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+{"text":"\\section{Introduction}\n\\label{section1}\n\\footnote{This version corrects an error in the published version.}Deep neural networks (DNNs) have been demonstrated to be vulnerable to adversarial examples \\cite{szegedy2013intriguing, goodfellow2014explaining,liao2018defense,shen2017ape,ma2018characterizing,wu2020adversarial, pmlr-v139-zhou21e}. The adversarial samples are typically generated by adding imperceptible but adversarial noise to natural samples. The vulnerability of DNNs seriously threatens many decision-critical deep learning applications, such as image processing \\cite{lecun1998gradient, Zagoruyko2016WRN, 2017Mask, xia2020part,  ma2021understanding, xia2021robust}, natural language processing \\cite{sutskever2014sequence} and speech recognition \\cite{sak2015fast}. The urgent demand to reduce these security risks prompts the development of adversarial defenses.\n\nMany researchers have made extensive efforts to improve the adversarial robustness of DNN. A major class of adversarial defense focuses on exploiting adversarial samples to help train the target model to achieve adversarially robust performance \\cite{madry2017towards,ding2019sensitivity,zhang2019theoretically,wang2019improving,zhou2021modeling,zheng2021regularizing,yang2021class}. However, the dependence between the output of the target model and the input adversarial sample has not been well studied yet. Explicitly measuring this dependence could help train the target model to make predictions that are closely relevant to the given objectives \\cite{belghazi2018mutual,sanchez2020learning}.\n\nIn this paper, we investigate the dependence from the perspective of information theory. Specifically, we exploit the mutual information (MI) to explicitly measure the dependence of the output on the adversarial sample. MI is an entropy-based measure that can reflect the dependence degree between variables. A larger MI typically indicates stronger dependence between the two variables. However, directly exploiting MI between the input and its corresponding output (called \\textit{standard MI}) to measure the dependence has limitations in improving classification accuracy for adversarial samples.\n\nNote that adversarial samples contain natural and adversarial patterns. As shown in \\cref{fig1}, given a target model and an adversarial sample, the natural pattern is derived from the corresponding natural sample, and the adversarial pattern is derived from the adversarial noise in the adversarial sample. The adversarial pattern controls the flip of the prediction from the correct label to the wrong label \\cite{ilyas2019adversarial}. The standard MI of the adversarial sample reflects the confused dependence of the output on the natural pattern and the adversarial pattern. Maximizing the standard MI of the adversarial sample to guide the target model may result in a larger dependence between the output and the adversarial pattern. This may cause more interference with the prediction of the target model. Therefore, directly maximizing the standard MI to help train the target model cannot surely promote the adversarial robustness.\n\nTo handle this issue, we propose to disentangle the standard MI to explicitly measure the dependence of the output on the natural pattern and the adversarial pattern, respectively. As shown in \\cref{fig1}, we disentangle the standard MI into the \\textit{natural MI} (i.e., the MI between the output and the natural pattern) and the \\textit{adversarial MI} (i.e., the MI between the output and the adversarial pattern). To present the reasonability of the disentanglement, we theoretically demonstrate that standard MI is closely related to the linear sum of natural MI and adversarial MI. In addition, how to effectively estimate the natural MI and the adversarial MI is a crucial problem. Inspired by the \\textit{mutual information maximization} in \\citet{hjelm2018learning,zhu2020learning}, we design a neural network-based method to train two MI estimators to estimate the natural MI and adversarial MI respectively. The detailed discussion can be found in \\cref{section3.2}. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=3.0in]{fig1.pdf}}\n\\caption{A visual illustration of disentangling the standard MI into the natural MI and the adversarial MI. The \\textit{longdash} lines show that the adversarial sample is disassembled into the natural pattern (derived from the natural sample) and the adversarial pattern (derived from the adversarial noise). The \\textit{dotted} lines denote the operation of disentangling the standard MI into the natural MI and the adversarial MI.}\n\\label{fig1}\n\\end{center}\n\\vskip -0.3in\n\\end{figure}\n\nBased on the above MI estimation, we develop an adversarial defense algorithm called \\textit{natural-adversarial mutual information-based defense} (NAMID) to enhance the adversarial robustness. Specifically, we introduce an optimization strategy using the natural MI and the adversarial MI on the basis of the adversarial training manner. The optimization strategy is to maximize the natural MI of the input adversarial sample and minimize its adversarial MI simultaneously. By iteratively executing the procedures of generating adversarial samples and optimizing the model parameters, we can learn an adversarially robust target model. \n\nThe main contributions in this paper are as follows:\n\\begin{itemize}\n    \\item Considering the adversarial sample has the natural pattern and the adversarial pattern, we propose the natural MI and the adversarial MI to explicitly measure the dependence of the output on the different patterns.\n    \\item We design a neural network-based method to effectively estimate the natural MI and the adversarial MI. By exploiting the MI estimation networks, we develop a defense algorithm to train a robust target model. \n    \\item We empirically demonstrate the effectiveness of the proposed defense algorithm on improving the classification accuracy. The evaluation experiments are conducted against multiple adversarial attacks.\n\\end{itemize}\n\nThe rest of this paper is organized as follows. In Section~\\ref{section2}, we introduce some preliminary information and briefly review related works. In Section~\\ref{section3}, we propose the natural MI and the adversarial MI, and develop an adversarial defense method. Experimental results are provided in Section~\\ref{section4}. Finally, we conclude this paper in Section~\\ref{section5}.\n\n\\section{Preliminaries}\n\\label{section2}\nIn this section, we introduce some preliminary about notation, the problem setting and mutual information (MI). We also review some representative literature on adversarial attacks and defenses.\n\n\\noindent\\textbf{Notation.} \nWe use \\textit{capital} letters such as $X$ and $Y$ to represent random variables, and \\textit{lower-case} letters such as $x$ and $y$ to represent realizations of random variables $X$ and $Y$, respectively. For norms, we denote by $\\|x\\|$ a generic norm, by $\\|x\\|_{\\infty}$ the $L_{\\infty}$ norm of $x$, and by $\\|x\\|_{2}$ the $L_{2}$ norm of $x$. In addition, let $\\mathbb{B}(x, \\epsilon)$ represent the neighborhood of $x$: $\\{\\tilde{x}:\\|\\tilde{x}-x\\| \\leq \\epsilon$\\}, where $\\epsilon$ is the perturbation budget. We define the \\textit{classification function} as $f: \\mathcal{X} \\rightarrow \\{1,2,\\ldots,C\\}$. It can be parameterized, e.g., by a deep neural network $h_{\\theta}$ with model parameter $\\theta$.\n\n\n\\noindent\\textbf{Problem setting.} \nIn this paper, we focuses on a classification task under the adversarial environment. Let $X$ and $Y$ be the variables for natural instances and ground-truth labels respectively. We sample natural data $\\{(x_i, y_i)\\}_{i=1}^{n}$ according to the distribution of variables $(X,Y)$. Given a pair of natural data $(x,y)$, the adversarial instance $\\tilde{x}$ satisfies the following constraint:\n\\begin{equation}\n\\label{eq1}\nf\\left(\\tilde{x}\\right) \\neq y \\quad \\text { s.t.} \\quad\\left\\|x-\\tilde{x}\\right\\| \\leq \\epsilon \\text{,}\n\\end{equation}\nwhere $\\tilde{x}=x+n$, $n$ denotes the adversarial noise. Our aim is to develop an adversarial defense method to help train the classification model $h_{\\theta}$ to make normal predictions.\n\n\\noindent\\textbf{Mutual information.}\nMI is an entropy-based measure that can reflect the dependence degree between variables. A larger MI typically indicates a stronger dependence between the two variables. Various methods have been proposed for estimating MI \\cite{moon1995estimation,darbellay1999estimation,kandasamy2015nonparametric,moon2017ensemble}. A representative and efficient estimator is the mutual information neural estimator (MINE) \\cite{belghazi2018mutual}. MINE is empirically demonstrated its superiority in estimation accuracy and efficiency, and proved that it is strongly consistent with the true MI. Besides, the work in \\citet{hjelm2018learning} points that using the complete input to estimate MI is often insufficient for classification task. Instead, estimating MI between the high-level feature and local patches of the input is more suitable. Therefore, in this paper, we refer the local DIM estimator \\cite{hjelm2018learning} to estimate MI. The definition of MI and other details are presented in \\cref{appendix_1}\n\n\\noindent\\textbf{Adversarial attacks.} \nAdversarial noise can be crafted by optimization-based attacks, such as PGD \\cite{madry2017towards}, AA \\cite{croce2020reliable}, CW \\cite{carlini2017towards} and DDN \\cite{rony2019decoupling}. Besides, some attacks such as FWA \\cite{wu2020stronger} and STA \\cite{xiao2018spatially} focus on mimicking non-suspicious vandalism by exploiting the geometry and spatial information. These attacks constrain the perturbation boundary by a small norm-ball $\\|\\cdot\\|_{p} \\leq \\epsilon$, so that their adversarial instances can be perceptually similar to natural instances. \n\n\\noindent\\textbf{Adversarial defenses.} The issue of adversarial attacks promotes the development of adversarial defenses. A major class of adversarial defense methods is devoted to enhance the adversarial robustness in an adversarial training manner \\cite{madry2017towards,ding2019sensitivity,zhang2019theoretically,wang2019improving}. They augment training data with adversarial samples and use a min-max formulation to train the target model \\cite{madry2017towards}. However, these methods do not explicitly measure the dependence between the adversarial sample and its corresponding output. In addition, some data pre-processing based methods try to remove adversarial noise by learning denoising functions \\cite{liao2018defense,naseer2020self,zhou2021removing} or feature-squeezing functions \\cite{guo2017countering}. However, these methods may suffer from the problems of human-observable loss \\cite{xu2017feature} and residual adversarial noise \\cite{liao2018defense}, which would affect the final prediction. To avoid the above problems, we propose to exploit the natural MI and the adversarial MI to learn an adversarially robust classification model in the adversarial training manner.\n\\vskip -0.1in\n\n\\section{Methodology}\n\\label{section3}\nIn this section, we first illustrate the motivation for using mutual information (MI) and disentangling the standard mutual information into the \\textit{natural MI} and the \\textit{adversarial MI} (\\cref{section3.1}). Next, we theoretically prove the reasonability of the disentanglement and introduce how to effectively estimate the natural MI and the adversarial MI (\\cref{section3.2}). Finally, we propose an adversarial defense algorithm which contains an MI-based optimization strategy (\\cref{section3.3}). The code is available at \\url{https:\/\/github.com\/dwDavidxd\/MIAT}.\n\n\\subsection{Motivation}\n\\label{section3.1}\nFor adversarial samples, The predictions of the target model are typically significantly irrelevant to the given objectives in the inputs. Studying the dependence between the adversarial sample and its corresponding output is considered to be beneficial for improving the adversarial robustness. The dependence could be exploited as supervision information to help train the target model to make correct predictions.\n\nEstimating the standard MI of the adversarial sample (i.e., the MI between the adversarial sample and its corresponding output in a target model) is a simple strategy to measure the dependence. However, different from natural samples, adversarial samples have two patterns, i.e., the natural pattern and the adversarial pattern. The standard MI cannot respectively consider the dependence of the output on the different patterns, which may limit its performance in helping the target model improve the adversarial robustness. \n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.8in]{fig2.pdf}}\n\\caption{The visualization of the proof-of-concept experiment. Given the natural instance $x$, its adversarial instance $\\tilde{x}$ and a target model $h$, the logit output of $h$ for $\\tilde{x}$ is denoted by $h(\\tilde{x})$. We respectively estimate the standard MI of the natural instance $\\widehat{I}(x, h(x))$, the standard MI of the adversarial instance $\\widehat{I}(\\tilde{x}, h(\\tilde{x}))$ and the MI between $h(\\tilde{x})$ and $x$. The \\textit{shaded} area represents the difference between $\\widehat{I}(\\tilde{x}, h(\\tilde{x}))$ and $\\widehat{I}(x, h(\\tilde{x}))$.}\n\\label{fig2}\n\\vskip -0.3in\n\\end{center}\n\\end{figure}\n\nSpecifically, the natural pattern is derived from the original natural sample. It provides available information for the target model to produce the right output. The adversarial pattern is derived from the adversarial noise. It controls the flip of the prediction from the correct label to the wrong label \\cite{ilyas2019adversarial}. Both the natural and adversarial patterns cause important impacts on the output, but they are mutually exclusive. Therefore, the standard MI actually measures a confused dependence.\n\nTo clearly illustrate the confused dependence, we conduct a proof-of-concept experiment. We randomly select a set of natural instances and use five attacks to generative adversarial instances. We use a classification model as the target model. The MI estimator is trained on natural instances and their outputs via the MI maximization \\cite{hjelm2018learning}. By exploiting the estimator, we respectively estimate the standard MI of the natural instance, the standard MI of the adversarial instance and the MI between the natural instance and the output for the corresponding adversarial instance. The details of the experiment are presented in \\cref{appendix_2}\n\nAs shown in \\cref{fig2}, the result shows that the standard MI of the adversarial sample is indeed smaller than that of the natural sample. However, it is still significantly larger than the MI between the output and natural pattern only (see the pink line). This shows that the standard MI of the adversarial sample contains the dependence of the output on the adversarial pattern. Thus, maximizing the standard MI may increase the dependence of the output on the adversarial pattern and cause more disturbance to the prediction. Directly maximizing the standard MI cannot comprehensively promote the target model to make more accurate predictions for adversarial samples. \n\nTo solve this problem, in this paper, we propose to disentangle the standard MI into two parts related to the natural and adversarial patterns respectively.\n\n\\subsection{Natural MI and adversarial MI}\n\\label{section3.2}\nWe define two new concepts: \\textit{natural mutual information} (natural MI) and \\textit{adversarial mutual information} (adversarial MI). The natural MI is MI between the output and the natural pattern of the input. The adversarial MI is MI between the output and the adversarial pattern of the input. To explicitly measure the dependence of the output on different patterns of the input, we need to disentangle the standard MI into the natural MI and the adversarial MI. \n\n\\subsubsection{Disentangling the standard MI}\n\\label{section3.2.1}\nIn this section, we introduce how to disentangle the standard MI and describe the reasonability of the disentanglement. We first provide Theorem 1 to illustrate the transformation relationship of MI among four variables.\n\n\\noindent \\textbf{Theorem 1.} \nLet $X, \\widetilde{X}, N, Z$ denote four random variables respectively, where $\\widetilde{X}=X+N$. Let $\\widetilde{\\mathcal{X}}$ be the feature space of $\\widetilde{X}$ and $\\mathcal{Z}$ be the feature space of $Z$. \nThen, for any function $h: \\widetilde{\\mathcal{X}} \\rightarrow \\mathcal{Z}$, we have \n\\begin{equation}\n\\label{eq2}\n\\begin{aligned}\n I(\\widetilde{X};Z) &= I(X;Z) + I(N;Z) - I(X;N;Z) \\\\ &+ H(Z|X,N) - H(Z|\\widetilde{X}) \\text{,}\n\\end{aligned}\n\\end{equation}\nwhere $I(\\cdot;\\cdot)$ denotes the MI between two variables and $I(\\cdot;\\cdot;\\cdot)$ denotes the MI between two three variables. A detailed proof is provided in \\cref{appendix_3}. \n\nThen, we apply Theorem 1 to the adversarial learning setting and obtain Corollary 1.\n\n\\noindent \\textbf{Corollary 1.}\nLet $X, \\widetilde{X}, N$ denote the random variables for the natural instance, adversarial instance and adversarial noise respectively, where $\\widetilde{X}=X+N$. Given a function parameterized by a target model $h_{\\theta}$ with model parameter $\\theta$, the logit output of $h_{\\theta}$ for $\\widetilde{X}$ is denoted by $h_{\\theta}(\\widetilde{X})$. Considering that the effects of the natural instance and the adversarial noise on the output are mutually exclusive, we assume that the MI between $X, N$ and $h_{\\theta}(\\widetilde{X})$ (i.e., $I(X;N;h_{\\theta}(\\widetilde{X}))$) is small. We also assume that the difference between $H(Z|X,N)$ and $H(Z|\\widetilde{X})$ is small (see \\cref{appendix_4} for more details), then we have:\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq3}\nI(\\widetilde{X};h_{\\theta}(\\widetilde{X})) \\approx \\underbrace{I(X;h_{\\theta}(\\widetilde{X}))}_{I_{N}}+\\underbrace{I(N;h_{\\theta}(\\widetilde{X}))}_{I_{A}} \\text{,}\n\\end{equation}\n\\vskip -0.15in\nwhere $I_{N}$ denote the MI between the output and the natural instance and $I_{A}$ denote the MI between the output and the adversarial noise. \n\nActually, we could use the original natural instance and the adversarial noise to represent the natural pattern and the adversarial pattern of the input respectively. In this way, $I_{N}$ and $I_{A}$ could denote the natural MI and the adversarial MI respectively.\n\nAccording to \\cref{eq3}, we can approximately disentangle the standard MI into the natural MI and the adversarial MI. The latter two can not only reflect the dependency between input and output as the former, but also provides independent measurements for different patterns. This is more conducive to designing specific optimization strategies for the two patterns to better alleviate the negative effects of the adversarial noise.\n\n\\subsubsection{Estimating the natural MI and the adversarial MI}\n\\label{section3.2.2}\nThe local DIM estimation method has been demonstrated to be efficient for estimating MI \\cite{hjelm2018learning,zhu2020learning}. We thus first use this method to train a estimation network for the natural MI and the adversarial MI respectively. Let $E_{\\phi_{n}}$ denote the estimation network for the natural MI and $E_{\\phi_{a}}$ denote the estimation network for the adversarial MI. Considering the inherent close relevance between the natural\/adversarial pattern and the output for the natural\/adversarial sample, we naturally use the natural\/adversarial samples to train the $E_{\\phi_{n}}$\/$E_{\\phi_{a}}$. The optimization goals for $\\phi_{n}$ and $\\phi_{a}$ are as follows:\n\n\\vskip -0.1in\n\\begin{equation}\n\\label{eq4}\n\\begin{aligned}\n&\\widehat{\\phi}_{n}=\\underset{\\phi_{n} \\in \\Phi_{N}}{\\arg \\max } \\,E_{\\phi_{n}}(h_{\\theta_{0}}(X)) \\text{,} \\\\\n&\\widehat{\\phi}_{a}=\\underset{\\phi_{a} \\in \\Phi_{A}}{\\arg \\max } \\, E_{\\phi_{a}}(h_{\\theta_{0}}(\\widetilde{X})) \\text{,}\n\\end{aligned}\n\\end{equation}\n\\vskip -0.1in\n\nwhere $\\Phi_{N}$ and $\\Phi_{A}$ denote the sets of model parameters, $h_{\\theta_0}$ denotes the pre-trained target model. It can be naturally trained or be adversarially trained. We use a ResNet-18 optimized by standard AT \\cite{madry2017towards} as the pre-trained model $h_{\\theta_0}$.  $E_{\\widehat{\\phi}_{n}}\\left(\\cdot \\right)$ is the estimated natural MI and $E_{\\widehat{\\phi}_{a}}\\left(\\cdot \\right)$ is the estimated adversarial MI, i.e.,  $E_{\\widehat{\\phi}_{n}}\\left(h_{\\theta}(X)\\right)=\\widehat{I}_{N}(X; h_{\\theta}(X))$, $E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\widetilde{X}))=\\widehat{I}_{N}(X; h_{\\theta}(\\widetilde{X}))$ and $E_{\\widehat{\\phi}_{a}}\\left(h_{\\theta}(X)\\right)=\\widehat{I}_{A}(N; h_{\\theta}(X))$,  $E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\widetilde{X}))=\\widehat{I}_{A}(N; h_{\\theta}(\\widetilde{X}))$.\n\nBy exploiting the two MI estimation network, we estimate the natural MI of the natural sample and the adversarial sample (i.e., $\\widehat{I}_{N}(X, h_{\\theta_{0}})$ and $\\widehat{I}_{N}(\\widetilde{X}, h_{\\theta_{0}})$), and the adversarial MI of the natural sample and the adversarial sample (i.e., $\\widehat{I}_{A}(X, h_{\\theta_{0}})$ and $\\widehat{I}_{N}(\\widetilde{X}, h_{\\theta_{0}})$), respectively. We find that adversarial samples usually have larger adversarial MI and smaller natural MI compared with those \\textit{w.r.t.} natural samples, which is consistent with our intuitive cognition. However, the change is relatively insignificant, and thus has limitations in reflecting the difference between the adversarial sample and the natural sample in the natural MI and the adversarial MI. We will show this observation later in \\cref{section4.2}. \n\nTo adequately represent the inherent difference in the natural MI and the adversarial MI between the natural sample and the adversarial sample, we design an optimization mechanism for the MI estimation. This is, we minimize the natural MI of the adversarial sample and minimize the adversarial MI of the natural sample during training the estimators. In addition, to estimate the two MI more accurately, we select samples that are correctly predicted by the target model and the corresponding adversarial samples are wrongly predicted, to train the estimation networks. The reformulated optimization goals are as follows:\n\\begin{equation}\n\\label{eq5}\n\\begin{aligned}\n&\\widehat{\\phi}_{n}=\\underset{\\phi_{n} \\in \\Phi_{N}}{\\arg \\max } \\,[E_{\\phi_{n}}( h_{\\theta_{0}}(X^{\\prime})) - E_{\\phi_{n}}(h_{\\theta_{0}}(\\widetilde{X}^{\\prime}))] \\text{,} \\\\\n&\\widehat{\\phi}_{a}=\\underset{\\phi_{a} \\in \\Phi_{A}}{\\arg \\max } \\, [E_{\\phi_{a}}(h_{\\theta_{0}}(\\widetilde{X}^{\\prime})) - E_{\\phi_{a}}\\left(h_{\\theta_{0}}(X^{\\prime}\\right))] \\text{,}\n\\end{aligned}\n\\end{equation}\n\nwhere $X^{\\prime}$ is the selected data:\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq6}\n    X^{\\prime}=\\arg _{X}\\left[\\delta\\left(h_{{\\theta}_0}(X)\\right)=Y \\& \\delta(h_{{\\theta}_0}(\\widetilde{X})) \\neq Y\\right]  \\text{,} \n\\end{equation}\n\nwhere $\\delta$ is the operation that transforms the logit output into the prediction label.\n\n\\subsection{Adversarial defense algorithm}\n\\label{section3.3}\nBased on the above two MI estimation networks, we develop an adversarial defense algorithm called \\textit{natural-adversarial mutual information-based defense} (NAMID) to enhance the adversarial robustness. In this section, we first introduce the natural-adversarial MI-based optimization strategy and then illustrate the training algorithm.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=5.6in]{fig3.pdf}}\n\\caption{The overview of our proposed NAMID adversarial defense algorithm.}\n\\label{fig3}\n\\vskip -0.2in\n\\end{center}\n\\end{figure*}\n\n\\subsubsection{natural-adversarial MI-based defense}\n\\label{section3.3.1}\nAccording to the observations and analysis in \\cref{section3.1} and \\cref{section3.2}, we plan to use the natural MI and the adversarial MI to help train the target model. We aim to guide the target model to increase the attention to the natural pattern while reducing the attention to the adversarial pattern. \n\nSpecifically, The optimization strategy is to maximize the natural MI of the input adversarial sample and minimize its adversarial MI simultaneously. The optimization goal for the target model is as follows:\n\n\\begin{equation}\n\\label{eq7}\n\\widehat{\\theta}=\\underset{\\theta \\in \\Theta}{\\arg \\max }\\left[E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\widetilde{X}))-E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\widetilde{X}))\\right] \\text{.}\n\\end{equation}\n\nTo achieve the optimization goal, we can directly utilize the natural and adversarial MI of the adversarial sample to construct a loss function. However, this loss function does not consider the difference in natural\/adversarial MI between the natural sample and the adversarial sample. Thus, we transform this absolute metric-based loss to a relative metric-based loss. In addition, as described in \\cref{section3.2.2}, we use the selected samples to compute the loss function. The loss function is formulated as:\n\n\\vskip -0.2in\n\\begin{equation}\n\\label{eq8}\n\\begin{gathered}\n\\mathcal{L}_{m i}(\\theta)=\\frac{1}{m} \\sum_{i=1}^{m}\\{\\mathcal{L}_{\\cos }(E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime})), E_{\\widehat{\\phi}_{n}}(h_{\\theta}(x_{i}^{\\prime}))) \\\\ + \\mathcal{L}_{\\cos }(E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}), E_{\\widehat{\\phi}_{a}}(h_{\\theta}(x_{i}^{\\prime}))) \\\\ + \\lambda \\cdot [E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}))-E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}_{i}^{\\prime}))]\\} \\text{,}\n\\end{gathered}\n\\end{equation}\n\nwhere $m$ is the number of the selected data $X^{\\prime}$ and $\\lambda$ is a hyperparameter. $\\mathcal{L}_{\\cos}(\\cdot, \\cdot)$ is the cosine similarity-based loss function, i.e., $\\mathcal{L}_{\\cos}(a, b) = ||1 - sim (a,b)||_1$, $sim(\\cdot,\\cdot)$ denotes the cosine similarity measure.\n\nThe MI-based optimization strategy can exploited together with the adversarial training manner. The overall loss function for training the target model is as follows:\n\n\\begin{equation}\n\\label{eq9}\n    \\mathcal{L}_{all}(\\theta)= \\mathcal{L}_{adv}(\\theta) + \\alpha \\cdot \\mathcal{L}_{mi}(\\theta) \\text{,}\n\\end{equation}\n\\vskip 0.1in\n\nwhere $\\mathcal{L}_{adv}(\\theta)$ is the loss function of the adversarial training method, which is typically the cross-entropy loss between the adversarial outputs and the ground-truth labels: $\\mathcal{L}_{adv}(\\theta)= -\\frac{1}{n} \\sum_{i=1}^{n} [\\boldsymbol{y_i} \\cdot \\log(\\sigma(h_{\\theta}(\\tilde{x}_{i})))]$. $n$ is the number of training samples and $\\sigma$ denotes the softmax function. $\\alpha$ is a trade-off hyperparameter. We provide the overview of the adversarial defense method in \\cref{fig3}.