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+{"text":"\\section{Introduction}\nThe generative adversarial network (GAN) framework \\cite{Goodfellow2014} is a relatively new type of framework that introduces a generator \\textit{G} and a discriminator \\textit{D}, both of which are often chosen to be a type of multilayer perceptron (MLP) network, and are trained in an adversarial manner. \\textit{D} is trained to label training examples as true, and outputs of \\textit{G} as false, while \\textit{G} is trained to maximize the probability of \\textit{D}'s classification errors.\n\nRecent works have reported that models trained using the GAN framework (especially for the image generation task) can generate excellent samples \\cite{RadfordMC15}. Nevertheless, it is widely reported that GAN models are difficult to train and different techniques are proposed (which are summarized in \\cite{Salimans2016,OPENAI2016,Oriol2016}).\nMost of these work try to deal with the difficulty of optimizing the \\textit{G} and \\textit{D} simultaneously in a synchronized manner and their joint convergence\\cite{Goodfellow2015}. \\cite{Salimans2016} proposes to train the generator by a new objective that guides \\textit{G} to match the statistics of features on an intermediate layer of the discriminator. This helps to avoid over-training of the current discriminator and discourages learning mappings that are not useful in generating realistic data. Another major difficulty in training GANs is the tendency for \\textit{G} to collapse by converging to a parameter setting which maps all $z$'s to the same data. \\cite{Salimans2016} uses minibatch discrimination so that \\textit{D} can generate different gradients for data samples and continue guiding \\textit{G} via backpropagation and force \\textit{G} to diversify its outputs and avoid collapsing.\n\n\nIn this paper, we argue that using \\textit{noise} for the generator contributes to the difficulty of training GANs. The generator is assigned the task of learning the mapping from the input signal $z$, which is a uniformly distributed noise to data space. In \\cite{Bengio2013}, a strong evidence is presented for better disentangling of the underlying factors of variation in data by higher-level representation spaces, and this might explain why a uniformly distributed random noise as input to \\textit{G} works. However, learning the mapping from a flat representation space to data space is difficult. This can be explained with an example from human-face image generation task. A completely disentangled representation space for human faces would likely consist of features including pose, illumination, gender, mood, facial expressions, 3-D model of face etc. The generator's task is then to find good mappings from this flat representation space to face images, which requires significant training. It is difficult for the generator network to achieve this goal and this might be one of the reasons why generator collapses during training. However, mapping a higher-level representation that does not assume complete disentanglement of factors of variation, such as mapping facial areas to a face image, can be expected to be an easier task. This higher-level representation-space features will be correlated (\\emph{e.g.}, the existence of vertical symmetry in the facial parts or the location of distinct facial areas depending on the pose). By using an associative memory on $z$ and generating samples from this memory as an input to \\textit{G}, our goal is to alleviate \\textit{G}'s learning task. Also, mapping data samples to a lower dimensional but higher-level representation space by using \\textit{D} and then learning an associative memory model causes the Markov chain (used in the model training) to mix faster - thanks to a more uniform distribution in this new space \\cite{Bengio2013}.\n\nThe associative adversarial networks (AANs) uses an associative memory that is located in between the generator and the discriminator networks, connecting the two networks together. We explore the special case that uses a Restricted Boltzmann Machine (RBM) as an associative memory, which learns a probability distribution over the features of an intermediate layer of the discriminator. Gibbs sampling is used to sample from this distribution, and the samples are fed to \\textit{G}. Figure~\\ref{fig1} depicts an example AAN, which we present in this work.\n\nThe main contributions of this work are the following: \n\\begin{itemize}\n  \\setlength{\\itemsep}{1pt}\n  \\setlength{\\parskip}{0pt}\n  \\setlength{\\parsep}{0pt}\n\\item We introduce a higher-level associative memory, a stochastic generative model that connects the two non-stochastic GAN models; the discriminator and the generator. \n\\item Training on CelebA face image dataset, we show convincing evidence that the associative memory can learn a probabilistic model on the higher-level representation space of the discriminator. It can then produce samples to be used as an input to the generator thereby alleviating its learning task. \n\\item We further evaluate the performance of the assoicative memory, RBM's performance, by investigating the held-out likelihood of the model for varying parameters.\n\\end{itemize}\n\n\\section{Related Work}\nNeural network models that generate data have been getting more attention recently due to their ability to learn reusable feature representations from large unlabeled data sets as well as generate realistic data samples. The generative adversarial networks (GANs) \\cite{Goodfellow2014}, variational autoencoders (VAE) \\cite{Kingma2014}, generative maximum mean discrepeancy networks \\cite{Dziugaite}, deep generative models\\cite{Rezende2014}, generative moment matching networks \\cite{Yujia2016}, etc., \nhave shown that a deep generative network can learn a distribution over samples. Specifically, GANs have been a promising family of generative models in the computer vision field, because they can produce sharp images by applying stochastic gradient descent on well-defined loss functions. Some GAN extensions have looked at laplacian pyramid extensions \\cite{Denton2015} showing higher quality images, a recurrent network\napproach \\cite{Gregor2015} and a de-convolution network approach \\cite{Dosovitskiy2014} demonstrating reasonable success with generating natural images.\nSeveral recent papers focus on improving the stability of training and the quality of generated GAN samples \\cite{Denton2015,RadfordMC15,Im2016,Yoo2016}. \nAmong recent ones (as recently summarized in \\cite{Oriol2016})  the following pop out for stabilizing GAN training: \\textit{balancing\/freezing}: to prevent the generator or discriminator to outpace one another, \\textit{minibatch discrimination}: to prevent the generator to collapse on to a single sample and enable back-propagation of gradients to improve weights, \\textit{historical averaging} \\cite{Salimans2016}, a technique common used in game theory that uses a historical average of parameters as a regularization term in optimization.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth]{gan2.png}\n\\end{center}\n\\caption{AAN: The Associative Adversarial Network Model. An intermediate layer of the discriminator (depicted in dark green) is clamped as a visible layer to the RBM network (the associative memory), which is then sampled to generate inputs for the generator network (as opposed to noise sampling). This layer that is visible to the associative memory represents a feature space that can capture latent factors of variations in the data. The following layers in the discriminator mostly serve in the classification task embedded into GAN objective functions. However, there is no actual distinction between the network layers, and no such constraints are imposed. There are three separate networks, of which two are non-stochastic (D and G) models and one is a stochastic generative (RBM) model. All networks are jointly learned.}\n\\label{fig1}\n\\end{figure}\n\nIn this work we build up on these techniques. We use some of the DCGAN architectural suggestions proposed in Radford et al. \\cite{RadfordMC15}, which uses strided convolutions in the initial layers of the discriminator and fractional-strided convolutions in the later layers of the generator, which will be discussed in the experiments section. In a way, our proposed technique resembles the learning algorithm for deep belief nets presented in \\cite{Hinton06}, which utilizes a contrastive version of the wake-sleep algorithm proposed in \\cite{Hinton2002}. Deep belief nets have a top-level undirected associative memory together with directed generative models in the lower levels. To generate samples from a model, Gibbs sampling is used in the top-level associative memory and a sample from this memory is passed down stochastically through the directed generative connections in the lower layers. The model is learned by performing a contrastive version of the wake-sleep algorithm after a greedy layerwise initialization process. In the wake phase, a stochastic \"up-pass\" starts and generative weights between two adjacent layers are updated locally to maximize the likelihood of upper layer samples generating lower level samples. In the sleep phase, a stochastic \"down-pass\" starts and recognition weights between two adjacent layers are updated similarly. Our proposed method uses D for the up-pass and G for the down-pass.\n\n\\section{Generative Adversarial Networks (GAN)}\nGANs \\cite{Goodfellow2014} are a class of generative models that pose the training process as a game between a generator network ($G$) and a discriminator network ($D$) both of which are non-stochastic models. The generator network, $G(z;\\theta^{(G)})$, is typically chosen as a feed forward or convolutional neural network depending on the task. It produces samples, by transforming vectors of noise $z$ as $x=G(z;\\theta^{(G)})$. The discriminator $D$, is trained by taking the samples from the generator, $p_{G}(x)$, as negative instances and from the real data $p_{data}(x)$ as positive instances and is trained to distinguish generated samples from the real (training) data. \n\n\\textit{D} takes the output of \\textit{G} and maps to a binary classification probability. Generator then tries to \"trick\" the discriminator by generating \\textit{fake} samples. The learning framework is a two-player game and is cast as a minmax optimization of a differentiable objective and solved greedily by iteratively performing gradient descent steps to improve $G$ and $D$ and eventually reaching a Nash equilibrium \\cite{wiki}. The GAN problem can be formulated as a zero-sum game (minimax) which has a \\textit{distinguishability game value function}, $V(D,G)$ \\cite{Goodfellow2015}:\n\\[\n\\min_{G}\\max_{D}=\\mathbb{E}_{x\\sim p_{data}(x)}[\\log D(x)]+\\mathbb{E}_{z\\sim p_{z}(z)}[\\log (1 - D(G(z)))] \n\\]\n\nThe first term in the cost function forces \\textit{D} to label real data samples as one, while the second term forces \\textit{D} to label fake data samples as zero. \\textit{G} tries to fake \\textit{D} into labeling its output as one so minimizes the given cost function, while \\textit{D} tries to maximize it.\n\n\n\\section{Using RBMs as an Associative Memory for GANs} \nAn RBM is an energy based model for unsupervised learning with an underlying undirected graph. It consists of two layers of binary stochastic units: one \\textit{visible} layer $v$ representing the data and one \\textit{hidden} layer $h$ for the latent variables. $w_{ij}$ determines the strength of the interaction between the hidden $h_j$ and visible $v_i$ units. An energy function between visible and hidden variables $E(v,h)$, the probability $P(v,h)$ and the partition function $Z$ is defined as below:\n\\[\nE(v,h) = \\frac{1}{2}\\sum_i v_i^2-\\sum_{i,j}v_ihjw_{ij}-\\sum_i{v_i b_i}-\\sum_j{h_j c_j} \\ ;\\ P(v,h)=\\frac{e^{-E(v,h)}}{Z} \\ ; \\ Z=\\sum_{x,y}e^{-E(x,y)}\n\\]\nThe probability of a data point (represented by the visible state $v$) is defined by marginalizing over the hidden variables $P(v)=\\sum_hP(v,h)$.\nThe training data log-likelihood for one sample is:\n\\[\n\\phi=log P(v)=\\phi^++\\phi^- ; \\ \\ \\phi^+=log\\sum_he^{-E(v,h)} ; \\ \\ \\ \\phi^-=logZ=\\sum_{x,y}e^{-E(x,y)}\n\\]\nThe gradient of the log-likelihood involves a positive and a negative term. The positive gradient is $\\frac{\\partial\\phi^+}{\\partial w_{ij}}=v_i \\cdot P(h_j=1|v)$ but the negative gradient $\\frac{\\partial\\phi^-}{\\partial w_{ij}}$ is intractable and requires summation of all values of hidden and visible variables which grows exponentially. A common method is to approximate the expectation in the second term by generating samples. Contrastive Divergence (CD) algorithm \\cite{Hinton2010} runs the Markov chain for a few steps after clamping the visible layer to data examples. Another technique is to use persistent chains without clamping the visible layer \\cite{persistent}. In our experiments, we use the CD algorithm with two steps of alternating Gibbs sampling for learning the RBM model. \n\n\\section{Associative Adversarial Networks (AAN)}\n\\label{aanm}\nThe discriminator model \\textit{D} tries to discriminate between the real and fake data. In doing so, \\textit{D} learns features that can explain factors of variation in data and uses these features to achieve its classification goal. On the other hand \\textit{G} tries to map a low-dimensional input to data. Contrary to common approaches, instead of representing the \\textit{G}'s input as a flat space, we think of it as an intermediate but higher-level representation corresponding to one of the intermediate layers of \\textit{D}. We use an RBM to learn a distribution in this space. The samples generated via contrastive divergence for updating the RBM model is used an input to \\textit{G}. Therefore we connect \\textit{D} and \\textit{G} through a high-level feature space.\n\nLet $F(x)$ denote an intermediate layer activations of \\textit{D}, while $C(y)$ denotes the operations in the remaining layers in \\textit{D}. Then, $D(x)=C(F(x))$. The associative memory learns the distribution $p_f$ of $f=F(x)$. Similar to the regular GANs, AAN optimization becomes:\n\n\\begin{equation}\n\\min_{G} \\max_{\\hat{p}_{f}} \\max_{D}=\\mathbb{E}_{x\\sim p_{data}(x)}[\\log D(x)]+\\mathbb{E}_{f\\sim \\hat{p}_{f}(f)}[\\log (1 - D(G(f)))] + \\mathbb{E}_{f\\sim p_{f}(f)}[\\log \\hat{p}_{f}]\\label{eq:cost}\n\\end{equation}\nThe second expectation is over the estimated $\\hat{p}_{f}$ since G is inputted with samples from the associative memory. Minimizing the third expectation forces the associative memory to estimate the probability distribution function of $f$.\n\n\\section{Details of Associative Adversarial Training}\nWe trained the AANs on Large-scale CelebFaces Attributes (CelebA) and the MNIST dataset. Images in the dataset are linearly scaled to the [-1, 1] range. GAN models are trained with mini-batch sizes of 256. Similar to \\cite{RadfordMC15}, we used an Adam optimizer. For the RBM learning, we used the same mini-batch size with a learning rate of 0.001 and stochastic gradient descent (SGD) with momentum. We picked a momentum rate of 0.8. A contrastive divergence with two steps is used to create the negative samples. We used leaky rectified linear activations (LeakyReLU) \\cite{Maas2013} for \\textit{D} with one exception to be discussed below and rectified linear activation \\cite{Hinton2010relu} for \\textit{G} as activation functions. Using an intermediate layer of \\textit{D} with LeakyReLU activation as a visible layer to the RBM requires it to be a Gaussian RBM which does not work well in practice. Hence, we used \\textit{tanh} activation for the layer that connects as a visible layer to a binary RBM. We chose binary variable states as (-1) and (+1) similar to spin states in an Ising model \\cite{ising}. The expected value of a unit in the RBM is $p*1 + (1-p)*-1=2p-1$ where $p$ is the conditional probability of a variable to be (1), which is a sigmoid. Thus, the expected value $2p-1$ becomes a \\textit{tanh} non-linearity. The negative samples created for RBM's contrastive divergence learning are used as inputs for the \\textit{G}.\n\n\\begin{figure}[t]\n\\subfloat[$1000\\times1000$ RBM for modeling $p_f$]{\\label{fig:conva}\\includegraphics[clip,width=0.50\\columnwidth]{convergence_1000_1000.png}}\n\\subfloat[$100\\times100$ RBM for modeling $p_f$]{\\label{fig:convb}\\includegraphics[clip,width=0.50\\columnwidth]{convergence_100_100.png}}\n\\caption{\\label{fig:conv} AAN achieves a better adversarial training performance compared to feeding noise to G, since G more closely follows D in terms of maximizing $\\log D()$ on real\/generated images.