thin region, the weak near-infrared \nexcess and the stronger $10\\hbox{$\\mu$m}$ excess in the system indicates \nthat the optically thin region is much smaller ($< 1$\\,AU) than the \noptically thin region in well-studied transition objects such as \nGM Aur and DM Tau (Calvet et al.\\ 2005), where the optically \nthin region is 3--20\\,AU in radius. \nThus CO fundamental emission, which probes the region $\\lesssim 2$\\,AU \n(Najita et al.\\ 2003), can potentially map out the radial structure of \nthe inner gaseous disk in V836 Tau to determine, for example, if \nthe inner disk has been truncated by a companion orbiting within 1\\,AU. \n\nAlong these lines, we find that the $v$=1--0 CO fundamental line \nprofiles of V836 Tau are unusual compared to those of \nother T Tauri stars in being markedly double-peaked. \nThe strength and shape of the line emission is consistent with \nemission from a Keplerian disk over a limited range of \nradii ($\\sim 0.05 - 0.4$\\,AU). \nFurther work is needed to determine whether the outer radius \nof the emission results from the physical truncation of the disk \nbeyond $\\sim 0.4$\\,AU or the truncated excitation of the \n$v$=1--0 CO fundamental transitions beyond this radius.\n\nA theoretical approach to this problem requires \nstudies of the thermal, chemical, and excitation \nstructure of disks at a level of detail that is appropriate for \ncomparison with observations. Studies of the possibility \nof non-LTE CO level populations in low accretion rate systems \nwould be particularly welcome. \nFor a more empirical approach to the problem, we might compare \nthe observed line profiles of V836 Tau with other low accretion \nrate systems in which the disk is expected to be radially continuous \nwithin a few AU. \nAs an example of the latter approach, we discussed the \nCO fundamental line profiles of LkCa15, a T Tauri star with a \nlow accretion rate similar to that of V836 Tau.\nThe more centrally peaked line profiles of LkCa15, if \nrepresentative of other low accretion rate systems, \nwould suggest that the double-peaked emission profiles of \nV836 Tau arise from a physically truncated inner disk. \n\nSuch a physically truncated inner disk might arise if the \nsystem has formed a Jovian mass planet that has cleared a gap in \nthe disk. A simple fit to the SED of V836 Tau is \nconsistent with a flared disk that has an optically thin \nregion within $\\sim 1$\\,AU.\nThus, a possible interpretation of the data is that \nan orbiting companion has created a gap between \na gaseous inner disk within 0.4\\,AU \nand an optically thick outer disk beyond 1\\,AU. \nSince the above fit to the SED is non-unique, it would be \nuseful to use both improved SED modeling techniques \n(e.g., Calvet et al.\\ 2005) \nand infrared interferometry (e.g., Ratzka et al.\\ 2007) \nto test the hypothesis that the dust disk has an \noptically thin gap or inner hole. \n\nIn systems that have formed a Jovian mass planet, \nsmall grains may be filtered out of the inward accretion flow \nat the outer edge of the gap (Rice et al.\\ 2006), rendering the \ndust distribution a poor tracer of the physical structure \nof the disk at smaller radii. \nGaseous disk tracers, like the CO fundamental emission \ndiscussed here, may then be {\\it needed} to probe disk structure \nat these smaller radii.\nThus, there is considerable motivation to expand the study of \ngaseous disk diagnostics beyond the present case, to \nunderstand more generally \nwhether and how well diagnostics such as CO fundamental emission \ncan probe the radial structure of gaseous disks. \n\n\n\n\n\n\n\n\n\n\n\n\\acknowledgments\n\nWe are grateful to Steve Strom for stimulating and \ninsightful discussions on this topic. \nWe also thank Lisa Prato for communicating her radial velocity \nresults in advance of publication. \nFinancial support for this work was provided by the NASA Origins \nof Solar Systems program (NNH07AG51I) and \nthe NASA Astrobiology Institute\nunder Cooperative Agreement No.\\ CAN-02-OSS-02 issued through the\nOffice of Space Science. This work was also supported by the Life and\nPlanets Astrobiology Center (LAPLACE).\nBasic research in infrared astronomy at the Naval Research Laboratory\nis supported by 6.1 base funding. \nThe authors wish to recognize and acknowledge the very significant\ncultural role and reverence that the summit of Mauna Kea has always\nhad within the indigenous Hawaiian community. We are most fortunate\nto have the opportunity to conduct observations from this mountain.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:level1}Introduction}\n\nThe possible existence of new interactions in nature with ranges of mesoscopic scale (millimeters to microns), corresponding to exchange boson masses in the 1 meV to 1 eV range and with very weak couplings to matter has been discussed for some time~\\cite{Leitner,Hill} and has recently begun to attract renewed scientific attention. Particles which might mediate such interactions are sometimes referred to generically as WISPs (Weakly-Interacting sub-eV Particles)~\\cite{Jae10} in recent theoretical literature. Many theories beyond the Standard Model, including string theories, possess extended symmetries which, when broken at a high energy scale, lead to weakly-coupled light particles with relatively long-range interactions such as axions, arions, familons, and Majorons~\\cite{Arvanitaki2010, PDG14}. The well-known Goldstone theorem in quantum field theory guarantees that the spontaneous breaking down of a continuous symmetry at scale $M$ leads to a massless pseudoscalar mode with weak couplings to massive fermions $m$ of order $g=m\/M$. The mode can then acquire a light mass (thereby becoming a pseudo-Goldstone boson) of order $m_{boson}=\\Lambda^{2}\/M$ if there is also an explicit breaking of the symmetry at scale $\\Lambda$~\\cite{Weinberg72}. New axial-vector bosons such as paraphotons~\\cite{Dobrescu2005} and extra Z bosons~\\cite{Appelquist2003} appear in certain gauge theories beyond the Standard Model. Several theoretical attempts to explain dark matter and dark energy also produce new weakly-coupled long-range interactions. The fact that the dark energy density of order (1 meV)${^4}$ corresponds to a length scale of $~100$ $\\mu$m also encourages searches for new phenomena on this scale~\\cite{Ade09}. \n\nA general classification of interactions between nonrelativistic fermions assuming only rotational invariance~\\cite{Dob06} reveals 16 operator structures involving the spins, momenta, interaction range, and various possible couplings of the particles. Of these sixteen interactions, one is spin-independent, six involve the spin of one of the particles, and the remaining nine involve both particle spins. Ten of these 16 possible interactions depend on the relative momenta of the particles. The addition of the spin degree of freedom opens up a large variety of possible new interactions to search for which might have escaped detection to date. Powerful astrophysical constraints on exotic spin-dependent couplings~\\cite{Raffelt1995a, Raffelt1995b, Raffelt2012} exist from stellar energy-loss arguments, either alone or in combination with the very stringent laboratory limits on spin-independent interactions from gravitational experiments~\\cite{Adelberger:2014}. However, a chameleon mechanism could in principle invalidate some of these astrophysical bounds while having a negligible effect in cooler, less dense lab environments~\\cite{Jain2006}, and the astrophysical bounds do not apply to axial-vector interactions~\\cite{Dob06}. These potential loopholes in the astrophysical constraints, coupled with the intrinsic value of controlled laboratory experiments and the large range of theoretical ideas which can generate exotic spin-dependent interactions, has led to a growing experimental activity to search for such interactions in laboratory experiments. \n\nMany experiments search for a monopole-dipole interaction~\\cite{Moody84} involving an exchange of spin-zero bosons with scalar and pseudoscalar couplings. This interaction violates $P$ and $T$ symmetry and in the nonrelativistic limit is proportional to $g_{S}g_{P}\\vec{\\sigma} \\cdot \\hat{r}$ where $g_{S}$ and $g_{P}$ are the scalar and pseudoscalar couplings, $\\vec{\\sigma}$ is the spin of one of the particles, and $\\vec{r}$ is the separation between the particles. Contrary to some expectations, experimental upper bounds on electric dipole moments, which are also $P$-odd and $T$-odd, do not in general rule out the existence of such bosons with masses in the meV to eV range~\\cite{Mantry2014}. Many of the experiments which have been performed to search for such interactions using polarized gases~\\cite{You96} and paramagnetic salts~\\cite{Chui93, Ni94, Ni99} are sensitive to ranges $\\lambda \\geq 1$ cm. Constraints on monopole-dipole interactions involving nucleons at smaller range have come from experiments using slow neutrons~\\cite{Bae07, Ser09, Ig09, Fed13, Jen14, Afach2015} and polarized helium and xenon gas~\\cite{Pok10, Pet10, Fu11, Zhe12, Chu13, Bul13, Tul13, Guigue15}. Many experiments have also sought exotic spin-spin interactions proportional to $g_{P}^{2} \\vec{\\sigma}_{1} \\cdot \\vec{\\sigma}_{2}$, where $g_{P}$ is the pseudoscalar coupling and $\\vec{\\sigma}_{1}$ and $\\vec{\\sigma}_{2}$ are the spins of the two particles. Such a spin-dependent potential with a dipole-dipole form is one of the three velocity-independent spin-spin interactions which can come from 1-boson exchange between two nonrelativistic spin-$1\/2$ fermions~\\cite{Dob06}. Separated ensembles of polarized atoms~\\cite{Wine91, Gle08, Vas09, Hunter2013} have set limits on long-range spin-dependent nucleon interactions, and analysis of high precision spectroscopy in molecular hydrogen~\\cite{Ram79, Ledbetter2013} has set limits on atomic-range spin-dependent nucleon interactions. Torsion balance measurements have recently set new stringent limits on both monopole-dipole interactions and dipole-dipole interactions involving polarized electrons with macroscopic ranges~\\cite{Rit93, Ham07, Heckel2008, Hoedl11, Heckel2013, Terrano2015}. Comparison of precision QED calculations with atomic physics data~\\cite{Karshenboim2011} has set strong limits on exotic spin-dependent electron interactions with ranges at the atomic scale. Ion traps~\\cite{Kotler2015} have recently constrained exotic spin-spin interactions between polarized electrons of the form $g_{A}^{2} \\vec{\\sigma}_{1} \\cdot \\vec{\\sigma}_{2}$ from spin-$1$ boson exchange at micron distance scales. New experimental methods to search for polarized electron couplings using rare earth-based ferrimagnetic test masses~\\cite{Leslie2014}, paramagnetic insulators~\\cite{Chu2015}, and spin-exchange relaxation-free (SERF) magnetometers~\\cite{Chu2016} have been proposed. \n\nLaboratory constraints on possible new interactions of mesoscopic range which depend on {\\it both} the spin {\\it and} the relative momentum are less common, since the polarized electrons or nucleons in most experiments employing macroscopic amounts of polarized matter typically possess $\\langle\\vec{p}\\rangle=0$ in the lab frame. Some limits exist for spin-$0$ boson exchange. Kimball {\\it et al.}~\\cite{Kim10} used measurements and calculations of cross sections for spin exchange collisions between polarized $^{3}$He and Na atoms to constrain possible new spin-dependent interactions between neutrons and protons. Hunter~\\cite{Hunter2014} exploited the existence of a small but nonzero polarization of the electrons in the Earth combined with atomic magnetometry to place very stringent constraints on a large number of spin and velocity-dependent interactions involving polarized electrons for macroscopic force ranges. \n\nSpin and velocity-dependent interactions from spin-$1$ boson exchange can be generated by a light vector boson $X_{\\mu}$ coupling to a fermion $\\psi$ with an interaction of the form $\\mathcal{L}_{I}=\\bar{\\psi}(g_{V}\\gamma^{\\mu}+g_{A}\\gamma^{\\mu}\\gamma_{5})\\psi X_{\\mu}$, where $g_{V}$ and $g_{A}$ are the vector and axial couplings. In the nonrelativistic limit, this interaction gives rise to two interaction potentials of interest depending on both the spin and the relative momentum~\\cite{Pie11}: one proportional to $g_{A}^{2}\\vec{\\sigma}\\cdot(\\vec{v}\\times\\hat{r})$ and another proportional to $g_{V}g_{A}\\vec{\\sigma}\\cdot\\vec{v}$. As noted above, many theories beyond the Standard Model can give rise to such interactions. For example, spontaneous symmetry breaking in the Standard Model with two or more Higgs doublets with one doublet responsible for generating the up quark masses and the other generating the down quark masses can possess an extra U(1) symmetry generator distinct from those which generate $B$, $L$, and weak hypercharge $Y$. The most general U(1) generator in this case is some linear combination $F=aB + bL +cY + dF_{ax}$ of $B$, $L$, $Y$, and an extra axial U(1) generator $F_{ax}$ acting on quark and lepton fields, with the values of the constants $a,b,c,d$ depending on the details of the theory. The new vector boson associated with this axial generator can give rise to $\\mathcal{L}_{I}$ above~\\cite{Fayet:1990}. \n\nNeutrons have recently been used with success to tightly constrain possible weakly coupled spin-dependent interactions of mesoscopic range~\\cite{Dubbers11}. A polarized beam of slow neutrons can have a long mean free path in matter and is a good choice for such an experimental search~\\cite{Nico05b}. Piegsa and Pignol~\\cite{Pie12} recently reported improved constraints on the product of axial vector couplings $g_{A}^{2}$ in this interaction. Polarized slow neutrons which pass near the surface of a plane of unpolarized bulk material in the presence of such an interaction experience a phase shift which can be sought using Ramsey's well-known technique of separated oscillating fields~\\cite{Ramsey:1950}. Other experiments have constrained $g_{V}g_{A}^{n}$. Yan and Snow reported constraints on $g_{V}g_{A}^{n}$ using data from a search for parity-odd neutron spin rotation in liquid helium~\\cite{Yan13}. Adelberger and Wagner~\\cite{Adelberger:2014} combined experimental constraints on $g_{V}^{2}$ from searches for violations of the equivalence principles and $g_{A}^{2}$ from other sources to set much stronger constraints on $g_{V}g_{A}^{n}$ for interactions with ranges beyond 1 cm. Yan~\\cite{Yan15} analyzed the dynamics of ensembles of polarized $^{3}$He gas coupled to the Earth to constrain $g_{V}g_{A}^{n}$ for interactions with ranges beyond 1 cm with laboratory measurements.\n\nThe strength of nearly all of these constraints is very weak compared to spin-independent interactions. Very stringent constraints exist on spin-independent Yukawa interactions arising from light scalar or vector boson exchange. The present constraints on the dimensionless coupling constants are $g^2_{S,V}$ $ \\leq 10^{-40}$ for an exchange boson with a mass between 10 meV and 100 $\\mu$eV \\cite{Decca}, which corresponds to a length scale between 10 $\\mu$m and 1 mm. Experimental constraints on possible new interactions of mesoscopic range which depend on the spin of one or both of the particles are much less stringent than those for spin-independent interactions~\\cite{Leslie2014, Antoniadis11}. Several facts contribute to this situation. First of all such experiments require one or both of the particles under investigation to be polarized. Even if one can achieve perfect polarization, only the valence fermions in the ground states of bound electrons and nucleons are accessible. Experimental polarization techniques are often specific to particular atoms or nuclei and vary widely in their efficiency. Macroscopic objects with large nuclear or electron polarization are not easy to arrange without an environment that includes large external magnetic fields. Even if one succeeds to polarize ensembles of particles in low ambient magnetic fields, the magnetic moments of the spin-aligned particles themselves generate magnetic fields which eventually interact with and depolarize other members of the ensemble. Both internal and external magnetic fields can produce large systematic effects in delicate experiments.\nAnother reason for the differing sensitivities follows from the fact that, for the small momentum transfers accessed in interactions between two nonrelativistic massive Dirac fermions, the amplitude for a helicity flip associated with a spin-dependent interaction at the fermion-boson vertex can be suppressed by a factor $(\\mu\/m)^n$, where $\\mu$ is the mass of the exchanged boson, $m$ is the fermion mass and $n$ = 1, 2, or 3 depending on the type of interaction. This suppression arises at parity-odd vertices such as $i g_{P} \\gamma_5$, $g_{V}\\boldsymbol{\\gamma}$, and $g_{A} {\\gamma_0}{\\gamma_5}$ where in order for parity to be conserved the boson must be emitted with nonzero angular momentum relative to the initial and final nonrelativistic fermions, thus giving rise to an angular momentum suppression of order $(\\mu\/m)^n$. The only case of a spin-dependent interaction with no mass suppressions arises in the ``dipole-dipole'' interaction mediated by an axial boson with even-parity coupling $g_{A} \\boldsymbol{\\gamma}{\\gamma_5}$.\nThis is one of the reasons why, for example the constraint on an electron axial vector coupling $(g^{e}_{A})^2 \\sim 10^{-40}$ for $\\mu \\geq 1$ $\\mu$eV \\cite{Heckel2013} is orders of magnitude stronger than the constraint on $(g^{N}_{A})^2 \\sim 10^{-13}$ for $\\mu \\sim 100$ $\\mu$eV \\cite{Pie12}, where the latter was obtained from a ``monopole-dipole'' interaction arising from parity-odd vertices. \\\\\nThe huge difference in the strength of these constraints on spin-dependent and spin-independent interactions motivated us to investigate whether or not limits on spin-dependent couplings can be improved using the constraints from existing spin-independent data. Exchange of two bosons can flip the helicity of the fermions twice and generate a spin-{\\it independent} contribution to the interaction energy between two fermions. Although two boson exchange between fermions generates an interaction energy of order $g^{4}$ and direct spin-dependent experiments look for effects from single boson exchange of order $g^{2}$, the strong constraints from spin-independent experiments can still be better than direct experiments in certain situations. Since searches for new spin-independent interactions span a broader range of exchange boson masses than the spin-dependent searches, such an analysis can extend constraints on spin-dependent interactions to new length scales where experimental coverage is either poor or nonexistent. Many experiments to search for spin-independent interactions are probing the smaller distance scales where limits on spin-dependent interactions are poor \\cite{kamiya, tanya}.\n\nSimilar analyses motivated by the same considerations have been conducted in the past. The functional form for 2-boson exchange with pseudoscalar couplings has been derived before and applied in different contexts~\\cite{ferrer,Drell,mostepanenko,grifols} such as tests of the inverse square law of gravity (ISL) and the weak equivalence principle~\\cite{2-pseudoscalar} to derive the first direct limits on $g^N_{P}$. The most recent constraints on spin-0 boson exchange with pseudoscalar couplings $g^N_{P}$ to nucleons~\\cite{klimchitskaya} span bosons masses between $ 0.01$ $\\mu$eV and $1$~eV. \n\nTo the best of our knowledge, no similar analysis has been performed for other spin-dependent couplings and no functional forms for the spin-independent component of the interaction energy arising from other types of 2-boson exchange have been exhibited in the nonrelativistic limit of interest to us. The aim of this paper is to calculate the dominant long-range contribution to the interaction energy between two nonrelativistic spin-1\/2 Dirac fermions from double boson exchange of spin-0 and spin-1 bosons with spin-dependent couplings of the form $g_S^{2}g_P^{2}$, and $g_V^{2}g_A^{2}$. The case of two axial vector exchange requires a special treatment and will be explored in another paper. In addition, we use the existing 2-boson calculation for pseudoscalar exchange in a reanalysis of data from a short-range gravity experiment to derive an improved constraint on $(g^N_{P})^2$, the pseudoscalar coupling for nucleons, in the range between $40$ and $200~\\mu$m of about a factor of 5 compared to previous limits. This analysis constitutes an existence proof that sensitive experimental searches for spin-independent interactions can also yield the most stringent constraints on spin-dependent interactions at certain distance scales. \n\nThe rest of this paper is organized as follows. In section~\\ref{Problem and Method Section} we define the problem and specify the method of calculation. The calculation itself along with the results are outlined in section~\\ref{Interaction Energy Section}. In section~\\ref{Constraints Section}, we present our derivation of a new limit on nucleon pseudoscalar couplings from analysis of an experiment to probe violations of the inverse square law in short-range gravity. Natural units with $\\hbar = c = 1 $ are used throughout the paper. \\\\\n\n\n\\section{Definition of the Problem and Method} \n\\label{Problem and Method Section}\n\nSome groups have undertaken exact calculations of the amplitudes for double boson exchange valid in the relativistic limit~\\cite{Guichon}. It is not our purpose here to attempt a complete calculation of this type. We are interested in determining the leading long-range contributions to the spin-independent component of the interaction energy associated with the exchange of two massive spin-0 and spin-1 bosons between two massive spin-1\/2 Dirac fermions with various types of spin-dependent couplings. The distance scale regime we are interested in is $r \\geqslant \\frac{1}{\\mu} \\gg \\frac{1}{m}$, where $r=|\\boldsymbol{r}_1-\\boldsymbol{r}_2|$ is the separation between the two fermions, $\\mu$ is the exchange boson mass, and $m$ is the fermion mass. \n\n\n\nMany authors have performed similar calculations for various purposes using different approaches. Iwasaki studied this problem using noncovariant perturbation theory \\cite{iwasaki}. Feinberg and Sucher used dispersion methods in covariant perturbation theory~\\cite{sucher88} to extract long-range effects from loop corrections. Holstein examined this problem using effective field theory (EFT)~\\cite{holstein}. In this paper we shall use a nonrelativistic approach based on ``old fashioned'' perturbation theory (OFPT) using time-ordered diagrams. The reason we are pursuing this approach is that it suffices for the direct identification of spin-independent long-range terms in the nonrelativistic limit that we are interested in. In dispersion methods obtaining long-range effects from loop corrections amounts to calculating $t$-channel discontinuities in Feynman diagrams and performing a Laplace transformation which, although doable in principle, is not necessary for our purposes. A similar procedure could be realized in EFT by recognizing that long-range components are associated with pieces in the scattering amplitude that are non-analytic in momenta transfer~\\cite{holstein}. \n\\\\\n\\\\\nWe first consider the elastic scattering of two spin-1\/2 Dirac fermions of masses $m_1$ and $m_2$. We denote the incoming momenta by $\\boldsymbol{p}_1$ and $\\boldsymbol{p}_2$ and the outgoing momenta by $\\boldsymbol{p}_{1}'$ and $\\boldsymbol{p}_{2}'$. The on-shell transition amplitude is given by \n\\begin{equation}\nT_{fi}(Q) = (2 \\pi)^3 \\delta( \\boldsymbol{p}'_1 + \\boldsymbol{p}'_2 - \\boldsymbol{p}_1 - \\boldsymbol{p}_2) N_f M_{fi}(Q) N_i .\n\\label{eq:1}\n\\end{equation} \nwhere $\\boldsymbol{Q}$ is the momentum transfer to the fermion of mass $m$. Here $M_{fi}$ is the Feynman scattering amplitude and $N_f$ and $N_i$ are normalization factors associated with the incoming and outgoing particles in the initial and final states which in the nonrelativistic limit are taken to be unity~\\cite{1}. We define the interaction energy corresponding to the long-range contribution from $M^{(2)}(Q)$ by ~\\cite{2}.\n\n\\begin{equation}\nV^{(2)}(r) = \\int \\frac{d^{3}Q}{(2\\pi)^3} e^{-i\\boldsymbol{Q}\\cdot \\boldsymbol{r}} \\: M^{(2)}_{fi}(Q) .\n\\label{eq:2}\n\\end{equation} \n\n\n\\section{ Calculation of the Interaction Energy}\n\\label{Interaction Energy Section}\nWe start with the Hamiltonian density \n\\begin{equation}\nH= \\overline{\\psi}(\\boldsymbol{\\gamma}\\cdot \\boldsymbol{p}+m) \\psi + H_{\\rm int},\n\\label{eq:4}\n\\end{equation}\nwhere $\\psi$ is the 4-component fermion field. The first term is the free fermion Hamiltonian density and $H_{\\rm int}$ is the interaction Hamiltonian density given by \n\\begin{equation}\nH_{\\rm int}=\\overline{\\psi} [ (g_S + i g_P \\gamma_5) \\phi + (g_V \\gamma^{\\mu}+ g_A \\gamma^{\\mu}\\gamma_5) A_{\\mu}] \\psi,\n\\label{eq:5}\n\\end{equation}\nwhere $\\phi$ and $A_{\\mu}$ are the massive spin-$0$ and spin-$1$ boson fields, respectively. The nonrelativistic limit of the Hamiltonians in Eqs.~\\eqref{eq:4} and \\eqref{eq:5} are derived by performing a Foldy-Wouthuysen unitary transformation~\\cite{cohen} to eliminate all pair production diagrams associated with higher energies which are subdominant in our limit. For our purposes we need only expand the effective Hamiltonian to order $p\/m$: \n\\begin{subequations}\n\\begin{eqnarray}\nH^{\\rm eff}_{S} & = & g_S\\psi^+ \\psi \\phi, \\label{Heff S} \\\\\nH^{\\rm eff}_{P} & = & \\psi^+[-i \\frac{g_P}{2m}\\boldsymbol{\\sigma}\\cdot \\boldsymbol{k} \\phi +\\frac{g_P^2}{2m}\\phi^2 ]\\psi, \\label{Heff P} \\\\\nH^{\\rm eff}_{V}& = & \\psi^+ [g_VA_0-\\frac{g_V}{2m}(\\pmb{p}+\\pmb{p}')\\cdot \\pmb{A} -i\\frac{g_V}{2m}\\pmb{\\sigma}\\cdot \\boldsymbol{k}\\times \\pmb{A} \\nonumber \\\\\n&& \\mbox{} +\\frac{g_V^2}{2m}\\pmb{A}^2] \\psi, \\label{Heff V} \\\\\nH^{\\rm eff}_{A}& = & \\psi^+ [-g_A \\pmb{\\sigma}\\cdot\\pmb{A}+\\frac{g_A}{2m}\\pmb{\\sigma}\\cdot(\\pmb{p}+\\pmb{p}')A_0+\\frac{g_A^2}{2m} A_0^2]\\psi, \\nonumber \\\\\n\\label{Heff A}\n\\end{eqnarray} \n\\end{subequations}\nwhere $\\psi$ is now a 2-component fermion field associated with a positive energy spinor. Here $\\boldsymbol{p}$ and $\\boldsymbol{p}'$ are the incoming and outgoing momenta of the fermion in each vertex, $\\boldsymbol{k}$ is the boson momentum, and $\\boldsymbol{A}$ and $A_0$ are the space and time components of the massive spin-1 field, respectively. \n\n\nIn OFPT momentum (but not energy) is conserved at the vertices. The propagator for internal lines $\\frac{1}{E_i\\: - \\: E_n}$, where $E_i$ is the energy of the initial state and $E_n$ is the energy of the intermediate state, is multiplied by a sum over the transverse and longitudinal modes, $\\delta_{ij} \\: - \\: \\frac{ k_i k_j}{\\mu^2}$ or $-1 + \\frac{{\\omega}^2}{\\mu^2}$, for each massive spin-1 exchange boson present in the diagram. The internal momenta are summed over in the usual way. \n\n\nWe will only derive the spin-independent long-range contributions to the interaction energy arising from the following three cases: exchanges with two pseudoscalar couplings, exchanges with one scalar and one pseudoscalar coupling, and exchanges with one vector coupling and one axial vector coupling. The case of two axial vector exchange requires insertions from higher order corrections in the small momentum expansion of the Hamiltonian and will be explored in detail in another paper. Although exotic spin-$0$ and spin-$1$ boson exchange could appear together in box and cross box diagrams we are not interested in this case for our purposes. \n\n\\begin{figure*}\n\\includegraphics[scale=0.70]{LoopDiagrams.pdf}\n\\caption{\\label{feynman diagram figure} The relevant 2-boson exchange time-ordered diagrams. Solid lines represent the fermions while wavy lines represent massive spin-0 or spin-1 bosons.}\n\\end{figure*}\nFor exchanges with two pseudoscalar couplings the leading effect comes from the double seagull diagrams (a) and (b) in Fig.~\\ref{feynman diagram figure}. Effects arising from diagrams (c)--(h) are suppressed by a factor of $(\\mu\/m)^3$ as can be inferred from the form of $H^{\\rm eff}_{P}$ in Eq.~\\eqref{Heff P}. The transition amplitude is \n\\begin{eqnarray}\nT^{(2)}_{P-P} & = &- \\frac{g^2_{P,1}g^2_{P,2} }{4 m_1 m_2} \\: \\int \\: \\frac{d^3k d^3q}{(2\\pi)^6} \\left[ \\frac{1}{ \\omega_k \\omega_q (\\omega_k \\:+\\: \\omega_q) }\\right.\\nonumber \\\\\n& & \\left.{ \\delta( \\boldsymbol{p}'_1 - \\boldsymbol{k} - \\boldsymbol{q} - \\boldsymbol{p}_1 ) \\delta (\\boldsymbol{p}'_2 + \\boldsymbol{k} + \\boldsymbol{q} - \\boldsymbol{p}_2) }\\right]. \n\\label{eq:7}\n\\end{eqnarray} \nFrom Eqs.~\\eqref{eq:1} and~\\eqref{eq:2}, the interaction energy is related to $T^{(2)}_{P-P}$ via \n\\begin{equation}\n\\begin{aligned} \n&{V}^{(2)}_{P-P}(r)= - \\frac{g^2_{P,1}g^2_{P,2}}{4 m_1 m_2}\\: \\int \\: \\frac{d^{3}Q}{(2\\pi)^3} e^{-i\\boldsymbol{Q}\\cdot \\boldsymbol{r}} \\\\\n& \\times \\int \\frac{d^3k d^3q}{(2\\pi)^3} \\frac{ \\delta( \\boldsymbol{p}'_1 - \\boldsymbol{k} - \\boldsymbol{q} - \\boldsymbol{p}_1 ) \\delta (\\boldsymbol{p}'_2 + \\boldsymbol{k} + \\boldsymbol{q} - \\boldsymbol{p}_2) }{ \\omega_k \\omega_q (\\omega_k \\:+\\: \\omega_q) } .\n\\label{eq:8}\n\\end{aligned}\n\\end{equation} \nNow by carrying out the integral over $\\boldsymbol{Q}$ first we obtain \n\\begin{eqnarray}\n{V}^{(2)}_{P-P}(r) &= & - \\frac{g^2_{P,1}g^2_{P,2}}{4 m_1 m_2} \\: \\int \\: \\frac{d^3k d^3q}{(2\\pi)^6} \\frac{e^{-i({\\boldsymbol{k} + \\boldsymbol{q})\\cdot \\boldsymbol{r}}}}{ \\omega_k \\omega_q (\\omega_k \\:+\\: \\omega_q) } \\nonumber \\\\\n&= & -\\frac{g^2_{P,1}g^2_{P,2}}{4m_1 m_2}\\frac{\\mu K_1(2\\mu r)}{8 \\pi^3 r^2},\n\\label{eq:9}\n\\end{eqnarray}\nwhere $K_1(x)$ is the modified Bessel function of the second kind. This agrees with the result previously derived by Drell and Huang \\cite{Drell} and Ferrer and Nowakowski~\\cite{ferrer}. This result, however, is not correct for the exchange of two pseudoscalar bosons which have a derivative coupling of the form $ \\frac{g_P}{m} \\overline{\\psi} \\gamma_{\\mu} \\gamma_5 \\psi \\partial^{\\mu} \\phi $. Derivative and non-derivative pseudoscalar couplings give the same interaction energy in first order perturbation theory but not on second order. The long-range behavior arising from two massless boson exchange with pseudoscalar derivative couplings to matter have been calculated in the limit as the exchange boson mass goes to zero and shown to be highly suppressed relative to the analogous case with non-derivative pseudoscalar couplings\\cite{grifols}. This is also expected to follow for non-massless bosons, but we have not calculated this case in this paper. Since the case of pseudoscalar boson exchange is especially interesting from a physics point of view we plan to calculate this case and present the results in a later paper. \n\n\nFor interactions with one scalar coupling and one pseudoscalar coupling, the leading spin-independent contribution arises from diagrams (c)--(h) of Fig.~\\ref{feynman diagram figure} with two orders of $g_{S} \\phi$ and one order of $(g^2_{P}\/2m) \\phi^2$. The transition amplitude is \n\n\\begin{widetext}\n\\begin{eqnarray}\nT^{(2)}_{S-P} &= &\\frac{g^2_{S,1}g^2_{P,2}}{2 m_2} \\int \\frac{d^3k d^3q d^3l_1}{(2 \\pi)^6} \\Bigg\\{\\frac{1}{ 4 \\omega_k \\omega_q} \\Bigg[ \\frac{\\delta(\\boldsymbol{p}'_2 + \\boldsymbol{k} + \\boldsymbol{q} - \\boldsymbol{p}_2) \\delta(\\boldsymbol{p}'_1 -\\boldsymbol{q} - \\boldsymbol{l}_1) \\delta(\\boldsymbol{l}_1 -\\boldsymbol{k} - \\boldsymbol{p}_1)}{( \\omega_q+X_1)(\\omega_k \\:+\\: \\omega_q)}+ \\nonumber \\\\\n&& \n\\frac{\\delta(\\boldsymbol{p}'_2 - \\boldsymbol{k} - \\boldsymbol{q} - \\boldsymbol{p}_2) \\delta(\\boldsymbol{p}'_1 +\\boldsymbol{q} - \\boldsymbol{l}_1) \\delta(\\boldsymbol{l}_1 +\\boldsymbol{k} - \\boldsymbol{p}_1)}{ (\\omega_k+X_1)(\\omega_k \\:+\\: \\omega_q)} + \\frac{\\delta(\\boldsymbol{p}'_2 + \\boldsymbol{q} - \\boldsymbol{k} - \\boldsymbol{p}_2) \\delta(\\boldsymbol{p}'_1 -\\boldsymbol{q} - \\boldsymbol{l}_1) \\delta(\\boldsymbol{l}_1 +\\boldsymbol{k} - \\boldsymbol{p}_1)}{ (\\omega_k+X_1)( \\omega_q+X_1)} \\Bigg] \\nonumber \\\\\n&& \\mbox{} + 1 \\leftrightarrow 2 , \\boldsymbol{k} \\leftrightarrow \\boldsymbol{-k}, \\boldsymbol{q} \\leftrightarrow \\boldsymbol{-q} \\}.\n\\label{eq:18}\n\\end{eqnarray}\n\\end{widetext}\nExpanding in the limit $X \\ll \\omega_k$ and taking advantage of symmetry under $\\boldsymbol{k}$ and $\\boldsymbol{q}$ gives\n\\begin{eqnarray}\nT^{(2)}_{S-P} &=& {\\frac{g^2_{S,1}g^2_{P,2}}{2 m_2} } \\int \\frac{d^3k d^3qd^3l_1}{(2\\pi)^6} \\Bigg\\{\\frac{1}{ 4 \\omega_k \\omega_q} \\nonumber \\\\\n&& \\delta(\\boldsymbol{p}'_2 + \\boldsymbol{k} + \\boldsymbol{q} - \\boldsymbol{p}_2) \\delta(\\boldsymbol{p}'_1 -\\boldsymbol{q} - \\boldsymbol{l}_1) \\delta(\\boldsymbol{l}_1 -\\boldsymbol{k} - \\boldsymbol{p}_1) \\nonumber \\\\\n&& \\Bigg[ \\frac{1}{\\omega_q(\\omega_k \\:+\\: \\omega_q)}+\\frac{1}{ \\omega_k(\\omega_k \\:+\\: \\omega_q)} + \\frac{1}{ \\omega_k \\omega_q} \\Bigg] \\nonumber \\\\\n&& \\mbox{}+ 1 \\leftrightarrow 2 , \\boldsymbol{k} \\leftrightarrow \\boldsymbol{-k}, \\boldsymbol{q} \\leftrightarrow \\boldsymbol{-q}\\}. \n\\label{eq:19}\n\\end{eqnarray}\nThe interaction energy is then given by \n\\begin{equation} \n{V}^{(2)}_{S-P}(r)= \\Bigg({\\frac{g^2_{S,1}g^2_{P,2}}{2 m_2} + \\frac{g^2_{S,2}g^2_{P,1}}{2 m_1} }\\Bigg) \\frac{e^{-2 \\mu r}}{32 \\pi^2 r^2}.\n\\label{eq:20}\n\\end{equation} \nThe leading spin-independent contribution for the case of one vector coupling and one axial vector coupling also follows from diagrams (c)--(h) of Fig.~\\ref{feynman diagram figure}. Two different processes give rise to this interaction at this order: one from two factors of $- g_{A} {\\boldsymbol{\\sigma}} \\cdot {\\boldsymbol{A}} $ with one factor of $(g^2_{V}\/2m)$ ${\\boldsymbol{A}^2}$ and the other from two factors of $g_{V} A_0$ with one factor of $(g^2_{A}\/2m) A^2_0$. In the limit $X \\ll \\omega_k$, the vector-axial interaction energy is given by\n\\begin{equation}\n\\begin{aligned}\n&{V}^{(2)}_{V-A}(r)= \\: \\int \\: \\frac{d^3k d^3q}{(2 \\pi)^6} \\Bigg[ \\Bigg( {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_1} } +{\\frac{g^2_{V,2}g^2_{A,1}}{2 m_2} } \\Bigg) \\frac{\\boldsymbol{k}^2 \\boldsymbol{q}^2}{\\mu^4}+ \\\\\n&\\Bigg( {\\frac{g^2_{V,2}g^2_{A,1}}{2 m_1} } + {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_2} }\\Bigg) \\Bigg({3-\\frac{\\boldsymbol{q}^2}{\\mu^2}-\\frac{\\boldsymbol{k}^2}{\\mu^2}+\\frac{(\\boldsymbol{k}\\cdot \\boldsymbol{q})^2}{\\mu^4} } \\Bigg) \\Bigg]\\\\ \n& \\frac{e^{-i({\\boldsymbol{k} + \\boldsymbol{q})\\cdot \\boldsymbol{r}}} }{ 2 \\omega_k \\omega_q}\\Bigg[ \\frac{1}{\\omega_q(\\omega_k \\:+\\: \\omega_q)}+\\frac{1}{ \\omega_k(\\omega_k \\:+\\: \\omega_q)} + \\frac{1}{ \\omega_k \\omega_q} \\Bigg].\n\\label{eq:21}\n\\end{aligned}\n\\end{equation}\n\nIntegration over $\\boldsymbol{k}$ and $\\boldsymbol{q}$ gives\n\\begin{eqnarray}\n{V}^{(2)}_{V-A}(r) & = & \\Bigg[ {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_1} }+{\\frac{g^2_{V,2}g^2_{A,1}}{2 m_2} } \\nonumber \\\\\n&& \\mbox{}+ 2 \\Bigg( {\\frac{g^2_{V,2}g^2_{A,1}}{2 m_1} }+ {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_2} }\\Bigg) \\nonumber \\\\\n &&\\Bigg( 3+ \\frac{2}{\\mu r}+ \\frac{5}{(\\mu r )^2} + \\frac{6}{(\\mu r )^3} + \\frac{3}{(\\mu r )^4} \\Bigg)\\Bigg] \\nonumber \\\\ \n && \\times \\frac{e^{-2 \\mu r}}{ 16 \\pi^2 r^2},\n\\label{eq:22}\n\\end{eqnarray}\nwhich is the same as Eq.~\\eqref{eq:20} except for extra terms due to the sum over polarization states. These terms possess singularities in the $\\mu \\rightarrow 0$ limit due to the inclusion of the longitudinal component of the massive spin-1 field in the absence of a conserved current~\\cite{Karshenboim2011, malta, Fischback, Fayet}. As we never let $\\mu \\to 0$ by assumption this infrared singularity is not realized in our case. The range of validity of Eq.~(\\ref{eq:22}) is $r \\gg 1\/\\mu \\gg 1\/m_1, 1\/m_2$ with $\\mu$ finite, in which case it simplifies to\n\\begin{eqnarray}\n{V}^{(2)}_{V-A}(r) & \\simeq & \\left[ {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_1} }+{\\frac{g^2_{V,2}g^2_{A,1}}{2 m_2} } \\right. \\nonumber \\\\\n&& \\mbox{} + \\left. 6 \\left( {\\frac{g^2_{V,2}g^2_{A,1}}{2 m_1} }+ {\\frac{g^2_{V,1}g^2_{A,2}}{2 m_2} }\\right)\\right] \\frac{e^{-2 \\mu r}}{ 16 \\pi^2 r^2}.