\n\n\\subsubsection{Training algorithm} \n\\label{section3.3.2}\nWe conduct the adversarial training on the procedures of generating adversarial samples and optimizing the target model parameter. The details of the overall procedure are presented in \\cref{alg1}.\n\n Specifically, the procedure requires the target model $h_{\\theta}$ with parameter $\\theta$, the natural MI estimation network with parameter $\\widehat{\\phi}_{n}$, the adversarial MI estimation network with parameter $\\widehat{\\phi}_{a}$ and perturbation budget $\\epsilon$. For the natural instance $x$ in mini-batch $\\mathcal{B}=\\{x_i\\}_{i=1}^{n}$ sampled from natural training set, we first craft adversarial noise $n$ and generate adversarial instance $\\tilde{x}$ via the powerful PGD adversarial attack \\cite{madry2017towards}. Then, we input the natural and adversarial training data into the target model $h_{\\theta}$ and obtain the selected instance $x^{\\prime}$, $\\tilde{x}^{\\prime}$ according to \\cref{eq6}. Next, we estimate the natural MI $E_{\\widehat{\\phi}_{n}}(h_{\\theta}(x^{\\prime})), E_{\\widehat{\\phi}_{n}}(h_{\\theta}(\\tilde{x}^{\\prime}))$ and the adversarial MI $ E_{\\widehat{\\phi}_{a}}(h_{\\theta}(x^{\\prime})), E_{\\widehat{\\phi}_{a}}(h_{\\theta}(\\tilde{x}^{\\prime})$. Finally, we optimize the parameter $\\theta$ according to \\cref{eq9}. By iteratively conduct the adversarial training, $\\theta$ is expected to be optimized well.\n\n\\begin{algorithm}[t]\n   \\caption{\\small Natural-adversarial mutual information-based defense (NAMID) algorithm}\n   \\label{alg1}\n\\begin{algorithmic}[1]\n   \\begin{small}\n   \\REQUIRE Target model $h_{\\theta}(\\cdot)$ parameterized by $\\theta$, natural MI estimation network $E_{\\widehat{\\phi}_{n}}$, adversarial MI estimation network $E_{\\widehat{\\phi}_{a}}$, batch size $n$, and the perturbation budget $\\epsilon$;\n   \\REPEAT\n   \\STATE Read mini-batch $\\mathcal{B}=\\{x_i\\}_{i=1}^{n}$ from training set;\n   \\FOR{$i=1$ to $n$ (in parallel)}\n   \\STATE Craft adversarial noise $n_{i}$ and generate adversarial instance $\\tilde{x}_i$ at the given perturbation budget $\\epsilon$ for $x_i$;\n   \\STATE Forward-pass $x_i$, $\\tilde{x}_i$ through $h_{\\theta}(\\cdot)$ and obtain $h_{\\theta}(x_i)$, $h_{\\theta}(\\tilde{x}_i)$;\n   \\STATE Select samples according to \\cref{eq6};\n   \\ENDFOR\n   \\STATE Calculate $\\mathcal{L}_{all}$ using \\cref{eq9} and optimize $\\theta$;\n   \\UNTIL training converged.\n   \\end{small}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\\label{section4}\nIn this section, we first introduce the experiment setups including datasets, attack setting and defense setting in \\cref{section4.1}. Then, we show the effectiveness of our optimization mechanism for evaluating MI in \\cref{section4.2}. Next, we evaluate the performances of the proposed adversarial defense algorithm in \\cref{section4.3}. Finally, we conduct ablation studies in \\cref{section4.4}.\n\n\\subsection{Experiment setups}\n\\label{section4.1}\n\\noindent\\textbf{Datasets.}\nWe verify the effective of our defense algorithm on two popular benchmark datasets, i.e., \\textit{CIFAR-10} \\cite{krizhevsky2009learning} and \\textit{Tiny-ImageNet} \\cite{wu2017tiny}. \\textit{CIFAR-10} has 10 classes of images including 50,000 training images and 10,000 test images. \\textit{Tiny-ImageNet} has 200 classes of images including 100,000 training images, 10,000 validation images and 10,000 test images. Images in the two datasets are all regarded as natural instances. All images are normalized into [0,1], and are performed simple data augmentations in the training process, including random crop and random horizontal flip. \n\n\\noindent\\textbf{Model architectures.} We use a ResNet-18 \\cite{he2016deep} as the target model for both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}. For the MI estimation network, we utilize the same neural network as in \\citep{zhu2020learning}. The estimation networks for the natural MI and the adversarial MI have same model architectures.\n\n\\noindent\\textbf{Baselines.} (1) \\textit{Standard AT} \\cite{madry2017towards}; (2) TRADES \\cite{zhang2019theoretically}; (3) MART \\cite{wang2019improving}; and (4)\\textit{WMIM}: A defense that refers to \\citet{zhu2020learning}. The first three are representative adversarial training methods, and the last one combines adversarial training with standard MI maximization (on adversarial samples).\n\n\\noindent\\textbf{Attack settings.}\nAdversarial data for evaluating defense models are crafted by applying state-of-the-art attacks. These attacks are divided into two categories: $L_{\\infty}$-norm attacks and $L_{2}$-norm attacks. The $L_{\\infty}$-norm attacks include PGD \\cite{madry2017towards}, AA \\cite{croce2020reliable}, TI-DIM \\cite{dong2019evading,xie2019improving}, and FWA \\cite{wu2020stronger}. The $L_{2}$-norm attacks include PGD, CW$_2$ \\cite{carlini2017towards} and DDN \\cite{rony2019decoupling}. Among them, the AA attack algorithm integrates three non-target attacks and a target attack. Other attack algorithms are utilized as non-target attacks. The iteration number of PGD and FWA is set to 40 with step size $\\epsilon\/4$. The iteration number of CW$_2$ and DDN are set to 20 respectively with step size 0.01. For \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}, the perturbation budgets for $L_{2}$-norm attacks and $L_{\\infty}$-norm attacks are $\\epsilon=0.5$ and $8\/255$ respectively.  \n\n\\noindent\\textbf{Defense settings.} For both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}, the adversarial training data for $L_{\\infty}$-norm and $L_{2}$-norm is generated by using $L_{\\infty}$-norm PGD-10 and $L_{2}$-norm PGD-10 respectively. The step size is $\\epsilon\/4$ and the perturbation budget is $8\/255$ and $0.5$ respectively. The epoch number is set to 100. For fair comparisons, all the methods are trained using SGD with momentum 0.9, weight decay $2 \\times 10^{-4}$, batch-size 1024 and an initial learning rate of 0.1, which is divided by 10 at the 75-th and 90-th epoch. In addition, we adjust the hyperparameter settings of the defense methods so that the natural accuracy is not severely compromised and then compare the adversarial accuracy. We set $\\alpha=5, \\lambda= 0.1$ for our algorithm. \n\n\\subsection{Effectiveness of MI estimation networks}\n\\label{section4.2}\nIn \\cref{section3.2.2}, we point out that training the MI estimation network directly by MI maximization may not clearly reflect the difference between the adversarial sample and the natural sample in natural MI and adversarial MI. We thus design an optimization mechanism for training the MI estimation network. To demonstrate the effectiveness of the optimization mechanism, we compare the performance of the estimation networks trained by \\cref{eq4} and \\cref{eq5} in \\cref{fig4}. The performances of the defenses based on the two different estimators are shown in \\cref{appendix_5_1}\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.5 in]{fig4.pdf}}\n\\caption{The performances of MI estimation networks trained by \\cref{eq4} (MIM) and \\cref{eq5} (Our). The left half is the estimated natural MI, and the right half is the adversarial MI.}\n\\label{fig4}\n\\vskip -0.35in\n\\end{center}\n\\end{figure}\n\n\\begin{table*}[hbtp]\n\\caption{Adversarial accuracy (percentage) of defense methods against white-box attacks on \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}. The target model is ResNet-18.}\n\\label{tab1}\n\\renewcommand\\tabcolsep{6pt}\n\\renewcommand\\arraystretch{1.1}\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{l|l|ccccc|cccc}\n\\hline\n\\multirow{2}{*}{Dataset} &\\multirow{2}{*}{Defense}  &\\multicolumn{5}{c|}{$L_{\\infty}$-norm} &\\multicolumn{4}{c}{$L_{2}$-norm} \\\\\n& & None & PGD-40 & AA & FWA-40 &TI-DIM & None &PGD-40& CW & DDN \\\\ \\hline\n\\multirow{7}{*}{CIFAR-10} &Standard & 83.39 & 42.38 & 39.01 & 15.44 &55.63 &83.97 &61.69 & 30.96 &29.34 \\\\\n&WMIM & 80.32 & 40.76 & 36.05 & 12.14 &53.10 &81.29 &58.36 & 28.41 &27.13 \\\\\n&NAMID & \\textbf{83.41} & \\textbf{44.79} & \\textbf{39.26} & \\textbf{15.67} &\\textbf{58.23} &\\textbf{84.35} &\\textbf{62.38} & \\textbf{34.48} &\\textbf{32.41}\\\\ \\cdashline{2-11}[3pt\/5pt]\n&TRADES & \\textbf{80.70} & 46.29 & 42.71 & 20.54 &57.06 &83.72 &63.17 & 33.81 &32.06 \\\\\n&NAMID\\_T &80.67  &\\textbf{47.53}  &\\textbf{43.39}  &\\textbf{21.17} &\\textbf{59.13} &\\textbf{84.19} &\\textbf{64.75} &\\textbf{35.41} &\\textbf{34.27} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&MART & 78.21 & 50.23 & 43.96 & 25.56 &58.62  &83.36 &65.38 & 35.57 &34.69  \\\\\n&NAMID\\_M &\\textbf{78.38} &\\textbf{51.69}  &\\textbf{44.42}  &\\textbf{25.64} &\\textbf{61.26} &\\textbf{84.07} &\\textbf{66.03} &\\textbf{36.19} &\\textbf{35.76} \\\\ \\hline\n\\multirow{7}{*}{Tiny-ImageNet} &Standard & 48.40 & 17.35 & 11.27 & 10.29 &27.84 &49.57 &26.19 &12.73 &11.25 \\\\\n&WMIM & 47.43 & 16.50 & 9.87 & 9.25 & 25.19 &48.16 &24.10 &11.35 &10.16  \\\\\n&NAMID & \\textbf{48.41} & \\textbf{18.67} & \\textbf{12.29} & \\textbf{11.32} &\\textbf{29.37}  &\\textbf{49.65} &\\textbf{28.13} & \\textbf{14.29} &\\textbf{12.57} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&TRADES & \\textbf{48.25} & 19.17 & 12.36 & 10.67 &29.64  & 48.83 & 27.16 & 13.28 &12.34 \\\\\n&NAMID\\_T &48.21  &\\textbf{20.12}  &\\textbf{12.86}  &\\textbf{14.91} &\\textbf{30.81}  &\\textbf{49.07} &\\textbf{28.83} &\\textbf{14.47} &\\textbf{13.91} \\\\ \\cdashline{2-11}[3pt\/5pt]\n&MART & \\textbf{47.83} & 20.90 & 15.57 & 12.95 &30.71 &48.56 &27.98 & 14.36 &13.79  \\\\\n&NAMID\\_M &47.80  &\\textbf{21.23}  &\\textbf{15.83}  &\\textbf{15.09} & \\textbf{31.59} &\\textbf{48.72} &\\textbf{29.14} &\\textbf{15.06} &\\textbf{14.23} \\\\ \\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\vskip -0.15in\n\\end{table*}\n\nWe use the test data from \\textit{CIFAR-10} to evaluate the performance. For the natural MI, we offset the estimated MI so that the worst-case of the natural MI equals 0, and calculate the average of all samples. Similarly, we offset the estimated MI so that the worst-case of the adversarial MI equals 0. Note that for a fair comparison, we use selected samples to train the estimation networks for both methods. As shown in \\cref{fig4}, the results demonstrate that the optimization mechanism could help adequately represent the inherent difference in the natural MI and the adversarial MI between the natural sample and the adversarial sample.\n\n\\subsection{Robustness evaluation and analysis}\n\\label{section4.3}\n\nTo demonstrate the effectiveness of our adversarial defense algorithm, we evaluate the adversarial accuracy using white-box and black-box adversarial attacks, respectively.\n\n\\noindent\\textbf{White-box attacks.} \nIn the white-box setting, all attacks can access the architectures and parameters of target models. We evaluate the robustness by exploiting six types of adversarial attacks for both \\textit{CIFAR-10} and \\textit{Tiny-ImageNet}: $L_{\\infty}$-norm PGD, FWA, AA, TI-DIM attacks and $L_{2}$-norm PGD, DDN, CW attacks. The average natural accuracy (i.e., the results in the third column) and the average adversarial accuracy of defenses are shown in \\cref{tab1}.\n\nThe results show that our method (i.e., NAMID) can achieve better robustness compared with \\textit{Standard AT}. The performance of our method on the natural accuracy is competitive (83.39\\% vs. 83.41\\%), and it provides more gains on adversarial accuracy (e.g., 5.69\\% against PGD-40). Compared with \\textit{WMIM}, the results show that our proposed strategy of disentangling the standard MI into the natural MI and the adversarial MI is effective. The standard deviation is shown in \\cref{appendix_5_2}.\n\nNote that the default adversarial training loss in our method (i.e., $\\mathcal{L}_{adv}$ in \\cref{eq9}) is the same as \\textit{Standard AT}. To avoid the bias caused by different adversarial training methods, we apply the adversarial training losses of \\textit{TRADES} and \\textit{MART} to our method respectively (i.e., NAMID\\_T and NAMID\\_M). As shown in \\cref{tab1}, the results show that our method can improve the adversarial accuracy (e.g., the accuracy against PGD is improved by 2.68\\% and 2.91\\% compared with \\textit{TRADES} and \\textit{MART} on \\textit{CIFAR-10}).\n\n\\noindent\\textbf{Black-box attacks.}\nBlock-box adversarial instances are crafted by attacking a surrogate model. We use a VggNet-19 \\cite{he2016deep} as the surrogate model. The surrogate models and the defense models are trained separately. We use \\textit{Standard AT} method to train the surrogate model and use PGD, AA and FWA to generate adversarial test data. The performances of our defense method is reported in \\cref{tab2}. The results show that our method is a practical strategy for real scenarios, which can protect the target model from black-box attacks by malicious adversaries.\n\n\\begin{table}[t]\n\\caption{Adversarial accuracy (percentage) of defense methods against black-box attacks on \\textit{CIFAR-10}. The target model is ResNet-18 and the surrogate model is adversarially trained VggNet-19. We show the most successful defense with \\textbf{bold}.}\n\\label{tab2}\n\\renewcommand\\tabcolsep{8pt}\n\\renewcommand\\arraystretch{1.05}\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{l|cccc}\n\\hline\nDefense  & None & PGD-40 & AA & FWA-40 \\\\ \\hline\nStandard & 83.39    & 65.88    & 60.93   & 56.42       \\\\\nWMIM     & 80.32     & 62.79       & 57.86   &  53.05      \\\\\nNAMID      & \\textbf{83.41}     & \\textbf{69.57}       & \\textbf{63.72}   & \\textbf{59.30}       \\\\ \\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\vskip -0.25in\n\\end{table}\n\n\n\\subsection{Ablation study}\n\\label{section4.4}\nTo clearly elucidate the role of each component of our method in improving adversarial robustness, we conduct ablation studies in three different settings: (i) removing the adversarial MI; (ii) removing the natural MI; and (iii) setting the hyperparameter $\\lambda$ (in \\cref{eq8}) to 0. We use $L_{\\infty}$-norm PGD and FWA attacks to evaluate the performances of these variants. As shown in \\cref{fig5}, the results demonstrate that each component of our method contributes positively to improving adversarial accuracy.\n\n\\begin{figure}[t]\n\\begin{center}\n\\vskip 0.1in\n\\centerline{\\includegraphics[width=2.5 in]{fig5.pdf}}\n\\caption{The ablation study.The bars with different colors represent the performance under different settings. Among them, 'Ori' denotes our method NAMID and 'Standard' denotes \\textit{Standard AT}.}\n\\label{fig5}\n\\vskip -0.35in\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\n\\label{section5}\nTo the best of our knowledge, the dependence between the output of the target model and input adversarial samples have not been well studied. In this paper, we investigate the dependence from the perspective of information theory. Considering that adversarial samples contain natural and adversarial patterns, we propose to disentangle the standard MI into the natural MI and the adversarial MI to explicitly measure the dependence of the output on the different patterns. We design a neural network-based method to train two MI estimation networks to estimate the natural MI and the adversarial MI. Based on the above MI estimation, we develop an adversarial defense algorithm called natural-adversarial mutual information-based defense (NAMID) to enhance the adversarial robustness. The empirical results demonstrate that our defense method can provide effective protection against multiple adversarial attacks. Our work provides a new adversarial defense strategy for the community of adversarial learning. In future, we will design more efficient mechanisms for training MI estimators and further optimize the natural-adversarial MI-based defense to improve the performance against stronger attacks. In addition, \n\n\\section{Acknowledgements}\nThis work was supported in part by the National Key Research and Development Program of China under Grant 2018AAA0103202, in part by the National Natural Science Foundation of China under Grant 61922066, 61876142, 62036007, 62006202, 61922066, 61876142, 62036007, and 62002090, in part by the Technology Innovation Leading Program of Shaanxi under Grant 2022QFY01-15, in part by Open Research Projects of Zhejiang Lab under Grant 2021KG0AB01, in part by the RGC Early Career Scheme No. 22200720, in part by Guangdong Basic and Applied Basic Research Foundation No. 2022A1515011652, in part by Australian Research Council Projects DE-190101473, IC-190100031, and DP-220102121, in part by the Fundamental Research Funds for the Central Universities, and in part by the Innovation Fund of Xidian University. The authors thank the reviewers and the meta-reviewer for their helpful and constructive comments on this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a neutrinoless double-beta ($0\\nu\\beta\\beta$) decay an atomic nucleus decays into another one with two more protons and two fewer neutrons, emitting two electrons. In other words, two leptons are created. Such violation of the lepton number conservation is only possible if neutrinos---unlike any other fundamental particle---are its own antiparticle, a possibility first suggested by Ettore Majorana in the 1930's. In spite of the challenges associated with the detection of a process that involves new physics, $0\\nu\\beta\\beta$ decay is being pursued by several experimental collaborations~\\cite{KamLAND-Zen16,EXO18,GERDA18,MAJORANA18,CUORE18,NEXT18,CUPID18}. A major advantage is that the parameter $m_{\\beta\\beta}$ that controls the $0\\nu\\beta\\beta$ decay half-life is fully fixed by the the known neutrino mass differences and mixing angles, in such a way that $m_{\\beta\\beta}$ only depends on the ordering ---``normal\" or ``inverted\"---of the neutrinos masses~\\cite{engelmen_review}:\n\\begin{equation}\n\\label{eq:half-life}\n[T^{0\\nu}_{1\/2}]^{-1}=G^{0\\nu}\n\\left|M^{0\\nu\\beta\\beta}\\right|^2  m_{\\beta\\beta}^2\\,.\n\\end{equation}\nThere is, however, a catch. The $0\\nu\\beta\\beta$ decay half-life also depends on the value of an associated nuclear matrix element (NME), $M^{0\\nu\\beta\\beta}$, like any other {\\it nuclear} decay---$G^{0\\nu}$ is a known phase-space factor. NMEs need to be calculated theoretically, and their value is key to assess the prospects to observe $0\\nu\\beta\\beta$ decay in present and next-generation experiments.\n\nAt present, predicted NME values vary by a factor two or three depending on the many-body method used to calculate them. In addition, the results may need to be ``quenched\" as is common for $\\beta$ decays, but since the momentum transfer in $0\\nu\\beta\\beta$ decay is much larger, the necessity of such ``quenching\" is unclear~\\cite{engelmen_review}. These proceedings discuss recent ideas towards a more reliable determination of the \nNMEs, with focus on improved many-body calculations, and on the relation between $0\\nu\\beta\\beta$ decay and double Gamow-Teller (GT) transitions.\n\n\n\\section{Shell model nuclear matrix elements in two harmonic oscillator shells}\n\nAmong the nuclear many-body methods used to study $0\\nu\\beta\\beta$ decay, the nuclear shell model plays a prominent role, as one of the most successful approaches to nuclear structure~\\cite{cau05}. Nonetheless the main drawback of shell-model NMEs is that they are typically calculated limiting the configuration space to one harmonic oscillator shell. While, in general, such restriction works very well to describe the nuclear structure and spectroscopy of stable nuclei, it has been claimed that such a configuration space may not be large enough to obtain converged $0\\nu\\beta\\beta$ decay NMEs~\\cite{vogel12}.\n\nThe lightest $\\beta\\beta$ emitter is $^{48}$Ca. This is therefore the nucleus for which shell model calculations beyond one harmonic oscillator shell are  less demanding computationally. Ref.~\\cite{iwata16} calculated the NME for the $0\\nu\\beta\\beta$ decay in a configuration space consisting of two harmonic oscillator shells, the $sd$ and $pf$ shells. Previous shell model calculations were restricted to the $pf$ shell, while Ref.~\\cite{iwata16} was able to include up to $2\\hbar\\omega$ $sd$-$pf$ excitations. The calculation was validated by reproducing the excitation spectra of the initial and final nuclei of the decay, $^{48}$Ca and $^{48}$Ti~\\cite{iwata16}. In addition, the shell model calculation of Ref.~\\cite{iwata16} showed a good description of the GT strength, including the GT giant resonance (GR), of $^{48}$Ca and $^{48}$Ti into $^{48}$Sc~\\cite{iwata15}---these GT strengths had been measured in charge-exchange experiments~\\cite{yako-ca48}---, and reproduced the two-neutrino $\\beta\\beta$ decay matrix element of $^{48}$Ca as well. For the transition operators, the agreement to experiment was only possible after a ``renormalization\", or ``quenching\", of the theoretical predictions by a factor $q=0.71$ for each spin-isospin ${\\bm \\sigma}\\tau$ term present in the corresponding operator. This is, once for the GT strength and twice for two-neutrino $\\beta\\beta$ decay matrix element.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=.82\\textwidth]{nme_shells.pdf}\n\\end{center}\t\n\\caption{\\label{fig:ca48_nme}\n$^{48}\\textrm{Ca}$ and $^{76}$Ge $0\\nu\\beta\\beta$ decay\nNME calculated in a one-shell (shaded) and two-shell (solid) configuration space. One-shell~\\cite{menendez09} and $^{48}\\textrm{Ca}$~\\cite{iwata16} calculations are performed with the full shell model, while the $^{76}$Ge two shell calculation~\\cite{jiao17} use the approximate generator-coordinate method.}\n\\end{figure}\n\nThe result of Ref.~\\cite{iwata16} is shown in Fig.~\\ref{fig:ca48_nme}. Note that the NME does not include a possible ``renormalization\" of the NMEs, even though such ``renormalization\" is required to reproduce the $^{48}$Ca two-neutrino $\\beta\\beta$ decay half-life. The main conclusion found in Ref.~\\cite{iwata16} is that, in spite of performing the calculation in a significantly larger configuration space, the $^{48}$Ca NMEs was enhanced by only $\\sim30\\%$ in the two-shell calculation compared to the one-shell one. The main reason for such a relatively small effect is the competition between two type of contributions: on the one hand, pair-type two-particle--two-hole excitations into the additional harmonic oscillator shell tend to enhance the value of the NMEs~\\cite{caurier08}; on the other hand, one-particle--one-hole type excitations---which are generally related to two decaying nucleons coupled to angular momentum $J>0$---tend to reduce the value of the NME~\\cite{caurier08}. Overall, the competition between these two kinds of contributions results in a moderate enhancement of the NME in the expanded configuration space. Since the competition is expected to be general, similar effects are expected for extending the configuration space of shell model NME calculations in heavier $\\beta\\beta$ emitters.\n\nThis expectation is consistent with the recent result of Ref.~\\cite{jiao17}, which calculated the nuclear matrix element of $^{76}$Ge in a configuration space consisting of two harmonic oscillator shells. \nThis calculation is based on the generator-coordinated method, which does not include all the many-body correlations present in the shell model, because a full shell model diagonalization of the two-shell configuration space is beyond present computing capabilities. Fig.~\\ref{fig:ca48_nme} shows the result of Ref.~\\cite{jiao17} in comparison with a shell model calculation in one oscillator shell. Similarly to the findings in $^{48}$Ca, the impact of increasing the size of the configuration space is small, with the NME even slightly reducing its value in the larger space.\n\n\n\n\\section{$\\beta\\beta$ decay and double Gamow-Teller transitions}\n\nIn the absence of a $0\\nu\\beta\\beta$ detection, theoretical calculations of the NMEs have to be tested against different nuclear structure data. First, all calculations compare their predictions to the nuclear structure of the initial and final states of the decay. In addition, an obvious observable to test calculations is the two-neutrino $\\beta\\beta$ decay, which shares initial and final states with $0\\nu\\beta\\beta$ decay and has similar spin-isospin structure. However, the momentum transfers in the two $\\beta\\beta$ modes are very different: while in the two-neutrino case the momentum transfer is limited by the $Q$-value---a couple of MeV---in the neutrinoless case momentum is transferred via the not-to-be-emitted virtual neutrinos. A test of the relevant momentum-transfer regime---about $q\\sim100$~MeV---would involve a comparison to muon capture or neutrino scattering. Unfortunately, data on these observables is limited.\nGT transition strengths measured in charge-exchange experiments are also typically used to test calculations. Described by the same operator as GT $\\beta$ decays, GT transitions are not limited by the $Q$-value, and can be studied to energies even past the GT GR at $E\\sim10-15$~MeV.\n\nA closer connection to $0\\nu\\beta\\beta$ decay can be expected to come from double GT (DGT) transitions that are being looked for in double charge-exchange experiments~\\cite{takaki-aris,uesaka-nppac,cappuzzello,takahisa-17}. The operator structure $0\\nu\\beta\\beta$ decay of DGT transitions is very similar, with the corresponding matrix elements given by\n\\begin{eqnarray}\n&M^{0\\nu\\beta\\beta}(i\\rightarrow f)&\n=M_{GT}^{0\\nu}+\\frac{M_F^{0\\nu}}{g_A^2}+M_T^{0\\nu}= \\sum_{X=GT,F,T}\n\\bra{f} \\sum_{a,b} H_X(r_{ab})\\,S_X\\, \\tau^+_a \\tau^+_b \\ket{i}\\,,\\\\\n\\label{eq:2GT}\n&M^{DGT}(i\\rightarrow f)&\n=\\bra{f} \\sum_{a,b}\n[{\\bm \\sigma}_a \\tau^+_a \\times {\\bm \\sigma}_b \\tau^+_b]^{\\lambda} \\ket{i}\\,,\n\\end{eqnarray}\nwhere $\\bm \\sigma$, $\\tau$ denote spin and isospin, respectively, and $g_A$ is the axial coupling. The labels $F$ and $T$ stand for the subleading Fermi and tensor parts of the $0\\nu\\beta\\beta$ NME, much smaller---less than 20\\%---than the dominant GT piece associated to the spin structure $S_{GT}={\\bm\\sigma}_1\\cdot{\\bm\\sigma}_2$. Therefore besides the small effect of the $F$ and $T$ terms, for DGT transitions to the ground state of the final nucleus---where the DGT operator can only couple to $\\lambda=0$---the $0\\nu\\beta\\beta$ and DGT operators only differ by the presence of the neutrino potential $H$, which depends on the internucleon distance $r_{ab}$. The form of the neutrino potentials is given in detail in Ref.~\\cite{menendez_heavy}.