}\n\\end{figure}\n\n\\begin{figure}[thp]\n\\subfloat[Using a $1000\\times1000$ RBM]{\\label{fig:a}\\includegraphics[clip,width=\\columnwidth]{gibbs_faces.png}}\\\\\n\\subfloat[Using a $100\\times100$ RBM]{\\label{fig:b}\\includegraphics[clip,width=\\columnwidth]{gibs_faces_10x10.png}}\n\\caption{\\label{fig:faces} Gibbs steps on faces - Face images generated by the model by running RBM's Markov chain with varying number of Gibbs sampling steps. Original images are in the leftmost column, while the other images are generated by increasing the number of Gibbs steps from one to ten with the last column corresponding to images generated by ten Gibbs steps. Note that images generated are not reconstructed versions of the original images since the discriminator and the generator networks are not designed to satisfy such a constraint. Images generated using a 1000-dimensional RBM change slowly with increasing number of Gibbs steps; facial features, expression and pose change smoothly. However, using a 100-dimensional RBM, generated images do not look correlated, implying that the Markov chain jumps from one mode of the distribution to another faster compared to a 1000-dimensional distribution. }\n\n\\end{figure}\n\\subsection{CelebA}\nDiscriminator for this dataset has four strided convolutional layers with depth 64, 128, 256, 512 as the first four layers. Stride size is set to match the filter width and height both of which is five. The last convolutional layer's outputs are reshaped into a one-dimensional representation $f$ which is fully connected to a layer that has the same dimension as the RBM's visible layer. After this layer there are two more layers of size 500. The two final layers can be thought as hidden layers of a classification network $C$ as defined in Section~\\ref{aanm}, which has $f$ as input so that their cascaded application composes D (\\textit{i.e.} $D(x)=C(F(x))$). $F$ defines a mapping to a feature space, and the RBM model learns the distribution of data set samples in this space.\n\nFigure \\ref{fig:faces} shows some face images generated using negative samples from the RBM generated by increasing the number of Gibbs steps from one to ten. RBM's Markov chain is initialized with $f$ created by real images from the data set which are given in the leftmost column. Figure \\ref{fig:a} uses an RBM with 1000-dimensional visible and hidden layer units and Figure \\ref{fig:b} uses an RBM with 100-dimensional visible and hidden layer units. With a $1000 \\times 1000$ RBM, the generated images using the Gibbs samples change slowly; as more Gibbs steps are performed facial features, expressions change gradually. For example a pose or gender change does not happen quickly. This shows it requires many steps for the Gibbs sample to jump to other modes of the distribution. However, using a $100 \\times 100$ RBM one sampling step is sufficient to change gender, race etc. as can be seen in Figure \\ref{fig:b}. This shows that $f$ features are distributed more uniformly in the space, helping the Gibbs sample jump from one mode of the distribution to another. Compared to a 1000-D $f$, a 100-D $f$ has to be more efficient in packing the information to pass on to $C$ for the correct classification, so $F$ ends up learning a mapping that more uniformly distributes mapped dataset samples in this space.\n\n\n\\subsection{Convergence Analysis of the Generator and the Discriminator}\n\\textit{G} implicitly defines a probability distribution $p_G$ as the distribution of samples $G(f)$ it generates when $f \\sim p_f$. The goal is to obtain a $p_G$ that is a good approximation to $p_{data}$. The minimax game that underlines the adversarial learning has a global optimum which is $p_G=p_{data}$ and $\\log D(x)=1\/2$. Therefore the value function given in \\ref{eq:cost} is $\\log 4$ at the optimal solution. As discussed in the introduction section, one of the difficulties with adversarial training is that \\textit{D} improves its cost significantly faster than \\textit{G}, and \\textit{G} lags behind so much so that it can not receive good gradients back-propagating from \\textit{D} and loses the game. For a good adversarial training, one would expect \\textit{G} to lag behind at an acceptable level so that \\textit{D} can guide \\textit{G}. \n\nTo analyze the convergence of adversarial models, we monitored $\\mathbb{E}_{x\\sim p_{data}(x)}[\\log D(x)]$ and $\\mathbb{E}_{f\\sim \\hat{p}_{f}(f)}[\\log D(G(f))]$ during training. The first expectation measures the objective associated with real images and the second with generated images. If \\textit{D} and \\textit{G} both converge to the global optimal solution than both expectations should converge to $log 1\/2$. However, in practice the first expectation becomes extremely close to zero while the second expectation diverges from zero. We analyzed the ratio $\\mathbb{E}_{x\\sim p_{data}(x)}[\\log D(x)] \/ \\mathbb{E}_{f\\sim \\hat{p}_{f}(f)}[\\log D(G(f))]$ for AAN and GAN where noise $z$ is sampled uniformly to feed \\textit{G}. Learning to map 1000-dimensional $f$ or $z$ to an image is more difficult as there is more learning associated with each dimension of \\textit{G}'s input. Hence \\textit{G} lags behind significantly. However, using a 100-dimensional input, \\textit{G}'s learning task is less difficult. This can be seen in Figure \\ref{fig:conv}. Similarly, AAN alleviates \\textit{G}'s learning task by producing inputs from the manifold that $f$ lies in.\n\n\n\\section{Conclusion and Future Work}\nWe proposed associate adversarial networks for improving the training of generative adversarial networks (GAN). GANs are a promising class of generative models. However, previous works have reported several issues pertaining to its stability during training. In this work, we argue that inputting noise to the generator makes \\textit{G}'s learning task difficult. Instead, we propose using an additional network, the RBM associative memory network, that connects the two networks of the GANs, the discriminator and generator. The associative memory networks can learn a probabilistic model using a higher level representation discovered by the discriminator and can be sampled to produce inputs for the generator network.\n\nAlthough we empirically tested the efficacy of the proposed associative adversarial networks, we hope to develop a more rigorous theoretical understanding in future work. Inspecting equation (\\ref{eq:cost}), one can see that it is possible for the associative memory to collapse to a degenerate probability distribution, similar to \\textit{G}'s collapsing when it lags behind \\textit{D} significantly. As a future work, we are planning to study entropy-maximizing regularizers for the associative memory.\n\nAnother future work would be to study probabilistic objectives for the generator, which can be obtained from the associative network. This is in agreement with the idea proposed in \\cite{Salimans2016}, which suggest changing \\textit{G}'s objective to match an intermediate discriminator layer's statistics.\n\n\\small\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nNetworked systems form the backbone of critical infrastructures such as power grids, transportation systems, home automation systems. Other examples of interconnected networked systems include robotic systems and autonomous aerial systems. These are interconnected by design separated geographically and functionally. Distributed coordination of multiple robotic vehicles, including unmanned aerial vehicles, unmanned ground vehicles, and unmanned underwater vehicles, has been a very active research subject by the systems and control community \\cite{ren2005survey,cao2012overview}. The distributed control of multiple agents can offer more flexibility, scalability, adaptivity, and more robustness to individual failure. Additionally, they find applications in distributed networks including smart grids \\cite{dorfler2013synchronization}, autonomous under water vehicles, unmanned aerial vehicles, etc.\n\nConsensus problems normally deal with cooperative systems. In these types of cooperative systems, the consensus is achieved through the communication of agents which are characterized by edge weights that are non-negative. There can be situations wherein the agents in a system need to repel each other or maintain a certain formation in which we will be forced to use a negative weight for interaction. Therefore, in short, non-negative weights corresponds to a system with cooperative interactions and negative weights implies antagonistic interactions. The formation of several clusters is generally referred to as a multi-partite consensus. If the number of clusters in a multi-partite consensus is two, then they are referred to as bipartite consensus where the agent forms two different clusters and they converge to a point which is different only in the sign of the value \\cite{altafini2012consensus, altafini2012dynamics}. We consider the problem of a set of agents trying to achieve bipartite consensus in the presence of an adversary.  \n\nThe networked multi-agent system can suffer from malicious interference which can disrupt the performance of the system. The robust performance of a networked multi-agent system requires physical attacks and cyber-attacks to be detected and countermeasures to be taken. Several attacks on networked control systems have been reported in the near past. These attacks can be classified as denial-of-service attacks, replay attacks, and fault data injection attacks \\cite{teixeira2015secure}. The authors in \\cite{dibaji2019systems} overview the existence of several categories of attacks on power and transportation fields on a control-theoretic framework. The study was conducted on the basis of prevention, resilience, and detection $\\&$ isolation of the different types of attacks. Authors in \\cite{amin2009safe} studied the problem of security on a linear dynamical system that is prone to DoS attacks and design a controller that minimizes the effect of this attack. A more comprehensive attack and system model considering fault detector mechanism and model uncertainties were proposed by \\cite{teixeira2015secure}. The work by Pasqualetti et. al. \\cite{pasqualetti2015control} explored the challenges of detection and identification of attack, using a descriptor system model. \n\nPreliminary work in the area of linear consensus of multi-agent systems under DoS attack was considered in \\cite{khanafer2012consensus}. They model the problem with cooperative interactions among the agents and finds the optimal strategy to slow down the consensus. The extension of the work was made in \\cite{khanafer2013robust} by introducing the effects caused by the presence of a defender. This system was also modeled based on Pontryagin's Maximum Principle and discussed the existence of saddle point equilibrium. In \\cite{khanafer2013robust} the authors considered only the cooperative interactions among the agents. \n\nTo the best of our knowledge, the performance of a network with cooperative and antagonistic interactions in the presence of an adversary remains unexplored. Here, we consider a set of agents that are interconnected by a signed graph that is structurally balanced. The set of agents are trying to achieve bipartite consensus. This system is disrupted by an external adversary which can cut links between agents. There is a constraint on the maximum number of links that can be broken at a certain time instant. The adversary has the ability to break both positive and negative links. The problem is formulated as an optimal problem and the optimal attack strategy for the adversary for slowing down the bipartite consensus is derived. The results are illustrated by numerical simulations. \n\nThe rest of the paper is organized as follows, section \\ref{sec:bipartite} talks about the basic preliminaries and section \\ref{problemform} is about the problem formulation used in this article. Section \\ref{sec:simulation} presents the simulation results with a numerical example and the article is concluded by section \\ref{sec:conclusion} which also provides the future scope of the work.\n\n\\section{BIPARTITE CONSENSUS} \\label{sec:bipartite}\n\nWhen a group of agents interacts over a signed graph, they can form a bipartite consensus where values of agents converge to two clusters. In this section we summarize some fundamental results on bipartite consensus from works \\cite{altafini2012consensus} and \\cite{altafini2012dynamics}.\n\nConsider a signed graph $G=\\left(V, E, A \\right)$, where $V=\\left\\{v_1, v_2, \\ldots, v_n\\right\\}$ is the set of nodes or agents, $E$ is the set of edges and $A$ is the adjacency matrix that contains or represents the interconnection between the agents. The non-diagonal elements in the matrix $A$ represent the weights between the agents and the diagonal entries in the matrix are considered to be zero. The Laplacian matrix for the system can be defined as $L^G=C-A$, where $C$ is the connectivity matrix which will be a diagonal matrix with the values as the sum of the absolute value of the adjacent links connected to that particular node. Therefore, the Laplacian matrix $L^G$ can be represented as\n\\begin{eqnarray}\n    l^{G}_{ik} = \\left\\{\n    \\begin{array}{ll}\n        \\sum_{j \\rightsquigarrow i}\\lvert{a_{ij}}\\rvert  & k\\ =\\ i \\\\ \n        -a_{ij} & k\\ \\neq\\ i.\\\\\n    \\end{array}\n    \\right. \\label{eq:Laplacian}\n\\end{eqnarray}\n\nThe dynamics of $n$ agents $x_i\\in \\mathbb{R}$ connected by the graph $G$ is given by,\n\\begin{align}\n    \\dot{x}=-L^{G}x(t)\n    \\label{MathDynamics}\n\\end{align}\nwhere $x =[x_1, \\cdots , x_n]^\\top$ and $L_G$ is calculated as in \\eqref{eq:Laplacian}. The idea of structurally balanced graph is defined below. \n\n\\begin{definition}\n\\label{strbalanced} \\cite{altafini2012consensus}\nA graph $G$ is structurally balanced if and only if \n\\begin{enumerate}\n    \\item All cycles of $G$ are positive\n    \\item $0$ is an eigenvalue of $L^G$ which is the Laplacian matrix of the graph $G$ as defined in \\eqref{eq:Laplacian}.\n\\end{enumerate}\n\\end{definition}\n\nA structurally balanced graph that satisfies the properties as in definition \\ref{strbalanced} corresponds to the partitioning of a signed graph into two sub-communities in such a manner all the edges within a sub-community have positive weights and all the edges connecting these sub-communities with other sub-communities has a negative weight. \n\n\\begin{lemma}\nThe system \\eqref{MathDynamics} admits a bipartite consensus solution if and only if the graph $G$ is structurally balanced \\cite{altafini2012consensus}.\n\nHere, bipartite consensus implies $\\lim_{t \\to \\infty} x_i = | \\beta | $ where $\\beta \\neq 0$, for all $i= 1, \\cdots, n$.\n\\label{lemma structurally balanced}\n\\end{lemma}\n\\begin{proof}\nThe proof is provided in \\cite{altafini2012consensus} and is omitted here for brevity. \n\\end{proof}\n\nConsider the example in fig \\ref{fig1}. The system has $11$ agents that form two different groups. The agents within a sub-community have positive weights thus showing cooperative interaction and the weights between the agents of sub-community A and sub-community B are negative thus showing an antagonistic interaction. As shown in \\cite{altafini2012consensus}, the agents of two sub-communities 'repel' each other, while moving towards agents of the same sub-community forming a bipartite consensus, essentially two clusters. \n\\begin{figure*}[h] \n    \\centering\n    \\includegraphics[scale=1]{bipartite.eps}\n    \\caption{Bipartite consensus of a multi-agent system; The black circles indicate the different agents, blue connections (links) represent the cooperative interactions and the red connections indicate the antagonistic interaction }\n    \\label{fig1}\n\\end{figure*}\n\nIn the next section we consider the problem of bipartite consensus under attack.\n\n\\section{PROBLEM FORMULATION}\n\\label{problemform}\nConsider a network of interconnected systems of weighted undirected graphs with $n$ agents or nodes. This can be represented using  $G=(V,E)$ where $V$ represents the vertices which are agents $\\left[v_{1}, v_{2},..., v_{n}\\right]$ and $E$ represents the edges. It is important that the network graph must be structurally balanced as in definition \\ref{strbalanced}. It is assumed that the adversary has complete knowledge of the system. \n  \nThe state of the nodes at time $t$ is represented as $x=\\left[x_{1}(t), x_{2}(t),...,x_{n}(t)\\right]^{T}$. Let the dynamics of the system be \n\\begin{align}\n    \\dot{x}=-L^{G}(t)x(t)\n    \\label{dynamics}\n\\end{align}\nwhere $L^{G}$ is defined as the difference between the connectivity matrix $C$ and the adjacency matrix $A$. The matrix $A$ contains the weights of that particular links which are any scalar values and the self loops such as $a_{11} $ or $a_{22}$, etc are not considered. These values are considered in the connectivity matrix $C$. Therefore, as a result the matrix $C$ will be a diagonal matrix with the values being the sum of the absolute values of adjacent links connected to that node, $C_{ii}=\\sum_{j \\in adj(i)}\\lvert{a_{ij}}\\rvert$. The weights of the system can be either zero, positive value or a negative value based on the following condition wherein $C_k$ and $C_l$ represents the set of nodes that belongs to the two different clusters such that $C_k \\cup C_l = V$ and $C_k\\ \\cap\\ C_l\\ =\\ \\emptyset$. $a_{ij}$ follows the standard rules if $(i,j) \\in E$ and if $(i,j)$ belong to the same cluster, then $a_{ij}>0$ and if $(i,j)$ belongs to different cluster, then $a_{ij}<0$. $a_{ij}=0$ if $(i,j) \\notin E$. From Definition \\ref{strbalanced} when the graph is structurally balanced, the agents of the same cluster will have positive connections and the connection between the inter-cluster agents will be negative. From Lemma \\ref{lemma structurally balanced}, this will result in agents achieving bipartite consensus, where each cluster of agents converging to values differing only in sign. Let us consider a structurally balanced network graph with the system dynamics as \n\\begin{equation}\n    \\dot{x}=-L^{G}(t)x(t)\n\\end{equation}\nwhere $L^{G}=C-A$.