\n\\phantom{spa}\n\\label{eq:23}\n\\end{eqnarray}\n\\section{Constraints from the Indiana Short-Range Gravity Experiment} \n\\label{Constraints Section}\n\n\nTo illustrate the potential power of these results, we have used existing data\nfrom a previous short-range gravity experiment to constrain the couplings in\nthe interaction energies in Eqs.~\\eqref{eq:9}, \\eqref{eq:20}, and \\eqref{eq:23}. \nThis experiment is optimized for sensitivity to macroscopic, spin-independent\nforces beyond gravity at short range, which in turn could\narise from exotic elementary particles or even extra spacetime\ndimensions. It is described in detail elsewhere~\\cite{Long03,Yan14};\nhere we concentrate on the essential features. \n\nThe experiment is illustrated in Fig.~\\ref{feynman diagram figure} of Ref.~\\cite{Long15}. The\ntest masses consist of 250~$\\mu$m thick planar tungsten \noscillators, separated by a gap of 100~$\\mu$m, with a stiff conducting\nshield in between them to suppress \nelectrostatic and acoustic backgrounds. Planar geometry\nconcentrates as much mass as possible at\nthe scale of interest, and is nominally null with\nrespect to $1\/r^{2}$ forces. This is effective in suppressing the\nNewtonian background relative to exotic short-range effects, and is\nwell-suited for testing interactions of the form $e^{-\\mu r}$, and\n$K_{1}(\\mu r)$. The force-sensitive\n``detector'' mass is driven by the force-generating ``source'' mass\nat a resonance near 1~kHz, placing a heavy burden on\nvibration isolation. The 1~kHz operation is chosen since at this\nfrequency it is possible to construct a simple vibration\nisolation system. This design has proven effective for suppressing all\nbackground forces to the level of the thermal noise due to dissipation\nin the detector mass~\\cite{Yan14}. After a run in 2002, the experiment set the\nstrongest limits on forces beyond gravity between 10 and\n100~$\\mu$m~\\cite{Long03}. The experiment has since been optimized to\nexplore gaps below 50~$\\mu$m, and new force data were acquired in\n2012. These data have been used to set limits on Lorentz invariance\nviolation in gravity~\\cite{Long15, Shao16}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{V1gPgP_v2.jpg}\n\\caption{\\label{fig:limits} Limits on the pseudoscalar coupling for nucleons. Red dashed curve is from this work. Black solid and black dashed curves follow from Ref.\\ \\cite{2-pseudoscalar} and \\cite{klimchitskaya}, respectively. }\n\\end{figure}\nAnalysis of the 2012 data for evidence of double boson exchange\nfollows that in Ref.~\\cite{Long03} for Yukawa-type mass-coupled\nforces. Eqs.~\\eqref{eq:9}, \\eqref{eq:20}, and \\eqref{eq:23} are converted to forces and\nintegrated numerically by Monte Carlo techniques over the 2012\nexperimental geometry, using the parameters in Refs.~\\cite{Long03}\nand~\\cite{Long15} and their errors. Systematic errors from the\ndimensions and positions of the test masses are determined at this stage, by\ncomputing a population of force values generated from a spread of\ngeometries based on the metrology errors. Gaussian likelihood\nfunctions for the experiment are constructed using the difference\nbetween the measured force and the numerical expressions for the\ndouble exchange forces as the means.\n\n\nLimits on the double boson exchange interactions are determined by\nintegration of the likelihood functions over the spin-dependent\ncouplings (which are free parameters in the likelihood functions), for\nseveral values of the range $\\lambda = 1\/\\mu$. Results for the\n2$\\sigma$ limits on the coupling $(g_{P}^{N})^{2}$ in Eq.~\\eqref{eq:9} are shown\nin Fig.~\\ref{fig:limits}. The constraints are more sensitive than previously\npublished limits \\cite{2-pseudoscalar, klimchitskaya} by about a factor of 5 in the range near\n100~$\\mu$m. Analysis of Eqs.~ \\eqref{eq:20}, and \\eqref{eq:23} is still in progress with the understanding that \\eqref{eq:23} is only applicable for $\\mu r \\gg 1$. \n\n\\section{Conclusion}\nWe have derived the leading-order spin-independent contribution to the interaction energy arising from the exchange of two light massive bosons between two spin-1\/2 Dirac fermions in the nonrelativistic limit. Our expressions agree with previous calculations in the literature where they exist. The functional forms derived in this paper open up an opportunity to constrain, using existing spin-independent data, spin-dependent couplings over new length scales that are outside the sensitivity of current spin-dependent experiments. We also used our expressions to reanalyze data from a short-range gravity experiment. From this analysis we derive a new limit on pseudoscalar couplings for nucleons which is more sensitive than direct constraints from other existing spin-dependent experiments. These limits can be further improved by reconfiguring existing experiments to make them more sensitive to the 2-BEP functional forms. \n\n\n\\begin{acknowledgments}\nThe work of S. A., J. C. L., and W. M. S. was supported\nby the U.S. National Science Foundation Grants No. PHY-\n1306942 and No. PHY-1614545, and by the Indiana\nUniversity Center for Spacetime Symmetries. S. A.\nacknowledges support from a King Abdullah Fellowship.\nWe also thank E. Fischbach for useful discussions.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Epsilon expansion: Some details}\n\\subsection{Conventions}\\label{secConv}\nIn section \\ref{Sec:epsilonbootstrap}, we use\n\\begin{equation}\\label{NRdef}\n\\begin{split}\n\\mathcal{N}_{\\Delta, \\ell}&=\\frac{2^{\\ell}(\\Delta+\\ell-1) \\Gamma^{2}(\\Delta+\\ell-1) \\Gamma(\\Delta-h+1)}{\\Gamma(\\Delta-1) \\Gamma^{4}\\left(\\frac{\\Delta+\\ell}{2}\\right) \\Gamma^{2}\\left(\\Delta_{\\phi}-\\frac{\\Delta-\\ell}{2}\\right) \\Gamma^{2}\\left(\\Delta_{\\phi}-\\frac{2 h-\\Delta-\\ell}{2}\\right)}\\,,\\\\ \\mathcal{R}_{\\Delta,\\ell}^{(k)}&=\\frac{\\Gamma^{2}\\left(\\frac{\\Delta+\\ell}{2}+\\Delta_{\\phi}-h\\right)\\left(1+\\frac{\\Delta-\\ell}{2}-\\Delta_{\\phi}\\right)_{k}^{2}}{k ! \\Gamma(\\Delta-h+1+k)}\\,.\n\\end{split}\n\\end{equation}\nNotice that $\\mathcal{N}_{\\Delta,\\ell}$ has zeros when $\\Delta=2{\\Delta_\\phi}+2n+\\ell$, {\\it i.e.,} the GFF values. Finding a general form for conformal blocks in Mellin space is a formidable challenge, see e.g. \\cite{Dolan:2011dv, Chen:2019gka}. A suitable form for the Mack polynomial that we will use can be found in \\cite{Gopakumar:2018xqi, Gopakumar:2021dvg}\n\\begin{equation}\nP^{}_{\\Delta,\\ell}(s_1,s_2)=\\sum_{m=0}^\\ell\\sum_{n=0}^{\\ell-m}\\mu^{(\\Delta,\\ell)}_{n,m}~~\\left(\\frac{\\Delta-\\ell}{2}-s_1-\\frac{2{\\Delta_\\phi}}{3}\\right)_{m}~~\\left(\\frac{{\\Delta_\\phi}}{3}-s_2\\right)_{n}\\,,\n\\end{equation}\nwhere \n\\begin{equation}\\label{Ap:mu}\n\\begin{split}\n&\\mu^{(\\Delta,\\ell)}_{n,m}\n=\\frac{2^{-\\ell} \\ell! (-1)^{m+n} (h+\\ell-1)_{-m} \\left(\\frac{\\bar\\tau }{2}-m\\right)_m (\\bar\\tau -1)_{n-\\ell} \\left(\\frac{\\tau}{2}+n\\right)_{\\ell-n} \\left(\\frac{\\tau}{2}+m+n\\right)_{\\ell-m-n} \\,}{m! n! (\\ell-m-n)!}\\\\\n&\\times {}_4F_3\\left(-m,-h+\\frac{\\tau }{2}+1,-h+\\frac{\\tau}{2}+1,n+\\Delta -1;\\frac{\\bar\\tau}{2}-m,\\frac{\\tau}{2}+n,-2 h+\\tau +2;1\\right)\\,.\n\\end{split}\n\\end{equation}\nHere $\\tau=\\Delta-\\ell, \\bar\\tau=\\Delta+\\ell$ and $(a)_b\\equiv \\Gamma(a+b)\/\\Gamma(a)$ is the Pochhammer symbol.\nWe will further use the following \"very well poised\" ${}_7F_6$ hypergeometric function which shows up in our calculations. \n\\begin{eqnarray}\\label{Wdef}\n&&W(a;b,c,d,e,f)\\equiv \\nonumber\\\\\n&&{}_7F_6\\bigg{(}\\begin{matrix} a, & 1+\\frac{1}{2}a, & b, & c, & d, & e, & f\\\\ ~&\\frac{1}{2}a,& 1+a-b, & 1+a-c, & 1+a-d, & 1+a-e, & 1+a-f\\end{matrix};1\\bigg{)}\\nonumber\\\\\n&=&\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-e)\\Gamma(1+a-f)}{\\Gamma(1+a)\\Gamma(b)\\Gamma(c)\\Gamma(d)\\Gamma(1+a-c-d)\\Gamma(1+a-b-d)\\Gamma(1+a-b-c)\\Gamma(1+a-e-f)}\\nonumber\\\\\n&\\times&\\frac{1}{2\\pi i}\\int_{-i\\infty}^{i\\infty}d\\sigma\\, \\frac{\\Gamma(-\\sigma)\\Gamma(1+a-b-c-d-\\sigma)\\Gamma(b+\\sigma)\\Gamma(c+\\sigma)\\Gamma(d+\\sigma)\\Gamma(1+a-e-f+\\sigma)}{\\Gamma(1+a-e+\\sigma)\\Gamma(1+a-f+\\sigma)}\\,.\\nonumber\\\\\n\\end{eqnarray}\nHere we have $a_\\ell=1-{\\Delta_\\phi}+(\\Delta-\\ell)\/2$ and \n\\begin{eqnarray}\\label{params}\na&=&\\ell'+2(a_\\ell+m+s_1+\\frac{2{\\Delta_\\phi}}{3}-1)\\,,\\quad b=e=a_\\ell+m,\\nonumber\\\\ c&=&d=a_\\ell+m+s_1+\\frac{2{\\Delta_\\phi}}{3}-1\\,,\\quad f=2(s_1-\\frac{{\\Delta_\\phi}}{3})+h+m+\\ell'-\\ell\\,.\n\\end{eqnarray}\nFor the above expression to be finite, we need $4a-2(b+c+d+e+f-2)>0$. When this is not satisfied, we need to analytically continue the expression (for instance in the epsilon expansion for non-zero spins in the $t$-channel, this condition is not respected for $m=\\ell$).\n\\subsection{Lengthy formulas}\\label{seclen}\n\\begin{equation}\nq_{\\ell'}^{diss}(s_1)=-\\frac{(s_1-\\frac{\\Delta_\\phi}{3})}{\\Gamma^2(\\frac{\\Delta_\\phi}{3}-s_1+1)} \\frac{2^{-\\ell'}\\Gamma(2s_1+2\\ell+\\frac{4{\\Delta_\\phi}}{3})}{\\Gamma^2(s_1+\\frac{2{\\Delta_\\phi}}{3})\\Gamma^2(s_1+\\ell+\\frac{2{\\Delta_\\phi}}{3})} \\ {}_3F_2\\bigg[\\begin{matrix} -\\ell,2s_1+\\ell-1+\\frac{4{\\Delta_\\phi}}{3},s_1-\\frac{{\\Delta_\\phi}}{3}\\\\\ns_1+\\frac{2{\\Delta_\\phi}}{3}, s_1+\\frac{2{\\Delta_\\phi}}{3}\n\\end{matrix};1\\bigg]\\,,\n\\end{equation}\nis the contribution from the identity operator which we have added by hand. Here, we have\n\\begin{eqnarray}\\label{qsgen}\nq^{(s)}_{\\Delta, \\ell' |\\ell}(s_1) &=& \\sum_{m,n} \\mu_{m,n}^{(\\ell)}(\\frac{\\Delta-\\ell}{2}-s_1-\\frac{2{\\Delta_\\phi}}3{})_m\\chi^{(n)}_{\\ell'}(s_1)\n \\frac{\\Gamma^2 \\left(\\frac{\\Delta+\\ell }{2}+\\Delta_\\phi -h\\right)}{(\\frac{\\Delta-\\ell}{2}-s_1-\\frac{2{\\Delta_\\phi}}{3}) \n \\Gamma (\\Delta-h +1)} \\,\\\\\n&\\times & _3F_2\\left[\\begin{matrix}\\frac{\\Delta -\\ell}{2}-s_1-\\frac{2{\\Delta_\\phi}}{3},1+\\frac{\\Delta-\\ell }{2}-\\Delta_\\phi ,1+\\frac{\\Delta-\\ell }{2}-\\Delta_\\phi \\\\1+\\frac{\\Delta-\\ell }{2}-s_1-\\frac{2{\\Delta_\\phi}}{3},\\Delta-h +1\\end{matrix};1\\right]\\,,\\nonumber \n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{schi1}\n\\chi_{\\ell'}^{(n)}\\!(s_1)=(-1)^{\\ell'}2^{-\\ell'}\\frac{\\Gamma(2s_1+2\\ell'+\\frac{4{\\Delta_\\phi}}{3})\\Gamma(s_1+\\frac{2{\\Delta_\\phi}}{3}+n)^2}{\\ell'! \\Gamma(\\ell'+s_1+\\frac{2{\\Delta_\\phi}}{3})^2\\Gamma(2s_1+n+\\frac{4{\\Delta_\\phi}}{3})}\\,{}\\frac{(-n)_{\\ell'}}{(2s_1+n+\\frac{4{\\Delta_\\phi}}{3})_{\\ell'}}\\,.\n\\end{equation}\nThe crossed-channel expression is given by:\n\\begin{eqnarray}\\label{qcoefft2s}\n&& q^{(t)}_{\\Delta, \\ell' |\\ell}(s_1) = \\frac{2^{-\\ell'}}{\\ell'!}\\frac{\\Gamma(2s_1+2\\ell'+\\frac{4{\\Delta_\\phi}}{3})}{\\Gamma^2(s_1+\\ell'+\\frac{2{\\Delta_\\phi}}{3})\\Gamma(a_\\ell)} \\frac{\\Gamma \\left(2\\Delta_\\phi +\\ell -h\\right)}{(a_\\ell+\\ell+2\\Delta_{\\phi}-h-1)} \\,, \\nonumber\\\\\n&\\times & \\sum_{p=0}^{\\ell'}\\sum_{n=0}^{\\ell}\\sum_{m =0}^{\\ell-n} \\mu_{m,n}^{(\\ell)}(\\frac{{\\Delta_\\phi}}{3}-s_1)_n \\frac{\\Gamma^2(s_1+\\frac{2{\\Delta_\\phi}}{3}+m+a_\\ell-1)}{\\Gamma(2s_1+\\frac{4{\\Delta_\\phi}}{3}+p+m+a_\\ell-1)}\\frac{(-\\ell')_p(2s_1+\\frac{4{\\Delta_\\phi}}{3}+\\ell'-1)_p}{p!} \\nonumber \\\\ \n& \\times & \\int_{0}^{1} dy \\,\n y^{s_1+\\frac{2{\\Delta_\\phi}}{3}-1}(1-y)^{a_\\ell-1}{}_2F_1[1, a_\\ell, a_\\ell+\\ell+(2\\Delta_{\\phi}-h);y]\\nonumber \\\\ &&~~~~~\\times {}_2F_1[s_1+\\frac{2{\\Delta_\\phi}}{3}+p,s_1+\\frac{2{\\Delta_\\phi}}{3}+m+a_\\ell-1, 2s_1+\\frac{4{\\Delta_\\phi}}{3}+p+m+a_\\ell-1;1-y] \\nonumber\\,,\\\\\n\\end{eqnarray}\nwhere $a_\\ell=1+\\frac{\\Delta-\\ell}{2}-{\\Delta_\\phi}$.\nRemarkably, this admits a closed form expression in terms of ${}_7F_6$ hypergeometric functions.\n\n\\begin{eqnarray}\\label{qtchann}\nq^{(t)}_{\\Delta,\\ell'|\\ell}(s_1)&=&\\sum_{n=0}^\\ell\\sum_{m=0}^{\\ell-n} (-1)^{\\ell'+m}2^{-\\ell'}\\mu_{m,n}^{(\\ell)} (\\frac{{\\Delta_\\phi}}{3}-s_1)_n (a_\\ell)_m^2 \\Gamma(2s_1+2\\ell'+\\frac{4{\\Delta_\\phi}}{3})\\nonumber \\\\ &\\times& \\Gamma^2(d)\\Gamma(\\frac{a}{2})\\Gamma(a+1)\\Gamma^2(1+a-f-b)\\tilde W(a;b,c,d,e,f)\\,,\n\\end{eqnarray}\n the parameters $a,b,c$ etc. are given in eq.(\\ref{params}) and the $\\tilde W$ is the regularized version of a special (``very well poised\") ${}_7F_6$ hypergeometric function as defined in eq.(\\ref{Wdef}). For $\\ell>\\ell'$ there are a finite set of terms that need to be added to the above expression \\cite{fgsz}. We will use eq.(\\ref{qcoefft2s}) for performing calculations.\n\n\\section{Bosonic components of $\\frac{1}{2}$-BPS multiplets}\\label{App:multipletcomponents}\nIn this appendix, we give more details of the $\\frac{1}{2}$-BPS multiplets which correspond to supergravity and supersymmetric gauge theory fields in AdS. We only keep the relevant bosonic fields which can appear in four-point functions. But the multiplets themselves contain more components. For a comprehensive discussion on superconformal multiplets in various spacetime dimensions and the complete set of superconformal descendants, see \\cite{Cordova:2016emh}. The tables below for the bosonic components are reproduced from \\cite{Alday:2020dtb,Behan:2021pzk}, and we now explain these tables.\n\n\\vspace{0.5cm}\n\\noindent {\\bf The cases with sixteen Poincar\\'e supercharges (maximally superconformal)}\n\\vspace{0.3cm}\n\n{\\begin{center}\n \\begin{tabular}{||c| c | c | c | c | c | c ||} \n \\hline\ncomponent field & $s_p$ & $A_{p,\\mu}$ & $\\varphi_{p,\\mu\\nu}$ & $C_{p,\\mu}$ & $t_p$ & $r_p$ \\\\ [0.5ex] \n \\hline\\hline\nLorentz spin $\\ell$ & 0 & 1 & 2 & 1& 0 & 0\\\\ \n \\hline\nconformal dimension $\\Delta$ & $\\epsilon p$ & $\\epsilon p+1$ & $\\epsilon p+2$ & $\\epsilon p+3$ & $\\epsilon p+4$ & $\\epsilon p+2$ \\\\\n \\hline\n$d_1$ & $p$ & $p-2$ & $p-2$ & $p-4$ & $p-4$ & $p-4$ \\\\ \n\\hline $d_2$ & $0$ & $2$ & $0$ & $2$ & $0$ & $4$ \\\\ [0.5ex] \n \\hline\n\\end{tabular}\n\\end{center}}\n\nFor supergravity theories with maximal superconformal symmetry, the supergravity fields are all organized into such $\\frac{1}{2}$-BPS multiplets. The parameter $\\epsilon=\\frac{d-2}{2}$ takes value $\\frac{1}{2}$, $1$, $2$, corresponding to the three maximal superconformal cases. The multiplets are labelled by an integer $p$ which corresponds to the Kaluza-Klein level with $p=2,3,\\ldots$. The lowest value $p=2$ corresponds to the stress tensor multiplet, and the massless graviton field $\\varphi_{2,\\mu\\nu}$ is dual to the stress tensor. In the table, the quantum numbers $d_1$, $d_2$ are associated with the R-symmetry representation of the component fields. They parameterize the Dynkin labels of the R-symmetry groups as follows\n\\begin{equation}\nSO(5):\\;\\; [d_1,d_2]\\;,\\quad\\quad SU(4):\\;\\; [\\tfrac{d_2}{2},d_1,\\tfrac{d_2}{2}]\\;,\\quad\\quad SO(8):\\;\\; [d_1,\\tfrac{d_2}{2},0,0]\\;.\n\\end{equation} \nNote that for $p<4$, some of the $d_1$ values in the table are negative. In this case the corresponding components are absent from the multiplet. Therefore, a generic multiplet with $p\\geq 4$ contains six bosonic fields which can be exchanged in the four-point function. For $p=2,3$ the multiplets are extra-short and contain only three such fields. \n\n\\vspace{0.5cm}\n\\noindent {\\bf The cases with eight Poincar\\'e supercharges}\n\\vspace{0.3cm}\n\n{\\begin{center}\n \\begin{tabular}{||c| c | c | c ||} \n \\hline\ncomponent field & $s^I_p$ & $A^I_{p,\\mu}$ & $r^I_p$ \\\\ [0.5ex] \n \\hline\\hline\nLorentz spin $\\ell$ & 0 & 1 & 0\\\\ \n \\hline\nconformal dimension $\\Delta$ & $\\epsilon p$ & $\\epsilon p+1$ & $\\epsilon p+2$ \\\\\n \\hline\n$SU(2)_R$ spin $j_R$ & $\\frac{p}{2}$ & $\\frac{p}{2}-1$ & $\\frac{p}{2}-2$ \\\\ [0.5ex] \n \\hline\n $SU(2)_L$ spin $j_L$ & $\\frac{p-2}{2}$ & $\\frac{p-2}{2}$ & $\\frac{p-2}{2}$ \\\\ [0.5ex] \n \\hline\n\\end{tabular}\n\\end{center}}\n\n\nThe cases with eight Poincar\\'e supercharges are relevant for our discussion of super gluons on $AdS_{d+1}\\times S^3$, and all super gluon fields as well as their superconformal descendants reside in the $\\frac{1}{2}$-BPS multiplets. Multiplets with different Kaluza-Klein levels are labelled by the integer $p=2,3,\\ldots$. The lowest value $p=2$ corresponds to the flavor current multiplet. As in the maximally superconformal cases, fields with negative $SU(2)$ quantum numbers are absent. \n\n\\section{Properties of Witten Diagrams}\\label{App:WittenDiagrams}\nIn this appendix we review several properties of tree-level Witten diagrams which are useful in applications to holographic correlators and in various analytic conformal bootstrap methods. In Appendix \\ref{Subapp:dfun}, we focus on the contact Witten diagrams, {\\it i.e.}, the $D$-functions. In Appendix \\ref{Subapp:ivi}, we review vertex identities obtained from integrating out an internal bulk-to-bulk propagator. These identities express integrated cubic vertices with a bulk-to-bulk propagator to contact vertices with only bulk-to-boundary propagators, and are useful for computing higher-point exchange diagrams. In Appendix \\ref{Subapp:eomid}, we review how the boundary two-particle Casimir equation translates to the equation of motion identity in the bulk, which relates exchange Witten diagrams and contact Witten diagrams. We also discuss several applications of this identity. In Appendix \\ref{Subapp:ivi} we discuss recursion relations satisfied by Witten diagrams, and demonstrate some general properties in a few explicit examples. \n\n\n\\subsection{$D$-functions}\\label{Subapp:dfun}\nThe $D$-functions are a class of special functions defined as \n\\begin{equation}\\label{defDf}\nD_{\\Delta_1\\ldots \\Delta_n}(x_i)=\\int \\frac{d^d\\vec{z}dz_0}{z_0^{d+1}}\\prod_{i=1}^nG^{\\Delta_i}_{B\\partial}(z,x_i)\\;,\\quad G^{\\Delta_i}_{B\\partial}(z,x_i)=\\left(\\frac{z_0}{z_0^2+(\\vec{z}-\\vec{x}_i)^2}\\right)^{\\Delta_i}\\;,\n\\end{equation}\nwhich are $n$-point contact Witten diagrams in $AdS_{d+1}$ with no derivatives. Contact diagrams with derivatives can also be expressed as $D$-functions with shifted weights by using the identity\n\\begin{equation}\n\\nabla^\\mu G^{\\Delta_1}_{B\\partial} \\nabla_\\mu G^{\\Delta_2}_{B\\partial}=\\Delta_1\\Delta_2(G^{\\Delta_1}_{B\\partial}G^{\\Delta_2}_{B\\partial}-2x_{12}^2G^{\\Delta_1+1}_{B\\partial}G^{\\Delta_2+1}_{B\\partial})\\;.\n\\end{equation}\nIt is convenient to write the $D$-functions as functions of cross ratios by extracting a kinematic factor. For $n=4$, one defines the $\\bar{D}$-functions as \n\\begin{equation}\\label{dbar}\n\\frac{ \\prod_{i=1}^4\\Gamma(\\Delta_i)}{\\Gamma(\\frac{1}{2}\\Sigma_\\Delta-\\frac{1}{2}d)}\\frac{2}{\\pi^{\\frac{d}{2}}}D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)=\\frac{(x_{14}^2)^{\\frac{1}{2}\\Sigma_\\Delta-\\Delta_1-\\Delta_4}(x^2_{34})^{\\frac{1}{2}\\Sigma_\\Delta-\\Delta_3-\\Delta_4}}{(x^2_{13})^{\\frac{1}{2}\\Sigma_\\Delta-\\Delta_4}(x^2_{24})^{\\Delta_2}}\\bar{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4} (U,V)\\, ,\n\\end{equation}\nwhere $\\Sigma_\\Delta=\\sum_{i=1}^n\\Delta_i$.\n\nWe can also represent the $D$-functions using the Feynman parameter representation\n\\begin{equation}\nD_{\\Delta_1\\ldots \\Delta_n}(x_i)=\\frac{\\pi^{\\frac{d}{2}}\\Gamma(\\frac{1}{2}\\Sigma_\\Delta-\\frac{1}{2}d)\\Gamma(\\frac{1}{2}\\Sigma_\\Delta)}{2\\prod_i\\Gamma(\\Delta_i)}\\int \\prod_j \\frac{d\\alpha}{\\alpha_j}\\alpha_j^{\\Delta_j}\\frac{\\delta(\\sum_j\\alpha_j-1)}{(\\sum_{k0$ from $Q_{\\ell,0}(u)$.\n\nAs another application, let us prove that the difference of two exchange Witten diagrams with opposite quantizations ({\\it i.e.}, with conformal dimension $\\Delta$ versus $d-\\Delta$) is proportional to the conformal partial wave\n\\begin{equation}\n\\mathcal{W}_{\\Delta,\\ell}-\\mathcal{W}_{d-\\Delta,\\ell}\\propto \\Psi_{\\Delta,\\ell}\\;.\n\\end{equation}\nThe conformal partial wave $\\Psi_{\\Delta,\\ell}$ is defined to be the linear combination of a conformal block and its shadow such that it is single-valued in Euclidean space ({\\it i.e.}, when $\\bar{z}=z^*$).\\footnote{Note that each conformal block is not single-valued.} To prove this relation, we act on the combination $\\mathcal{W}_{\\Delta,\\ell}-\\mathcal{W}_{d-\\Delta,\\ell}$ with the operator $({\\rm Cas}-C_{\\Delta,\\ell})$. The contact term on the RHS of (\\ref{eomidgen}) does not distinguish the two quantizations, and therefore \n\\begin{equation}\n\\big( {\\rm Cas}-C_{\\Delta,\\ell}\\big) (\\mathcal{W}_{\\Delta,\\ell}-\\mathcal{W}_{d-\\Delta,\\ell})=0\\;.\n\\end{equation}\nThis equation tells us that the double-trace conformal blocks in each exchange Witten diagram have been precisely cancelled, and the difference is a linear combination of the single-trace conformal blocks with dimensions $\\Delta$ and $d-\\Delta$. On the other hand, single-valuedness is obvious. It follows from the fact that each exchange Witten diagram is single-valued. \n\nFinally, let us mention that the equation of motion identity also implies efficient recursion relations that can be used to obtain the crossed channel conformal block decomposition coefficients of exchange Witten diagrams or conformal partial waves. The latter is related to the crossing kernel (also known as the $6j$ symbol) of the conformal group. The idea is that the equation of motion turns an exchange Witten diagram into contact Witten diagrams (or a conformal partial wave into zero) which can be easily decomposed into conformal blocks in the crossed channel. On the other hand, the conformal Casimir operator acts nicely on crossed channel conformal blocks, and its action can be expressed as a linear combination of finitely many conformal blocks with shifted dimensions and spins. This gives rise to relations among the crossed channel conformal block decomposition coefficients which can be recursively solved. We refer the reader to \\cite{Zhou:2018sfz} for details of this recursive approach. For other approaches to this problem, see \\cite{Hogervorst:2017sfd,Sleight:2018epi,Liu:2018jhs,Sleight:2018ryu}.\n\n\\subsection{Recursion relations}\\label{Subapp:recur}\nIt is well known that conformal blocks satisfy various intricate recursion relations (see, {\\it e.g.}, \\cite{Dolan:2003hv,Dolan:2011dv}). These recursion relations are very useful for studying the properties of conformal blocks and for performing conformal block decomposition for conformal correlators. Exchange Witten diagrams are intuitively very similar to conformal blocks. They contain a single-trace conformal block which is associated with the exchange of a particle in AdS. But at the same time they also contain infinitely many double-trace conformal blocks which are two-particle states. Because of the infinitely many conformal blocks involved, at first sight it seems rather unlikely that similar recursion relations can exist for Witten diagrams. However, it was pointed out in \\cite{Zhou:2020ptb} that their existence is always {\\it guaranteed}. There is an intimate connection between the recursion relations of conformal blocks and Witten diagrams, and one can easily generate Witten diagram recursion relations from known recursion relations of conformal blocks.\n\nLet us demonstrate this correspondence in the simplest situation where the conformal block recursion relations have the form of a linear combination of conformal blocks with constant coefficients. The prime examples in this category are the dimensional reduction formulae. It was found in \\cite{Hogervorst:2016hal} that a $d$-dimensional conformal block can be expressed in terms of infinitely many $(d-1)$-dimensional ones \n\\begin{equation}\ng^{(d)}_{\\Delta,\\ell}=\\sum_{n=0}^\\infty\\sum_{j} A_{n,j}\\, g^{(d-1)}_{\\Delta+2n,j}\\;,\\quad j=\\ell\\;,\\ell-2\\;,\\ldots\\;, \\ell\\;\\text{mod}\\;2\\;.\n\\end{equation}\nOn the other hand, for conformal blocks in $d$ and $d-2$ dimensions it is possible to find a relation with finitely many terms \\cite{Kaviraj:2019tbg}\n\\begin{equation}\ng^{(d-2)}_{\\Delta,\\ell}=g^{(d)}_{\\Delta,\\ell}+c_{2,0}g^{(d)}_{\\Delta+2,\\ell}+c_{1,-1}g^{(d)}_{\\Delta+1,\\ell-1}+c_{0,-2}g^{(d)}_{\\Delta,\\ell-2}+c_{2,-2}g^{(d)}_{\\Delta+2,\\ell-2}\\;.\n\\end{equation}\nHere $A_{n,j}$ and $c_{i,j}$ are numerical constants whose explicit expressions are not important for our discussion and can be found in \\cite{Hogervorst:2016hal} and \\cite{Kaviraj:2019tbg}. To obtain recursion relations for Witten diagrams from these identities, a simple prescription was pointed out in \\cite{Zhou:2020ptb}. One just needs to replace the conformal blocks $g^{(D)}_{\\Delta,\\ell}$ by the corresponding $AdS_{D+1}$ exchange Witten diagrams $W^{AdS_{D+1}}_{\\Delta,\\ell}$ which contain $g^{(D)}_{\\Delta,\\ell}$ as the single-trace conformal block.\\footnote{The normalization is chosen such that the single-trace conformal block appears with the unit coefficient.} Note that for spin $\\ell\\geq 1$, one has multiple choices for the contact terms in the exchange diagrams. Therefore, one further needs to choose appropriate contact terms in order for the identities to hold. But such choices turn out to always exist as we shall see. Let us first write down the corresponding Witten diagram relations following from the above prescription\n\\begin{equation}\nW^{AdS_{d+1}}_{\\Delta,\\ell}=\\sum_{n=0}^\\infty\\sum_{j} A_{n,j}\\, W^{AdS_d}_{\\Delta+2n,j}\\;,\\quad j=\\ell\\;,\\ell-2\\;,\\ldots\\;, \\ell\\;\\text{mod}\\;2\\;,\n\\end{equation}\n\\begin{equation}\nW^{AdS_{d-1}}_{\\Delta,\\ell}=W^{AdS_{d+1}}_{\\Delta,\\ell}+c_{2,0}W^{AdS_{d+1}}_{\\Delta+2,\\ell}+c_{1,-1}W^{AdS_{d+1}}_{\\Delta+1,\\ell-1}+c_{0,-2}W^{AdS_{d+1}}_{\\Delta,\\ell-2}+c_{2,-2}W^{AdS_{d+1}}_{\\Delta+2,\\ell-2}\\;,\n\\end{equation}\nwhere we have left the choice of the contact terms implicit. Note that identities of the second kind are responsible for the Parisi-Sourlas dimensional reduction structure found in the super graviton and super gluon correlators (with $\\ell=0$), as we mentioned in Section \\ref{Subsec:MRV} and \\ref{Subsec:supergluons}. To understand why this simple prescription works and also to see how to choose the contact terms, it is most convenient to go to the Mellin space. We recall that the Mellin amplitude of an exchange Witten diagram is a sum over simple poles plus a polynomial regular term. On the other hand, under conformal block decomposition, an exchange Witten diagram contains a single-trace conformal block and infinitely many double-trace conformal blocks. The single-trace conformal block is determined by the singular terms, and is produced when we take the residues at these simple poles. By contrast, the double-trace conformal blocks in the Witten diagram are produced when we take residues at the poles of the Gamma function factor. Note that crucially there is no freedom left to change the double-trace conformal blocks once the Mellin amplitude is determined. With this observation, it is easy to see why this prescription gives the correct answer. The original conformal block recursion relation, which yields the equality of the single-trace conformal blocks, guarantees that the singular part of the Mellin amplitudes are the same on both sides. The remaining task is to match the polynomial terms, which are the sums of the contact terms in the exchange Witten diagrams. Since each spin-$\\ell$ exchange Witten diagram can accommodate a contact term which is a degree-$(\\ell-1)$ polynomial, clearly this is always possible. \n\nIn Mellin space the existence of these Witten diagram relations is almost obvious following the above reasoning. However, from the position space perspective such identities are rather remarkable, as they require intricate cancellations of infinitely many double-trace conformal blocks. One can also take these Witten diagram identities and decompose them in the crossed channel. These identities then give rise to highly nontrivial relations which constrain the crossed channel conformal block decomposition coefficients of exchange Witten diagrams. \n\nIn the above, we have only discussed the simplest scenario. More generally, conformal block relations may have cross ratio dependence in their linear combination coefficients. Such relations also induce Witten diagram relations although sometimes additional correction terms are needed. The simplest example in this class is the Casimir equation for conformal blocks\n\\begin{equation}\n\\big({\\rm Cas}-C_{\\Delta,\\ell}\\big) g_{\\Delta,\\ell}=0\\;.\n\\end{equation}\nIt is mapped to the equation of motion identity for exchange Witten diagrams encountered in the previous subsection\n\\begin{equation}\n\\big({\\rm Cas}-C_{\\Delta,\\ell}\\big) W_{\\Delta,\\ell}=W_{\\rm con}\\;.\n\\end{equation}\nThat we can generate Witten diagram relations from conformal block relations in the more general case is essentially guaranteed by the same fact as before, namely, the double-trace conformal blocks are fully determined by the Mellin amplitudes. However, an important difference to note is the extra term on the RHS. This contact term cannot be absorbed by redefining the contact part in the exchange diagram.\\footnote{It is easiest to convince oneself of this fact in the example which has $\\ell=0$. The scalar exchange Witten diagram admits no contact term. However, the equation of motion identity has a zero-derivative contact term on the RHS.} This represents a general feature when the coefficients of the conformal blocks are no longer just constants. These cross ratio dependent coefficients translate into difference operators in Mellin space. Such operators generically shift the simple poles of the Mellin amplitude. But at the same time they can also generate new poles or multiply the Mellin amplitudes by polynomials. As a result, whenever this happens we need to add a finite number of extra exchange Witten diagrams or contact Witten diagrams in order to match the Mellin amplitudes. Many examples of such relations were given in \\cite{Zhou:2020ptb} and were verified by explicit computations. However, they are a bit too technical to be included here and we will not discuss this further. The interested reader can read \\cite{Zhou:2020ptb} for more details. \n\n\n\n\n\n\n\\section{Introduction}\\label{sec_intro}\nA ``bootstrap'' method or process is one that is self-generating or self-sustaining. As such, the bootstrap philosophy in quantum field theory refers to an ambitious program to use only basic symmetries and consistency conditions such as Poincar\\'e invariance, unitarity, crossing symmetry and analyticity to constrain observables like the S-matrix elements \\cite{heisenberg}. In the 1960s, the bootstrap program was pursued with the hope of understanding the strong interactions \\cite{chew}. In the 1970s, a similar program was initiated to understand the physics of second order phase transitions, described by quantum field theories with conformal symmetries, {\\it i.e.,} Conformal Field Theories (CFTs). This program is called the Conformal Bootstrap \\cite{Ferrara:1973yt, Polyakov:1974gs}. In addition to the familiar Poincar\\'e symmetries, CFTs enjoy scale symmetry as well as special conformal symmetries. These extra symmetries completely fix the structure of two- and three-point correlators \\cite{bigfatyellow}. One of the goals of the conformal bootstrap is to constrain the dynamical content appearing in four-point correlators in CFTs. \n\n\nConformal symmetry allows one to classify operators annihilated by the special conformal generators as ``primaries''. There are an infinite class of operators called ``descendants'' which are derivatives of these primary operators. The central idea of the conformal bootstrap program is to fix the operator product expansion (OPE) of any pair of local primary operators in the theory. Once this is accomplished, any $n$-point correlation function of local operators can be recursively calculated, at least in principle. In addition to conformal invariance, one uses crossing symmetry in a judicious manner. In the context of Euclidean correlators, crossing symmetry arises due to operator associativity. This leads to the notion of different channels, which in an overlapping region of convergence are set equal, leading to the so-called crossing conditions. Naively, these are an infinite number of conditions and finding any consistent solution seems to be a Herculean task. In fact, while the idea of the conformal bootstrap framework has been around since the 1970s, the main success it encountered, until recently, was restricted to two dimensional CFTs \\cite{bpz, bigfatyellow}. In 2008, the work of \\cite{rrtv} introduced a new numerical paradigm in the game. This paradigm enables us to extract, arguably, some of the most numerically accurate critical exponents for the 3d Ising model \\cite{3dising1, 3dising2, 3disingkos}. In addition to this flagship result, numerical methods have enabled a systematic study of ``islands'' of CFTs allowed by unitarity and crossing symmetry. These developments have been recently reviewed in \\cite{numrev}.\n\nIn addition to these remarkable numerical results, it is worthwhile to develop analytic tools. There are several reasons for this. First, establishing potentially universal results for generic CFTs would require an analytic handle. Second, there is a plethora of results, both old and new, that the Feynman diagrammatic approach has produced; one would like to see how the bootstrap method compares to the successes of the diagrammatic approach. Finally, it is important to identify and establish techniques that can produce results that are hard using other established methods.\n\n\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{fig_flow.png}\n\\caption{The relation of sections in this review.