\n\nReference~\\cite{2GT0nbb} predicted the DGT strength of $^{48}$Ca, including the DGT GR using large-scale shell model calculations up to two oscillator shells. Interestingly, the energy of the resonance was found to be correlated---in the shell model calculation---to the value of the $0\\nu\\beta\\beta$ decay NME. This relation is due to the dependence of the two observables to particle-like pairing correlations. As a consequence, a measurement of the DGT GR in $^{48}$Ca could provide an indication of the value of the NME of the same nucleus.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{nme_linear_ge-xe.pdf}\t\n\t\\end{center}\t\n\t\\caption{\\label{fig:nme_linear}\t\n\t\tCorrelation between\n\t\t$0\\nu\\beta\\beta$ decay NMEs\n\t\t$M^{0\\nu\\beta\\beta}(0^+_{gs,i} \\rightarrow 0^+_{gs,f})$\n\t\tand the DGT matrix elements\n\t\t$M^{\\rm DGT}(0^+_{gs,i} \\rightarrow 0^+_{gs,f})$.\n\t\tShell model results for germanium, tellurium, tin, tellurium, and xenon isotopes (black) including the $\\beta\\beta$ emitters $^{76}$Ge, $^{82}$Se, $^{124}$Sn, $^{130}$Te and $^{136}$Xe (blue) are compared to EDF theory~\\cite{rodriguez} (green) and QRPA predictions~\\cite{simkovic-11} (open red symbols).\n\t\tThe calculations use several shell model interactions for each isotope~\\cite{menendez09,JUN45,qi-12}.  Adapted from Ref.~\\cite{2GT0nbb}.\n\t}\n\\end{figure}\n\nIn addition, Ref.~\\cite{2GT0nbb} studied DGT transitions to the ground state, and compared the results to the $0\\nu\\beta\\beta$ decay NMEs. Note that the initial and final states of both processes are the same, and also the transition operator is very similar, as discussed above. Instead of limiting to one particular case---as in the study of the DGT GR---the calculations included a set of nuclei ranging from calcium to xenon isotopes, with nuclear mass number $42\\leq A\\leq 136$. Therefore several $\\beta\\beta$ emitters but also many isotopes not relevant for $0\\nu\\beta\\beta$ decay searches were studied. Nonetheless, these additional calculations are very useful to illuminate systematic effects.\n\nFigure~\\ref{fig:nme_linear} summarizes the results of Ref.~\\cite{2GT0nbb}. A good linear correlation is found between the DGT transitions to the ground state and $0\\nu\\beta\\beta$ NMEs. The linear correlation does not depend on the details of the shell-model interaction used, or in the correlations included in the shell-model initial and final states---as long as particle-like pairing correlations are present.  Furthermore, the correlation between $0\\nu\\beta\\beta$ and DGT matrix elements is valid for $\\beta\\beta$ emitters, shown in blue in Fig.~\\ref{fig:nme_linear}, and for all the other nuclei---seventeen isotopes in total. Moreover, the correlation observed in the shell model is consistent with the results of energy-density functional (EDF) theory~\\cite{rodriguez}---also shown in Fig.~\\ref{fig:nme_linear}---even if for the latter approach $0\\nu\\beta\\beta$ and DGT matrix elements are much larger than the shell model ones. In contrast, quasiparticle random-phase approximation (QRPA) results~\\cite{simkovic-11}---shown in Fig.~\\ref{fig:nme_linear} as well---do not support the linear correlation found for the shell model.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=.95\\textwidth]{density_se_te_DGT.pdf}\t\n\t\\end{center}\t\n\t\\caption{\\label{fig:radial_density}\t\n$^{82}$Se (left panel) and $^{130}$Te (right) normalized radial density distributions $C(r)$ of the GT $0\\nu\\beta\\beta$ (red) and DGT (orange) matrix elements. Shell model interactions from Ref.~\\cite{menendez09} are used.\n\t}\n\\end{figure}\n\nThe linear correlation shown in Fig.~\\ref{fig:nme_linear} relates the $0\\nu\\beta\\beta$ decay NME, driven by the weak interaction, and the DGT matrix element, a result of the strong interaction. It therefore opens the door to exploring $0\\nu\\beta\\beta$ decay NMEs in nuclear double charge-exchange experiments~\\cite{takaki-aris,uesaka-nppac,cappuzzello,takahisa-17}. This is, however, a formidable challenge at the experimental and theoretical level. First, the DGT transition is a tiny---0.03 per mil---piece of the DGT sum rule. In addition, dedicated reaction theory efforts are needed to establish the relation between double charge-exchange cross-sections and DGT matrix elements.\n\nWhat is the origin of the linear correlation between $0\\nu\\beta\\beta$ decay and DGT transitions? To address this question, Fig.~\\ref{fig:radial_density} shows the normalized radial densities of the $0\\nu\\beta\\beta$ and DGT matrix elements, defined as\n\\begin{eqnarray}\nC_{GT}^{0\\nu}(r)= \\langle f | \\sum_{ab} \\delta(r-r_{ab}) \\,H_{GT}(r_{ab})\\,\n{\\bm \\sigma}_a\\cdot{\\bm \\sigma}_b \\,\\tau_a \\tau_b | i \\rangle \/ M_{GT}^{0\\nu}\\,, \\\\\nC^{DGT}(r)= \\langle f | \\sum_{ab} \\delta(r-r_{ab}) \\,\n[{\\bm \\sigma}_a\\times{\\bm \\sigma}_b]^0 \\,\\tau_a \\tau_b | i \\rangle \/ M^{DGT}\\,.\n\\label{eq:density_r}\n\\end{eqnarray}\nFigure~\\ref{fig:radial_density} shows that the two matrix elements are dominated by the contribution of nucleons that are relatively close to each other, $r_{ab}\\lesssim3$~fm. In the case of DGT transitions this is the result of the partial cancellation of the longer-range contributions. This short-range dominance is non trivial, as Fig.~\\ref{fig:radial_density} shows that the shell model calculation naturally probes internucleon distances up to twice the nuclear radius.\n\nThe short-range character provides an explanation for the existence of the linear correlation between the two matrix elements. The work of Bogner et al.~\\cite{bogner-10,bogner-12} shows that when an operator probes only the short-range physics of low-energy states, the corresponding matrix elements factorize into a universal operator-dependent constant times a state-dependent number which is common to all short-range operators. Since both $0\\nu\\beta\\beta$ decay and DGT shell-model matrix elements fulfill these conditions, a linear relation between them is predicted. In contrast, the QRPA DGT matrix elements receive contributions from longer range, so that the correlation is not predicted in their case, in agreement with Fig.~\\ref{fig:nme_linear}.\n\n\n\\section{Conclusions}\nWe have summarized two advances that improve our understanding of $0\\nu\\beta\\beta$ decay. On the one hand, shell model calculations in a configuration space comprising two oscillator shells suggest that the NME obtained in standard shell model calculations are reasonably converged. On the other hand, the finding of a good linear correlation between the NMEs and DGT transitions, valid across the nuclear chart, brings the opportunity to obtain precious information on $0\\nu\\beta\\beta$ decay in double charge-exchange nuclear reactions. These advances pave the way towards a more reliable determination of the $0\\nu\\beta\\beta$ NMEs in the mid-term future.\n\n\n\\section*{Acknowledgments}\nI would especially like to thank Prof. T. Otsuka for many stimulating discussions and for his support, as well as for his kind introduction to research in Tokyo and Japanese culture. I am grateful to my co-authors T. Abe, M. Honma, Y. Iwata, T. Otsuka, N. Shimizu, Y.~Utsuno, and K. Yako for using in these proceedings results of our common research.\nThis work was supported by the\nCNS-RIKEN joint project for large-scale nuclear structure calculations, and by MEXT and JICFuS as a priority issue \n(Elucidation of the fundamental laws and evolution of the universe, hp170230) \nto be tackled by using Post K Computer.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the early days of the holographic correspondence \\cite{Maldacena:1997re,Witten:1998qj,Gubser:1998bc}, $(2+1)$-dimensional gravitational systems have played a central role in testing and exploring the ideas behind the duality. In fact, with the benefit of hindsight, one can see that the work of Brown and Henneaux on the asymptotic symmetries of three-dimensional spacetimes with a negative cosmological constant \\cite{Brown:1986nw} displayed some basic features of the correspondence as early as a decade before it was proposed. The 3$d$ gravitational systems of interest in the framework of holography are special in that their field theory duals enjoy an infinite-dimensional (local) conformal symmetry; via the powerful techniques of conformal field theory (CFT), one then has a better grasp of the boundary theory structure which is often lacking in higher-dimensional examples. A beautiful example of this fact is the precise connection between the CFT spectrum and retarded Green's functions, and black hole quasinormal modes in the bulk \\cite{Birmingham:2001pj}.\n\nThe feature that makes the gauge\/gravity correspondence outstanding is that it postulates the equivalence between gravitational weakly-coupled degrees of freedom propagating in the bulk spacetime, and strongly-coupled degrees of freedom in a dual quantum field theory in one less dimension (``the boundary\"). A pivotal ingredient in the proposal is the dilatation symmetry of the boundary CFT; broadly speaking, using this symmetry one can relate the masses of the bulk fields to the conformal dimensions of operators in the quantum theory on the boundary, as first established in \\cite{Witten:1998qj}. However, for a given bulk field, the spectrum of conformal dimensions in the dual quantum field theory is not entirely determined by the masses of the fields. This is intimately related to the fact that the boundary conditions that yield well-defined dynamics are not unique. In fact, in the so-called ``bottom-up\" holography where the bulk theory is phenomenologically devised, the operator content of the possible dual theories is completely specified only after the boundary conditions for bulk fields have been properly chosen.\n\nIn the present article, we will focus on the study of the Abelian Maxwell-Chern-Simons (MCS) theory, frequently referred to as ``Topologically Massive Electrodynamics\" \\cite{Deser:1981wh,Deser:1982vy}, in three-dimensional asymptotically-AdS spacetimes. The emphasis will be on determining a set of ``admissible\" boundary conditions, in a sense that will be made precise below; this is crucial to the dictionary problem in the context of the AdS\/CFT correspondence, as discussed above. One of the motivations to study Chern-Simons terms in the bulk is that these arise naturally in the context of string theory compactifications, and endow the bulk black hole solutions with $U(1)$ charge (see \\cite{Kraus:2006nb,Kraus:2006wn}, for example). It is worth mentioning, however, that the MCS system plays a central role in condensed matter physics as well, in particular in the study of fermionic systems in two spatial dimensions, where it describes the low-energy effective theory of the Fractional Quantum Hall Effect (FQHE). Furthermore, even in flat space the MCS theory is often said to be holographic, albeit in a different sense from the above: in the topological limit (where the bulk quasiparticles become infinitely massive), the degrees of freedom are effectively localized on the boundary \\cite{Wen:1991mw}.\\footnote{The key difference being that the AdS\/CFT correspondence is an \\textit{equivalence} between bulk and boundary degrees of freedom, while in the topological theory the bulk degrees of freedom are gapped, and the low-energy excitations propagate exclusively on the boundary.} From a mathematical point of view, this is the well-known correspondence between three-dimensional Chern-Simons gauge theory and a chiral rational CFT \\cite{Witten:1988hf, Moore:1989yh, Elitzur:1989nr,Balachandran:1991dw, Bos:1989kn, Bos:1989wa,Schwarz:1979ae}. More recently, the latter correspondence has been refined to reconcile the modular transformation properties of the string theory partition function on $AdS_3$ and those of the Chern-Simons theory which dominates its infrared dynamics \\cite{Gukov:2004id}, and the potential relevance of these observations for condensed matter physics was also pointed out. This provides yet another motivation to carefully study the holographic dictionary of the full (finite coupling) MCS theory; here we will do so from a bottom-up perspective, in the hope that our results could be useful in the study of other models which might be interesting for applications of holography to condensed matter physics.\n\nOur analysis starts by determining a broad set of boundary conditions under which the bulk theory is expected to have well-posed dynamics. We find it convenient to approach this problem using the covariant phase space formalism, along the lines of \\cite{Marolf:2006nd,Amsel:2006uf,Amsel:2008iz}. Within this framework, the first requirement on the boundary conditions is that they lead to a conserved symplectic structure (in the sense of timelike evolution). In the context of holography, this condition can be conveniently rephrased as the vanishing of the symplectic flux on the (radial) boundary. Roughly speaking, the bulk gauge field splits into a ``massive\", gauge-invariant piece, and the flat connections. Accordingly, the boundary theory operators organize themselves into two sectors: a vector operator dual to the massive part of the connection, and the well-known $U(1)$ chiral currents (which also arise in the pure Chern-Simons theory). We will obtain a variety of boundary conditions that correspond to double-trace deformations from the dual field theory perspective. In particular, we shall note the possibility of coupling the vector operator and the chiral currents via this mechanism. To our knowledge, these ``hybrid\" boundary conditions intertwining the massive and topological sectors have not been discussed in the literature; their existence was anticipated in \\cite{Gukov:2004id}, however, where the topology of the spacetime manifold was chosen in such a way that the two sectors effectively decouple.\n\nIt is worth emphasizing that all of this physics occurs at finite Maxwell coupling. It is often argued in the literature that the Maxwell coupling should be irrelevant in the infrared; this is certainly true from the bulk perspective. However, the Maxwell coupling is not irrelevant in the UV, and so is an important parameter holographically. One also notes in parallel that in condensed matter systems such as quantum Hall, the Maxwell coupling sets the cut-off scale where quasi-particle excitations live, going away only in the topological limit. It seems quite plausible that such excitations will exist in the holographic theory as well, a subject that we will explore elsewhere.\n\nHaving obtained the class of boundary conditions that lead to a conserved symplectic structure, one can examine in detail which of these are consistent with unitarity. Our motivation to consider this restriction comes primarily from the existence of the unitarity bound in conformal field theories (see \\cite{Mack:1975je,Minwalla:1997ka}, for example), which dictates that the presence of operators whose dimension is ``too low\" leads to negative norm states (ghosts). Via the holographic correspondence, this fact should manifest in the bulk physics as well, which is the question we address. As first noted in \\cite{Balasubramanian:1998sn,Witten:2001ua}, a closely analogous concern arises when considering bulk scalar fields with sufficiently high masses if one imposes boundary conditions that allow the slow-decaying branch to fluctuate. As a result of this choice, the conformal dimension of the dual operator lies below the unitarity bound and one expects the bulk theories to be ill-defined. Recently, these setups were considered in \\cite{Andrade:2011dg}, which confirmed that such bulk theories are indeed pathological and that, generically, they suffer from ghosts.\n\n We will address the question of unitarity by studying the dynamics of the MCS system in $AdS_{3}$ in both global and Poincar\\'e coordinates; in particular, we will discuss the resulting spectrum and symplectic products for the various boundary conditions for which the symplectic structure is conserved. Our main result is that the only boundary conditions consistent with unitarity do not mix the massive and topological sectors, and in particular they require to hold fixed the slower fall-off of the massive mode (i.e. they are of Dirichlet type). In short, the class of permissible boundary conditions is severely restricted by unitarity considerations. Interestingly, we will also find additional ghosts in the flat sector whose presence cannot be linked to unitarity bounds in an obvious way. We will also include an analysis of the symmetries we expect to be present in the dual theory as a result of various choices of boundary conditions.\n\nThe three-dimensional MCS theory has been previously considered in the context of AdS\/CFT. We refer the reader to \\cite{Gukov:2004id,Kraus:2006wn, Kraus:2006nb,Jensen:2010em} for work which focuses on the flat (topological) sector of the theory. The massive sector has also received some attention and the holographic dictionary problem has been studied to some extent \\cite{Minces:1999tp,DHoker:2010hr,Yee:2011yn}. Our results agree with the references above as far as the operator content is concerned. The novelty of our analysis lies in the fact that we have considered a wider class of boundary conditions, including ``hybrid\" boundary conditions that mix the massive and topological sectors, and analyzed their consistency with unitarity in detail. Additional related work includes \\cite{Carlip:2008jk,Fujita:2009kw,Balasubramanian:2010sc,Fosco:2011ra}.\n\nThis paper is organized as follows. In section \\ref{sec: prelim} we review the MCS system, the solution of the asymptotic equations of motion on asymptotically $AdS_3$ backgrounds, and the corresponding conformal dimensions of dual operators. In section \\ref{section:symplectic} we briefly describe the covariant phase space formalism, and use the conservation of the symplectic structure as a criterion to determine a wide class of \\textit{a priori} admissible boundary conditions in the holographic setup. For all the boundary conditions of interest, we construct the appropriate action principles and compute the one-point functions of the dual operators holographically. We also review the notion of symplectic product, which will play a central role in the analysis of unitarity. In section \\ref{section:spectrum} we discuss the spectrum of excitations in the dual field theory for the class of boundary conditions previously found, and discuss the normalizability of the various bulk modes. In section \\ref{section:norms} we present the calculation of the symplectic product for the various normalizable modes, focusing on the existence of ghosts; the requirement of unitarity in the dual theory then leads to a restricted class of permissible boundary conditions, which constitutes our main result. We conclude in section \\ref{section:discussion} with a discussion of our findings, along with possible extensions and applications. Some useful results used in the body of the paper have been collected in the appendices, as well as a brief discussion of the $U(1)$ symmetries in the dual field theory for the different boundary conditions under consideration.\n\n\\section{The Maxwell-Chern-Simons system}\\label{sec: prelim}\nWe consider the Maxwell-Chern-Simons (MCS) system in $(2+1)$ spacetime dimensions,\n\\begin{align}\\label{TME bulk action}\n I\n={}&\n- \\frac{1}{4\\gc^2}\\int_{M}d^{3}x\\,\\sqrt{|g|}F_{\\mu\\nu}F^{\\mu\\nu} -\\frac{\\hat{\\alpha}}{4} \\int_{M}d^{3}x\\, \\varepsilon^{\\mu\\nu\\rho}A_{\\mu}F_{\\nu\\rho}\n \\, ,\n\\end{align}\n\\noindent where $\\gc^2$ is the gauge coupling (with units of $[length^{-1}]$) and $\\hat{\\alpha}$ is the (dimensionless) Chern-Simons (CS) coupling. Throughout this paper we work in a fixed background in which we neglect the backreaction of the gauge field on the metric, {\\it i.e.} $G_N\/\\gc^2 \\to 0$, where  $G_N$ is the three dimensional gravitational coupling (which has units of $[length]$). Where appropriate, we will occasionally comment on issues of backreaction, and will consider them in a subsequent publication. The background metrics we consider are solutions of the Einstein equations in the presence of a negative cosmological constant $\\Lambda = -1\/L^2$, and the normalization is chosen such that pure $AdS_3$ space is a vacuum solution of the decoupled gravitational sector with radius $L$ and scalar curvature $R=-6\/L^2$. As usual, in a holographic context the action \\eqref{TME bulk action} must be supplemented by a collection of boundary terms that render the variational problem well defined and remove divergent contributions; these will be fully specified later on in the paper.\n\nThe equations of motion that follow from \\eqref{TME bulk action} are\\footnote{Our convention for the Levi-Civita tensor is $\\epsilon^{\\mu\\nu\\rho} = -\\frac{1}{\\sqrt{|g|}}\\varepsilon^{\\mu\\nu\\rho}$, where $\\varepsilon^{\\mu\\nu\\rho}$ is the Levi-Civita symbol.}\n\\begin{align}\n\\nabla_{\\nu}F^{\\nu\\mu} + \\frac{\\alpha}{2L}\\epsilon^{\\mu\\nu\\rho}F_{\\nu\\rho}=0\\, ,\\label{MaxCS equation}\n\\end{align}\n\\noindent where we have defined the rescaled CS coupling $\\alpha$ as\n\\begin{equation}\\label{rescaled CS coupling}\n\\alpha = \\gc^2 L\\hat{\\alpha}\\, ,\n\\end{equation}\n\\noindent which is also dimensionless. Without loss of generality, we will assume $\\alpha > 0$. When taking backreaction on the metric into account, asymptotically AdS solutions exist only for $\\alpha<1$, and we will restrict our discussions in the present paper to that range.\n\nIn form language, the Maxwell-CS equation \\eqref{MaxCS equation} can be written as\\footnote{On a $D$-dimensional spacetime, our convention for the Hodge dual is $*(dx^{\\nu_{1}}\\wedge\\cdots \\wedge dx^{\\nu_{r}}) = \\frac{1}{(D-r)!}\\epsilon^{\\nu_{1}\\ldots \\nu_{r}}_{\\phantom{\\nu_{1}\\ldots \\nu_{r}}\\mu_{1}\\ldots \\mu_{D-r}}dx^{\\mu_{1}}\\wedge\\cdots \\wedge dx^{\\mu_{D-r}}$.}\n\\begin{equation}\\label{Maxwell-CS eom}\nd^{\\dagger}F = \\frac{\\alpha}{L}*F\n\\end{equation}\n\\noindent where $d^\\dagger$ is the adjoint exterior derivative, which in our conventions acts on $F$ as $d^\\dagger F = -*d(*F) = -\\nabla_{\\mu}F^{\\mu}_{\\phantom{\\mu}\\nu}\\, dx^{\\nu}$. Hence, the equation of motion implies\n\\begin{equation}\\label{gauge field splitting}\n    A = A^{(0)} + B\\, ,\n\\end{equation}\n\\noindent  where $A^{(0)}$ is a flat connection and we have defined\n\\begin{equation}\\label{definition B}\nB \\equiv  -\\frac{L}{\\alpha}*F\\,.\n\\end{equation}\n\\noindent We note that $B$ is, by definition, invariant under the $U(1)$ gauge symmetry of the theory. In a later section we will study the consequences of the splitting \\eqref{gauge field splitting} at the level of the symplectic structure and the boundary conditions in a holographic context.\n\nSince $dB=dA=F$, the equation of motion \\eqref{Maxwell-CS eom} becomes a first order equation for $B$:\n\\begin{equation}\\label{equation of motion for B}\n*dB + \\frac{\\alpha}{L}B=0\\, ,\n\\end{equation}\n\\noindent which is the familiar equation for a massive vector field. In components, this equation reads\n\\begin{equation}\\label{first order equation for B}\n \\epsilon^{\\mu\\nu\\rho}\\partial_{\\nu}B_{\\rho} + \\frac{\\alpha}{L}B^{\\mu}=0\\, .\n\\end{equation}\n\\noindent Notice also that the definition \\eqref{definition B} implies a consistency condition:\n\\begin{equation}\nd^\\dagger B = 0\\, ,\n\\end{equation}\n\\noindent i.e. $B$ is a co-closed form ($\\nabla^{\\mu}B_{\\mu}=0$); naturally, this also follows from the equation of motion \\eqref{equation of motion for B}.  Acting on \\eqref{equation of motion for B} with $*d$ we can write a second-order equation for $B$,\n\\begin{equation}\n0 = d^{\\dagger}d B +\\frac{\\alpha^2}{L^2}B=\\Delta B + \\frac{\\alpha^2}{L^2}B\\, ,\n\\end{equation}\n\\noindent where $\\Delta = d^\\dagger d + d d^\\dagger$ is the Laplacian.\n\n\\subsection{Asymptotic solutions}\\label{subsection:asymptotic sols}\nFor the sake of concreteness,  we will write the metric of the asymptotically AdS spacetimes we are interested in  as\n\\begin{equation}\\label{asympt expansion of metric}\nds^{2} \\xrightarrow[r\\to \\infty]{} L^{2}\\frac{dr^{2}}{r^{2}} + \\frac{r^{2}}{L^{2}}g^{(0)}_{ij}(x)dx^{i}dx^{j} + \\ldots\n\\end{equation}\n\\noindent Restricting ourselves to flat connections which are finite at the conformal boundary, the asymptotic form of the solution for the gauge field is then of the form\\footnote{As we will see in appendix \\ref{section:sym bndy}, any finite $r$-dependent piece in the near-boundary behavior of the flat connection can be removed with the appropriate gauge transformation.}\n\\begin{equation}\\label{gen asympt}\nA(r,x) \\xrightarrow[r\\to \\infty]{} A^{(0)}(x) + r^{\\alpha}\\left(B^{(+)}(x) +\\mathcal{O}(r^{-2})\\right)+ r^{-\\alpha}\\left(B^{(-)}(x)+\\mathcal{O}(r^{-2})\\right)  \\,,\n\\end{equation}\nwhere $A^{(0)}$ is flat, i.e. $F^{(0)}=dA^{(0)}=0$. Similarly, solving the equations of motion asymptotically  one finds that the radial component $B_{r}$ of the gauge-invariant mode is subleading with respect to the $B_{i}$ components, which are moreover constrained by\n\\begin{equation}\\label{asymptotic constraint}\nP_{\\pm}^{ij}B_{j}^{\\left(\\mp\\right)} = 0\\, ,\\qquad \\mbox{where } \\quad  P_{\\pm}^{ij} = \\frac{1}{2}\\left(g^{(0) ij} \\pm  \\epsilon^{ij}\\right) .\n\\end{equation}\n\\noindent We have adopted the convention that $ \\epsilon^{ij} = -\\varepsilon^{ij}\/\\sqrt{|g^{(0)}|}$, where $\\varepsilon^{ij}$ is the two-dimensional Levi-Civita symbol, related to its three-dimensional counterpart by $\\varepsilon^{ij} = \\varepsilon^{rij}$. Notice that the projectors $P_{\\pm}^{ij}$ satisfy the usual properties: $\\left(P_{+}P_{-}\\right)^{ij} = P_{+}^{ik}g^{(0)}_{kl}P_{-}^{lj}=0$, $\\left(P_{\\pm}^{2}\\right)^{ij} = P_{\\pm}^{ik}g^{(0)}_{kl}P_{\\pm}^{lj} = P_{\\pm}^{ij}\\,$.\n\n\\subsection{Conformal dimensions}\\label{subsection:conf dims}\nGiven the asymptotic expansion \\eqref{gen asympt} and noting that the pullback to the boundary of the bulk vector field is simply a boundary vector, we conclude that the standard AdS\/CFT dictionary relates $B^{(+)}$ and $B^{(-)}$ with vector operators of dimensions $\\Delta_- = 1 - \\alpha$ and $\\Delta_+ = 1 + \\alpha$, respectively. On the other hand, the components of $A^{(0)}$ have scaling dimension one. As we shall review below, the components of $A^{(0)}$ along the boundary directions correspond to chiral currents that live on the boundary theory, \\cite{Witten:1988hf, Moore:1989yh, Elitzur:1989nr,Wen:1991mw,Balachandran:1991dw, Bos:1989kn, Bos:1989wa,Schwarz:1979ae}. We note that the lower scaling dimension is positive as long as $\\alpha < 1$, which implies that we can allow both fall-offs to fluctuate while preserving locally AdS asymptotics\\footnote{Here we use the terminology of \\cite{Skenderis:2002wp}, i.e., we mean that the curvature near the conformal boundary is that of AdS plus subleading corrections.