\n\\begin{eqnarray}\n    l^{G}_{ik} = \\left\\{\n    \\begin{array}{ll}\n        \\sum_{j \\rightsquigarrow i}\\lvert{a_{ij}}\\rvert  & k\\ =\\ i \\\\ \n        -a_{ij} & k\\ \\neq\\ i.\\\\\n    \\end{array}\n    \\right.\n\\end{eqnarray}\nwhere $j \\rightsquigarrow i$ implies $j$ is connected to $i$. \n\n$L^{G}$ is a matrix that represents the system before the attack. The adversary aims to slow down the consensus by breaking the links that connect different agents. Let $\\alpha$ be the maximum number of links that the adversary can disrupt at each time instant. The control is given by $u(t)=[u_{12}(t), u_{23}(t),...,u_{(n-1)n}(t)]^{T}$, where $u_{ij}(t) \\in \\{0, 1\\}$. When $u_{ij}(t) =1$, the link $(i,j)$ is broken at time $t$. The weight of each link $(i, j)$ can be redefined as $a^{a}_{ij}=a_{ij}(1-u_{ij})$ and the graph under attack can be modelled as \n\\begin{eqnarray}\n    l^{Ga}_{ik} = \\left\\{\n    \\begin{array}{ll}\n        \\sum_{j \\rightsquigarrow i}\\lvert{a_{ij}(1-u_{ij})}\\rvert  & k\\ =\\ i \\\\ \n        -a_{ij}(1-u_{ij})& k\\ \\neq\\ i. \\\\\n    \\end{array}\n    \\right.\n\\end{eqnarray}\nThe objective function of the adversary can be defined as\n\\begin{equation}\n    J(u)=\\int ^{T}_{0}k(t)\\ x(t)^{T}M(t)\\ x(t)\\ dt,\n\\end{equation}\nwhere $k(t)$ is the positive kernel integrable over $[0,T]$ and $M$ is the connection matrix which indicates all the connections between the sub-communities of the bipartite network graph. \n\\begin{align}\n    M_{ij}=\\left\\{ \n    \\begin{array}{cll}\n    -\\sgn(a_{ij}) & ; & i \\ \\neq \\ j\\\\\n    n-1 & ; & i \\ = \\ j  \\\\ \n    \\end{array}\\right.\n\\end{align}\nThe problem can be formulated in a optimal control framework as follows\n\\begin{align}\n    & \\underset{u(t) \\in U}{\\max}\\ J(u) \\nonumber \\\\\n    s.t.\\ \\ \\ & \\dot{x}(t)\\ =\\ -L^{G}(t)x(t). \n\\end{align}\nThe Hamiltonian is defined as\n\\begin{eqnarray}\n    H(x, p, u) = k(t)x(t)^{T}Mx(t)-p(t)^{T}L^{G}(t)x(t) \\label{Hamiltonian}\n\\end{eqnarray}\nConsidering the first order necessary conditions for optimality \\cite{naidu2002optimal} can be formulated wherein the state and co-state dynamics becomes \n\\begin{align}\n    \\frac{\\partial H}{\\partial p}  & = \\dot{x}(t) = -L_{G}(t)x(t) \\label{eq:opti1} \\\\\n    \\frac{-\\partial H}{\\partial x} & = \\dot{p}(t) = -k(t)Mx(t)+L_{G}(t)^{T}p(t) \\label{eq: opti2}\\\\\n    \\frac{\\partial H}{\\partial u} & = 0 \\label{eqq:opti3}\n\\end{align}\nThe optimal control can be obtained as \n\\begin{align}\nu^{*} & = \\underset{u \\in U}{\\arg \\max}\\ H(x,p,u) \\label{opti:4}\n\\end{align}\nTo solve for $u^{*}$, we need to maximize the $H$. Solving for $p^{T}L^{G}x$ and substituting in (\\ref{Hamiltonian})\n\\begin{eqnarray}\n    p^{T}L^{G}x & = & \\sum_{j=2}^{n}\\sum_{i=1}^{n-1}\\lvert{a_{ij}}\\rvert(1-u_{ij})f_{ij}\n\\end{eqnarray}\n where $f_{ij}=\\big(p_{i}-\\sgn(a_{ij})p_{j}\\big)\\big(x_{i}-x_{j}\\big)$.\n\\begin{equation}\n     \\max H(x, p, u)=k(t)x(t)^{T}Mx(t) - \\sum_{j=2}^{n}\\sum_{i=1}^{n-1}(1-u_{ij})f_{ij}\n\\end{equation}\n\\begin{eqnarray}\n    u^{*}_{ij} = \\left\\{\n    \\begin{array}{cll}\n        \\{0, 1\\} & ; & \\text{if}\\ f_{ij} = 0 \\\\\n        1 & ; & \\text{if}\\ (i,j)\\ \\in I_{t} \\\\\n        0 & ; & \\text{if}\\ f_{ij}>0.\n    \\end{array}\n    \\right.\n\\end{eqnarray}\n where $I_{t}$ is the ordered set of values of $f_{ij}$ that are less than zero and can be attacked, that is, \n \\begin{align}\n     I_{t} & = \\big\\{ (i,j) \\in V : f_{ij}<0,  \\text{ and } f_{ij}<f_{\\alpha + 1} \\big\\}\n \\end{align}\n \nThe function $f_{ij}$ depends on both the state $x(t)$ and co-state $p(t)$ variables. The objective of the adversary is to slow down the consensus and for this he can break the links that connect these agents with each other. Therefore, we need to find an optimal strategy for the adversary so that the $J(u)$ increases because of the disruption caused by him. We need to look at the cost function of the adversary to determine the optimal strategy which is explained using the following theorem.\n\n\\begin{theorem}\nFor a particular time $t$, the optimal strategy of the adversary is to break the $\\alpha$ number of links with the highest $w_{ij}$ value where $w_{ij}=\\lvert{a_{ij}}\\rvert \\big(x_{j}-\\sgn(a_{ij})x_{i} \\big)^{2}.$ \n\\end{theorem}\n\\textbf{Proof.} Let us consider $\\alpha$ be the number of links that are break by the adversary at time instant $t$. The value of $f_{ij}$ cannot change in a finite interval as it solely depends on $x(t)$ and $p(t)$ which does not change in that interval which results in the control to remain unchanged. \n\nLet $[s, s+\\Delta s]$ be a sub-interval of $[0, T]$ in which the adversary applies the strategy $u^{A}$ with a system matrix as $L_{1}$. Therefore, the state trajectory is given as\n\\begin{align*}\n    x(t)=e^{-L_{1}(t-s)}x(s) \\ ;\\ \\ t \\in [s, s+\\Delta s]\n\\end{align*} \nLet $P(t)$ be $e^{-L_{1}t}$. Let $\\nu$ be the eigenvector corresponding to eigenvalue $0$ of the matrix $L_{1}$. Let $\\nu\\nu^{T}$ be $K$. Then the connection matrix $M$ can be expressed as $ (nI-K) $  where $n$ is the number of agents in the network. Let $u^{B}$ be the strategy that is similar to the $u^{A}$ except at link $(i, j)$ with a system matrix as $L_{2}$. At $ (i, j) $, $u^{A}_{ij}=0$ where as $u^{B}_{ij}=1$. Let $Q(t)$ be a doubly stochastic matrix such that  $Q(t)=e^{-L_{2}t}$,\\ $t\\geq$0. We need to show that strategy $u^{B}$ is better than $u^{A}$. The aim of the adversary is to maximize the cost function. We can obtain the conditions for which the cost for $u^{A}$ greater than the cost for $u^{B}$. \n\\begin{align}\n& \\int_{s}^{s+\\Delta s} k(t)x(s)^{T}P(t-s)MP(t-s)x(s) dt    \\nonumber \\\\\n        & < \\int_{s}^{t^{*}} k(t)x(s)^{T}P(t-s)MP(t-s)x(s) dt +\\int_{t^{*}}^{s+\\Delta s}k(t) \\nonumber \\nonumber \\\\\n        & x(s)^{T}P(t^{*}-s)Q(t-t^{*})MQ(t-t^{*})P(t^{*}-s)x(s) dt. \n\\end{align}\n\nThe cost of strategy corresponding to $u^{B}-u^{A}$ can be written as\n\\begin{equation*}\n    \\begin{array}{l}\n        \\int_{s}^{s+\\Delta s} kx^{T}[P(t^{*}-s)Q(t-t^{*})MQ(t-t^{*})P(t^{*}-s)- \\\\\n        P(t-s)MP(t-s)]x dt >0 \\\\\n    \\end{array}\n\\end{equation*}\n\\begin{equation}\n    \\int_{s}^{s+\\Delta s} kx(s)^{T}\\Lambda(t,t^{*}) x(s) dt > 0\\ ,\n\\end{equation}\n \nwhere $\\Lambda=P(t^{*}-s)Q(t-t^{*})MQ(t-t^{*})P(t^{*}-s)- P(t-s) M P(t-s)$. As $t$ decreases, $P(t)=I+tL_{1}+\\mathcal{O}(t^{2})$ and $Q(t)=I+tL_{2}+\\mathcal{O}(t^{2})$. Substituting for $P(t), Q(t)$ and $M$, the $\\Lambda(t,t^{*})$ can be simplified as,\n\\begin{equation}\n    \\Lambda(t,t^*)=-2n(t-t^*)(L_2-L_1)+\\mathcal{O}(t^2).\n    \\label{lam}\n\\end{equation}\n \nFor sufficiently small $t$ and $t^*$, the first term is very much larger than the second term, therefore it dominates. At the link $(i, j)$, $L_{1_{ij}}>L_{2_{ij}}=0$, and at all the links $L_{1_{kl}}=L_{2_{kl}}$ where $(k, l) \\neq (i, j)$.\n\\begin{equation}\n    \\begin{array}{lll}\n        h(t,x(s)) & = & 2n(t-t^{*}) \\times \\\\\n        & & \\sum_{l=1}^{n} \\sum_{k=1}^{l-1} (L_{1kl}-L_{2kl})(x_{k}-\\sgn(L_{ij})x_{l})^2  \\\\\n        & = & 2n(t-t^{*})L_{1ij}(x_{i}-\\sgn(L_{ij})x_{j})^2 \n    \\end{array}\n    \\label{eq:finalH}\n\\end{equation}\n\nIn a bipartite consensus, the adversary can either break the link with a positive weight or he can break the link with a negative weight. When the link with a positive weight is attacked, (\\ref{lam}) becomes $h(t,x(t))=2n(t-t^{*})\\lvert{a_{ij}}\\rvert(x_{i}-x_{j})^{2}$ and if the link with a negative weight is attacked, then $h(t,x(t))=2n(t-t^{*})\\lvert{a_{ij}}\\rvert(x_{i}+x_{j})^{2}$. This is possible only when the graph has multiple connected components and at least one link exists between them. If the graph at time $s$ is a single connected component, then the breaking of the link by the adversary to slow down the convergence is not possible as the system has already reached bipartite consensus. This proof provides an easy way of calculating optimal strategy and picturing the DoS attack. In the cost $J(u)$, each term corresponds to the value of $w_{ij}$. Therefore, it is inferred that the optimal strategy for the attacker to break the bipartite consensus and thus increase or maximize the cost function is by breaking the link with the highest value of $w_{ij}$. The attacker has complete knowledge of the system and that he is able to disconnect a link for the entire process. It will be interesting to see how the solution would change if there was a constraint on the amount of time an attacker can break a link.\n \n\\section{SIMULATION RESULTS} \\label{sec:simulation}\nWe consider a structurally balanced graph with $n=4$ as shown in fig \\ref{fig3}(a) for illustration. \n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=1.5]{with_wt_0.1.eps}\n    \\includegraphics[scale=0.9]{bipartitewithattack.eps}\n    \\caption{(a) Example of a four agent structurally balance graph with cooperative and antagonistic interactions; (b) The graph under attack of the adversary where the link $(3, 4)$ was attacked and therefore the weight of that particular link was changed to $0$.}\n    \\label{fig3}\n\\end{figure}\nTherefore, according to the theory discussed above, the Laplacian matrix $L_1$ of the system can be written as, \n\\begin{eqnarray*}\n    L_{1}=\\left[\\begin{array}{cccc}\n         0.2 & 0.1 & 0 & -0.1 \\\\\n         0.1 & 0.2 & -0.1 & 0 \\\\\n         0 & -0.1 & 0.2 & 0.1 \\\\\n         -0.1 & 0 & 0.1 & 0.2 \n    \\end{array}\\right]\n\\end{eqnarray*}\nThe initial conditions are chosen randomly between any interval. Here, let the initial conditions be $[-20, -5, 25, 10]^{T}$. This is the bipartite consensus where the four agents formed two different clusters. Now the adversary acts upon. In order for the adversary to give the best attack, he tries to attack that link that has the highest $w_{ij}$ value. Now, the $w_{ij}$ value is calculated for all the links. This value is found to be the highest for the link $(3, 4)$, with $w_{34}=122.5$. Therefore, the optimal strategy of the adversary to slow the bipartite consensus is to break this link at that particular time. Therefore, the $L_{1}$ matrix changes to a new matrix $L_{2}$.\n \\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.7]{bpconsolidated.eps}\n    \\includegraphics[scale=0.7]{bpcost.eps}\n    \\caption{(a) Simulation result of the graph before and after the attack by the adversary; (b) The cost of the bipartite consensus $(J)$ with and without the attack}\n    \\label{fig4}       \n\\end{figure} \nFrom fig \\ref{fig4}(a), we can infer that before the presence of an adversary the four agent network graph described in the illustration moves to bipartite consensus. These converging values are only different in sign. The speed of convergence can be adjusted by changing the weight of each link. After the attack by the adversary at the link $(3, 4)$, the bipartite consensus is slowed. From the simulation results, it is clear that the agents $1$ and $2$ still moves to the said consensus point whereas the adversary fails the bipartite consensus by breaking the link $(3, 4)$. The agents $3$ and $4$ deviates from the original trajectory. From \\ref{fig4}(b), it is clear that the cost of the bipartite consensus $J(u)$ before the presence of the attack is nearly zero but when the adversary acts upon the graph by breaking the link $(3, 4)$, the cost increases.  The cost of bipartite consensus is thus maximized, slowing down the bipartite consensus.\n \n\\section{CONCLUSION} \\label{sec:conclusion}\nWhen a set of agents interact with each other using both cooperative and antagonistic interactions, and the interaction graph is structurally balanced, the agents end up with a bipartite consensus. The effect of the denial of service attack on a bipartite consensus problem was studied here. The optimal plan of action that can be followed by an adversary at the time of the attack on a structurally balanced multi-agent system was found out. It has been demonstrated by MATLAB simulation results. The future work will explore the case of having an attacker with incomplete knowledge and the additional constraint on time. \n\n\\bibliographystyle{ieeetran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSupersymmetry (SUSY) has been the top candidate for beyond standard model (BSM) physics since many years. This has several reasons: SUSY solves the hierarchy problem of the standard model (SM), provides a dark matter candidate, leads to gauge coupling unification, and gives an explanation for electroweak symmetry breaking (EWSB). Before the LHC has turned on, the main focus has been on the minimal supersymmetric extensions of the SM, the MSSM. \nHowever, the negative results from SUSY searches at the LHC as well as the measured Higgs mass of about 125~GeV put large pressure on the simplest scenarios. Wide regions of the parameter space, which had been considered as natural before LHC has started, have been ruled out. This has caused more interest in non-minimal SUSY models. Also non-SUSY extensions of the SM became more popular again. Other BSM models can provide many advantages compared the MSSM:\n\\begin{itemize}\n \\item {\\bf Naturalness}: about one third of the Higgs mass has to be generated radiatively in the MSSM to explain the observation. Heavy SUSY masses are needed to gain sufficiently large loop corrections, i.e. a soft version of the hierarchy problem appears again. The need for large loop corrections gets significantly\n softened if $F$- or $D$-terms are present which already give a push to the tree-level mass.\n \\item {\\bf SUSY searches}: the negative results from all SUSY searches at the LHC have put impressive limits on the Sparticle masses. \n However, the different searches are based on certain assumptions. As soon as these conditions are no longer given like in models with broken $R$-parity, the limits become much weaker. \n \\item {\\bf Neutrino data}:  neutrino masses are not incorporated in the MSSM. To do that, either one of the different seesaw mechanisms can be utilised or $R$-parity \n must be broken to allow a neutrino--neutralino mixing.\n \\item {\\bf Strong CP-problem}: the strong CP problems remains not only an open question in the SM but also in the MSSM. In principle, for both models the same solution exists to explain the smallness of the $\\Theta$ term in QCD: the presence of a broken Peccei-Quinn symmetry.\n \\item {\\bf $\\mu$-problem}: the superpotential of the MSSM involves one parameter with dimension mass: the $\\mu$-term. This term is not protected by any symmetry, i.e. the natural values would be either exactly 0 or $O(M_{GUT})$.  However, for phenomenological reasons it should be similar to the electroweak scale.  This could be obtained if the $\\mu$-term is actually not a fundamental parameter but generated dynamically like in singlet extensions. .\n \\item {\\bf UV completion}: it is not clear that only the gauge sector and particle content of the MSSM is the low energy limit of a GUT or String theory. Realistic UV completions predict often many additional matter at the TeV scale. In many cases also additional neutral and even charged gauge bosons are present.\n \\item {\\bf $R$-symmetry}: if one considers $R$-symmetric models, Majorana  masses for gauginos are forbidden. To give masses to the gauginos in these models, a coupling to a chiral superfield in the adjoint representation is needed. This gives rise to Dirac masses for the gauginos which are in agreement with $R$-symmetry. \n Dirac gauginos are also attractive because they can weaken LHC search bounds, and flavour constraints. \n\\end{itemize}\n\nDespite the large variety and flexibility of SUSY, many dedicated public computer tools like {\\tt SoftSUSY}, {\\tt SPheno}, {\\tt Suspect}, {\\tt Isajet}  or {\\tt FeynHiggs} are restricted to the simplest realization of SUSY, the MSSM, or small extensions of it. Therefore, more generic tools are needed to provide the possibility to study many other models with the same precision as the MSSM. This precision is needed to confront also these models with the strong limits from collider searches, flavour observables, dark matter observations, and Higgs measurements.\nSuch a tool is the \\Mathematica package \\SARAH. I'll given in the next section an introduction to \\SARAH, before I explain the interface to \\SPheno  in sec.~\\ref{sec:SPheno}, to Monte-Carlo (MC) tools in sec.~\\ref{sec:MC}, and to other tools in sec.~\\ref{sec:tools}. I summarize in sec.~\\ref{sec:summary}. Interested readers find many more details about the presented framework in Ref.~\\cite{Staub:2015kfa}.