}\n \\label{fig:flow}\n\\end{figure}\n\nIn this present review, we will guide the reader on a journey through certain selected topics covering modern techniques in {\\it analytic} conformal bootstrap in spacetime dimensions $d\\geq 3$. The road map of the journey that we will take the reader on is depicted in Figure \\ref{fig:flow}. It begins with an ``appetizer'' section \\ref{Sec:BCFT}, which discusses boundary conformal field theories (BCFT). These are CFTs in the presence of a boundary or a co-dimension 1 defect. In nature, such systems may occur at the surface of a crystal. The two-point functions in such a scenario carry dynamical information, both of the bulk properties and of the new data due to the presence of a boundary. For technical reasons (lack of positivity in the so-called ``bulk channel''), setting up numerics in this scenario is hard. However, BCFTs allow for a rich phase structure corresponding to different boundary conditions. It is therefore important to develop analytic techniques. For our purpose, the case of BCFTs also serves as a simplifying example where we clarify some of the general ideas used in the analytic methods. Kinematically, this setup is very similar to CFTs placed on a real projective space. We will therefore also discuss analytic techniques for real projective space CFTs in the same section. \n\nWe then discuss three possible routes. The first route begins in section \\ref{Sec:largespin} and discusses large spin perturbation theory (LSPT). This is arguably the standard example in any discussion of the analytic conformal bootstrap. The main idea here is to reproduce contributions of certain known operators in one channel in the crossing equation in terms of the other channel. Typically, this needs an infinite number of operator contributions. One can argue that to reproduce the contribution of the identity operator, there have to be generalized free field (GFF) operators in the spectrum. This is done by analyzing the large spin tail of such contribution. As we will review, this strategy works when there is a twist gap between the identity operator and other operators in the spectrum. We will study this canonical example in some detail and show how one can further go on to deriving leading order anomalous dimensions for the GFF spectrum. A natural continuation of this route is to discuss the now-famous Lorentzian inversion formula. This remarkable formula enables us to express the OPE coefficients as a convolution of the so-called double discontinuity of the position space correlator against an analytically continued (in spin) conformal block. This formula can then be used in the context of AdS\/CFT to extract information about tree-level and loop-level AdS Witten diagrams. \n\n\n\nBoth the second and the third routes embark on perturbing away from the GFF spectrum (section \\ref{Sec:largeN}). The perturbation parameter, by anticipating a connection with the AdS\/CFT correspondence, is generically denoted by $1\/N$, where $N$ is related to the central charge and taken to be large. Calculations along these routes are facilitated by a transition to Mellin space (section \\ref{Sec:MellinFormalism}). Using Mellin techniques one can either continue the journey by discussing correlators in the $\\epsilon$-expansion (the second route) or in the $1\/N$ expansion (the third route). \n\nIn the second route (sections \\ref{Sec:epsilon}-\\ref{Sec:epsilonbootstrap}), the $\\epsilon$-expansion makes contact with the Wilson-Fisher fixed point \\cite{Wilson:1971dc} and extracts the anomalous dimensions of certain operators in a perturbative expansion in $\\epsilon$ where the spacetime dimension is written as $d=4-\\epsilon$. Quite remarkably, not only can all the results of the famous Wilson-Kogut review \\cite{Wilson:1973jj} be reproduced, but one can also easily get novel results for OPE coefficients which are difficult to calculate using the diagrammatic approach. In order to extract OPE data analytically, it is convenient to use Polyakov's 1974 seminal idea \\cite{Polyakov:1974gs}, where he postulated that the bootstrap equations can be solved analytically by starting with a basis that is manifestly crossing symmetric. As we will review, this approach, in modern parlance, is tied with the crossing symmetric Witten diagrams in AdS space. The crossing symmetric AdS Witten diagrams provide a convenient kinematical basis for expanding the Mellin space correlator. Since the basis is crossing symmetric, constraints arise on demanding consistency with the OPE, leading to the so-called Polyakov conditions. This needs a discussion of crossing symmetric dispersion relations (section \\ref{Sec:dispersionPolyakov}) which enables one to fix the so-called contact term ambiguities. \n\nIn the third route (sections \\ref{Sec:PositionSpace}-\\ref{Sec:openproblemscorrelators}), we discuss efficient modern techniques to compute holographic correlators in various top-down string theory\/M-theory models. We will focus on the regime where the bulk dual descriptions are weakly coupled and local. The basic observables are holographic correlators which correspond to on-shell scattering amplitudes in AdS. From these objects we can extract analytic data of the strongly coupled boundary theories by performing standard CFT analysis. The models which we will consider include the paradigmatic example of the strongly coupled 4d $\\mathcal{N}=4$ super Yang-Mills, which is dual to IIB supergravity on $AdS_5\\times S^5$, along with others preserving a certain amount of supersymmetry. Due to the presence of a compact internal manifold in these models, the Kaluza-Klein reduced effective theory in AdS contains infinitely many particles. The extreme complexity of the bulk effective action together with the proliferation of curved-space diagrams render the standard diagrammatic expansion method practically useless beyond just a few simplest cases. However, as we will see, using symmetries and consistency conditions allows us to fix the correlators completely and therefore circumvents these difficulties. After a brief review of the superconformal kinematics in section \\ref{Subsec:scfkinematics}, we will discuss in detail three complementary bootstrap methods to compute tree-level correlators (sections \\ref{Sec:PositionSpace}, \\ref{Sec:MellinSpace}). These methods yield all four-point tree-level correlators of arbitrary Kaluza-Klein modes in all maximally superconformal theories, and reveal remarkable simplicity and structures hidden in the Lagrangian description. We also discuss various extensions: higher-point correlators (section \\ref{Subsec:5ptfunctions}), correlators corresponding to super gluon scattering in AdS (section \\ref{Subsec:supergluons}), and loop-level correlators (section \\ref{Sec:loops}). The results and techniques which we will review in this part of the review also bear great resemblance with the on-shell scattering amplitude program in flat space, as we will point out along the way. \n\n\nIn organizing this review, we have presented the material in a way such that these routes are relatively independent and can be read separately. We also accompanied the discussions with many pedagogical examples. All the journeys along these different routes end with a brief discussion of open questions in this research area. We also conclude in section \\ref{Sec:omissions} with a discussion of further reading material which covers a broader range of topics. Where possible, we will delegate lengthy formulas and algebraic steps to the appendices. The third appendix (appendix \\ref{App:WittenDiagrams}) also constitutes a self-contained review of various properties of Witten diagrams which make appearances at multiple places in this review. We will assume some familiarity with CFTs on the part of the reader. For introductory material on this topic, we refer the reader to \\cite{bigfatyellow,slavaepfl,Simmons-Duffin:2016gjk, numrev}. For introductory material on the AdS\/CFT correspondence we refer the reader to \\cite{Aharony:1999ti,DHoker:2002nbb,Penedones:2016voo}. \nThere will be special functions like Gauss and generalized hypergeometric functions used in several places. Most of these functions are inbuilt in $\\mathtt{Mathematica}$. For authoritative references, we ask the reader to consult \\cite{AnAsRo99, 10.5555\/1830479}. \n\n\n\n\n\n\\section*{\\small Abstract}}}\nThis review aims to offer a pedagogical introduction to the analytic conformal bootstrap program via a journey through selected topics. We review analytic methods which include the large spin perturbation theory, Mellin space methods and the Lorentzian inversion formula. These techniques are applied to a variety of topics ranging from large-$N$ theories, to the epsilon expansion and holographic superconformal correlators, and are demonstrated in a large number of explicit examples.\n\n\n\\vspace{2.6cm}\n\n\\begin{center}\n{\\it Invited review for Physics Reports}\n\\end{center}\n\n\n\n\n\n\\newpage\n\\tableofcontents\n\n\n\\newpage\n\n\\markboth{1\\quad INTRODUCTION}{}\n\\input introduction.tex \n\\newpage\n\\markboth{2\\quad OVERTURE: BOOTSTRAP WITH TWO-POINT FUNCTIONS}{}\n\\input section_BCFT.tex\n\\newpage\n\\markboth{3\\quad LARGE SPIN ANALYTIC BOOTSTRAP}{}\n\\input section_largespin.tex\n\\newpage\n\\markboth{4\\quad LARGE N}{}\n\\input section_largeN.tex\n\\newpage\n\\markboth{5\\quad LORENTZIAN INVERSION FORMULA}{}\n\\input section_inversion.tex\n\\newpage\n\\markboth{6\\quad MELLIN SPACE}{}\n\\input section_Mellin.tex\n\\newpage\n\\markboth{7\\quad THE EPSILON EXPANSION}{}\n\\input section_epsilonexpansion1.tex\n\\newpage\n\\markboth{8\\quad POLYAKOV BOOTSTRAP FROM DISPERSION RELATION}{}\n\\input section_epsilonexpansion2.tex\n\\newpage\n\\markboth{9\\quad EPSILON EXPANSION FROM BOOTSTRAP}{}\n\\input section_polyakov.tex\n\\newpage\n\\markboth{10\\quad BOOTSTRAPPING TREE-LEVEL CORRELATORS: POSITION SPACE}{}\n\\input section_positionspace.tex\n\\newpage\n\\markboth{11\\quad BOOTSTRAPPING TREE-LEVEL CORRELATORS: MELLIN SPACE}{}\n\\input section_mellinspace.tex\n\\newpage\n\\markboth{12\\quad BOOTSTRAPPING LOOP-LEVEL HOLOGRAPHIC CORRELATORS}{}\n\\input section_loops.tex\n\\newpage\n\\markboth{13\\quad HOLOGRAPHIC CORRELATORS: OTHER ASPECTS AND OPEN PROBLEMS}{}\n\\input section_openproblemscorrelators.tex\n\\newpage\n\\markboth{14\\quad FURTHER READING}{}\n\\input section_omissions.tex\n\\newpage\n\\begin{appendices}\n\\markboth{A\\quad EPSILON EXPANSION: SOME DETAILS}{}\n\\input app_conventions.tex\n\\input app_formulas.tex\n\\newpage\n\\markboth{B\\quad BOSONIC COMPONENTS OF $\\frac{1}{2}$-BPS MULTIPLETS}{}\n\\input app_multipletcomponents.tex\n\\newpage\n\\markboth{C\\quad PROPERTIES OF WITTEN DIAGRAMS}{}\n\\input app_wittendiagrams.tex\n\n\n\\end{appendices}\n\n\\newpage\n\n\\setlength{\\bibsep}{4pt plus 0.2ex}\n\n{\\small\n\n\\section{Overture: Bootstrap with two-point functions}\\label{Sec:BCFT}\nThis section serves as an appetizer for the reader to get a taste of the kind of analytic conformal bootstrap techniques which we will present in the review. To this end, we would like to choose systems which are as simple as possible (yet still nontrivial). One toy example that comes to mind is the one dimensional CFT where the simplest nontrivial observables are the four-point functions. Though conceptually closer to the higher dimensional case as it deals with the same kind of observables, the application of 1d CFTs is quite limited. Therefore, we choose to investigate instead two closely related but perhaps less familiar setups, namely, CFTs with a conformal boundary and CFTs on real projective space. These setups are equally simple compared to CFT$_1$ but can be discussed in arbitrary spacetime dimensions. This gives them a wider range of physical applicability. In particular, BCFTs have important applications in various condensed matter systems. Therefore, we believe that the greater effort needed to get acquainted with these new CFT systems is justified and will be rewarding in the end. The most noticeable feature of these setups is that conformal symmetry is only partially preserved. But as a result, there are new observables. The simplest nontrivial observables are the two-point functions. We will use these two-point functions to demonstrate the power of analytic conformal bootstrap without too much technical complexity. Note that this section is structured to be independent from the other sections. Therefore, if the reader wishes to go directly to the three routes of the review, skipping it will not affect their understanding.\n\n The rest of this section is organized as follows. In Section \\ref{Subsec:BCFTkinematics} we introduce the setups and discuss the kinematics. In Section \\ref{Subsec:analyticmethods} we review analytic bootstrap methods for studying two-point functions. In Section \\ref{Subsec:otherbackgrounds} we give a short discussion of CFTs in other backgrounds. As we already mentioned, the two setups which we will study in this section are also interesting in their own right. For readers who are interested in learning more about these topics, we refer them to the original papers. An incomplete sampling of the literature on BCFTs from the bootstrap perspective includes \\cite{Liendo:2012hy,Gliozzi:2015qsa,Liendo:2016ymz,Rastelli:2017ecj,Bissi:2018mcq,Mazac:2018biw,Kaviraj:2018tfd,Dey:2020lwp,Dey:2020jlc,Giombi:2020rmc,Giombi:2021cnr}. For works on CFTs on real projective space, see \\cite{Nakayama:2015mva,Verlinde:2015qfa,Nakayama:2016cim,Nakayama:2016xvw,Hasegawa:2016piv,Hogervorst:2017kbj,Hasegawa:2018yqg,Giombi:2020xah,Wang:2020jgh,Tsiares:2020ewp,Nakayama:2021muk}. \n\n\n\n\n\\subsection{Kinematics}\\label{Subsec:BCFTkinematics}\nTo discuss the conformal symmetry these systems preserve, it is most convenient to use the embedding space formalism. We can represent each point $x^\\mu\\in\\mathbb{R}^{d-1,1}$ by a null ray in the embedding space $\\mathbb{R}^{d,2}$\n\\begin{equation}\nP^A\\;,\\quad A=1,2,\\ldots, d+2\\;,\\quad P\\cdot P=0\\;,\\quad P\\sim \\lambda P\\;.\n\\end{equation}\nOperators are defined on the null rays with the condition \n\\begin{equation}\\label{opsca}\n\\mathcal{O}_\\Delta(\\lambda P)=\\lambda^{-\\Delta}\\mathcal{O}_\\Delta(P)\\;.\n\\end{equation}\nLet us choose the signature of the embedding space to be $(-,+,-,+,\\ldots,+)$. Then we can choose a particular $\\lambda$ to parameterize the null vector as \n\\begin{equation}\\label{Pemb}\nP^A=\\big(\\frac{1+x^2}{2},\\frac{1-x^2}{2},x^\\mu \\big)\n\\end{equation}\nwhere $x^2=x^\\mu x_\\mu$. Conformal group transformations correspond to $SO(d,2)$ rotations on $P^A$. Their actions on $x^\\mu$ are obtained by further rescaling $P^0+P^1$ of the rotated embedding vector to 1. \n\nLet us now introduce two fixed vectors\n\\begin{equation}\nN_b=(0,0,0,\\ldots,1)\\;,\\quad\\quad N_c=(1,0,0,\\ldots,0)\\;,\n\\end{equation}\nwhich will correspond to two different systems. Either vector partially breaks the conformal group. In the first case, the surface with $N_b\\cdot P=0$ gives rise to a planar boundary located at $x_d=0$. We will often denote $x_d$ as $x_\\perp$ as is common in the BCFT literature. Therefore, this case is related to boundary CFTs, and the residual conformal symmetry is $SO(d-1,2)$. This $SO(d-1,2)$ symmetry is just the conformal group of the $d-1$ dimensional boundary. \nNote that we can also perform a conformal transformation to change the planar boundary into a sphere. This can be accomplished by choosing the fixed vector $\\tilde{N}_b=(0,1,0,\\ldots,0)$, and the boundary is a unit sphere centered at $x=0$. Now we consider the second case. The symmetry group preserving the vector $N_c$ is $SO(d,1)$. Using this vector, we can define a transformation \n\\begin{equation}\\label{invP}\nP\\to -2(P\\cdot N_c)N_c-P\\;,\n\\end{equation}\nwhich upon rewriting in the form (\\ref{Pemb}) by rescaling gives the conformal inversion transformation\n\\begin{equation}\\label{invx}\nx^\\mu\\to -\\frac{x^\\mu}{x^2}\\;.\n\\end{equation}\nUpon identifying $x^\\mu\\sim -x^\\mu\/x^2$, we obtain the real projective space $\\mathbb{RP}^d$.\\footnote{A more familiar definition of the real projective space is to take the $\\mathbb{Z}_2$ quotient of a sphere $S^d$\n\\begin{equation}\n\\mathbf{X}^2=1\\;,\\quad \\mathbf{X}\\in\\mathbb{R}^{d+1}\\;,\\quad \\mathbf{X}\\sim-\\mathbf{X}\\;.\n\\end{equation}\nSince we are considering CFTs, we can perform a Weyl transformation to map it to the flat space by $x^\\mu=\\frac{X^\\mu}{1-X^{d+1}}$, $\\mu=1,\\ldots,d$ and $d s^2_{\\mathbb{R}^d}=\\frac{(1+x^2)^2}{4}ds^2_{S^d}$. In two dimensions, this is also known as a crosscap.} To consider CFTs on this quotient space, we also need to identify the operators inserted at points related by inversion. For scalar operators, we have\n\\begin{equation}\\label{invopid}\n\\mathcal{O}^\\pm_\\Delta(x)\\leftrightarrow \\pm x^{2\\Delta} \\mathcal{O}^\\pm_\\Delta(x')\\;,\\quad x'^\\mu=-\\frac{x^\\mu}{x^2}\\;.\n\\end{equation}\nwhere we have two choices for the parity of the operator. \n\nLet us now discuss correlators of local operators. For these systems, the simplest correlators are one-point functions. Because there are residual Lorentz symmetries, spinning operators cannot have one-point functions and we only need to consider scalar operators. The only invariants which we can construct from the embedding vector $P$ and the fixed vectors are $P\\cdot N_b$ and $P\\cdot N_c$. Moreover, using the scaling behavior (\\ref{opsca}) we can fix the one-point functions up to an overall constant. For BCFTs this gives\n\\begin{equation}\n\\langle O_\\Delta \\rangle_b=\\frac{a_{b,\\Delta}}{(2P\\cdot N_b)^\\Delta}=\\frac{a_{b,\\Delta}}{(2x_\\perp)^\\Delta}\\;,\n\\end{equation} \nand for $\\mathbb{RP}^d$ CFTs we have\n\\begin{equation}\n\\langle O_\\Delta^+ \\rangle_c=\\frac{a_{c,\\Delta}}{(-2P\\cdot N_c)^\\Delta}=\\frac{a_{c,\\Delta}}{(1+x^2)^\\Delta}\n\\end{equation} \nfor operators with $+$ parity and zero for the other choice.\\footnote{When we perform a Weyl transformation and map it to $S^d$, the one-point functions are just constants. Identifying operators on antipodal points with a minus sign forces their expectation values to be zero.} The coefficients $a_{b,\\Delta}$ and $a_{c,\\Delta}$ are new CFT data defining the theories.\\footnote{To see the coefficients correspond to new data, let us try to absorb them by changing the normalization of the operators. However, this would change the normalization of two-point functions. Note that when the points are very close to each other, we can ignore the presence of the boundary or the identification under inversion. The limiting two-point functions to should approach those in the CFT in infinite flat space with the same normalization.} \n\nLet us move on to two-point functions. In this case, one can construct cross ratios which are invariant under the residual conformal symmetry and independent rescalings of the embedding vectors. The cross ratio for the BCFT case is \n\\begin{equation}\\label{defcrxi}\n\\xi=\\frac{(-2P_1\\cdot P_2)}{(2N_b\\cdot P_1)(2N_b\\cdot P_2)}=\\frac{(x_1-x_2)^2}{4x_{1,\\perp}x_{2,\\perp}}\\;,\n\\end{equation} \nand the cross ratio for the real projective space case is \n\\begin{equation}\\label{defcreta}\n\\eta=\\frac{(-2P_1\\cdot P_2)}{(-2N_c\\cdot P_1)(-2N_c\\cdot P_2)}=\\frac{(x_1-x_2)^2}{(1+x_1^2)(1+x_2^2)}\\;.\n\\end{equation} \nThe two-point functions can be written as functions of the cross ratios after extracting a kinematic factor \n\\begin{eqnarray}\n\\langle \\mathcal{O}_{\\Delta_1}(x_1)\\mathcal{O}_{\\Delta_2}(x_2)\\rangle_b&=&\\frac{\\mathcal{G}(\\xi)}{|2N_b\\cdot P_1|^{\\Delta_1}|2N_b\\cdot P_2|^{\\Delta_2}}=\\frac{\\mathcal{G}(\\xi)}{|2x_{1,\\perp}|^{\\Delta_1}|2x_{2,\\perp}|^{\\Delta_2}}\\;,\\\\\n\\langle \\mathcal{O}^\\pm_{\\Delta_1}(x_1)\\mathcal{O}^\\pm_{\\Delta_2}(x_2)\\rangle_c&=&\\frac{\\mathcal{G}^\\pm(\\eta)}{(-2N_c\\cdot P_1)^{\\Delta_1}(-2N_c\\cdot P_2)^{\\Delta_2}}=\\frac{\\mathcal{G}^\\pm(\\eta)}{(1+x_1^2)^{\\Delta_1}(1+x_2^2)^{\\Delta_2}}\\;.\n\\end{eqnarray}\nNote that for two-point functions to be nonzero in real projective space CFTs, the two operators must have the same parity so that the two-point function is neutral under the parity $\\mathbb{Z}_2$. We can expand the two-point functions in the limits of operator product expansion (OPE), and the contributions are organized by the residual conformal symmetry into conformal blocks. We look at these two cases separately. \n\n\n\\begin{figure}\n \\centering\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\linewidth]{fig_BCFT_OPEB}\n \\caption{Bulk channel.}\n \\label{fig:BCFTOPEB}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\linewidth]{fig_BCFT_OPEbdr}\n \\caption{Boundary channel.}\n \\label{fig:BCFTOPEbdr}\n\\end{subfigure}\n\\caption{Two OPE limits in a BCFT two-point function. The horizontal lines represent the conformal boundary.}\n\\label{fig:de}\n\\end{figure}\n\nIn BCFTs we have two distinct OPEs. The first one is usually referred to as the {\\it bulk channel} OPE (Figure \\ref{fig:BCFTOPEB}) where the two operators are taken to be close to each other\n\\begin{equation}\\label{OPEB}\n\\mathcal{O}_{\\Delta_1}(x_1)\\mathcal{O}_{\\Delta_2}(x_2)=\\frac{\\delta_{12}}{(x_1-x_2)^{2\\Delta_1}}+\\sum_k C_{12k} D[x_1-x_2,\\partial_{x_2}]\\mathcal{O}_{\\Delta_k}(x_2)\n\\end{equation}\nwhere $C_{12k}$ are OPE coefficients and the differential operators $D[x_1-x_2,\\partial_{x_2}]$ are determined by conformal symmetry. For simplicity, we have only displayed in the OPE the scalar operators which contribute to the two-point function. Using this OPE, two-point functions can be expressed as an infinite sum of one-point functions.The contribution of each primary operator and its descendants can be resummed into a bulk-channel conformal block \\cite{McAvity:1995zd}\n\\begin{equation}\\label{BCFTbulkg}\ng_{b,\\Delta}^{\\rm bulk}(\\xi)=\\xi^{\\frac{\\Delta-\\Delta_1-\\Delta_2}{2}}{}_2F_1\\big(\\frac{\\Delta+\\Delta_1-\\Delta_2}{2},\\frac{\\Delta+\\Delta_2-\\Delta_1}{2};\\Delta-\\frac{d}{2}+1;-\\xi\\big)\\;,\n\\end{equation}\nand the two-point function can be written as \n\\begin{equation}\n\\mathcal{G}(\\xi)=\\delta_{12}\\xi^{-\\Delta_1}+\\sum_k \\mu_{b,12k}\\, g_{b,\\Delta_k}^{\\rm bulk}(\\xi)\\;,\n\\end{equation}\nwith $\\mu_{b,12k}=a_{b,\\Delta_k}C_{12k}$. The second OPE is the so-called {\\it boundary channel} OPE (Figure \\ref{fig:BCFTOPEbdr}) where operators are taken near the boundary and expressed in terms of operators living on the boundary at $x_\\perp=0$\n\\begin{equation}\n\\mathcal{O}_\\Delta(x)=\\frac{a_{b,\\Delta}}{|2x_\\perp|^\\Delta}+\\sum_l \\rho_l C[x]\\widehat{O}_{\\widehat{\\Delta}_l}(x)\\;.\n\\end{equation}\nHere $\\rho_l$ are OPE coefficients and the differential operators $C[x]$ are fixed by conformal symmetry. Using this OPE we can write the two-point function as an infinite sum of two-point functions on the boundary which are fixed by the residual conformal symmetry. The contribution of each operator is resummed into a {\\it boundary channel} conformal block \\cite{McAvity:1995zd}\n\\begin{equation}\\label{BCFTbdrg}\ng_{b,\\widehat{\\Delta}}^{\\rm boundary}(\\xi)=\\xi^{-\\widehat{\\Delta}}{}_2F_1\\big(\\widehat{\\Delta},\\widehat{\\Delta}-\\frac{d}{2}+1;2\\widehat{\\Delta}+2-d;-\\frac{1}{\\xi}\\big)\\;.\n\\end{equation}\nIn terms of the boundary channel conformal blocks, we can write the two-point function as \n\\begin{equation}\n\\mathcal{G}(\\xi)=a_{b,\\Delta}^2\\delta_{12}+\\sum_l \\rho_{1,l}\\rho_{2,l} g_{b,\\widehat{\\Delta}_l}^{\\rm boundary}(\\xi)\\;.\n\\end{equation}\nSimilar to four-point conformal blocks in CFTs without boundaries, the bulk channel and the boundary channel conformal blocks are also more conveniently computed as the eigenfunctions of conformal Casimir operators \\cite{Liendo:2012hy}. The two ways of expanding two-point functions are equivalent, and the equivalence gives rise to the BCFT crossing equation\n\\begin{equation}\n\\delta_{12}\\xi^{-\\Delta_1}+\\sum_k \\mu_{b,12k}\\, g_{b,\\Delta_k}^{\\rm bulk}(\\xi)=a_{b,\\Delta}^2\\delta_{12}+\\sum_l \\rho_{1,l}\\rho_{2,l} g_{b,\\widehat{\\Delta}_l}^{\\rm boundary}(\\xi)\\;.\n\\end{equation}\nHere in both channels we have explicitly singled out the identity operator exchange. We can also absorb them into the sums by extending the sums to include operators with dimension zero. \n\n\nIn real projective space CFTs, the situation is slightly different. We still have the bulk channel OPE (\\ref{OPEB}), which allows us to express the two-point function as a sum of one-point functions. The contribution of an operator resums into the conformal block \\cite{Nakayama:2016xvw}\n\\begin{equation}\\label{RPdCFTg}\ng_{c,\\Delta}(\\eta)=\\eta^{\\frac{\\Delta-\\Delta_1-\\Delta_2}{2}}{}_2F_1\\big(\\frac{\\Delta+\\Delta_1-\\Delta_2}{2},\\frac{\\Delta+\\Delta_2-\\Delta_1}{2};\\Delta-\\frac{d}{2}+1;\\eta\\big)\\;,\n\\end{equation}\nand the two-point function can be written as \n\\begin{equation}\\label{RPdcfdecom}\n\\mathcal{G}^\\pm(\\eta)=\\delta_{12}\\eta^{-\\Delta_1}+\\sum_k \\mu_{c,12k}\\, g_{c,\\Delta_k}(\\eta)\n\\end{equation}\nwhere $\\mu_{c,12k}=a_{c,\\Delta_k}C_{12k}$. On the other hand, we no longer have the boundary channel OPE since there is no boundary.\\footnote{For this reason, there are a lot more data in the BCFT case which are associated to the operators living on the boundary.} Instead, we can move $\\mathcal{O}_2$ towards the inversion image of $\\mathcal{O}_1$. Due to the operator identification \\ref{invopid}, we can apply the same OPE (\\ref{OPEB}). This gives rise to a new channel which we will refer to as the {\\it image channel}. These two OPE channels are illustrated in Figure \\ref{fig:RPdOPE}. The image channel conformal blocks are given by \\cite{Nakayama:2016xvw}\n\\begin{equation}\\label{RPdCFTgmirror}\n\\bar{g}_{c,\\Delta}(\\eta)=(1-\\eta)^{\\frac{\\Delta-\\Delta_1-\\Delta_2}{2}}{}_2F_1\\big(\\frac{\\Delta+\\Delta_1-\\Delta_2}{2},\\frac{\\Delta+\\Delta_2-\\Delta_1}{2};\\Delta-\\frac{d}{2}+1;1-\\eta\\big)\\;,\n\\end{equation}\nand the two-point function can be written as \n\\begin{equation}\n\\mathcal{G}^\\pm(\\eta)=\\delta_{12}(1-\\eta)^{-\\Delta_1}+\\sum_k \\mu_{c,12k}\\, \\bar{g}_{c,\\Delta_k}(\\eta)\\;.\n\\end{equation}\nThe conformal blocks in the two channels can also be obtained from solving Casimir equations. Equating these two conformal block decompositions, we arrive at the following crossing equation\n\\begin{equation}\\label{RPdcrossingeqn}\n\\delta_{12}\\eta^{-\\Delta_1}+\\sum_k \\mu_{c,12k}\\, g_{c,\\Delta_k}(\\eta)=\\pm\\big(\\delta_{12}(1-\\eta)^{-\\Delta_1}+\\sum_k \\mu_{c,12k}\\, \\bar{g}_{c,\\Delta_k}(\\eta)\\big)\n\\end{equation}\nwhere $\\pm$ is the common parity of the two operators. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{fig_RPd_OPE}\n\\caption{Two OPE channels in real projective space CFTs. The circle represents the unit sphere at $x=0$, and $\\mathcal{O}'_1(x'_1)$ is the inversion image of $\\mathcal{O}_1(x_1)$ with respect to the unit sphere.}\n \\label{fig:RPdOPE}\n\\end{figure}\n\n\nFinally, let us comment that we can complexify the cross ratios, and study the analytic property of correlators on the complex plane. For BCFTs, the complex $\\xi$-plane is shown in Figure \\ref{fig:xiplane}. The two special points $\\xi=0$ and $\\xi=\\infty$ correspond to the bulk channel and boundary channel OPE limits respectively. In Euclidean spacetime, the cross ratio is restricted to the semi-infinite real axis $\\xi\\in[0,\\infty)$. However, there is another interesting point at $\\xi=-1$ which can be reached via analytic continuation. In Lorentzian signature, $\\xi=-1$ corresponds to one operator approaching the lightcone of the other operator's image with respect to the boundary (Figure \\ref{fig:xiLorentzian}). This limit is referred to as the Regge limit. In a unitary BCFT, one can prove that the growth of the two-point function in the Regge limit is bounded by the exchange of the operator with the lowest conformal dimension in the bulk channel. The proof takes advantage of the so-called $\\rho$ coordinate \\cite{Hogervorst:2013sma}, and the positivity of the conformal block decomposition coefficients in the boundary channel. Details of the proof can be found in Appendix A of \\cite{Mazac:2018biw}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig_BCFT_xiplane}\n\\caption{The $\\xi$-plane for a BCFT two-point function. Three interesting points on this plane are the bulk channel OPE limit ($\\xi=0$), the boundary channel OPE limit ($\\xi=\\infty$), and the Regge limit ($\\xi=-1$).}\n \\label{fig:xiplane}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{fig_BCFT_Lorentzianxi}\n\\caption{The Regge limit of a BCFT two-point function in Lorentzian spacetime. The vertical line represents the conformal boundary, and time is one of the dimensions of the boundary. The Regge limit $\\xi=-1$ is reached when $\\mathcal{O}_2$ approaches the lightcone of the image of $\\mathcal{O}_1$.}\n \\label{fig:xiLorentzian}\n\\end{figure}\n\nThe complex $\\eta$-plane for real projective space CFTs is shown in Figure \\ref{fig:etaplane}. There are also three points of special interest. The points $\\eta=0$ and $\\eta=1$ respectively correspond to the bulk channel OPE and the image channel OPE limits, and $\\eta\\in[0,1]$ for Euclidean space. The point $\\eta=\\infty$ plays a similar role as the Regge limit in BCFT two-point functions \\cite{Giombi:2020xah}, and can only be reached via analytic continuation in Euclidean signature. However, unlike in the BCFT case, there is no analogue of the boundary channel where the conformal block decomposition coefficients are positive. Therefore, one cannot adapt the proof for BCFTs to prove boundedness of two-point functions in the Regge limit. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig_RPd_etaplane}\n\\caption{The $\\eta$-plane for a real projective space CFT two-point function. Three interesting points on this plane are the bulk channel OPE limit ($\\eta=0$), the image channel OPE limit ($\\eta=1$), and the Regge limit ($\\eta=\\infty$).}\n \\label{fig:etaplane}\n\\end{figure}\n\n\n\n\\subsection{Analytic methods}\\label{Subsec:analyticmethods}\nIn this subsection we discuss analytic methods for BCFTs \\cite{Kaviraj:2018tfd,Mazac:2018biw} and real projective space CFTs \\cite{Giombi:2020xah} which are based on ``analytic functionals''. Such functional methods were originally introduced for four-point functions in 1d CFTs \\cite{Mazac:2016qev,mazacpaulos1,mazacpaulos2}, and later generalized to higher dimensions in \\cite{Paulos:2019gtx,MRZ19,Caron-Huot:2020adz}. While the level of technical sophistication varies greatly in these different setups, the essential ideas remain the same. Here we will exploit the simpler kinematics of two-point functions to demonstrate the main features of such an approach. \n\nTo help the reader navigate through this subsection, let us give below a quick summary of these features and also point out the connections. We will argue that the ``double-trace'' conformal blocks, from both the direct and the crossed channels, form a new basis for expanding the correlators. These double-trace conformal blocks are associated with special product operators of which the conformal dimensions are the sums of the elementary building operators. This should be contrasted with the standard conformal block decomposition which exploits only one channel at a time and does not require the spectrum to be discrete. The dual of the double-trace conformal blocks are the analytic functionals. Their actions on the crossing equation turn it into sum rules for the CFT data. We will develop this functional approach both from a dispersion relation, and by exploiting the structure of Feynman diagrams (Witten diagrams) in certain holographic setups. The first argument can be viewed as a toy example of the CFT dispersion relation for four-point functions \\cite{Carmi:2019cub}. The second argument is closely related to Polyakov's original version of the conformal bootstrap \\cite{Polyakov:1974gs}, which will be reviewed later in Section \\ref{Sec:dispersionPolyakov}. As we will see, the Witten diagrams also give rise to another set of basis which are essentially those used in \\cite{Polyakov:1974gs}. Moreover, the sum rules from the functionals are just a modern paraphrase of the consistency conditions imposed by Polyakov in his approach. \n\n\n\\subsubsection{Real projective space CFTs}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig_RPd_contours}\n\\caption{Deformation of contours in the dispersion relation.}\n \\label{fig:RPdcontours}\n\\end{figure}\n\nLet us first consider a simplified example of a two-point functions in a 2d real projective CFT where the external dimensions are equal $\\Delta_1=\\Delta_2=\\Delta_\\varphi$, following the discussion in \\cite{Giombi:2020xah}. The two-point function can be written as \n\\begin{equation}\n\\mathcal{G}(\\eta)=\\oint \\frac{d\\zeta}{2\\pi i} \\frac{\\mathcal{G}(\\zeta)}{\\zeta-\\eta}\n\\end{equation}\nusing Cauchy's integral formula. To lighten the notation we will suppress the parity choice $\\pm$ of the operators in this subsection. We can deform the contour to wrap it around the two branch cuts $[1,\\infty)$, $(-\\infty,0]$ as in Figure \\ref{fig:RPdcontours}. Assuming that the two-point function has the following behavior in the Regge limit\n\\begin{equation}\\label{Reggebounded}\n|\\mathcal{G}(\\eta)|\\lesssim |\\eta|^{-\\epsilon}\\;,\\quad \\eta\\to \\infty\\;,\n\\end{equation}\nwhere $\\epsilon$ is an infinitesimal positive number, we can drop the contribution from the arcs at infinity. The two-point function becomes\n\\begin{equation}\n\\mathcal{G}(\\eta)=\\mathcal{G}_1(\\eta)+\\mathcal{G}_2(\\eta)\n\\end{equation}\nwhere \n\\begin{equation}\\label{G1G2int}\n\\begin{split}\n\\mathcal{G}_1(\\eta)=&\\int_{C_1} \\frac{d\\zeta}{2\\pi i} \\frac{\\mathcal{G}(\\zeta)}{\\zeta-\\eta}=\\int_1^\\infty \\frac{d\\zeta}{2\\pi i} \\frac{{\\rm Disc}_1[\\mathcal{G}(\\zeta)]}{\\zeta-\\eta}\\;,\\\\\n\\mathcal{G}_2(\\eta)=&-\\int_{C_2} \\frac{d\\zeta}{2\\pi i} \\frac{\\mathcal{G}(\\zeta)}{\\zeta-\\eta}=\\int_{-\\infty}^0 \\frac{d\\zeta}{2\\pi i} \\frac{{\\rm Disc}_2[\\mathcal{G}(\\zeta)]}{\\zeta-\\eta}\\;,\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n{\\rm Disc}_1[\\mathcal{G}(\\zeta)]=&\\mathcal{G}(\\zeta+i0^+)-\\mathcal{G}(\\zeta-i0^+)\\;,\\quad \\zeta\\in(1,\\infty)\\;,\\\\\n{\\rm Disc}_2[\\mathcal{G}(\\zeta)]=&\\mathcal{G}(\\zeta+i0^+)-\\mathcal{G}(\\zeta-i0^+)\\;,\\quad \\zeta\\in(-\\infty,0)\\;.\n\\end{split}\n\\end{equation}\nThe two functions $\\mathcal{G}_1(\\eta)$ and $\\mathcal{G}_2(\\eta)$ are related by crossing symmetry\n\\begin{equation}\\label{RPdG1G2}\n\\mathcal{G}_2(\\eta)=\\pm \\mathcal{G}_1(1-\\eta)\\;.\n\\end{equation}\nTo proceed, let us define a function \n\\begin{equation}\nk_h(\\eta)=\\eta^h {}_2F_1(h,h;2h,\\eta)\n\\end{equation}\nwhich has the following orthonormal property\n\\begin{equation}\n\\oint_{|\\eta|=\\epsilon} \\frac{d\\eta}{2\\pi i}\\eta^{-2}k_{x+n}(\\eta)k_{1-x-m}(\\eta)=\\delta_{nm}\\;.\n\\end{equation}\nWe note that the conformal blocks with $d=2$, $\\Delta_1=\\Delta_2=\\Delta_\\varphi$ are related to $k_h(\\eta)$ by \n\\begin{equation}\ng_{c,\\Delta}(\\eta)=\\eta^{-\\Delta_\\varphi} k_{\\frac{\\Delta}{2}}(\\eta)\\;.\n\\end{equation}\nWe will now show that the two-point function $\\mathcal{G}(\\eta)$ can be decomposed in terms of a special class of conformal blocks with dimensions $\\Delta_n^{\\rm d.t.}=2\\Delta_\\varphi+2n$ in {\\it both} OPE channels. Here the superscript ${\\rm d.t.}$ stands for {\\it double-trace} as $\\Delta_n^{\\rm d.t.}$ is the conformal dimension of a double-trace operator of the schematic form $:\\varphi\\square^n\\varphi:$. These are operators which appear universally in the mean field theory, and their dimensions are just the sums of the dimensions of the building blocks.\\footnote{The operator $\\varphi$ has dimension $\\Delta_\\varphi$ and $\\square$ has dimension 2. Therefore, $:\\varphi\\square^n\\varphi:$ has dimension $2\\Delta_\\varphi+2n$. The terminology ``double-trace'' is borrowed from gauge theory to denote the fact such an operator is made of {\\it two} ``single-trace'' operators $\\phi$. Here ``trace'' refers to the trace over gauge group indices because a single-trace operator in gauge theories has the form ${\\rm tr}(X_1X_2\\ldots X_n)$, with operators $X_i=X_i^aT^a$ in the adjoint representation of the gauge group. For the moment, these terminologies can just be regarded as names if they are not familiar to the reader.} To show this, we note that the kernel in the Cauchy integral admits the following expansion in terms double-trace conformal blocks\n\\begin{equation}\\label{Ckernalindt}\n\\frac{1}{\\zeta-\\eta}=\\sum_{n=0}^\\infty H_n(\\zeta) g_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\n\\end{equation}\nwith coefficients which are functions of $\\zeta$. These coefficients can be computed using the orthonormal property of $k_h(\\eta)$\n\\begin{equation}\nH_n(\\zeta)=\\oint_{|\\eta|=\\epsilon}\\frac{d\\eta}{2\\pi i} \\frac{\\eta^{\\Delta_\\varphi-2}}{\\zeta-\\eta}k_{1-\\Delta_\\varphi-n}(\\eta)\\;,\n\\end{equation}\nand gives\n\\begin{equation}\nH_n(\\zeta)=\\frac{(-4)^{-n}(\\Delta_\\varphi)_n(2\\Delta_\\varphi-1)_n}{n!(\\Delta_\\varphi-\\frac{1}{2})_n}\\zeta^{-1}{}_3F_2(1,-n,2\\Delta_\\varphi+n-1;\\Delta_\\varphi,\\Delta_\\varphi;\\zeta^{-1})\\;.\n\\end{equation}\nInserting (\\ref{Ckernalindt}) into (\\ref{G1G2int}), we find that $\\mathcal{G}_1(\\eta)$ can be expanded in terms of double-trace conformal blocks\n\\begin{equation}\n\\mathcal{G}_1(\\eta)=\\sum_{n=0}^\\infty r_{n,1}\\, g_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;,\n\\end{equation}\nwith\n\\begin{equation}\\label{rn1}\nr_{n,1}=\\int_{C_1}\\frac{d\\zeta}{2\\pi i} H_n(\\zeta)\\mathcal{G}(\\zeta)\\;.\n\\end{equation}\nNow using crossing symmetry (\\ref{RPdG1G2}), we find that $\\mathcal{G}_2(\\eta)$ can be expanded in terms of double-trace conformal blocks in the image channel\n\\begin{equation}\n\\mathcal{G}_2(\\eta)=\\sum_{n=0}^\\infty r_{n,2}\\, \\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;,\n\\end{equation}\nwhere $r_{n,2}=\\pm r_{n,1}$. This proves that any two-point function $\\mathcal{G}(\\eta)$, suitably bounded in the Regge limit as in (\\ref{Reggebounded}), can be decomposed as a linear combination of double-trace conformal blocks $\\{g_{c,\\Delta_n^{\\rm d.t.}}(\\eta),\\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\}$ from both channels. Note this is quite different from the standard conformal block decomposition where we use only one OPE channel and the conformal dimensions of the conformal blocks are not forced to take discrete values. \n\nThe conclusions we reached in this simple example in fact generalize to the general case. Let us consider a two-point function in a $d$-dimensional real projective space CFT with dimensions $\\Delta_1$ and $\\Delta_2$. If the two-point function satisfies the boundedness condition (\\ref{Reggebounded}), then a basis is given by the conformal blocks in the bulk channel and the image channel\n\\begin{equation}\n\\{g_{c,\\Delta_n^{\\rm d.t.