} if $\\alpha < 1$. We have verified this statement explicitly by studying the effect of backreaction on a general asymptotically locally AdS metric of the form \\eqref{asympt expansion of metric}.\n\nIt should be noted that the operator of dimension $\\Delta_-$ violates the unitarity bound $\\Delta_V = 1$ for vector operators in two dimensions for all $\\alpha>0$\n\\cite{Mack:1975je,Minwalla:1997ka}, see also \\cite{ElShowk:2011gz} for the explicit expression. This suggests that boundary conditions that allow this degree of freedom to fluctuate should yield pathologies in the bulk; in subsequent sections we shall verify that this is indeed the case.\n\n\n\\section{Symplectic structure and boundary conditions}\\label{section:symplectic}\nIn the present section we study the issue of boundary conditions in the holographic description of the MCS system. We find it convenient to work within the covariant phase space formalism, which we will review shortly. The motivation for employing this formalism is two-fold: first, the classification of the allowed boundary conditions is nicely encoded in a simple vanishing-flux condition; and second, it allows us to classify the spectrum of excitations in a clean way. We emphasize however that this decision is just a matter of personal preference, and the results obtained within this framework should indeed be equivalent to the ones arrived at by more familiar, say canonical, methods.\n\nWe now proceed to briefly review the covariant phase space techniques; more detailed discussions can be found in \\cite{Lee:1990nz, Wald:1995yp,Wald:1999wa,Iyer:1994ys,Ashtekar:1990gc}. First, we stress that the construction is inherently Lorentzian, so we shall assume that the spacetime is endowed with a Lorentzian metric. Now, the ingredient that lies at the heart of this construction is the identification of the phase space with the space of solutions of the equations of motion which satisfy certain boundary conditions. This is possible since in any well-defined setup the specification of a point in canonical phase-space, i.e. of initial data, completely determines the subsequent evolution of the system. The other main ingredient is an algebraic structure that determines the dynamics once a Hamiltonian function is given, or crudely speaking, something that contains information about the Poisson brackets. This is nothing but the pre-symplectic structure of the theory, $\\Omega$, which can be thought of as a (possibly degenerate) two-form in the tangent space of (linearized) solutions. In other words, $\\Omega$ maps a pair of tangent vectors in the space of solutions to the real numbers. Given a background solution $\\bar{s}$ and two linearized solutions $\\delta_1 s$ and $\\delta_2 s$, we denote the symplectic product of $\\delta_1 s$ with $\\delta_2 s$ by $\\Omega(\\delta_1 s, \\delta_2 s; \\bar{s})$. Quite conveniently, this object can be constructed algorithmically given a Lagrangian \\cite{Lee:1990nz}, and we will illustrate this below.\n\nFrom the discussion above, it follows that the pre-symplectic structure must indeed be conserved in order for the identification of the initial data with the space of solutions to be independent of the surface on which the initial data is specified. This conservation condition is what we shall take as a guiding principle to classify the allowed boundary conditions for the MCS system. It is worth emphasizing here that the boundary conditions are in fact a crucial part of the definition of the phase space of a given theory. As pointed out above, the covariant phase space formalism also provides a useful way to classify the spectrum of excitations of the theory. In particular, we mention that in the presence of gauge symmetries the pre-symplectic structure is degenerate, the gauge orbits being precisely its null directions. Thus, we shall refer to any solution of the equations of motion whose symplectic product with an arbitrary solution vanishes as ``pure gauge\".\\footnote{We mention that the prefix ``pre\" makes reference to the degeneracy of $\\Omega$: by definition, a symplectic structure is non-degenerate. In a slight  abuse of notation we drop the prefix from now on, even when the kernel of $\\Omega$ is non-empty.} Further nomenclature will be discussed in section \\ref{sec: symp prod}.\n\nAfter taking the quotient by the gauge directions, the symplectic structure has a unique inverse and this corresponds to the Poisson bracket defined for gauge-invariant quantities. As discussed in detail in \\cite{Wald:1995yp}, this relation can be written succinctly as\n\\begin{equation}\\label{PB omega}\n    \\{ \\Omega(\\delta_1 s, \\cdot; \\bar{s}) , \\Omega(\\delta_2 s, \\cdot; \\bar{s}) \\}_{PB} = - \\Omega(\\delta_1 s, \\delta_2 s; \\bar{s}) \\, .\n\\end{equation}\n\\noindent Here $\\Omega(\\delta_1 s, \\cdot; \\bar{s})$ is to be understood as a linear function in covariant phase space. Then, the fact that the Poisson bracket and $\\Omega$ are the inverse of each other follows trivially by writing \\eqref{PB omega} in component notation. Finally, we mention that, at the classical level, one can construct conserved charges directly in terms of $\\Omega$. More precisely, given an infinitesimal transformation $\\delta_\\lambda s$ and an arbitrary linearized solution $\\delta s$, the infinitesimal variation of the generator $Q_\\lambda$ along $\\delta s$ is given by\n\\begin{equation}\\label{d Q}\n    \\delta Q_\\lambda  = \\Omega(\\delta_\\lambda s, \\delta s;\\bar{s}) \\, ,\n\\end{equation}\n\\noindent which, once again, is most easily visualized by translating \\eqref{d Q} into component notation. We stress that the charge $Q_\\lambda$ is only defined if \\eqref{d Q} is finite and satisfies the appropriate integrability conditions, see e.g. \\cite{Wald:1993nt}. Expression \\eqref{d Q} also makes it clear that gauge transformations, i.e. null directions of $\\Omega$, have a vanishing generator. This is just the familiar statement that the generators of gauge symmetries are constraints, and as such vanish on-shell.  On the other hand, global symmetries are associated to a non-zero charge.\n\n\\subsection{The symplectic flux}\\label{subsection:symplectic flux}\nIn this section we apply the method of \\cite{Lee:1990nz} to construct the symplectic structure of the MCS theory and determine the expression for the symplectic flux, which serves as a first step in classifying the allowed boundary conditions. Under an infinitesimal variation $\\delta A_{\\mu}$ of the gauge field  (and assuming a fixed background metric), the first order variation of the bulk action\\footnote{Note that we have not included the boundary terms in the action here. We will come back to them later, and confirm that they do not contribute to the symplectic structure.} is\n\\begin{equation}\\label{first variation action}\n\\delta I = \\int _{M}d^{3}x\\sqrt{|g|}\\, \\mbox{EOM}(A)^{\\mu}\\delta A_{\\mu} - \\int_{\\partial M}d^{2}x\\,\\sqrt{|\\gamma|}\\, \\rho_{\\mu}\\left(\\frac{1}{q^2}F^{\\mu\\nu} + \\frac{\\hat{\\alpha}}{2}\\epsilon^{\\mu\\rho\\nu}A_{\\rho}\\right)\\delta A_{\\nu}\\, ,\n\\end{equation}\n\\noindent where $\\mbox{EOM}(A)^{\\mu}=0$ is the equation of motion of the background gauge field, $\\gamma$ is the determinant of the induced metric on the timelike boundary (a ``constant radius\" slice), and $\\rho^{\\mu}$ denotes the corresponding unit normal.  From the above variation we read off the symplectic 1-form (see \\cite{Lee:1990nz,Compere:2008us})\n\\begin{align}\n\\theta^{\\mu}\n={}&\n - \\left(\\frac{1}{q^2}F^{\\mu\\nu} + \\frac{\\hat{\\alpha}}{2}\\epsilon^{\\mu\\rho\\nu}A_{\\rho}\\right)\\delta A_{\\nu}\\, .\n\\end{align}\n\\noindent Next, denoting by $\\delta_{1}A$ and $\\delta_{2}A$ two independent solutions of the linearized equations of motion\\footnote{We note that in the probe approximation the equations of motion for the background gauge field and its fluctuation have the same form, because the MCS system is linear.} we define the symplectic 2-form\n\\begin{align}\n\\omega^{\\mu}(\\delta_1 A, \\delta_2 A; \\bar{A})\n\\equiv{}&\n\t\\delta_{1}\\theta[\\delta_{2}A] - \\delta_{2}\\theta[\\delta_{1}A]\n\t\\nonumber\\\\\n={}&\n\t-\\frac{1}{q^2}\\Bigl(\\delta_{1}F^{\\mu\\nu}\\delta_{2} A_{\\nu}-\\delta_{2}F^{\\mu\\nu}\\delta_{1} A_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_{1}A_{\\rho}\\delta_{2} A_{\\nu}\\, .\n\\end{align}\n\\noindent Using the equation of motion for $\\delta F_{\\mu\\nu}$ (which is the same as \\eqref{MaxCS equation}, because we are ignoring backreaction on the metric) one can then show the crucial property\n\\begin{equation}\\label{symplectic continuity eq}\n\\nabla_{\\mu}\\omega^{\\mu} = 0\\, .\n\\end{equation}\n\nAs stated above, we assume that the $(2+1)$ manifold is Lorentzian, with the topology $X\\times \\mathds{R}$, where the $\\mathds{R}$ factor is parameterized by the timelike coordinate ($t$, say). The boundary $\\partial M$ is a surface of constant $r$. We now define the symplectic structure by\n\\begin{equation}\\label{OM bulk}\n\\Omega(\\delta_1 A, \\delta_2 A; \\bar{A}) = \\int_{\\Sigma}d^{2}x\\sqrt{h }\\, n_{\\mu}\\omega^{\\mu}\\, ,\n\\end{equation}\n\\noindent where $\\Sigma$ is a spacelike hypersurface (a $t={\\rm constant}$ slice, for example) with unit normal $n^{\\mu}$ and induced metric determinant $h$. Since the theory under consideration is linear, we can take the background to be the trivial configuration, i.e. $\\bar{A} = 0$, without loss of generality. We shall do so henceforth and omit the explicit reference to the background as an argument of the symplectic structure. We mention that, in principle, the bulk expression \\eqref{OM bulk} may require renormalization; the appropriate counterterms can be read off from a well-defined action principle as explained in \\cite{Compere:2008us}. However, working in the range $0 < \\alpha < 1$, no (UV) divergences arise in \\eqref{OM bulk} even if we allow the slow fall-off of the field to fluctuate, as we will verify by explicit computation in section \\ref{section:norms}. This is intimately related to the fact that, for $0 < \\alpha < 1$, the counterterms that render the variational principle well-defined do not include derivatives along the timelike direction, see section \\ref{sec:1pt}.\n\nAs discussed above, in order to obtain a well-defined phase space it is necessary to impose boundary conditions on our solutions in such a way that the symplectic structure is \\textit{conserved} (i.e. independent of $\\Sigma$). Integrating equation \\eqref{symplectic continuity eq} over a ``pillbox\" bounded by two spacelike hypersurfaces $\\Sigma_{1}$ and $\\Sigma_{2}$ and a region $R \\subset \\partial M$ (i.e. $R$ is an open subset of the boundary slice at constant $r$, see figure \\ref{pillbox}), one learns that the symplectic structure is independent of $\\Sigma$ provided the symplectic flux $\\Phi$ through $R$ vanishes, i.e.\n\\begin{figure}[htb]\n\\center\n\\includegraphics[width=0.65\\linewidth]{pillbox.pdf}\n\\label{pillbox}\n\\caption{The symplectic structure is conserved, i.e. $\\Omega(\\Sigma_{1})=\\Omega(\\Sigma_{2})$, when the symplectic flux through the region $R \\subset \\partial M$ vanishes.}\n\\end{figure}\n\\begin{equation}\\label{vanishing of flux}\n\\Phi = \\int_{R}d^{2}x\\sqrt{|\\gamma|}\\,\\rho_{\\mu}\\omega^{\\mu} =0\\, ,\n\\end{equation}\n\\noindent where, as before, $\\rho^{\\mu}$ and $\\gamma$ are the unit normal and the determinant of the induced metric on $R$, respectively. We suppose that this is attained {\\it locally}, so that the flux through the boundary vanishes through any open subset $R$. We mention that, from the point of view of the dual theory, these local boundary conditions correspond to the insertion of local operators. In the presence of additional boundaries, e.g. the Poincar\\'e horizon, one must also require the flux to vanish there. Given our assumption of locality, the boundary conditions at the extra boundaries are of course independent of the ones at the conformal boundary. It is worth noting that, for black hole spacetimes, the phase space is typically defined including the interior of the black hole, so a non-vanishing flux through the horizon is not in conflict with conservation of $\\Omega$.\n\nIn the coordinates introduced in \\eqref{asympt expansion of metric} the only non-vanishing component of $\\rho$ is $\\rho_{r} = \\sqrt{g_{rr}} = N_{r}$, where $N_{r}$ is the lapse in a radial foliation. Since $\\sqrt{|g|} = N_{r}\\sqrt{|\\gamma|}$, we have\n\\begin{equation}\n\\Phi  = \\int_{R}d^{2}x\\sqrt{|g|}\\,\\bar{\\rho}_{\\mu}\\omega^{\\mu}\\, ,\n\\end{equation}\n\\noindent where $\\bar{\\rho}_{\\mu}\\, dx^{\\mu}=dr$ and $g$ is the determinant of the full $(2+1)$ metric, as before. If we now split the connection as in \\eqref{definition B}, so that in an obvious notation the gauge field fluctuation is $\\delta A  = \\delta B + \\delta A^{(0)}$, we find\n\\begin{equation}\n\\omega^{\\mu} = \\omega^{\\mu}_{B} + \\omega^{\\mu}_{0} + \\omega^{\\mu}_{mix}\\, ,\n\\end{equation}\n\\noindent where we have defined\n\\begin{align}\n\\omega^{\\mu}_{B}\n\\equiv{}&\n\t-\\frac{1}{q^2}\\Bigl(f_{1}^{\\mu\\nu}\\delta_2 B_{\\nu}-f_{2}^{\\mu\\nu}\\delta_1 B_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_1 B_{\\rho} \\delta_2 B_{\\nu}\n\\\\\n\\omega^{\\mu}_{0}\n\\equiv {}&\n\t-\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\delta_1 A^{(0)}_{\\rho} \\delta_2 A^{(0)}_{\\nu}\n\\\\\n\\omega^{\\mu}_{mix}\n\\equiv {}&\n\t-\\frac{1}{q^2}\\Bigl(f_{1}^{\\mu\\nu}\\delta_2 A^{(0)}_{\\nu}-f_{2}^{\\mu\\nu}\\delta_1 A^{(0)}_{\\nu}\\Bigr) -\\hat{\\alpha}\\, \\epsilon^{\\mu\\rho\\nu}\\bigl(\\delta_1 A^{(0)}_{\\rho} \\delta_2 B_{\\nu} + \\delta_1 B_{\\rho} \\delta_2 A^{(0)}_{\\nu}\\bigr)\n\\end{align}\n\\noindent with $f$ the field strength of $\\delta B$. We now notice that contracting equation \\eqref{first order equation for B} with the Levi-Civita tensor results in $0 = F_{\\mu\\nu} - q^{2}\\hat{\\alpha}\\,\\epsilon_{\\mu\\nu\\rho}B^{\\rho}$. Consequently, the fluctuations of the gauge-invariant mode satisfy\n\\begin{equation}\\label{massive mode fluctuation eq}\nf^{\\mu\\nu} =   q^{2}\\hat{\\alpha}\\,\\epsilon^{\\mu\\nu\\rho}\\delta B_{\\rho}\\, .\n\\end{equation}\n\n\\noindent Using this on-shell condition in the above expression for $\\omega^{\\mu}$ we find\n\\begin{align}\\label{omegas flat and non-flat}\n\\omega^{\\mu}_{B}\n=\n\t\\hat{\\alpha}\\,\\epsilon^{\\mu\\nu\\rho}\\delta_1 B_{\\nu}\\delta_2 B_{\\rho}\\, ,\\qquad\n\\omega^{\\mu}_{0}\n=\n\t-\\hat{\\alpha}\\, \\epsilon^{\\mu\\nu\\rho}\\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\rho}\n\\,, \\qquad\n\\omega^{\\mu}_{mix}\n=\n\t0\\, .\n\\end{align}\n\nAs a result of the splitting \\eqref{omegas flat and non-flat}, the symplectic structure can be written as\n\\begin{equation}\\label{omega split}\n    \\Omega = \\int_\\Sigma d^2 x \\sqrt{h}\\, n_\\mu \\omega^{\\mu}_{B} + \\int_\\Sigma d^2 x \\sqrt{h}\\, n_\\mu \\omega^{\\mu}_{0} \\, .\n\\end{equation}\n\\noindent This suggests that the space of solutions is a direct product of the flat and non-flat sectors. However, a more detailed analysis reveals that this is only true if the boundary conditions do not mix modes in the various sectors, see section \\ref{sec:bc}.\n\nLet us now find an expression for the symplectic flux that will allow us to determine the allowed boundary conditions. In order to do so, it is important to keep in mind that the modes $\\delta B^{(\\pm)}$ are constrained by the asymptotic equations of motion, and therefore obey \\eqref{asymptotic constraint}. For example, in light-cone coordinates $(u,v)$ in which the boundary metric takes the form\n\\begin{equation}\\label{bndy light cone}\n    g^{(0)}_{ij} = \\left(\n                     \\begin{array}{cc}\n                       0 & 2 \\\\\n                       2 & 0 \\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\n\\noindent these lead to\n\\begin{equation}\n\\delta B_{v}^{(+)} = \\delta B_{u}^{(-)}=0\\,.\n\\end{equation}\n\\noindent Taking the asymptotic constraints \\eqref{asymptotic constraint} into account then, we find that the symplectic flux through $R$ is given by\n\\begin{align}\\label{flux coeff}\n\\Phi\n  ={}&\n \\hat{\\alpha}\\int_{R}d^{2}x\\,\n\t\\varepsilon^{r\\nu\\lambda}\\left( \\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\lambda}-\\delta_1 B_{\\nu}^{(+)}\\delta_2 B_{\\lambda}^{(-)} + \\delta_2 B_{\\nu}^{(+)} \\delta_1 B_{\\lambda}^{(-)}  \\right)\n\t\\nonumber\\\\\n\t={}&\n\t\t\\hat{\\alpha}\\int_{R}d^{2}x\\,\\varepsilon^{ij}\\left( \\delta_1 A^{(0)}_{i} \\delta_2 A^{(0)}_{j}-\\delta_1B_{i}^{(+)}\\delta_2 B_{j}^{(-)} + \\delta_2 B_{i}^{(+)} \\delta_1 B_{j}^{(-)}  \\right).\n\\end{align}\n\n\\subsection{Boundary conditions}\\label{sec:bc}\nAs discussed above, demanding the vanishing of the symplectic flux gives us a useful way of classifying the boundary conditions. Momentarily giving up covariance in the boundary directions, in light-cone coordinates \\eqref{bndy light cone} we find that possible local boundary conditions include\n\\begin{eqnarray}\\label{bcs flat sector super}\nA^{(0)}_{u} = W\\bigl[A^{(0)}_{v}\\bigr],\\qquad  B_{u}^{(+)} = V\\bigl[ B_{v}^{(-)}\\bigr].\n\\end{eqnarray}\n\n\\noindent For general ``potentials\" $W$ and $V$, such boundary conditions would correspond to multi-trace deformations in the dual CFT. For simplicity, let us focus on the linear case\n\\begin{eqnarray}\\label{bcs flat sector}\n\\delta A^{(0)}_{u} = \\bar{\\beta}\\, \\delta A^{(0)}_{v},\\qquad \\delta B_{u}^{(+)} = \\beta\\, \\delta B_{v}^{(-)},\n\\end{eqnarray}\n\\noindent for any constants $\\beta,\\bar\\beta$. Note that $\\beta=0,\\infty$ correspond to chiral boundary conditions, while other values mix the modes and break covariance. We will refer to $\\delta B_{u}^{(+)} = 0$ as Dirichlet and to $\\delta B_{v}^{(-)} = 0$ as Neumann boundary conditions, in close analogy to the terminology commonly used for scalar fields in AdS. We will term the boundary condition $\\delta B_{u}^{(+)} = \\beta\\, \\delta B_{v}^{(-)}$ as ``mixed\" when $\\beta$ is finite. As usual, the boundary conditions with finite $\\beta$ and $\\bar\\beta$ are related to double-trace deformations of the boundary theory \\cite{Witten:2001ua,Berkooz:2002ug}, as we will review later on. Furthermore, we notice that, because $B_{u}^{(+)}$ and $B_{v}^{(-)}$ have scaling dimensions $\\Delta_- = 1- \\alpha$ and $\\Delta_+ = 1 + \\alpha$, respectively, the constant $\\beta$ has dimension $\\Delta_\\beta = - 2\\alpha$. The RG flow interpretation of double-trace deformations has been discussed in, for example, \\cite{Witten:2001ua,Gubser:2002zh,Gubser:2002vv,Leigh:2003gk,Hartman:2006dy}.\\footnote{This interpretation requires both end points of the RG flow to be well-defined, e.g. as in the case of scalar fields with masses close to Breitenlohner-Freedman bound in AdS. We shall see below that in the present case the Neumann theories are ill-defined so this picture does not strictly hold.} On the other hand, since $A^{(0)}_u$ and $A^{(0)}_v$ both have dimension one, the constant $\\bar{\\beta}$ is dimensionless.\nInterestingly, we also note the possibility of a ``hybrid\" boundary condition\n\\begin{equation}\\label{gen hyb}\n\\delta A^{(0)}_{u} = \\kappa\\, \\delta B_{u}^{(+)} \\qquad \\mbox{\\textbf{and}}\\qquad \\delta A^{(0)}_{v} = \\frac{1}{\\kappa}\\delta B_{v}^{(-)}\\, ,\n\\end{equation}\n\\noindent that mixes the flat connections with the massive sector. Here, $\\kappa$ is a constant of scaling dimension $\\Delta_\\kappa = \\alpha$. Notice that, in view of the flatness condition on $\\delta A^{(0)}$, \\eqref{gen hyb} implies\n\\begin{equation}\\label{hyb on b}\n    \\kappa^2\\, \\partial_v \\delta B_u^{(+)} = \\partial_u \\delta B_v^{(-)}\\, .\n\\end{equation}\n\\noindent In analogy with the linear boundary conditions discussed above, this hybrid boundary condition has the interpretation of a double-trace deformation. To our knowledge, the possibility of such boundary conditions has not been explicitly discussed in the literature.\n\nIt is now clear from the decomposition \\eqref{omega split} and the analysis of the boundary conditions above that, as anticipated in \\cite{Gukov:2004id}, the flat and massive sectors do not always decouple. In fact, for our hybrid boundary conditions \\eqref{gen hyb} both sectors indeed interact with one another. The decoupling only occurs if one imposes boundary conditions which do not mix both sectors, i.e. if we impose  boundary conditions like those in \\eqref{bcs flat sector}. This is because it is only in this case that the symplectic structure effectively splits as a direct sum of two independent pieces.\n\n\\subsection{One-point functions}\n\\label{sec:1pt}\n\nAs usual in the context of holography, the Maxwell-Chern Simons action \\eqref{TME bulk action} must be supplemented by a series of boundary terms that serve two purposes: achieving a well-defined variational principle for a chosen set of boundary conditions, and removing divergences. We will refer to the latter as counterterms. We recall now that the first variation of the bulk action is given by \\eqref{first variation action}. Evaluating this expression on-shell we find\n\\begin{equation}\n\\left.\\delta I \\right|_{os} = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\sqrt{|\\gamma|}\\, \\rho_{\\mu}\\epsilon^{\\mu\\rho\\nu}\\left(B_{\\rho}- A^{(0)}_{\\rho}\\right)\\delta A_{\\nu}\\, ,\n\\end{equation}\n\n\\noindent where the gauge field fluctuations are understood to be evaluated on the solution of the linearized equations of motion.\\footnote{Since we are ignoring backreaction, the various metric quantities are always understood to be evaluated on their (fixed) background values.}  Employing the notation established above, we find\n\\begin{align}\n\\left.\\delta I \\right|_{os}\n={}&\n -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B_{i}- A^{(0)}_{i}\\right)\\left( \\delta A^{(0)}_{j} + \\delta B_{j}\\right)\n  \\nonumber\\\\\n ={}&\n  -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B^{(+)}_{i}\\delta B_{j}^{(-)} + B_{i}^{(-)}\\delta B_{j}^{(+)}- A^{(0)}_{i}\\delta A^{(0)}_{j}  \\right)\n  \\nonumber\\\\\n &\n  -\\frac{\\hat{\\alpha}}{2}\\lim_{r\\to \\infty} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij} r^{\\alpha}\\left(B_{i}^{(+)}\\delta A^{(0)}_{j} + A_{j}^{(0)}\\delta B_{i}^{(+)}\\right)\\, ,\n\\end{align}\n\\noindent where in the last equality we used the restrictions placed by the asymptotic equations of motion on the $B^{(\\pm)},\\delta B^{(\\pm)}$ modes (c.f. section \\ref{subsection:asymptotic sols}). We note the presence (for any finite Maxwell coupling $q^2$) of the divergent term, which we cancel by the addition of a counterterm. Noticing that $\\varepsilon^{ij} r^{\\alpha}\\left(B_{i}^{(+)}\\delta A^{(0)}_{j} + A_{j}^{(0)}\\delta B_{i}^{(+)}\\right) = \\delta \\left(r^{\\alpha}\\varepsilon^{ij}A_{j}^{(0)}B_{i}^{(+)}\\right)$ it is easy to check that the desired counterterm is given by the covariant expression\n\\begin{align}\nI_{ct} = \\frac{1}{2q^2} \\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}F^{i}A_{i}\\, ,\n\\end{align}\n\\noindent where, as before, $\\gamma$ is the determinant of the induced metric on the $r={\\rm constant}$ surface, and we have defined\n\\begin{equation}\nF^{i}\\equiv \\rho_{\\mu}F^{\\mu i}\\, ,\n\\end{equation}\n\\noindent with $\\rho_{\\mu}$ the unit normal 1-form on the radial slices. Therefore, we have that\n\\begin{align}\n\\left.\\delta\\left( I + I_{ct}\\right) \\right|_{os}\n ={}&\n  -\\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}\\left(B^{(+)}_{i}\\delta B_{j}^{(-)} + B_{i}^{(-)}\\delta B_{j}^{(+)}- A^{(0)}_{i}\\delta A^{(0)}_{j}  \\right)\n \\end{align}\n\\noindent is finite as $r\\to\\infty$.\n\\subsubsection{Covariant boundary conditions}\nIn order to proceed further we need to discuss the additional \\textit{finite} boundary terms needed in order to enforce different boundary conditions of interest. Confining ourselves to covariant terms for the moment, we consider the following quantities:\n\\begin{align}\nB_{\\pm}\n ={}&\n  \\mp \\frac{1}{4\\gc^{4}\\hat{\\alpha}}\\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}F^{i}\\gamma_{ij}F^{j}\\, ,\n\\\\\nB_{(0)} ={}&\n  \\frac{1}{2q^{2}}\\int_{\\partial M}d^{2}x\\, \\varepsilon^{ij}F_{i}A_{j}+\\frac{1}{4}\\int_{\\partial M}d^{2}x\\sqrt{|\\gamma|}\\gamma^{ij}\\left(\\frac{1}{q^4\\hat\\alpha}F_{i}F_{j}-\\hat\\alpha A_{i}A_{j}\\right)\\, .\n\\end{align}\n\n\\noindent Evaluating on-shell we find\n\\begin{align}\\label{on shell covariant counter-terms}\n\\left. B_{\\pm}\\right|_{os}\n ={}&\n \\pm  \\frac{\\hat\\alpha}{2}\\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}g^{(0)ij}B_{i}^{(-)}B_{j}^{(+)}\n\\nonumber\\\\\n={}&\n\\pm  \\frac{\\hat\\alpha}{2}\\int_{\\partial M}d^{2}x\\,\\varepsilon^{ij}B_{i}^{(+)}B_{j}^{(-)}\n\\\\\n \\left. B_{(0)} \\right|_{os}\n ={}&\n-\\frac{\\hat\\alpha}{4}\\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}g^{(0)ij}A_{i}^{(0)}A_{j}^{(0)}\\, .\n\\end{align}\n\\noindent By taking linear combinations of these finite boundary terms we can achieve a variational principle well-suited for the various boundary conditions \\eqref{bcs flat sector} of interest in the flat and gauge-invariant (massive) sectors. For example, we find\n\\begin{align}\\label{variation with Bplus fixed}\n\\delta \\left. \\left(I + I_{ct}\\pm B_{(0)} + B_{+}\\right) \\right|_{os}\n={}&\n     \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\left[\\epsilon^{ij}B_{i}^{(-)}\\delta B_{j}^{(+)} \\mp A_{i}^{(0)}P_{\\pm}^{ij}\\delta A^{(0)}_{j}\\right]\\, ,\n\\end{align}\n\\noindent and\n\\begin{align}\\label{variation with Bminus fixed}\n\\delta \\left. \\left(I + I_{ct}\\pm B_{(0)} + B_{-}\\right) \\right|_{os}\n={}&\n     \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\left[\\epsilon^{ij}B^{(+)}_{i}\\delta B_{j}^{(-)}\\mp A_{i}^{(0)}P_{\\pm}^{ij}\\delta A^{(0)}_{j}\\right].\n\\end{align}\n\\noindent Now that we have identified the sources for the covariant boundary conditions, i.e. $\\delta B^{(\\pm)}_{i}$ and $\\left(P_{\\pm}\\delta A^{(0)}\\right)_{i} = g^{(0)}_{ij}P_{\\pm}^{jk}\\delta A^{(0)}_{k}$, we write the variation of the renormalized action $I_{ren}$ generically as\n\\begin{equation}\\label{I ren gen}\n\\left.\\delta I_{ren}\\right|_{os} =  \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|}\\biggl[\\langle \\mathcal{O}^{(\\pm)i}\\rangle \\delta B^{(\\pm)}_{i} + \\langle\\mathcal{O}_{\\pm}^{(0)\\, i}\\rangle \\left(P_{\\pm}\\delta A^{(0)}\\right)_{i}\\biggr].\n\\end{equation}\n\\noindent Comparing with \\eqref{variation with Bplus fixed} and \\eqref{variation with Bminus fixed} and using the properties of $P_{\\pm}$ we can read-off the one-point functions of the dual operators, and we obtain\n\\begin{align}\\label{O b}\n\\langle \\mathcal{O}^{(\\pm)i}\\rangle\n ={}&\n  \\pm \\hat{\\alpha}\\,g^{(0)ij}B_{j}^{(\\mp)}\\, ,\n  \\\\ \\label{O A0}\n \\langle\\mathcal{O}_{\\pm}^{(0)\\, i}\\rangle\n ={}&\n \\mp \\hat{\\alpha} P_{\\mp}^{ij}A_{j}^{(0)}\\, .