\n\n\n\\section{BSM models and \\SARAH}\n\\label{sec:sarah}\n\\subsection{Model definition}\n\\subsubsection{Input needed by \\SARAH to define a model}\n\\SARAH \\cite{Staub:2008uz,Staub:2009bi,Staub:2010jh,Staub:2012pb,Staub:2013tta} is optimized for the handling of a wide range of BSM models. The basic aim of \\SARAH is to give the user the possibility to implement models in an easy, compact and straightforward way. Most tasks to get the Lagrangian are fully  automatized; it is sufficient to define just the fundamental properties of the model:\n\\begin{enumerate}\n \\item Global symmetries\n \\item Gauge symmetries\n \\item Chiral superfields\n \\item (Super)potential\n \\item Field rotations\n\\end{enumerate}\nThat means that \\SARAH automatizes many steps to derive the Lagrangian from that input:\n\\begin{enumerate}\n \\item All interactions of matter fermions and the $F$-terms are derived from the (super)potential\n \\item All vector boson and gaugino interactions as well as $D$-terms are derived from gauge invariance\n \\item All gauge fixing terms are derived by demanding that scalar--vector mixing vanishes in the kinetic terms\n \\item All ghost interactions are derived from the gauge fixing terms\n \\item All soft-breaking masses for scalars and gauginos as well as the soft-breaking counterparts to the superpotential couplings are added automatically  for SUSY models\n \\item All rotations from the gauge to the mass eigenstates are performed\n\\end{enumerate} \nThe range of models which can be handled by \\SARAH is very broad: \n\\begin{itemize}\n\\item {\\bf Global symmetries}: \\SARAH can handle an arbitrary number of global symmetries which are either $Z_N$ or $U(1)$ symmetries. Also a continuous $R$-symmetry $U(1)_R$ is possible.  \n\\item {\\bf Gauge sector}: \\SARAH is not restricted to the SM gauge sector, but many more gauge groups can be defined. To improve the power in dealing with gauge groups, \\SARAH has linked routines from the \\Mathematica package \\Susyno \\cite{Fonseca:2011sy}. \\SARAH together with \\Susyno take care of all group-theoretical calculations: the Dynkin and Casimir invariants are calculated, and the needed representation matrices as well as Clebsch-Gordan coefficients are derived. Also gauge kinetic mixing is fully supported for an arbitrary number of Abelian gauge groups. \n\\item {\\bf Matter sector}: there is no practical restriction how many matter states --either defined as chiral superfields in SUSY models, are as component fields in non-SUSY models-- are present. All states can come with an arbitrary number of generations and can transform as any irreducible representation with respect to the defined gauge groups.\n\\item {\\bf (Super)potential}: the main restriction is that only renormalizable terms are supported in the (super)potential. However, there is no restriction how many terms are present.\n\\end{itemize}\nAs example how short a model implementation in \\SARAH is, I show in Fig.~\\ref{fig:MSSM} the model file for the MSSM. \n\n\\begin{figure}[hbt]\n\\begin{center}\n \\includegraphics[width=0.9\\linewidth]{MSSMmodelfile}\n\\end{center}\n\\caption{The model file to implement the MSSM in \\SARAH}\n\\label{fig:MSSM}\n\\end{figure}\n\n\n\\subsubsection{Checks of implemented models}\n\\label{sec:checks}\n\\SARAH provides functions to check the (self-) consistency of a model addressing the following questions:\n\\begin{itemize}\n \\item Causes the particle content gauge anomalies?\n \\item Leads the particle content to the Witten anomalies?\n \\item Are all terms in the (super)potential in agreement with global and local symmetries?\n \\item Are there other terms allowed in the (super)potential by global and local symmetries?\n \\item Are all unbroken gauge groups respected?\n \\item Are there terms in the Lagrangian of the mass eigenstates which can cause additional mixing between fields?\n \\item Are all mass matrices irreducible?\n \\item Are the properties of all particles and parameters defined correctly?\n\\end{itemize}\n\n\n\\subsection{Calculations and output}\n\\label{sec:analytical}\n\\SARAH can perform in its natural \\Mathematica environment many calculations for a model on the analytical level. For an exhaustive numerical analysis usually one of the dedicated interfaces  to other tools discussed in the next sections is the best approach. I start with some details about the analytical calculations which \\SARAH can perform. \n\n\\subsubsection{Tree-level properties}\n\\paragraph{Tadpole equations} During the evaluation of a model, \\SARAH calculates 'on the fly' all minimum conditions of the tree-level potential, the so called tadpole equations.\n\n\\paragraph{Masses} \\SARAH calculates for all states which are rotated to mass eigenstates the mass matrices during the evaluation of a model. In addition, it calculates also the masses for the states for which no field rotation has taken place. \n\n\\paragraph{Vertices} \\SARAH has functions to extract in an efficient way all tree-level vertices from the Lagrangian. These vertices are saved in different \\Mathematica arrays according to their generic type. \n\n\\subsubsection{Renormalization group equations}\n\\label{sec:RGEs}\n\\SARAH calculates for SUSY and non-SUSY models the full two-loop RGEs including the full CP and flavour structure. I give here a short summary of the underlying, generic results which are used in these calculations.  \n\\paragraph*{SUSY RGEs} The calculation of the SUSY RGEs is mainly based on Ref.~\\cite{Martin:1993zk}. However, this work does not cover all possible subtleties which can appear in SUSY models. Therefore, \\SARAH has implemented also some more results from literature which became available in the last few years: \n\\begin{itemize}\n \\item In the case of several $U(1)$'s, gauge-kinetic mixing can arise if the groups are not orthogonal. Substitution rules to translate the results of Ref.~\\cite{Martin:1993zk} to those including gauge kinetic mixing where presented in Ref.~\\cite{Fonseca:2011vn} and are used by \\SARAH \n \\item The calculation of the RGEs in the presence of Dirac gaugino is based on Ref.~\\cite{Goodsell:2012fm}\n \\item The results of Refs.~\\cite{Sperling:2013eva,Sperling:2013xqa} are used to get the gauge dependence in the running of the VEVs.\n\\end{itemize}\n\n\\paragraph*{Non-SUSY RGEs} \\SARAH uses the expressions of Refs.~\\cite{Machacek:1983tz,Machacek:1983fi,Machacek:1984zw,Luo:2002ti} for the calculation of the RGEs in a general quantum field theory. These results are completed by Ref.~\\cite{Fonseca:2013bua} to cover gauge kinetic mixing and again by Refs.~\\cite{Sperling:2013eva,Sperling:2013xqa} to include the gauge-dependence of the running VEVs also in the non-SUSY case.\n\n\\subsubsection{One- and two-loop corrections to tadpoles and self-energies}\n\\label{sec:loopcorrections}\n\\paragraph{One-loop corrections} \\SARAH calculates the analytical expressions for the one-loop corrections to the tadpoles and the one-loop self-energies for all particles. For states which are a mixture of several gauge eigenstates, the self-energy matrices are calculated. The calculations are performed in ${\\ensuremath{\\overline{\\mathrm{DR}}}}$-scheme using 't Hooft gauge. In the case of non-SUSY models \\SARAH switches to ${\\ensuremath{\\overline{\\mathrm{MS}}}}$-scheme. All generic diagrams which are covered at one-loop are depicted in Fig.~\\ref{fig:1loopDiagrams}. \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{1-loop} \n\\caption{Generic diagrams included by \\SARAH to calculate one-loop tadpoles and self-energies.}\n\\label{fig:1loopDiagrams}\n\\end{figure}\n\n\\paragraph{Two-loop corrections} It is even possible to go beyond one-loop with \\SARAH and to calculate two-loop contributions to the self-energies of real scalars. There are two equivalent approaches implemented in the \\SPheno interface of \\SARAH to perform these calculations: an effective potential approach, and a diagrammatic approach with vanishing external momenta. More details about these calculations are given in sec.~\\ref{sec:TwoLoop}.\n\n\n\\subsection{Output}\nThe information derived by \\SARAH can be outputted in different ways\n\\begin{itemize}\n \\item Model files for Monte-Carlo tools like \\CalcHep, \\MG, or \\WHIZARD\n \\item Model files for other tools like \\FeynArts and \\Vevacious \n \\item Fortran code to implement a model in \\SPheno \\footnote{A similar interface to generate a spectrum generator written in {\\tt C+} became available via \\FlexibleSUSY \\cite{Athron:2014yba}.}. \n \\item \\LaTeX\\ files \n\\end{itemize}\nA few more details about the different outputs are given in the next three sections. \n\n\\section{Linking \\SPheno and \\SARAH}\n\\label{sec:SPheno}\n\\SARAHv{3} has been the first `spectrum--generator--generator': SARAH writes Fortran source code\nfor \\SPheno using the derived information about the \nmass matrices, tadpole equations, vertices, loop corrections and RGEs for the given model. \nBased on this code the user gets a fully functional spectrum generator for a model. \nThe features of a spectrum generator created in this way are \n\\begin{itemize}\n\\item Full {two-loop running} of all parameters and {all masses at one-loop}\n\\item {two-loop} corrections to {Higgs} masses\n\\item Complete {one-loop thresholds} at $M_Z$\n\\item calculation of {flavour} and {precision observables} at full 1-loop\n\\item calculation of {decay widths} and {branching ratios}\n\\item interface to {\\tt HiggsBounds} and {\\tt HiggsSignals}\n\\item estimate of electroweak {Fine-Tuning}\n\\end{itemize}\nI give a few more details about the different calculations. \n\n\\subsection{Mass calculation with \\SPheno}\n\\subsubsection{Threshold corrections}\nFor a precise calculation of the masses, it is necessary to have an accurate input for all parameters which enter the calculation. In general, the running SM parameters depend on the masses of BSM states. The reason are the thresholds to match the running parameters to the measured ones. The routines generated by \\SARAH perform a full one-loop matching in the given model to calculate the SM gauge and Yukawa couplings. \n\n\\subsubsection{One-loop masses}\nThe one-loop mass spectrum is calculated from the information about the one-loop self-energies and tadpole equations. The procedure is a generalisation of the one explained in detail for the MSSM in Ref.~\\cite{Pierce:1996zz}. The main features are \n\\begin{itemize}\n \\item Any one-loop contribution in a given model to all fermions, scalars and vector bosons is included \n \\item The full $p^2$ dependence in the loop integrals is included\n \\item An iterative procedure is applied to find the on-shell masses $m(p^2=m^2)$. \n\\end{itemize}\n\n\\subsubsection{Two-loop Higgs masses}\n\\label{sec:TwoLoop}\n\\SARAH can also generate Fortran code to calculate the two-loop corrections to real scalars with \\SPheno. The same approximation as usual done for the MSSM are also applied here: (i) all calculations are performed in the gaugeless limit, i.e. the electroweak contributions are dropped, and  (ii) the momentum dependence is neglected. Using these routines,the theoretical uncertainty in the Higgs mass prediction for many models has shrunken now to the level of the MSSM. In general, there are two different techniques to calculate the two-loop corrections with \\SARAH\\SPheno:\n\\begin{itemize}\n \\item {\\bf Effective potential calculation}: \\SARAH makes use of the generic two-loop results for the effective potential given in Ref.~\\cite{Martin:2001vx}. To get the values for the two-loop self-energies and two-loop tadpoles, the derivatives of the potential with respect to the VEVs are taken numerically as proposed in Ref.~\\cite{Martin:2002wn}. There are two possibilities for this derivation implemented in \\SARAH\/\\SPheno: (i)  a fully numerical procedure which takes the derivative of the full effective potential with respect to the VEVs. A semi-analytical derivation which takes analytical the derivative of the loop functions with respect to involved masses, but derives the masses and coupling numerically with respect to the VEVs. More details about both methods and the numerical differences are given in Ref.~\\cite{Goodsell:2014bna}. \n \\item {\\bf Diagrammatic calculation}: A fully diagrammatic calculation for two-loop contributions to scalar self-energies with \\SARAH--\\SPheno became available with Ref.~\\cite{Goodsell:2015ira}. In this setup a set of generic expressions first derived in  Ref.~\\cite{Goodsell:2015ira} is used. All two-loop diagrams shown in Fig.~\\ref{fig:2loopDiagram} are included in the limit $p^2=0$. The advantage of the diagrammatic approach is that no numerical derivation is needed.\n\\end{itemize}\n\\begin{figure}[hbt]\n \\includegraphics[width=1.0\\linewidth]{2-loop_diagrammatic}\n \\caption{Generic diagrams included to calculate the two two-loop corrections to tadpoles and scalars. These are the diagram which don't vanish in the gaugeless.}\n\\label{fig:2loopDiagram}\n\\end{figure}\nA comparison between the results using the new routines generated by \\SARAH and widely used routines based on Refs.~\\cite{Brignole:2001jy,Degrassi:2001yf,Brignole:2002bz,Dedes:2002dy,Dedes:2003km} for the MSSM and Ref.~\\cite{Degrassi:2009yq} for the NMSSM is shown in Fig.~\\ref{fig:TwoLoop}. Obviously, a perfect agreement is found for both models.   \n\\begin{figure}[hbt]\n\\includegraphics[width=0.45\\linewidth]{MSSM_m0} \\hfill \n  \\includegraphics[width=0.45\\linewidth]{MSSM_M12} \\\\ \n \\includegraphics[width=0.45\\linewidth]{NMSSM_2L_AlphaS_lam} \\hfill \n \\includegraphics[width=0.45\\linewidth]{NMSSM_2L_AlphaS_kap}\n \\caption{Comparison of the Higgs mass calculation at two-loop level using the automatically generated routines of \\SARAH (full lines) and the routines based in \n Refs.~\\cite{Brignole:2001jy,Degrassi:2001yf,Brignole:2002bz,Dedes:2002dy,Dedes:2003km} for the MSSM and Ref.~\\cite{Degrassi:2009yq} for the NMSSM. The green lines include only the corrections $O(\\alpha_s (\\alpha_t+\\alpha_b))$, while for the blue lines all other, available corrections are included. The black line corresponds to the one-loop results.}\n\\label{fig:TwoLoop} \n\\end{figure}\nSince there are no further possibilities to cross check the results obtained for the Higgs mass with \\SARAH\/\\SPheno, the above described implementation of two independent approaches (effective potential, diagrammatic) was necessary to have the possibility to double check results. In the meantime, this setup has been used to analyse the Higgs mass also in other non-minimal SUSY models at the two-loop level:\n\\begin{itemize}\n \\item Contributions of trilinear RpV, see Ref.~\\cite{Dreiner:2014lqa}\n \\item Missing corrections for NMSSM, see Ref.~\\cite{Goodsell:2014pla}\n \\item Contributions from non-holomorphic soft-terms, see Ref.~\\cite{Un:2014afa}\n \\item MRSSM at full two-loop, see Ref.~\\cite{Diessner:2015yna}\n \\item Contributions from vectorlike stops, see Ref.~\\cite{Nickel:2015dna}\n\\end{itemize}\n\n\n\\subsection{Decay widths and branching ratios}\n\\SPheno modules created by \\SARAH calculate all two-body decays for SUSY and Higgs states as well as for additional gauge bosons. In addition, the three-body decays of a fermion into three other fermions, and of a scalar into another scalar and two fermions are included. \\\\\nIn the Higgs sector, possible decays into two SUSY particles, leptons and massive gauge bosons are calculated at tree-level. For two quarks in the final state the dominant QCD corrections due to gluons are included \\cite{Spira:1995rr}. The loop induced decays into two photons and gluons are fully calculated at leading-order (LO) with the dominant next-to-leading-order corrections known from the MSSM. For the LO contributions all charged and coloured states in the given model are included in the loop, i.e. new contributions rising in a model beyond the MSSM are fully covered at one-loop. In addition, in the Higgs decays also final states with off-shell gauge bosons ($Z Z^*$, $W W^*$) are included. \n\n\\subsection{Flavour observables}\nThe \\SPheno modules written by \\SARAH contain already out of the box the routines to calculate many quark and lepton flavour violating observables:\n\\begin{itemize}\n \\item Lepton flavour violation:\n\\begin{itemize}\n  \\item Br($l_i \\to l_j \\gamma$), Br($l \\to 3l'$), Br($Z\\to l l'$)\n  \\item CR($\\mu-e,N$) { (N=Al,Ti,Sr,Sb,Au,Pb)}, Br($\\tau\\to l+P$) {  (P=$\\pi$, $\\eta$,$\\eta'$)}\n \\end{itemize} \n \\item Quark flavour violation:\n\\begin{itemize}\n  \\item Br($B\\to X_s\\gamma$), Br($B_{s,d}^0 \\to l \\bar{l}$), Br($B \\to s l \\bar{l}$), Br($K \\to \\mu \\nu$)\n  \\item Br($B \\to q \\nu\\nu$), Br($K^+ \\to \\pi^+ \\nu\\nu$), Br($K_L \\to \\pi^0 \\nu\\nu$)\n  \\item $\\Delta M_{B_s,B_d}$, $\\Delta M_K$, $\\epsilon_K$, Br($B \\to K \\mu \\bar{\\mu}$)\n  \\item Br($B\\to l \\nu$), Br($D_s \\to l \\nu$)\n \\end{itemize} \n\\end{itemize}\nThe calculation is based on the \\FlavorKit functionality \\cite{Porod:2014xia} which is a connection  \\FeynArts--\\SARAH--\\SPheno. This provides a full one-loop calculation in a given model. In addition, this interface can be used to derive Wilson coefficients for new operators, and to relate coefficients to calculate new observables with \\SPheno. In order to demonstrate that this fully automatized calculation reproduces the results from dedicated MSSM codes, a comparison is shown in Fig.~\\ref{fig:flavour}.