}}(\\eta),\\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\}\\;,\\quad\\quad n=0,1,2,\\ldots\n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta_n^{\\rm d.t.}=\\Delta_1+\\Delta_2+2n\\;.\n\\end{equation}\nWith this basis of functions, we can define a dual basis whose elements are the {\\it functionals}\n\\begin{equation}\n\\{\\omega_{c,n},\\bar{\\omega}_{c,n}\\}\\;,\\quad\\quad n=0,1,2,\\ldots\\;.\n\\end{equation}\nThese functional are defined to have the following orthonormal action on the basis vectors\n\\begin{equation}\\label{omegaong}\n\\begin{split}\n&\\omega_{c,m}(g_{c,\\Delta_n^{\\rm d.t.}})=\\delta_{nm}\\;,\\quad\\quad \\omega_{c,m}(\\bar{g}_{c,\\Delta_n^{\\rm d.t.}})=0\\;,\\\\\n&\\bar{\\omega}_{c,m}(g_{c,\\Delta_n^{\\rm d.t.}})=0\\;,\\quad\\quad \\bar{\\omega}_{c,m}(\\bar{g}_{c,\\Delta_n^{\\rm d.t.}})=\\delta_{nm}\\;.\n\\end{split}\n\\end{equation}\n\nTo fully specify these functionals, we need to know how they act on a generic conformal block, {\\it i.e.}, computing\n\\begin{equation}\\label{omegaaction}\n\\omega_{c,n}(g_{c,\\Delta})\\;,\\quad \\omega_{c,n}(\\bar{g}_{c,\\Delta})\\;,\\quad \\bar{\\omega}_{c,n}(g_{c,\\Delta})\\;,\\quad \\bar{\\omega}_{c,n}(\\bar{g}_{c,\\Delta})\\;,\n\\end{equation}\nfor a general conformal dimension $\\Delta$. Let us consider decomposing a conformal block in the above double-trace basis\n\\begin{equation}\\label{gcdecomp}\ng_{c,\\Delta}(\\eta)=\\sum_n M_n\\, g_{c,\\Delta_n^{\\rm d.t.}}(\\eta)+\\sum_n N_n\\, \\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;.\n\\end{equation}\nActing on it with the basis functionals and using the orthonormal relation (\\ref{omegaong}), we find \n\\begin{equation}\nM_n=\\omega_{c,n}(g_{c,\\Delta})\\;,\\quad \\quad N_n=\\bar{\\omega}_{c,n}(g_{c,\\Delta})\\;.\n\\end{equation}\nSimilarly, the actions $\\omega_{c,n}(\\bar{g}_{c,\\Delta})$, $\\bar{\\omega}_{c,n}(\\bar{g}_{c,\\Delta})$ appear in the decomposition coefficients of the image channel conformal block $\\bar{g}_{c,\\Delta}$. Once we know the actions of these functionals, we can act with them on the crossing equation of two-point functions (\\ref{RPdcrossingeqn}) to systematically extract the constraints on the CFT data in the form of {\\it sum rules}\\footnote{Here we have absorbed the identity exchange into the infinite sum. Moreover, we have assumed that we are allowed to swap the infinite summation with the action of the functionals. However, this may not always be true. For a detailed discussion on this swapping subtlety, see \\cite{Qiao:2017lkv,mazacpaulos2}.}\n\\begin{equation}\\label{PRdsumrule}\n\\sum_k \\mu_{c,12k}\\, \\omega_n(g_{c,\\Delta_k})=\\pm \\sum_k \\mu_{c,12k}\\, \\omega_n(\\bar{g}_{c,\\Delta_k})\\;.\n\\end{equation}\nIn the 2d example considered above, these actions can be computed as contour integrals (\\ref{rn1}) with $\\mathcal{G}(\\eta)$ taken to be a conformal block. However, these coefficients can also be computed in a different way, by considering a seemingly unrelated problem of conformal block decomposition of tree-level Witten diagrams in AdS space, as we now explain. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{fig_qAdS}\n\\caption{The quotient AdS space obtained by identifying points related by the inversion with respect to the unit radius hemisphere. The point $Z'$ is the inversion image of $Z$. The north pole $N_c$ of the hemisphere is invariant under inversion.}\n \\label{fig:qAdS}\n\\end{figure}\n\nWe consider the following simple setup that realizes the kinematics of a real projective space CFT in AdS space. We first extend the inversion (\\ref{invx}) in $\\mathbb{R}^d$ to $AdS_{d+1}$ by\n\\begin{equation}\\label{AdSinv}\nz_0\\to \\frac{z_0}{z_0^2+\\vec{z}^2}\\;,\\quad \\quad \\vec{z}\\to -\\frac{\\vec{z}}{z_0^2+\\vec{z}^2}\n\\end{equation}\nwhere $z=(z_0,\\vec{z})$ are the Poincar\\'e coordinates of AdS and $z_0$ is the radial direction. Note that at the conformal boundary $z_0=0$, (\\ref{AdSinv}) reduces to (\\ref{invx}). This transformation can also be obtained from (\\ref{invP}) by replacing the embedding space vector $P$ by the embedding space vector $Z$ of an AdS point\n\\begin{equation}\nZ^A=\\frac{1}{z_0}\\bigg(\\frac{1+z_0^2+\\vec{z}^2}{2},\\frac{1-z_0^2-\\vec{z}^2}{2},\\vec{z}\\bigg)\\;.\n\\end{equation}\nGeometrically, (\\ref{AdSinv}) corresponds to an inversion with respect to a unit radius hemisphere located at $z_0=0$, $\\vec{z}=0$, as is illustrated in Figure \\ref{fig:qAdS}. The kinematics of real projective space CFTs can be realized in the quotient space $AdS_{d+1}\/\\mathbb{Z}_2$ which is defined by identifying points under the inversion (\\ref{AdSinv})\n\\begin{equation}\nz_0\\leftrightarrow \\frac{z_0}{z_0^2+\\vec{z}^2}\\;,\\quad \\quad \\vec{z}\\leftrightarrow -\\frac{\\vec{z}}{z_0^2+\\vec{z}^2}\\;.\n\\end{equation}\nNote that (\\ref{AdSinv}) has a special fixed point at $z_0=1$, $\\vec{z}=0$, which corresponds to the north pole of the hemisphere. In fact, written in terms of the embedding space coordinates, this point is nothing but the fixed vector $N_c$. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{fig_RPd_WD}\n\\caption{Tree-level Witten diagrams. On the LHS, we have a single exchange Witten diagram in the quotient AdS space. On the RHS, we use the method of images to express it in terms of the exchange Witten diagram in the full AdS space and its image diagram where one boundary point $x_1$ has been moved to its inversion image $x'_1$.}\n \\label{fig:RPdWD}\n\\end{figure}\n\nLet us now consider scalar fields on this quotient AdS space, and require the fields to have the same value at points related by inversion \n\\begin{equation}\\label{Phicondi}\n\\Phi(Z)=\\pm \\Phi(Z')\\;,\n\\end{equation}\nup to a sign which corresponds to the parity of the dual CFT operator. Here $Z'$ is the inversion image of $Z$. We further assume that the effective action of these fields contains a cubic term $\\int dZ \\Phi_1(Z) \\Phi_2(Z) \\Phi(Z)$, and a linear term $\\Phi(N_c)$ that is localized at the fixed point $N_c$. We can then consider an exchange Witten diagram $V^{\\rm exchange}_{\\Delta}$ in the quotient AdS space as is shown on the LHS of Figure \\ref{fig:RPdWD}. The propagators in this diagram need to be consistent with the condition (\\ref{Phicondi}) on the hemisphere. The insertion point of the cubic vertex is integrated over the quotient AdS space, while the end $N_c$ is held fixed. By using the method of images, we can express this exchange diagram in terms of exchange Witten diagrams defined on the full AdS space without taking the quotient (Figure \\ref{fig:RPdWD})\n\\begin{equation}\\label{VqAdS}\nV^{\\rm exchange}_{\\Delta}(P_1,P_2)=W^{\\rm exchange}_{\\Delta}(P_1,P_2)\\pm (x_1^2)^{-\\Delta_1}\\bar{W}^{\\rm exchange}_{\\Delta}(P_1,P_2)\\;.\n\\end{equation}\nHere, the Witten diagrams\n\\begin{equation}\\label{Wcexchange}\nW^{\\rm exchange}_{\\Delta}(P_1,P_2)=\\int_{AdS_{d+1}}d^{d+1}Z G^\\Delta_{BB}(N_c,Z)G^{\\Delta_1}_{B\\partial}(Z,P_1)G^{\\Delta_2}_{B\\partial}(Z,P_2)\\;,\n\\end{equation}\n\\begin{equation}\n\\bar{W}^{\\rm exchange}_{\\Delta}(P_1,P_2)=W^{\\rm exchange}_{\\Delta}(P'_1,P_2)\\;,\n\\end{equation}\nare defined with the standard AdS bulk-to-boundary propagator\n\\begin{equation}\nG^{\\Delta}_{B\\partial}(Z,P)=\\frac{1}{(-2Z\\cdot P)^\\Delta}\\;,\n\\end{equation}\nand the bulk-to-bulk propagator satisfying \n\\begin{equation}\n\\big(\\square_Z-\\Delta(\\Delta-d)\\big)G^\\Delta_{BB}(Z,W)=\\delta(Z,W)\\;.\n\\end{equation}\nThe integration region of the cubic vertex insertion points is the entire $AdS_{d+1}$ space. There appears to be two more diagrams where the sources are inserted at $(P'_1,P'_2)$ and $(P_1,P'_2)$. but one can show that they are the same as the two diagrams above. Let us also extract the kinematic factor from (\\ref{VqAdS}), and then we have \n\\begin{equation}\nV^{\\rm exchange}_{\\Delta}(P_1,P_2)=\\frac{1}{(1+x_1^2)^{\\Delta_1}(1+x_2^2)^{\\Delta_2}}\\left(\\mathcal{W}^{\\rm exchange}_{\\Delta}(\\eta)\\pm \\bar{\\mathcal{W}}^{\\rm exchange}_{\\Delta}(\\eta)\\right)\\;,\n\\end{equation}\nwith \n\\begin{equation}\\label{WWbar}\n\\bar{\\mathcal{W}}^{\\rm exchange}_{\\Delta}(\\eta)=\\mathcal{W}^{\\rm exchange}_{\\Delta}(1-\\eta)\\;.\n\\end{equation}\n\nAfter this long detour into AdS, let us finally get to our point: the functional actions (\\ref{omegaaction}) can be extracted from the conformal block decomposition coefficients of the Witten diagrams $\\mathcal{W}^{\\rm exchange}_{\\Delta}(\\eta)$, $\\bar{\\mathcal{W}}^{\\rm exchange}_{\\Delta}(\\eta)$. One can show that \nboth diagrams obey the boundedness condition (\\ref{Reggebounded}) in the Regge limit. Moreover, under conformal block decomposition the exchange Witten diagram $\\mathcal{W}^{\\rm exchange}_{\\Delta}(\\eta)$ is comprised of a single-trace conformal block and infinitely many double-trace conformal blocks in the same channel\n\\begin{equation}\n\\mathcal{W}^{\\rm exchange}_{\\Delta}(\\eta)=A\\, g_{c,\\Delta}(\\eta)+\\sum_{n=0}^\\infty A_n\\, g_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;,\n\\end{equation} \nand infinitely many double-trace conformal blocks in the crossed channel\n\\begin{equation}\n\\mathcal{W}^{\\rm exchange}_{\\Delta}(\\eta)=\\sum_{n=0}^\\infty B_n\\, \\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;.\n\\end{equation}\nWe are stating here these decomposition properties merely as facts to avoid going into unnecessary technicalities. But they follow directly from a study of these integrals and the details of the analysis can be found in \\cite{Giombi:2020xah}. Comparing these two expansions with (\\ref{gcdecomp}), one finds that the functional actions can be expressed in terms of the conformal block decomposition coefficients of exchange Witten diagrams as\n\\begin{equation}\n\\omega_{c,n}(g_{c,\\Delta})=-\\frac{A_n}{A}\\;,\\quad \\quad \\bar{\\omega}_{c,n}(g_{c,\\Delta})=\\frac{B_n}{A}\\;.\n\\end{equation}\nAs was shown in \\cite{Giombi:2020xah}, one can explicitly evaluate the exchange Witten diagram integral (\\ref{Wcexchange}) in terms of hypergeometric functions, and recursively compute all the conformal block decomposition coefficients. Here we do not give the explicit expressions of these coefficients, and refer the reader to \\cite{Giombi:2020xah} for details. Similarly, the image diagram decomposes as \n\\begin{equation}\n\\bar{\\mathcal{W}}^{\\rm exchange}_{\\Delta}(\\eta)=\\sum_{n=0}^\\infty B_n\\, g_{c,\\Delta_n^{\\rm d.t.}}(\\eta)=A\\, \\bar{g}_{c,\\Delta}(\\eta)+\\sum_{n=0}^\\infty A_n\\, \\bar{g}_{c,\\Delta_n^{\\rm d.t.}}(\\eta)\\;,\n\\end{equation}\nwhich follows from the crossing relation (\\ref{WWbar}). From these identities we find \n\\begin{equation}\\label{oobarfWbar}\n\\omega_{c,n}(\\bar{g}_{c,\\Delta})=\\bar{\\omega}_{c,n}(g_{c,\\Delta})\\;,\\quad \\bar{\\omega}_{c,n}(\\bar{g}_{c,\\Delta})=\\omega_{c,n}(g_{c,\\Delta})\\;.\n\\end{equation}\nAll in all, these Witten diagrams give us an efficient holographic method to obtain these functionals. \n\nIn fact, there is a further use of these Witten diagrams. As we now show, they also furnish a new basis of functions to decompose conformal correlators. The decomposition reads\n\\begin{equation}\\label{RPdWDexp}\n\\mathcal{G}(\\eta)=\\sum_k \\frac{\\mu_{c,12k}}{A}(\\mathcal{W}_{\\Delta_k}^{\\rm exchange}(\\eta)\\pm\\bar{\\mathcal{W}}_{\\Delta_k}^{\\rm exchange}(\\eta))\n\\end{equation}\nwhere we sum over the same spectrum appearing in the conformal block decomposition (\\ref{RPdcfdecom}) and $\\mu_{c,12k}$ are the same coefficients. To prove it, we expand both $\\mathcal{W}_{\\Delta_k}^{\\rm exchange}(\\eta)$ and $\\bar{\\mathcal{W}}_{\\Delta_k}^{\\rm exchange}(\\eta)$ in the $\\eta\\to 0$ channel. This gives\n\\begin{equation}\n\\mathcal{G}(\\eta)=\\sum_k \\mu_{c,12k} g_{c,\\Delta_k}(\\eta)+\\sum_k\\sum_{n=0}^\\infty \\mu_{c,12k}\\left(-\\omega_{c,n}(g_{\\Delta_k})\\pm \\bar{\\omega}_{c,n}(g_{c,\\Delta_k})\\right)g_{c,\\Delta_n^{\\rm d.t.}}\\;.\n\\end{equation}\nInterchanging the order of the sums and using (\\ref{oobarfWbar}), we find the second term vanishes when the sum rules (\\ref{PRdsumrule}) are used. The expansion in terms of Witten diagrams then reduces to the conformal block decomposition in the bulk channel. While this new expansion is very similar to the conformal block expansion, we must note the important difference that it exploits building blocks from both channels at the same time. Basis of this kind first appeared in the original work of Polyakov \\cite{Polyakov:1974gs}. Finally, we can also reverse the logic starting from (\\ref{RPdWDexp}). Requiring that double-trace conformal blocks vanish in the conformal block decomposition gives rise to sum rules (\\ref{PRdsumrule}).\n\n\\vspace{0.5cm}\n\n\\noindent{\\bf An application}\n\n\\vspace{0.2cm}\n\n\\noindent The zeros in the functional actions (\\ref{omegaong}) at the double-trace conformal dimensions can considerably simplify the sum rules (\\ref{PRdsumrule}) if the theory spectrum contains such operators. The simplest (and almost trivial) example is the mean field theory. However, we can also consider CFTs which can be viewed as small perturbations around the mean field theory. This special feature of the analytic functionals therefore makes them particularly suitable for studying such theories. As a simple application, let us show how to use functionals to bootstrap the one-point function coefficients of the $O(N)$ model on real projective space. We will only outline the computation, and refer the reader to \\cite{Giombi:2020xah} for the explicit details. \n\nThe CFT of interest is the Wilson-Fisher fixed point of the Lagrangian theory \n\\begin{equation}\nS=\\frac{\\Gamma(\\frac{d}{2}-1)}{4\\pi^{\\frac{d}{2}}}\\int d^dx \\bigg(\\frac{1}{2}(\\partial_\\mu\\varphi^I)^2+\\frac{\\lambda}{4}(\\varphi^I\\varphi^I)^2\\bigg)\\;,\\quad I=1,\\ldots,N\\;,\n\\end{equation}\nat $d=4-\\epsilon$ dimension. We consider the $\\langle\\varphi^I\\varphi^J\\rangle$ two-point function. To order $\\epsilon^2$, the only operators that can be exchanged are the identity and the double-trace operators $[\\varphi\\varphi]_n=\\varphi\\square^n\\varphi$, and we parameterize the deviations from the mean field theory values as follows\n\\begin{equation}\n\\mu_{\\varphi\\varphi n}=\\mu^{(0)}_{\\varphi\\varphi n}+\\epsilon \\mu^{(1)}_{\\varphi\\varphi n}+\\epsilon^2 \\mu^{(2)}_{\\varphi\\varphi n}\\;,\\quad \\Delta_\\varphi=\\frac{d}{2}-1+\\epsilon^2\\gamma_\\varphi^{(2)}\\;,\\quad \\Delta_{[\\varphi\\varphi]_n}=\\Delta_n^{\\rm d.t.}+\\epsilon\\gamma_n^{(1)}+\\epsilon^2\\gamma_n^{(2)}\\;.\n\\end{equation}\nWe have used the well known fact that the anomalous dimension of $\\varphi$ starts at $\\epsilon^2$. \n\nTo proceed, we act on crossing equation with the functionals and expand the sum rules in powers of $\\epsilon$\n\\begin{equation}\n\\omega_n(g_{c,0})+\\sum_n \\mu_{\\varphi\\varphi n}\\, \\omega_n(g_{c,\\Delta_{[\\varphi\\varphi]_n}})=\\pm \\big(\\omega_n(\\bar{g}_{c,0}) +\\sum_n \\mu_{\\varphi\\varphi n}\\,\\omega_n(\\bar{g}_{c,\\Delta_{[\\varphi\\varphi]_n}})\\big)+\\mathcal{O}(\\epsilon^3)\\;.\n\\end{equation}\nAt the zeroth order, we have just the mean field theory and one can check that the sum rules \n\\begin{equation}\n\\omega_n(g_{c,0})+\\mu_{\\varphi\\varphi n}^{(0)}=\\pm \\omega_n(\\bar{g}_{c,0})\n\\end{equation}\ngives the correct mean field theory coefficients\n\\begin{equation}\n\\mu_{\\varphi\\varphi n}^{(0)}=\\pm \\delta_{n,0}\\;.\n\\end{equation}\nMoreover, one finds $\\omega_n(g_{c,0})=0$. Using these results in the next order and we find that the sum rules at $\\mathcal{O}(\\epsilon)$ are given by\n\\begin{equation}\n\\mu_{\\varphi\\varphi n}^{(1)}\\pm \\big[\\omega_{c,n}(g_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}=\\pm \\big[\\omega_{c,n}(\\bar{g}_{c,0})\\big]_{\\mathcal{O}(\\epsilon)}+\\big[\\omega_{c,n}(\\bar{g}_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}\\;.\n\\end{equation}\nSolving these equations, we find\n\\begin{equation}\n\\mu_{\\varphi\\varphi 0}^{(1)}=-\\frac{\\gamma_0^{(1)}}{2}\\;,\\quad \\mu_{\\varphi\\varphi 1}^{(1)}=\\frac{\\gamma_0^{(1)}}{4}\\;,\\quad \\mu_{\\varphi\\varphi\\, n\\geq 2}^{(1)}=0\\;,\n\\end{equation}\nwhich agrees with \\cite{Hasegawa:2018yqg}. Note that the fact that only finitely many coefficients are nonzero at this order is very useful. It implies that at the next order the sum rules will continue to have only finitely many terms. Explicitly, we find at $\\mathcal{O}(\\epsilon^2)$\n\\begin{equation}\n\\begin{split}\n&\\mu_{\\varphi\\varphi n}^{(2)}\\pm \\big[\\omega_{c,n}(g_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}+\\epsilon^2\\gamma_0^{(2)}})\\big]_{\\mathcal{O}(\\epsilon^2)}+\\mu_{\\varphi\\varphi 0}^{(1)}\\big[\\omega_{c,n}(g_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}\\\\&+\\mu_{\\varphi\\varphi 1}^{(1)}\\big[\\omega_{c,n}(g_{c,\\Delta_1^{\\rm d.t.}+\\epsilon \\gamma_1^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}=\\pm\\big[\\omega_{c,n}(\\bar{g}_{c,0})\\big]_{\\mathcal{O}(\\epsilon^2)}+\\big[\\omega_{c,n}(\\bar{g}_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}+\\epsilon^2\\gamma_0^{(2)}})\\big]_{\\mathcal{O}(\\epsilon^2)}\\\\\n&\\pm \\mu_{\\varphi\\varphi 0}^{(1)} \\big[\\omega_{c,n}(\\bar{g}_{c,\\Delta_0^{\\rm d.t.}+\\epsilon \\gamma_0^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}\\pm \\mu_{\\varphi\\varphi 1}^{(1)} \\big[\\omega_{c,n}(\\bar{g}_{c,\\Delta_1^{\\rm d.t.}+\\epsilon \\gamma_1^{(1)}})\\big]_{\\mathcal{O}(\\epsilon)}\\;.\n\\end{split}\n\\end{equation}\nFrom these equations, we can solve the coefficients $\\mu_{\\varphi\\varphi n}^{(2)}$ in terms of the bulk data of anomalous dimensions. After using their values in the $O(N)$ model, we find, for example\n\\begin{equation}\n\\begin{split}\n&\\mu_{\\varphi\\varphi 0}=\\pm 1-\\frac{N+2}{2(N+8)}\\epsilon-\\frac{3(N+2)(2N+6\\pm(N+8))}{2(N+8)^3}\\epsilon^2+\\mathcal{O}(\\epsilon^3)\\;,\\\\\n&\\mu_{\\varphi\\varphi 1}=\\frac{N+2}{4(N+8)}\\epsilon-\\frac{(N+2)}{4(N+8)^2}\\big(\\frac{76+N(N+10)}{2(N+8)}\\pm(N-2)\\big)\\epsilon^2+\\mathcal{O}(\\epsilon^3)\\;.\n\\end{split}\n\\end{equation}\nThese results are consistent with the analytic results obtained from large $N$ analysis \\cite{Giombi:2020xah}, and also with the numerical bootstrap results \\cite{Nakayama:2016cim} which considered $\\epsilon=1$, $N=1$. Proceeding to $\\mathcal{O}(\\epsilon^3)$ and higher orders however is difficult. Due to the fact that all $\\mu_{\\varphi\\varphi n}^{(2)}$ are nonzero, the functional sum rules at the next order inevitably contain infinitely many terms, making them difficult to solve analytically. \n\n\n\n\n\\subsubsection{Boundary CFTs}\nTwo-point functions in BCFTs also admit a similar functional treatment \\cite{Kaviraj:2018tfd,Mazac:2018biw}, which is closely related to mean field theories with boundaries. Analogous to the choice of parity in the real projective CFT case, here one can choose either Neumann or Dirichlet boundary conditions for the associated mean field theory. For definiteness, we will only discuss the Neumann boundary condition case here. The Dirichlet case is similar and its discussion can be found in \\cite{Kaviraj:2018tfd}. \n\nWe start with the conformal block decomposition of the mean field theory two-point function with Neumann boundary condition\n\\begin{equation}\n\\langle \\varphi(x_1) \\varphi(x_2) \\rangle_{\\rm Neumann}=\\frac{1}{(2x_{1,\\perp})^{\\Delta_\\varphi}(2x_{2,\\perp})^{\\Delta_\\varphi}}\\big(\\xi^{-\\Delta_\\varphi}+(\\xi+1)^{-\\Delta_\\varphi}\\big)\\;.\n\\end{equation}\nIn the bulk channel, we find infinitely many double-trace operators with dimensions $\\Delta_n^{\\rm d.t.}=2\\Delta_\\phi+2n$, $n=0,1,\\ldots$. In the boundary channel, we find an infinite tower of boundary modes $\\widehat{\\varphi}_n$ with dimensions $\\widehat{\\Delta}_n=\\Delta_\\varphi+2n$, $n=0,1,\\ldots$. If we had considered the Dirichlet boundary condition, we would have found a different tower with dimensions $\\widehat{\\Delta}_n=\\Delta_\\varphi+2n+1$. \n\nLet us now consider a two-point function with $\\Delta_1\\neq \\Delta_2$. We will also make a technical assumption that the two-point function satisfies the following boundedness condition in the Regge limit\n\\begin{equation}\\label{BCFTsRbound}\n|\\mathcal{G}(\\xi)|\\lesssim |\\xi+1|^{-\\frac{\\Delta_1+\\Delta_2-1}{2}+\\epsilon}\\;,\\quad \\quad \\xi\\to-1^+\\;,\n\\end{equation}\nfor some $\\epsilon>0$. This behavior was referred to as {\\it Regge super-boundedness} in \\cite{Mazac:2018biw}, and here we assume it to simplify the discussion. The claim is that the following set of conformal blocks in {\\it both} bulk and boundary channels, which are closely related to the mean field theory spectrum, furnishes a basis for Regge super-bounded functions\n\\begin{equation}\\label{BCFTbasis}\n\\begin{split}\n&g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}\\;,\\quad \\text{with}\\quad \\Delta_n^{\\rm d.t.}=\\Delta_1+\\Delta_2+2n\\;,\\quad n=0,1,\\ldots\\;,\\\\\n&g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}\\;,\\quad \\text{with}\\quad \\widehat{\\Delta}_n^i=\\Delta_i+2n\\;,\\quad n=0,1,\\ldots\\;,\\; i=1,2\\;.\n\\end{split}\n\\end{equation}\nThe dual basis is given by the set of functionals $\\{\\omega_n, \\widehat{\\omega}_n^{(i)}\\}$ defined by the orthonormal relations\n\\begin{equation}\n\\begin{split}\n&\\omega_m(g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk})=\\delta_{mn}\\;,\\quad\\quad \\omega_m(g_{c,\\widehat{\\Delta}_n^i}^{\\rm boundary})=0\\;,\\\\\n&\\widehat{\\omega}_m^{(j)}(g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk})=0\\;,\\quad\\quad \\widehat{\\omega}_m^{(j)}(g_{c,\\widehat{\\Delta}_n^i}^{\\rm boundary})=\\delta_{mn}\\delta_{ij}\\;.\n\\end{split}\n\\end{equation}\nSimilar to the real projective CFT case, a convenient way to see that $\\{g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}, g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}\\}$ provides a basis is to use holography. It also allows us to obtain the actions of the dual functionals. \n\n\\begin{figure}\n \\centering\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\linewidth]{fig_hAdS_bulk}\n \\caption{Bulk channel diagram.}\n \\label{fig:hAdSbulk}\n\\end{subfigure}\n\\begin{subfigure}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\linewidth]{fig_hAdS_boundary}\n \\caption{Boundary channel diagram.}\n \\label{fig:hAdSboundary}\n\\end{subfigure}\n\\caption{Exchange Witten diagrams in the half AdS space. Here spacetime stops after the vertical wall.}\n\\label{fig:hAdSWD}\n\\end{figure}\n\nLet us consider the following holographic setup where we take half of the $AdS_{d+1}$ space by requiring $z_\\perp\\geq 0$. This amounts to extending the boundary of the BCFT at $x_\\perp=0$ into a wall in $AdS_{d+1}$. The mean field theory boundary condition is also extended by requiring scalar fields in the half AdS space to obey Neumann boundary condition on the wall. We can then consider the following two types of diagrams: the bulk channel exchange Witten diagram \\ref{fig:hAdSbulk} and the boundary channel exchange Witten diagram \\ref{fig:hAdSboundary}. Here both the bulk-to-bulk and the bulk-to-boundary propagators need to obey the Neumann boundary condition at the $AdS_d$ subspace $z_\\perp=0$. In the bulk channel diagram, the cubic vertex insertion point is integrated over the half $AdS_{d+1}$ space, and the other end of the bulk-to-bulk propagator is integrated over the entire $AdS_d$ wall. In the boundary channel diagram, the bulk-to-bulk propagator lives in $AdS_d$ and both vertex insertion points are integrated over $AdS_d$. Again, by using the method of images, we can express these diagrams in terms of diagrams defined in the full $AdS_{d+1}$ (Figure \\ref{fig:ICFTWD}). In this new setup, we have a $AdS_d$ probe brane located at $z_\\perp=0$ which is just an interface. There are localized degrees of freedom living on this subspace, but the brane does not back-react to the geometry. The half AdS space diagram \\ref{fig:hAdSbulk} is equivalent to the sum of a bulk channel exchange diagram in the full AdS space and its mirror diagram in which $\\mathcal{O}_1$ is inserted at $-x_{1,\\perp}$. These two diagrams are shown in Figure \\ref{fig:ICFTbulk}, and we denote them by $\\mathcal{W}^{\\rm bulk}_\\Delta$, $\\mathcal{W}^{\\rm mirror}_\\Delta$ respectively. The integration over the cubic vertex insertion points is now over the entire $AdS_{d+1}$ space. On the other hand, $x_{1,\\perp}\\to-x_{1,\\perp}$ in an $AdS_{d+1}$ boundary channel exchange diagram does not change its value. Therefore, doubling the space does not affect the boundary channel exchange Witten diagrams and the two diagrams \\ref{fig:hAdSboundary} and \\ref{fig:ICFTboundary} are the same. We denote \\ref{fig:ICFTboundary} by $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$.\n\n\\begin{figure}\n \\centering\n\\begin{subfigure}{0.63\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{fig_ICFT_bulk}\n \\caption{Bulk and mirror channel diagrams.}\n \\label{fig:ICFTbulk}\n\\end{subfigure}\n\\begin{subfigure}{0.34\\textwidth}\n \\centering\n \\includegraphics[width=\\linewidth]{fig_ICFT_boundary}\n \\caption{Boundary channel diagram.}\n \\label{fig:ICFTboundary}\n\\end{subfigure}\n\\caption{Exchange Witten diagrams in the full AdS space. Here the $AdS_d$ subspace is just an interface which hosts localized degrees of freedom and does not back-react to the geometry.}\n\\label{fig:ICFTWD}\n\\end{figure}\n\nThe crucial property we need to make progress is how these Witten diagrams decompose into conformal blocks. Using for example the Mellin representation for BCFTs \\cite{Rastelli:2017ecj}, one can show that $\\mathcal{W}^{\\rm bulk}_\\Delta+\\mathcal{W}^{\\rm mirror}_\\Delta$ decomposes into single-trace and double-trace conformal blocks in the bulk channel\n\\begin{equation}\n\\mathcal{W}^{\\rm bulk}_\\Delta(\\xi)+\\mathcal{W}^{\\rm mirror}_\\Delta(\\xi)= E\\, g_{b,\\Delta}^{\\rm bulk}(\\xi)+\\sum_{n=0}^\\infty E_n\\, g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}(\\xi)\\;,\n\\end{equation}\nand only double-trace conformal blocks in the boundary channel\n\\begin{equation}\n\\mathcal{W}^{\\rm bulk}_\\Delta(\\xi)+\\mathcal{W}^{\\rm mirror}_\\Delta(\\xi)= \\sum_{n=0}^\\infty\\sum_{i=1,2} F_n^{(i)}\\, g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}(\\xi)\\;,\n\\end{equation}\n Similarly, the boundary channel exchange diagram $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$ decomposes as\n\\begin{equation}\n\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}(\\xi)=K g_{b,\\widehat{\\Delta}}^{\\rm boundary}(\\xi)+\\sum_{n=0}^\\infty\\sum_{i=1,2} K_n^{(i)}\\, g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}(\\xi)\n\\end{equation}\nin the boundary channel, and \n\\begin{equation}\n\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}(\\xi)=\\sum_{n=0}^\\infty L_n\\, g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}(\\xi)\n\\end{equation}\nin the bulk channel. Equating the two decompositions in each case, we find that conformal blocks $g_{b,\\Delta}^{\\rm bulk}(\\xi)$, $g_{b,\\widehat{\\Delta}}^{\\rm boundary}(\\xi)$ with arbitrary conformal dimensions $\\Delta$, $\\widehat{\\Delta}$ can be expanded in terms of the double-trace conformal blocks (\\ref{BCFTbasis}). This almost leads to our claim that (\\ref{BCFTbasis}) is a basis. However, we need to check if the Regge behaviors of the Witten diagrams satisfy the condition (\\ref{BCFTsRbound}). One can show that as $\\xi\\to -1^+$, these diagrams behave as \\cite{Mazac:2018biw}\n\\begin{equation}\n|\\mathcal{W}^{\\rm bulk}_\\Delta(\\xi)+\\mathcal{W}^{\\rm mirror}_\\Delta(\\xi)|\\sim |\\xi+1|^{-\\frac{\\Delta_1+\\Delta_2-1}{2}}\\;,\\quad |\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}(\\xi)|\\sim |\\xi+1|^{-\\frac{\\Delta_1+\\Delta_2-3}{2}}\\;.\n\\end{equation}\nTherefore, only $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$ is Regge super-bounded and $\\mathcal{W}^{\\rm bulk}_\\Delta+\\mathcal{W}^{\\rm mirror}_\\Delta$ is only {\\it Regge bounded} in the parlance of \\cite{Mazac:2018biw}. To see why this point is important, we note that there is another Regge-bounded diagram $\\mathcal{W}^{\\rm contact}$ (Figure \\ref{fig:ICFTcontact}) which decomposes into only $\\{g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}, g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}\\}$ in both channels\n\\begin{equation}\n\\mathcal{W}^{\\rm contact}(\\xi)=\\sum_{n=0}^\\infty R_n\\, g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}(\\xi)=\\sum_{n=0}^\\infty\\sum_{i=1,2} S_n^{(i)}\\, g_{b,\\widehat{\\Delta}_n^i}^{\\rm boundary}(\\xi)\\;.\n\\end{equation}\nThis implies a linear relation among the basis vectors. However, we can avoid this relation by insisting that we are in the smaller space of functions defined by (\\ref{BCFTsRbound}). It turns out that there is a unique combination of the exchange diagrams and the contact diagram\n\\begin{equation}\n\\mathcal{V}^{\\rm bulk}_\\Delta(\\xi)=\\mathcal{W}^{\\rm bulk}_\\Delta(\\xi)+\\mathcal{W}^{\\rm mirror}_\\Delta(\\xi)+\\theta\\, \\mathcal{W}^{\\rm contact}(\\xi)\n\\end{equation}\nsuch that $\\mathcal{V}^{\\rm bulk}_\\Delta$ has improved Regge behavior $|\\xi+1|^{-\\frac{\\Delta_1+\\Delta_2-3}{2}}$ and is therefore super-bounded. Then in this Regge super-bounded space a basis is given by (\\ref{BCFTbasis}). Moreover, the actions of the dual functionals can be read off from the conformal block decomposition coefficients of the combination $\\mathcal{V}^{\\rm bulk}_\\Delta$ and $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$\n\\begin{equation}\n\\begin{split}\n&\\omega_m(g_{b,\\Delta}^{\\rm bulk})=-\\frac{1}{E}(E_m+\\theta R_m)\\;,\\quad \\widehat{\\omega}^{(j)}_m(g_{b,\\Delta}^{\\rm bulk})=\\frac{1}{E}(F_m^{(j)}+\\theta S_m^{(j)})\\;,\\\\\n& \\omega_m(g_{b,\\widehat{\\Delta}}^{\\rm boundary})=\\frac{L_m}{K}\\;,\\quad \\widehat{\\omega}^{(j)}_m(g_{b,\\widehat{\\Delta}}^{\\rm boundary})=-\\frac{K_m^{(j)}}{K}\\;.\n\\end{split}\n\\end{equation}\nSimilar to the real projective case, one can also show that $\\mathcal{V}^{\\rm bulk}_\\Delta$ and $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$ form a Polyakov style basis for expanding correlators.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{fig_ICFT_contact}\n\\caption{The contact Witten diagram. This diagram is given by the product of two bulk-to-boundary propagators with the vertex point integrated over the $AdS_d$ brane.}\n \\label{fig:ICFTcontact}\n\\end{figure}\n\nThe above discussion assumed $\\Delta_1\\neq \\Delta_2$. However, the story of the equal weight case $\\Delta_1=\\Delta_2=\\Delta_\\varphi$ is similar and requires only minor modifications. In this case the two towers of boundary channel conformal blocks in the basis (\\ref{BCFTbasis}) become degenerate, but the degeneracy can be compensated by turning one tower into derivative conformal blocks\n\\begin{equation}\n\\begin{split}\n&g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk}\\;,\\quad \\text{with}\\quad \\Delta_n^{\\rm d.t.}=2\\Delta_\\varphi+2n\\;,\\quad n=0,1,\\ldots\\;,\\\\\n&g_{b,\\widehat{\\Delta}_n}^{\\rm boundary}\\;,\\partial g_{b,\\widehat{\\Delta}_n}^{\\rm boundary}\\;,\\quad \\text{with}\\quad \\widehat{\\Delta}_n=\\Delta_\\varphi+2n\\;,\\quad n=0,1,\\ldots\\;,\n\\end{split}\n\\end{equation}\nHere $\\partial g_{b,\\widehat{\\Delta}}^{\\rm boundary}=\\partial_{\\widehat{\\Delta}} g_{b,\\widehat{\\Delta}}^{\\rm boundary}$. This basis again can be found by examining the conformal block decomposition of Witten diagrams with equal external weights, where the derivative conformal blocks are related to anomalous dimensions. The dual functional basis is then defined to be $\\{\\omega_n,\\widehat{\\omega}_n,\\widetilde{\\omega}_n\\}$, which acts on the basis vectors $\\{g_{b,\\Delta_n^{\\rm d.t.}}^{\\rm bulk},g_{b,\\widehat{\\Delta}_n}^{\\rm boundary},\\partial g_{b,\\widehat{\\Delta}_n}^{\\rm boundary}\\}$ in the orthonormal way. Their actions on general conformal blocks (and their derivatives) can be read off from the conformal block decomposition coefficients of $\\mathcal{V}^{\\rm bulk}_\\Delta$ and $\\mathcal{W}^{\\rm boundary}_{\\widehat{\\Delta}}$.\n\nFinally, the functionals discussed in this subsection can be applied to a variety of analytic bootstrap problems. For example, \\cite{Kaviraj:2018tfd} used the functionals to recover the Wilson-Fisher BCFT data to order $\\epsilon^2$. In \\cite{Mazac:2018biw}, the functionals were used to study a deformation of the mean field theory which interpolates the Neumann and Dirichlet boundary conditions. These applications are similar to the $O(N)$ model example we studied in the real projective space CFT subsection, and therefore will not be further discussed. We refer the reader to the original papers for the details.\n\n\\subsection{CFTs on other backgrounds}\\label{Subsec:otherbackgrounds}\nThe two situations we reviewed in this section can be viewed more generally as special cases of CFTs on backgrounds which are not conformally equivalent to (empty) $\\mathbb{R}^d$. There has been a lot of progress in applying bootstrap techniques to study such CFTs. \n\nClosely related to boundary CFTs are CFTs with conformal defects of various codimensions. There is a vast literature on this topic in the context of conformal bootstrap, see, {\\it e.g.}, \\cite{Billo:2013jda,Gaiotto:2013nva,Gliozzi:2015qsa,Bianchi:2015liz,Billo:2016cpy,Gadde:2016fbj,Giombi:2017cqn,Soderberg:2017oaa,Lauria:2017wav,Lemos:2017vnx,Liendo:2018ukf,Isachenkov:2018pef,Lauria:2018klo,Liendo:2019jpu,Wang:2020seq,Drukker:2020swu,Komatsu:2020sup,Lauria:2020emq,Giombi:2021uae}. Another important background is $\\mathbb{R}^{d-1}\\times S^1$ and is related to CFTs at finite temperature. There the simplest nontrivial observable is also the two-point function, and the Kubo-Martin-Schwinger condition is cast into a crossing equation. Therefore, the situation is quite similar to the cases of BCFTs and CFTs on real projective space. For works in this direction, see \\cite{El-Showk:2011yvt,Iliesiu:2018fao,Manenti:2019wxs,Alday:2020eua,Dodelson:2020lal}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Mellin space}\\label{Sec:MellinFormalism}\n\n\\subsection{General comments}\n\nThe Mellin space formalism was introduced in \\cite{Mack:2009mi,Penedones:2010ue} (see also \\cite{Paulos:2011ie,Fitzpatrick:2011ia}) and is a natural language for discussing holographic correlators. In position space, these are rather complicated functions of the conformal cross ratios. However, in this formalism the analytic structure of holographic correlators becomes drastically simplified, and manifests the underlying scattering amplitude nature of these objects. \n\nConsider the correlation function of $n$ scalar operators.\\footnote{For spinning correlators, the Mellin formalism is more difficult to define. See \\cite{Goncalves:2014rfa} for the case of $n$-point function with one spinning operator, and \\cite{Sleight:2018epi, Chen:2017xdz} for correlators of four spinning operators. Mellin formalism can also be developed for boundary CFT \\cite{Rastelli:2017ecj} and defect CFTs \\cite{Goncalves:2018fwx}.} The correlator can be written as a multi-dimensional inverse Mellin transformation\n\\begin{equation}\\label{defMellinnpt}\n\\langle \\mathcal{O}_1(x_1)\\ldots \\mathcal{O}_n(x_n)\\rangle=\\int [d\\delta_{ij}] \\bigg(\\prod_{id+2$ as for $n\\leq d+2$ there is a nontrivial stability group. To see this explicitly, we can use a conformal transformation to send two points to $0$ and $\\infty$. The remaining $n-2$ points define an $n-2$ dimensional hyperplane. The rotation group $SO(d+2-n)$ in directions orthogonal to the plane is a stability group, and we should subtract this group when it is nontrivial. Adding back its dimension, we get \n\\begin{equation}\\label{cftcounting2}\n\\frac{n(n-3)}{2}\\;.\n\\end{equation}\nAnother way to understand this change of counting behaviors is that for $n>d+2$ an $M\\times M$ matrices with elements $\\{P_i\\cdot P_j\\}$ and $d+2D+1$, we have similarly\n\\begin{equation}\n\\underbrace{n (D-1)}_{\\text{coordinates of }n\\text{ on-shell momenta}}-\\underbrace{\\frac{1}{2}D(D+1)}_{\\text{dimension of the Poincar\\'e group}}\\;,\n\\end{equation} \nwhere we note that the on-shell condition eliminates one degree of freedom for each particle. For $n\\leq D+1$, there is also a stability group which can be seen as follows. We can go to the frame in which the total momentum is zero\n\\begin{equation}\n\\sum_i \\vec{p}_i=0\\;.