\n\\end{align}\n\nSince $A_{i}^{(0)}$ is constrained by the flatness condition, the variational derivatives with respect to its components are ill-defined, and, as a consequence, the one-point functions \\eqref{O A0} suffer from an ambiguity. However, this ambiguity is nothing but the one associated to the $U(1)$ gauge transformations. In other words, \\eqref{O A0} are only defined up to the transformations $\\delta A_i  = \\partial_i \\lambda$ that preserve the boundary conditions in the variational principle. See \\cite{Marolf:2006nd} for a related discussion in the context of (pure) Maxwell fields.\n\\subsubsection{Symmetry-breaking boundary conditions}\nLet us now turn to the less symmetric scenarios. First, we consider the case of ``mixed\" boundary conditions, i.e. $B^{(+)}_u - \\beta B^{(-)}_v = 0$, where $\\beta$ is a finite dimensionful constant. It is clear that this requirement breaks both conformal and Poincar\\'e symmetry, so we are allowed to write down the appropriate boundary terms simply in terms of the coefficients of the asymptotic expansion. As we shall see shortly, it is useful to generalize the above boundary condition and consider instead\n\\begin{equation}\\label{mix bc J}\n    B^{(+)}_u - \\beta B^{(-)}_v = J_\\beta\\, ,\n\\end{equation}\n\\noindent where $J_\\beta$ is an arbitrary fixed function of the boundary coordinates.  We ignore the contribution from the flat sector momentarily. Starting from the Neumann theory, i.e. the theory in which $B^{(-)}_i$ is fixed and whose action we denote by $I_N$, the boundary term we need to add in order to attain the mixed boundary condition is\n\\begin{equation}\\label{I def mix}\n    I_{def,\\beta} = - \\frac{\\hat{\\alpha}}{4 \\beta} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left(B^{(+)}_u \\right)^2\\, .\n\\end{equation}\n\\noindent In fact, using the variation of the Neumann action (\\ref{variation with Bminus fixed}) and the explicit boundary term \\eqref{I def mix}, we obtain\n\\begin{equation}\\label{dI mixed}\n    \\delta\\left(I_N + I_{def,\\beta}\\right) = \\frac{\\hat{\\alpha}}{2 \\beta} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} B^{(+)}_u\\left( \\beta\\, \\delta B^{(-)}_v - \\delta B^{(+)}_u \\right),\n\\end{equation}\n\\noindent which is finite and stationary when the boundary condition \\eqref{mix bc J} holds. Comparing \\eqref{dI mixed} with \\eqref{mix bc J}, we note that the quantity that is being held fixed in the variational principle is in fact $J_\\beta$. This means that $J_\\beta$ is to be interpreted as the source for the dual operator in the boundary theory. Given this, it follows from \\eqref{dI mixed} that the one-point function in the presence of sources for the dual operator in the deformed theory is given by\n\\begin{equation}\\label{1pt mix}\n    \\langle {\\cal O}^{(-)}_u \\rangle_{\\beta} = - \\frac{\\hat{\\alpha}}{2 \\beta} B^{(+)}_u,\n\\end{equation}\n\\noindent where, as usual, $B^{(+)}_u$ must be thought of as a function of the source $J_\\beta$ defined in \\eqref{mix bc J}. Before constructing variational principles suitable for the remaining boundary conditions, we comment that the computation above provides a simple illustration of the well-known fact that linear boundary conditions of the form \\eqref{mix bc J} correspond to double-trace deformations in the dual theory. The argument is as follows. First, we recall that in AdS\/CFT the Neumann action $I_N$ is interpreted as the generating function for the operator associated to $B^{(+)}_u$ in the dual CFT. Then, the boundary term \\eqref{I def mix} is transparently identified with a double-trace deformation for this operator. Moreover, the inclusion of \\eqref{I def mix} implies that the original Neumann boundary condition needs to be shifted in such a way that the the modified action has an extremum. As noted above, the new boundary condition is nothing but the linear relation \\eqref{mix bc J}, which completes the argument. It is worth commenting on the possibility of thinking of the \\eqref{mix bc J} as a deformation of the Dirichlet theory. In such case, the boundary term that implements the shift in the boundary condition is quadratic in $B^{(-)}_v$, so it has dimension $2(1+\\alpha)$. We see that the deformation is then irrelevant.\n\nWe now construct an appropriate action for the boundary condition\n\\begin{equation}\\label{mix bc A0 J}\n    A^{(0)}_u - \\bar{\\beta} A^{(0)}_v = J_{\\bar \\beta}\n\\end{equation}\n\\noindent where $\\bar\\beta$ is a non-zero dimensionless constant and $J_{\\bar \\beta}$ is a fixed arbitrary function of the boundary coordinates. In analogy with the previous case, $J_{\\bar \\beta}$ corresponds to the source of the dual operator. Note that since $\\bar \\beta$ is dimensionless the boundary condition \\eqref{mix bc A0 J} for $J_{\\bar \\beta} = 0$ preserves scale invariance, yet it breaks Lorentz invariance. Once again, as a consequence of this, it is licit to write extra boundary terms which are not Lorentz densities. Moreover, because this boundary condition does not mix the flat and massive sectors, we concentrate on the flat connections and temporarily drop the contribution from the massive modes. Now, assuming that we start with an action $I^{(1)}_{ren}$ which attains an extremum when $P_- A^{(0)}$ is fixed, we find\n\\begin{equation}\\label{dI1}\n    \\delta I^{(1)}_{ren} = \\hat{\\alpha} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\langle\\mathcal{O}_{-}^{(0)\\, i}\\rangle (P_- \\delta A^{(0)})_i = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\delta A^{(0)}_v\n\\end{equation}\n\\noindent as follows from \\eqref{I ren gen} and \\eqref{O A0}.  In this case, the boundary term that we need to add to $I^{(1)}_{ren}$ in order for \\eqref{mix bc A0 J} to hold can be written as\n\\begin{equation}\\label{I def bar beta}\n    I_{def, \\bar \\beta} = - \\frac{\\hat{\\alpha}}{4 \\bar{\\beta}} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left( A^{(0)}_u \\right)^2\\, .\n\\end{equation}\n\\noindent In fact, with this choice the on-shell variation of the action reads\n\\begin{equation}\\label{d I mix A0}\n    \\delta \\left(I^{(1)}_{ren} + I_{def, \\bar \\beta}\\right) = \\frac{\\hat{\\alpha}}{2 \\bar{\\beta}} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\left( \\bar{\\beta} \\delta A^{(0)}_v - \\delta A^{(0)}_u \\right),\n\\end{equation}\n\\noindent as desired. As discussed above, the boundary condition \\eqref{mix bc A0 J} is in one-to-one correspondence with the inclusion of the double-trace deformation \\eqref{I def bar beta} in the dual theory. The relevant one-point function is given by\n\\begin{equation}\\label{1pt mix A0}\n    \\langle {\\cal O}^{(0)}_{+ u} \\rangle_{\\bar \\beta} = - \\frac{\\hat{\\alpha}}{2 \\bar{\\beta}} A^{(0)}_u\\, .\n\\end{equation}\n\n\\noindent Once again, we mention that the one-point function \\eqref{1pt mix A0} is only defined up to the appropriate $U(1)$ transformation.\n\nFinally, we consider the ``hybrid\" boundary conditions defined in \\eqref{gen hyb}, which admit the obvious generalization\n\\begin{equation}\\label{hyb bc J}\n    A^{(0)}_u - \\kappa B^{(+)}_u = J_\\kappa\\,, \\qquad A^{(0)}_v - \\kappa^{-1} B^{(-)}_v = \\tilde{J}_\\kappa \\, ,\n\\end{equation}\n\\noindent where we take $J_\\kappa$, $\\tilde{J}_\\kappa $ to be the sources of the dual operators. It is convenient to start with a renormalized action $I^{(2)}_{ren}$ such that\n\\begin{equation}\\label{I ren for hyb}\n    \\delta I^{(2)}_{ren} = \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\left( A^{(0)}_u \\delta A^{(0)}_v + B^{(-)}_v \\delta B^{(+)}_u \\right).\n\\end{equation}\n\\noindent With $I^{(2)}_{ren}$ as a starting point, the boundary term that implements hybrid boundary conditions is given by\n\\begin{equation}\\label{I def hyb}\n    I_{def, \\kappa} = - \\frac{\\hat{\\alpha}}{2 \\kappa} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} B^{(-)}_v A^{(0)}_u,\n\\end{equation}\n\\noindent as it follows from\n\\begin{align}\\label{dI hyb}\n    \\delta\\left(I^{(2)}_{ren} + I_{def, \\kappa}\\right)\n    ={}&\n     \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} A^{(0)}_u \\left( \\delta A^{(0)}_v - \\kappa^{-1} \\delta B^{(-)}_v \\right)\n     \\nonumber \\\\\n    &-\n     \\frac{\\hat{\\alpha}}{2} \\int_{\\partial M}d^{2}x\\sqrt{|g^{(0)}|} \\kappa^{-1} B^{(-)}_v \\left( \\delta A^{(0)}_u - \\kappa\\, \\delta B^{(+)}_u \\right).\n\\end{align}\n\n\\noindent As pointed out before, the hybrid boundary conditions correspond to a double-trace deformation in the dual theory. Note that, in this case, the deformation \\eqref{I def hyb} explicitly mixes the flat and massive sectors, so indeed these do not decouple in the theory defined by the hybrid boundary conditions. It follows from \\eqref{dI hyb} that the one-point functions in the dual theory are given by\n\\begin{align}\\label{O kappa 1}\n    \\langle {\\cal O}^{(0)}_{+ u} \\rangle_\\kappa\n    =\n     \\frac{\\hat{\\alpha}}{2} A^{(0)}_u \\qquad \\mbox{and}\\qquad\n    \\langle {\\cal O}^{(0)}_{- v} \\rangle_\\kappa\n    =\n     - \\frac{\\hat{\\alpha}}{2 \\kappa} B^{(-)}_v\\, .\n\\end{align}\n\\noindent It is worthwhile noting that, since the $U(1)$ transformations do not preserve the boundary conditions \\eqref{gen hyb}, the one-point functions \\eqref{O kappa 1} are unambiguously defined.\n\nBefore closing this section, we emphasize that, provided $0 < \\alpha < 1$, the boundary terms involved do not contain derivatives along the timelike direction. Given the results of \\cite{Compere:2008us}, this strongly suggests that the bulk symplectic structure does not need to be supplemented by additional boundary contributions. This is indeed the case, as we will explicitly verify below. Specifically, we will check that, for $0 < \\alpha < 1$, the bulk symplectic structure is finite and conserved for all the boundary conditions under scrutiny.\n\n\\subsection{The symplectic product}\\label{sec: symp prod}\nRecall that the symplectic structure is given by \\eqref{omega split} with \\eqref{omegas flat and non-flat}, i.e.\n\\begin{equation}\\label{omega recap}\n    \\Omega = \\hat{\\alpha} \\int_\\Sigma d^2 x \\sqrt{h} n_\\mu \\epsilon^{\\mu\\nu\\rho}\\delta_1 B_{\\nu}\\delta_2 B_{\\rho} - \\hat{\\alpha}  \\int_\\Sigma d^2 x \\sqrt{h} n_\\mu \\epsilon^{\\mu\\nu\\rho}\\delta_1 A^{(0)}_{\\nu} \\delta_2 A^{(0)}_{\\rho}\\, .\n\\end{equation}\n\\noindent As mentioned above, it turns out that the restriction $0 < \\alpha < 1$ ensures that \\eqref{omega recap} is finite for all the boundary conditions of interest, provided one imposes additional requirements on the solutions in the deep interior.\n\nQuite generally, given a symplectic structure it is possible to endow the space of solutions with an inner product defined in terms of $\\Omega$, as we now review briefly. A more detailed discussion can be found in \\cite{Wald:1995yp}, for example. We start by complexifying the space of solutions and consider\\footnote{The reader uneasy with the use of the complex conjugates in \\eqref{ip} can think of using a basis of solutions in momentum space in which the modes are generically complex despite the fact that the field is real.}\n\\begin{equation}\\label{ip}\n    (A_1, A_2) = - i \\Omega(A_1^*, A_2)\\, .\n\\end{equation}\n\\noindent We will refer to \\eqref{ip} as the symplectic product of the theory. One can verify that \\eqref{ip} satisfies the expected properties of bi-linearity and Hermiticity, although in general it fails to be positive definite.\n\nThe inner product \\eqref{ip} allows us to introduce some useful terminology. First, we shall term a given solution $A_0$ as \\textit{normalizable} if $(A_0, A)$ is finite for all $A$. As stated above, in our particular setup this translates into a requirement on the fields in the deep interior. Second, we define a \\textit{ghost} to be an excitation of definite positive(negative) frequency with negative(positive) norm. Here, we will use the definition of positive frequency associated to the timelike Killing vector of the relevant background geometry under consideration. For example, if $\\partial_t$ is a vector field which is timelike everywhere, the solution $A$ is said to be a positive frequency solution if $\\partial_t A = - i \\omega A $ with $\\omega > 0$. Third, we will refer to a solution $A_{gauge}$ as \\textit{pure gauge} if $(A_{gauge}, A) = 0$ for all $A$.\n\nIt should be noted that the presence of ghosts in a given system is correlated with the lack of unitarity in the associated quantum theory. At the classical level, the presence of ghosts also signals pathologies since these give negative contributions to the energy.\n\n\n\\section{The dual field theory spectrum}\\label{section:spectrum}\nIn this section we determine the spectrum of normalizable solutions of the MCS system in $AdS_3$, in both global and Poincar\\'e coordinates, for the various boundary conditions of interest. As explained in section \\ref{sec: symp prod}, by ``normalizable\" we mean excitations that have finite symplectic product with all the modes. We mention that, while normalizability at the conformal boundary is guaranteed by restricting the coupling $\\alpha$ defined in \\eqref{rescaled CS coupling} to satisfy $0 < \\alpha < 1$, normalizability at the interior is achieved by restricting the wave functions appropriately. More precisely, when the geometry is global AdS we shall require the wave functions to be smooth at the origin, as is customary. In the Poincar\\'e AdS case, in addition to smoothness in the interior, we restrict the wave functions in such a way that no symplectic flux can leak through the Poincar\\'e horizon.\n\nAs explained in section \\ref{sec: prelim}, the connection splits into flat and ``massive\" pieces, and we can solve the bulk equations of motion separately for each sector. Moreover, as discussed in section \\ref{sec:bc}, these sectors decouple unless we impose the ``hybrid\" boundary condition \\eqref{gen hyb}. Our strategy to find the spectrum will be to focus on the massive and flat sectors separately, and incorporate the effects of the mixing only when we discuss the hybrid boundary conditions. For the sake of simplifying the exposition, we display the general solution to the equations of motion of the massive mode in appendix \\ref{section:solutions}, while here we focus exclusively on imposing the appropriate boundary conditions.\n\n\\subsection{Global $AdS_{3}$} \\label{sec: spec global adS}\nWe first consider the MCS theory in global $AdS_{3}$, whose line element is given by \\eqref{Global AdS3}. Since the spacetime is topologically trivial, there is no room for holonomies and the connection must be smooth at the origin, where the vector field $\\partial_x =(1\/L)\\partial \\varphi$ becomes singular. As a result, in addition to normalizability we must impose $A[\\partial_x] = A_x = 0$ at $\\rho = 0$. It should be stressed that setting $A^{(0)}_\\rho = 0$ \\textit{everywhere} in the bulk is generically in conflict with smoothness. To see why, we note that this implies that the components of $A^{(0)}$ along the boundary directions are independent of $\\rho$ everywhere, so any boundary condition other than $A^{(0)}_x |_{\\partial M} = 0$ would yield singular configurations. Having said this, we initiate the study of the spectrum for all the boundary conditions of interest.\n\\subsubsection{Flat sector}\n\\label{spec flat sector global}\n\nWe first consider the flat sector. Since there are no holonomies, the flat connections can be written as\n\\begin{equation}\\label{soln flat global}\n    \\delta A^{(0)}_\\mu = \\partial_\\mu \\lambda\\, ,\n\\end{equation}\n\\noindent where $\\lambda$ is smooth everywhere, with  $\\partial_x \\lambda = 0 $ at $\\rho=0$. Recall that in our analysis of the symplectic flux, we encountered the allowed boundary condition  \\eqref{bcs flat sector}, which in terms of the $(t,x)$ coordinates defined in appendix \\ref{section:solutions} takes the form\n\\begin{equation}\\label{bc flat t x}\n   \\left.\\left( \\delta A^{(0)}_t - \\hat{\\beta}\\, \\delta A^{(0)}_x \\right)\\right|_{\\partial M} = 0\\, ,\n\\end{equation}\n\\noindent where $\\hat{\\beta} = (\\bar{\\beta}-1)\/(\\bar{\\beta} +1)$ is a (possibly vanishing or infinite) constant. Fourier-decomposing $\\lambda$ as $\\lambda = e^{-i \\omega t + i k x} \\hat \\lambda(k,\\omega)$ with $k\\in\\mathds{Z}$, and using \\eqref{soln flat global} in \\eqref{bc flat t x}, we learn that the frequencies must satisfy\n\\begin{equation}\\label{freq global}\n    \\omega = - \\hat{\\beta} k\\, ,\n\\end{equation}\n\\noindent which determines the spectrum of the flat sector. As is well-known \\cite{Witten:1988hf, Moore:1989yh,Elitzur:1989nr,Wen:1991mw,Balachandran:1991dw,Bos:1989kn,Bos:1989wa,Schwarz:1979ae}, the degrees of freedom of the flat sector reside exclusively on the spacetime boundary,\\footnote{In particular, if $\\lambda$ goes to zero at the boundary the flat connections are pure gauge.} a fact that we will briefly review in appendix \\ref{section:sym bndy}.   We will consider the flat solutions for hybrid boundary conditions in the next subsection.\n\n\\subsubsection{Massive sector}\nFocusing now on the massive sector we use the ansatz \\eqref{ansatz global AdS}, in which case the solution is given by \\eqref{solution b in global AdS3}-\\eqref{the cs 2} in terms of functions $F(\\omega,\\pm k,\\pm\\alpha;\\rho)$. We observe that only the $F(\\omega,|k|,\\alpha ;\\rho)$ profiles are regular in the interior ($\\rho \\to 0$). Hence, for $k<0$, we take the $F(\\omega,-k,\\alpha ;\\rho)$ solution. Consequently, we will write the general solution which is smooth in the interior of $AdS_3$ as\n\\begin{align}\\label{regular solution b in global AdS3}\nb_{u} ={}&\nC_{u}F( \\omega, |k|,\\alpha;  \\rho)\n\\\\\nb_{v} ={}&\nC_{v}F(  \\omega, |k|, -\\alpha;  \\rho)\\, ,\n\\end{align}\n\\noindent where $(u,v)$ are the light-cone coordinates defined in \\eqref{nc}, $F( \\omega, k,\\alpha;  \\rho)$ is defined as in \\eqref{F hyp}, and\n\\begin{equation}\\label{relation coeff global AdS}\n   \\frac{C_{v}}{C_{u}} = \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha}\\, .\n\\end{equation}\n\n\\noindent Here, $s$ denotes the sign function, i.e. $s(k) = 1 $ for $k \\geq 0$ and $s(k) = - 1$ for $k < 0$. The $b_{\\rho}$ component is obtained from $b_{u}$ and $b_{v}$ via \\eqref{radial constraint global ads} and it is subleading with respect to them near the conformal boundary of $AdS_{3}$. Expanding $F(\\omega,k,\\alpha ;\\rho)$ near $\\rho = \\infty$ and using \\eqref{regular solution b in global AdS3}-\\eqref{relation coeff global AdS}, we learn that the relevant coefficients in the asymptotic expansion are\n\\begin{align}\\label{explicit coeffs b+}\n    b^{(+)}_u\n     ={}&\n      C_u C(\\alpha, |k|, \\omega)\\,,\n      &    b^{(+)}_v ={}&0 \\, ,\n\\\\\n\\label{explicit coeffs b-}\n    b^{(-)}_v\n    ={}&\n      C_u \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega)\\,,\n     &   b^{(-)}_u ={}&0\\, ,\n\\end{align}\n\\noindent where\n\\begin{equation}\\label{C alpha}\n    C(\\alpha, k, \\omega) = \\frac{\\Gamma(k+1) \\Gamma(1+\\alpha)}{\\Gamma\\left(\\displaystyle{1+\\frac{ k + \\alpha - \\omega}{2}}\\right) \\Gamma\\left(\\displaystyle{1+\\frac{ k + \\alpha + \\omega}{2}}\\right)}\\, .\n\\end{equation}\n\nA choice of asymptotic boundary conditions will constrain the allowed values of $(\\omega,k)$, corresponding to the normal modes of the system. In the Dirichlet case $(\\beta=0)$ the source is identified with $b^{(+)}_{u}$ and the normal modes are given by the zeros of $C(\\alpha, |k|, \\omega)$, located at\n\\begin{equation}\\label{omega D2}\n    \\omega^{\\pm}_{nk} = \\pm \\left(2 n + |k|  +\\alpha \\right)\\, ,\\qquad   n = 1,2,\\ldots\n\\end{equation}\n\\noindent and the zeros of the denominator in \\eqref{relation coeff global AdS}, located at\n\\begin{alignat}{2}\\label{omega D zero modes}\n    \\omega^{+}_{0k}\n    ={}&\n   +(  |k| + \\alpha)    &\\phantom{aaaa}&{\\rm for} \\ k > 0\n     \\\\\n     \\omega^{-}_{0k}\n     ={}&\n      -(|k| + \\alpha)  &\\phantom{aaaa}&{\\rm for} \\ k < 0\\, .\n      \\label{omega D zero modes 2}\n\\end{alignat}\n\\noindent We notice that the normal modes with $n \\geq 1$ are doubly degenerate, with each frequency attained for both $k$ and $-k$, while $\\omega^\\pm_{0k}$ occur only once.\n\n\\noindent Similarly, for Neumann boundary condition $(b^{(-)}_v = 0)$ we find the eigenfrequencies\n\\begin{equation}\\label{omega N2}\n    \\omega^{\\pm}_{nk} = \\pm \\left(2 n + |k|  - \\alpha \\right)\\, ,\\qquad   n = 1,2,\\ldots\n\\end{equation}\n\\noindent in addition to\n\\begin{alignat}{2}\n    \\omega^{-}_{0k}\n    ={}& \\alpha - |k|  &\\phantom{aaaa}&{\\rm for} \\ k > 0\n     \\\\\n     \\label{omega N zero modes k <0}\n     \\omega^{+}_{0k}\n     ={}&\n      |k| - \\alpha &\\phantom{aaaa}&{\\rm for} \\ k < 0\\, .\n\\end{alignat}\n\\noindent More generally, the boundary condition $b^{(+)}_u = \\beta\\, b^{(-)}_v $ for finite $\\beta$ gives\n\\begin{equation}\\label{bc eq for w}\n    C(\\alpha, |k|, \\omega) - \\beta \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega) = 0\\, .\n\\end{equation}\n\\noindent For generic $\\beta$, we will proceed numerically, examining the structure of the solutions of \\eqref{bc eq for w} in the complex-$\\omega$ plane as a function of $k$, $\\beta$ and $\\alpha$. For $\\beta > 0$ and all values of $k$, we find an infinite discrete set of real frequency solutions, in analogy to the Dirichlet and Neumann cases. Now, while for $\\beta < 0$ and $k>0$ all frequencies are real, for $\\beta < 0$ and $k<0$ a pair of complex solutions occurs in addition to the series of real solutions. Notice that, with the exception of $\\omega$, all the parameters in \\eqref{bc eq for w} are real, which implies that complex solutions must appear in complex conjugate pairs. These complex solutions go off to $\\pm i \\infty$ as $\\beta \\rightarrow 0$, in agreement with our analysis for Dirichlet boundary conditions.  See figures \\ref{w mix bc} and \\ref{w mix bc k neg}. The complex frequency solutions signal an instability of the system, since some perturbations can grow exponentially with time. This instability is associated with ghosts, as we will see in section \\ref{ip massive global}. We stress that, aside from the existence of complex frequencies, there is nothing particularly special about $\\beta < 0$. In fact, we will see below that all values of $\\beta \\neq 0$ are qualitatively equivalent, since they all yield ghosts.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{w mix bc}\n\\includegraphics[width=0.4\\linewidth]{wmixbcGkpos.pdf}\n}\\qquad\\qquad\n\\subfigure[][]{\n\\label{w mix bc k neg}\n\\includegraphics[width=0.4\\linewidth]{wmixbcGkneg.pdf}}\n\\caption{\\ref{w mix bc}:  We plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{bc eq for w} in the complex-$\\omega$ plane for $\\{k =2,\\,\\alpha = 0.2,\\,\\beta = 1.7\\}$. We observe that these solutions only intersect for $\\mbox{Im}(\\omega) = 0$, which illustrates the fact  that \\eqref{bc eq for w} has only real solutions for $\\beta > 0$. \\ref{w mix bc k neg}: For $\\{k =-2,\\,\\alpha = 0.2,\\, \\beta = -1.7\\}$, we plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{bc eq for w} in the complex-$\\omega$ plane. We note that in this case there are complex frequency solutions. }\\end{figure}\n\nFinally, we consider the hybrid boundary conditions \\eqref{gen hyb}. As noted in section \\ref{sec:bc}, the condition \\eqref{gen hyb} along with the flatness of $\\delta A^{(0)}$ imply the extra requirement \\eqref{hyb on b}, which in view of our mode decomposition translates into\n\\begin{equation}\\label{hyb bc coeff}\n    \\kappa^2 \\frac{k + \\omega}{k -\\omega} C(\\alpha, |k|, \\omega) - \\frac{k+\\omega- s(k) \\alpha}{k-\\omega+ s(k) \\alpha} C(-\\alpha, |k|, \\omega) = 0\\, .\n\\end{equation}\nThus, the spectrum of frequencies is given by the solutions of \\eqref{hyb bc coeff} provided the flat components of the connection are related to the massive ones by \\eqref{gen hyb}. Lacking an analytic solution of \\eqref{hyb bc coeff} for finite $\\kappa$, we proceed numerically. Studying \\eqref{hyb bc coeff} for various values of the parameters, we find that generically there is an infinite set of real solutions. Additionally, a pair of complex solutions occurs when $k > 0$ and $|\\kappa| > |\\kappa_c|$, where $\\kappa_c$ is an increasing function of $\\alpha$ and $k$. See figures \\ref{tach hyb real}, \\ref{tach hyb cplex} for an illustration of this fact. We have also verified numerically that the complex solutions go off to $\\pm i \\infty$ as $|\\kappa|$ approaches infinity, consistent with the Dirichlet result. As in the case of mixed boundary conditions, the complex frequency solutions correspond to a dynamical instability of the system that is associated to ghosts. We shall also find that the all finite values of $\\kappa$ yield ghosts, in agreement with the CFT unitarity bound.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{tach hyb real}\n\\includegraphics[width=0.4\\linewidth]{tachhybcplex.pdf}\n}\\qquad\\qquad\n\\subfigure[][]{\n\\label{tach hyb cplex}\n\\includegraphics[width=0.4\\linewidth]{tachhybreal.pdf}}\n\\caption{\\ref{tach hyb real}: We plot in red\/blue the solution of the real\/imaginary part of equation \\eqref{hyb bc coeff} in the complex-$\\omega$ plane for $\\{k =1,\\,\\alpha = 0.8,\\,\\kappa = 1.0\\}$. We note that there are real solutions but also a pair of complex solutions near $|\\omega| = 0$.\n\\ref{tach hyb cplex}: Solutions for $\\{k =1,\\,\\alpha = 0.8,\\, \\kappa = 0.9\\}$. We observe that the complex solutions become real, which shows that for $\\alpha = 0.8$ the critical value of $\\kappa$ is near $|\\kappa_c| = 0.95\\,$.}\n\\end{figure}\n\nAs mentioned above, given a solution of \\eqref{hyb bc coeff} the components of the flat connection are uniquely determined by \\eqref{gen hyb}.\nIt is worth mentioning that with these boundary conditions the chiral currents acquire a non-vanishing expectation value. See section \\ref{sec:1pt}.\n\n\\subsection{Poincar\\'e patch of  $AdS_{3}$}\\label{sec: spec PP}\nWe now carry out the study of the spectrum of normalizable excitations for the boundary conditions of interest in the Poincar\\'e patch of $AdS_{3}$. As in the global AdS case, normalizability at the conformal boundary is guaranteed by the restriction $0 < \\alpha < 1$. On the other hand, the treatment of the Poincar\\'e horizon turns out to be more delicate as we will discuss in detail below.\n\\subsubsection{Flat sector}\nLet us first consider the flat sector. As mentioned in section \\ref{subsection:symplectic flux}, when the geometry is the Poincar\\'e patch of $AdS_3$, symplectic flux can generically leak through the Poincar\\'e horizon. In the flat sector, the easiest way to see this is to note that in this sector the theory is actually topological, so there is no difference between the Poincar\\'e horizon and the conformal boundary. From our experience with the latter, we conclude that good boundary conditions in the flat sector correspond to fixing half of the connection on the Poincar\\'e horizon. We will impose the condition\n\\begin{equation}\\label{bc A0 at P hor}\n    \\left. \\delta A^{(0)}_x \\right|_{z = \\infty} = 0\\, .