\n\\begin{figure}[hbt]\n \\includegraphics[width=0.45\\linewidth]{Bs_to_e_e_tb}\n \\hfill \n \\includegraphics[width=0.45\\linewidth]{Kp_to_pip_nu_nu_m0} \\\\\n \\includegraphics[width=0.45\\linewidth]{b_to_s_gamma_M12}\n \\hfill \n \\includegraphics[width=0.45\\linewidth]{Delta_MBs_M12}  \n\\caption{Comparison of different flavour observables in the CMSSM. The colour code is: {\\tt FlavorKit} (red), {\\tt SPheno 3.3} (blue), {\\tt SUSY\\_Flavor 1} (brown), {\\tt SUSY\\_Flavor 2} (green), {\\tt MicrOmegas} (orange), {\\tt SuperIso} (plum).}\n\\label{fig:flavour} \n\\end{figure}\n\n\n\\subsection{Checking Higgs constraints}\n\\HB \\cite{Bechtle:2008jh,Bechtle:2011sb,Bechtle:2013wla} and \\HS \\cite{Bechtle:2013xfa} are dedicated tools to study the Higgs properties of a given parameter point in a particular model. While \\HB checks the parameter point against exclusion limits from Higgs searches, \\HS gives a $\\chi^2$ value to express how good the point reproduces the Higgs measurements. In general, \\HS and \\HB can handle different inputs: either the cross sections for all necessary processes can be given at the parton or hadron level, or the effective couplings of the Higgs states to all SM particles are taken as input. In addition, the masses and widths of all neutral as well as charged Higgs states are always needed. \n\\SPheno provides all input for the effective coupling approach. The information is given in the SLHA spectrum file and in addition in separated files (called {\\tt MH\\_GammaTot.dat}, {\\tt  MHplus\\_GammaTot.dat}, {\\tt BR\\_H\\_NP.dat}, {\\tt BR\\_Hplus.dat}, {\\tt BR\\_t.dat}, {\\tt effC.dat}). While SLHA files can be used with \\HB for models with up to five neutral scalars, the  separated files can be used with even up to nine neutral and nine charged scalars.\n\n\n\\section{MC Tools and \\SARAH}\n\\label{sec:MC}\n\\SARAH writes all necessary files to implement a new model in different MC tools. I give a short overview of the main features and explain, how parameter values between \\SPheno and the MC can be exchanged. \n\\subsection{Output of models files}\n\\paragraph{\\CalcHep}\nThe model files for \\CalcHep \\cite{Pukhov:2004ca,Boos:1994xb} can also be used with \\MO \\cite{Belanger:2014hqa} for dark matter studies. \n\\CalcHep is  able to read the numerical\n values of the masses and mixing matrices from a SLHA spectrum file based on the \n \\texttt{SLHA+} functionality \\cite{Belanger:2010st}. \\SARAH makes use of this functionality to generate the model files in a way that they read the spectrum files written by a \\SARAH generated \\SPheno version.\n\n\\paragraph{UFO format}\n\\SARAH can write model files in the UFO format \\cite{Degrande:2011ua} to implement new models in\n \\MGv{5}. This format is also supported by other tools like {\\tt GoSam}\n \\cite{Cullen:2011ac}, {\\tt Aloha} \\cite{deAquino:2011ub}, \n \\texttt{Herwig++} \\cite{Gieseke:2003hm,Gieseke:2006ga} or \\texttt{MadAnalysis 5} \\cite{Conte:2012fm}. \\\\\nOne can give the spectrum file written by \\SPheno as parameter card to \\MG. \n\n\n\\paragraph{\\WHIZARD\/\\OMEGA}\n\\SARAH writes all necessary files to implement a model in \\WHIZARD and \\OMEGA  \\cite{Kilian:2007gr,Moretti:2001zz}. Since the SLHA reader of \\WHIZARD is at the moment restricted to the MSSM and NMSSM, \\SPheno versions generated by \\SARAH write all information about the spectrum and parameters also in an additional file in the \\WHIZARD specific format. This file can be then read by \\WHIZARD. \n\n\\subsection{Interplay \\SARAH--\\SPheno--MC-Tool} \nThe tool chain \\SARAH--\\SPheno--MC-Tools has one very appealing feature: the implementation of a model in the spectrum generator (\\SPheno) as well as in a MC tool is based on just one single implementation of the model in \\SARAH. Thus, the user has not to worry that the codes might use different conventions to define the model. In addition, \\SPheno also provides all widths for the particles and this information can be used by the MC-Tool as well to save time. \n\n\n\\section{Other output}\n\\label{sec:tools}\n\n\\subsection{\\FeynArts\/\\FormCalc}\n\\FeynArts\/\\FormCalc   \\cite{Hahn:2000kx,Hahn:2009bf} is a very powerful combination of two \\Mathematica packages to perform tree-level and loop calculations. \\FeynArts is a Mathematica package for the {generation and visualization} of Feynman {diagrams} and {amplitudes}. \\FormCalc makes use of Feynman diagrams generated by \\FeynArts, and of {\\tt Form} for {analytical manipulations}: it calculates the amplitude, squares it, performs {colour simplifications}, makes the {polarisation sums}, and so on. \\FormCalc can also write {\\tt Fortran} or {\\tt C} code for further {numerical calculations}. \\SARAH can be used to generate the model files for these tools to implement a new model. \n\n\n\\subsection{\\Vevacious}\n\\Vevacious  is a tool to check for the global minimum of the one-loop effective potential for a given model allowing for a particular set of non-zero VEVs. For this purpose \\Vevacious finds first all tree-level minima by using {\\tt HOM4PS2} \\cite{lee2008hom4ps}. Afterwards, it minimizes the one-loop effective potential starting from these minima using {\\tt minuit}  \\cite{James:1975dr}. If the input minimum turns out not to be the global one, life-time of meta-stable vacua can be calculated using {\\tt Cosmotransitions} \\cite{Wainwright:2011kj}.\\\\\n\\Vevacious takes the tadpole equations, the polynomial part of the  scalar potential and all mass \nmatrices as input. All of this information has to be expressed including all VEVs which should be tested. That means, in order to check for charge and colour breaking minima the stop and stau have to show up in the potential and mass matrices and the entire mixing triggered by these VEVs should be included. To take care of all that, the corresponding input files can be generated by \\SARAH. Possible applications when these checks become important are:\n\\begin{itemize}\n \\item Charge and colour breaking minima, see for instance Refs.~\\cite{Camargo-Molina:2013sta,Camargo-Molina:2014pwa}\n \\item Spontaneous R-parity violation, see for instance Ref.~\\cite{CamargoMolina:2012hv} \n \\item NMSSM with large $A$ terms, see for instance Refs.~\\cite{Kanehata:2011ei,Kobayashi:2012xv}\n\\end{itemize}\n\n\n\\subsection{\\LaTeX}\nAll derived information (mass matrices, vertices, RGEs, loop corrections) can be exported to \n\\LaTeX\\ to get the expressions in a human readable form. \nThere exists the possibility to draw Feynman diagrams with {\\tt FeynMF} \\cite{Ohl:FeynMF} \nfor all vertices. \n\n\\section{Summary}\n\\label{sec:summary}\nI have given in this lecture a short summary how the study of a new particle physics models can be automatized to a large extent. The first step is to implement a new model in \\SARAH. \\SARAH then calculates all analytical properties of this model and passes this information to other tools for the numerical evaluation. The main work-flow of this setup is summarized in Fig.~\\ref{fig:summary}. \n\n\\begin{figure}[hbt]\n\\begin{minipage}{\\linewidth}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\colorbox{blue!70}{\\parbox{2.cm}{\\centering \\bf collider pheno}} &\n\\colorbox{blue!70}{\\parbox{2.cm}{\\centering \\bf dark\\\\ matter}} &\n\\colorbox{blue!70}{\\parbox{2.cm}{\\centering \\bf Higgs constraints}}  &\n\\colorbox{blue!70}{\\parbox{2.cm}{\\centering \\vspace{0.35cm} \\dots\n\\\\\\vspace{0.35cm}}}   \\\\ \n\\(\\uparrow\\) & \\(\\uparrow\\) &  \\(\\uparrow\\) &  \\(\\uparrow\\) \\\\\n\\end{tabular}\n\\colorbox{blue!50}{\\parbox{10cm}{\\centering \\bf calculate observables; decays of sparticles and Higgs}} \\\\\n\\(\\uparrow\\)\\\\\n\\colorbox{blue!50}{\\parbox{10cm}{\\centering {\\bf calculate running parameters; loop corrected masses; fine-tuning}}} \\\\\n\\(\\uparrow\\)\\\\\n\\colorbox{blue!30}{\\parbox{10cm}{\\centering {\\bf create model files for MC tools and Fortran code for \\SPheno}}} \\\\\n\\(\\uparrow\\)\\\\\n\\colorbox{blue!30}{\\parbox{10cm}{\\centering {\\bf derive expressions for masses, vertices, RGEs, \\dots}}} \\\\\n\\(\\uparrow\\)\\\\\n\\colorbox{blue!10}{\\parbox{10cm}{\\centering {\\bf create \\SARAH input file}}} \\\\\n\\(\\uparrow\\)\\\\\n\\colorbox{blue!10}{\\parbox{10cm}{\\centering {\\bf Idea for a new model}}}\n\\end{center} \n\\begin{picture}(0,0)\n\\put(375,18){\\mbox{\\rotatebox{90}{\\colorbox{blue!10}{\\parbox[c][1.5cm]{1.24cm}{\\centering User}}}}}\n\\put(375,70){\\mbox{\\rotatebox{90}{\\colorbox{blue!30}{\\parbox[c][1.5cm]{1.65cm}{\\centering \\SARAH}}}}}\n\\put(375,132){\\mbox{\\rotatebox{90}{\\colorbox{blue!50}{\\parbox[c][1.5cm]{1.72cm}{\\centering \\SPheno}}}}}\n\\put(375,212){\\mbox{\\rotatebox{0}{\\colorbox{blue!70}{\\parbox[c][1.00cm]{1.5cm}{\\centering \\tiny \\CalcHep, \\WHIZARD, {\\tt HB}, {\\tt HS}, {\\tt MG5},  \\MO, \\FeynArts,\\dots}}}}}\n\\end{picture}\n\\end{minipage} \n\\caption{Schematic framework to study all aspects of a new model in a highly automatized way.}\n\\label{fig:summary}\n\\end{figure}\n\n\n\\section*{Acknowledgements}\nI thank the organizers of ``Summer School and Workshop on the Standard Model and Beyond'', Corfu2015, for the invitation and the hospitality during the stay. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nWith the arrival of the GRAVITY and MATISSE instruments after 2012, and VSI after 2015, the VLTI will routinely work with 4 or possibly 6 telescopes simultaneously. As ground-based interferometric devices, they will be extremely sensitive to the atmospheric turbulence and in particular to the randomly varying optical path difference (OPD) between the telescopes. The random movement of the fringes it involves quickly blurs the interferometric signal, preventing for integration time longer than the coherence time of the atmosphere, typically a few tens of milliseconds in the infrared. The limiting magnitude and precision of ground-based interferometers are therefore dramatically reduced. To reach their ultimate performance, it is then mandatory to dispose of a fringe tracker, that is a device measuring in real time the position of the fringes within a fraction of wavelength and finally stabilizing them.\n\n\nIn the context of the next generation of interferometric instruments, the 2$^{nd}$ Generation Fringe Tracker (2GFT) is intended to cophase an array up to 6 telescopes, formed of Auxiliary and\/or Unit Telescopes (ATs and UTs respectively). The Planar Optics Phase Sensor (POPS) is a fringe sensor  proposition for the 2GFT resulting from the collaboration between LAOG (PI institute), IAGL, OCA and GIPSA-Lab. \nWe present here the results of the related system studies. They aim at defining the most efficient conceptual design within the VLTI environment, providing precise measurements of the fringe position as well as robust operations, that is a stable fringe tracking whatever the observationnal or atmospheric conditions. To do so, we propose 4-telescope and 6-telescope (4T and 6T hereafter) concepts based on the integrated optics (IO) technology, providing a single-mode spatial filtering as well as pairwise co-axial combinations of the beams, and coding the fringes with a spatial ABCD estimator. These system choices are based on theoretical studies and realistic simulations as well as experience gained in operation with real interferometers, especially VINCI at VLTI\\cite{lebouquin_2004, lebouquin_2006} , IONIC-3TH at IOTA\\cite{berger_2003}~, and the fringe-sensor FINITO\\cite{lebouquin_2008} at VLTI.\n\nWe briefly summarize the results of a theoretical analysis of modal filtering influence on the phase measurement in section \\ref{part:IO_choice}. The best conceptual architectures for 4- and 6-telescope fringe sensors are determined in relation to their sensitivity, robustness and some additional operational considerations in section \\ref{part:combination_choice}. The performance of the most common phase and group delay estimators are then compared in part \\ref{part:sensing_concepts}, taking into account atmospheric disturbances. The instrument conceptual design and the results of realistic simulations for the 4T concept are finally presented in sections \\ref{part:proposed_concept} and \\ref{part:expected_perf} respectively.\n\n\n\n\n\t\\section{CHOICE OF THE INTEGRATED OPTICS} \\label{part:IO_choice}\n\n The initial idea of POPS is to propose an instrument based on the IO technology. IO components provide very compact interferometric instruments, only a few centimeters long, with low sensitivity to external constraints (vibrations or temperature fluctuations) and they simplify the alignement procedure. Transmission of IO is however lower than their equivalent in bulk optics, by $20$ to $30\\%$ for a co-axial pairwise combination for instance, which corresponds to a limiting magnitude $0.3$\\,mag lower.\n\n  \nadditionally, this technology provides by design a single-mode (SM hereafter) propagation of light, which spatially filters the incoming wavefront and selects its coherent part. Thanks to a theoretical description of the signal, we compare the performance of multi-mode (MM) and SM interferometry for the estimation of the phase (see Tatulli et al\\cite{tatulli_2010} (these proceedings) for details). The conclusion of this study is that despite the loss of flux occurring when injecting the light into the SM component (i.e. SM fibers, integrated optics), spatial filtering globally improves fringe sensing performance. \nThis study also demonstrates SM interferometry is more efficient for off-axis fringe tracking and equally efficient for the stability of the beams injection. \n\nConsequently the POPS concept, based on single-mode integrated optics, is a great alternative to classical bulk concepts, from the performance and operational points of view.\n\n\n\n\n\n\\section{THE COMBINATION CONCEPT} \\label{part:combination_choice}\n\nA fringe tracker can cophase a N-telescope interferometer by measuring only N-1 OPDs, thanks to N-1 baselines. However because of the noisy measurements and varying observationnal conditions during a night, it can be profitable to dispose of some information redundancy with additional baselines. It is then possible to retrieve the phase on a baseline with several independant ways, insuring a better fringe tracking stability.\n\n\nAll-in-one multi-axial schemes are therefore interesting concepts since they provide all the possible baselines. However they are not currently compatible with the specific requirements of fringe tracking, that is very fast measurements and high  limiting magnitudes. Because of the high number of pixels they require (95 in the 4T case) and of the performance of available detectors (Hawaii II RG, with a read-out noise of $10e^-$\/pixel and a low acquisition frequency), their performance is limited in this context. A detailed discussion of the properties and ultimate performance of IO multi-axial combination shemes is presented in Ref.~\\citenum{tarmoul_2010} (these proceedings).\n\n\n\t\t\\subsection{Study of the co-axial pairwise schemes}\n\nThe POPS concept therefore concentrates on co-axial and pairwise solutions, which allow the creation of  the desired number of baselines, thereby limiting the number of pixels to read (to the maximum 24 pixels for a 4T concept with an ABCD coding), and enhancing both sensitivity and speed of the fringe sensor. However when the number of baselines increases each one is less sensitive because the incoming flux has to be divided between more baselines. The sensitivity of the fringe sensor then depends on a competition between the information redundancy between the baselines and their sensitivity. To determine the optimal pairwise co-axial architectures in the 4T and 6T cases, we study the intrinsic performance of several schemes and also point out some important operational aspects (the detailed study is presented in Ref.~\\citenum{blind_2010}, in preparation). The schemes studied are presented in the Figure \\ref{fig:schemes} with the associated nomenclature.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\includegraphics[width=0.14\\textwidth]{4TO}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{4TR2}   &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{4TR3}   &\n\t\t                                      \t\t\t\t\t                                               &\n\t\t                                                        \t\t\t\t\t                           &\n\t\t                                                                                      \\\\\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TO}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TR2}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TR3}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TR3A}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TR4}     &\n\t\t\t\\includegraphics[width=0.14\\textwidth]{6TR5}     \\\\\n\t\t\\end{tabular}\n\t\t\\caption{The conceptual combination schemes studied in the 4-telescope (top) and 6-telescope (bottom) cases, with the associated nomenclature below each one.