\n\\end{equation}\nThen these momenta span an $n-1$ dimensional subspace, which remains invariant under an $SO(D-n+1)$ rotation group which is orthogonal to it. Adding back the dimension of the stability group, we again arrive at \n\\begin{equation}\n\\frac{n(n-3)}{2}\\;.\n\\end{equation}\nNote that the answers from the two counting problems coincide precisely if $D=d+1$. This indicates that a correlation function in CFT$_d$ can be mapped into a scattering amplitude in a $d+1$ dimensional spacetime. Of course, this is not surprising as we know that the AdS\/CFT correspondence is a way to establish such a relation. \n\nBefore we move on, let us make a quick comment regarding (\\ref{defMellinnpt}). In writing (\\ref{defMellinnpt}), we are implicitly thinking that we are in the case of (\\ref{cftcounting2}) where the spacetime dimension is sufficiently high with respect to the number of operators and we are free of the determinant relations. While (\\ref{defMellinnpt}) remains a valid representation even when we are in the case of (\\ref{cftcounting1}), one might imagine that there is an alternative formalism better suited for this situation with fewer Mellin variables. This is particularly relevant for the case of CFT$_1$ where starting from the first nontrivial case with $n=4$ one encounters only the case (\\ref{cftcounting1}), and the case (\\ref{cftcounting2}) never shows up. On the other hand, it is also useful to think of this problem from the dual perspective. It is well known that remarkable properties of S-matrices such as integrability crucially rely on the special kinematics in 2d. The redundant parameterization in (\\ref{defMellinnpt}), however, does not reflect these special kinematic features. It would therefore be of great interest to find another Mellin representation that is intrinsic to the CFT$_1$ kinematics. Relevant works on the 1d Mellin representation include \\cite{fgsz,Bianchi:2021piu}, but it is not yet clear how to establish such a formalism for general $n$-point functions.\n\nLet us get back to the definition (\\ref{defMellinnpt}). Operators exchanged in a CFT correlator are manifested as poles in the Mellin formalism. To see this, let us consider the OPE\n\\begin{equation}\n\\mathcal{O}_1(x_1)\\mathcal{O}_2(x_2)=\\sum_k C_{12k}(x_{12}^2)^{\\frac{\\Delta_k-\\Delta_1-\\Delta_2}{2}}(\\mathcal{O}_k(x_2)+hx_{12}^2 \\partial^2\\mathcal{O}_k(x_2)+\\ldots)\\;.\n\\end{equation}\nHere we have restricted to the scalar operators for simplicity. The constant $h$ is fixed by conformal symmetry and multiplies the first of the descendant terms with the others collectively denoted by $\\ldots$. We now perform this OPE in the $n$-point function and compare it with the Mellin representation. In the limit of $x_{12}^2\\to 0$, it is convenient to integrate over $\\delta_{12}$ by closing the contour to the left in the complex plane. In order to match the OPE, it is clear that the integrand of the inverse Mellin transformation must have poles at \n\\begin{equation}\n\\delta_{12}=\\frac{\\Delta_1+\\Delta_2-\\Delta_k-2m}{2}\\;,\\quad m=0\\;,1\\;,2\\;,\\ldots.\n\\end{equation}\nThe residues of these poles are proportional to the product of the OPE coefficient $C_{12k}$ and the Mellin amplitude of the lower-point correlator. In other words, OPE in the CFT correlator leads to factorization in the Mellin amplitude. The precise relation was derived in \\cite{Goncalves:2014rfa}. Similar reasoning also applies to the general case where we exchange a spinning operator with dimension $\\Delta_k$ and spin $\\ell_k$. The corresponding poles are at\n\\begin{equation}\\label{nptOPEpole}\n\\delta_{12}=\\frac{\\Delta_1+\\Delta_2-(\\Delta_k-\\ell_k)-2m}{2}\\;,\\quad m=0\\;,1\\;,2\\;,\\ldots.\n\\end{equation}\n\nNote that in the definition (\\ref{defMellinnpt}),we have included a factor of Gamma functions which contain poles at integer locations. These poles correspond to the ``double-trace'' operators which are ubiquitous in holographic theories (in the strict central charge $c\\to\\infty$ limit they are just mean field theories). It turns out that separating out their contributions in this way is convenient when considering holographic correlators, as we will explain in more detail in the next subsection in the four-point function context. \n\n\\subsection{Four-point function case}\nSince the focus of most of this review will be on four-point functions, here we spell out the details of the Mellin representation formalism for $n=4$.\n\nIn this case, the general definition (\\ref{defMellinnpt}) reduces to \n\\begin{equation}\n\\langle\\mathcal{O}_1(x_1)\\ldots \\mathcal{O}_4(x_4)\\rangle=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^a\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^b \\mathcal{G}(U,V)\\;,\n\\end{equation}\nwith $a=\\frac{1}{2}(\\Delta_2-\\Delta_1)$, $b=\\frac{1}{2}(\\Delta_3-\\Delta_4)$, and \n\\begin{equation}\\label{defMellin4pt}\n\\begin{split}\n\\mathcal{G}(U,V)=&\\int_{-i\\infty}^{i\\infty}\\frac{dsdt}{(4\\pi i)^2}U^{\\frac{s}{2}}V^{\\frac{t}{2}-\\frac{\\Delta_2+\\Delta_3}{2}}\\mathcal{M}(s,t)\\,\\Gamma(\\tfrac{\\Delta_1+\\Delta_2-s}{2})\\Gamma(\\tfrac{\\Delta_3+\\Delta_4-s}{2})\\\\\n&\\quad\\quad\\quad\\times \\Gamma(\\tfrac{\\Delta_1+\\Delta_4-t}{2})\\Gamma(\\tfrac{\\Delta_2+\\Delta_3-t}{2}) \\Gamma(\\tfrac{\\Delta_1+\\Delta_3-u}{2})\\Gamma(\\tfrac{\\Delta_2+\\Delta_4-u}{2})\\;.\n\\end{split}\n\\end{equation}\nHere to make it more symmetric we have also introduced the third Mandelstam variable $u$ satisfying \n\\begin{equation}\ns+t+u=\\sum_{i=1}^4\\Delta_i\\;.\n\\end{equation} \nThe integration contours of $s$ and $t$ run parallel to the imaginary axis and separate semi-infinite series of poles running to the left and to the right. Bose symmetry acts by permuting $s$, $t$, $u$ and the operator labels. Therefore the Mellin amplitude has the same symmetry properties as a flat-space amplitude. For example, for four identical operators the Mellin amplitude satisfies\n\\begin{equation}\n\\mathcal{M}(s,t)=\\mathcal{M}(s,u)=\\mathcal{M}(t,u)\\;.\n\\end{equation} \n\nLet us consider again the OPE of the four-point function in the s-channel. Then (\\ref{nptOPEpole}) gives rise to poles in $s$ and the Mellin amplitude takes the following form\n\\begin{equation}\\label{4ptOPEinMellin}\n\\mathcal{M}(s,t)\\supset \\sum_{m=0}^{\\infty} C_{12k}C_{34k} \\frac{Q_{\\ell_k,m}(t)}{s-(\\Delta_k-\\ell_k)-2m}\\;.\n\\end{equation}\nThe numerators $Q_{\\ell,m}(t)$ are {\\it kinematic} polynomials of degree $\\ell$ in $t$ and are known as the {\\it Mack polynomials}. They can be obtained by, for example, matching the position space expressions of the conformal blocks after evaluating the Mellin integrals. \n\nLet us note that the above form (\\ref{4ptOPEinMellin}) of OPE in Mellin space is reminiscent of flat-space tree-level scattering. Such a behavior strengthens the analogy between correlators and amplitudes which we argued about in the previous subsection. To further appreciate this analogy, let us examine the structure of tree-level Witten diagrams in Mellin space. These tree-level Witten diagrams are the leading corrections in the $1\/N$-expansion of correlators in a local holographic theory. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{fig_Mellin_disc}\n\\caption{Disconnected Witten diagrams.}\n \\label{fig:Mellindisc}\n\\end{figure}\n\n\nFor simplicity, let us consider four identical operators of dimension $\\Delta_\\varphi$. The four-point function can be expanded in the following way\n\\begin{equation}\n\\mathcal{G}=\\mathcal{G}_{\\rm disc}+\\frac{1}{N^2}\\mathcal{G}_{\\rm tree}+\\frac{1}{N^4}\\mathcal{G}_{\\rm 1-loop}+\\ldots\\;.\n\\end{equation}\nIn the $N\\to\\infty$ limit, the leading order contribution to the correlator is given by the mean field theory, and is comprised of products of two-point functions\n\\begin{equation}\n\\mathcal{G}_{\\rm disc}=1+U^{\\Delta_\\varphi}+\\left(\\frac{U}{V}\\right)^{\\Delta_\\varphi}\\;.\n\\end{equation}\nThis corresponds to the disconnected diagrams depicted in Figure \\ref{fig:Mellindisc}. Decomposing it into conformal blocks we find only double-trace operators of the schematic form \n\\begin{equation}\n:\\varphi \\square^n \\partial^\\ell \\varphi:\\;,\n\\end{equation}\nwhich have conformal dimension $\\Delta_{n,\\ell}=2\\Delta_\\varphi+2n+\\ell$ and spin $\\ell$. They correspond to poles in the integrand at $s=2\\Delta_\\varphi+2n$ which are precisely the poles of the s-channel Gamma functions. At the next order, we have connected tree-level diagrams depicted in Figure \\ref{fig:Mellintrees}. These tree-level diagrams can further be divided into exchange diagrams and contact diagrams. They are built out of propagators following Feynman rules similar to those in flat space. For example, the contact diagram following from a quartic vertex without derivatives is defined as, see, {\\it e.g.}, \\cite{Witten:1998qj,DHoker:1999mqo,DHoker:2002nbb}\n\\begin{equation}\\label{defWcon0der}\nW_{\\rm con}(x_i)=\\int_{AdS_{d+1}} dz \\prod_{i=1}^4 G_{B\\partial}^{\\Delta_\\varphi}(x_i,z)\\;,\n\\end{equation} \nand the s-channel exchange diagram of a scalar field with dimension $\\Delta$ is defined as \n\\begin{equation}\nW_{\\Delta,\\ell=0}(x_i)=\\int_{AdS_{d+1}} dz dw G_{B\\partial}^{\\Delta_\\varphi}(x_1,z)G_{B\\partial}^{\\Delta_\\varphi}(x_2,z)G_{BB}^\\Delta(z,w)G_{B\\partial}^{\\Delta_\\varphi}(x_3,w)G_{B\\partial}^{\\Delta_\\varphi}(x_4,w)\\;.\n\\end{equation} \nHere $G_{B\\partial}^{\\Delta}(x,z)$ and $G_{BB}^{\\Delta}(z,w)$ are the bulk-to-boundary and the bulk-to-bulk propagators in AdS respectively. Since these tree diagrams are corrections to the mean field theory correlator, the double-trace operators appearing in the OPE at the disconnected order will also appear at this order. Their presence is conveniently captured by the Gamma function factor. In addition, in the exchange diagrams there is a ``single-trace'' operator which is dual to the exchanged field in AdS. This requires the Mellin amplitude of an s-channel exchange diagram to contain the contribution (\\ref{4ptOPEinMellin}). Since there are no other operators exchanged in the OPE at this order, we conclude that the Mellin amplitude of the exchange Witten diagram is just \n\\begin{equation}\n\\mathcal{M}_{\\Delta,\\ell}(s,t)=\\sum_{m=0}^{\\infty} \\frac{Q_{\\ell,m}(t)}{s-(\\Delta-\\ell)-2m}+R_{\\ell-1}(s,t)\n\\end{equation}\nwhere $R_{\\ell-1}(s,t)$ is a degree-$(\\ell-1)$ polynomial free of poles and we have set $C_{12k}$ and $C_{34k}$ to 1 for convenience. By contrast, the contact diagrams contain only double-trace operators in the conformal block decomposition. Therefore, their Mellin amplitudes are regular in the Mandelstam variables. For example, the Mellin amplitude of the zero-derivative contact diagram (\\ref{defWcon0der}) is just a constant. More generally, the Mellin amplitude of a contact diagram with $2L$ contracted derivatives in the quartic vertex is a degree-$L$ polynomial\n\\begin{equation}\n\\mathcal{M}_{\\text{con, }2L\\text{-der}}=P_{L}(s,t)\\;.\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{fig_Mellin_trees}\n\\caption{Tree-level Witten diagrams.}\n \\label{fig:Mellintrees}\n\\end{figure}\n\n\nEvidently, the Mellin amplitudes of AdS tree-level diagrams are highly similar to the flat-space tree-level amplitudes. This similarity makes the Mellin formalism a very useful tool to study holographic correlators and allows us to apply many intuitions from flat-space scattering. \n\n\\subsection{Flat-space limit}\nWe have considered scattering processes of a relativistic theory placed in an AdS space, and we have implicitly set its radius $R$ to be 1 when discussing the diagrams. Here let us restore the $R$ dependence and make it tunable. If we take $R$ to be much larger than any length scales in the theory, then clearly the curvature effects should be negligible. The AdS scattering amplitude should correspondingly reduce to the flat-space scattering amplitude in this limit. The Mellin formalism provides a convenient way to extract the flat-space limit of AdS scattering amplitudes. \n\nThe precise relation was given in \\cite{Penedones:2010ue}. On the one hand, we have the Mellin amplitude $\\mathcal{M}(\\delta_{ij})$ of an $n$-point scalar correlator where the conformal dimension of each operator is $\\Delta_i$. On the other hand, we have the scattering amplitude $\\mathcal{T}_n$ of $n$ massless particles in flat space. The flat-space amplitude is effectively reproduced from the high-energy limit of the Mellin amplitude\n\\begin{equation}\\label{MtoT}\n\\mathcal{T}_n(s_{ij})=\\mathcal{N}R^{\\frac{n(d-1)}{2}-d-1}\\lim_{R\\to\\infty}\\int_{-i\\infty}^{i\\infty} d\\alpha e^\\alpha \\alpha^{\\frac{d-2-\\sum_i\\Delta_i}{2}}\\mathcal{M}\\left(\\delta_{ij}=-\\frac{R^2s_{ij}}{4\\alpha},\\Delta_a=R m_a\\right)\\;.\n\\end{equation}\nHere $s=2\\vec{p}_i\\cdot \\vec{p}_j$ are the Mandelstam variables in the flat space, and $\\mathcal{N}$ is an overall factor depending on the external dimensions $\\Delta_i$. The dimensions $\\Delta_a$ belong to exchanged internal fields. They scale linearly with $R$ if we wish to assign a nonzero mass $m_a$ in the flat-space limit.\\footnote{This follows from the mass relation $m^2R^2=\\Delta(\\Delta-d)$ in the large $R$ limit.} Finally, the integration contour of $\\alpha$ runs to the right of all poles in the Mellin amplitude and the branch cut from $\\alpha^{\\frac{d-2-\\sum_i\\Delta_i}{2}}$. The relation (\\ref{MtoT}) was presented as a conjecture in \\cite{Penedones:2010ue} and was checked in many explicit examples, including contact and exchange diagrams at tree level, and four-point one-loop amplitudes. It was also derived in \\cite{Fitzpatrick:2011hu} using wavepackets where the scattering was limited to a small flat region of AdS. \n\n\n\n\n\n\n\n\n\\section{The epsilon expansion}\\label{Sec:epsilon}\n\\subsection{A brief review}\nThe epsilon expansion was introduced in \\cite{Wilson:1971dc, Wilson:1973jj} as an approximate technique to compute critical exponents for the 3d Ising and $O(N)$ models. The idea is to start with the $O(N)$ model\n\\begin{equation}\nS=\\sum_{i=1}^{N}\\int d^{d} x\\, \\left(\\partial_\\mu \\phi^i \\partial^\\mu \\phi^i-m^2 \\phi^i\\phi^i +\\lambda (\\phi^i\\phi^i)^2\\right)\\,,\n\\end{equation}\nwith $d=4-\\epsilon$\nand compute various scaling dimensions in this model at the fixed point, where $m$ and $\\lambda$ are tuned appropriately, pretending $\\epsilon$ to be small using the Feynman diagram approach \\cite{Wilson:1973jj} to some loop order. Then at the end of the calculation either $\\epsilon$ is set equal to unity or some resummation technique is used \\cite{Kleinert:2001ax} to obtain physical answers. For reasons not completely well understood, the results are remarkably close to both Monte Carlo simulations of the 3d Ising\/O(N) models as well as experimental measurements.\n\nIn terms of $\\tau=\\frac{T-T_c}{T_c}$, the specific heat $C\\propto \\tau^{-\\alpha}$ defines the critical exponent $\\alpha$. Similarly at the critical point $T=T_c$, the correlator is expected to behave like $\\langle \\phi(r)\\phi(0)\\rangle\\propto r^{-d+2+\\eta}$, defining the exponent $\\eta$. Let us focus for now on $N=1$ which is relevant for the 3d Ising model. In terms of the scaling dimension $\\Delta_\\phi$ of $\\phi$ and $\\Delta_{\\phi^2}$ of $\\phi^2$, we have\n\\begin{equation}\n\\alpha=2-\\frac{d}{d-\\Delta_{\\phi^2}}\\,,\\quad \\eta=2\\Delta_\\phi-d+2\\,.\n\\end{equation}\nIn a free theory $\\Delta_\\phi=1-\\epsilon\/2, \\Delta_{\\phi^2}=2\\Delta_\\phi=2-\\epsilon$. The $\\lambda (\\phi^i \\phi^i)^2$ interaction induces a flow to the Wilson-Fisher fixed point where operators get anomalous dimensions.\nThe results for $\\Delta_\\phi$ and $\\Delta_{\\phi^2}$ are \\cite{Wilson:1973jj}:\n\\begin{eqnarray}\\label{WF}\n\\Delta_\\phi&=& 1-\\frac{\\epsilon}{2}+\\frac{\\epsilon^2}{108}+\\frac{109 \\epsilon^3}{11664}+O(\\epsilon^4)\\,,\\\\\n\\Delta_{\\phi^2}&=&2-\\frac{2\\epsilon}{3}+\\frac{19\\epsilon^2}{162}+O(\\epsilon^3)\\,.\n\\end{eqnarray}\nThe results for double field higher gradient operators of the form $\\mathcal{O}_\\ell=\\phi\\partial_{\\mu_1}\\cdots \\partial_{\\mu_\\ell}\\phi$ are also known and were worked out to $O(\\epsilon^2)$ in \\cite{Wilson:1973jj}. Using the Feynman diagram approach, the $O(\\epsilon^4)$ anomalous dimensions of ${\\mathcal O}_\\ell$ \\cite{gracey}, the $O(\\epsilon^5)$ anomalous dimension of $\\phi^2$ and $O(\\epsilon^6)$ anomalous dimension of $\\phi$ \\cite{panzer} have been worked out. \n\nIf we take the Wilson-Fisher results in eq.(\\ref{WF}) above and naively substitute $\\epsilon=1$, we get $\\Delta_{\\phi}=0.5186$ which is in a remarkable agreement with numerical results which give 0.5181 as the answer to 4 significant figures \\cite{numrev}. For $\\phi^2$ we find $\\Delta_{\\phi^2}=1.45$ while numerical results give 1.41 to 2 decimal places. However, despite these encouraging findings, it is difficult to compute OPE coefficients using this approach. Also it is known that the $\\epsilon^k$ term in the anomalous dimension calculations grows as $k^{k+4} e^{-k}(\\frac{\\epsilon}{3})^k$ \\cite{Brezin:1976vw} necessitating the use of resummation techniques. Furthermore, a more crucial drawback is that this approach does not use the conformal symmetry of the critical point and is inherently perturbative. In what follows, we will review how conformal field theory techniques can be used to extract OPE data including OPE coefficients in the epsilon expansion.\n\n\n\\subsection{CFT derivation of leading order anomalous dimension}\nIn this section, we will review the derivation of the leading order anomalous dimension of the operators $\\phi^4$ using CFT techniques using the elegant method of \\cite{rychkovtan}. We will need this information in what follows. Following \\cite{rychkovtan}, we assume that\n\\begin{enumerate}\n\\item The WF fixed point is invariant under the full conformal symmetry.\n\\item Each local operator in the free theory at $\\epsilon=0$ has a counterpart at the WF fixed point. In particular:\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow 0} V_n=\\phi^n\\,.\n\\end{equation}\nThis enables us to refer to $\\phi^n$ in the WF theory unambiguously. The conformal dimension of $V_n$ is denoted by $\\Delta_n$. We will further define\n\\begin{equation}\n\\Delta_n=n(1-\\frac{\\epsilon}{2})+\\gamma_n\\,.\n\\end{equation}\n\\item $V_3$ is a descendant. Namely\n\\begin{equation}\n\\partial^2 V_1=\\beta V_3\\,.\n\\end{equation}\nHere $\\beta=\\beta(\\epsilon)$ will be fixed later. This equation also means that $\\Delta_3=\\Delta_1+2$.\n\\end{enumerate}\nWe will choose the normalizations such that \n\\begin{equation}\n\\langle \\phi(x)\\phi(0)\\rangle=\\frac{1}{|x|^2}\\,,\\quad \\langle V_1(x) V_1(0)\\rangle=\\frac{1}{|x|^{2\\Delta_1}}\\,.\n\\end{equation}\nThis will enable us to fix $\\beta$. To do this we compare $\\langle \\partial^2 V_1(x)\\partial^2 V_1(0)\\rangle$ and $\\langle V_3(x) V_3(0)\\rangle$. This leads to\n\\begin{equation}\\label{alf}\n\\beta=4 \\left(\\frac{\\gamma_1}{3}\\right)^{1\/2}\\,.\n\\end{equation}\nConsider first the OPE in the free theory\n\\begin{equation}\\label{phiop}\n\\phi^n(x)\\times \\phi^{n+1}(0)\\supset (n+1)! |x|^{-2n}\\left(\\phi(0)+\\frac{n}{2} |x|^2\\phi^3(0)\\right)\\,.\n\\end{equation}\nThe RHS is obtained using Wick contractions. Next we need the WF OPE\n\\begin{equation} \\label{Vop}\nV_n(x)\\times V_{n+1}(0)\\supset \\tilde f |x|^{\\Delta_1-\\Delta_n-\\Delta_{n+1}}\\left(1+q_1 x^\\mu\\partial_\\mu+q_2 x^\\mu x^\\nu \\partial_\\mu \\partial_\\nu+q_3 x^2 \\partial^2+\\cdots\\right)V_1(0)\\,.\n\\end{equation}\nHere $q_1,q_2,q_3$ are fixed in terms of $\\Delta_1, \\Delta_n,\\Delta_{n+1}$. By considering $\\langle V_n(x) V_{n+1}(0) V_1(z)\\rangle$ and matching with $\\langle \\phi^n(x)\\phi^{n+1}(0)\\phi(z)\\rangle$ in the $\\epsilon\\rightarrow 0$ limit, we will find that $\\tilde f=(n+1)!+O(\\epsilon)$. Next using eq.(\\ref{Vop}) we have\n\\begin{equation}\n\\langle V_n(x) V_{n+1}(0) V_3(z)\\rangle \\approx (n+1)!|x|^{\\delta}\\left(1+q_1 x^\\mu\\partial_\\mu+q_2 x^\\mu x^\\nu \\partial_\\mu \\partial_\\nu+q_3 x^2 \\partial^2+\\cdots\\right)\\langle V_1(0) V_3(z)\\rangle\\,,\n\\end{equation}\nwhere $\\delta=\\Delta_1-\\Delta_n-\\Delta_{n+1}.$\nWe have to match this in the limit $|x|\\ll |z|$ with\n\\begin{equation}\n\\langle \\phi^n(x)\\phi^{n+1}(0)\\phi^3(z)\\rangle \\approx (n+1)! \\frac{n}{2} |x|^{-2n+2}\\langle \\phi^3(0)\\phi^3(z)\\rangle\\,,\n\\end{equation}\nin the $\\epsilon\\rightarrow 0$ limit which follows from eq.(\\ref{phiop}). We consider first $n=1$ or $n\\geq 4$. The key step is to match the $O(x^2)$ terms for which we need $q_3 \\beta\\rightarrow n\/2$. Since one can show \\cite{rychkovtan} that $q_3\\approx (\\gamma_{n+1}-\\gamma_n+\\gamma_1)\/(16\\gamma_1)$ and $\\beta\\sim O(\\epsilon)$, this would need $q_3$ to be singular in the $\\epsilon\\rightarrow 0$ limit. This leads to $\\gamma_1=O(\\epsilon^2)$. Writing $\\gamma_1=\\delta_1^{(2)}\\epsilon^2$ and $\\gamma_n=\\delta_n^{(1)}\\epsilon$ we find \n\\begin{equation}\n(\\delta_{n+1}^{(1)}-\\delta_n^{(1)})\\epsilon = \\frac{\\beta}{6} n\\,.\n\\end{equation}\nUsing eq.(\\ref{alf}), we conclude that $\\beta=O(\\epsilon)$. \nOne can further argue that this relation holds for $n=2,3$ as well and hence for all integer $n$. Matching $\\delta_1^{(1)}=0$ and $\\Delta_3-\\Delta_1=2$ fixes \n\\begin{equation}\\label{phin}\n\\delta_n^{(1)}=\\frac{n(n-1)}{6}\\,.\n\\end{equation}\nThese agree with the Feynman diagram calculation of the anomalous dimensions of $\\phi^n$ operators in the WF theory. \nIn particular we have $\\Delta_{4}=4+O(\\epsilon^2)$ which we will need in the next section. In order to go beyond leading order and also compute corrections to OPE coefficients, we will need to use bootstrap equations. Further applications of this technique to evaluate leading order anomalous dimensions in the $\\phi^6$ theory in $2+\\epsilon$ dimensions can be found in \\cite{fep1}, for the Gross-Neveu model in $2+\\epsilon$ dimensions can be found in \\cite{fep2}, while $\\phi^3$ theory in $6-\\epsilon$ dimensions was examined in \\cite{fep3}.\n\n\n\n\\section{Polyakov Bootstrap from dispersion relation}\\label{Sec:dispersionPolyakov}\nIn his seminal 1974 work \\cite{Polyakov:1974gs}, Polyakov postulated a crossing symmetric way to solve the dynamical content of the conformal bootstrap program. In this paper, he looked at momentum space consistency conditions in the context of the leading order epsilon expansion, as well as a non-perturbative version of these conditions. In order to frame the non-perturbative conditions, Polyakov used a spectral function representation of the conformal correlator and argued that in order to have better convergence in the spectral variable, one needs to incorporate spurious double poles, corresponding to operators that are absent from the spectrum. Since crossing symmetry is in-built in this formalism, consistency conditions arise on demanding that the OPE does not include contributions from these spurious double poles. The modern incarnation of the Polyakov bootstrap was discussed in \\cite{Sen:2015doa, Gopakumar:2016wkt, Gopakumar:2016cpb, Dey:2017fab}. Mellin space was found to be suitable in understanding Polyakov's seminal paper in the language of exchange Witten diagrams. Note that these provide a convenient kinematical basis and we are not assuming any knowledge of the dual gravity theory.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{wittendiag_csb.png}\n\\caption{Polyakov's 1974 \\cite{Polyakov:1974gs} idea in its modern incarnation. The Mellin amplitude can be expanded in a basis of crossing symmetric AdS exchange Witten diagrams and contact diagrams. The crossing symmetric dispersion relation fixes this basis.}\n \\label{fig:witttendiag_cs}\n\\end{figure}\n\n\nWe will focus on identical external scalars for which we have\n\\begin{equation}\nG(x_i)=\\frac{1}{(x_{12}^2 x_{34}^2)^{\\Delta_\\phi}}{\\mathcal G}(U,V)\\,,\n\\end{equation}\nwith\n\\begin{equation}\n{\\mathcal G}(U,V)=\\int_{-i\\infty}^{i\\infty}\\frac{ ds dt}{(4\\pi i)^2} U^{\\frac{s}{2}}V^{\\frac{t}{2}-\\Delta_\\phi} {\\mathcal M}(s,t)\\mu(s,t)\\,,\n\\end{equation}\nwhere the measure factor $\\mu$ is given by\n\\begin{equation}\n\\mu(s,t)=\\Gamma^2(\\Delta_\\phi-\\frac{s}{2})\\Gamma^2(\\Delta_\\phi-\\frac{t}{2})\\Gamma^2(\\Delta_\\phi-\\frac{u}{2})\\,,\n\\end{equation}\nand \n\\begin{equation}\ns+t+u=4\\Delta_\\phi\\,.\n\\end{equation}\nWe will call ${\\mathcal M}(s,t)$ as the Mellin amplitude. \nThe double poles in the measure factor, if not canceled, would correspond to operators in the spectrum with exact dimensions $\\Delta=2\\Delta_\\phi+2n+\\ell$. Since in generic non-supersymmetric CFTs, we expect operators to gain anomalous dimensions, such exact operators would be spurious. The original Polyakov conditions in \\cite{Polyakov:1974gs} are then the cancellation of such contributions in the Mellin amplitude. In \\cite{Sen:2015doa, Gopakumar:2016wkt, Gopakumar:2016cpb}, $O(\\epsilon^2)$ anomalous dimension for the scalar $\\phi^2$ operator, $O(\\epsilon^3)$ anomalous dimensions for the higher spin operators $\\phi\\partial_{a_1}\\cdots \\partial_{a_\\ell}\\phi$ as well as the corresponding OPE coefficients to one higher order in $\\epsilon$ were calculated. The anomalous dimensions were in perfect agreement with existing Feynman diagram calculations, while the OPE coefficients were new. For the stress tensor OPE, alternative arguments (see appendix B in \\cite{Dey:2016mcs}) give rise to the same answer, giving credence to such calculations. Nevertheless, in spite of these successes, in \\cite{Gopakumar:2018xqi}, it was realized that there are contact term ambiguities in the kinematical basis being used\\footnote{The simpler case of 1d CFTs where there are no spins was discussed in \\cite{mazacpaulos1, mazacpaulos2}; for a Mellin space discussion see \\cite{fgsz}.}. These ambiguities resulted in a mismatch at $O(\\epsilon^3)$ for the anomalous dimension of $\\phi^2$ compared to the Feynman diagram results. Thus the question becomes how to fix such ambiguities. This requires understanding the non-perturbative existence of Mellin amplitudes.\n\n\nThe non-perturbative existence of Mellin amplitudes was discussed in detail in \\cite{Penedones:2019tng}. The main criteria are analyticity in a sectorial domain (arg $[U]$, arg $[V]$)$\\in \\Theta_{CFT}$ and polynomial boundedness for ${\\mathcal G}(U,V)$. It was shown that quite generally the latter condition does not hold, and in order to define Mellin amplitudes, one will need to perform subtractions. Our interest is in the epsilon expansion to the first few orders, where subtracting off the disconnected contribution arising from the exchange of the identity operator is sufficient. After subtractions, one can write down fixed-$t$ dispersion relations for the Mellin amplitude, much like how one writes dispersion relations for flat space scattering amplitudes. The non-perturbative origin of the Polyakov conditions is subtle and has been explained in \\cite{Penedones:2019tng, Caron-Huot:2020adz}. In \\cite{Caron-Huot:2020adz}, these conditions originate from demanding consistency between dispersion relations and the $s$-channel OPE. For the epsilon expansion, the conclusion from such analyses is that we can continue to use the Polyakov conditions as discussed in \\cite{Gopakumar:2016wkt, Gopakumar:2016cpb, Gopakumar:2018xqi}.\n\nIn order to make connection with Polyakov's original idea of a manifestly crossing symmetric approach, we need to start with a crossing symmetric dispersion relation. This was done in \\cite{Gopakumar:2021dvg}. We will now summarize the derivation. For ease of notation, we will use\n\\begin{equation}\\label{stuvar}\ns=2s_1+\\frac{4\\Delta_\\phi}{3},\\quad t=2s_2+\\frac{4\\Delta_\\phi}{3},\\quad u=2s_3+\\frac{4\\Delta_\\phi}{3}\\,,\n\\end{equation}\nso that we have $s_1+s_2+s_3=0$. Full crossing symmetry means invariance under the permutations of the $s_i$'s. In order to write a crossing symmetric dispersion relation, we use an old but forgotten idea given by Auberson and Khuri in 1972 \\cite{Auberson:1972prg}. For QFT, this was resurrected in \\cite{Sinha:2020win} and then developed for CFT in \\cite{Gopakumar:2021dvg}. Rather than working with Mandelstam variables, we will use a different parametrization, namely\n\\begin{equation}\ns_k= a\\left(1-\\frac{(z_k-z)^3}{z^3-1}\\right)\\,,\\quad {k=1,2,3}\\,,\n\\end{equation}\nwhere $z_k=\\exp(2\\pi i (k-1)\/3)$ are the cube-roots of unity. The parameter $a$ works out to be \n\\begin{equation}\\label{adef}\na=\\frac{s_1 s_2 s_3}{s_1 s_2+s_1 s_3+s_2 s_3}=\\frac{y}{x}\\,,{\\qquad} y=-s_1 s_2 s_3\\,,\\quad x=-(s_1 s_2+s_1 s_3+s_2 s_3)\\,,\n\\end{equation}\nso that in terms of the $s_i$'s, $a$ is manifestly crossing symmetric. \nThe idea now is to write a dispersion relation in the variable $z$ keeping $a$ fixed. Notice that since \\begin{equation} s_1+s_2+s_3=0\\,,\\end{equation} the above equation for fixed $a$ gives two roots for $s_2$ in terms of $s_1$, namely\n\\begin{equation}\ns_2^{\\pm}=-\\frac{s_1}{2}\\left[1\\mp (\\frac{s_1+3a}{s_1-a})^{1\/2}\\right]\\,.\n\\end{equation}\nIn the fully crossing symmetric case of interest, both roots give the same result so we will work with $s_2^+$. \nIn terms of the $z$ variable, the poles on the real $s_1$ axis gets mapped to the boundary of the disc $|z|=1$. The region where $|s_i|$'s are small is the neighborhood of $z=0$ while the Regge limit, for example, $s_1\\rightarrow \\infty$, keeping $s_2$ fixed gets mapped to $z\\rightarrow z_2$.\nIn order to proceed, we have to make assumptions about the fall-off of ${\\mathcal M}(s,t)$ as $|z|\\rightarrow 1$. We will first assume that two subtractions suffice, i.e., ${\\mathcal M}(s,t)\\rightarrow o(s^2)$ for fixed-$t$.\nThe final form of this dispersion relation is given by \\cite{Gopakumar:2021dvg}:\n\\begin{equation}\\label{disp}\n{\\mathcal M}(s_1,s_2)={\\mathcal M}(0,0)+\\frac{1}{\\pi}\\int_{\\sigma}^\\infty \\frac{ds_1'}{s_1'}{\\mathcal A}(s_1';s_2^+(s_1',a)) H_2(s_1'; s_1,s_2,s_3)\\,,\n\\end{equation}\nwhere \n\\begin{equation}\nH_2(s;s_1,s_2,s_3)= \\left(\\frac{s_1}{s-s_1}+\\frac{s_2}{s-s_2}+\\frac{s_3}{s-s_3}\\right)\\,,\n\\end{equation}\nis a manifestly crossing symmetric kernel. The lower limit of the integrand $\\sigma$ is where the chain of poles in the Mellin variable $s_1$ starts. ${\\mathcal A}(s_1;s_2)$ is the s-channel discontinuity. Since the Mellin amplitude is meromorphic, this will generally be a sum of delta functions. Denoting these poles by \n\\begin{equation}\n\\tau_k=\\frac{\\Delta-\\ell}{2}+k-\\frac{2{\\Delta_\\phi}}{3}\\,,\n\\end{equation}\nwe can explicitly write\n\\begin{equation}\\label{poly}\n{\\mathcal M}(s_1,s_2)={\\mathcal M}(0,0)+\\sum_{\\Delta,\\ell,k}^\\infty \\frac{c_{\\Delta,\\ell}}{\\tau_k}{\\mathcal Q}_{\\ell,k}^{(\\Delta)}(a)H_2(\\tau_k; s_1,s_2,s_3)\\,,\n\\end{equation}\nwhere $c_{\\Delta,\\ell}={\\mathcal N}_{\\Delta,\\ell} C_{\\Delta,\\ell}$, with $C_{\\Delta,\\ell}$'s being the OPE coefficient square, given in appendix (\\ref{secConv}) and\n\\begin{equation}\n{\\mathcal Q}_{\\ell,k}^{(\\Delta)}(a)=\\mathcal{R}_{\\Delta,\\ell}^{(k)} P_{\\Delta,\\ell}(\\tau_k, s_2^+(\\tau_k,a))\\,,\n\\end{equation}\nwith $P_{\\Delta,\\ell}$'s being the Mack polynomials and $\\mathcal{R}_{\\Delta,\\ell}^{(k)}$ being some normalization factors, whose explicit expressions can be found in appendix (\\ref{secConv}).\nWe will refer to this as the Polyakov block expansion.\n\nOne can argue \\cite{Gopakumar:2021dvg} that the conformal partial wave expansion converges in the neighbourhood of $a=0$ so that we can consider Taylor expanding around $a=0$. Now notice that in terms of $x,y$ defined in eq.(\\ref{adef}), the kernel is\n\\begin{equation}\nH_2(s_1'; x,a)=\\frac{x(2s_1'-3a)}{x a-x s_1'+ (s_1')^3}\\,,\n\\end{equation}\nso that if we Taylor expand around $a=0$ followed by $x=0$, we will only get positive powers of $x$ and $y$. In other words, the kernel is ``local''. On the other hand, ${\\mathcal A}$ is a function of $a$ and $s_1'$ only so that Taylor expanding around $a=0$ will generically lead to arbitrary powers of $a$ and hence would lead to inverse powers of $x$ in the expansion. Specifically, the form of the integrand is\n\\begin{equation}\n{\\mathcal A}(s_1';s_2^+(s_1',a)) \\times H_2(s_1'; x,a)=\\left(\\sum_{p=0}^\\infty d_p a^p\\right)\\times\\left(\\sum_{m=0}^\\infty \\sum_{n=0}^m a^n x^m c_{nm}\\right)\\,,\n\\end{equation}\nwhere $d_p, c_{nm}$ are functions of $s_1'$. Assuming the Mellin amplitude to be meromorphic, $d_p$ would be proportional to $\\delta(s_1'-s_k)$ where $s_k$'s are the location of the $s_1$ poles. After integration over $s_1'$ we would get an expression with a sum over these poles.\nAs an example, let us consider $x^2$. We see that every term in $\\left(\\sum_{p=3}^\\infty d_p a^p\\right)$ when multiplied by $c_{02} x^2+c_{12} a x^2+ c_{22} a^2 x^2$ would lead to negative powers of $x$ after using $a=y\/x$. Further for $p\\leq 2$ in the sum, we would have $d_1 c_{22} a^3 x^2+ d_2 c_{12} a^3 x^2+d_2 c_{22} a^4 x^2$ which would also give negative powers of $x$. In a local theory, one should expect to see only positive powers of $x$ when expanded around $a=0$, $x=0$. Thus we expect that the sum over the Mellin poles would give a cancellation of such negative powers. Cancellation of negative powers of $x$ leads to the ``locality'' constraints \\cite{Gopakumar:2021dvg}. A non-trivial example worked out in \\cite{Gopakumar:2021dvg} shows how this works for the 2d-Ising model. It was further argued in \\cite{Gopakumar:2021dvg} that these locality constraints are identical to the crossing symmetry conditions that one would impose in the fixed-$t$ dispersion relation.\n\nIn addition to these locality constraints, we impose the Polyakov conditions. For the purpose of epsilon expansion, we will use the original Polyakov conditions as discussed in \\cite{Gopakumar:2018xqi}. These read\n\\begin{equation}\\label{con1}\n{\\mathcal M}(s_1=\\frac{\\Delta_\\phi}{3}+p,s_2)=0\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{con2}\n\\partial_{s_1}{\\mathcal M}(s_1=\\frac{\\Delta_\\phi}{3}+p,s_2)=0\\,.\n\\end{equation}\nThe validity of these conditions in this context was established in \\cite{Carmi:2020ekr}.\nThese have to hold for any integer $p\\geq 0$ and for any $s_2$ inside the region shown in the figure.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{convreg2.png}\n\\caption{Convergence regions in the Mellin-Mandelstam $stu$-plane eq.(\\ref{stuvar}). The non-derivative condition converges in the yellow region (which overlaps with the blue region on the right). The derivative condition converges in the blue region. $\\tau_{gap}$ is the minimum twist of the operator that appears in the $\\phi\\times\\phi$ OPE. Figure adapted from \\cite{Carmi:2020ekr}.}\n \\label{fig:convreg}\n\\end{figure}\n\n\n\n Notice that in eq.(\\ref{disp}), there is an unfixed constant ${\\mathcal M}(0,0)$. Thus, to make use of the constraints eq.(\\ref{con1}) we will work with subtracted equations in which this unfixed constant drops out. From a different perspective, this was also discussed in \\cite{Gopakumar:2018xqi}. \n\\section{Lorentzian inversion formula}\\label{Sec:inve}\nIn this section we are going to review a method that goes under the name of Lorenzian inversion formula, first introduced by Caron-Huot in \\cite{Caron-Huot:2017vep}. This approach serves as a proof of the fact that the large spin expansion is analytic, down to spin one. Thus the resummations that we presented in the previous sections are not accidental but are instead solidly based on this fact. More importantly, it provides us with an alternative way of computing OPE data from the singularities of the correlators. The plan of this section is to first present the main formula. Then we will motivate it and discuss a few applications of this formula.\n\\subsection{Main formula}\nFor simplicity, let us consider the correlator of four identical scalar operators.\\footnote{The discussion of \\cite{Caron-Huot:2017vep} is valid for any external scalar operator.