\n\\end{equation}\nAs reviewed in appendix \\ref{section:sym bndy}, when fixing the spatial part of $A^{(0)}_i$, the degrees of freedom that reside at the Poincar\\'e horizon become pure gauge, which allows us to focus on the physics at the boundary. Note however that \\eqref{bc A0 at P hor} can be generalized in the same way as the boundary conditions discussed in section \\ref{sec:bc}. Also, in analogy with the global case, we see that $U(1)$ transformations that set $A_z = 0$ everywhere in the bulk generically do not preserve the boundary condition \\eqref{bc A0 at P hor}, so they are not allowed symmetries of the system.\n\nFrom the above discussion, it is clear that the spectrum of the flat connections in the Poincar\\'e case is analogous to the one in global AdS discussed in section \\ref{sec: spec global adS}. In particular, the frequencies are fixed as \\eqref{freq global} as a consequence of the boundary conditions at the conformal boundary, which are identical to the ones we consider in the Poincar\\'e patch. Note however that in the present case the spatial momentum $k$ is not quantized, so the spectrum of eigenfrequencies is continuous.\n\\subsubsection{Massive sector}\nLet us now focus on the massive sector. In order to solve the equations of motion, we use the mode decomposition $\\delta B_\\mu =  e^{i( k_u u + k_v v )} b_\\mu\\, $; see appendix \\ref{sec: soln PP} for the explicit solutions. We classify the modes according to the value of $m^2:= -k_u k_v = \\omega^2 - k^2$ as: timelike ($m^2 > 0$), lightlike ($m^2 = 0$), and spacelike $(m^2 < 0)$.\n\nFrom \\eqref{gen asympt} it follows that the asymptotic expansion of the solution for the massive mode reads (after noting that near the boundary we have $r = 1\/z$)\n\\begin{equation}\\label{asympt PP}\n    b_\\mu = z^{-\\alpha} b^{(+)}_\\mu + z^{\\alpha} b^{(-)}_\\mu + \\mathcal{O}\\left(z^{1-\\alpha}\\right).\n\\end{equation}\nHere $z$ is the radial variable defined in \\eqref{ds2}. Note that under the isometry \\eqref{dilation PP}, the coefficients in \\eqref{asympt PP} scale as\n\\begin{equation}\\label{scaling}\n    b^{(+)}_\\mu \\rightarrow c^{\\alpha-1} b^{(+)}_\\mu~,  \\qquad b^{(-)}_\\mu \\rightarrow c^{-\\alpha-1} b^{(-)}_\\mu~,\n\\end{equation}\n\\noindent in agreement with our discussion of section \\ref{subsection:conf dims} regarding the conformal dimensions of the dual operators.\n\nHaving said this, let us consider the spectrum of timelike modes, whose radial profile is given by \\eqref{soln tl PP}. Comparing \\eqref{soln tl PP} with \\eqref{asympt PP}, we read-off\n\\begin{align}\\label{}\n    b^{(+)}_{u}\n     ={}&\n      k_u C(\\vec{k}) \\frac{2^{1+\\alpha} m^{-(\\alpha+1)}}{\\Gamma(-\\alpha)}\\,,&\n       b^{(+)}_v ={}&\n        b^{(+)}_z = 0\\, ,\n\\\\\n    b^{(-)}_v\n    ={}&\n     k_v A (\\vec{k}) \\frac{2^{1-\\alpha} m^{\\alpha-1}}{\\Gamma(\\alpha)}\\,,&\n      b^{(-)}_u ={}&\n       b^{(-)}_z = 0\\, .\n\\end{align}\n\\noindent Thus, $C(\\vec{k}) = 0$ corresponds to Dirichlet and $A(\\vec{k})= 0$ to Neumann boundary conditions. We also find that mixed boundary conditions imply\n\\begin{equation}\\label{mix bc}\n    C(\\vec{k}) = \\beta \\frac{k_v}{k_u} \\frac{\\Gamma(-\\alpha)}{4^\\alpha \\Gamma(\\alpha)} m^{2\\alpha} A(\\vec{k})\\,,\n\\end{equation}\n\\noindent while hybrid boundary condition translate into\n\\begin{equation}\\label{hybrid bc}\n    C(\\vec{k}) = \\kappa^{-2} \\frac{\\Gamma(-\\alpha)}{4^\\alpha \\Gamma(\\alpha)} m^{2\\alpha} A(\\vec{k})\\, .\n\\end{equation}\n\nWe stress that the timelike modes above oscillate rapidly near $z= \\infty$. As a result, one can construct wave packets that behave smoothly near the Poincar\\'e horizon. Alternatively, one can work with the modes as they stand and treat their symplectic products in the appropriate distributional sense, and this is the strategy we adopt below. More precisely, in section \\ref{sec ip PP} we find that the timelike modes are in fact (plane wave-)normalizable for all the boundary conditions of interest.\n\nWe now study the existence of spacelike solutions, whose profiles are given by \\eqref{soln sl PP}. Taking $\\mbox{Re}( p) >0$ by convention, we see that unless we set $C(\\vec{k}) = 0$ in \\eqref{soln sl PP}, the solutions blow up exponentially at the horizon ($z=\\infty$) and are thus non-normalizable. Therefore, we set  $C(\\vec{k}) = 0$ which implies that the coefficients of the asymptotic expansion for the spacelike solution can be written as\n\\begin{align}\\label{bplus tach}\n    b^{(+)}_u\n     ={}&\n       A(\\vec{k}) k_u 2^{\\alpha} p^{-\\alpha-1} \\Gamma(1+\\alpha)\\,,& b^{(+)}_v ={}& b^{(+)}_z = 0\\, ,\n    \\\\\n\\label{bminus tach}\n    b^{(-)}_v ={}&\n     A(\\vec{k}) k_v  2^{-\\alpha} p^{\\alpha-1} \\Gamma(1-\\alpha)\\,,& b^{(-)}_u ={}& b^{(-)}_z = 0\\, .\n\\end{align}\n\\noindent Both Dirichlet and Neumann boundary conditions require $A(\\vec{k}) = 0$, so in these cases spacelike solutions do not exist. Mixed boundary conditions $b^{(+)}_u = \\beta b^{(-)}_v$, in turn, imply the relation\n\\begin{equation}\\label{mix bc tach PP}\n    \\tilde{\\beta} = \\frac{(k-\\omega)^{1-\\alpha}}{(k+\\omega)^{1+\\alpha}}\\, ,\n\\end{equation}\n\\noindent where we have defined $\\tilde{\\beta} = 4^{-\\alpha} \\frac{\\Gamma(1-\\alpha)}{\\Gamma(1+\\alpha)} \\beta$. Spacelike solutions are then in one-to-one correspondence with the solutions of \\eqref{mix bc tach PP}, which we now study. First, we observe that regularity at transverse infinity, $|x| \\rightarrow \\infty$, requires $k \\in \\mathbb{R}$. On the other hand, recall that we derived \\eqref{mix bc tach PP} only under the assumption $\\mbox{Re}(p) >0$, so in principle complex frequency solutions are allowed and their existence is exclusively dictated by \\eqref{mix bc tach PP}. Examining \\eqref{mix bc tach PP} it is not hard to conclude that for all $\\beta >0$ there are real solutions in the region $k-\\omega>0$, $k+\\omega >0$; see figure \\ref{w real tach}. On the other hand, if $\\beta < 0$ real solutions are ruled out, but we find instead a pair of complex-frequency solutions that are conjugate to each other, see figure \\ref{w cplex tach}. The fact that our results depend on the sign of $\\beta$ only can be easily understood in terms of the scaling symmetry \\eqref{dilation PP}, which acts non-trivially on $\\beta$.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{w real tach}\n\\includegraphics[width=0.42\\linewidth]{tachrealw.pdf}\n} \\qquad \\qquad\n\\subfigure[][]{\n\\label{w cplex tach}\n\\includegraphics[width=0.4\\linewidth]{complexwtach.pdf}}\n\\caption{\\ref{w real tach}: We plot in red the real solutions of \\eqref{mix bc tach PP} in the $(k,\\omega)$ plane for $\\{\\tilde{\\beta} = 0.5\\,,\\alpha = 0.6\\}$. The dashed line corresponds to the light-cone in momentum space. \\ref{w cplex tach}: For $\\{\\tilde{\\beta} = -0.5,\\,\\alpha = 0.6,\\, k = -1\\}$, we plot in red\/blue the solutions to the real\/imaginary part of \\eqref{mix bc tach PP} in the complex-$\\omega$ plane. Complex solutions are given by the intersection of both lines at $\\omega \\approx 1.45 \\pm i 2.01\\,$. This implies $p \\approx 2.18 - i 1.34$ so $\\mbox{Re}(p) > 0$, consistent with the assumption under which the solution is regular at the Poincar\\'e horizon. }\n\\end{figure}\n\nSimilarly, for spacelike solutions satisfying the ``hybrid'' boundary condition \\eqref{hyb on b} we have\n\\begin{equation}\\label{hyb bc tach PP}\n    \\tilde{\\kappa}^2 = (k^2 - \\omega^2)^\\alpha\\, ,\n\\end{equation}\n\\noindent where we defined $\\tilde{\\kappa}^2 = 4^\\alpha \\frac{\\Gamma(1+\\alpha)}{\\Gamma(1-\\alpha)} \\kappa^2 $. Since $\\tilde{\\kappa}^2 > 0$, it follows that $ \\omega^2 = k^2 - \\tilde{\\kappa}^{2\/\\alpha}\\,$. Now, because $k$ can be arbitrarily small, we find real as well as imaginary frequency solutions for all values of $\\kappa$. In analogy to the mixed boundary conditions studied above, we can use the scaling symmetry \\eqref{dilation PP} to set $\\kappa$ to any desired value. Furthermore, in this case the spectrum is insensitive to the sign of $\\kappa$ due to the structure of the boundary condition \\eqref{hyb on b}.\n\nFinally, we discuss the lightlike modes. For the right-moving modes, i.e. those with $k_{v}=0$, the general solution is given in \\eqref{soln PP kv=0}. Examining the expression for the inner product, we conclude that the norm of the right-moving modes diverges if $b_z \\neq 0$. Therefore, we find that right-moving modes are only allowed for Neumann boundary conditions. In this case, they read\n\\begin{equation}\\label{RM Neumann}\n    \\delta B = A(k_u) z^{-\\alpha} e^{i k_u u} du\\, .\n\\end{equation}\n\n\\noindent We emphasize that the solution \\eqref{RM Neumann} is smooth at the Poincar\\'e horizon. Similarly, the left-moving modes \\eqref{soln PP ku=0} are only normalizable only for Dirichlet boundary conditions, in which case they can be written as\n\\begin{equation}\\label{LM Dirichlet}\n    \\delta B = C(k_v) z^{\\alpha} e^{i k_v v} dv\\,.\n\\end{equation}\n\\noindent Note however that in this case they fail to be smooth at $z = \\infty$, which removes them from the spectrum.\n\n\n\\section{Evaluating the symplectic product}\\label{section:norms}\nNext, we compute the symplectic product of the various solutions found in section \\ref{section:spectrum}. The emphasis will be on determining the existence of ghosts, which, as stated above, correspond to positive (resp. negative) frequency modes having negative (resp. positive) norm. According to CFT considerations regarding unitarity bounds for vector operators, we expect the theories in which $B^{(+)}$ fluctuates to contain ghosts. Up to certain subtleties present in the Poincar\\'e patch, we will verify that the expected ghosts arise in the bulk, consistent with the field theory result. In addition, we will also find ghosts in the flat sector for a certain class of double-trace boundary conditions; the latter are not related to unitarity bounds of the kind mentioned above. The presence of these ghosts should not be at all surprising, however, since the symplectic structure restricted to the flat sector is not manifestly positive definite, see e.g. \\eqref{omega split}. We find it convenient to study first the flat sector separately, assuming that we have chosen boundary conditions which decouple this sector from the massive one. The results for the massive sector and the mixed hybrid case will be presented later in this section.\n\n\\subsection{Flat sector}\\label{ip flat sector}\nWe start by discussing the case of global AdS. The symplectic product is evaluated on a slice of constant $t$, so we have $\\sqrt{h} n_\\mu = \\sqrt{g} \\delta_\\mu^t$. Then, using \\eqref{omega recap} and \\eqref{ip} the symplectic product reads\n\\begin{equation}\\label{ip flat 1}\n    (A_1,A_2) = - i \\hat{\\alpha} \\int d^2x\\, \\varepsilon^{t \\lambda \\nu} \\left(\\delta_1 A^{(0)}_{\\lambda}\\right)^* \\delta_2 A^{(0)}_{\\nu}\\,.\n\\end{equation}\n\\noindent Using the solution \\eqref{soln flat global} and the mode decomposition $\\lambda = e^{-i \\omega t + i k x} \\hat \\lambda$, it is straightforward to arrive at\n\\begin{equation}\\label{ip flat 2}\n    (A_1,A_2) = - 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2} k_1 e^{i t (\\omega_1 - \\omega_2)} \\int_0^\\infty d\\rho \\left(\\hat{\\lambda}_2\\partial_\\rho \\hat{\\lambda}_1^*  -  \\hat{\\lambda}_1^* \\partial_\\rho \\hat{\\lambda}_2\\right).\n\\end{equation}\n\\noindent \nUpon using \\eqref{freq global} in \\eqref{ip flat 2} we see that the time dependence in the symplectic product cancels out, as required by conservation of the symplectic structure. Finally, integrating by parts the first term in \\eqref{ip flat 2} and using the smoothness condition $\\delta A_x^{(0)} = 0 $ at $\\rho = 0\\,$, we conclude\n\\begin{equation}\\label{ip flat 3}\n    (A_1,A_2) = 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2} \\frac{\\omega_1}{\\hat{\\beta}} \\bigl| \\hat{\\lambda}_{1, \\partial}  \\bigr|^2\\, .\n\\end{equation}\n\\noindent where $\\hat{\\lambda}_{\\partial} = \\hat{\\lambda} \\bigr|_ {\\partial M}$ are the (finite) boundary values of the Fourier components of $\\lambda$. Note that \\eqref{ip flat 3} is manifestly finite and conserved, as promised.\nWe observe that the symplectic product is local on the boundary values of $\\lambda$, as expected in a topological theory with a boundary. In other words, flat connections for which $\\lambda$ vanishes on the boundary are pure gauge degrees of freedom. Moreover, for the boundary condition $\\bigl.\\delta A^{(0)}_x \\bigr|_{\\partial M} = 0$, i.e. $k=0$, we also find that the flat sector becomes pure gauge. We refer the reader to appendix \\ref{section:sym bndy} for a discussion on gauge symmetries.\nRecall that ghost excitations are defined as positive(negative) frequency solutions with negative(positive) norm. Thus, with the assumption that $\\hat \\alpha > 0$, we conclude that there are ghosts in the flat sector for $\\hat\\beta < 0$. Although in this case there is no obvious violation of unitarity bounds (recall that $A^{(0)}$ has scaling dimension one), the mere fact that the symplectic product \\eqref{ip flat 1} is not positive definite is an indication that such ghosts might occur.\n\nLet us now focus on the case of Poincar\\'e coordinates. As discussed in section \\ref{sec: spec PP}, with our choice of boundary conditions at the Poincar\\'e horizon, the flat sector largely resembles that of global $AdS_3$. Carrying out a calculation analogous to the one above we find that the symplectic product for flat modes in the Poincar\\'e patch is given by (\\ref{ip flat 3}), with the replacement of the Kronecker-$\\delta$ by a Dirac $\\delta$-function since $k$ is no longer quantized.\n\n\n\\subsection{Massive sector in global AdS}\n\\label{ip massive global}\n\nNext we  evaluate the symplectic products \\eqref{ip} for the positive frequency modes found in section \\ref{sec: spec global adS}. We first focus on the non-flat sector, and at the end of this section we consider the hybrid boundary conditions which introduce a mixing with the flat sector. We choose to evaluate the symplectic product on a surface $\\Sigma$ in which $t = {\\rm const}$, in which case we obtain\n\\begin{equation}\\label{ip global nf1}\n    (A_1, A_2) = i \\hat{\\alpha} \\int dz dx\\, \\varepsilon^{t \\lambda \\nu} \\delta_1 B^*_{ \\lambda} \\delta_2 B_{ \\nu}\\, .\n\\end{equation}\n\\noindent Using the mode decomposition $\\delta B_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} b_\\mu (k) $ in \\eqref{ip global nf1} and computing the integral over $x$, we get\n\\begin{equation}\\label{ip global}\n    (A_1,A_2) = - 2 \\pi i\\hat{\\alpha}\\, \\delta_{k_1, k_2} e^{i\\frac{t}{L} (\\omega_1- \\omega_2)} \\int_0^\\infty \\left( b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\right) d \\rho\\, .\n\\end{equation}\n\\noindent It will prove convenient to express \\eqref{ip global} in terms of $b_u$ and $b_v$. To do so, we recall that $b_x = \\frac{1}{2}(b_u + b_v)$ along with the fact that the first order equation for $b$ yields\n\\begin{equation}\\label{elim brho}\n    b_\\rho = \\frac{i \\alpha \\rho}{ 2 k(1+\\rho^2)} (b_u - b_v) - \\frac{i}{2 k} (b_u + b_v)' .\n\\end{equation}\n\\noindent Therefore, we have\n\\begin{align}\\label{brho bx}\n\\nonumber\n    - i \\int_0^\\infty d \\rho \\left(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\right)\n     ={}&\n      \\frac{\\alpha}{2 k_1}\\Bigl(\\langle b_{1v}, b_{2v} \\rangle - \\langle b_{1u}, b_{2u} \\rangle\\Bigr)\n      + \\frac{1}{4 k_1}\\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\bigg| ^\\infty_0\n       \\\\\n\\nonumber\n    ={}&\n     \\frac{\\alpha \\rho(1+ \\rho^2)}{2 k_1(\\omega_1 - \\omega_2)} \\Bigl[\\bigl(b_{1v} b'_{2v} - b_{2v} b'_{1v}\\bigr) - \\bigl(b_{1u} b'_{2u} - b_{2u} b'_{1u}\\bigr)  \\Bigr] \\bigg| ^\\infty_0\n      \\\\\n    &\n    + \\frac{1}{4 k_1}\\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\bigg| ^\\infty_0\\, .\n\\end{align}\n\\noindent Here, $\\langle \\cdot, \\cdot \\rangle$ is the Sturm-Liouville (SL) product defined in appendix \\ref{SL gen}. It is straightforward to verify that regularity of the modes at the origin guarantees that the contribution to \\eqref{brho bx} from $\\rho = 0$ vanishes, so the solutions found in \\ref{sec: spec global adS} are indeed normalizable, as promised. For generic frequencies $\\omega_1$ and $\\omega_2\\,$, the contribution from $\\rho = \\infty$ is finite and it evaluates to\n\\begin{equation}\\label{IP gen omega}\n    - i \\int_0^\\infty d \\rho \\bigl(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\bigr) = \\frac{b^{(+)}_{2 u} b^{(-)}_{1 v} - b^{(+)}_{1 u} b^{(-)}_{2 v} }{2(\\omega_1 - \\omega_2)}\\, .\n\\end{equation}\n\\noindent It is not hard to see that \\eqref{IP gen omega} vanishes for Dirichlet, Neumann and mixed boundary conditions if $\\omega_1 \\neq \\omega_2$. Using this fact in \\eqref{ip global} we conclude that the inner product is conserved (i.e. independent of $t$) for all of the above boundary conditions, in agreement with our analysis of the symplectic flux. In order to calculate the norms, we take the limit $\\omega_2 = \\omega_1$ in \\eqref{IP gen omega} and set $\\omega_1$ to its quantized value at the end of the calculation. Since \\eqref{IP gen omega} vanishes for $\\omega_1 \\neq \\omega_2$, we can write\n\\begin{equation}\\label{IP gen omega2}\n    - i \\int_0^\\infty d \\rho \\bigl(b^*_{1 \\rho} b_{2 x} - b^*_{1 x} b_{2 \\rho}\\bigr) = \\delta_{\\omega_1, \\omega_2}\n    \\frac{1}{2} \\left(b^{(+)}_{1 u} \\partial_{\\omega_1} b^{(-)}_{1 v} - b^{(-)}_{1 v} \\partial_{\\omega_1} b^{(+)}_{1 u} \\right)\\, .\n\\end{equation}\n\\noindent Plugging \\eqref{IP gen omega2} into \\eqref{ip global} we find the following general expression for the symplectic products:\n\\begin{equation}\\label{ip global formal}\n    (A_1,A_2) = \\pi \\hat{\\alpha} \\delta_{\\vec{k}_1, \\vec{k}_2} \\left(b^{(+)}_{1 u} \\partial_{\\omega_1} b^{(-)}_{1 v} - b^{(-)}_{1 v} \\partial_{\\omega_1} b^{(+)}_{1 u} \\right).\n\\end{equation}\n\\noindent We now specialize to the various boundary conditions of interest. For Dirichlet boundary conditions, the spectrum of eigenfrequencies is given by \\eqref{omega D2}, \\eqref{omega D zero modes}. The positive frequency solutions can be expressed more succinctly as\n\\begin{equation}\\label{omega D positive}\n   \\omega^{+}_{n,k} =  2 \\bigl[n + \\theta(-k)\\bigr] + |k| + \\alpha \\qquad  n = 0,1,2,\\ldots\\,,\n\\end{equation}\n\\noindent where $\\theta(x)$ is the Heaviside function, defined as $\\theta(x) = 1 $ for $x \\geq 0$ and $\\theta(x) = 0$ for $x < 0$. Using \\eqref{omega D positive} in \\eqref{ip global formal} we find\n\\begin{equation}\\label{ip D global}\n    (A_1,A_2) = \\pi \\hat{\\alpha}  \\frac{(-1)^n \\pi  \\alpha  \\csc(\\pi  \\alpha) n!\\, \\Gamma \\bigl(2\\theta(-k)+|k|+n \\bigr) \\Gamma\\bigl(- s(-k) -n-\\alpha\\bigr)}{4 \\Gamma\\bigl(1-\\alpha\\bigr)^2\\, \\Gamma\\bigl(1+|k|+n+\\alpha\\bigr)}\\, ,\n\\end{equation}\n\\noindent where $n$ is a non-negative integer. Here we have normalized the modes in such a way that the leading term has coefficient $1$. We shall continue to do so henceforth, unless explicitly otherwise stated. Similarly, the spectrum of positive frequency solutions for Neumann boundary conditions can be written as\n\\begin{equation}\\label{omega N positive}\n    \\omega_N =  2 \\bigl[n + \\theta(k)\\bigr] + |k| - \\alpha \\qquad  n = 0,1,2,\\ldots\n\\end{equation}\n\\noindent as it follows from \\eqref{omega N2} and \\eqref{omega N zero modes k <0}. Inserting \\eqref{omega N positive} in \\eqref{ip global formal} we conclude that the Neumann norms read\n\\begin{equation}\\label{ip N global}\n    (A_1,A_2) = \\pi \\hat{\\alpha}  \\frac{(-1)^{n+1} \\pi  \\alpha  \\csc(\\pi  \\alpha) n! \\Gamma \\bigl(2\\theta(k)+|k|+n \\bigr) \\Gamma\\bigl(-s(k) -n+\\alpha\\bigr)}{4 \\Gamma\\bigl(1+\\alpha\\bigr)^2\\, \\Gamma\\bigl(1+|k|+n-\\alpha\\bigr)}\\, ,\n\\end{equation}\n\\noindent where $n$ is a non-negative integer. Inspecting \\eqref{ip D global} and \\eqref{ip N global}, we note that all the modes have positive norm with the exception of the Neumann modes characterized by $n=0$, $k<0$, whose frequencies are given by \\eqref{omega N zero modes k <0}. Therefore, we conclude that the theory contains ghosts for Neumann boundary conditions, as expected.\n\nLet us now focus on mixed boundary conditions. In this case, the lack of a closed expression for the frequencies prevents us from displaying the norm explicitly. However, we find substantial evidence that ghosts must be present in the system for generic values of the deformation parameter $\\beta$. For $\\beta < 0$, the existence of ghosts follows immediately from the existence of complex frequency solutions, see for example \\cite{Andrade:2011dg}. The argument is as follows. First we recall that, as all the parameters are real, the complex frequencies always occur in pairs of complex conjugate values;  c.f. figure \\ref{w mix bc k neg}. Second, denoting the aforementioned solutions by $\\psi_1$, $\\psi_2$, we can verify that $(\\psi_1, \\psi_1) = (\\psi_2, \\psi_2) = 0$. A simple way to see this is to note that the norms have the overall time-dependent factor $\\exp(- 2 t\\, \\mbox{Im}(w))$. Since this is in conflict with conservation, the norms must vanish. Third, the definition of the norm guarantees that cross-terms satisfy $(\\psi_1, \\psi_2) = (\\psi_2, \\psi_1)^*$, and we can explicitly verify that they are non-vanishing. Finally, diagonalizing the symplectic structure we find that one of the excitations has negative norm, signaling the presence of ghosts. We turn now to the case $\\beta > 0$. In this situation we did not find evidence of complex frequency solutions, so we need to examine the norms in more detail. Indeed, we found numerical evidence that ghosts should be present for this case as well, c.f. figure \\ref{ghostsmixbc}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.75]{ghostmixbckappapos.pdf}\n\\caption{For $\\{\\alpha = 0.7,\\, k = -2,\\, \\beta = 1.1\\}$, we plot the formal expression for the norm \\eqref{ip global formal} with a red dashed line and the left hand side of \\eqref{bc eq for w}, whose zeroes correspond to the allowed frequencies, with a red solid line. Note that the smallest frequency corresponds to a ghost. Exploring the norm numerically for various values of the parameters, we find that this behavior is generic.}\n\\label{ghostsmixbc}\n\\end{center}\n\\end{figure}\n\nFinally, we consider the hybrid boundary conditions. Recall that in section \\ref{sec: spec global adS} we found that there is a pair of complex frequency solutions if the absolute value of the deformation parameter $\\kappa$ is large enough. Following the conventions in section \\ref{sec: spec global adS}, we denote the critical value by $|\\kappa_c|$. As mentioned above, these complex frequency solutions correspond to ghost\/anti-ghost pairs, so we do not discuss this case any further. Let us now examine the norms of the real frequency solutions. We expect the pair of complex frequency solutions that occur for $|\\kappa| > |\\kappa_c|$ to remain as a ghost\/anti-ghost pair when we tune $|\\kappa|$ below $|\\kappa_c|$. We will exhibit ample numerical evidence that this is indeed the case, and thus conclude that ghosts are present for all values of $\\kappa$.\n\nRecall that the symplectic structure splits into the contributions from the flat and non-flat sectors as in \\eqref{omega recap}, and that the symplectic product is given in terms of the symplectic structure by \\eqref{ip}. Choosing the Cauchy slice $\\Sigma$ on which we evaluate the product to be a surface of constant $t$, we can write the inner product as\n\\begin{equation}\\label{ip hyb bc}\n    (A_1, A_2) = (A_1, A_2)_{nf} + (A_1, A_2)_{f}\\, ,\n\\end{equation}\n\\noindent where\n\\begin{align}\\label{ip nf exp}\n    (A_1, A_2)_{nf}\n    ={}&\n     - i \\hat{\\alpha} \\int_{\\Sigma} d^{2}x\\Bigl(\\delta B^*_{1 \\rho} \\delta B_{2x} - \\delta B^*_{1x} \\delta B_{2 \\rho}\\Bigr)\n\\\\\n\\label{ip f exp}\n    (A_1, A_2)_{f}\n    ={}&\n     i \\hat{\\alpha} \\int_{\\Sigma}d^{2}x \\Bigl(\\delta_1 A^{(0) *}_{ \\rho} \\delta_2 A^{(0)}_{ x} - \\delta_1 A^{(0) *}_{ x} \\delta_2 A^{(0)}_{\\rho}\\Bigr).\n\\end{align}\n\\noindent Next, we introduce Fourier decompositions as $\\delta B_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} b_\\mu (k) $, $\\delta A^{(0)}_\\mu = e^{\\frac{i}{L}(-\\omega t + k x)} a^{(0)}_\\mu$ and proceed as above by computing \\eqref{ip nf exp} and \\eqref{ip f exp} for generic frequencies $\\omega_1$ and $\\omega_2$. The expression \\eqref{ip nf exp} is then given by \\eqref{IP gen omega} and it only remains to compute \\eqref{ip f exp}. We manipulate \\eqref{ip f exp} noting that the flatness condition implies that the modes satisfy\n\\begin{equation}\\label{flat fourier 1}\n    a^{(0)}_\\rho = - \\frac{i}{k}\\left( a^{(0)}_x\\right)'\\, ,\n\\end{equation}\n\\noindent where the prime denotes a radial derivative. Using \\eqref{flat fourier 1} in \\eqref{ip f exp} we thus find\n\\begin{equation}\\label{ip flat fourier}\n    (A_1, A_2)_f = \\left. - 2 \\pi \\hat{\\alpha}\\, \\delta_{k_1, k_2}\\, k^{-1} a^{(0)}_{1 x} a^{(0)}_{2 x} \\right|^{\\rho = \\infty}_{\\rho = 0}\\, .\n\\end{equation}\n\\noindent From the regularity condition $a^{(0)}_x =0$ at $\\rho  = 0$ we conclude that only the term at $\\rho = \\infty$ contributes to \\eqref{ip flat fourier}. Thus, gathering the results \\eqref{IP gen omega} and \\eqref{ip flat fourier} we find for generic frequencies\n\\begin{equation}\\label{IP hyb generic}\n    (A_1, A_2) = 2 \\pi \\hat{\\alpha}\\, e^{it(\\omega_1 - \\omega_2)} \\delta_{k_1, k_2} \\left [ \\frac{b^{(+)}_{2 u} b^{(-)}_{1 v} - b^{(+)}_{1 u} b^{(-)}_{2 v}  }{2(\\omega_1 - \\omega_2)} - \\frac{\\bigl(a^{(0)}_{1 u} + a^{(0)}_{1 v}\\bigr) \\bigl(a^{(0)}_{2 u} +  a^{(0)}_{2 v}\\bigr) }{4 k} \\right].\n\\end{equation}\n\\noindent We can readily verify that the symplectic structure is conserved by noting that \\eqref{IP hyb generic} vanishes for $\\omega_1 \\neq \\omega_2$ when the boundary conditions \\eqref{gen hyb}, \\eqref{hyb on b}, hold. Therefore, the symplectic product can be written in terms of the coefficients of the asymptotic expansion as\n\\begin{equation}\\label{IP hyb final}\n    (A_1, A_2) = \\pi \\hat{\\alpha}\\, \\delta_{\\vec{k}_1, \\vec{k}_2} \\biggl[ b^{(+)}_u \\partial_{\\omega_1} b^{(-)}_{v} - \\kappa^2 \\left( \\frac{k+\\omega}{k-\\omega} b^{(+)}_u \\partial_{\\omega_1} b^{(+)}_u  + \\frac{2 k}{(k-\\omega)^2} (b^{(+)}_u)^2 \\right) \\biggr],\n\\end{equation}\n\\noindent where $\\omega$ is implicitly given by the solutions of \\eqref{hyb bc coeff}. Studying \\eqref{IP hyb final} numerically, we find that there is always a ghost among the lowest real frequency modes that occur for $|\\kappa| < |\\kappa_c|$; see figures \\ref{ghost hyb bc 1}, \\ref{ghost hyb bc 2}. Furthermore, we find that this picture continues to hold true for all $|\\kappa|$ in the range $0 \\leq |\\kappa| < |\\kappa_c|$. This must indeed be the case since $\\kappa = 0$ corresponds to Neumann boundary conditions, which were found above to induce violations of unitarity in the bulk.