}\n\t\t\\label{fig:schemes}\n\\end{figure}\n\n\n\\paragraph{Performance study} The performance study is based on theoretical models of the error on the phase measurement\\cite{tatulli_2010}, and is similar to the one led by Houairi et al\\cite{houairi_2008} in the context of the GRAVITY fringe tracker. This study consists in computing the optimal optical path estimators used to drive the delay lines, from the noisy phase measurements made on each baselines. When computing the optimal optical paths, the measurements are weighted depending on the measurement error, so that the importance of the noisiest baselines is lowered with respect to the others. \n\nThe quantity of interest is finally the relative error $\\epsilon$ on the corrected differential piston which are calculated with respect to a reference noise. The expression of this latter only depends on the considered regime (i.e. detector or photon noise regime), so that results for the different schemes are perfectly comparable within the same regime. More precisely we study three different cases:\n\\begin{enumerate}\n\t\\item \\textbf{Ideal case}: all the baselines are strictly equivalent in term of flux and visibility.\n\t\\item \\textbf{Resolved source case}: we assume the source is highly resolved by one baseline (the $\\{12\\}$ for instance), so that the fringe visibility is set to 0.1 on it, and to 1 on the others.\n\t\\item \\textbf{Low coupling efficiency case}: because of quick variations of the instantaneous Strehl ratio, the coupling efficiency on one pupil can be very low compared to the others. To study the behavior of the schemes with respect to this effect, the flux of the telescope $1$ is set to one tenth of the others.\n\\end{enumerate}\nFor each case, looking at the relative error $\\epsilon$ on the corrected OPD on the different baselines, we are able to compare the combination schemes and to make a choice from a performance point of view. The results of the performance study are on the Table \\ref{tab:scheme_perf} for the 4T schemes.\n\n\n\\paragraph{Extracting the photometry} Some schemes also allow to extract the instantaneous photometry on each pupils without the need of dedicated outputs. We show that it is possible only when the schemes have a particular structure: to evaluate the photometry from a pupil, this latter has to be part of a closed sub-array constituted with an odd number of pupils. Otherwise the system linking the fringe signals to the photometries is degenerated. Thus, the 4T2 cannot extract the photometry since it is a cycle of 4 telescopes, whereas the 4T3 can, since there are triangular sub-arrays. The results of this study are presented in Table \\ref{tab:photometry}.\n\n\\begin{table}[t!]\n\t\\begin{center}\n\t\t\\caption{Results of the performance calculations made for the 4T schemes. The relative error is given in detector ($\\epsilon^{det}$) and photon ($\\epsilon^{phot}$) noise regimes, in the three cases. In the second case, the resolving baseline is the $\\{12\\}$, and is therefore not equivalent to the others (noted $\\{ij\\}$). In the third case the baselines affected by the low coupling are noted $\\{1j\\}$.}\n\t\t\\label{tab:scheme_perf}\n\t\t\\begin{tabular}{cccccccccccccccccccc}\n\t\t\\\\\n\t\t\n\\hline\n\\hline\n\t\t\t   &&& \\multicolumn{2}{c}{Ideal case} &&& \\multicolumn{5}{c}{Resolved case} &&& \\multicolumn{5}{c}{Low coupling efficiency case} \\\\\n Schemes &&&  $\\epsilon_{ij}^{det}$ & $\\epsilon_{ij}^{phot}$ &&& $\\epsilon_{12}^{det}$ & $\\epsilon_{ij}^{det}$ && $\\epsilon_{12}^{phot}$&  $\\epsilon_{ij}^{phot}$ &&& $\\epsilon_{1j}^{det}$ & $\\epsilon_{ij}^{det}$ && $\\epsilon_{1j}^{phot}$&  $\\epsilon_{ij}^{phot}$\\\\\n\\hline\n4TO &&& 1.6 & 1.3 &&& 16.2 & 1.6 && 13.1 & 1.3 &&& 5.1 & 1.6   && 3.1 & 1.3 \\\\\n4T2 &&&  1.7 & 1.2 &&& 3.4  &  2.0 &&  2.4  &  1.4 &&&  4.7  &  1.9  &&  2.5  &  1.4 \\\\\n4T3 &&&  2.1 & 1.2 &&&  3.0  &  $2.2$ && 1.7  &  $1.3$ &&&  5.6  &  2.4 &&  2.5  &  1.4 \\\\\n\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\\begin{table}[b!]\n\\centering\n\\caption{\\label{tab:photometry} Capability of the considered schemes to provide the inputs photometries without dedicated outputs.}\n\\begin{tabular}{cccccccccccc}\n\\\\\n \\hline \\hline\n Schemes       && 4TO & 4T2 & 4T3 && 6TO & 6T2 & 6T3A & 6T3B & 6T4 & 6T5 \\\\\n Photometries? && no & no  & yes  && no & no  & no  & yes    & yes  & yes \\\\ \n \\hline\n\\multicolumn{10}{c}{}\\\\\n\\end{tabular}\n\\end{table}\n\n\n\n\n\t\t\\subsection{Choice for the 4T case}\n\nLooking at the performance (Table \\ref{tab:scheme_perf}), the open scheme (4TO) appears less interesting than the cyclic one (4T2, with the minimum redundancy) because of its lower redundancy and of the flux imbalance on the extreme baselines, inducing a sub-optimal use of the input photons\n\\footnote{This observation leads us not to consider schemes with intrinsically imbalanced photometric inputs.}\n. The 4T2 and 4T3 appear roughly equivalent: the former provides slightly better performance in the first and third cases, whereas in the second one the 4T3 shows better performance thanks to a more important redundancy. This higher degree of redundancy also provides the following additional advantages, mainly related to operational aspects:\n\\begin{itemize}\n\t\\item The photometry of each input beam can be directly recovered from the fringe signal (see Table \\ref{tab:photometry}). The knowledge of the photometry is theoretically not mandatory to measure the fringe phase, but operation without photometric information is practically possible only for perfectly symmetric combiners. Otherwise, the fringe signal is polluted by photometric fluctuation residuals and the phase and group delay measurements are biased.\n\nAdditionally, a real-time photometric monitoring is very useful during operation, providing an additional diagnosis in case of flux drop-outs and allowing the injection to be optimized in all beams simultaneously.\n\t\\item As seen from the VLTI (and so from the sky), all the baselines of a fully redundant pairwise combination scheme are perfectly identical. Whatever the observed target and the geometry of the telescope array, the way to map the interferometer beams into the fringe sensor inputs is always optimal. This is a critical point since during a night unpredictable baselines can exhibit very low visibilities, changing with a time scale of less than one hour (see a typical example in Figure~\\ref{fig:contrast_on_resolved_binary}).\n\t\\item Finally, such a strategy also maximizes the cophasing stability in case of frequent flux drop-outs since unaffected baselines are kept locked and thus only the extinguished beam(s) have to be cophased back after extinction.\n\\end{itemize}\nTherefore the fully redundant pairwise combination 4T3 has been chosen. This result is in agreement with the conclusions of Houairi et al\\cite{houairi_2008} for the GRAVITY fringe tracker, also based on IO technology.\n\n\n\\begin{figure}[b!]\n\t\\centering\n  \t\\includegraphics[width=0.49\\textwidth]{binaire_12mas}\n  \t\\includegraphics[width=0.49\\textwidth]{binaire_12mas_data}\n  \t\\caption[Predicted fringe contrast when observing a binary]{ Predicted fringe contrast when observing a binary with equal fluxes and a separation of about 10mas with the four UTs and a fringe sensor working in the K band. The left panel shows the (u,v) tracks overlaid on the fringe contrast from the model. Right panel shows the fringe contrast versus time for 4h for each baseline. The figures have been made with the \\texttt{aspro} public software from JMMC.}\n  \t\\label{fig:contrast_on_resolved_binary}\n\\end{figure}\n\n\n\t\t\\subsection{Choice for the 6T case}\n\nThe performance study in the 6T case leads to the same kind of conclusion than previously. As for the 4TO scheme, the 6TO open scheme is still penalized because of the imbalance of the inputs and of a lower redundancy. And when the number of baselines exceeds 6, the more important redundancy roughly counterbalances the loss of signal-to-noise ratio on the individual baselines. The 6T3B scheme is finally favored to the 6T2, for the same reasons as in the 4T case (i.e. nearby performance, the possibility to directly extract the photometry from the fringe signal and a higher number of baselines enhancing the robustness).\n\n\n\n\n\n\n\n\\section{THE FRINGE SENSING CONCEPTS CHOICES} \\label{part:sensing_concepts}\n\nFringe tracking consists in: \\textit{1)} Ensuring the measurements are done in the center of the fringe packet, that is in the highest signal-to-noise ratio (SNR) area, thanks to the group delay measurement; \\textit{2)} Keeping the fringes locked thanks to the real-time knowledge of the phase, then allowing long integration times and improved SNR on the scientific instrument. We consider in the following that a fringe tracking regime requires a precision on the phase better than $\\lambda\/10$, since the contrast loss due to residual fringe blurring is kept below 5\\% on the scientific instrument. To fringe track with the highest SNR, it is then mandatory to dispose of phase and group delay estimators as precise as possible, whatever the conditions.\nBecause POPS is a co-axial combiner based on the IO technology, we consider the ABCD estimator\\cite{colavita_1999b}\\,, which can be implemented in different ways to measure the phase and the group delay. For  both measurements, we compare  in realistic conditions the performance of two different implementations. These studies are detailled in Ref.~\\citenum{blind_2010} (in preparation). We also determine the optimal spectral band and spectral resolution.\n\n\n\t\\subsection{Phase sensing} \\label{part:phase_sensing}\n\nFor the phase we compare two different implementations of the ABCD algorithm: \\textit{1)} Four phase states are recorded simultaneously over four different outputs (spatial ABCD); \\textit{2)} The phase measurement is made in two steps (temporal ABCD): two phase opposite outputs are first recorded with a zero delay and then with an additional $\\lambda\/2$ delay. The comparison is based on a theoretical analysis of the signal in both cases, and takes into account atmospheric disturbances (i.e. piston and injection fluctuations). The result in the case of the ATs and an integration time of 1\\,ms is presented in Figure \\ref{fig:snr_phase}.\n\nThis study shows that the non-simultaneity of the measurements in the temporal concept introduces an important, additional noise, mainly due to piston. Because of it, the temporal ABCD is always less efficient in a phase tracking regime, so that the spatial ABCD appears as the best choice. Especially, in bad seeing conditions or on UTs, the phase error can be higher than $\\lambda\/10$, limiting the phase tracking capabilities. \n\n\n\t\\subsection{Group delay sensing} \\label{part:group_sensing}\n\nStill in the context of the ABCD coding, we consider two concepts to measure the group delay : \\textit{1)} The scanned-fringes concept where the fringe contrast is measured on some points inside the main lobe of the fringe packet; \\textit{2)} The dispersed-fringes concept, where the fringe phase is measured over a few spectral channels.\n\nGiven the impossibility to carry a realistic analytical description of these group delay estimators, we proceed with Monte-Carlo simulations taking into account atmospheric disturbances. The comparison of the dispersed-fringes and the scanned-fringes concepts, in various atmospheric conditions, shows the first is the most efficient (the results of simulations in the case of the ATs with an integration time of 1\\,ms are presented on the Figure \\ref{fig:snr_gd}). Equally as the temporal ABCD for the phase, the scanned-fringes concept exhibits a very high sensitivity to atmospheric disturbances (piston mainly) so that the dispersed-fringes concept appears as the best choice. For instance, even for integration times as low as 0.5\\,ms, the precision of the temporal concept is worse than $\\lambda\/10$ for the ATs in bad atmospheric conditions and for the UTs whatever the conditions.\n\n\n\\begin{figure*}[t]\n\t\\begin{minipage}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8\\linewidth]{AT-ABCDvsAC-BD_tint1}\n\t\t\\caption{Phase measurement statistical error functions of the incoming flux in the ATs case with an integration time of 1\\,ms, for the spatial ABCD (solid) and temporal ABCD (dashed) phase sensing concepts. From black to red, the atmopsheric conditions vary from excellent to bad.}\n\t\t\\label{fig:snr_phase}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8\\linewidth]{AT-rmsGD-DIT2}\n\t\t\\caption{Statistical error for the group delay estimation with the dispersed-fringes (solid) and the scanned-fringes (dashed) group sensing concepts. Curves are given functions of the incoming flux in the ATs case for an integration time of 1\\,ms. From black to red, the atmopsheric conditions vary from excellent to bad.}\n\t\t\\label{fig:snr_gd}\n\t\\end{minipage}\n\\end{figure*}\n\n\n\t\\subsection{Spectral disperion} \\label{part:spectral_disp}\n\nThe number of spectral channels directly determines the efficiency of the dispersed group delay sensing. A trade-off should be made between an accurate determination (larger spectral dispersion) and an optimal sensitivity when limited by detector noise (smaller dispersion). We decide to disperse the light over 5 channels, corresponding to a spectral resolution $\\sim 20$ in order to provide a non-ambiguous range of $\\pm 20 \\mu m$ in which the group delay can be found. This non-ambiguous range is larger than the OPD typically introduced by the turbulence over few seconds ($\\approx 10\\mu m$ at Paranal). Therefore the group delay can be quickly recovered after a temporary flux loss (low Strehl or telescope chopping for MATISSE). This non-ambiguous range is also significantly larger than the size of the fringe packet seen in broad band. The detection of the main lobe of the fringe packet is much easier, and thus avoids locking the fringes on a secondary lobe, with a low SNR.\n\n\n\n\n\t\\subsection{Fringe sensing in the K band} \\label{part:Kband}\n\nAll required optical devices (IO, fibers, bulk) exist in both the H and K astronomical bands. We used the simulation tool \\texttt{sim2GFT}{} (see section \\ref{part:sim2GFT} for details) to compute the expected performance in both bands, taking into account the differences in atmospheric turbulence and transmission, sky brightness and expected instrumental transmission.\n\nWe finally choose the K band from $1.9$ to $2.4\\,\\mu m$ because it presents a higher limiting magnitude ($\\sim 0.3$\\,mag). The effective wavelength is also 35\\% longer enhancing the non-ambiguous phase delay and group delay measurement range, and the atmospheric turbulence strength is smaller, leading to a higher and more stable flux injection in the optical fibers. Furthermore, longitudinal and transversal atmospheric dispersion compensation (LADC and TADC) is not required considering the low losses that are expected (typical coupling losses of about $20\\%$ and typical fringe contrast loss of $10\\%$ for a zenith distance Z$=60\\deg$).\n\n\nOn the other hand, the K band suffers from an increased thermal background with respect to the H band. The IO component has to be kept cooler than about $240$K to limit its emission in the $1.9-2.4\\mu m$ spectral range.\n\n\n\n\t\n\\begin{figure}[tb]\n\t\\centering\n\t\t\\includegraphics[width=.5\\textwidth]{concept}\n\t\\caption[Schematic view of the POPS concept]{\\label{fig:concept_scheme} Schematic view of the 4T POPS concept (top) with the 4 modules injecting the VLTI beams into single-mode fibers, and combining them into the IO component. This latter is placed with the camera inside the cryogenic dewar. At the bottom is a detailled view of an injection module.}\n\t\\label{fig:concept_design}\n\\end{figure}\n\n\n\\section{PROPOSED CONCEPT} \\label{part:proposed_concept}\n\nBased on the conclusions of the previous sections, the 4-telescope POPS concept has finally the following characteristics: \\textit{a)} The beam combination is performed thanks to a single-mode integrated optics component using a fully redundant 4T3 scheme with internal spatial ABCD coding; \\textit{b)} The phase is estimated thanks to the ABCD outputs, and the group delay is measured by additionaly dispersing them over 5 channels; \\textit{c)} The concept is optimized for the K spectral range ($1.9$ to $2.4 \\mu m$) and needs no TADC nor LADC.\n\n\n\t\\subsection{Concept overview} \\label{part:overview}\n\nFigure \\ref{fig:concept_design} presents a schematic view of the overall concept as well as one of the injection module in the bottom part. The VLTI beams are collected through 4 injection modules, insuring several functions, before to be injected into the SM fibers and the IO beam combiner. The outputs of the IO chip are imaged and dispersed over 5 pixels of the Hawaii II RG camera. Since the fringe coding is based on an internal ABCD modulation, there is no particular need for an external metrology. \n\n\nWith this design, the transmission of POPS is estimated to roughly 23\\%, and the instrumental contrast to 87\\%. The concept footprint is 1000$\\times$2500\\,mm in the 6T configuration.\n\n\n\n\n\t\\subsection{Injection unit} \\label{part:injection}\n\nThe injection assembly is based on the PIONIER\\cite{berger_2010} visitor instrument that is currently under integration at LAOG and provides the following functions (see Figure \\ref{fig:concept_design}, bottom):\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Selection of the K band} \\quad The VLTI beams are collected by dichroics (in green) reflecting the K band in POPS while the H band is transmitted to the IRIS guiding camera.