} The correlator can be decomposed into conformal blocks as\n\\begin{equation}\n\\mathcal{G}(z,\\bar{z})=\\sum_{\\Delta, \\ell}a_{\\Delta, \\ell} G_{\\Delta, \\ell}(z,\\bar{z})\n\\end{equation}\nwhere $G_{\\Delta, \\ell}(z,\\bar{z})=(z \\bar z)^{\\frac{\\Delta-\\ell}{2}}g_{\\Delta, \\ell}(z, \\bar z)$ and $g_{\\Delta, \\ell}(z, \\bar z)$ is the conformal block associated to the exchange of an operator of dimension $\\Delta$ and spin $\\ell$. The main goal of this section is to write down a relation that can be used to invert the OPE decomposition, meaning that it gives the OPE data from the analytic structure of the four-point correlator. To arrive at such a formula, we first need to analytically continue to the Lorentzian regime and define the double-discontinuity \n\\begin{equation}\n\\text{dDisc}\\left[ \\mathcal{G}(z,\\bar{z})\\right]=\\mathcal{G}_{\\text{Eucl}}(z,\\bar{z})-\\frac{1}{2}\\mathcal{G}^{\\circlearrowright}(z,\\bar z)-\\frac{1}{2}\\mathcal{G}^{\\circlearrowleft}(z,\\bar z)\\;.\n\\end{equation}\nHere $\\mathcal{G}_{\\text{Eucl}}(z,\\bar{z})$ is the Euclidean correlator and the other terms are the two possible analytic continuations around the branch point $\\bar z=1$. We have all the ingredients to write down the inversion formula which reads\n\\begin{equation}\n c_{\\ell,\\Delta}= c^{(t)}_{\\ell, \\Delta}+(-1)^{\\ell}c^{(u)}_{\\ell, \\Delta}\n\\end{equation}\nwhere\n\\begin{equation} \\label{inv}\nc^{(t)}_{\\ell, \\Delta}=\\frac{1}{4}k_{\\frac{\\Delta+\\ell}{2}}\\int_{0}^{1}d z d\\bar{z} \\left(\\frac{z-\\bar z }{z \\bar z} \\right)^2 \\frac{G_{\\ell+3,\\Delta-3}(z, \\bar{z})}{z^2 \\bar z^2}\\text{dDisc}\\left[ \\mathcal{G}(z,\\bar{z})\\right]\\;,\n\\end{equation}\nwith $k_{\\alpha}=\\frac{\\Gamma(\\alpha)^4}{2 \\pi^2 \\Gamma(2 \\alpha -1) \\Gamma(2 \\alpha)}$ and $c^{(u)}_{\\ell, \\Delta}$ has the same form as $c^{(t)}_{\\ell, \\Delta}$ but with $x_1 \\leftrightarrow x_2$. This relation is fully analytic in the spin, except the term $(-1)^{\\ell}.$\\footnote{In the case discussed here $(-1)^{\\ell}=1$ since the spin of the intermediate operators is always even.}. \nThe spectral functions $c_{\\ell, \\Delta}$ are related to $a_{\\Delta, \\ell}$ In particular, it is related to the s-channel OPE data in this way\n\\begin{equation}\nc_{\\ell, \\Delta} \\xrightarrow[\\Delta \\to \\Delta_k]{} \\frac{a_{\\Delta_k,\\ell}}{\\Delta_k-\\Delta}\\;.\n\\end{equation}\n\n\\subsection{Motivation and a sketchy proof}\nIt is clear from the previous discussions that OPE coefficients in a unitary CFT are not arbitrary. In particular, they are not independent of each other but give rise to an analytic function which fixes completely their structure. Based on that, it is possible to exploit the analytic properties of the correlator to extract the OPE data, such as the conformal dimensions and three-point function coefficients. The reason behind analyticity in spins resides in the fact that generically Euclidean physics needs to resum into a function which is sensible at high energy, so it is intimately related to causality. Following \\cite{Caron-Huot:2017vep}, we would like to present a simple example which shows the logic behind the inversion formula. \n\nLet us consider a function\n\\begin{equation}\nf(x)=\\sum_{j=1}^{\\infty}f_j x^j\\;,\n\\end{equation}\nwith the properties that \n\\begin{itemize}\n \\item $f(x)$ is analytic in the whole complex plane, except for the branch cuts at real $|x| >1$,\n \\item $\\left|\\frac{f(x)}{x}\\right|\\to 0$ as $x \\to \\infty$.\n\\end{itemize}\nThis allows us to use Cauchy's theorem to extract the coefficients $f_j$ as\n\\begin{equation}\n f_j=\\frac{1}{2 \\pi i} \\oint \\frac{dx}{x} x^{-j} f(x)\\;.\n\\end{equation}\nBy deforming the contour and using the second property above we can write \n\\begin{equation}\n f_j=\\frac{1}{2 \\pi i} \\int_{1}^{\\infty} \\frac{dx}{x} x^{-j} \\text{Disc} f(x)\n\\end{equation}\nwhere $\\text{Disc} f(x)=-i[f(x(1+i0))-f(x(1-i0))]$. From this relation, it is clear that the coefficients $f_j$ are analytic for $\\text{Re}(j)\\geq 1$ and fully determined by the imaginary part of $f(x)$. \n\nIn spirit, this is the same that happens for the Froissart-Gribov formula \\cite{Gribov:1961ex,Donnachie:2002en}, which forms the foundation of the Regge theory by proving that the relativistic S-matrix is analytic in spins. In that case, the role of $x^j$ in the simple example is played by Legendre polynomials, and the formula is proven by ``inverting'' these polynomials. It turns out that Euclidean CFTs admit a similar treatment and the idea of \\cite{Caron-Huot:2017vep} was to adapt such a reasoning to the case of CFT four-point functions. In particular, it is possible to start with the usual Euclidean decomposition of the four-point function $G(z,\\bar{z})$ into conformal blocks, or more precisely into conformal partial waves. This step has been achieved in \\cite{Costa:2012cb}\\footnote{Notice that we specialised this formula to $d=4$, the bound of integration is also related to $d$.}\n\\begin{equation}\n G(z,\\bar{z})=\\delta_{12}\\delta_{34}+\\sum_{\\ell=0}^{\\infty} \\int_{2-i \\infty}^{2+i \\infty}\\frac{d \\Delta}{2 \\pi i} c(\\ell,\\Delta)F_{\\ell,\\Delta}(z, \\bar{z})\n\\end{equation}\nwhere the first term is the contribution of the identity operator and the functions $F_{\\ell,\\Delta}$ is a single-valued combination of the conformal blocks and their shadows. Notice that in this decomposition the spin $\\ell$ takes integer values while the dimension $\\Delta$ is continuous. Now the idea is to use the orthogonality of the functions $F_{\\ell,\\Delta}$ to invert such integral and obtain $c(\\ell,\\Delta)$.\nThis can be achieved and it reads\n\\begin{equation} \\label{inveucl}\n c(\\ell,\\Delta)= N(\\ell, \\Delta) \\int \\frac{d^2 z}{z^2 \\bar{z}^2} \\left(\\frac{z -\\bar{z}}{z \\bar{z}} \\right)^2 F_{\\ell,\\Delta}(z, \\bar{z}) G(z,\\bar{z})\n\\end{equation}\nThe function $ N(\\ell, \\Delta)$ can be computed by using the behaviour of the functions $F_{\\ell,\\Delta}(z, \\bar{z})$ around $z=0$. Notice that the expression \\eqref{inveucl} is valid in the Euclidean signature, so $\\bar{z}=z^*$. The integration contour is responsible for making contact with the OPE, and in particular the conformal block decomposition is satisfied if the spectral function $c(\\ell,\\Delta)$ has poles and residues related to the conformal dimensions and OPE coefficients of the exchanged operator respectively.\\footnote{There are subtleties related to the convergence of this integral and on the precise location of the shadow poles. We refer the interested reader to Section 3 and Appendix A of \\cite{Caron-Huot:2017vep}} \n\nIt may seem that we have reached our goal but actually we still have only a Euclidean relation, which is not yet analytic in the spin. In order to make it Lorenzian, we need to introduce appropriate variables in this way \\cite{Hogervorst:2013sma}\n\\begin{equation}\n z=\\frac{4\\rho}{\\left( 1+\\rho\\right)^2}\\;,\n\\end{equation}\nand then \n\\begin{equation}\n w=\\sqrt{\\frac{\\rho}{\\bar{\\rho}}}=e^{i \\theta}\\;.\n\\end{equation}\nThe integration then can be rewritten in this way \n\\begin{equation}\n \\int d^2 z \\to \\int_0^1 d|\\rho| \\oint \\frac{d w}{w}\n\\end{equation}\nLet us recall that for any $d$, the conformal blocks are eigenfunctions of the quadratic and quartic Casimir operators, and it is possible to see that generically solving the differential equations associated to these eigenvalue problems leaves us with 8 solutions. These solutions can be built starting from pure power laws in the configuration $0 \\ll z \\ll \\bar{z} \\ll 1$, and they are given by\\footnote{The symmetries are $\\ell \\leftrightarrow 2-d-\\ell$, $\\Delta \\leftrightarrow d-\\Delta$ and $\\Delta \\leftrightarrow 1-\\ell$. Thus by using them, it is possible to generate all the solutions.}\n\\begin{equation} \n g_{\\Delta, \\ell}(z,\\bar{z}) \\sim z^{\\frac{\\Delta-\\ell}{2}}\\bar{z}^{\\frac{\\Delta-\\ell}{2}}\n\\end{equation}\nThen in principle the functions $F_{\\ell, \\Delta}$ are complicated linear combinations of these 8 solutions. In order to close the $w$-contour we would like to roughly decompose the function $F_{\\ell, \\Delta}=F^{+}(\\ell,\\Delta)+F^{-}(\\ell,\\Delta)$ such that \n\\begin{equation}\n F^{+}(\\ell,\\Delta)\\to w^{\\ell} \\quad \\text{as} \\,\\,w \\to 0\n\\end{equation}\nand \n\\begin{equation}\n F^{-}(\\ell,\\Delta)\\to w^{-\\ell} \\quad \\text{as} \\,\\,w \\to \\infty\n\\end{equation}\nThis is not straightforward and {\\it a priori} it is not guaranteed to be possible. Quite remarkably, as shown in \\cite{Caron-Huot:2017vep}, with some manipulations it is possible to find a precise linear combination which brings \\eqref{inveucl} to the formula already introduced in \\eqref{inv}, properly integrated on the Lorentzian diamond. Now it becomes clear that we have a relation which gives the s-channel OPE data as an integral of the $\\text{dDisc}$ which is convergent in the t-channel and a kernel which is essentially the Lorenzian counterpart of the conformal blocks, and it is convergent for spin larger than 1. What happens for spins smaller or equal than 1 is that the contribution of the arc in the $w$-plane cannot be dropped.\\footnote{Notice that in the discussion presented here the spin of intermediate operators can only be even, since we started from a four-point function of identical scalar operators. Thus we can say that the inversion formula is valid for $j>0$.} This ends our sketchy motivation and derivation of the Lorentzian inversion formula. We refer to the original paper \\cite{Caron-Huot:2017vep} and to \\cite{Simmons-Duffin:2017nub} for a more detailed and rigorous proof. \n\n\\subsection{Examples}\nLet us analyse this formula \\eqref{inv}, in particular in connection with the discussion about large spin reconstruction. The first point to make is that this formula is analytic up to spin one, so it is possible to invert the four-point correlator up to this value of the spin. The second is that it turns out that the information contained in the double discontinuity is the same as the one obtained when considering the singularities as $V \\to 0$ of the correlators. To understand these points, let us list the double discontinuity of some useful functions:\n\\begin{align} \\label{doub}\n\\text{dDisc}[\\log(1-\\bar{z})]&=0\\;,\\\\\n\\text{dDisc}[\\log^2(1-\\bar{z})]&=4\\pi^2\\;,\\\\\n\\text{dDisc}\\left[\\left( \\frac{1-\\bar z}{\\bar z} \\right)^p \\right] &=\\left( \\frac{1-\\bar z}{\\bar z} \\right)^p 2\\sin ^2(\\pi p)\\;.\n\\end{align}\nThis set of functions are the ones that we have encountered in the previous sections, in particular in the discussion of large $N$ CFTs. We have seen that the functions that perform a singularity as $V \\to 0$ are the last two in \\eqref{doub} which appear at order $N^{-4}$ and $N^{0}$ respectively. Let us discuss more in details the last line. If we consider the four-point correlator introduced in \\eqref{MFT}, we see that written in terms of $z$ and $\\bar z$ the function has a non-vanishing double discontinuity only due to the presence of the term \n\\begin{equation}\n\\left( \\frac{z \\bar z}{(1-z)(1-\\bar z)}\\right)^{\\Delta_{\\varphi}}\\;.\n\\end{equation}\nThus we have that \n\\begin{align}\n\\nonumber c_{\\ell, \\Delta}&=\\frac{1+(-1)^{\\ell}}{4}k_{\\frac{\\Delta+\\ell}{2}}\\int_{0}^{1}d z d\\bar{z} \\left(\\frac{z-\\bar z }{z \\bar z} \\right)^2 \\frac{G_{\\ell+3,\\Delta-3}(z, \\bar{z})}{z^2 \\bar z^2}\\text{dDisc}\\left[ \\left(\\frac{z \\bar z}{(1-z)(1-\\bar z)}\\right)^{\\Delta_{\\varphi}} \\right] \\\\\n\\nonumber &=\\frac{1+(-1)^{\\ell}}{4}k_{\\frac{\\Delta+\\ell}{2}}\\int_{0}^{1}d z d\\bar{z} \\left(\\frac{z-\\bar z }{z \\bar z} \\right)^2 \\frac{G_{\\ell+3,\\Delta-3}(z, \\bar{z})}{z^2 \\bar z^2}\\left( \\frac{z\\bar z}{ (1-\\bar z)(1- z)}\\right)^{\\Delta_{\\varphi}} 2\\sin ^2(\\pi \\Delta_{\\varphi})\\\\\n\\nonumber &=\\frac{2^{2-\\Delta} (\\Delta-2) \\Delta_{\\varphi} \\left((-1)^\\ell+1\\right) (\\ell+1) \\Gamma (2-\\Delta_{\\varphi})^2 \\Gamma (-\\Delta_{\\varphi}) \\Gamma (-\\Delta+\\ell+4) }{\\Gamma (1-\\Delta_{\\varphi}) \\Gamma (\\Delta_{\\varphi})^2 \\Gamma\n \\left(\\frac{1}{2} (-\\Delta+\\ell+4)\\right)^2 \\Gamma (\\Delta+\\ell-1) \\Gamma \\left(\\frac{1}{2} (-\\Delta-2 \\Delta_{\\varphi}+\\ell+8)\\right)} \\\\\n \\times& \\frac{\\Gamma \\left(\\frac{\\Delta+\\ell}{2}\\right)^2 \\Gamma\n \\left(-\\frac{\\Delta}{2}+\\Delta_{\\varphi}+\\frac{\\ell}{2}\\right) \\Gamma \\left(\\frac{1}{2} (\\Delta+2 \\Delta_{\\varphi}+\\ell-4)\\right)}{ \\Gamma \\left(\\frac{1}{2} (\\Delta-2 \\Delta_{\\varphi}+\\ell+4)\\right)}\\;.\n\\end{align}\nWe can see that \n\\begin{equation}\n\\text{Res}_{\\Delta=2\\Delta_{\\varphi}+2n+\\ell}c_{\\ell, \\Delta} =a_{n,\\ell}^{\\text{MF}}\\;.\n\\end{equation}\nThis confirms our previous observation that the presence of the term $\\left(\\frac{U}{V} \\right)^{\\Delta_{\\varphi}}$ in the four-point function fully fixes the OPE data. Using the inversion formula, one recovers from it both the dimensions of the exchanged operators $\\Delta=2\\Delta_{\\varphi}+2n+\\ell$ which correspond to the poles, and the squared three-point functions coefficients which correspond to the residues at the pole. Let us also point out that the third equation in (\\ref{doub}) is responsible for the double discontinuity of a conformal block where $p=\\tau-2\\Delta_\\varphi$. Note that when $\\tau=2\\Delta_\\phi+2n$ for $n\\in \\mathbb{Z}_{\\geq 0}$, {\\it i.e.}, when the exchanged operator is a double-trace operator, the double discontinuity vanishes and it does not contribute to the spectral function. This is a welcome feature of the Lorentzian inversion formula, in particular in applications to tree-level correlators in AdS where we only need to consider the contribution of single-trace operators.\n\n\n\\section{Large N}\\label{Sec:largeN}\n\nIn this section we are going to review the perturbative expansion around large $N$, which corresponds to the limit of the large central charge. This is a setup which is most interesting when studying holographic theories, where $N$ plays the role of the degrees of freedom. In particular this study has been pioneered to understand the family of large $N$ CFTs that can have weakly coupled and local gravity duals. For concreteness, in this section we will implicitly identify $N$ with the rank of $SU(N)$ gauge group in four dimensional gauge theories. However, the discussion of large $N$ expansion is universal and $N$ can take other meanings. It should be noted that the expansion powers may differ depending on the context. The main reference of this topic is \\cite{Heemskerk:2009pn}. We will review the content and results of the paper and also discuss the expansion at subleading orders \\cite{Aharony:2016dwx}. \n\n\\subsection{Setup}\nWe consider a setup in which we have a generic CFT with a large $N$ expansion and a large gap in conformal dimensions. Holographically, this corresponds to a local quantum field theory in AdS with a large mass gap. More precisely, we assume that there exists a ``single-trace'' type\\footnote{Here and below, when writing ``single-trace'', ``double-trace'', {\\it etc}, we are borrowing the terminology from gauge theories. It should be noted, however, that in a generic large $N$ theory we do not necessarily need to have the notion of traces. Roughly speaking, we may think of single-trace and double-trace as single-particle and double-particle in AdS space. } of scalar field $\\varphi$ which has a fixed dimension $\\Delta_{\\varphi}$. We consider the four-point function \n\\begin{equation}\n\\langle \\varphi(x_1) \\varphi(x_2) \\varphi(x_3) \\varphi(x_4) \\rangle=\\frac{\\mathcal{G}(U,V)}{x_{12}^{2\\Delta_{\\varphi}} x_{34}^{2\\Delta_{\\varphi}}}\\;,\n\\end{equation}\nand its large $N$ expansion which takes the form\n\\begin{equation}\n\\mathcal{G}(U,V)=\\mathcal{G}^{(0)}(U,V)+\\frac{1}{N^2}\\mathcal{G}^{(1)}(U,V)+\\frac{1}{N^4}\\mathcal{G}^{(2)}(U,V)+\\cdots\\;.\n\\end{equation}\nThe displayed first three orders of the expansion will respectively correspond to the disconnected, tree-level and one-loop level contributions in AdS.\nWe will assume that the OPE content of $\\varphi \\varphi$ is \n\\begin{equation}\n\\varphi \\varphi= 1+\\varphi+T_{\\mu \\nu}+[\\varphi \\varphi]_{n,\\ell}+[T T]_{n,\\ell}+[\\varphi T]_{n,\\ell}+[\\varphi \\varphi \\varphi]_{n,\\ell}+\\dots\n\\end{equation}\nwhere the dots denote higher-traces operators. The stress tensor $T_{\\mu\\nu}$ is dual to the graviton field in AdS.\n\n\\subsection{Leading order: $N^0$}\nTo simplify even further the setup, let us ignore the presence in the OPE of single-trace operators including the stress tensor. To do so we can assume that there is a $\\mathbb{Z}_2$ symmetry which will allow for only double-trace operators $[\\varphi \\varphi ]_{n, \\ell}$. Notice however that as we have seen, double-trace operators are necessary since the identity operator in one channel requires their presence in the crossed channel. \n\nAt this order in $N$, the only contributions come from the disconnected part of the four-point correlator, thus practically this is a mean field theory correlator \\eqref{MFT}. For completeness, let us reproduce it\n\\begin{equation}\n\\mathcal{G}^{(0)}(U,V)= 1+ \\left( \\frac{U}{V} \\right)^{\\Delta_{\\varphi}}+U^{\\Delta_{\\varphi}}\\;.\n\\end{equation}\nThe OPE data are the ones discussed in \\eqref{MFTa}. In particular, the intermediate operators are double-trace operators (besides the unit operator) with dimensions and squared OPE coefficients \n\\begin{eqnarray}\n&&\\Delta^{(0)}_{n,\\ell}=2 \\Delta_{\\varphi}+ 2n +\\ell\\;,\\\\\n&&a_{n,\\ell}^{(0)}=a_{n,\\ell}^{\\text{MF}}= \\frac{2^{\\ell+1} (\\ell+1) (\\ell+2 (\\Delta_\\varphi+n-1)) \\Gamma (n+\\Delta_\\varphi-1)^2 }{(\\Delta_\\varphi-1)^2 n! \\Gamma (\\Delta_\\varphi-1)^4} \\nonumber\\\\ &&\\quad\\quad\\quad\\quad\\quad\\quad\\times\\frac{\\Gamma\n (n+2\\Delta_\\varphi-3) \\Gamma (\\ell+n+\\Delta_\\varphi)^2 \\Gamma (\\ell+n+2 \\Delta_\\varphi-2)}{ \\Gamma (\\ell+n+2)\n \\Gamma (2 n+2 \\Delta_\\varphi-3) \\Gamma (2 \\ell+2 n+2 \\Delta_\\varphi-1)}\\;.\n\\end{eqnarray}\nAs we have discussed, even if we did not know the structure of the four-point correlator, we could have arrived at this answer by using the fact that the identity operator is exchanged in one channel. Under crossing, this generates a power law divergence that requires an infinite number of double-trace operators in the OPE. \n\n\\subsection{First order: $N^{-2}$} \nWe would like to understand how to fix the corrections to the OPE data at order $N^{-2}$ . In particular notice that crossing symmetry should be satisfied at each order. Also, there are two scenarios that can be studied now. One situation is to consider corrections to the OPE data, in the absence of any other operators appearing at order $N^{-2}$. The other situation is to consider the corrections to the OPE data of the double-trace operators in the presence of a new operator appearing at order $N^{-2}$. It requires the OPE coefficient of the new operator to scale as $N^{-1}$. \n\\subsubsection{Absence of new operators}\nLet us study the first scenario, which has been extensively analysed in \\cite{Heemskerk:2009pn}. In this case the OPE expansion looks like\n\\begin{equation}\n\\varphi \\varphi = 1+ [\\varphi \\varphi].\n\\end{equation}\nThus we will focus on the correction to the dimensions of the double-trace (or double-twist) operators and to their squared three-point functions. They can be expanded to this order as \n \\begin{eqnarray}\n \\label{expansions}\n \\Delta_{n,\\ell} &=&\\Delta_{n,\\ell}^{(0)}+ \\frac{1}{N^2} \\gamma_{n,\\ell}^{(1)} + \\cdots,\\\\\n a_{n,\\ell} &=& a_{n,\\ell}^{(0)}+ \\frac{1}{N^2} a_{n,\\ell}^{(1)} + \\cdots\\;.\n \\end{eqnarray}\nIf we insert them into the four-point function and expand to order $N^{-2}$, we obtain\n\\begin{align} \\label{expN}\n{\\cal G}^{(1)}(U,V) = \\sum_{n,\\ell} U^{\\Delta_{\\varphi}+n} \\left(a^{(1)}_{n,\\ell} + \\frac{1}{2} a^{(0)}_{n,\\ell} \\gamma^{(1)}_{n,\\ell} \\left(\\log U+\\frac{\\partial}{\\partial n}\\right)\\right) g_{2\\Delta_{\\varphi}+2n+\\ell,\\ell}(U,V)\\;.\n\\end{align}\nCrossing symmetry would require then a term of the form\n\\begin{equation}\n{\\cal G}^{(1)}(V,U)=\\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}}}\\sum_{n,\\ell} V^{\\Delta_{\\varphi}+n} \\left(a^{(1)}_{n,\\ell} + \\frac{1}{2} a^{(0)}_{n,\\ell} \\gamma^{(1)}_{n,\\ell} \\left(\\log V+\\frac{\\partial}{\\partial n}\\right)\\right) g_{2\\Delta_{\\varphi}+2n+\\ell,\\ell}(V,U)\\;.\n\\end{equation}\nLet us study the limit of small $V$. Different from the previous order, there is no power-law divergence due to the fact that we have only double-trace operators. The consequence of this simple observation is striking, there is no need for infinitely many operators with large spins since they would otherwise produce an enhanced divergence in the small $V$ limit. Thus the correction to the OPE data are different from zero only for a finite range of spins. In the language of twist conformal blocks, we have exactly the same structure. In particular, since there is no divergence in $V$, there cannot be any twist conformal block $H^{m}(U,V)$ with $m=0,1, \\dots \\Delta_{\\varphi}-1$. On the other hand, since $\\Delta_{\\phi}$ is an integer, all the higher $m$ terms would produce terms of the form $(\\log V)^2$ which are incompatible with crossing, due to the fact that the only possible logarithmic term is $\\log(U)$. Another option would have been to have $H^{m}(V,U)$ but those are also absent due to the fact that there is no divergence in $V$ to allow for them. \nThus it is possible to state that \n\\begin{eqnarray}\na^{(1)}_{n, \\ell} & \\neq 0 \\qquad \\forall \\quad \\ell=0,2, \\dots, L\\;,\\\\\n\\gamma^{(1)}_{n, \\ell} & \\neq 0 \\qquad \\forall \\quad \\ell=0,2, \\dots, L\\;.\n\\end{eqnarray}\nThe precise structure of this solution can be found by studying the small $U$ and $V$ limits of the crossing equations, and using projectors to isolate the contribution of only a finite number of spins. In particular there are $ \\frac{(L+2)(L+4)}{8}$ undetermined constants for each spin $L$, this means that the structure of the conformal block decomposition together with crossing symmetry is not enough to fix completely the OPE data. The details can be found in \\cite{Heemskerk:2009pn}. As an example, one finds\n\\begin{equation}\n\\gamma^{(1)}_{n, 0}=\\alpha \\frac{\\left(2 \\Delta_{\\varphi} -1\\right)\\left(n+1\\right)\\left(2 \\Delta_{\\varphi} +n-3\\right)\\left( \\Delta_{\\varphi}+n -1\\right)}{\\left( \\Delta_{\\varphi}-1\\right)\\left( 2\\Delta_{\\varphi}+2n -3\\right) \\left( 2\\Delta_{\\varphi}+2n -1\\right)}\n\\end{equation}\nwhere $\\alpha$ is an unfixed parameter corresponding to the freedom we discussed before. Generically, for the squared OPE coefficient it is possible to find a derivative relation, meaning that \n\\begin{equation}\na^{(1)}_{n,\\ell}=\\frac{1}{2}\\partial_n \\left(a^{(0)}_{n,\\ell} \\gamma^{(1)}_{n, \\ell} \\right) \\qquad \\ell=0,2, \\dots L\\;.\n\\end{equation}\nWe can now make contact with the AdS physics. In particular, these solutions correspond to quartic vertex of the kind $\\varphi^4$, $\\varphi^2 \\nabla^2 \\varphi^2$ and so on.\\footnote{Here we are abusing the notation slightly by using $\\varphi$ to denote both the CFT operator and the field in AdS. Notice also that there are no cubic vertices since we are in the simplest setup where we imposed a $\\mathbb{Z}_2$ symmetry.} \n\\begin{figure}[ht]\n\t\\centering\n \\includegraphics[width=0.3\\textwidth, height=0.3\\textwidth]{Fig1c}\n\t\\caption{The quartic tree-level contact diagram, which is the only one at order $N^{-2}$ in the case where only double-trace operators are exchanged.}\n\t\\label{figvertex}\n\\end{figure}\n\nAlso in this case, we can count how many interactions with $2L$ derivatives are present which contribute a spin up to $L$ and we have exactly the same number $ \\frac{(L+2)(L+4)}{8}$. \n\\subsubsection{Presence of new operators}\nThe situation changes when the OPE contains another operator $\\varphi_{\\tau,s}$ which contributes to the four-point function at order $N^{-2}$\n\\begin{equation}\n\\varphi \\varphi =1+[ \\varphi \\varphi]+\\frac{1}{N}\\varphi_{\\tau,s}\\;.\n\\end{equation}\nThe new operator has conformal twist $\\tau$ and spin $s$. As a result, corrections to the OPE data of the double-trace operators will depend also on the presence of $\\varphi_{\\tau,s}$. We are interested in understanding how crossing symmetry fixes the corrections of the form \\eqref{expansions}. In this case, the situation is very different from the previous case. In particular, the four-point function receives a contribution corresponding to the conformal block associated with the exchange of the new operator \n\\begin{equation}\na_{\\tau, s} U^{\\tau\/2}g_{\\tau,s}(U,V)\n\\end{equation}\nwhere $a_{\\tau, s}$ is the squared three-point function coefficient $\\langle \\varphi \\varphi \\varphi_{\\tau,s}\\rangle^2 $. If we use crossing, we observe that it requires the presence of a term of the form\n\\begin{equation}\n\\mathcal{G}^{(1)}(U,V) =\\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}-\\frac{\\tau}{2}}} a_{\\tau, s} g_{\\tau,s}(V,U)+\\dots\n\\end{equation}\nThis already signals that for any positive non-integer $\\Delta_\\varphi-\\frac{\\tau}{2}$ the corrections to the CFT data need to have an infinite support in the spin. This is because, differently from the previous section, there is a divergence as $V\\to 0$ that needs to be reproduced by the divergent part of this sum\n\\begin{equation}\n\\mathcal{G}^{(1)}(U,V)= \\frac{1}{2}\\sum_{n,\\ell}a_{n,\\ell}^{(0)}\\gamma_{n,\\ell}U^{\\Delta_{\\varphi}+n}g_{n,\\ell}(U,V) \\log U +\\dots\n\\end{equation}\nwhere the dots denote terms which are analytic as $z$ goes to zero. This means that \n\\begin{equation}\\label{crosin}\n\\frac{1}{2}\\sum_{n,\\ell}a_{n,\\ell}^{(0)}\\gamma_{n,\\ell}U^{\\Delta_{\\varphi}+n}g_{n,\\ell}(U,V) \\sim \\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}-\\frac{\\tau}{2}}}a_{\\tau,s}g_{\\tau,s}(V,U)|_{ \\log U}\n\\end{equation}\nwhere this equation means that we need to consider the divergence on the LHS as $V \\to 0$. To control this problem, we need to construct the twist conformal blocks and in particular\n\\begin{equation} \\label{crossingle}\n\\sum_{m, n} B_{m,n}H^{(m)}_{n}(U,V)|_{div}=\\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}-\\frac{\\tau}{2}}} a_{\\tau, s} g_{\\tau,s}(V,U)|_{\\log U}\n\\end{equation}\nwhere it is assumed that $\\gamma_{n,\\ell}=2\\sum_m \\frac{B_{mn}}{J^{2m}}$ and $J^2$ is the conformal spin.\n\\paragraph{Intermezzo on twist conformal blocks.}\nLet us fill in more details regarding the twist conformal blocks. Similarly to Section \\ref{Subsec:TCB}, we can define them as \n\\begin{equation} \\label{twconf}\nH^{(m)}_{n}(z,\\bar{z})=\\sum_{\\ell}a_{n,\\ell} \\frac{(z \\bar{z})^{\\Delta_{\\varphi}+n}}{J^{2m}}g_{n,\\ell}(z,\\bar{z})\n\\end{equation}\nwhere $J^2=(\\ell+n+\\Delta_{\\varphi})(\\ell+n+\\Delta_{\\varphi}-1)$. To solve the problem above we are interested in to the divergent contribution of such blocks in the limit in which $\\bar{z}\\to 1$. To this end, we can construct \n\\begin{equation} \\label{serex}\n\\sum_{n}H^{(0)}_n(z,\\bar{z})=\\left( \\frac{z \\bar{z}}{(1-z)(1-\\bar{z})}\\right)^{\\Delta_{\\varphi}}\\;.\n\\end{equation}\nIn addition, the structure of the conformal blocks fixes the form to be\n\\begin{equation}\n\\sum_{n}H^{(0)}_n(z,\\bar{z})=\\frac{1}{\\bar{z}-z} z^{\\Delta_{\\varphi}+n}F_{\\Delta_{\\varphi}+n-1}(z)\\bar{H}^{(0)}_n(\\bar{z})\n\\end{equation}\nwhere $F_{\\beta}(z)= {}_2F_1(\\beta, \\beta, 2\\beta,z)$. By matching the series expansion of both sides of \\eqref{serex}, it is possible to find the full structure for $\\bar{H}^{(0)}_n(\\bar{z})$ which reads \n\\begin{equation}\n\\bar{H}^{(0)}_n(\\bar{z})=\\left(\\frac{\\bar{z}}{1-\\bar{z}} \\right)^{\\Delta_\\varphi}d_n(1+b_n (1-\\bar{z}))\\;.\n\\end{equation}\nHere\n\\begin{equation}\nd_n=-\\frac{\\sqrt{\\pi } 2^{-2 \\Delta_{\\varphi} -2 n+4} \\Gamma (n+\\Delta_{\\varphi} -1) \\Gamma (n+2 \\Delta_{\\varphi} -3)}{\\Gamma (\\Delta_{\\varphi} -1)^2 \\Gamma (n+1) \\Gamma \\left(n+\\Delta_{\\varphi} -\\frac{3}{2}\\right)}\\;,\n\\end{equation}\nand\n\\begin{equation}\nb_n=-\\frac{(\\Delta_{\\varphi} -1)^2+n^2+(2 \\Delta_{\\varphi} -3) n}{(\\Delta_{\\varphi} -1)^2}\\;.\n\\end{equation}\nWe can use then the recurrence relation to extrapolate this result to any positive $m$. The idea is to use the fact that the Casimir operator acts on the twist conformal blocks in the following way\n\\begin{equation}\n\\mathcal{C}_\\tau H^{(m+1)}_n(z,\\bar{z})=H^{(m)}_n(z,\\bar{z})\n\\end{equation}\nwhich, due to the factorization in $z$ and $\\bar{z}$, leads to a recurrence relation for $\\bar{H}^{(m)}_n(z,\\bar z)$ as\n\\begin{equation}\n\\mathcal{D}\\bar{H}^{(m+1)}_n(z,\\bar z)=\\bar{H}^{(m)}_n(z,\\bar z)\\;.\n\\end{equation}\nHere $\\mathcal{D}=\\bar{z} \\bar{D} \\bar{z}^{-1}$ and $D$ is defined in \\eqref{casimirdef}. For a fixed twist, it is possible to write down an expansion of the form \n\\begin{equation}\n\\bar{H}^{(m)}_n(z,\\bar z)=\\left(\\frac{\\bar{z}}{1-\\bar z} \\right)^{\\Delta_{\\varphi}-m}h_0^{(m)}(1+h_1^{(m)}(1-\\bar{z})+h_2^{(m)}(1-\\bar{z})^2+\\dots)\\;,\n\\end{equation}\nand the coefficients $h_n^{(m)}$ can be found iteratively. \n\nWith this piece of information we can tackle the main problem \\eqref{crossingle}. Due to the factorisation property of \\eqref{twconf}, it is possible to see that also the functions $B_{mn}$ satisfy a similar equation. In particular, by inserting the expansion of the anomalous dimension in \\eqref{crossingle} one gets\n\\begin{eqnarray} \n \\sum_{mn} B_{mn} F_{\\Delta_{\\varphi}+n-1}(z) \\frac{z^{\\Delta_{\\varphi}+n}}{z-\\bar z}\\bar{H}_n^{(m)}(\\bar z)&=&\\frac{a_{\\tau,s} (z \\bar z)^{\\Delta_{\\varphi}}}{\\left( (1-z)(1-\\bar{z}) \\right)^{\\Delta_{\\varphi}-\\tau\/2}} \\label{crosag}\\\\\n\\nonumber &\\times & \\frac{(1-\\bar z)^{s+1} F_{\\tau\/2+s}(1-\\bar z) F_{\\tau\/2-1}(1- z)}{z-\\bar z}\\big|_{\\log z}\\;. \n\\end{eqnarray}\nThen we can factor out the dependence on $m$ in the following way\n\\begin{equation}\nB_{mn}=\\kappa_{\\tau-2}(n) \\rho^{(\\tau+2s)}_m(J)-\\kappa_{\\tau+2s}(n) \\rho^{(\\tau-2)}_m(n)\\;.\n\\end{equation}\nInputting this expression in \\eqref{crosag} we get two decoupled equations for $\\kappa$ and $\\rho$ respectively\n\\begin{align} \\label{finaleq}\n\\nonumber \\sum_{n} \\kappa_{\\tau-2}(n) z^{n+\\Delta_{\\varphi}}F_{\\Delta_{\\varphi}+n-1}(z)&=\\frac{1}{d_n}\\frac{\\Gamma(\\tau-2)}{\\Gamma^2\\left(\\frac{\\tau-2}{2} \\right)}\\frac{z^{\\Delta_{\\varphi}}}{(1-z)^{\\Delta_{\\varphi}}}(1-z)^{\\tau\/2}{}_2F_1\\left(\\frac{\\tau-2}{2}, \\frac{\\tau-2}{2},1,z\\right)\\;,\\\\\n\\sum_m \\rho_m^{(\\tau+2s)}(n)\\bar{H}^{(m)}_n (\\bar{z})&= a_{\\tau,s} d_n \\frac{z^{\\Delta_{\\varphi}}}{(1-\\bar z)^{\\Delta_{\\varphi}}}(1-\\bar z)^{\\tau\/2+s+1}F_{\\tau\/2+s}(1-\\bar z)\\;.\n\\end{align}\nExpanding order by order \\eqref{finaleq} in $z$ and $1-\\bar z$ it is possible to find all $\\kappa$ and $\\rho$, for any twist $\\tau$ and spin $s$. Moreover, it is possible to show that crossing fixes the range for $m$ to be integer and $m=\\tau\/2+s+1, \\tau\/2+s+2, \\dots$. \n\nTo avoid cumbersome equations we give the solution for $\\tau=2$. This case is particularly important also because it corresponds to, when $s=2$, the stress-energy tensor which is always present in consistent CFTs. The solution of $\\eqref{finaleq}$ gives all the coefficients in the large $J$ expansion. For this case, the expansion can be resummed to give \n\\begin{equation}\\label{gamas}\n\\gamma_{n,\\ell}^{as}=-a_{2,s}\\frac{2 \\kappa_{2+2s}(n)(\\Delta_{\\varphi}-1)^2}{(\\ell+1)(\\ell+2\\Delta_{\\varphi}+2n-2)}\\;.\n\\end{equation}\nThe function $\\kappa_{2+2s}(n)$ is a degree $2s$ polynomial in $n$. For the exchange of a scalar operator, with spin $s=0$, $\\kappa_2(n)=1$ while for $s=2$ such as the stress-energy tensor we have\n\\begin{equation}\n\\kappa_6(n)=30\\frac{6 n^4+12(2\\Delta_{\\varphi}-3)n^3+6 (5 \\Delta_{\\varphi}^2-14\\Delta_{\\varphi}+11)n^2+6(2\\Delta_{\\varphi}^3-7\\Delta_{\\varphi}^2+10\\Delta_{\\varphi}-6)n}{\\Delta_{\\varphi}^2(\\Delta_{\\varphi}-1)^2}+30\\;.\n\\end{equation}\nBy plugging the solution for $\\tau=2$ back into \\eqref{crosin} it is possible to show that such anomalous dimension do not solve crossing. In fact, it needs to be supplemented with a correction $\\gamma_{n,\\ell}^{fin}$ which is different from zero only for $\\ell=0,1,\\dots, s$. The precise structure can be found in a very similar way as in \\cite{Heemskerk:2009pn}. As an example, for the scalar exchange of twist two, {\\it i.e.,} $\\tau=2$ and $s=0$, this extra piece differs from zero only for $\\ell=0$ and is given by \n\\begin{equation}\n\\gamma_{n,0}^{fin}=\\frac{1}{2}a_{2,0}\\frac{(n+1)(2\\Delta_{\\varphi}+n-3)(\\Delta_{\\varphi}-1)^2}{(\\Delta_{\\varphi}+n-1)(2\\Delta_{\\varphi}+2n-3)(2\\Delta_{\\varphi}+2n-1)}\\;.\n\\end{equation}\nIn this way we have that the full anomalous dimension is given by $\\gamma_{n,\\ell}^{as}+\\gamma_{n,0}^{fin}$. In addition, it is always possible to add solutions truncated in the spin, which are crossing symmetric by themselves and for which \\eqref{crosin} does not put any constraints. More examples of such anomalous dimensions can be found in \\cite{Alday:2017gde}.\n\nThe corrections to the three-point functions $a_{n,\\ell}^{(1)}$ can be found in a very similar way as the correction to the anomalous dimensions. The only difference is that they are not proportional to $\\log U$, as it is clear from \\eqref{expN}.\n\nQuite nicely, the structure of $a_{n,\\ell}^{(1)}$ is simple and it is given by \n\\begin{equation}\na_{n,\\ell}^{(1)}=\\frac{1}{2}\\partial_n \\left(a_{n,\\ell}^{(0)}\\gamma_{n,\\ell} \\right)+a_{n,\\ell}^{(0)} \\hat{a}_{n,\\ell}^{(1)}\\;,\n\\end{equation}\nand generically $\\gamma_{n,\\ell}=\\gamma_{n,\\ell}^{as}+\\gamma_{n,\\ell}^{fin}$ and $\\hat{a}_{n,\\ell}^{(1)}=0$ for $\\Delta_{\\varphi}=2,3,\\dots ,\\tau\/2+1+s$. This form is reminiscent of the situation in which there are no new operator, except for the extra piece. Explicit results for this term can be found in \\cite{Alday:2017gde}.\n\n\\begin{figure}[ht]\n\t\\centering\n \\includegraphics[width=0.8\\linewidth]{singleop}\n\t\\caption{In the presence of a single-trace intermediate operator exchange, there are two contributions to the anomalous dimension and three-point functions of double-trace operators at order $N^{-2}$, that come from the quartic and cubic vertices depicted. }\n\t\\label{figvertex}\n\\end{figure}\n\n\\subsection{Second order: $N^{-4}$}\nIn this section we would like to understand the corrections to order $N^{-4}$ of the anomalous dimension and of the squared three-point functions \\cite{Aharony:2016dwx}. In particular, we assume that at order $N^{-2}$ there are no new operators, thus we only have corrections to the anomalous dimensions and OPE coefficients of double-trace operators which have support on finitely many spins. We also implicitly assume that there is only one operator with the same quantum numbers, which are the dimension and the spin. Thus the OPE data admit the following expansion\n\\begin{align}\n\\Delta&=2\\Delta_{\\varphi}+2n+\\ell+\\frac{1}{N^2}\\gamma^{(1)}_{n,\\ell}+\\frac{1}{N^4}\\gamma^{(2)}_{n,\\ell}\\;,\\\\\na_{n,\\ell}&=a_{n,\\ell}^{(0)}+\\frac{1}{N^2}a_{n,\\ell}^{(1)}+\\frac{1}{N^4}a_{n,\\ell}^{(2)}\n\\end{align}\nwhere $\\gamma^{(1)}_{n,\\ell}\\neq 0$ and $a_{n,\\ell}^{(1)}$ for $\\ell=0,2,\\dots, L$. \nThis expansion together with the conformal block decomposition imply that the correction to the four-point function at order $N^{-4}$ has the form\n\\begin{align} \\label{crossing4}\n \\mathcal{G}^{(2)}(U,V)=\\sum_{n,\\ell}U^{\\Delta_{\\varphi}+n} &\\left(a_{n,\\ell}^{(2)}+\\frac{1}{2}a_{n,\\ell}^{(0)}\\gamma^{(2)}_{n,\\ell}\\left(\\log U+ \\frac{\\partial}{\\partial n} \\right) \\right.\\\\\n\\nonumber &\\left. +\\frac{1}{2}a_{n,\\ell}^{(1)}\\gamma^{(1)}_{n,\\ell}\\left( \\log U+\\frac{\\partial}{\\partial n}\\right)\\right.\\\\\n\\nonumber &\\left.+\\frac{1}{8}a_{n,\\ell}^{(1)} \\left(\\gamma^{(1)}_{n,\\ell}\\right)^2 \\left( \\log^2 U+2 \\log U \\frac{\\partial}{\\partial n}+\\frac{\\partial^2}{\\partial n^2}\\right)\\right)g_{2\\Delta_{\\varphi}+2n+\\ell,\\ell}(U,V)\\;.