\n\n\\begin{figure}[htb]\n\\center\n\\subfigure[][]{\n\\label{ghost hyb bc 1}\n\\includegraphics[width=0.42\\linewidth]{ghosthybbc1.pdf}\n} \\qquad \\qquad\n\\subfigure[][]{\n\\label{ghost hyb bc 2}\n\\includegraphics[width=0.42\\linewidth]{ghosthybbc2.pdf}}\n\\caption{\\ref{ghost hyb bc 1}: For $\\{\\alpha = 0.8,\\,  k = 1,\\, \\kappa = 0.9\\}$ we plot the left hand side of \\eqref{hyb bc coeff} (solid line), whose zeros correspond to the allowed frequencies, and the expression for the norm \\eqref{IP hyb final} (dashed line). We notice that the second to lowest frequency solution is a ghost. By slightly increasing the value of $\\kappa$ the solutions move to the complex plane, as seen in figures \\ref{tach hyb real} and \\ref{tach hyb cplex}. \\ref{ghost hyb bc 2}: For $\\{\\alpha = 0.8$,  $k = 1$, $\\kappa = 0.1\\}$, we plot the left hand side of \\eqref{hyb bc coeff} (solid line) and the expression for the norm \\eqref{IP hyb final} (dashed line). We observe that the lowest frequency mode found for higher values of $|\\kappa|$ disappears, but there is still a ghost in the system. }\n\\end{figure}\n\n\\subsection{Massive sector in Poincar\\'e AdS}\\label{sec ip PP}\nWe now proceed to compute the symplectic product for the Poincar\\'e AdS modes, focusing on the non-flat piece of the connection. We start with the timelike modes. It should be noted that, since the spectrum is continuous, the products should be understood in the sense of distributions. As usual, in order to evaluate \\eqref{ip} we choose $\\Sigma$ to be a surface of $t = {\\rm const}$, so we have\n\\begin{equation}\\label{ip nf1}\n    (A_1, A_2) = i \\hat{\\alpha} \\int dz dx\\, \\varepsilon^{t \\lambda \\nu} \\delta_1 B^*_{ \\lambda} \\delta_2 B_{ \\nu}\\,.\n\\end{equation}\nWe find it convenient to use the mode decomposition $    \\delta B_\\mu = e^{i(-\\omega t + k x)} b_\\mu (k)$ in \\eqref{ip nf1} and computing the integral over $x$ we get\n\\begin{equation}\\label{ip nf2}\n    (A_1, A_2) = 2 \\pi i \\hat{\\alpha}\\, \\delta(k_{1} - k_{2}) e^{i t (\\omega_{1} - \\omega_{2})} \\int_0^\\infty dz \\bigl(b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z} \\bigr),\n\\end{equation}\n\\noindent where we have used $\\varepsilon^{ztx} = - 1$, in consistency with the convention $\\varepsilon^{zuv} = - 1$ employed in appendix \\ref{sec: soln PP}.\nIt is convenient to express \\eqref{ip nf2} in terms of the components $b_u$ and $b_v$. To this end we note that the first order equation for $b$ implies\n\\begin{equation}\\label{bz in bu bv}\n    b_z = -\\frac{i \\alpha}{2 k z} (b_u - b_v) - \\frac{i}{2 k} \\left(b_u + b_v\\right)'\\, ,\n\\end{equation}\n\\noindent and we recall $ b_x = \\frac{1}{2}(b_u + b_v)$, $b_t = \\frac{1}{2}(b_u - b_v)$. It follows that we can write\n\\begin{equation}\\label{rad int PP}\n    i \\int_0^\\infty dz (b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z} ) = \\frac{\\alpha}{2 k} \\int dz\\, z^{-1}\\bigl(b^*_{1 v} b_{2 v} - b^*_{1 u} b_{2 u}\\bigr) - \\frac{1}{4 k} \\Bigl[ \\bigl(b_{1 u} + b_{1 v}\\bigr)^*\\bigl(b_{2 u} + b_{2 v}\\bigr) \\Bigr] \\Big|_{0}^\\infty\\, .\n\\end{equation}\n\\noindent The first two terms in \\eqref{rad int PP} correspond to the SL inner product associated to \\eqref{decoupled v} and \\eqref{decoupled u}, respectively. Thus, from the results of appendix \\ref{SL gen} it follows that\n\\begin{align}\\label{rad int PP Wrons}\n\\nonumber\n    i \\int_0^\\infty dz \\bigl(b^*_{1 z} b_{2 x} - b^*_{1 x} b_{2 z}\\bigr)\n     ={}&\n      \\frac{\\alpha z^{-1}}{2 k (m_1^2 - m_2^2)} \\Bigl[ \\bigl(b_{1 v} b'_{2 v}- b_{2 v} b'_{1 v}\\bigr)  - \\bigl( b_{1 u} b'_{2 u} - b_{2 u} b'_{1 u} \\bigr) \\Bigr] \\Big|_{0}^\\infty\n       \\\\\n     &\n     - \\frac{1}{4 k} \\Bigl[ \\bigl(b_{1 u} + b_{1 v} \\bigr)^*\\bigl(b_{2 u} + b_{2 v}\\Bigr)] \\Big|_{0}^\\infty\\, ,\n\\end{align}\n\\noindent where we have used the explicit form of the SL coefficients \\eqref{SL bu} and \\eqref{SL bv}. Next, using the near-boundary expansion one can readily verify that the contribution from $z = 0$ to \\eqref{rad int PP Wrons} vanishes for Dirichlet, Neumann and mixed boundary conditions. The contribution at the Poincar\\'e horizon can be evaluated by introducing a regulator $z_\\infty$ at large $z$ and using\n\\begin{equation}\\label{J large x}\n    J_\\nu(x) \\to \\sqrt{\\frac{2}{\\pi x}} \\cos \\left( x - \\frac{\\nu  \\pi }{2} - \\frac{\\pi }{4} \\right) \\quad\\mbox{for }\\,\\, x \\gg 1 \\, .\n\\end{equation}\n\\noindent From this point on the calculation proceeds in close analogy to that for a scalar field in Poincar\\'e AdS. We refer the reader to \\cite{Andrade:2011dg} for details.\\footnote{The present calculation exhibits one additional complication: roughly speaking, the third term in \\eqref{rad int PP Wrons} has the structure $\\sim (m_1 m_2)^{-1\/2} z_\\infty \\cos[(m_1 - m_2) z_\\infty]$, so it is indeed power-counting divergent as $z_\\infty \\rightarrow \\infty\\,$. However, integrating this against test functions $f(m_1)$ and $f(m_2)$ of compact support, one can show that the contribution from this type of terms vanishes as we remove the regulator.} Up to terms that vanish in the distributional sense, we find that the general expression for the inner product \\eqref{ip nf2} reads\n\\begin{equation}\\label{ip PP gen}\n    (A_1,A_2) = 2 \\pi \\hat{\\alpha}\\, \\delta^{(2)}(k_1^i - k_2^i) {\\cal Q}(\\alpha, k)\\, ,\n\\end{equation}\n\\noindent where\n\\begin{equation}\\label{Q PP gen}\n   {\\cal Q}(\\alpha, k_i) =  2 \\alpha \\left|A(\\vec{k}) + e^{i \\pi \\alpha} C(\\vec{k}) \\right|^2\\, .\n\\end{equation}\n\\noindent Here we have used that $C$ and $A$ satisfy the relation \\eqref{mix bc}. Clearly, the norm \\eqref{ip PP gen} is manifestly positive definite for Dirichlet, Neumann and mixed boundary conditions.\n\nLet us now calculate the products of the spacelike excitations that are present for mixed boundary conditions. As discussed in section \\ref{sec: spec PP}, their radial profile is given by \\eqref{soln sl PP} with $C(\\vec{k}) = 0$, which ensures normalizability since they vanish exponentially at the horizon. Furthermore, the mixed boundary condition holds provided the frequencies satisfy \\eqref{mix bc tach PP}. Recall also that for all $\\beta < 0$ the spectrum contains a pair of solutions $\\psi_1$, $\\psi_2$ whose frequencies are complex conjugate to each other. As argued above, there is always a ghost among these degrees of freedom, so we do not consider this case any further. For $\\beta >0$ we found real frequency spacelike solutions, whose norm we now compute.\n\nSince the real frequency spacelike solutions form a discrete set, we compute their norms in analogy to the calculation of the inner product in global coordinates. Our starting point is the general expression \\eqref{rad int PP Wrons}. Because the radial profiles decay exponentially at the horizon, the only non-vanishing contribution to \\eqref{rad int PP Wrons} comes from the boundary asymptotics. A simple computation reveals that the norm of the real frequency spacelike solutions can be written in terms of the coefficients of the asymptotic expansion as\n\\begin{equation}\\label{norm tach1}\n    (A_1, A_2)_{T} = \\pi \\hat{\\alpha}\\, \\delta(k_1 - k_2) \\Bigl(b^{(+)}_u \\partial_\\omega b^{(-)}_v - b^{(-)}_v \\partial_\\omega b^{(+)}_u\\Bigr)\\, ,\n\\end{equation}\n\\noindent where $\\omega$ satisfies \\eqref{mix bc tach PP}. Plugging in \\eqref{norm tach1} the explicit expressions for the coefficients $b^{(\\pm)}$ found previously in \\eqref{bplus tach} and \\eqref{bminus tach}, we find that the norm of the spacelike solution is\n\\begin{equation}\\label{norm tach2}\n    (A_1, A_2)_{T} =  \\pi \\hat{\\alpha}\\, \\delta(k_1 - k_2)  |A|^2 \\frac{\\pi \\alpha (k- \\alpha \\omega) \\csc (\\pi  \\alpha)}{p^2}\\, .\n\\end{equation}\n\\noindent Note that the sign of the norm \\eqref{norm tach2} is controlled by the factor $(k - \\alpha \\omega)$. Now, it follows from \\eqref{mix bc tach PP} that positive norm solutions occur for positive and negative frequencies, see also figure \\ref{w real tach}. Thus, we conclude that there are ghosts in the theory for mixed boundary conditions and $\\beta > 0$.\n\nIt only remains to discuss hybrid boundary conditions. In this case, the spectrum consists of both real and imaginary frequencies, regardless of the value of the deformation parameter $\\kappa$. As argued above, the existence of non-real frequencies is associated with ghosts on general grounds. Therefore, no detailed calculation of the norms is required to show that this class of theories violate unitarity in the bulk.\n\nAs pointed out in \\cite{Andrade:2011dg}, bulk theories dual to CFT's in which the unitarity bound is violated do not necessarily contain ghosts when the geometry is that of Poincar\\'e AdS. Alternatively, the two-point function suffers from a divergence near the light-cone, which implies that the theory does not exist. This motivates us to inspect the near light-cone structure of the Neumann correlators.\n\nThe boundary (Wightman) two-point function can be easily computed given the matrix of symplectic products, see e.g. \\cite{Andrade:2011dg}. For the timelike modes in the Neumann theory we find\n\\begin{equation}\\label{2pt N}\n    \\langle 0| b^{(+)}_u (-k_i) b^{(+)}_u (k_i) |0 \\rangle = (A_1,A_2)^{-1} |_{Neumann} = \\frac{4^\\alpha q^2 L}{\\pi \\alpha^2 \\Gamma(-\\alpha)^2}  \\frac{(\\omega-k)^{1-\\alpha}}{(\\omega+k)^{1+\\alpha}}\\, .\n\\end{equation}\n\\noindent In order to obtain \\eqref{2pt N} we have normalized the radial profiles such that the leading term is $1$. As expected, the Fourier transform does not converge due to the behavior near $\\omega = -k$; this behavior is to be contrasted with the Dirichlet case, in which we find\n\\begin{equation}\\label{2pt D}\n    \\langle 0| b^{(-)}_v (-k_i) b^{(-)}_v (k_i) |0 \\rangle = (A_1,A_2)^{-1} |_{Dirichlet} =   \\frac{4^{-\\alpha} q^2 L}{\\pi \\alpha^2 \\Gamma(\\alpha)^2}\\frac{(\\omega+k)^{1+\\alpha}}{(\\omega-k)^{1-\\alpha}}\\, .\n\\end{equation}\n\\noindent This is clearly finite as we approach $\\omega = - k$. In the parameter range of interest, namely $0 < \\alpha < 1$, the divergence near $\\omega = k$ is mild enough so that the Fourier transform of \\eqref{2pt D} converges.\n\nThe parallel with \\cite{Andrade:2011dg} extends beyond the existence of the light-cone divergence discussed above, in that this divergence can be related to the appearance of lightlike gauge modes. In fact, recall that in section \\ref{sec: spec PP} we found that the Neumann theory admits the lightlike solution \\eqref{RM Neumann}, which in position space can be written as\n\\begin{equation}\\label{RM Neumann gen}\n    \\delta B = f(u) z^{-\\alpha}  du\\, ,\n\\end{equation}\n\\noindent where $f$ is an arbitrary function of $u$. From \\eqref{ip nf2}, it is clear that the aforementioned solution has zero norm since its $z$-component vanishes. Moreover, it is straightforward to verify that the inner product of \\eqref{RM Neumann gen} with the timelike modes vanishes in the distributional sense. Thus, assuming that the spectrum of the Neumann theory we found in \\ref{sec: spec PP} is complete,\\footnote{In principle, there could be solutions with anharmonic time dependence, which lie outside of the class of modes we consider here. Although we have not studied this possibility in detail, the present setup is self-consistent in that it provides the correct physical results, namely that the Neumann theory is sick.} i.e. that there are only timelike and lightlike modes, we conclude that the lightlike solution \\eqref{RM Neumann gen} is a null direction of the symplectic structure and is thus pure gauge. The reader might be somewhat puzzled by the fact that there is a gauge mode which is not flat. However, one can argue that this must be the case by noting the large arbitrariness in \\eqref{RM Neumann gen} parametrized by the function $f(u)$, which is unconstrained by the equations of motion.\n\n\n\\section{Discussion}\\label{section:discussion}\nBy studying the bulk symplectic structure, we have obtained a class of admissible boundary conditions for the MCS system in asymptotically-AdS spaces. According to the holographic dictionary, these boundary conditions determine the operator content in the possible dual theories. In agreement with the existing literature, we find that there is a vector operator of conformal dimension $1 \\pm \\alpha\\,$, in addition to the well-known chiral currents which are also present in the pure Chern-Simons theory. The vector operator is associated with the Hodge dual of the bulk field strength, which behaves as a massive vector with a mass proportional to the Chern-Simons coupling. It is worth mentioning that the components of these operators satisfy a constraint, so they have less degrees of freedom than naively expected. This feature is reminiscent of the situation in topologically massive gravity, where similar constraints exist \\cite{Skenderis:2009nt}. The chiral currents, on the other hand, are associated to the flat piece of the connection, and are in that sense topological.\n\nOur analysis reveals that, whereas it is possible to impose boundary conditions such that the topological and massive sectors decouple, it is also in principle valid to introduce a mixing between them. In particular, we studied a class of boundary conditions that corresponds to double-trace deformations that couple the chiral currents with the vector operators. Regarding the boundary conditions in which both sectors decouple, we have also considered boundary conditions that yield double-trace deformations within each sector. In this case, it is even possible to generalize these to incorporate multi-trace deformations in the usual way. Our main result is that this apparently large freedom in the choice of boundary conditions is severely restricted once we impose unitarity as an extra requirement.\n\nWe have addressed the issue of unitarity by studying the MCS theory both in Poincar\\'e and global AdS. In these setups, the violations of unitarity generically manifest themselves as ghost excitations in the spectrum of the theories defined with given boundary conditions. The boundary conditions that pass the test of unitarity correspond to fixing the leading behavior of the massive sector (Dirichlet boundary conditions), while separately specifying a linear relation between the two components of the flat connection along the boundary directions. It is worth mentioning that the latter also requires a specific choice of sign in the proportionality constant. Furthermore, we mention that for boundary conditions that fix the spatial part of the boundary connection, the topological degrees of freedom become pure gauge (in the absence of holonomies).\n\nFor the boundary conditions corresponding to double-trace deformations which involve the massive sector, we contented ourselves with numerical results and the reader may wonder whether our analysis was exhaustive enough to rule out the existence of a non-trivial phase diagram. In particular, since the Dirichlet theory is well defined and we can in principle approach it by continuously tuning the deformation parameters, it is reasonable to ask whether there is an open set of unitarity-preserving values near the Dirichlet point. The answer to this question is negative, as it is most easily seen when the geometry is taken to be the Poincar\\'e patch of $AdS_{3}$. In this case, the presence of a scaling symmetry dictates that, up to sign changes, all non-zero values of the coupling constants are equivalent. One can use this fact to draw conclusions regarding the mixed boundary condition $\\delta B^{(+)}_u = \\beta\\, \\delta B^{(-)}_v$. For the reason we just mentioned, it suffices to study the cases $\\beta = 0,\\, \\infty,\\, \\pm 1$, where $\\beta = 0 $ corresponds to Dirichlet boundary conditions. Then, finding ghosts for $\\beta = \\pm 1$ implies that these remain for all non-zero $\\beta$, forbidding a non-zero critical value. Moreover, noting that the Poincar\\'e patch theory captures the high-momentum dynamics of the theory in global AdS, \\footnote{This is most easily seen by noting that, for short characteristic lengths, a cylinder is equivalent to a plane.} one can extend this result to the global case. Clearly, the analogous statement holds true for our hybrid boundary conditions parametrized by the constant $\\kappa$.\n\nIn many scenarios, the presence of the ghosts is in one-to-one correspondence with violations of the unitarity bound in the dual theory, which establishes that the scaling dimension of vector operators must be greater than one. In fact, the operator of dimension $1-\\alpha$ violates the bound for all $\\alpha$ and, accordingly, we find ghosts whenever the corresponding slow-decaying branch fluctuates. The only exception are the conformally invariant Neumann boundary conditions in the Poincar\\'e patch, which set to zero the faster fall-off. In analogy with the scalar case discussed in \\cite{Andrade:2011dg}, we have found in this case that the spectrum is ghost-free, and that the expected pathologies arise instead in the 2-point function, which is ill-defined even at large distances. Interestingly, we also found ghosts in the flat sector, which occur for some choices of the parameter that controls the double-trace deformation. Since the chiral currents have dimension one, these unitarity violations cannot be linked to the bound on the scaling dimension. These pathologies are indeed to be expected, however, because the expression for the symplectic product restricted to the flat sector is not positive definite in any obvious way.\n\nIt is worth commenting in more detail on the mixed boundary conditions in relation to the unitarity bound. Above, we obtained these as a deformation of the Neumann theory, as it is customary for bulk scalars whose mass lies in the Breitenlohner-Freedman window. Had the Neumann theory been well defined, the inclusion of the relevant double-trace operator could have been thought of as triggering an RG flow towards the Dirichlet theory. However, as we have seen, the Neumann theory is sick, and inclusion of the double-trace operator does not cure its pathologies. Thus, the aforementioned flow is not well defined. Attempting to remedy this, one might try to understand the mixed boundary conditions as triggering a flow from the Dirichlet theory. In this case, unfortunately, the deformation term one needs to add is of the form $\\sim \\Bigl(B^{(-)}_v\\Bigr)^2$, so it corresponds to an irrelevant operator of dimension $2(1+\\alpha)$. It follows that the resulting theory is non-renormalizable and the ghosts that arise can be understood as being the result of our loss of control of the theory in the UV.\n\nIt is interesting to contrast our results with those of \\cite{DHoker:2010hr}, in which the authors found an effective three-dimensional MCS theory in the context of holographic RG flows. More precisely, they constructed five-dimensional solutions in the Einstein-Maxwell-Chern-Simons theory which have the interpretation of magnetic branes. Perturbations around these backgrounds turn out to describe RG flows from four-dimensional field theories in the UV to two-dimensional ones in the IR, and the dynamics of the latter are captured by a 3$d$ MCS theory. Requiring the usual Dirichlet boundary conditions in the UV and imposing matching conditions in the bulk, the effective IR theory contains a double-trace for the vector operators which we denoted by $B^{(\\pm)}$. Our analysis reveals that this theory must contain ghosts, and indeed, the results of \\cite{DHoker:2010hr} indicate that violations of unitarity are present if one tries to extend the domain of validity of the IR description to the entire bulk. Then, what saves the theory is the existence of an effective cut-off associated to the domain wall solution, whose presence implies that the IR description breaks down at some intermediate value of the radial coordinate. This is to be expected since Dirichlet boundary conditions were imposed in the UV, and these respect the dual unitarity bounds. The issue of removing the bulk ghosts by introducing the appropriate cut-offs will be discussed in an upcoming publication\n\\cite{T.Andrade}.\n\nWe now briefly comment on the implications of our results in the context of potential condensed matter applications. For illustrative purposes, we first review the relevant results of the pure Maxwell theory and then move on to describe how the addition of the Chern-Simons term changes the picture. In terms of the radial variable of \\eqref{Global AdS3}, the asymptotics of the gauge field in the pure Maxwell theory are of the form\n\\begin{equation}\\label{maxwell asympt}\n    A_i = \\log r A^{(1)}_i + A^{(0)}_i + \\ldots \\qquad {\\rm with} \\quad \\nabla^{(0)}_i A^{(1) i} = 0\\, ,\n\\end{equation}\n\\noindent where $i$ is a boundary index and $\\nabla^{(0)}$ is the covariant derivative associated with the conformal boundary metric. The conservation equation satisfied by the coefficient $A^{(1)}_i$ indicates that it should be interpreted as the $U(1)$ current. This fact was overlooked in\n\\cite{Ren:2010ha,Nurmagambetov:2011yt,Liu:2011fy,Lashkari:2010ak}, in which the authors discussed the construction of a holographic $1+1$ dimensional superfluid\/superconductor incorrectly interpreting $A^{(0)}$ as the boundary current. We mention that this confusion was resolved in \\cite{Jensen:2010em} using the conservation argument given above. However, there is still an obstruction to the study of such holographic theory, since the boundary conditions that allow for a fluctuating current yield ghosts \\cite{Andrade:2011dg}. Thus, the applicability of the by now standard procedure \\cite{Hartnoll:2008vx,Hartnoll:2008kx} to the study of holographic $1+1$ superconductors remains, at least, unclear. Given this, it is compelling to ask ourselves what are the implications of adding the Chern-Simons term to the Maxwell theory and the possible AdS\/CMT applications of the resulting setup.\\footnote{This possibility was suggested in \\cite{Lashkari:2010ak}, with a different motivation.} As we have seen, the inclusion of the Chern-Simons term drastically modifies the scenario, as the $U(1)$ vector current is replaced by the topological chiral currents associated to the flat connections, and one can imagine introducing an order parameter (dual to a minimally coupled bulk charged scalar, say) which could potentially break the associated symmetry spontaneously.\\footnote{We thank Per Kraus for pointing out this possibility.} We leave the exploration of this line of research for future work.\n\n\\vskip 1cm\n\\centerline{\\bf Acknowledgments}\n\nWe are grateful to Geoffrey Comp\\`ere, Eduardo Fradkin, Gary Horowitz, Per Kraus, Mukund Rangamani, Simon Ross and Jorge Santos for helpful conversations, and specially to Don Marolf for many useful discussions on these and related topics. T.A. was partly supported by a Fulbright-CONICYT fellowship, by the US National Science Foundation under grant PHY08-55415 and by funds from the University of California. T.A. is also pleased to thank the Department of Physics of the University of Illinois at Urbana-Champaign and the Department of Mathematics of the University of California, Davis, for their hospitality during the completion of this work. RGL is partially supported by the US Department of Energy under contract FG02-91-ER40709. The work of J.I.J. is supported by the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe main theorem of the Bass-Serre theory of groups acting on\ntrees states that a group $G$ acting on a tree $T$ is the\nfundamental group of a graph of groups whose vertex and edge\ngroups are the stabilizers of certain vertices and edges of $T$.\nThis tells that $G$ can be obtained by successively forming\namalgamated free products and ${\\rm HNN}$-extensions.\nThe pro-$p$ version of this theorem does not hold in general\n({\\it cf.} Example \\ref{ex-4.3}),\nnamely a pro-$p$ group acting on a pro-$p$ tree does not have\nto be isomorphic to the fundamental pro-$p$ group of\na graph of finite $p$-groups (coming from the stabilizers).\nMoreover, the fundamental pro-$p$ group of a profinite graph\nof pro-$p$ groups does not have to split as\nan amalgamated free pro-$p$ product or as\na pro-$p$ ${\\rm HNN}$-extension over some edge stabilizer\n(the reason is that by deleting an edge\nof the profinite graph one may destroy its compactness).\nThese two facts are usually the main obstacles for proving\nsubgroup theorems of free constructions in the category of\npro-$p$ groups.\n\nWe show that the two Bass-Serre theory principal results\nmentioned above hold for finitely generated infinite\npro-$p$ groups acting {\\it virtually freely} on pro-$p$ trees, \n{\\it i.e.} such that the restriction of the action \non some open subgroup is free.\nSuch a group is then virtually free pro-$p$. \n\n\\begin{thm}\\label{t-treeacting_intro}\nLet $G$ be a finitely generated infinite pro-$p$ group acting\nvirtually freely on a pro-$p$ tree $T$.\nThen\n\n\\begin{enumerate}\n  \\item[\\rm (a)] $G$ splits either as\n  an amalgamated free pro-$p$ product or as\n  a pro-$p$ ${\\rm HNN}$-extension over some edge stabilizer;\n\n \\item[\\rm (b)] $G$ is isomorphic to\n the fundamental pro-$p$ group $\\Pi_1(\\cG,\\Gamma)$ of\n a finite graph of finite $p$-groups.\n\\end{enumerate}\n\\end{thm}\n\n\nOne should say that in contrast to the classical theorem from\nBass-Serre theory our $\\Gamma$ in item (b) is not $T\/G$.\nThe graph $\\Gamma$ is constructed in a special way by first modifying $T$\nwithout loosing the essential information of the action.\n\nAs a corollary we deduce the following subgroup theorem.\n\n\\begin{thm}\\label{t-subgrouptheorem_intro}\nLet $H$ be a finitely generated subgroup of a\nfundamental pro-$p$ group $G$ of a finite graph of\nfinite $p$-groups.\nThen $H$ is the fundamental pro-$p$ group of a finite graph of\nfinite $p$-groups which are intersections of $H$\nwith some conjugates of vertex and edge groups of $G$.\n\\end{thm}\n\n\nMoreover, as an application of Theorem \\ref{t-treeacting_intro},\nwe obtain the following result.\nIt is a pro-$p$ analogue of a classical result of\nG. Baumslag \\cite[Thm. 2]{Baumslag:62}\nthat gave an impulse to the theory known now as the theory of limit groups. Note that our theorem also generalizes \nthe pro-$p$ {\\it ipsis litteris} version of\n\\cite{BBaumslag:68}, as well as \\cite[Thm. 7.3]{KZ:11}.\n\n\\begin{thm} \\label{t:freeorabelian_intro}\n  Let $G=A\\amalg_{C} B$ be a free pro-$p$ product of\n  $A$ and $B$ with procyclic amalgamating subgroup $C$.