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Tip-tilt correction} \\quad A tip-tilt mirror provides a real-time compensation of the tip-tilt variations created in the VLTI tunnels.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Calibration} \\quad An optical path modulator and a shutter are used for internal calibration of the beam combiner (instrumental contrast, ABCD phase shifts, etc.). We use a long-stroke modulation in order to have a sufficient spectral resolution.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Injection} \\quad The VLTI beams are coupled into a SM fiber through an off-axis parabola. The fiber and the parabola are mounted on a common support, so that the image quality on the fiber head can be adjusted conveniently. The mechanical support of the injection module embeds a manual OPD adjustment to cophase all arms.\n\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.65\\textwidth]{4TR3-concept}\n\t\\caption[Design of the 4T3-ABCD GRAVITY beam combiner. ]{Design of the 4T3-ABCD beam combiner. The 24 interferometric outputs that are imaged on the camera are on the right hand side of the figure.}\n\t\\label{fig:IO-4TR3}\n\\end{figure}\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\\caption{Expected performance of the 4T3 IO component in K band.}\n\t\\label{tab:IOperf}\t\t\\begin{tabular}{ll}\n\t\\\\\n\t\t\\hline\n\t\tAverage transmission  & $>55\\%$ \\\\\n\t\tBroadband contrast level  &\t$> 90 \\% \\pm 5\\%$, with a Wollaston prism.\\\\\n \t\tNarrowband contrast level &  $95\\%$, with a Wollaston prism. \\\\\n \t\tPhase shifting value & $90\\deg\\pm25\\deg$ \\\\\n \t\tPhase shift Chromaticity & $5$ to $15\\deg$ \\\\\n\t\tPhase shift between polarization & $5\\deg$ to $10\\deg$ \\\\\n \t\tContrast stability & $\\leq 0.5 \\%$ \\\\\n\t\tPhase shift stability & better than $1\\deg$ \\\\\n\t\t\\hline\n\\\\\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\t\\subsection{Beam combination unit} \\label{part:IOC}\n\nThis assembly is composed of the integrated beam combiner and its feeding fibers. Working in the K band, the design of this assembly is based on the one currently developped for GRAVITY. It is consequently based on fluoride fibers to maximize the throughput of POPS for K band operation, especially for wavelengths above $2.3\\,\\mu m$. They are mounted in a fiber array V-groove glued to the IO component. This assembly is then integrated in the cryostat in order to insure the required stability between the component output and the detector itself. It allows also cooling down the component and the imaging optics at 240K to achieve a negligible background emission.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{IO combiner} The concept chosen for POPS is the 4-beam pairwise ABCD combining 6 baselines (4T3 scheme), resulting in 24 interferometric outputs. The circuit, designed in collaboration with our partner CEA\/LETI, is presented in Figure~\\ref{fig:IO-4TR3}. The combiner includes a series of achromatic phase shifters to produce four phase states in quadrature, resulting in a spatial ABCD coding for each baseline.\n\nA prototype in the H band has been fully characterized by Benisty et al\\cite{benisty_2009} . The equivalent circuit for the K band is currently under manufacturing for GRAVITY and the expected performance are evaluated from a set of prototypes tested in laboratory (Table~\\ref{tab:IOperf}). The current spectral transmission in K band also presents a strong OH absorption peak at $2.2\\, \\mu m$, so that the average transmission over the K band should be $>55\\%$.\n\n\\vspace{0.1cm}\n\\noindent \\textbf{Polarization control}\tUsing low birefringence fibers, polarization control is required to optimize the contrast level. Each fiber arm would therefore implement a fibered polarization controller motorized with a stepper motor, similar to those designed for GRAVITY.\n\n\n\n\n\t\\subsection{Cryogenic unit} \\label{part:cryo}\n\nThe cryogenic mechanical structure does not include any manual or motorized adjustment to ensure a high stability level. The cryogenic dewar is based on the last generation of cryostats developped by ESO. It cools down the beam combination optics, the imaging optics and the Hawaii II RG camera. Since POPS is designed for operation with 4 or 6 telesopes, two IO components (4T3 and 6T3B) have to be implemented, mounted upside-down on a bracket support in order to have the two outputs slits imaged on the detector as close as possible. \n\n\n\n\n\n\\section{EXPECTED PERFORMANCE OF THE 4T CONCEPT} \\label{part:expected_perf}\n\nThis final section presents the expected performance of the POPS proposition in the 4-telescope case. We first describe our end-to-end simulation tool \\texttt{sim2GFT}{} and then present the computed performance for the ATs, the UTs and the specific case of mixed arrays ATs+UTs.\n\n\n\t\\subsection{\\texttt{sim2GFT}} \\label{part:sim2GFT}\n\nIn order to assess the performance of our conceptual designs, and to compare the performance of various designs (i.e., combination of a hardware solution and a phase estimator) under various assumptions in terms of environmental and observing conditions, we have developed a dedicated software simulation tool called \\texttt{sim2GFT}{}.\n\n\\texttt{sim2GFT}{} is largely based on the \\texttt{GENIEsim} software, which is described in detail by Absil et al.\\cite{absil_2006} , and therefore follows the same architecture and philosophy. The simulations are taking into account all major contributors to the final performance, from the atmosphere and the telescopes down to the fringe sensor and delay lines.\nIn order to properly estimate the amount of coherent and incoherent photons, all the VLTI and POPS subsystems are described by their influence on the intensity, piston, and wavefront quality of the light beams collected by each telescope. The implementation of each subsystem in the simulator has been extensively validated during the GENIE study\\cite{Absil_master} .\n\nSome additional physical phenomena (visibility loss due to piston jitter, atmospheric refraction, intensity mismatch between the beams due to atmospheric turbulence, and longitudinal dispersion in the delay lines) have been implemented in the fringe tracking loop of \\texttt{sim2GFT}{} with respect to what was implemented in \\texttt{GENIEsim}. \n\n\\begin{figure*}[t]\n\t\\begin{minipage}[t]{0.49\\linewidth}\n \t\t\\centering\n \t\t\\includegraphics[viewport=0 10 510 370,clip,width=0.95\\textwidth]{Sim2GFT_FSUnoise}\n\t\t\\caption[Fringe sensing noise as a function of the target magnitude]{Fringe sensing noise functions of the K-magnitude of the K0\\,III star source in the 4T3-ABCD case. The corresponding stellar flux in photons per second at the entrance of POPS is on the top scale. The respective contributions of shot and detector noises are represented by dotted and dashed lines.}\n\t\t\\label{fig:fsunoise_4TABCD}\n\t\\end{minipage}\n\\hfill\n\t\\begin{minipage}[t]{0.49\\linewidth}\n   \t\\centering\n  \t\t\\includegraphics[viewport=5 10 500 370,clip,width=0.95\\textwidth]{Sim2GFT_loopres}\n  \t\t\\caption[Repetition time, time delay and noise residuals as a function of the magnitude]{Noise residuals at the output of the closed-loop for the fringe sensing, the delay line and the corrected atmospheric piston.}\n  \t\t\\label{fig:closeloop_4TABCD}\n \\end{minipage}\n\\end{figure*}\n\n\n\t\\subsection{ATs case}\n\nIn the case of the ATs, simulations have been done for 4 different concepts (4T2 and 4T3 schemes associated with a spatial or a temporal ABCD for the phase estimation). They show good agreement with the previous studies, that is the 4T3 with a spatial ABCD concept is the most efficient in various atmospheric or observationnal conditions. They particularly highlight the contribution of the redundancy, providing a more stable fringe tracking: in the case of a resolved binary (see Figure \\ref{fig:contrast_on_resolved_binary}), the fully redundant scheme (4T3) outperforms the cyclic one (4T2).\n\nTherefore we present here the results of simulation for the selected concept that is the 4T3 scheme with a spatial ABCD coding of the fringes. The fringe sensing and closed-loop control performance are summarized in Figures~\\ref{fig:fsunoise_4TABCD} and \\ref{fig:closeloop_4TABCD}, for a range of magnitudes from $K=-1.7$ to $K=10.5$. These simulations have been obtained for a K0\\,III star located at various distances ranging from 12\\,pc to 3200\\,pc, with the A0-G1-K0-I1 quadruplet of ATs, and in standard atmospheric conditions: seeing $\\varepsilon=0.85\"$, Fried parameter $r_0=12$\\,cm, coherence time $\\tau_0=3$\\,ms, average wind speed of 12\\,m\/s, outer scale ${\\cal L}_{\\rm out}=25$\\,m, and sky temperature $T_{\\rm sky}=285$\\,K. The target star is assumed to be located close to zenith.\n\nFigure~\\ref{fig:fsunoise_4TABCD} shows the fringe sensing noise (due to shot and detector noises in the fringe sensor) functions of the magnitude\u00d2. \nOn the bright-side end of the plot, shot noise dominates the noise budget. The increase in shot noise from $K=1.5$ to $K=-2$ is due to the star being (strongly) resolved, which reduces the available coherent flux. Detector noise becomes larger than shot noise around $K=3$, and the fringe sensing noise reaches its allowed limit (100\\,nm RMS) around $K=6$ . In this regime the loop can be operated at its maximum repetition frequency  of 3559\\,Hz. For fainter magnitudes, \\texttt{sim2GFT}\\ makes sure that the fringe sensing noise remains at the same level by gradually increasing the  closed-loop repetition frequency (i.e., increasing the integration time on the detector) to keep a sufficient SNR on each individual fringe measurement. This is possible only until magnitude $K\\sim9$ in the present case, where a phase sensing noise of 100\\,nm per baseline cannot be reached any more for any integration time, because of the strong fringe blurring that appears at long integration times. Above this limit, the integration time is adjusted to minimize the measurement noise. Also represented in Figure~\\ref{fig:fsunoise_4TABCD} is the fringe sensing noise per telescope, which results from the optimized use of the differential piston measured on all the baselines: it is significantly smaller than these latters thanks to the information redundancy.\n\n\nFinally, for an integration time of 10\\,ms, the limit above which the sensing noise cannot be maintained below 100\\,nm is $K=8.2$. If we consider the {\\it operational limiting magnitude} for stable fringe tracking to be the limit for which the SNR is larger than $\\approx 4$ for at least $95\\%$ of the time, we obtain a value $K = 7.2$ for the ATs in standard conditions. The limit for fringe detection (SNR$\\approx 4$) is $K=9.2$ in excellent conditions.\n \nThe noise residuals at the output of the closed-loop are represented in Figure \\ref{fig:closeloop_4TABCD}. It shows that the fringe sensing noise is always dominated by the residual atmospheric noise, which is due to the delay between the phase estimation and the moment the correction is applied to the delay lines as well as the finite acquisition frequency of the loop. The fringe sensing noise has however a strong influence on the fringe tracking residuals, as it sets the maximum loop repetition frequency that can be used. In particular, for magnitudes $K>6$, in order to keep a sufficient SNR on the measurements, the loop frequency is reduced, which results in increased closed-loop residuals.\n\n\n\n\t\\subsection{UTs case}\n\nThe same performance study has been carried out in the case of the UTs, showing a similar general behaviour as in the case of ATs. The main difference is that the {\\it operational limiting magnitude} reaches $K=9.2$, and the limit for fringe detection $K=11.2$. Furthermore, the coupling efficiency decreases for stars fainter than $V=10$, because of the reduced performance of the MACAO adaptive optics system. According to our simulations, one can consider that for V magnitudes higher than $\\sim12$, the MACAO correction becomes insufficient to feed the fringe sensor with a beam stable enough. However telescope vibrations are expected to strongly affect the residual piston jitter at the output of the closed-loop (an effect not simulated in \\texttt{sim2GFT}).\n\nThese expected limiting magnitude are in agreement with the results of instruments in operation. Indeed, PRIMA FSU detected but did not track noisy fringes on a $K=11.7$ target\\cite{sahlmann_2009} on the UTs.\n\n\n\n\t\\subsection{Case of mixed array ATs + UTs}\n\nCombining UTs and ATs allows to access to more apertures and to increase the SNR. From a technical point of view, mixing ATs with UTs is possible since the VLTI infrastructure is designed to allow the combination of apertures of different sizes. This case has not been simulated, but simple calculations on the fringe SNR show an increase in the limiting magnitude up to 1.2 mag compared to the ATs+ATs case.\n\n\n\n\t\\subsection{Influence of the atmospheric conditions}\n\t\nThe influence of atmospheric conditions on the fringe tracking performance is mainly twofold: on the one hand it determines the input atmospheric noise that needs to be corrected, and on the other hand it affects the amount of available coherent photons since it determines the injection efficiency into SM fibers. The POPS performance are simulated from bad atmospheric conditions (seeing $\\epsilon_0=1.1\"$ and coherence time $\\tau_0=2$\\,ms) to excellent ones ($\\epsilon_0=0.5\"$, $\\tau_0=10$\\,ms) for the ATs. The main conclusion is that the limiting magnitude increases by about 1~magnitude between the bad and the excellent conditions. With integration time of $10$\\,ms the limit to  maintain the sensing noise below $100$\\, nm varies between $K=7.7$ and $K=8.8$ depending on the conditions. \n\n\n\n\n\n\n\\section{CONCLUSION}\nPOPS is a K band fringe sensor based on the integrated optics technologies. It measures the fringe position (phase and group delay) thanks to a spatial ABCD fringe coding, dispersed over 5 spectral channels. For the moment only the 4T concept is clearly defined and will recombine the 6 possible baselines to improve the fringe tracking stability in varying observationnal and atmospherical conditions. Realistic simulations show the expected performance and limiting magnitude of POPS are similar to those obtained with the current instruments in operation, in particular the PRIMA FSU.\n\nPOPS will highly benefit from the technological breakthroughs in the coming years. In particular, the developments of the GRAVITY beam combiners will be directly applicable to the proposed POPS design, especially the improvement of the components transmission and the validation of the polarization controllers.\n\nPOPS has also an instrument layout similar to the PIONIER visiting instrument, which will be commissioned at VLTI at the end of 2010. POPS should then benefit from the on-sky experience feedback of PIONIER.\n\nFinally, the availability of fast new generation detectors with smaller read-out noise will impact several of the system choices of POPS. In particular, with such detectors, the multi-axial scheme is significantly less penalized by the number of pixels it requires, and becomes more attractive. Indeed, the multi-axial IO components are shorter and simpler, i.e. more transmissive and probably less chromatic. Furthermore a component designed for 6 telescopes can be used also to combine 4 telescopes, with a limited loss of SNR with respect to a design built for 4T only. Consequently multi-axial combination could be a great alternative with these future detectors. The final choice for the 6T combination concept is therefore still open. \n\n\n\n\n\n\n\\acknowledgments\n\\noindent This phase A study was supported by funds provided by ESO and has made use of the Jean-Marie Mariotti Center \\texttt{Aspro} service \\footnote{Available at http:\/\/www.jmmc.fr\/aspro}.\n\n\n\\bibliographystyle{spiebib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is known that in Maxwell's electrodynamics a point-like charge has an infinite electromagnetic energy. This problem of singularity was solved in the Born-Infeld (BI) electrodynamics \\cite{Born}, \\cite{Infeld}, \\cite{Plebanski} where a new parameter with the dimension of the length was introduced. The dimensional constant introduced in BI electrodynamics gives the upper bound on the possible electric field. Therefore, non-linear electrodynamics can give a finite electromagnetic energy of an electron. It should be noted that in QED one-loop quantum corrections give contributions to classical electrodynamics that result in the appearance of non-linear terms in the Lagrangian \\cite{Heisenberg}, \\cite{Schwinger}, \\cite{Adler}. Some examples of non-linear electrodynamics were considered in \\cite{Kruglov}, \\cite{Kruglov1}, \\cite{Kruglov2}, \\cite{Kruglov3}, \\cite{Gaete}, \\cite{Hendi} and \\cite{Kruglov5}. Thus, for strong electromagnetic fields in the vacuum we have to take into account non-linear effects. In addition, nonlinear electrodynamics can arise due to possible quantum gravity corrections.\n\nIn this paper we suggest and investigate a new model of nonlinear electrodynamics which has the same attractive feature as in BI electrodynamics: a finite electromagnetic energy of a point-like charge.