\n\\end{align}\nThe corrections to the CFT data appearing at order $N^{-4}$ are only in the first line of the equation above while the remaining two lines pertain to corrections to order $N^{-2}$ and $N^{0}$ that we already determined from solving constraints from crossing at previous orders. Most importantly, due to the order of the perturbation, there is a logarithmic singularity $\\log^2 U$, which correspondingly is mapped to $\\log^2 V$ under crossing. This simple observation already signals the fact that $\\gamma^{(2)}_{n,\\ell}$ and $a_{n,\\ell}^{(2)}$ need to be different from zero for arbitrarily large spins, because a finite number of conformal blocks can have a divergence which is at most $\\log V$. Let us now analyse the problem in more detail. \n\\begin{itemize}\n\\item The term proportional to $\\log^2 U$ corresponds to \n\\begin{equation} \\label{rhssum}\n\\frac{1}{8}\\sum_{n}\\sum_{\\ell=0}^{L} U^{\\Delta_{\\varphi}+n}a_{n,\\ell}^{(0)}\\left(\\gamma_{n,\\ell}^{(1)} \\right)^2 g_{2\\Delta_{\\varphi}+2n+\\ell,\\ell}(U,V)= U^{\\Delta_{\\varphi}} \\left(g_1(U,V) \\log V+g_{2} (U,V) \\right) \\end{equation}\nwhere the functions $g_1(U,V)$ and $g_2(U,V)$ can be expanded in positive integer powers in $U$ and $V$ and the presence of the $\\log V$ makes manifest the fact that the sum over the spin is truncated, up to spin $L$. For later convenience we define $\\lim_{V \\to 0}g_1(U,V)=\\tilde{g}_1(U)$.\n\\item Crossing symmetry implies that $\\mathcal{G}^{(2)}(U,V)$ contains \n\\begin{equation}\nU^{\\Delta_{\\varphi}} \\log^2 V \\left(g_1(V,U) \\log U+ g_2(V,U)\\right)\\;.\n\\end{equation}\nThis is the only term which contains a $\\log^2 V$. \n\\item The last two lines of \\eqref{crossing4} involve only finite sums over the spin, so they cannot reproduce the $\\log^2 V$ divergence. This implies that the only candidate is the first line of \\eqref{crossing4}, in particular\n\\begin{equation} \\label{fineq}\n\\frac{1}{2}\\sum_{n,\\ell}U^n a_{n,\\ell}^{(0)} \\gamma_{n,\\ell}^{(2)} g_{2\\Delta_{\\varphi}+2n+\\ell,\\ell}(U,V) |_{\\log^2 V}=g_1(V,U)\\;,\n\\end{equation}\nand similarly for $a_{n,\\ell}^{(2)}$. This provides an equation for $\\gamma_{n,\\ell}^{(2)}$, which is given in terms of $g_1(V,U)$, a fully specified function once we know $\\gamma^{(1)}_{n,\\ell}$ and $ a_{n,\\ell}^{(0)}$. \n\\end{itemize}\nIn order to solve this equation one would need to compute the sum \\eqref{rhssum}, which is an infinite sum over $n$. Instead of performing the sum directly, it is convenient to compute the contribution to the sum of a single conformal block and then sum these terms. From now on, we will consider the leading twist correction for the anomalous dimension $\\gamma^{(2)}_{0,\\ell}$. Despite its simplicity, this case already contains several interesting information that we can extract from this problem. To extract this contribution it is enough to focus on the leading $U \\to 0$ term of the LHS of \\eqref{fineq}. Conversely, \\eqref{fineq} tells us that we should focus on the leading term as $V \\to 0$ of \\eqref{rhssum}, corresponding to $\\tilde{g}_1(V)$. More compactly, the answer can be written as \n\\begin{equation} \\label{gammatwo}\n\\gamma^{(2)}_{0,\\ell}=\\frac{1}{8}\\sum_{n,s}a_{n,s}^{(0)} \\left(\\gamma^{(1)}_{n,s} \\right)^2 \\gamma_{0,\\ell}^{(2)}|_{(n,s)}\n\\end{equation}\nwhere $\\gamma_{0,\\ell}^{(2)}|_{(n,s)}$ denotes the contribution to the anomalous dimension for a single conformal block corresponding to exchanged operators with quantum number $n$, $s$. The procedure of computing $\\gamma_{0,\\ell}^{(2)}|_{(n,s)}$ is general but its final result depends on the quantum numbers. The idea is very similar to what we have seen in the previous section and it amounts to computing the contribution of a single conformal block to $\\tilde{g}_1(U)$ and then acting with the Casimir operator to probe higher orders terms in $J$. With this piece of information we can insert the specific $\\gamma^{(1)}_{n,s}$ and perform the sum.\n\nLet us report the results for a specific case in which the operator $\\varphi$ has dimension two and we have a $\\varphi^4$ type interaction at order $N^{-2}$. The corresponding anomalous dimension is\n\\begin{equation}\n\\gamma^{(1)}_{n,0}= \\frac{3(n+1)^3}{(1+2n)(3+2n)}\\alpha\\;,\n\\end{equation}\nand $\\alpha$ is a proportionality constant that cannot be fixed using solely crossing symmetry. By inserting this information in \\eqref{gammatwo}, we find for the first few values of the spin that \n\\begin{align}\n\\gamma_{0,0}^{(2)} & \\rightarrow \\text{divergent}\\;,\\\\\n\\gamma_{0,2}^{(2)} & =\\frac{2(174 \\pi^2-1925)}{3465}\\alpha^2\\;,\\\\\n\\gamma_{0,4}^{(2)} & =\\frac{150600 \\pi^2-1520519}{2252250}\\alpha^2\\;,\n\\end{align}\nand generically as a function of $J$ \n\\begin{equation} \\label{larges}\n\\gamma^{(2)}_{0,\\ell}=-\\frac{12}{J^4}\\left(1+\\frac{18}{5} \\frac{1}{J^2}+\\frac{96}{7} \\frac{1}{J^4}+\\frac{360}{7} \\frac{1}{J^6}+\\dots \\right)\\alpha^2\\;.\n\\end{equation}\nWith these results at hand it is possible to reconstruct the full Mellin amplitude as the polar terms can be reconstructed with \\eqref{larges} supplemented by crossing symmetry (the Mellin representation will be introduced in section \\ref{Sec:MellinFormalism}). The presence of the divergence at spin zero could be worrisome but actually it is consistent with the expectations from its AdS interpretation. In particular, we expect any bulk loop diagram in AdS that can be considered in the specific setup to have UV divergences. Since the curvature of AdS can be ignored in the UV, these divergences have to behave in the same way as the flat space ones. This necessitates the presence of counterterms that are contact diagrams and need be included in the effective field theory description to make the CFT data finite. In fact, each local bulk term comes with an arbitrary coefficient, that is expected to be responsible of the cancellation of the divergences. In particular, there is a divergent part which is precisely needed to cancel the divergence and an arbitrary finite part. From the CFT point of view, the same happens since each of the terms that we considered at order $N^{-2}$ comes with a coefficient that cannot be fixed with any consideration based on symmetries. \n\nThere are two main lessons that can be learnt from solving crossing up to order $N^{-4}$:\n\\begin{itemize}\n\\item It is possible to reconstruct fully the one-loop answer using lower order CFT data, supplemented with crossing symmetry, the structure of the OPE and the singularity pattern.\n\\item The CFT analysis contains all the ingredients that are expected from the dual AdS picture. \n\\end{itemize}\n\\begin{figure}[ht]\n\t\\centering\n \\includegraphics[width=0.8\\linewidth]{loopdiagram}\n\t\\caption{In the presence of only quartic vertices, this is the set of allowed Witten diagrams to the order $N^{-4}$. These diagrams are also supplemented by the crossing symmetric counterpart.}\n\t\\label{figvloop}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Large spin analytic bootstrap}\\label{Sec:largespin}\nIn this section we would like to discuss how crossing symmetry, the structure of the OPE and basic properties of the conformal blocks imply the presence of operators with large spins, and how to characterize them. These developments are based on \\cite{Alday:2007mf, Komargodski:2012ek, Fitzpatrick:2012yx}. For reader's convenience, we also offer a quick review of some basic concepts of CFT in Section \\ref{Subsec:basicCFTreview}. However, for the readers who already have a working knowledge of CFT, this subsection can be safely skipped.\n\\subsection{Important concepts: A lightning review}\\label{Subsec:basicCFTreview}\nIn this subsection, we will briefly summarize the important concepts needed in order to understand the rest of the review which deals with mostly four-point functions. For readers who have read Section \\ref{Sec:BCFT}, they will already find great familiarity with these concepts. Nevertheless, we will still go through them due to their essential importance and also to set up the notations that we will use in the review. It should be noted that this subsection is not intended to be a pedagogical introduction to CFT since these basic concepts have already been reviewed in great detail in many excellent reviews \\cite{numrev, slavaepfl,Penedones:2016voo,Simmons-Duffin:2016gjk}. Our discussion will be concise, and the reader is referred to these references for further details. For this subsection, we will focus on external scalar operators.\n\n\n\\begin{itemize}\n \n\\item {\\it Operator product expansion (OPE)}: The concept of OPE holds the center stage in the discussion of the conformal bootstrap. In quantum field theory, the idea of OPE enables us to replace the product of two operators which are close to each other by an infinite set of operators inserted at the midpoint. Unlike QFT, where OPE is asymptotic, in CFT the OPE has a finite radius of convergence. For scalar primary operators $\\varphi_1(x)$, $\\varphi_2(x)$, we have the following operator equation\\footnote{We already encountered this OPE in (\\ref{OPEB}).}\n\\begin{equation}\\label{OPE}\n \\varphi_1(x_1)\\varphi_2(x_2)=\\frac{\\delta_{12}}{(x_1-x_2)^{2\\Delta_{\\varphi_1}}}+\\sum_{\\mathcal{O}} C_{12\\mathcal{O}}\\,D[x_1-x_2,\\partial_{x_2}]\\mathcal{O}(x_2)\\,,\n \\end{equation}\nwhere the sum is over primary operators $\\mathcal{O}$. $C_{12\\mathcal{O}}$ are the OPE coefficients and $D[x_1-x_2,\\partial_{x_2}]$ are differential operators whose form is fixed by conformal invariance. The goal of the bootstrap is to constrain the OPE coefficients as well as the spectrum (scaling dimensions) of primary operators that appear in the OPE. In the CFT literature, the operator spectrum and the OPE coefficients are often referred to as the CFT data. If a theory is unitary then there are unitarity bounds that the scaling dimensions of operators have to obey, namely\n\\begin{eqnarray}\\label{unit}\n&&\\Delta\\geq \\frac{d-2}{2}\\,,\\quad \\ell=0\\,,\\\\\n&&\\Delta\\geq d-2+\\ell\\,,\\quad \\ell>0\\,,\n\\end{eqnarray}\nwhere $\\ell$ denotes the spin of the operator. The quantity $\\tau\\equiv\\Delta-\\ell$ is referred to as the twist of the operator.\n\\item {\\it Four-point functions}: The spacetime dependence of two- and three-point functions are completely fixed by conformal invariance. Starting at four points, however, there are quantities which are invariant under all conformal transformations.\\footnote{These statements are easy to see in the embedding space formalism introduced in Section \\ref{Subsec:BCFTkinematics}.} These are the conformal cross ratios\\footnote{They are the analogues of the cross ratios $\\xi$ and $\\eta$ for BCFTs and real projective space CFTs defined in (\\ref{defcrxi}) and (\\ref{defcreta}).}\n \\begin{equation}\nU=\\frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}\\,,\\quad V=\\frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}\\,,\n\\end{equation}\nAs a result, conformal symmetry can only determine a four-point function up to an arbitrary function of $U$ and $V$. For example, we can write the correlation function of four identical scalar primary operators $\\varphi$ with dimension $\\Delta_\\varphi$ as \n\\begin{equation}\n \\langle \\varphi(x_1) \\varphi(x_2) \\varphi(x_3) \\varphi(x_4) \\rangle=\\frac{1}{\\left(x^2_{12}x^2_{34}\\right)^{\\Delta_\\varphi}} \\mathcal{G}(U,V)\\,.\n\\end{equation}\n\n\\item {\\it Conformal blocks:} Four-point functions can be deconstructed by using the OPE. Performing the OPE \\eqref{OPE} for $\\varphi(x_1)$ and $\\varphi(x_2)$ we reduce the four-point function to a sum of three-point functions which are fixed by conformal symmetry up to the OPE coefficients. Equivalently, we can perform \\eqref{OPE} for $\\varphi(x_1)$, $\\varphi(x_2)$ and $\\varphi(x_3)$, $\\varphi(x_4)$ to reduce the four-point function as a sum of two-point functions of operators which are contained in both OPEs. In other words, the four-point function can be interpreted as the sum of infinitely many operator exchanges. The contribution to the four-point function from exchanging a conformal primary operator and its conformal descendants is known as a conformal block $g_{\\Delta,\\ell}(U,V)$.\\footnote{Recall that we had similar notions for BCFTs and real projective CFTs. Depending on the OPE which we use, we have the bulk channel conformal block (\\ref{BCFTbulkg}) and the boundary channel conformal block (\\ref{BCFTbdrg}) for BCFTs. Similarly, we have the bulk channel conformal block (\\ref{RPdCFTg}) and the image channel conformal block (\\ref{RPdCFTgmirror}) for real projective space CFTs.} It can be obtained by directly resumming these contributions contained in the RHS of \\eqref{OPE} for a specific primary operator $\\mathcal{O}$. But more efficiently, the conformal block can be obtained as the eigenfunction of the bi-particle quadratic conformal Casimir operator. Explicit expressions for $g_{\\Delta, \\ell}(U,V)$ in any spacetime dimensions can be found in \\cite{Dolan:2011dv} and they have a closed form expression in even spacetime dimensions. Using conformal blocks, we can write the decomposition of the four-point function more explicitly as follows\n\\begin{equation}\\label{ope12}\n\\mathcal{G}(U,V)=1+\\sum_{\\Delta,\\ell} a_{\\Delta,\\ell} U^{\\frac{\\Delta-\\ell}{2}}g_{\\Delta, \\ell}(U,V)\\;.\n\\end{equation}\nHere we have separated out the contribution of the identity operator, whose presence we shall assume. The coefficients $a_{\\Delta,\\ell}=C^2_{\\varphi \\varphi \\mathcal{O}_{\\Delta,\\ell}}$ are the squares of the OPE coefficients. For unitary theories $a_{\\Delta,\\ell}$ are positive as the OPE coefficients are real. It should be noted that the OPE coefficients depend on the normalizations which one chooses for the conformal blocks\n$g_{\\Delta, \\ell}(U,V)$. For a survey of different normalizations used in the literature, see \\cite{numrev}.\n\n\n\n\n\\item{\\it Crossing equation:} In \\eqref{ope12}, we made a particular choice of applying the OPE \\eqref{OPE} to $\\varphi(x_1)$, $\\varphi(x_2)$ and $\\varphi(x_3)$, $\\varphi(x_4)$. We could have also used the OPE for $\\varphi(x_1)$, $\\varphi(x_4)$ and $\\varphi(x_2)$, $\\varphi(x_3)$ instead. Equating the two cases leads to the following crossing equation\n\\begin{equation}\n\\mathcal{G}(U,V)=\\left( \\frac{U}{V}\\right)^{\\Delta_\\varphi}\\mathcal{G}(V,U)\\,.\n\\end{equation}\nNote that the crossing equation does not obviously follow from the conformal block decomposition (\\ref{ope12}). Instead, they together impose infinitely many constraints on the CFT data and form the cornerstone of the Numerical Conformal Bootstrap. The goal of the Analytic Conformal Bootstrap program is to develop analytic techniques to extract information from these equations.\n\n\n\\item{\\it Generalized Free Fields:} In this review, we will frequently refer to generalized free fields (GFF) or the mean field theories (MFT) which constitute the simplest examples of conformal theories. These theories also arise as the leading order approximation in the expansion of certain small parameters. GFF theories exhibit similar features as free theories. For example, if we consider the four-point function of identical scalars with scaling dimension $\\Delta_\\varphi$, the exchanged spectrum consists of operators with dimensions\n\\begin{equation}\n \\Delta_{n,\\ell}=2\\Delta_{\\varphi}+2n+\\ell\\,,\n\\end{equation}\nand spin $\\ell$. These operators are the normal ordered products with the schematic form $:\\varphi\\square \\partial^\\ell \\varphi:$ and their conformal dimensions are just the engineering dimensions. The motivation behind these operators is that one can use Wick contraction to get their contribution. In what we will discuss, we will consider corrections to the scaling dimensions of these operators (anomalous dimensions) and also their OPE coefficients. These small corrections are subleading in the expansion parameter.\\footnote{We encountered examples of GFF and studied perturbations around them in Section \\ref{Sec:BCFT}. The discussions of four-point functions will be similar in spirit.} \n\n\n \n\n\\item{\\it Holographic correlators:} Much of the discussion that will follow is motivated by the AdS\/CFT correspondence \\cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}. This correspondence is an equivalence between a specific string theory (or M-theory) in anti de Sitter (AdS) space in $d+1$ dimensions and a CFT in $d$ dimensions. Correlation functions in the CFT are mapped to scattering amplitudes in AdS space under this duality. In the bulk, the Feynman diagrams with external points anchored at the boundary of the AdS space are referred to as Witten diagrams. Similar to Feynman diagrams in flat space, we can classify Witten diagrams according to their topologies. For example, Witten diagrams relevant for four-point functions at tree level can be either contact diagrams or exchange diagrams.\\footnote{In Section \\ref{Subsec:analyticmethods}, we have seen similar Witten diagrams in more complicated setups, and they played an important role in the construction of the functional approach.}\n\\end{itemize}\n\n\n\n\\subsection{Euclidean vs Lorentzian}\nIn this section we need to discuss some important differences when discussing CFTs in Euclidean or Lorentzian kinematics. In the Lorentzian case, it is possible to define the so called lightcone limit, which amounts to sending $x^2 \\to 0$ while being on the lightcone. This is realised because it is possible to send one of the lightcone coordinates to zero while keeping at least another one fixed. Let us write the conformal cross ratios \nas\n\\begin{equation}\nU= z \\bar z\\,,\\quad V=(1-z)(1-\\bar z)\\,. \n\\end{equation}\nWhile in Euclidean signature $z^*=\\bar{z}$, in the Lorentzian case $z$ and $\\bar{z}$ are independent from each other. If we consider a four-point function in a space-like configuration in Minkowski signature, it is possible to use conformal symmetry to set the coordinates of the four points to be one at the origin $x_1=(0,0)$, one at $x_2=(z,\\bar{z})$, one at $x_3=(1,1)$ and $x_4$ is sent to infinity along both directions, see Fig.\\ref{figlorent}. \n\\begin{figure}[ht]\n\t\\centering\n \\includegraphics[width=0.8\\linewidth]{lorentzian.png}\n\t\\caption{Kinematics in Lorentzian signature.}\n\t\\label{figlorent}\n\\end{figure}\nThen the lightcone limit amounts to taking $z$ small with $\\bar{z}$ fixed. An interesting limit is the so called double lightcone limit, in which we send $z\\to 0$, and then with $\\bar{z} \\to 1$. The study of the conformal block decomposition, or of the OPE, in the Lorentzian regime necessarily probes the operators with small twists and large spins. \\footnote{Notice that in Lorentzian signature the value of the spin is continuous, differently from the Euclidean counterpart. Despite the fact that we deal with local operator having integer spin, this is essential in the context of the Lorentzian inversion formula \\cite{Caron-Huot:2017vep} which we will review in section \\ref{Sec:inve}, see also \\cite{Kravchuk:2018htv}.} This is exactly the spirit of the following section. Throughout the section, we will also use $U$, $V$ interchangeably with $z$, $\\bar{z}$.\n\\subsection{Necessity of a large spin sector}\nIn this section we would like to study the regime of large spins and understand how crossing symmetry applied to the four-point function of a scalar operator $\\varphi$ constrains the CFT data in this regime. Let us start with the simplest example of generalised free fields in four space-time dimensions\\footnote{The generalisation of this discussion to generic even dimensions is straightforward, see for instance\\cite{Fitzpatrick:2012yx}.}, which are the dual of free field theories in AdS. We will study the four-point correlator of four identical scalars of dimension $\\Delta_\\varphi$ in such a theory. Correlators in mean field theory are given by the sum of two-point contractions, giving\n\\begin{eqnarray} \\label{MFT}\n\\nonumber\\langle \\varphi(x_1) \\varphi(x_2) \\varphi(x_3) \\varphi(x_4) \\rangle&=& \\frac{1}{\\left(x^2_{12}x^2_{34}\\right)^{\\Delta_\\varphi}}+ \\frac{1}{\\left(x^2_{13}x^2_{24}\\right)^{\\Delta_\\varphi}}+\\frac{1}{\\left(x^2_{14}x^2_{23}\\right)^{\\Delta_\\varphi}}\\\\\n\\nonumber&=& \\frac{1}{\\left(x^2_{12}x^2_{34}\\right)^{\\Delta_\\varphi}}\\left( 1+ U^{\\Delta_\\varphi}+ \\left(\\frac{U}{V}\\right)^{\\Delta_\\varphi} \\right)\\\\\n&=& \\frac{1}{\\left(x^2_{12}x^2_{34}\\right)^{\\Delta_\\varphi}} \\mathcal{G}(U,V)\n\\end{eqnarray}\nwhere in the last line we have defined $\\mathcal{G}(U,V)$ for later convenience. \nIf we decompose the above correlator in conformal blocks, assuming that we are taking the OPE of $\\varphi(x_1) \\varphi(x_2)$ together with $\\varphi(x_3) \\varphi(x_4)$, we obtain that\n\\begin{equation}\n\\mathcal{G}(U,V)=1+\\sum_{\\Delta,\\ell} a_{\\Delta,\\ell} U^{\\frac{\\Delta-\\ell}{2}}g_{\\Delta, \\ell}(U,V)\\;.\n\\end{equation}\nWe observe that in addition to exchanging the identity operator, there is a tower of intermediate operators of the form $\\varphi \\partial^{2n}\\partial_{\\mu_1}\\dots \\partial_{\\mu_\\ell}\\varphi=[\\varphi, \\varphi]_{n,\\ell}$ being exchanged. Their dimensions are $\\Delta_{n,\\ell}=2\\Delta_\\varphi+2n+\\ell$ and the corresponding $a_{\\Delta,\\ell}$'s read \n\\begin{equation} \\label{MFTa}\n\\begin{split}\n a^{\\text{MF}}_{n,\\ell}={}&\\frac{2^{\\ell+1} (\\ell+1) (\\ell+2 (\\Delta_\\varphi+n-1)) \\Gamma (n+\\Delta_\\varphi-1)^2 }{(\\Delta_\\varphi-1)^2 n! \\Gamma (\\Delta_\\varphi-1)^4 \\Gamma (\\ell+n+2)}\\\\\n {}&\\times\\frac{\\Gamma\n (n+2\\Delta_\\varphi-3) \\Gamma (\\ell+n+\\Delta_\\varphi)^2 \\Gamma (\\ell+n+2 \\Delta_\\varphi-2)}{\\Gamma (2 n+2 \\Delta_\\varphi-3) \\Gamma (2 \\ell+2 n+2 \\Delta_\\varphi-1)}\\;.\n \\end{split}\n\\end{equation}\nWe can set up this problem more abstractly and consider the constraints of crossing symmetry which are \n\\begin{equation}\n\\mathcal{G}(U,V)=\\left( \\frac{U}{V}\\right)^{\\Delta_\\varphi}\\mathcal{G}(V,U)\\,.\n\\end{equation}\nThis relation implies that we can decompose both sides in conformal blocks, leading to \n\\begin{eqnarray} \\label{cross}\n1+ \\sum_{\\tau, \\ell}a_{\\tau, \\ell}U^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V)=\\left( \\frac{U}{V}\\right)^{\\Delta_\\varphi} \\left(1+ \\sum_{\\tau, \\ell}a_{\\tau, \\ell}V^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(V,U)\\right)\\,,\n\\end{eqnarray}\nwhere we have introduced the conformal twist $\\tau=\\Delta-\\ell$. Before proceeding, it is useful to discuss some properties of the conformal blocks \\cite{Dolan:2000ut,Dolan:2003hv,Dolan:2011dv}. While we are showing them explicitly only for four-dimensional conformal blocks, such properties are much more general and can be easily generalised to any dimension. We discuss three properties of the conformal blocks that will be relevant later on.\n\\begin{itemize}\n\\item {\\it Small $U$ limit:} This limit is already explicit and it is controlled by the twist of the operator. Specifically, the conformal block behaves in this limit as \n\\begin{equation}\nU^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V) \\xrightarrow{U \\ll 1} -2^{-\\ell} U^{\\tau\/2} (1-V)^{\\ell} {}_2F_1 \\left( \\frac{\\tau}{2}+\\ell ,\\frac{\\tau}{2}+\\ell ,\\tau+2\\ell,1-V \\right)+\\dots\\;.\n \\end{equation}\n\\item {\\it Small $V$ limit:} This limit is more subtle. We will discuss at length this limit later, but the structure is as follows \n\\begin{equation} \\label{vdiv}\ng_{\\tau, \\ell}(U,V) \\xrightarrow{V \\ll 1} a(U,V) \\log(V) + b(U,V) \n \\end{equation}\nwhere $a(U,V)$ and $b(U,V)$ admit a regular series expansion in the small $U$ and $V$ limits. In particular, the relation above should be understood as meaning that a small $V$ expansion of a single conformal block does not contain any power-law divergence and the only divergence appearing is logarithmic. \n\\item {\\it Casimir operator:} The conformal blocks are eigenfunctions of the quadratic and quartic Casimir operators of the conformal group, whose eigenvalues depend on the twist and spin of the intermediate operator. Specifically to four dimensions, we have\n\\begin{eqnarray}\n\\mathcal{D}_2 \\left(U^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V) \\right)&=& \\frac{1}{2} ((\\ell+\\tau -4) (\\ell+\\tau )+\\ell (\\ell+2)) \\left(U^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V) \\right)\\;,\\\\\n\\mathcal{D}_4 \\left(U^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V) \\right)&=& \\ell (\\ell+2) (\\ell+\\tau -3) (\\ell+\\tau -1) \\left(U^{\\frac{\\tau}{2}}g_{\\tau, \\ell}(U,V) \\right)\n\\end{eqnarray}\nwhere \n\\begin{eqnarray} \\label{casimirdef}\n{\\cal D}_2 &=&D+\\bar D + 2 \\frac{z \\bar z}{z-\\bar z}\\left( (1-z) \\partial -(1-\\bar z) \\bar \\partial \\right) \\label{quadrcas}\\;, \\\\\n{\\cal D}_4 &=& \\left( \\frac{z \\bar z}{z-\\bar z}\\right)^{2} (D-\\bar D)\\left( \\frac{z-\\bar z}{z \\bar z}\\right)^{2}(D-\\bar D)\\;.\n\\end{eqnarray}\nHere $D=(1-z)z^2 \\partial^2 - z^2 \\partial$.\n\\end{itemize}\n\nAfter this digression, let us come back to \\eqref{cross}. By taking the limit of $U \\ll 1$ on both sides of the relation, we note that there is a potential paradox. In particular, we observe that\n\\begin{eqnarray}\n1&\\sim &\\frac{U^{\\Delta_\\varphi}}{V^{\\Delta_\\varphi}}\\sum_{\\tau, \\ell} a_{\\tau, \\ell} V^{\\tau\/2}g_{\\tau, \\ell}(V,U)\\;,\\\\\n\\implies \\frac{1}{U^{\\Delta_\\varphi}} &\\sim& \\frac{1}{V^{\\Delta_\\varphi}}\\sum_{\\tau, \\ell} a_{\\tau, \\ell} V^{\\tau\/2}g_{\\tau, \\ell}(V,U)\\;. \\label{largespin}\n\\end{eqnarray}\nThe LHS has a divergence $U^{-\\Delta_{\\varphi}}$ as $U \\to 0$ while each conformal block on the RHS, following \\eqref{vdiv}, has a logarithmic divergence. Then the question becomes: how is it possible to reproduce a power-law divergence with a sum of logarithmic divergences? This is only possible by having an infinite sum of conformal blocks on the RHS, with twist $\\tau=2\\Delta_\\varphi$. This is the case because the sum does not converge for all real $U$. In particular, when $\\sqrt{U} <0$ the sum diverges and by analytically continuing the sum to the region of convergence, it can be seen that it contains a power-law behaviour which fixes the problem. The next step is to understand if there are any parameters controlling such divergence. It is possible to study the limit of large $\\tau=2 \\Delta_{\\varphi}+2n$, at fixed $\\ell$, and it is possible to see that \n\\begin{equation}\nU^{\\tau\/2} g_{\\tau, \\ell} (U,V) \\xrightarrow{U,V \\ll 1} U^{\\tau\/2} V^{\\tau\/2} +\\dots\\;.\n\\end{equation}\nMoreover, $a_{\\tau, \\ell}$ for large $\\tau$ are bounded \\cite{Pappadopulo:2012jk}, ensuring that the sum for small $U$ and $V$ converges.\n\nThe limit of large spin $\\ell$ and fixed $\\tau$ is instead different. Let us study it in a more detailed way. We would like to study the RHS of \\eqref{largespin}. In particular, if we consider the small $V$ limit of this term we have\n\\begin{equation}\n\\sum_{\\ell} \\left( -\\frac{1}{2}\\right)^{\\ell} a_{\\tau, \\ell} V^{\\tau\/2} \\left( 1-U\\right)^{\\ell} {} _2F_1\\left(\\ell+\\frac{\\tau}{2}, \\ell+\\frac{\\tau}{2},2\\ell+\\tau,1-U\\right)\\;.\n\\end{equation}\nIn this sum, most of the contribution comes from the region of $U$ goes to zero, when the spin $\\ell$ is large. Thus we can make the following change of coordinates \n\\begin{equation} \n\\ell=\\frac{x}{\\sqrt{U}},\n\\end{equation}\nwhere $x$ is a constant that does not depend on $U$. In this way, we can replace the sum with an integral over the parameter $x$. At the same time, we also consider the integral representation of the hypergeometric function\n\\begin{equation} \\label{hyper}\n{} _2F_1\\left(a,b,c,q\\right) = \\int_{0}^{1} dt \\frac{\\Gamma(c)}{\\Gamma(b)\\Gamma(c-b)}t^{b-1}(1-t)^{c-b-1}(1-t q)^{-a}\\;.\n\\end{equation}\nTo start with, we would like to see how the example of generalised free field works. Thus we can use $a^{\\text{MF}}_{\\tau, \\ell}$ as the squared OPE coefficients and by combining all the pieces together and performing the change of coordinates $t \\to 1-t \\sqrt{U}$ in the $U\\to 0$ limit we obtain\\footnote{Notice that there is a factor of $\\frac{1}{2}$ comes from the fact that we are summing only over even spins.}\n\\begin{eqnarray} \\label{transf}\n\\nonumber && \\frac{4 V^{\\tau\/2} U^{-\\Delta_{\\varphi}} \\Gamma \\left(\\frac{\\tau }{2}-1\\right)^2 \\Gamma \\left({-\\Delta_{\\varphi}}+\\frac{\\tau }{2}-3\\right)}{({-\\Delta_{\\varphi}}-1)^2 \\Gamma\n ({\\Delta_{\\varphi}}+1)^4 \\Gamma (\\tau -3) \\Gamma \\left(\\frac{1}{2} (\\tau -2 {-\\Delta_{\\varphi}})+1 \\right)}\n\\displaystyle \\int_0^{\\infty} d x x^{2 {-\\Delta_{\\varphi}}-1}\\text{K}_0(2 x) \\\\\n&&=\\frac{V^{\\tau\/2} U^{-\\Delta_{\\varphi}} \\Gamma (\\Delta_{\\varphi})^2 \\Gamma \\left(\\frac{\\tau }{2}-1\\right)^2 \\Gamma \\left(\\Delta_{\\varphi}+\\frac{\\tau }{2}-3\\right)}{(\\Delta_{\\varphi}-1)^2 \\Gamma\n (\\Delta_{\\varphi}-1)^4 \\Gamma (\\tau -3) \\Gamma\\left(\\frac{1}{2} (\\tau -2 \\Delta_{\\varphi})+1\\right)},\n\\end{eqnarray}\nwhere the function $\\text{K}_0$ is the modified Bessel function of the second kind. If we combine this result with \\eqref{largespin} we can see that it has several interesting features. Firstly, we have proven that the tail of large spin of the sum in \\eqref{largespin} is essential to reproduce the divergence as $U \\to 0$ that we were studying. In particular, we see that in order to reproduce the leading terms in a small $U,V$ expansion, the CFT under study needs to have infinitely many operators with twist that accumulates at $\\tau=2 \\Delta_{\\varphi}$. Remarkably\n\\begin{equation}\n\\frac{U^{-\\Delta_{\\varphi}} \\Gamma (\\Delta_{\\varphi})^2 \\Gamma \\left(\\frac{\\tau }{2}-1\\right)^2 \\Gamma \\left(\\Delta_{\\varphi}+\\frac{\\tau }{2}-3\\right)}{(\\Delta_{\\varphi}-1)^2 \\Gamma\n (\\Delta_{\\varphi}-1)^4 \\Gamma (\\tau -3) \\left(\\frac{1}{2} (\\tau -2 \\Delta_{\\varphi})\\right)!}\n\\xrightarrow{\\tau=2 \\Delta_{\\varphi}} V^{\\tau\/2} U^{-\\Delta_{\\varphi}}\\;,\n\\end{equation}\nso it exactly reproduce the LHS of \\eqref{largespin}. In addition, at any order in $V$ we need to have the same behaviour and thus the twist should accumulate around $\\tau=2 \\Delta_{\\varphi}+n$, with $n$ being an integer. Secondly, since we did not use any information about the specific theory, this result is completely general and it can be applied in any context. Notice that we have used as squared three-point function coefficients $a^{\\text{MF}}_{\\tau, \\ell}$, and this suggests that this subsector of the theory, namely the large spin sector, behaves as a generalised free theory. Actually, such a result has been shown to be completely general and it is possible to see that if we insist on reproducing the power-law divergence, the behaviour of the squared three-point function in the large spin limit should match the one of the generalised free field. An interesting remark is that the regime of $U$, $V$ going both to zero can only be reached in Minkowski spacetime. With this starting point it is also possible to study several cases, and in particular it is possible to see how to study corrections around large spins \\cite{Fitzpatrick:2012yx,Komargodski:2012ek}. In addition, in \\cite{Alday:2015eya} it has been shown that at any order in the perturbative series in large spin $\\ell$ it is possible to compute all the terms in such expansion of the squared of the three-point functions and of the dimension away from the degenerate point by matching all the divergences in the direct and crossed channels.\n\\subsubsection{Anomalous dimension at large spin}\nThe starting point is the situation that we reviewed in the previous subsection, in particular we consider a setup in which there exists a family of operators of a given twist, that is unbounded in the spin. This degenerate point can be perturbed and as we move away from this particular regime, the operators start gaining an anomalous dimension and the degeneracy is lifted. We parametrise this perturbation with the anomalous dimension $\\gamma_{\\ell}$ that we require to be small. Explicitly, we write\n\\begin{equation}\n\\Delta=2\\Delta_\\varphi+2n+\\ell+\\gamma_{n,\\ell}\\,,\\quad \\gamma_{0,\\ell}\\equiv \\gamma_\\ell\\,. \n\\end{equation}\nThe case $\\gamma_{\\ell}=0$ corresponds to the degenerate case of the previous subsection. Now we would like to understand the constraints coming from crossing symmetry, unitarity and the structure of the conformal block decomposition on the correction to the anomalous dimensions. In order to do so, we need to explore more orders in the small $U$ and $V$ expansion. In particular, in the OPE $\\varphi \\varphi$ there will be an operator with twist $\\tau_{min}$ and spin $\\ell_0$ with associated squared OPE coefficient $a_{\\tau_{min},\\ell_0}$ and thus the expansion in small $U$ reads\n\\begin{equation}\n{\\cal G}(U,V) =1+ a_{\\tau_{min},\\ell_0} U^{\\frac{\\tau_{min}}{2}} (V-1)^{\\ell_0} ~_2F_1(\\ell_0 + \\tau_{min}\/2,\\ell_0+\\tau_{min}\/2,2\\ell_0+\\tau_{min},1-V) +\\ldots\\;.\n\\end{equation}\nCrossing symmetry then implies a term of the form\n\\begin{align} \\label{cr11}\n{\\cal G}(U,V) &= \\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}}} \\left( 1+ a_{\\tau_{min},\\ell_0} V^{\\frac{\\tau_{min}}{2}} (U-1)^{\\ell_0} ~_2F_1(\\ell_0 + \\frac{\\tau_{min}}{2},\\ell_0+\\frac{\\tau_{min}}{2},2\\ell_0+\\tau_{min},1-U) +...\\right) \\nonumber \\\\\n& = \\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}}} \\left( 1+ a_{\\tau_{min},\\ell_0} V^{\\frac{\\tau_{min}}{2}} \\left( \\alpha \\log U+ \\beta+\\ldots \\right) +\\ldots\\right)\\,,\n\\end{align}\nwhere $\\alpha,\\beta$ are related to $a(U,V)$ and $b(U,V)$ in \\eqref{vdiv}. The dots stand for more suppressed powers in $U$ and $V$. As discussed in\n\\cite{Fitzpatrick:2012yx,Komargodski:2012ek, Alday:2013cwa,Alday:2015eya}, crossing symmetry together with the structure of conformal blocks imply that the powers of $V$ multiplying $V^{\\tau_{min}\/2}$ are integers. In a small $U$ limit, we have then\n\\begin{eqnarray}\n\\label{crosslarge}\n\\nonumber &&\\sum_{\\ell} a_{\\ell} U^{{\\Delta_{\\varphi}}+\\gamma_\\ell\/2}(1-V)^\\ell ~_2F_1\\left(\\Delta_{\\varphi}+\\ell+\\frac{\\gamma_{\\ell}}{2},\\Delta_{\\varphi}+\\ell+\\frac{\\gamma_{\\ell}}{2},2(\\Delta_{\\varphi}+\\ell+\\frac{\\gamma_{\\ell}}{2}),1-V\\right)\\\\\n&&=\\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}}} \\left( 1+ a_{\\tau_{min},\\ell_0} V^{\\frac{\\tau_{min}}{2}} \\left( \\alpha \\log U+ \\beta+\\ldots \\right)+\\ldots\\right)\\;.\n\\end{eqnarray}\nThe divergence in $\\frac{U^{\\Delta_{\\varphi}}}{V^{\\Delta_{\\varphi}}}$ fixes the behavior of $a_{\\ell}$ to be the same as the one of $a_{0,\\ell}^{\\text{MF}}$ at large $\\ell$, as we have already discussed. In order to study the consequences of (\\ref{crosslarge}) having only integer powers of $V$ times $V^{\\tau_{min}\/2-\\Delta_{\\varphi}}$, let us make a few remarks. The main idea is to go through similar steps compared to the previous subsection to obtain equations constraining the OPE data.\nFirstly, it is convenient to rescale the squared OPE coefficient in the following way\n\\begin{equation}\na_{\\ell}=\\frac{2^{\\ell+1}\\Gamma \\left(\\ell+\\frac{\\gamma_{\\ell} }{2}+\\Delta_{\\varphi} \\right)^2 \\Gamma \\left(\\ell+\\frac{\\gamma_{\\ell} }{2}+2 \\Delta_{\\varphi} -1\\right)}{\\Gamma (\\Delta_{\\varphi})^2 \\Gamma\n \\left(\\ell+\\frac{\\gamma_{\\ell} }{2}+1\\right) \\Gamma \\left(2 \\ell+\\frac{\\gamma_{\\ell} }{2}+2 \\Delta_{\\varphi} -1\\right)} \\hat{a}_{\\ell}\\;.\n\\end{equation}\nAt leading order in the expansion, we have $\\gamma_{\\ell}=0$, $\\hat{a}_{\\ell}=1$. This rescaling makes the manipulation in \\eqref{transf} less lengthy. \nThe second insight resides in the usage of the quadratic Casimir operator \\eqref{quadrcas}. In particular, since we are working in a small $U$ expansion, we need to compute the limit of the Casimir operator in \\eqref{quadrcas} and the corresponding eigenvalues can be written as \n\\begin{equation}\nJ^2=(\\ell+\\Delta_{\\varphi} +\\gamma_\\ell\/2)(\\ell+\\Delta_{\\varphi} +\\gamma_\\ell\/2-1)\\;.\n\\end{equation}\nThe most interesting point is that if we act with the Casimir operator on the RHS of \\eqref{cr11} it increases the power divergence as $V \\to 0$ of the equation, and correspondingly it follows that the LHS has an enhanced behaviour for large $\\ell$. This is crucial, and allows us to act repetitively to the crossing equation to explore more and more divergences as $V \\to 0$, and probe subleading corrections of the CFT data in the large spin limit. To do so, we can rewrite both $\\gamma_{\\ell}$ and $\\hat{a}_{\\ell}$ as functions of $J$, and expand them in inverse powers of $J$. Then we see that the leading behaviour at large $J$ is fixed by the divergence $V^{\\frac{\\tau_{min}}{2}-\\Delta_{\\varphi}}$ to be\n\\begin{align}\n\\gamma_\\ell &= \\frac{c_1}{J^{\\tau_{min}}}+\\ldots\\;,\\\\\n\\hat a_\\ell &= 1+ \\frac{d_1}{J^{\\tau_{min}}}+\\ldots\n\\end{align}\nwhere the coefficients $c_1,d_1$ can be fixed in terms of $\\alpha,\\beta$ in \\eqref{crosslarge}. The subleading corrections depend on the value of $\\tau_{min}$. For instance $\\tau_{min}=2$, which corresponds to the presence of the stress tensor, has an expansion of the form \n\\begin{align}\\label{largejnonperturbative}\n\\gamma_\\ell &= \\frac{c_1}{J^2}+ \\frac{c_2}{J^3}+ \\frac{c_3}{J^4}+ \\frac{c_4}{J^5}+\\ldots\\;, \\\\\n\\hat a_\\ell &= 1+ \\frac{d_1}{J^2}+ \\frac{d_2}{J^3}+ \\frac{d_3}{J^4}+ \\frac{d_4}{J^5}+\\ldots\\;. \n\\end{align}\nTo fix the coefficients $c_i$ and $d_i$, we plug the expressions \\eqref{largejnonperturbative} into \\eqref{crosslarge} and we follow a similar procedure to the one in the previous section. Now $J^2$ has to scale as $V^{-2}$, and then using \\eqref{hyper} and the same scalings as previously, we end up with integral relations containing $\\gamma_J$ and $\\hat{a}_J$. These expansions are valid also to subleading order as $V\\to 0$, and by requiring such expansion not to have half-integer divergent powers of $V$ we find arbitrarily many relations for the coefficients $c_i$ and $d_i$. The final results can be summarised by saying that the expansion of $\\gamma(J)$ for large $J$ contains only even powers of $1\/J$, and the expansion of $\\hat a(J)\\left(1 - \\frac{\\sqrt{1+4 J^2}}{4J} \\gamma'(J)\\right)$ for large $J$ contains only even powers of $1\/J$. These expansions provide all orders in the large spin expansion.\n\nThe same technology, as presented in \\cite{Alday:2015eya}, can be used in the case of perturbative theories. In this case the minimal twist that appears in the crossed channel is generically $\\tau_0$ but the same analysis can be carried over. A remarkable result of this approach brought to the proof of the so called reciprocity principle, which is a statement about the large spin expansion of the anomalous dimensions. It is also possible to use similar methods to compute anomalous dimensions \\cite{Kaviraj:2015cxa, Kaviraj:2015xsa} to operators with leading dimension $\\Delta=2\\Delta_\\varphi+2n+\\ell$, with $n\\neq 0$ and $n\\ll \\ell$. \nIn the next subsection we will discuss how to construct these corrections using a more powerful technology, which is based on the simple observations that we made so far.\n\n\n\\subsection{Twist conformal blocks and large spin perturbation theory}\\label{Subsec:TCB}\nIn this section we are going to review \\cite{Alday:2016njk}, from which the definition of the twist conformal blocks stems. This approach builds on what was described in the previous sections and most importantly, provides an algebraic way of solving the constraints of crossing symmetry which was also observed in \\cite{Alday:2015ewa}. The plan of this section is to review the abstract construction of the twist conformal blocks and their properties. In section \\ref{Sec:largeN} we will use this technology to study one of the most interesting applications of this method which are theories admitting a large central charge expansion. \n\n\nThe main idea of \\cite{Alday:2016njk} is to introduce a family of functions $H^{(\\rho)}_{\\tau}(U,V)$, called twist conformal blocks, that can be easily expanded both around small $U$ and $V$ in the Lorentzian regime. As we have seen, when perturbing around the degenerate point in which the twists of the operators are degenerate, there are infinitely many operators with unbounded spin. In particular it should be possible to rewrite the four-point function as an infinite sum of twist conformal blocks as\n\\begin{equation}\n\\mathcal{G}(U,V)=\\sum_{\\tau, \\rho} H^{(\\rho)}_{\\tau}(U,V)\\;.\n\\end{equation}\nFor the moment, let us think about this expansion as an abstract statement and we later discuss how to interpret the RHS. The most important point that we would like to make is that the functions appearing on the RHS have a controlled expansion in the small $U$, $V$ limit, differently from the conformal blocks. The task is then to understand how to construct these functions. \n\\subsubsection{Degenerate point}\nAs we have seen in the case of generalised free fields, when $g=0$ there are infinitely many double-trace operators whose spins are not bounded. We will then define \n\\begin{equation}\n\\sum_{\\ell} a^{(0)}_{\\tau, \\ell} U^{\\tau\/2} g_{\\tau, \\ell}(U,V)=H^{(0)}_{\\tau}(U,V)\n\\end{equation}\nwhere we have introduced the notation $a^{(0)}_{\\tau, \\ell}=a^{\\text{MF}}_{\\tau, \\ell}$ and at the $g=0$ point the intermediate operators are highly degenerate. The properties of the functions $H^{(0)}_{\\tau}(U,V)$ are\\footnote{Notice that we are specifying our discussion to the four dimensional case. But with minor modification it can be extended to any number of spacetime dimensions, as it is discussed in \\cite{Alday:2016njk}. }\n\\begin{itemize}\n\\item $H^{(0)}_{\\tau}(U,V) \\xrightarrow{U \\to 0} U^{\\tau\/2} $.\n\\item $H^{(0)}_{\\tau}(U,V) \\xrightarrow{V \\to 0} V^{-\\Delta_{\\varphi}} $.\n\\item $\\mathcal{H}_{\\tau} H^{(0)}_{\\tau}(U,V)=\\frac{\\tau}{4} (\\tau -6) (\\tau -4) (\\tau -2) H^{(0)}_{\\tau}(U,V) $ where $\\mathcal{H}_{\\tau}$ is a combination of the Casimir operators of the conformal group given by $\\mathcal{H}_{\\tau}=\\mathcal{D}_4-\\mathcal{D}_2^2+\\left( (\\tau -6) \\tau +6\\right)\\mathcal{D}_2$.\n\\end{itemize}\nThe first two properties come from the fact that we can decompose these objects into conformal blocks both in the direct and the crossed channels. In particular, they need to reproduce the identity conformal block in the crossed channel. On the other hand, the last property resides on the fact that since $H^{(0)}_{\\tau}(U,V)$ does not depend on the spin, it has to be the eigenfunction of a specific differential operator whose eigenvalues do not depend on the spin either. \n\nNow the idea is to use the second property to write down an expansion for the twist conformal blocks, and then plug it into the differential equation given by the Casimir and use the first property to fix the boundary conditions. \n\n\n\\subsubsection{Perturbation around the degenerate point}\nWhen we go away from the degenerate point, the operators start acquiring an anomalous dimension. Now, we turn on the parameter $g$, the squared OPE coefficients and the dimensions will get corrected. In the same spirit as in the previous subsection, we need to consider the effects of the Casimir operator on the conformal blocks and on the structure of the divergences. In particular, we define the following Casimir operator\n\\begin{equation}\n\\mathcal{C}_\\tau ={\\cal D}_2 +\\frac{1}{4} \\tau(6-\\tau)\\;.\n\\end{equation}\nThis Casimir operator is slightly different from the one in \\eqref{quadrcas}, and its eigenvalues can be easily computed as\n\\begin{equation}\nJ^2_{\\tau,\\ell} = \\frac{1}{4}(2\\ell+\\tau)(2\\ell+\\tau-2)\\;.\n\\end{equation}\nWe will assume that the dimensions and the squared of the OPE coefficients have the following structure in the expansion in inverse powers of $J$ around large spins $\\ell$\n\\begin{eqnarray}\n\\tau_\\ell &=& \\tau + g \\sum_\\rho \\frac{c^{(\\rho)}_{\\tau}}{J^{\\rho}_{\\tau,\\ell}}\\;, \\\\\na_{\\tau,\\ell} &=&a_{\\tau,\\ell}^{(0)}\\left( 1+ g \\sum_\\rho \\frac{d^{(\\rho)}_{\\tau}}{J^{\\rho}_{\\tau,\\ell}} \\right)\\;.\n\\end{eqnarray}\nNotice again that $\\rho$ parametrises the deviation from the degenerate point. The crucial point in the construction is in defining the twist conformal blocks in this way\n\\begin{equation}\n\\sum_{\\ell} a_{\\tau,\\ell}^{(0)} \\frac{U^{\\tau\/2}}{J_{\\tau,\\ell}^{2m}} g_{\\tau,\\ell}(U,V) = H^{(m)}_{\\tau}(U,V)\\;.\n\\end{equation}\nThe particular case of $m=0$ is the one studied before. These objects are not eigenfunctions of the quadratic Casimir $\\mathcal{C}_\\tau$. But its action defines a recurrence relation which relates twist conformal blocks associated to a given twist $\\tau$ but with different values of $m$\n\\begin{equation}\n\\label{recurrence}\n{\\cal C}_\\tau H^{(m+1)}_{\\tau}(U,V)= H^{(m)}_{\\tau}(U,V)\\;.\n\\end{equation}\nAnalogously to the small $U$ and $V$ limits of $H^{(0)}_{\\tau}(U,V)$, it is possible to see that\n\\begin{itemize}\n\\item $H^{(m)}_{\\tau}(U,V) \\xrightarrow{U \\to 0} U^{\\frac{\\tau}{2}}$.\n\\item $H^{(m)}_{\\tau}(U,V) \\xrightarrow{V \\to 0} V^{-\\Delta_\\varphi+m}$.\n\\end{itemize}\nNote that when $\\Delta_\\varphi-m$ is an integer, it is also possible to get a $\\log^2 V$. Accordingly, in the case where we are away from the degenerate point $g=0$, the four-point function can be expanded in the following way\n\\begin{equation}\n{\\cal G}(U,V) ={\\cal G}^{(0)}(U,V) + g \\,{\\cal G}^{(1)}(U,V)+\\cdots\n\\end{equation}\nwhere ${\\cal G}^{(1)}(U,V)$ admits the twist conformal block decomposition\n\\begin{equation}\n{\\cal G}^{(1)}(U,V) = \\sum_{\\tau,\\rho} \\left( P_{\\tau,\\rho} H^{(\\rho)}_{\\tau}(U,V)+\\log U Q_{\\tau,\\rho} H^{(\\rho)}_{\\tau}(U,V) \\right)\\;.\n\\end{equation}\nHere $\\tau$ runs over the spectrum of twists at the $g=0$ point and $\\rho$ over the spectrum of twists plus integers. The most powerful feature of this decomposition resides in the fact that the twist conformal blocks $H^{(\\rho)}_{\\tau}(U,V)$ constructed in this way have a controlled and computable expansion in the small $U$, $V$ limit, and in particular it makes the problem completely algebraic. This means that it is possible to compute all the coefficient $P_{\\tau,\\rho} $ and $Q_{\\tau,\\rho}$ and be able to reconstruct the four-point correlator by solving algebraic equations. In the next section we will see explicit examples of how to construct these blocks and how to reconstruct the four-point correlators. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Bootstrapping loop-level holographic correlators}\\label{Sec:loops}\nIn this section we discuss how to compute one-loop level correlators in full-fledged holographic models by incorporating the techniques discussed in Section \\ref{Sec:largeN}. A major complexity that arises in these supersymmetric theories is the so-called operator mixing, and this complexity requires us to modify the techniques of Section \\ref{Sec:largeN}. While conceptually not difficult to digest, a full discussion of the details of unmixing would require a great deal of additional technical knowledge and goes beyond the scope of this section. Therefore, we will keep this part of the discussion as schematic as possible, with the goal being only to explain the problem and to outline its solution. The main focus of this section is the explicit computation of the one-loop amplitude after taking the solution of the mixing problem as an input. The procedures of this calculation will be explained in detail. We will discuss the case of 4d $\\mathcal{N}=4$ SYM which is dual to IIB supergravity on $AdS_5\\times S^5$. To keep the discussion pedagogical, we will only consider the simplest correlator $\\langle \\mathcal{O}_2\\mathcal{O}_2\\mathcal{O}_2\\mathcal{O}_2\\rangle$.\n\n\n\\subsection{The mixing problem}\nAs discussed in Section \\ref{Sec:largeN}, in order to construct the one-loop answer, it is sufficient to know the tree-level anomalous dimension $\\gamma_{n,\\ell}^{(1)}$ and the OPE coeffient of leading order $a^{(0)}_{n,\\ell}$. Generically, to extract this piece of information it is enough to consider the correlator with the same external operators at order $N^{0}$ and $N^{-2}$ and decompose it in conformal blocks. However, this algorithm rests on the assumption that the intermediate operators are unique, meaning that there is a single operator with the same quantum numbers. \n\nFor $\\mathcal{N}=4$ SYM at strong coupling, this is not the case and the above algorithm needs modifications. In particular, the unprotected superconformal primaries that appear in $\\langle \\mathcal{O}_2\\mathcal{O}_2\\mathcal{O}_2\\mathcal{O}_2\\rangle$ as intermediate operators are singlets under the $SU(4)$ R-symmetry. Schematically, these operators are linear combinations of double-trace operators of the form $\\left[\\mathcal{O}_2\\mathcal{O}_2\\right]_{n,\\ell}$, $\\left[\\mathcal{O}_3\\mathcal{O}_3\\right]_{n-1,\\ell}$, $\\dots$ $\\left[\\mathcal{O}_{2+n}\\mathcal{O}_{2+n}\\right]_{0,\\ell}$. Each operator is neutral under R-symmetry, and has the same spin $\\ell$ and bare conformal twist $2n+4$. Therefore, in general there is mixing among all such double-trace operator. This adds an extra layer of complication to the loop-level computation. However, we can still use the method of Section \\ref{Sec:largeN} once we solve the mixing problem. More specifically, we need to diagonalize the dilatation operator and apply the squaring of anomalous dimension on each eigenstate. To unmix operators with conformal twist $2n+4$, one needs to consider the family of four-point functions $\\langle {\\cal O}_2{\\cal O}_2{\\cal O}_p{\\cal O}_p\\rangle$ for $p=2,\\dots, 2+n$. At order $N^0$ and $N^{-2}$ respectively, we extract from each correlator the averages \n \\begin{equation}\n\\langle a^{(0)}\\rangle=\\sum_{I=1}^{1+n} c_{22,I}c_{pp,I}\\;.\n \\end{equation}\n and \n \\begin{equation}\n \\langle a^{(0)}\\gamma^{(1)}\\rangle=\\sum_{I=1}^{1+n} c_{22,I}c_{pp,I} \\gamma_I^{(1)}\n \\end{equation}\nwhere $c$ are the three-point function. Notice that for simplicity, we removed the spin and conformal dimension labels. Here we have also chosen a normalization in which these eigenstates $\\Sigma^I$ are orthonormal, {\\it i.e.}, $\\langle \\Sigma^{I} \\Sigma^{J}\\rangle =\\delta^{IJ}$. This gives the mixing matrices which are diagonalized in \\cite{Aprile:2017bgs, Alday:2017xua}. The explicit results which are relevant for our purposes are recorded in the next subsection. Let us also mention that to compute one-loop correlators with higher Kaluza-Klein weights, we need to consider more general tree-level correlators. Diagonalizing these more complicated mixing matrices is discussed in \\cite{Aprile:2017xsp,Aprile:2017bgs, Aprile:2018efk} and in general the spectrum still has remaining degeneracy.\\footnote{For the singlet sector the degeneracy is lifted completely to order $1\/N^2$, see \\cite{Aprile:2018efk}.}\n\n\n\\subsection{Super graviton one-loop amplitude in $AdS_5\\times S^5$}\nIn the previous subsection, we explained how to solve the mixing problem. With the data from its solution we can proceed to compute the leading logarithmic singularity. Knowing it allows us to fix the full correlator. However, for $AdS_5\\times S^5$ correlators we can also define the reduced correlators. At tree level, we have also seen in Section \\ref{Subsec:AdS5Mellin} that using them leads to a lot of simplifications, as they automatically take into account superconformal symmetry. For this reason, we will continue to work with the reduced correlator at one-loop level. Note that in Section \\ref{Sec:largeN} we explained the principle of the calculation in position space. However, the computation of loop-level amplitudes is particularly simple in Mellin space and we find a very compact answer in this representation. Therefore, for pedagogical purpose, we will only give below a review of the Mellin method of \\cite{Alday:2018kkw,Alday:2019nin}. There are other complementary approaches such as the position space method \\cite{Aprile:2017bgs,Aprile:2017qoy,Aprile:2019rep} and a method based on the Lorentzian inversion formula \\cite{Alday:2017vkk,Caron-Huot:2018kta}. We will briefly comment on these approaches in Section \\ref{Sec:openproblemscorrelators}, but will refer the reader to these references for details.\n\nThe one-loop leading logarithmic singularity $\\mathcal{H}^{(2)}|_{\\log^2U}(U,V)$ is singular in the small $V$ limit, and has the form\n\\begin{equation}\\label{HlognV}\n\\mathcal{H}^{(2)}|_{\\log^2U}(U,V)=f_2(U,V)\\log^2V+f_1(U,V)\\log V+f_0(U,V)\\;,\n\\end{equation}\nwhere the coefficient functions $f_i(U,V)$ are regular in $U,V\\to 0$. This structure is expected from crossing symmetry as $\\{\\log^2U, \\log U,1\\}$ are mapped to $\\{\\log^2V, \\log V,1\\}$ under $U\\leftrightarrow V$. Closed form expressions of these functions can be found in \\cite{Aprile:2017bgs,Caron-Huot:2018kta}, but it is not necessary for our purpose. Instead we will only need them order by order in the power expansion with respect to $U$\n\\begin{equation}\nf_i(U,V)=U^2(f_i^{(0)}(V)+Uf_i^{(1)}(V)+\\ldots)\\;.\n\\end{equation}\nLet us explain how to compute them from the conformal block decomposition of the leading logarithmic singularity\\footnote{Note there is a shift of 4 in the dimension of the conformal block. This is because we are looking at the reduced correlator. See \\cite{Nirschl:2004pa} for details.}\n\\begin{equation}\\label{lls1loop}\n\\mathcal{H}^{(2)}|_{\\log^2U}(U,V)=\\frac{1}{2}\\sum_n\\sum_{\\ell\\text{ even}}\\langle a^{(0)}_{n,\\ell}(\\gamma^{(1)}_{n,\\ell})^2\\rangle U^{-2}G_{8+2n+\\ell,\\ell}(z,\\bar{z})\\;.\n\\end{equation}\nHere $G_{\\tau,\\ell}(z,\\bar{z})$ are the conformal blocks with the extra power of $U^{\\frac{\\tau}{2}}$\n\\begin{equation}\n G_{\\tau+\\ell,\\ell}(z,\\bar{z})=(z\\bar{z})^{\\frac{\\tau}{2}}g_{\\tau,\\ell}(z,\\bar{z})\\;,\n\\end{equation}\ncompared to the one used in Section \\ref{Sec:largespin}. For reader's convenience, we have reproduced the average $\\langle a^{(0)}_{n,\\ell}(\\gamma^{(1)}_{n,\\ell})^2\\rangle$ from \\cite{Aprile:2017bgs}\n\\begin{equation}\n\\langle a^{(0)}_{t-2,\\ell}(\\gamma^{(1)}_{t-2,\\ell})^2\\rangle=\\sum_{i=1}^{t-1}C^{(0)}_{t,\\ell}R_{t,\\ell,i}a_{t,i}\\;,\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\nonumber &&C^{(0)}_{t,\\ell}=\\frac{2((t+\\ell+1)!)^2(t!)^2(\\ell+1)(2t+\\ell+2)}{(2t)!(2t+2\\ell+2)!}\\;,\\\\\n&&R_{t,\\ell,i}=\\frac{2^{1-t}(2\\ell+3+4i)(\\ell+i+1)_{t-i-1}(t+\\ell+4)_{i-1}}{(\\frac{5}{2}+\\ell+i)_{t-1}}\\;,\\\\\n\\nonumber &&a_{t,i}=\\frac{2^{1-t}(2+2i)!(t-2)!(2t-2i+2)!}{3(i-1)!(i+1)!(t+2)!(t-i-1)!(t-i+1)!}\\;.\n\\end{eqnarray}\nNotice that conformal blocks can be power-expanded in $U$ and $(1-V)$. In particular, the leading $U$ power of $G_{8+2n+\\ell,\\ell}(z,\\bar{z})$ is $U^{4+n}$. As a result, for a fixed power of $U$ in the leading logarithmic singularity, there are only finitely values of $n$ that can contribute. On the other hand, the expansion of the conformal blocks in $(1-V)$ consists of finitely many terms of the form\n\\begin{equation}\n(1-V)^{a+\\ell}{}_2F_1(A,B;C;1-V)\\;,\n\\end{equation}\nwith the minimal power of $(1-V)$ controlled by the spin of the exchanged operator. Therefore, we can truncate the sum over $\\ell$ in (\\ref{lls1loop}) when obtaining the coefficients $B^{(n)}_i$ of $(1-V)^i$ with small $i$ since they are not affected by the large spins. With the help of $\\mathtt{Mathematica}$ we can easily find the general formula for the coefficients $B^{(n)}_i$ as a function of $i$ from a few low-lying values. Performing the infinite sum over $i$ gives $f^{(n)}_i(V)$\n\\begin{equation}\n\\sum_{i=0}^\\infty B^{(n)}_i (1-V)^i=f^{(n)}_2(V)\\log^2V+f^{(n)}_1(V)\\log V+f^{(n)}_0(V)\\;.\n\\end{equation}\nFor example, we have\\footnote{We do not keep track of the overall normalization of the correlator in this subsection.}\n\\begin{eqnarray}\n\\nonumber &&f^{(0)}_2(V)=\\frac{96(V^2+4 V+1)}{(V-1)^6}\\;,\\quad f^{(0)}_1(V)=-\\frac{288 (V+1)}{(V-1)^5}\\;,\\quad f^{(0)}_0(V)=0\\;,\\\\\n\\nonumber &&f^{(1)}_2(V)=\\frac{48 (5 V^3+37 V^2+37 V+5)}{(V-1)^8}\\;,\\quad f^{(1)}_1(V)=-\\frac{144 (7 V^2+22 V+7)}{(V-1)^7}\\;,\\\\\n&& f^{(1)}_0(V)=\\frac{576 (V+1)}{(V-1)^6}\\;,\\\\\n\\nonumber &&f^{(2)}_2(V)=\\frac{48 (59 V^4+706 V^3+1494 V^2+706 V+59)}{7 (V-1)^{10}}\\;,\\\\\n\\nonumber &&f^{(2)}_1(V)=-\\frac{144 (101 V^3+627 V^2+627 V+101) }{7 (V-1)^9}\\;,\\quad f^{(2)}_0(V)=\\frac{384 (5 V^2+14 V+5)}{(V-1)^8}\\;.\n\\end{eqnarray}\n\nNow let us focus on $f_2^{(n)}(V)$ which multiplies $U^{2+2n}\\log^2U\\log^2V$. To produce such terms from the Mellin representation we must have simultaneous cubic poles at $s=4+2n$ and $t=4+2m$. Note that the Gamma function factor in the definition\n\\begin{equation}\n\\mathcal{H}=\\int\\frac{dsdt}{(4\\pi i)^2}U^{\\frac{s}{2}}V^{\\frac{t}{2}-2}\\widetilde{\\mathcal{M}}(s,t)\\Gamma^2[\\frac{4-s}{2}]\\Gamma^2[\\frac{4-t}{2}]\\Gamma^2[\\frac{4-\\tilde{u}}{2}]\\;,\\quad s+t+\\tilde{u}=4\\;,\n\\end{equation}\nalready provides double poles at these locations. Therefore, the reduced Mellin amplitude must contain a pair of simultaneous pole. Our minimal assumption is \n\\begin{equation}\n\\widetilde{\\mathcal{M}}(s,t)\\supset \\sum_{n,m=0}^\\infty \\frac{c_{mn}}{(s-4-2n)(t-4-2m)}\\;,\n\\end{equation}\nwith {\\it constant} $c_{mn}$ coefficients which are independent of the Mandelstam variables. These coefficients are picked up by taking residues in the Mellin representation at $s=4+2n$, $t=4+2m$ and can be determined by comparing with the Taylor expansion coefficients of $f_2^{(n)}(V)$. In practice, we can proceed by first finding an expression for the coefficients for fixed $n$ and then obtain a list of these functions as we increase $n$. It is not difficult to find from these data points that the general expression is given by a symmetric function\n\\begin{equation}\nc_{mn}=\\frac{ p^{(6)}(m,n)}{5(m+n-1)_5}\\;,\n\\end{equation}\nwhere $p^{(6)}(m,n)$ is a degree 6 polynomial \n\\begin{equation}\n\\begin{split}\np^{(6)}(m,n)={}&32\\big(15 m^4 n^2+25 m^4 n+12 m^4+30 m^3 n^3+120 m^3 n^2+114 m^3 n+36 m^3\\\\\n{}&+15 m^2 n^4+120 m^2 n^3+216 m^2 n^2+77 m^2 n-8 m^2+25 m n^4+114 m n^3\\\\\n{}&+77 m n^2-76 m n-40 m+12 n^4+36 n^3-8 n^2-40 n\\big)\\;.\n\\end{split}\n\\end{equation}\n\nLet us now go back to check the assumption that $c_{mn}$ are constants. By crossing symmetry, the Mellin amplitude should also contain simultaneous poles in $s$, $\\tilde{u}$ and $t$, $\\tilde{u}$. Therefore, our minimal assumption corresponds to the following expression \n\\begin{equation}\\label{M1loop}\n\\begin{split}\n\\widetilde{\\mathcal{M}}(s,t)={}& \\sum_{n,m=0}^\\infty c_{mn}\\bigg(\\frac{1}{(s-4-2n)(t-4-2m)}+\\frac{1}{(s-4-2n)(\\tilde{u}-4-2m)}\\\\\n{}&\\quad\\quad\\quad\\quad\\quad +\\frac{1}{(t-4-2n)(\\tilde{u}-4-2m)}\\bigg)\\;.\n\\end{split}\n\\end{equation}\nTo check it, we again take the residue at $s=4+2n$ and select the term proportional to $\\log^2U$ corresponding to the leading logarithmic singularity. Notice that it receives contributions only from the first two pairs of simultaneous poles in (\\ref{M1loop}). We then perform the sum over $m$ and compute the residue of $t$ at $t=4+2\\mathbb{Z}_{\\geq 0}$. We find that $f_2(U,V)$ in (\\ref{HlognV}) is matched by construction. However, very nontrivially, both $f_1(U,V)$ and $f_0(U,V)$ are also fully reproduced. This tells us that there cannot be {\\it single poles} of the form $1\/(s-4-2n)$ in (\\ref{M1loop}). They do not modify the $\\log^2U\\log^2V$ coefficients but can change the $\\log^2U\\log^pV$ coefficient functions for $p=0,1$. By crossing symmetry we also rule out the existence of single poles in the other Mandelstam variables. Therefore, the only ambiguities are regular terms which correspond to contact interactions. These contact terms can be fixed by looking at the flat-space limit of the reduced Mellin amplitude (see \\cite{Alday:2018kkw,Alday:2019nin} for details). Therefore, we conclude that (\\ref{M1loop}) is the full one-loop amplitude. \n\nThe same computational strategy also applies to one-loop correlators with higher Kaluza-Klein weights and a closed form expression for all correlators of the form $\\langle22pp\\rangle$ was obtained in \\cite{Alday:2019nin}. It also can be used to compute one-loop super gluon amplitudes on $AdS_5\\times S^3$ \\cite{Alday:2021ajh} where the amplitudes have the same structure of simultaneous poles (see also \\cite{Behan:2022uqr} for related work on super gluon one-loop correlators in the so called S-fold theories). \n\\section{Bootstrapping tree-level correlators: Mellin space}\\label{Sec:MellinSpace}\nIn Section \\ref{Sec:PositionSpace} we showed how to bootstrap holographic correlators in position space. While the algorithm works in the same way for correlators of higher weights, the implementation becomes more and more cumbersome as the external conformal dimensions are increased. Therefore using this method to find a closed form formula for general correlators does not seem very feasible. In this section, we introduce alternative methods in Mellin space which allow us to obtain general correlators with arbitrary Kaluza-Klein weights. Crucially, these methods exploit the simple analytic structure of holographic correlators in Mellin space, which allows us to extend our intuition of flat-space scattering amplitudes. We first look at the case of IIB supergravity in $AdS_5\\times S^5$ in Section \\ref{Subsec:AdS5Mellin}, and review the method of \\cite{Rastelli:2016nze,Rastelli:2017udc}. In this approach the task of computing correlators can be translated into solving an algebraic bootstrap problem in Mellin space which is formulated by imposing symmetry constraints and consistency conditions. This algebraic problem can be solved in general. The solution is elegantly simple and gives all tree-level four-point amplitudes of super gravitons with arbitrary Kaluza-Klein levels. Unfortunately, this method implicitly relies on special features of the $AdS_5\\times S^5$ theory, and is not as effective for $AdS_7\\times S^4$. Moreover, it does not apply to $AdS_4\\times S^7$. Nevertheless, in the latter two cases superconformal symmetry is still constraining enough to uniquely fix super graviton correlators. This suggests the existence of a universal method which exploits superconformal symmetry in a dimension-independent way and treats all three backgrounds on an equal footing. Such a method was developed in \\cite{Alday:2020lbp,Alday:2020dtb}, building on earlier work \\cite{Zhou:2017zaw}, as we will review in Section \\ref{Subsec:Mellinscfwi} and Section \\ref{Subsec:MRV}. In Section \\ref{Subsec:Mellinscfwi} we first explain how the position space superconformal Ward identities can be exploited in Mellin space and translated into a system of difference equations. Then in Section \\ref{Subsec:Mellinscfwi} we examine a special kinematic limit where the Mellin amplitudes drastically simplify and can be easily computed. Having solved the correlators in this limit, we can then obtain amplitudes in generic kinematic configurations by using symmetries. This method was first developed in \\cite{Alday:2020lbp,Alday:2020dtb} for computing super graviton amplitudes in maximally superconformal theories. However, with small modifications it can also be used to compute tree-level super gluon amplitudes in a variety of non-maximally superconformal theories in different spacetime dimensions, as we will briefly discuss in Section \\ref{Subsec:supergluons}.\n\n\\subsection{A Mellin bootstrap problem for $AdS_5\\times S^5$ IIB supergravity}\\label{Subsec:AdS5Mellin}\nIn Section \\ref{Subsec:positionspace} we have seen that the superconformal Ward identities (\\ref{scfWardid}) play a central role in bootstrapping tree-level four-point functions. One might wonder if it is possible to directly solve these differential constraints, which will then automatically take the consequence of superconformal symmetry into account. While for generic spacetime dimensions the solutions to (\\ref{scfWardid}) are quite complicated \\cite{Dolan:2004mu}, in $d=4$ the answer is rather simple. One can show that the four-point functions split into two parts \\cite{Eden:2000bk,Nirschl:2004pa}\n\\begin{equation}\\label{solscfwi4d}\nG_{k_1k_2k_3k_4}=G_{{\\rm free},k_1k_2k_3k_4}+{\\rm R}\\, H_{k_1k_2k_3k_4}\\;,\n\\end{equation}\nwhere $G_{{\\rm free},k_1k_2k_3k_4}$ is the four-point correlator in the free $\\mathcal{N}=4$ SYM theory and $H_{k_1k_2k_3k_4}$ is the {\\it reduced} correlator containing all the dynamical information. The factor ${\\rm R}$ is determined by superconformal symmetry to be \n\\begin{equation}\n{\\rm R}=t_{12}^2t_{34}^2x_{13}^4x_{24}^4(1-z\\alpha)(1-\\bar{z}\\alpha)(1-z\\bar{\\alpha})(1-\\bar{z}\\bar{\\alpha})\\;.\n\\end{equation}\nSince ${\\rm R}$ carries nontrivial weights under conformal and R-symmetry transformations, the conformal dimensions and R-symmetry charges of the reduced correlator $H_{k_1k_2k_3k_4}$ are shifted\n\\begin{equation}\n\\text{conformal dimensions: }k_i\\to k_i+2\\;,\\quad\\quad \\text{R-symmetry charges: }k_i\\to k_i-2\\;.\n\\end{equation}\nCompared to the full correlator $G_{k_1k_2k_3k_4}$, the reduced correlator $H_{k_1k_2k_3k_4}$ is generally much simpler. For example, $G_{2222}$ contains six independent R-symmetry structures corresponding to the Wick contractions\n\\begin{equation}\nt_{12}^2t_{34}^2\\;,\\quad t_{13}^2t_{24}^2\\;,\\quad t_{14}^2t_{23}^2\\;,\\quad t_{12}t_{23}t_{34}t_{14}\\;,\\quad t_{13}t_{23}t_{24}t_{14}\\;,\\quad t_{12}t_{24}t_{34}t_{13}\\;.\n\\end{equation} \nBy contrast, $H_{2222}$ is independent of $t_i$ and therefore has only one R-symmetry structure. \n\nWe now translate the solution (\\ref{solscfwi4d}) into Mellin space. From the full correlator $G_{k_1k_2k_3k_4}$, we define the Mellin amplitude $\\mathcal{M}_{k_1k_2k_3k_4}$ in the standard way\n\\begin{equation}\\label{defMfull}\nG_{k_1k_2k_3k_4}=\\int_{-i\\infty}^{i\\infty}\\frac{dsdt}{(4\\pi i)^2} {\\rm K}(x_{ij}^2;s,t) \\mathcal{M}_{k_1k_2k_3k_4}(s,t;t_{ij}) \\Gamma_{\\{k_i\\}}(s,t)\\;.\n\\end{equation}\nHere to manifest Bose symmetry, we wrote the correlator without extracting the kinematic factor in contrast to what we did in (\\ref{GandcalG}). The factor ${\\rm K}(x_{ij}^2;s,t)$ is defined by\n\\begin{equation}\n{\\rm K}(x_{ij}^2;s,t)=(x_{12}^2)^{\\frac{s-k_1-k_2}{2}}(x_{34}^2)^{\\frac{s-k_3-k_4}{2}}(x_{14}^2)^{\\frac{t-k_1-k_4}{2}}(x_{23}^2)^{\\frac{t-k_2-k_3}{2}}(x_{13}^2)^{\\frac{u-k_1-k_3}{2}}(x_{24}^2)^{\\frac{u-k_2-k_4}{2}}\\;,\n\\end{equation}\nwith $s+t+u=k_1+k_2+k_3+k_4$, and \n\\begin{equation}\n\\Gamma_{\\{k_i\\}}(s,t)=\\Gamma[\\tfrac{k_1+k_2-s}{2}]\\Gamma[\\tfrac{k_3+k_4-s}{2}]\\Gamma[\\tfrac{k_1+k_4-t}{2}]\\Gamma[\\tfrac{k_2+k_3-t}{2}]\\Gamma[\\tfrac{k_1+k_3-u}{2}]\\Gamma[\\tfrac{k_2+k_4-u}{2}]\\;.\n\\end{equation}\nSimilarly, we define the {\\it reduced} Mellin amplitude $\\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}$ from the reduced correlators $H_{k_1k_2k_3k_4}$\n\\begin{equation}\\label{defMreduced}\nH_{k_1k_2k_3k_4}=\\int_{-i\\infty}^{i\\infty}\\frac{dsdt}{(4\\pi i)^2} \\widetilde{{\\rm K}}(x_{ij}^2;s,t) \\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}(s,t;t_{ij}) \\widetilde{\\Gamma}_{\\{k_i\\}}(s,t)\\;,\n\\end{equation}\nwhere \n\\begin{equation}\n\\widetilde{{\\rm K}}(x_{ij}^2;s,t)=(x_{12}^2)^{\\frac{s-k_1-k_2}{2}}(x_{34}^2)^{\\frac{s-k_3-k_4}{2}}(x_{14}^2)^{\\frac{t-k_1-k_4}{2}}(x_{23}^2)^{\\frac{t-k_2-k_3}{2}}(x_{13}^2)^{\\frac{\\tilde{u}-k_1-k_3}{2}}(x_{24}^2)^{\\frac{\\tilde{u}-k_2-k_4}{2}}\\;,\n\\end{equation}\n\\begin{equation}\n\\widetilde{\\Gamma}_{\\{k_i\\}}(s,t)=\\Gamma[\\tfrac{k_1+k_2-s}{2}]\\Gamma[\\tfrac{k_3+k_4-s}{2}]\\Gamma[\\tfrac{k_1+k_4-t}{2}]\\Gamma[\\tfrac{k_2+k_3-t}{2}]\\Gamma[\\tfrac{k_1+k_3-\\tilde{u}}{2}]\\Gamma[\\tfrac{k_2+k_4-\\tilde{u}}{2}]\\;,\n\\end{equation}\nand $s+t+\\tilde{u}=k_1+k_2+k_3+k_4-4$. Note that the shift $\\tilde{u}=u-4$ is needed because $H_{k_1k_2k_3k_4}$ has shifted conformal dimensions relative to $G_{k_1k_2k_3k_4}$. Moreover, we should note that Bose symmetry, which permutes $s$, $t$, $u$ in (\\ref{defMfull}), now permutes $s$, $t$, $\\tilde{u}$ in (\\ref{defMreduced}).\n\nOnce including the factor ${\\rm R}$, the combination ${\\rm R}\\, H_{k_1k_2k_3k_4}$ has the same weights as $G_{k_1k_2k_3k_4}$ and therefore should have the same Mellin representation (\\ref{defMfull}). This leads us to interpret ${\\rm R}$ as a difference operator in Mellin space. We note that ${\\rm R}$ is a polynomial in $x_{ij}^2$, and multiplicative $x_{ij}^2$ monomials outside of the inverse Mellin transformation can be absorbed into the $\\widetilde{{\\rm K}}(x_{ij}^2;s,t)$ factor by shifting $s$ and $t$. More precisely, let us write \n\\begin{equation}\n\\frac{{\\rm R}}{x_{13}^4x_{24}^4}=t_{12}^2t_{34}^2\\bigg(\\tau+(1-\\sigma-\\tau)V+(\\tau^2-\\tau-\\sigma\\tau)U+(\\sigma^2-\\sigma-\\sigma\\tau)UV+\\sigma V^2+\\sigma\\tau U^2\\bigg)\\;.\n\\end{equation}\nComparing (\\ref{defMfull}) and (\\ref{defMreduced}), we find that each monomial $U^mV^n$ in the RHS becomes a difference operator $\\widehat{U^mV^n}$ which acts as \n\\begin{equation}\n\\widehat{U^mV^n}\\circ \\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}(s,t;t_{ij})=\\frac{\\widetilde{\\Gamma}_{\\{k_i\\}}(s-2m,t-2n)}{\\Gamma_{\\{k_i\\}}(s,t)} \\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}(s-2m,t-2n;t_{ij})\\;.\n\\end{equation}\nThis substitution defines an operator $\\mathbb{R}$ acting on the reduced Mellin amplitude\n\\begin{equation}\n\\mathbb{R}\\circ\\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}\\;.\n\\end{equation}\nFinally, it can be argued that the free correlator $G_{{\\rm free},k_1k_2k_3k_4}$ does not contribute to the Mellin amplitudes \\cite{Rastelli:2016nze,Rastelli:2017udc}. When we multiply with the factor ${\\rm R}$, the contours in the inverse Mellin transformations are also shifted. In bringing the contours to the correct ones, we encounter situations where the contours are pinched at poles with a {\\it vanishing} Mellin amplitude. The ``zero times infinity'' contribution coming from contour pinching gives rise to rational terms which together become precisely the free correlator $G_{{\\rm free},k_1k_2k_3k_4}$. We will not keep track of the contours in this review. Therefore, as far as the Mellin amplitudes are concerned we can ignore the free correlators. To see how the free correlator is explicitly reproduced in an example with $k_i=2$, see \\cite{Rastelli:2017udc}. All in all, the solution to the superconformal Ward identity (\\ref{solscfwi4d}) implies the following difference relation \n\\begin{equation}\\label{MandMt}\n\\mathcal{M}_{k_1k_2k_3k_4}=\\mathbb{R}\\circ\\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}\\;,\n\\end{equation}\nwhich compactly packages the full Mellin amplitudes in terms of the reduced Mellin amplitudes. \n\nNote that (\\ref{MandMt}) has only exploited superconformal symmetry. The Mellin amplitudes $\\mathcal{M}_{k_1k_2k_3k_4}$ further need to satisfy a number of other consistency conditions in order to be physical. First of all, the Mellin amplitude should have Bose symmetry. This requirement means that the Mellin amplitude is invariant under exchanging external particle labels, which also permutes the Mandelstam variables. Secondly, the Mellin amplitude is local. It has simple poles at locations corresponding to the twist of exchanged single-trace particles, and the residues at these poles are polynomial in the other independent Mandelstam variable. Finally, the high energy limit of the Mellin amplitude with $s,t,u\\to\\infty$ at the same rate is proportional to the flat-space scattering amplitude of IIB super gravity. The latter grows linearly in energy. Therefore the high-energy growth of the Mellin amplitude must also have the same linear behavior. \n\nThese three conditions together with (\\ref{MandMt}) formulate a highly constraining bootstrap problem. For example, it is not difficult to convince oneself that for $k_i=2$ the reduced Mellin amplitude can only be proportional to \n\\begin{equation}\n\\frac{1}{(s-2)(t-2)(\\tilde{u}-2)}\\;,\n\\end{equation}\nin order to be compatible with all the above conditions. On the other hand, this reformulation of the problem places correlators with any choice of $k_i$ on the same footing, which makes it possible to find a general solution in one go. After studying a few explicit examples, \\cite{Rastelli:2016nze,Rastelli:2017udc} found the following ansatz for the reduced Mellin amplitudes\n\\begin{equation}\\label{solAdS5}\n\\widetilde{\\mathcal{M}}_{k_1k_2k_3k_4}=\\prod_{i