\nWe introduce the dimensional parameters $\\beta$, $\\gamma$ and the dimensionless parameter $C$.\nAt $C=0$, $\\gamma=0$ the model becomes Maxwell's electrodynamics. If $\\gamma\\neq 0$ the model possesses the phenomenon of birefringence. We note that in generalized BI electrodynamics \\cite{Kruglov4} birefringence takes place.\n\nThe paper is organized as follows. In section 2, we postulate the Lagrangian of a new model of non-linear electrodynamics. The field equations are represented in the form of Maxwell's equations with\nthe electric permittivity $\\varepsilon_{ij}$ and  magnetic permeability $\\mu_{ij}$ tensors depending on electromagnetic fields. It is shown that the electric field of a point-like charge is not singular at the origin. In section 3 we investigate the effect of vacuum birefringence in the presence of a constant and uniform magnetic field.\nThe canonical and symmetrical Belinfante energy-momentum tensors, the dilatation\ncurrent and its non-zero divergence are obtained in  section 4. The section 5 is devoted to the conclusion.\n\nWe use the Heaviside-Lorentz system with $\\hbar =c=\\varepsilon_0=\\mu_0=1$ , and the Euclidian metric. Greek\nletters run from $1$ to $4$ and Latin letters run from $1$ to $3$.\n\n\\section{The model and field equations}\n\nWe postulate the Lagrangian density of nonlinear electrodynamics\n\\begin{equation}\n{\\cal L} = -{\\cal F}-\\frac{C}{\\beta}\\arcsin(\\beta{\\cal F})+\\frac{\\gamma}{2}{\\cal G}^2,\n \\label{1}\n\\end{equation}\nwhere $\\beta$, $\\gamma$ are parameters with the dimension of length$^4$ ($\\beta{\\cal F}$, $\\gamma{\\cal G}$ are dimensionless) and $C$ is a dimensionless parameter. The Lorentz-invariants are ${\\cal F}=(1\/4)F_{\\mu\\nu}^2=(\\textbf{B}^2-\\textbf{E}^2)\/2$, ${\\cal G}=(1\/4)F_{\\mu\\nu}\\tilde{F}_{\\mu\\nu}=\\textbf{E}\\cdot \\textbf{B}$, and $F_{\\mu\\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu$ is the field strength tensor,  $\\tilde{F}_{\\mu\\nu}=(1\/2)\\varepsilon_{\\mu\\nu\\alpha\\beta}F_{\\alpha\\beta}$ is a dual tensor ($\\varepsilon_{1234}=-i$) and $A_\\mu$ is the 4-vector-potential.\nAt $C = 0$, $\\gamma=0$ the Lagrangian density (1) becomes the Maxwell Lagrangian density.\nWe expect that the non-linear model introduced is\nan effective non-linear model of electrodynamics which occurs at strong electromagnetic fields.\n\nFrom Eq. (1) and the Euler-Lagrange equations we obtain the equations of motion\n\\begin{equation}\n\\partial_\\mu\\left(F_{\\mu\\nu}+\\frac{CF_{\\mu\\nu}}{\\sqrt{1-\\left(\\beta{\\cal F}\\right)^2}}-\\gamma {\\cal G}\\tilde{F}_{\\mu\\nu}\\right)=0.\n\\label{2}\n\\end{equation}\nWith the help of the expression for the electric displacement field\n$\\textbf{D}=\\partial{\\cal L}\/\\partial \\textbf{E}$ ($E_j=iF_{j4}$), we obtain\nfrom Eq.(1)\n\\begin{equation}\n\\textbf{D}=\\left(1+\\frac{C}{\\sqrt{1-\\left(\\beta{\\cal F}\\right)^2}}\\right)\\textbf{E}+\\gamma{\\cal G}\\textbf{B}.\n\\label{3}\n\\end{equation}\nEq.(3) can be represented in the tensor form, $D_i=\\varepsilon_{ij}E_j$, where the electric permittivity tensor $\\varepsilon_{ij}$ is\n\\begin{equation}\n\\varepsilon_{ij}=\\varepsilon\\delta_{ij}+\\gamma B_iB_j,~~~~\\varepsilon=1+\\frac{C}{\\sqrt{1-\\left(\\beta{\\cal F}\\right)^2}}.\n\\label{4}\n\\end{equation}\nThe magnetic field is given by $\\textbf{H}=-\\partial{\\cal L}\/\\partial \\textbf{B}$ ($B_j=(1\/2)\\varepsilon_{jik}F_{ik}$, $\\varepsilon_{123}=1$), and we obtain from Eq.(1)\n\\begin{equation}\n\\textbf{H}= \\left(1+\\frac{C}{\\sqrt{1-\\left(\\beta{\\cal F}\\right)^2}}\\right)\\textbf{B}-\\gamma{\\cal G}\\textbf{E}.\n\\label{5}\n\\end{equation}\nIntroducing the magnetic induction field $\\textbf{B}_i=\\mu_{ij}\\textbf{H}_j$, one finds the inverse magnetic permeability tensor $(\\mu^{-1})_{ij}$\n\\begin{equation}\n(\\mu^{-1})_{ij}=\\varepsilon\\delta_{ij}-\\gamma E_iE_j.\n\\label{6}\n\\end{equation}\nField equations (2) can be rewritten, with the aid of Eqs. (3),(5), in the form of the first pair of Maxwell's equations\n\\begin{equation}\n\\nabla\\cdot \\textbf{D}= 0,~~~~ \\frac{\\partial\\textbf{D}}{\\partial\nt}-\\nabla\\times\\textbf{H}=0.\n\\label{7}\n\\end{equation}\nThe Bianchi identity\n\\begin{equation}\n\\partial_\\mu \\widetilde{F}_{\\mu\\nu}=0,\n\\label{8}\n\\end{equation}\ngives the second pair of Maxwell's equations\n\\begin{equation}\n\\nabla\\cdot \\textbf{B}= 0,~~~~ \\frac{\\partial\\textbf{B}}{\\partial\nt}+\\nabla\\times\\textbf{E}=0.\n\\label{9}\n\\end{equation}\nEquations (7),(9) are Maxwell's equations where the electric permittivity tensor $\\varepsilon_{ij}$ and magnetic permeability tensor $\\mu_{ij}$ depend on the fields $\\textbf{E}$ and $\\textbf{B}$.\nEqs. (3),(5) mimic a medium with complicated properties.\nFrom equations (3),(5), one can obtain the equality $\\textbf{D}\\cdot\\textbf{H}=(\\varepsilon^2+2\\gamma\\varepsilon {\\cal F}-\\gamma^2{\\cal G}^2)\\textbf{E}\\cdot\\textbf{B}$. As $\\textbf{D}\\cdot\\textbf{H}\\neq\\textbf{E}\\cdot\\textbf{B}$ the dual symmetry is broken \\cite{Gibbons}. At $C=0$ ($\\varepsilon=1$), $\\gamma=0$, we arrive at classical electrodynamics and the dual symmetry is recovered. It should be noted that BI electrodynamics is dual symmetrical but in QED, due to one loop quantum corrections (Heisenberg-Euler Lagrangian), the dual symmetry is broken.\n\nLet us consider electrostatics, $\\textbf{B}=\\textbf{H}=0$. Then the equation for the point-like charge is given by\n\\begin{equation}\n\\nabla\\cdot \\textbf{D}_0=e\\delta(\\textbf{r})\n\\label{10}\n\\end{equation}\nwith the solution\n\\begin{equation}\n\\textbf{D}_0=\\frac{e}{4\\pi r^3}\\textbf{r}.\n\\label{11}\n\\end{equation}\nEq. (11), taking into consideration (4), becomes\n\\begin{equation}\nE_0\\left(1+\\frac{C}{\\sqrt{1-\\beta^2 E_0^4\/4}}\\right)=\\frac{e}{4\\pi r^2}.\n\\label{12}\n\\end{equation}\nThe solution to Eq. (12) at $r\\rightarrow 0$ is given by\n\\begin{equation}\nE_0=\\frac{\\sqrt{2}}{\\sqrt{\\beta}}.\n\\label{13}\n\\end{equation}\nThus, the maximum electric field (13) has a finite value similar to BI electrodynamics. In linear electrodynamics the electric field strength possesses the singularity that results in an infinite electric energy of the point-like charged particle.\n\nIt is useful to define unitless values\n\\[\nx=\\frac{4\\sqrt{2}\\pi r^2}{e\\sqrt{\\beta}},~~~~y=\\sqrt{\\frac{\\beta}{2}}E_0.\n\\]\nThen Eq. (12) is rewritten as\n\\begin{equation}\nx=\\frac{\\sqrt{1-y^4}}{y\\left(\\sqrt{1-y^4}+C\\right)}.\n\\label{14}\n\\end{equation}\nAt $x\\rightarrow 0$ ($r\\rightarrow 0$), we have $y\\rightarrow 1$ and this is equivalent to Eq. (13). When distance $r$ approaches to infinity, $x\\rightarrow \\infty$, and $y\\rightarrow 0$.\nAs a result, the electric field of a point-like charged particle is finite at the origin.\nThe function $x(y)$ for different values of the parameter $C$ is presented in Fig.1.\n\\begin{figure}[h]\n\\includegraphics[height=3.0in,width=4in]{Fig2.eps}\n\\caption{\\label{fig.1}$x$ versus $y$ for different values of the parameter $C$.}\n\\end{figure}\n\n\\section{Vacuum birefringence}\n\nVacuum birefringence within QED was investigated in \\cite{Adler1}, \\cite{Biswas}.\nLet us consider the external constant and uniform magnetic induction field $\\textbf{B}_0=(B_0,0,0)$ and the plane electromagnetic wave $(\\textbf{e}, \\textbf{b}$)\n\\begin{equation}\n\\textbf{e}=\\textbf{e}_0\\exp\\left[-i\\left(\\omega t-kz\\right)\\right],~~~\\textbf{b}=\\textbf{b}_0\\exp\\left[-i\\left(\\omega t-kz\\right)\\right],\n\\label{15}\n\\end{equation}\npropagating in the $z$-direction.\nThe total electromagnetic fields are $\\textbf{E}=\\textbf{e}$, $\\textbf{B}=\\textbf{b}+\\textbf{B}_0$. We consider the case of the strong magnetic induction field. Then amplitudes of the electromagnetic wave, $e_0, b_0$, are small  compared to the magnetic induction field, and we have $e_0,b_0\\ll B_0$. Linearizing Eq. (3),(5) one finds the electric permittivity and magnetic permeability tensors\n\\begin{equation}\n\\varepsilon_{ij}=\\varepsilon\\delta_{ij}+\\gamma \\delta_{i1}\\delta_{j1}B_{0}^2,~~~\\varepsilon=1+\\frac{C}{\\sqrt{1-\\beta^2 B_0^4\/4}},~~~\\mu_{ij}=\\mu\\delta_{ij},~~~\\mu=\\varepsilon^{-1}.\n\\label{16}\n\\end{equation}\nAs $C\\ll 1$ and $\\beta^2 B_0^4\\ll 1$, the term $C\\beta^2 B_0^4$ in the expression for $\\mu_{ij}$ was neglected.\nFrom Maxwell's equations (7), (9) we obtain the wave equation\n\\begin{equation}\n\\partial_j^2 E_i-\\mu\\varepsilon_{ij}\\partial^2_tE_j-\\partial_i\\partial_jE_j=0.\n\\label{17}\n\\end{equation}\nIn the case when the polarization is parallel to the external magnetic field, $\\textbf{e}=e_0(1,0,0)$, and from Eq. (17) one finds the relation $\\mu\\varepsilon_{11}\\omega^2=k^2$. Then the index of refraction is given by\n\\begin{equation}\nn_\\|=\\sqrt{\\mu\\varepsilon_{11}}=\\sqrt{1+\\frac{\\gamma B_0^2\\sqrt{1-\\left(\\beta B_0^2\\right)^2\/4}}{C+\\sqrt{1-\\left(\\beta B_0^2\\right)^2\/4}}}.\n\\label{18}\n\\end{equation}\nIf the polarization of the electromagnetic wave is perpendicular to the external induction magnetic field, $\\textbf{e}=e_0(0,1,0)$, and the relation $\\mu\\varepsilon\\omega^2=k^2$ holds. Then the index of refraction is\n\\begin{equation}\nn_\\perp=\\sqrt{\\varepsilon\\mu}=1.\n\\label{19}\n\\end{equation}\nAs a result, the effect of vacuum birefringence takes place and the phase velocity depends on the polarization of the electromagnetic wave. When the polarization of the electromagnetic wave is parallel to the external magnetic field, $\\textbf{e}_0\\parallel \\textbf{B}_0$, the speed becomes $v_\\|=1\/n_\\|$ ($c=1$), and in the case $\\textbf{e}\\perp \\textbf{B}_0$ the speed of the electromagnetic wave is $v_\\perp=1$.\n\n\n\\section{The energy-momentum tensor and dilatation current}\n\nFrom Eq. (1) and the expression of the canonical energy-momentum tensor\n\\[\nT_{\\mu\\nu}^{c}=(\\partial_\\nu\nA_\\alpha)\\frac{\\partial{\\cal L}}{\\partial(\\partial_\\mu\nA_\\alpha)}-\\delta_{\\mu\\nu}{\\cal L},\n\\]\nwe obtain\n\\begin{equation}\nT_{\\mu\\nu}^{c}=-(\\partial_\\nu A_\\alpha)\\left(F_{\\mu\\alpha}\\varepsilon-\\gamma{\\cal G}\\tilde{F}_{\\mu\\alpha}\\right)-\\delta_{\\mu\\nu}{\\cal\nL}. \\label{20}\n\\end{equation}\nThe canonical energy-momentum tensor (20) is conserved,  $\\partial_\\mu T^c_{\\mu\\nu}=0$, but it is not\ngauge-invariant and symmetrical tensor. To obtain the symmetric\nBelinfante tensor we use the relation \\cite{Coleman}:\n\\begin{equation}\nT_{\\mu\\nu}^{B}=T_{\\mu\\nu}^{c}+\\partial_\\beta X_{\\beta\\mu\\nu},\n\\label{21}\n\\end{equation}\nwhere\n\\begin{equation}\nX_{\\beta\\mu\\nu}=\\frac{1}{2}\\left[\\Pi_{\\beta\\sigma}\\left(\\Sigma_{\\mu\\nu}\\right)_{\\sigma\\rho}\n-\\Pi_{\\mu\\sigma}\\left(\\Sigma_{\\beta\\nu}\\right)_{\\sigma\\rho}-\n\\Pi_{\\nu\\sigma}\\left(\\Sigma_{\\beta\\mu}\\right)_{\\sigma\\rho}\\right]A_\\rho,\n\\label{22}\n\\end{equation}\n\\begin{equation}\n\\Pi_{\\mu\\sigma}=\\frac{\\partial{\\cal L}}{\\partial(\\partial_\\mu\nA_\\sigma)}=-F_{\\mu\\sigma}\\varepsilon+\\gamma{\\cal G}\\tilde{F}_{\\mu\\sigma}.\n\\label{23}\n\\end{equation}\nBecause $X_{\\beta\\mu\\nu}=-X_{\\mu\\beta\\nu}$, the equation $\\partial_\\mu\\partial_\\beta X_{\\beta\\mu\\nu}=0$ holds. As a result, the symmetrical Belinfante tensor is conserved, $\\partial_\\mu T^B_{\\mu\\nu}=\\partial_\\mu T^c_{\\mu\\nu}=0$. The generators of the Lorentz transformations $\\Sigma_{\\mu\\alpha}$ have the matrix elements\n\\begin{equation}\n\\left(\\Sigma_{\\mu\\alpha}\\right)_{\\sigma\\rho}=\\delta_{\\mu\\sigma}\\delta_{\\alpha\\rho}\n-\\delta_{\\alpha\\sigma}\\delta_{\\mu\\rho},\n\\label{24}\n\\end{equation}\nand from Eqs. (22)-(24), we find\n\\begin{equation}\n\\partial_\\beta X_{\\beta\\mu\\nu}=\\Pi_{\\beta\\mu}\\partial_\\beta A_\\nu,\n\\label{25}\n\\end{equation}\nand from Eq. (2) one obtains $\\partial_\\mu\\Pi_{\\mu\\nu}=0$. Using\nEqs. (22),(25), we find the Belinfante tensor (21)\n\\begin{equation}\nT_{\\mu\\nu}^{B}=-F_{\\nu\\alpha}\\left(F_{\\mu\\alpha}\\varepsilon-\\gamma{\\cal G}\\tilde{F}_{\\mu\\alpha}\\right)-\\delta_{\\mu\\nu}{\\cal L}.\n\\label{26}\n\\end{equation}\nThe trace of the energy-momentum tensor (26) is\n\\begin{equation}\nT_{\\mu\\mu}^{B}=-\\frac{4C{\\cal F}}{\\sqrt{1-(\\beta{\\cal F})^2}}+\\frac{4C}{\\beta}\\arcsin(\\beta{\\cal F})+2\\gamma{\\cal G}^2.\n\\label{27}\n\\end{equation}\nIf $C=0$, $\\gamma=0$ we arrive at Maxwell's electrodynamics, and the trace of the energy-momentum tensor (27) is zero. According to \\cite{Coleman}, we use the modified dilatation current\n\\begin{equation}\nD_{\\mu}^{B}=x_\\alpha T_{\\mu\\alpha}^{B}+V_\\mu,\n\\label{28}\n\\end{equation}\nand the field-virial $V_\\mu$ is given by\n\\begin{equation}\nV_\\mu=\\Pi_{\\alpha\\beta}\\left[\\delta_{\\alpha\\mu}\\delta_{\\beta\\rho}\n-\\left(\\Sigma_{\\alpha\\mu}\\right)_{\\beta\\rho}\\right]A_\\rho.\n\\label{29}\n\\end{equation}\nOne can verify that $V_\\mu=0$, and the modified dilatation current (28) is $D_{\\mu}^{B}=x_\\alpha T_{\\mu\\alpha}^{B}$.\nThe divergence of the dilatation current becomes\n\\begin{equation}\n\\partial_\\mu D_{\\mu}^{B}=T_{\\mu\\mu}^B.\n\\label{30}\n\\end{equation}\nThus, the dilatation (scale) symmetry is violated because the dimensional parameters $\\beta$, $\\gamma$ were introduced. It should be noted that in BI electrodynamics the scale and conformal symmetries are broken \\cite{Kruglov4}, but linear Maxwell's electrodynamics is conformal symmetrical theory.\n\n\\subsection{Energy of the point-like charge}\n\nLet us consider the total electric energy of the charged point-like particle, for example electron's energy.\nOne can obtain the energy density of the electric field ($\\textbf{B}=0$) from Eq. (26)\n\\begin{equation}\n\\rho_E=T^B_{44}=E^2\\left(\\frac{1}{2}+\\frac{C}{\\sqrt{1-(\\beta E^2)^2\/4}}\\right)-\\frac{C}{\\beta}\\arcsin\\left(\\frac{\\beta E^2}{2}\\right).\n\\label{31}\n\\end{equation}\nThe variable $\\beta^{1\/4}{\\cal E}$, where ${\\cal E}=\\int \\rho_E dV$ is a total energy, becomes\n\\begin{equation}\n\\beta^{1\/4}{\\cal E}=\\frac{e^{3\/2}}{2^{11\/4}\\sqrt{\\pi}}\\int_0^\\infty \\sqrt{x}\\left[\\left(1+\\frac{2C}{\\sqrt{1-y^4}}\\right)y^2-C\\arcsin(y^2)\\right]dx,\n\\label{32}\n\\end{equation}\nwhere the function $x(y)$ is given by Eq. (14) ($e$ is the charge of the electron). Changing the variables in Eq. (32), one can represent (32) as\n\\[\n\\beta^{1\/4}{\\cal E}=\\frac{e^{3\/2}}{2^{11\/4}\\sqrt{\\pi}}\n\\]\n\\begin{equation}\n\\times\\int_0^1 \\frac{\\left[y^2\\left(\\sqrt{1-y^4}+2C\\right)-C\\sqrt{1-y^4}\\arcsin(y^2)\\right]\\left[C\\left(1+y^2\\right)+(1-y^4)^{3\/2}\\right]}\n{y^{5\/2}(1-y^4)^{3\/4}\\left(\\sqrt{1-y^4}+C\\right)^{5\/2}}dy.\n\\label{33}\n\\end{equation}\nNumerical calculations of the integral (33) give the values $\\beta^{1\/4}{\\cal E}$, for different parameters $C$, represented in Table 1.\n\\begin{table}[ht]\n\\caption{The normalized total energy $\\beta^{1\/4}{\\cal E}$ of a point-like charged particle }\n\\vspace{5mm}\n\\centering\n\\begin{tabular}{c c c c c c c c c c c c}\n\\hline \\hline\n$C$ & $0.00001$ & $0.0001$ & $0.001$ & $0.1$ & $0.2$ & $0.3$ & $0.4$ \\\\[0.5ex]\n\\hline\n$\\beta^{1\/4}{\\cal E}$ & $0.0271$ & $0.0277$ & $0.0279$ & 0.0263 & 0.0249 & 0.0238 & 0.0228\\\\\n[1ex]\n\\hline\n\\end{tabular}\n\\label{table:crit\n\\end{table}\nIf one assumes that the electron mass has a pure electromagnetic nature according to the Abraham and Lorentz idea \\cite{Born1}, \\cite{Rohrlich}, \\cite{Spohn}, then ${\\cal E}=0.51$~MeV. Taking the parameter $C$ to be $C=0.00001$, we obtain the length parameter $l_1=\\beta^{1\/4}=10.49$ fm. Thus, the electromagnetic nature of the electron may be realized in the model suggested.\n\n\\section{Conclusion}\n\nThus, we suggest a new model of nonlinear electrodynamics with three parameters $C$, $\\beta$ and $\\gamma$.  The values $\\beta^{1\/4}$, $\\gamma^{1\/4}$ possess the dimensions of the length and can be treated as affective constants due to some effects, for instance, quantum gravity (in electromagnetic units, according to Eq. (1), the constants $\\beta^{-1\/2}$ and $\\gamma^{-1\/2}$ have the same dimensions as the field strength). In the presence of the external constant and uniform induction magnetic field the phenomenon of vacuum birefringence takes place. In the particular case when $\\gamma=0$ birefringence vanishes. The indices of refraction for two polarizations of electromagnetic waves, parallel and perpendicular to the magnetic field, were calculated. As a result, phase velocities of electromagnetic waves depend on polarizations if $\\gamma\\neq 0$. The canonical and symmetrical Belinfante energy-momentum tensors and the dilatation current were obtained showing that the dilatation symmetry is broken in the model considered. The scale symmetry is violated because the dimensional parameters $\\beta$, $\\gamma$ were introduced. We have demonstrated that the electric field of a point-like charge is not singular, and the maximum possible electric field is equal to $E_{max}=\\sqrt{2}\/\\sqrt{\\beta}$ similar to BI electrodynamics. We show that the static electric self-energy of point-like particles is finite, and therefore, one can speculate that the mass of the electron possesses a pure electromagnetic energy according to old idea.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}