\n  Suppose that\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n  If each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group\n  then so is each $2$-generated pro-$p$ subgroup of $G$.\n\\end{thm}\n\nThe method of proof is to consider the standard pro-$p$ tree $T$\non which $G$ acts naturally; so $A$ and $B$ are stabilizers of\nvertices $v$ and $w$, and $C$ is the stabilizer of the edge\nconnecting $v$ and $w$.\nThen we decompose the pair $(G, T)$ as a\ninverse limit of $(G_U,T_U)$ satisfying the hypothesis of\nTheorem \\ref{t-treeacting_intro}.\n\n\\\n\n\\noindent {\\bf Notation.}\nThroughout this paper, $p$ is a fixed but arbitrary prime number.\nThe additive group of the ring of $p$-adic integers is ${\\mathbb Z}_p$;\nthe natural numbers, ${\\mathbb N}$.\nFor $x$, $y$ in a group we shall write $y^x \\!:= x\\inv y x$.\nAll groups are pro-$p$, subgroups are closed and\nhomomorphisms are continuous.\nFor $A\\subseteq G$ we denote by $\\gp A$ the subgroup of $G$\n(topologically) generated by $A$\nand by $A^G$ the normal closure of $A$ in $G$, {\\it i.e.},\nthe smallest closed normal subgroup of $G$ containing $A$.\nBy $d(G)$ we denote the smallest cardinality of a\ngenerating subset of $G$.\nRecall that a cyclic profinite group is always finite.\nThe Frattini subgroup of $G$ will be denoted by $\\Phi(G)$.\nBy $\\tor G$ we mean the set of all torsion elements of $G$.\n\nFor a pro-$p$ group $G$ acting continuously\non a space $X$ we denote the set of fixed points of $G$ by $X^G$\nand for each $x\\in X$ the {\\em point stabilizer} by $G_x$.\nWe define $\\widetilde{G}=\\gp{ G_x\\,|\\, x\\in X}$.\n\nThe rest of our notation is very standard and basically follows\n\\cite{RZ:00a}\nand\n\\cite{RZ:00b}.\n\n\n\n\\section{Preliminary Results}\n\nIn this section we collect properties of amalgamated free pro-$p$\nproducts, pro-$p$ {\\rm HNN}-extensions and pro-$p$ groups acting\non pro-$p$ trees to be used in the paper.\nFurther information on this subject can be found in\n\\cite{RZ:00a} and \\cite{RZ:00b}.\nRecall the following two notions.\nFirst, an amalgamated free pro-$p$ product\n$G\\!:=A\\amalg_CB$ is {\\em non-fictitious}\nif $C$ is a proper subgroup of both, $A$ and $B$.\nUnless differently stated we shall consider exclusively\nnon-fictitious free amalgamated products\nand we shall make use of the fact from\n\\cite{Ribes:71} that a free pro-$p$ product\nwith either procyclic or finite amalgamating subgroup\nis always {\\em proper}, {\\it i.e.},\nthe factors $A$ and $B$ embed in $G$ via the natural maps.\n\nSecond,\na pro-$p$ {\\rm HNN}-extension $G={\\rm HNN}(H,A,f,t)$\nis {\\em proper} if the natural map from $H$ to $G$ is injective.\nOnly such free pro-$p$ products and pro-$p$ {\\rm HNN}-extensions\nwill be used in this paper and they are therefore always proper.\n\nWe start with a simple general lemma.\n\n\\begin{lemma}\\label{l:invsys}\nLet $G\\!:=\\varprojlim G_i$ be the inverse limit of\nan inverse system $\\{G_i,\\varphi_{ij},I\\}$ of\npro-$p$ groups and $H_i\\le G_i$ so that $\\varphi_{ij}(H_i)\\le H_j$ holds whenever $j\\le i$.\nSuppose that there is a constant $d$ with $d(G_i)=d$ for all $i\\in I$.\nThe following statements hold:\n\\begin{itemize}\n\n\\item[{\\rm (a)}]\n If $d(G)=d$,\n then there exists $j\\in I$ such that\n the projection $G\\to G_j$ is surjective.\n\\item[{\\rm (b)}]\nFor the induced inverse limit $H\\!:=\\varprojlim H_i\\le G$,\nwe have equality $H^G=\\varprojlim H_i^{G_i}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFor each $i\\in I$, let $\\varphi_i\\colon G\\to G_i$ be the\nprojection.\n\n\\noindent (a)\nThere is an induced inverse system of Frattini\nquotients with $G\/\\Phi(G)=\\varprojlim_iG_i\/\\Phi(G_i)$.\nIf $\\varphi_{ij}(G_i)\\Phi(G_j)\/\\Phi(G_j)$ is a proper subgroup of\n$G_j\/\\Phi(G_j)$, for all $i,j$ belonging to a cofinal subset  of $I$,\nthen  $G\/\\Phi(G)=\\varprojlim_i \\varphi_{j}(G)\\Phi(G_j)\/\\Phi(G_j)$\nand so $G$ can be generated by $d-1$ elements.\nOtherwise $\\varphi_{ij}$ must be surjective for $i,j$\nbelonging to a cofinal subset of $I$, and so is $\\varphi_j$\n({\\it cf.} \\cite[Prop. 1.1.10]{RZ:00b}).\n\n\\medskip\n\n\\noindent (b)\nSet, for the moment, $K\\!:=\\varprojlim H_i^{G_i}$.\nSince $H\\le K$, we have $H^G\\le K$, as $K\\triangleleft G$.\nSo, it suffices to establish $K\\le H$.\nThis is certainly true when there is a bound on the orders of the $G_i$.\nFix $n\\in\\mathbb N$.\nThen, as $d(G_i)=d$, there is a bound on the\norders of all $G_i\/\\Phi^n(G_i)$.\nThen the statement reads\n$\\varprojlim H_i^{G_i}\\Phi^n(G_i)\\le H^G\\Phi^n(G)$\nand therefore $K\\le H^G\\Phi^n(G)$.\nSince, $d(G)\\le d$, $G$ is finitely generated, and so the set $(\\Phi^n(G))_{n\\ge1}$\nis a fundamental system of neighbourhoods of the identity in $G$\n({\\it cf.} \\cite[Prop. 2.8.13]{RZ:00b}).\nHence $K\\le H$, as needed.\n\\end{proof}\n\nWe recollect the following fundamental results\nfrom the theory of pro-$p$ groups acting on pro-$p$ trees\nand their consequences for\nan amalgamated free pro-$p$ product or\na pro-$p$ {\\rm HNN}-extension.\n\nRecall that for a pro-$p$ group $G$ acting on a pro-$p$ tree $T$,  the closed subgroup generated by all vertex stabilizers\nis denoted by $\\widetilde{G}$; also, \nthe (unique) smallest pro-$p$ subtree of $T$ containing\ntwo vertices $v$ and $w$ of $T$ is denoted by $[v,w]$\nand called the geodesic connecting $v$ to $w$ in $T$\n({\\it cf.} \\cite[p. 83]{RZ:00a}).\n\n\\begin{thm}\n\\label{t:trees1}\nLet $G$ be a pro-$p$ group acting on a pro-$p$ tree $T$.\n\\begin{itemize}\n\n \\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Prop. 3.5]{RZ:00a}})}\n  $T\/\\widetilde{G}$ is a pro-$p$ tree.\n\n \\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Cor. 3.6]{RZ:00a}})}\n  $G\/\\widetilde{G}$ is a free pro-$p$ group.\n\n \\item[{\\rm (c)}] {\\rm (\\protect{\\cite[Cor. 3.8]{RZ:00a}})}\n If $v$ and $w$ are two different vertices of $T$,\n then $E([v,w])\\neq\\emptyset$ and\n $(G_v\\cap G_w)\\leq G_e$ for every $e\\in E([v,w])$.\n\n \\item[{\\rm (d)}] {\\rm (\\protect{\\cite[Thm. 3.9]{RZ:00a}})}\n If $G$ is finite, then $G=G_v$, for some $v\\in V(T)$.\n \\end{itemize}\n\\end{thm}\n\n\\begin{thm}\n\\label{t:afpproperties}\nLet $G=G_1\\coprod_H G_2$ be a proper amalgamated free pro-$p$\n product of pro-$p$ groups.\n\\begin{itemize}\n\\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Thm. 4.2(b)]{RZ:00a}})}\n Let $K$ be a finite subgroup of $G$.\n Then  $K\\leq G_i^{g}$ for some $g\\in G$ and for some $i=1$ or $2$.\n\n\\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Thm. 4.3(b)]{RZ:00a}})}\n Let $g\\in G$. Then\n\n $G_i\\cap G_j^{g}\\leq H^{b}$\n\n for some $b\\in G_i$, whenever $1\\leq i\\neq j\\leq 2$ or $g\\not\\in G_i$.\n\\end{itemize}\n\\end{thm}\n\n\\begin{thm}\n\\label{t:hnnproperties}\n Let $G={\\rm HNN}(H,A,f)$ be a proper pro-$p$ {\\rm HNN}-extension.\n\\begin{itemize}\n\\item[{\\rm (a)}] {\\rm (\\protect{\\cite[Thm. 4.2(c)]{RZ:00a}})}\n Let $K$ be a finite subgroup of $G$.\n Then $K\\leq H^{g}$ for some $g\\in G$.\n\n\n\\item[{\\rm (b)}] {\\rm (\\protect{\\cite[Thm. 4.3(c)]{RZ:00a}})}\n Let  $g\\in G $. Then\n \\begin{equation*}\n H\\cap H^{g}\\leq A^{b}\n \\end{equation*}\n for some $b\\in H\\cup tH$, whenever $g\\not\\in H$.\n\\end{itemize}\n\\end{thm}\n\n\nThe next  technical results concern inverse systems that will play\nan essential role during the proof of\nTheorem \\ref{t:freeorabelian_intro} in section \\ref{s:2-generated}.\nUntil the end of this section the directed set $I$\nwill be assumed to be order isomorphic to $\\mathbb N$.\n\n\\begin{prop} \\label{p:decomphnn}\n Let $G$ be the inverse limit of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of pro-$p$ groups.\n Suppose that each $G_i={\\rm HNN}(H_i,A_i,t_i)$\n is an\n {\\rm HNN}-extension with\n $H_i$ finite and $\\varphi_{ij}(H_i)\\cong H_j$.\n Then\n there are inverse systems of groups\n $\\{H_i',\\varphi_{ij},I\\}$ and\n $\\{A_i'',\\varphi_{ij},I\\}$ such that\n $G=\\text{\\rm HNN}(H,A,t)$ with\n $H\\!:=\\varprojlim H_i'$, $A\\!:=\\varprojlim A_i''$\n where each $H_i'$ (resp. $A_i''$)\n is a conjugate of $H_i$ (resp. $A_i$)\n by an element of $G_i$.\n\\end{prop}\n\n\\begin{proof} \\\nFix $k\\in I$.\nBy Theorem \\ref{t:hnnproperties}(a)\nthere are $g_k\\in G_k$ with $\\varphi_{jk}(H_j)= H_k^{g_k}$\n(remember they are isomorphic by the hypothesis).\nPick $g_j\\in\\varphi_{jk}^{-1}(g_k)$ and define\n$H_j' \\!:= H_j^{g_j\\inv}$, $A_j' \\!:= A_j^{g_j\\inv}$, and,\n$t_j' \\!:= t_j^{g_j\\inv}$; clearly $\\varphi_{jk}(H_j')= H_k$.\nSince $A_j = H_j \\cap H_j^{t_j}$ and\n$\\varphi_{jk}$ is surjective, we have\n\\begin{equation*}\n\\varphi_{jk}(A_j')\n\\le H_k \\cap H_k^{\\varphi_{jk}(t_j')}\n\\le A_k^{t_k^{\\epsilon_k}h_k} \\, ,\n\\end{equation*}\nfor suitable $h_k\\in H_k$ and $\\epsilon_k=0$ or $1$,\nby Theorem \\ref{t:hnnproperties}(b).\nChoose $h_j\\in\\varphi_{jk}^{-1}(h_k)\\cap H_j'$ and\n$x_j\\in\\varphi_{jk}^{-1}(t_k)$.\nDefining\n$A_j'' \\!:= {A_j'}^{(x_j^{\\epsilon_k}h_j)\\inv}$\nwe obtain\n$\\varphi_{jk}(A_j'') \\le A_k$\nin both cases.\nContinuing inductively \nwe obtain the desired inverse systems\n $\\{H_i',\\varphi_{ij},I\\}$ and\n $\\{A_i'',\\varphi_{ij},I\\}$.\n\nIt is straightforward to check that the other associated subgroup\nalso ``fits'' into the inverse system, that is\n$\\varphi_{jk}({A_j''}^{t_j''}) \\le A_k ^{t_k}$ where $t_j'' \\!:=\n{t_j'}^{(x_j^{\\epsilon_k}h_j)\\inv}$.\n\nNow, let $H\\!:=\\varprojlim H_i'$,\n$A\\!:=\\varprojlim A_i''$ and $B\\!:=\\varprojlim {A_i''}^{t_i''}$.\nFor each $i\\in I$ let us consider the subset\n\\begin{equation*}\nX_i\\!:=\\{\\tau_i\\in G\\,|\\, A^{\\tau_i}\\!=\\!B\n{ \\, \\ and \\, \\ }\nG_i\\!:=\\langle H_i, \\tau_i\\rangle \\}\\,.\n\\end{equation*}\nClearly every $X_i$ is compact, and\nsince $X_{i+1}\\subseteq X_i$ for all $i\\in I$,\nthere exists $t\\in \\bigcap_i X_i$ so that $B=A^t$.\n\nThe desired isomorphism from\n$\\text{\\rm HNN}(H,A,t)$ onto $G$ follows now from\nthe universal property of {\\rm HNN}-extensions.\n\\end{proof}\n\n\\begin{prop} \\label{p:decompfp}\n Let $G$ be the inverse limit\n of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of pro-$p$ groups\n $G_i$ each a free pro-$p$ product\n $G_i=A_i\\amalg B_i$ whith $A_i$ cyclic and $B_{i}$ procyclic.\n\n Then\n there are inverse systems of pro-$p$ groups,\n $\\{A_{i}',\\varphi_{ij},I\\}$ and\n $\\{B_{i}',\\varphi_{ij},I\\}$,\n where each $A_{i}'$\n is a conjugate of $A_{i}$\n by an element of $G_i$, and $B_{i}'\\le G_i$, and,\n $G\\cong \\left(\\varprojlim A_{i}'\\right)\n \\amalg \\left(\\varprojlim B_{i}'\\right)$.\n\\end{prop}\n\n\\begin{proof}\nSuppose first that there exists $i_0\\in I$ such that\n$B_{i_0}\\cong {\\mathbb Z}_p$.\nThen, since each $\\varphi_{ij}$ is surjective,\nthe induced homomorphism between the continuous abelianizations\n$A_i\\times {\\mathbb Z}_p\\cong\nG_i\/[G_i,G_i]\\to G_{j}\/[G_{j},G_{j}]\\cong A_j\\times {\\mathbb Z}_p$\nis surjective for $i_0\\le j\\le i$.\nTherefore $\\varphi_{ij}(A_i)\\le A_j\\Phi(G_j)$ and\nby Theorem \\ref{t:afpproperties}(a)\nthere is $g_{j}\\in G_j$ with $\\varphi_{ij}(A_i)\\le A_j^{g_j}$\nshowing that $A_i$ maps onto a conjugate of $A_j$.\nNow, observing that $G=\\varprojlim G_i$ with\n$G_i\\cong {\\rm HNN}(A_{i},1,t_i)$,\nwhere $t_i$ generates $B_{i}$, we can apply\nProposition \\ref{p:decomphnn}\nto obtain the result.\n\nSuppose that each $B_{i}$ is finite.\nSince $\\varphi_{ij}$ are surjective, from\nTheorem \\ref{t:afpproperties}(a),\nwe obtain that distinct free factors of $G_i$ are mapped,\nup to conjugation, to distinct free factors of $G_j$.\nSo, there is $k_0$ in $I$ so that\nfor all $i,j$ we have\n\\begin{equation*}\n\\varphi_{ij}(A_{i})= A_{j}^{x_{j}}\n{ \\ \\, and \\, \\ }\n\\varphi_{ij}(B_{i})= B_{j}^{y_{j}} \\, ,\n\\end{equation*}\nfor some $x_{j},y_{j}\\in G_j$.\nThen inductively the desired inverse systems $\\{A_{i}',\\varphi_{ij},I\\}$ and\n $\\{B_{i}',\\varphi_{ij},I\\}$,\ncan be exhibited.\nThe result follows now from\n\\cite[Lemma 9.1.5]{RZ:00b}.\n\\end{proof}\n\n\n\\begin{lemma} \\label{l:2gen}\n Let $G$ be a $2$-generated pro-$p$ group.\n \\begin{itemize}\n \\item[{\\rm (a)}]\n If $G$ is a free pro-$p$ product with\n procyclic amalgamation,\n then one of its free factors is procyclic.\n \\item[{\\rm (b)}]\n If $G$ is a proper {\\rm HNN}-extension with\n procyclic associated subgroups,\n then its base subgroup $H$ is at most $2$-generated.\n Moreover, if $d(H)=2$ then $H$ is generated by the associated subgroups.\n \\item[{\\rm (c)}]\n If $G$ is the fundamental pro-$p$ group of a finite tree of finite  groups\n such that all edge groups are cyclic,\n then either $|G|< \\infty$ or $G=K\\amalg_{C} R$ with $K$ cyclic and $R$ finite,\n or $G=K\\amalg_CM\\amalg_DN$,\n with $K$ and $N$ cyclic and $M\\le \\Phi(G)$.\n \\end{itemize}\n\\end{lemma}\n\n\\begin{proof} \\\n\n\\noindent (a) Suppose that $G=A\\amalg_{C} B$ and let ``bar''\nindicate passing to the Frattini quotient. We have an obvious\nepimorphism from $G$ to the induced pushout \n$P\\!:= \\bar A\\amalg_{\\bar C} \\bar B$. Let\n$n\\!:=d(A)+d(B)$. Since $C$ is procyclic, the image $M$ of the\nkernel of the canonical map $\\bar A\\amalg \\bar B \\to \\bar G$ via\nthe cartesian map $\\bar A\\amalg\\bar B\\to \\bar A\\times \\bar B$ is\nalso procyclic. The latter map induces an epimorphism from $\\bar\nG$ to the at least $(n-1)$-generated elementary abelian pro-$p$\ngroup $(\\bar A\\times \\bar B)\/M$. Therefore, $n\\leq 3$ and the\nresult follows.\n\n\\noindent (b) Suppose that $G={\\rm HNN}(H,C,f,t)$ with $C=\\gp c$.\nIf $d(H)\\ge3$ then $d(G)\\ge3$ as can be seen by using the obvious\nepimorphism $G\\to (H\\times {\\langle t\\rangle})\/{\\langle tct^\n{-1}f(c)^ {-1}\\rangle}$. Thus $d(H)\\le2$.\n\nFinally suppose that $d(H)=2$. Now $G$ is the quotient of \n$Q\\!:=H\\amalg \\gp t$  modulo the relation\n$f(c)\\inv c^{t}$. Since $d(Q)=3$ we can conclude that $c\\not\\in\n\\Phi(G)$ and $f(c)\\not\\in\\Phi(G)$. Therefore neither $c\\in\\Phi(H)$\nnor $f(c)\\in\\Phi(H)$. So we cannot have $f(c)\\inv c\\in\\Phi(H)$\nelse $d(G\/\\Phi(G))$ turns out to be 3. Hence $H=\\langle C,\nC^{t}\\rangle$.\n\n\\noindent (c) Let $G=\\Pi_1({\\mathcal G},\\Gamma)$ with finite\nvertex groups ${\\mathcal G}(v)$ and cyclic edge groups ${\\mathcal\nG}(e)$. We claim that $|V(\\Gamma)|\\le 3$. By assumption\n$|V(\\Gamma)|\\ge2$, and therefore it has an edge $e$. Splitting $G$\nover $e$, we can assume that ${\\mathcal G}(d_0(e))$ is procyclic\nby (a); hence $d_0(e)$ is a pending vertex of $\\Gamma$. Suppose\nnow that $\\Gamma$ has at least $3$ vertices, and let $a$ be an\narbitrary edge $\\neq e$ of $\\Gamma$ having initial or terminal\nvertex $v=d_1(e)$. Without loss of generality, suppose that\n$d_0(a)=v$. Then $d_1(a)$ is a pending vertex with procyclic\nvertex group ${\\mathcal G}(d_1(a))$; for, otherwise, by splitting\n$G$ over the edge $a$ we would obtain that $d(G)>2$, a\ncontradiction. Now, if we have $r\\geq 2$ edges with initial or\nterminal vertex $v$ then it follows from the pro-$p$ presentation\nof $G$ that it has a free pro-$p$ abelian group ${\\mathbb Z}_p^r$\nas a quotient; this implies $r=2$, whence $|V(\\Gamma)|\\le 3$.\n\nIf $|V(\\Gamma)|=2$ then $G=K\\amalg_CM$ with $K$ and $M$ finite,\nand, by (a), we can assume that $K$ is cyclic.\n\nSuppose now that $|V(\\Gamma)|=3$. Then $G=K\\amalg_CM\\amalg_DN$\nwith $C$ and $D$ cyclic and $K$, $M$, and $N$ finite. By the\nproperness of our decomposition we have\n$d(K\\amalg_CM)=d(M\\amalg_DN)=2$ and, making use of (a), we can\nconclude that $K$ and $N$ must both be cyclic. Since $d(G)=2$ then\n$M\\le\\Phi(G)$ follows.\n\\ignor{\n\n}\n\\end{proof}\n\n\n\\begin{prop} \\label{p:treeprod}\n Let $G$ be the inverse limit of a surjective inverse system\n $\\{G_i,\\varphi_{ij},I\\}$ of  pro-$p$ groups\n $G_i$. Suppose $G_i$ decomposes as\n an amalgamated free pro-$p$ product\n $G_i=K_i\\amalg_{C_i} R_i$ with $K_i$ cyclic and $R_i$ finite or\n $G_i=K_i\\amalg_{C_i}M_i\\amalg_{D_i}N_i$,\n with $K_i$ and $N_i$ cyclic and $M_i\\le \\Phi(G_i)$. Then,\n passing to a cofinal subset of $I$, if necessary,\n there are inverse systems\n $\\{K_i',\\varphi_{ij},I\\}$ and $\\{C_i'',\\varphi_{ij},I\\}$ such\n that $C_i''\\le K_i'$,\n $\\varphi_{ij}(K_i')= K_j'$ and $\\varphi_{ij}(C_i'')\\le C_j''$\n where each $K_i'$ (resp. $C_i''$) is a conjugate of $K_i$ (resp. $C_i$)\n by an element of $G_i$.\n\\end{prop}\n\\begin{proof}\n\nUsing Theorem \\ref{t:afpproperties}(a) in both cases we can pass to a cofinal\nsubset $J$ of $I$ such that for all $i\\geq j$ in $J$ we have\n$\\varphi_{ij}(K_i)\\leq K_j^{g_j}$, for some $g_j\\in G_j$.\nIndeed, in the first case $\\varphi_{ij}$ sends factors to the\nfactors up to conjugation and in the second case $\\varphi_{ij}$\nsends cyclic factors to cyclic factors up to conjugation. Then in\nfact, since $K_j$ is cyclic $\\varphi_{ij}(K_i)=K_j^{g_j}$ (indeed,\notherwise $\\varphi_{ij}(K_i)^{G_j}\\neq K_j^{G_j}$ contradicting\nthe surjectivity of $\\varphi_{ij}$). Now  selecting\n$g_i\\in\\varphi_{ij}^{-1}(g_j)$ and letting\n$K_i'\\!:=K_i^{g_i\\inv}$, and using an induction argument, we\nobtain the desired inverse system $\\{K_i',\\varphi_{ij},J\\}$. Next,\nletting $C_i'\\!:=C_i^{g_i\\inv}$ we have $C_i'\\le K_i'\\cap\nM_i^{g_i}$; then, by Theorem \\ref{t:afpproperties}(b),\n$\\varphi_{ij}(C_i')\\le K_j\\cap \\varphi_{ij}(M_i)\\le {C_j'}^{b_j}$,\nfor some $b_j\\in K_j'$. Choosing $b_i\\in\\varphi_{ij}^{-1}(b_j)\\cap\nK_i'$ and letting $C_i''\\!:={C_i'}^{b_i\\inv}$ we obtain the other\ninverse system $\\{C_i'',\\varphi_{ij},I\\}$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:fp}\nLet $X$ be a $G$-space and $(\\widetilde U_n)_{n\\ge1}$ be a subset\nof normal subgroups of $G_n$ with $\\bigcap \\widetilde U_n=1$.\nWrite $X_n\\!:=X\/\\widetilde U_n$ and $G_n\\!:=G\/\\widetilde U_n$. Let\nthere be subgroups $S_n\\le G_n$ so that $\\varphi_{nm}(S_n)\\le S_m$\nand $S\\!:=\\varprojlim S_n$ be the inverse limit. If\n$X_n^{S_n}\\neq\\emptyset$ for all $n\\in\\mathbb N$ then\n$X^S\\neq\\emptyset$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi_n$ denote the canonical projection from $X\\cup G$\nonto $X_n\\cup G_n$. Then $X_n^{\\varphi_n(S)}\\supseteq\nX_n^{S_n}\\neq\\emptyset$. Therefore\n$Y_n:=\\varphi_n\\inv(X_n^{\\varphi_n(S)})\\neq\\emptyset$. Now\n$Y_n=\\{x\\in X\\mid xS\\subseteq x\\widetilde U_n\\}$ so that\n$Y_{n+1}\\subseteq Y_n$; by the compactness of $X$ we can deduce\nthat $\\emptyset\\neq \\bigcap Y_n\\subseteq X^S$.\n\\end{proof}\n\n\n\nWe end this section by quoting results to be used\nin the next section.\n\n\\begin{prop}[\\protect{\\cite[Thm. 1.1]{Scheiderer:99}}]\n\\label{p:scheiderer}\nLet $G$ be a finitely generated pro-$p$ group which\ncontains an open free pro-$p$ subgroup of index $p$.\nThen $G$ is isomorphic to a free pro-$p$ product\n\\begin{equation*}\nH_0 \\amalg (S_1\\times H_1 )\\amalg\\cdots\\amalg (S_m\\times H_m)\n\\end{equation*}\nwhere $m\\geq 0$, the $S_i$ are cyclic groups of order $p$\nand the $H_i$ are free pro-$p$ groups of finite rank.\n\\end{prop}\n\n\\begin{cor}[\\protect{\\cite[Cor. 1.3(a)]{Scheiderer:99}}]\n\\label{c:finiteconjugacy}\nEvery pro-$p$ group which contains an open\nfree pro-$p$ subgroup of finite rank\nhas, up to conjugation, only a finite number\nof finite subgroups.\n\\end{cor}\n\n\n\\begin{prop}\\label{p-freeedgeaction}\nA pro-$p$ group $G$ acting on a pro-$p$ tree $T$\nwith trivial edge stabilizers such that there\nexists a continuous section $\\sigma:V(T)\/G\\longrightarrow V(T)$ is\nisomorphic to a free pro-$p$ product\n\\begin{equation*}\n\\left(\\coprod_{\\dot v\\in V(T)\/G} G_{\\sigma(\\dot v)}\\right)\\amalg \\left(G\/\\gp{G_w\\mid w\\in V(T)}\\right) \\, .\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nThis follows from the proof of \\protect{\\cite[Thm. 3.6]{Z:96}}.\nSee also the last section of \\cite{Melnikov:90}.\n\\end{proof}\n\n\n\n\n\\section{Groups acting virtually freely on trees}\n\\label{s-proofs}\n\nIf a pro-$p$ group $G$ acts on a profinite graph $\\Gamma$ we shall\ncall sometimes $\\Gamma$ a $G$-graph.\n\n  \\begin{lemma}\\label{l-actionfree}\n  Let $G$ be a non-trivial finitely generated pro-$p$ group,\n  and let $\\Gamma$ be a connected $G$-graph.\n  Suppose that $\\Delta$ is a connected subgraph of $\\Gamma$\n  such that $\\Delta G=\\Gamma$.\n  Then there exists a minimal set of generators $X$ of $G$\n  such that $\\Delta\\cap \\Delta x\\neq \\emptyset$ holds for each $x\\in X$.\n  \\end{lemma}\n\n  \\begin{proof}\n  It is enough to prove the lemma under the\n  additional assumption that $G$ is elementary abelian.\n  Indeed, using ``bar'' to denote passing to the quotient modulo $\\Phi(G)$,\n  making $Z$  a minimal generating set of $G$, suppose that for each\n  $z\\in Z$ there exists a vertex $v_z\\in \\Delta$ with\n  $\\overline{v_z}\\in\\ol \\Delta\\cap \\ol \\Delta\\ol z\\neq\\emptyset$.\n  Then there exists $f_z\\in\\Phi(G)$ with $v_zz\\in \\Delta f_z$, so that\n  the set  $X\\!:=\\{zf_z\\inv\\mid z\\in Z\\}$ is a minimal set of\n  generators of $G$ for  which the assertion of the Lemma holds.\n\n  Suppose that the lemma is false for an elementary abelian group.\n  Then there is a counterexample $G$ with minimal $d(G)$.\n  Select a minimal generating subset $X$ of $G$.\n  If $d(G)=1$ then,\n  due to the connectedness  of $\\Gamma$,\n  there are $g_1,g_2\\in G$ with $g_1\\neq g_2$ such that\n  $\\Delta g_1\\cap \\Delta g_2\\neq\\emptyset$.\n  Replacing $X$ by $\\{g_1g_2\\inv\\}$ shows that\n  the conclusion of the lemma holds, a contradiction.\n  Hence $d(G)\\ge2$.\n  Select an element $z\\in X$ and let ``bar'' denote passing\n  to the quotient modulo $\\gp z$.\n  Since $d(\\ol G)=d(G)-1$, by the minimality assumption there is a\n  subset $\\ol Y$ of $\\ol G$ which is\n  a minimal set of generators of $\\ol G$ such that\n  $\\ol\\Delta\\cap \\ol\\Delta \\ol y\\neq\\emptyset$ for all $\\ol y\\in \\ol Y$.\n  Let $Y$ denote a transversal of $\\ol Y$ in $G$.\n  Then there are elements $z_y\\in \\gp z$ such that\n  $\\Delta\\cap \\Delta yz_y\\neq\\emptyset$ for all $y\\in Y$.\n  Set $W=\\{yz_y\\mid y\\in Y\\}$. Since\n  $\\Gamma$ can be viewed as a $\\gp z$-graph,\n  $\\Delta W \\cap \\Delta W z\\neq\\emptyset$ can be assumed by  the\n  minimality assumption.\n  This means that there exist $w_1,w_2\\in W$ such that\n  $\\Delta w_1\\cap \\Delta w_2z\\neq \\emptyset$, so\n  $X=W\\cup\\{w_1^{-1}w_2z\\}$ would satisfy the assertion\n  of the lemma, a contradiction.\n  \\end{proof}\n\n \\begin{lemma}\\label{l-action}\n Let $G$ be a finitely generated infinite pro-$p$ group.\n Suppose that $G$ acts on a pro-$p$ tree $T$ containing a\n pro-$p$ subtree $D$ such that $DG=T$.\n Then there exists a minimal set of generators $X$ of a retract $H$\n of $G$ such that $D\\cap Dx\\neq \\emptyset$ for each $x$ in $X$.\n \\end{lemma}\n\n  \\begin{proof}\n  Let ``bar'' denote passing to the quotient modulo\n  $\\widetilde G=\\gp{G_v\\mid v\\in V(T)}$.\n  By Theorem \\ref{t:trees1}(a)\n  the quotient graph $\\ol T\\!:=T\/\\widetilde G$ is a pro-$p$ tree.\n  Applying Lemma \\ref{l-actionfree} to $\\ol G$ acting on $\\ol T$\n  yields a subset $Z$ of $G$ such that $\\ol Z$ is\n  a minimal set of generators of $\\ol G$ and\n  for each $z$ there exists a vertex $v_z\\in D$ such that\n  $\\overline{v_z}\\ol z\\in\\ol D\\cap \\ol D\\ol z\\neq\\emptyset$ holds.\n  Hence there  exists $k_z\\in \\widetilde G$\n  with $v_zzk_z\\in Dk_z\\cap Dz$ and so $v_zz\\in D\\cap Dzk_z^{-1}$.\n  Now set $X\\!:=\\{zk_z^{-1}\\mid z\\in Z\\}$ and $H\\!:=\\gp X$.\n  Finally observe that by\n  Theorem \\ref{t:trees1}(b),\n  $\\ol G$ is a free pro-$p$ group, so that $H$ is indeed a retract.\n  \\end{proof}\n\n\n\n\\begin{lemma}\\label{l-rankformula}\nLet $G$ be a finitely generated pro-$p$ group\nacting on a pro-$p$ tree $T$.\nSuppose that all vertex stabilizers are finite and\nall edge stabilizers are pairwise conjugated.\nAssume further that there exist an edge $e\\in T$ and\na finite subset $V\\subseteq T^{G_e}$ such that:\n\\begin{itemize}\n\\item[\\rm (i)] for every $v_1, v_2\\in V$,\n$v_1G=v_2G$ implies $v_1=v_2$,\n\\item[\\rm (ii)]  $G$ is generated by the $G_v$, $v\\in V$.\n\\end{itemize}\nIf $F$ is a free pro-$p$ open normal subgroup of $G$, then \n\\begin{equation*}\n{\\rm rank}(F)-1=\n[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V} \\frac{1}{|G_v|}\\right) \\, .\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof} \nWe use induction on the index $[G:F]$.\nObviously $F\\neq G$, from hypothesis (ii); so,\nlet us consider the preimage $N$ in $G$ of a central subgroup\nof order $p$ of $G\/F$.\n\n\\medskip\n\\noindent {\\it Case 1}.  $N\\cap G_e=1$.\n\nIt follows that each non-trivial torsion element\n$t$ of $N$ generates a self-centralized subgroup.\nIndeed, by\nTheorem \\ref{t:trees1}(d)\n$t$ stabilizes some vertex $w$, so if $g$ centralizes\n$t$, the element $t$ also stabilizes $wg$.\nBut then by\nTheorem \\ref{t:trees1}(c)\n$t$ stabilizes\nthe geodesic $[wg,w]$.\nSince, however, $G_e\\cap N=1$,\nthe element $t$ cannot stabilize any edge,\nso $wg=w$, and therefore $g$ is a power of $t$.\n\nThus the decomposition of $N$ according to\nProposition \\ref{p:scheiderer}\nbecomes $N=\\left(\\coprod_{i\\in I}C_i\\right)\\amalg F_1$,\nwith $F_1$ a free pro-$p$ subgroup of $F$.\nTaking into account that $G$ acts upon\nthe conjugacy classes of subgroups of order $p$ we have\n\\begin{equation}\n\\label{e-scheiderer}\nN=\\coprod_{v\\in V}\\left(\\coprod_{r_v\\in G\/NG_v}\n(N\\cap G_{v})^{r_v}\\right) \\amalg F_1 \\, .\n\\end{equation}\nSet $V_1=\\{v\\in V\\mid N\\cap G_{v}\\neq 1\\}$.\nSince $NG_v=FG_v$ for every $v\\in V_1$ we can rewrite\nEq.~(\\ref{e-scheiderer}) as\n\\begin{equation*}\nN=\\coprod_{v\\in V_1}\\left(\\coprod_{r_v\\in G\/FG_v}\n(N\\cap G_{v})^{r_v}\\right) \\amalg F_1\\, .\n\\end{equation*}\nUsing this free decomposition and comparing it with\nProposition \\ref{p:scheiderer}\nwe find\n\\begin{equation}\n|I|=\\sum_{v\\in V_1}|G\/FG_v|=[G:F]\\sum_{v\\in V_1}\\frac{1}{|G_v|} \\, ,\\label{e-AinLemma}\n\\end{equation}\nand\n\\begin{equation}\n{\\rm rank}(F)-1=p\\ {\\rm rank}(F_1)+(p-1)(|I|-1)-1 \\, .\\label{e-rankF}\n\\end{equation}\n\nIf $N=G$ then $F_1=1$, since $G_v, v\\in V$ generate $G$.\nThen $G_e=1$ since otherwise $G$ is finite.\nSo $|V|=|I|$ and the last equation becomes exactly the needed one.\nThis gives the base of induction.\n\nSuppose now that $N\\not= G$.\nThen the product $p\\ {\\rm rank}\\of{F_1}$ can be computed\nby observing that passing to the quotient modulo $\\torgp{N}$\nand indicating it by ``bar'' we have\n${\\rm rank}\\of{\\ol F}={\\rm rank}\\of{F_1}$,\nso that using $[G:F]=p[\\ol G :\\ol F]$\nthe induction hypothesis yields\n\\begin{eqnarray*}\np\\  {\\rm rank}\\of{F_1}&=&p\\ {\\rm rank}\\of{\\ol F}\\\\\n&=&p[{\\ol G}:{\\ol F}]\\left(\\frac{|V|-1}{|\\ol G_e|}-\\sum_{v\\in V} \\frac{1}{|\\ol G_v|}\\right)+p\\\\\n&=&[G:F] \\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V_1} \\frac{1}{|\\ol G_v|}-\\sum_{v\\in V\\,-\\, V_1} \\frac{1}{|G_v|}\\right)+p\\\\\n&=&[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V_1} \\frac{p}{|G_v|}-\\sum_{v\\in V\\,-\\, V_1} \\frac{1}{|G_v|}\\right)+p\n\\end{eqnarray*}\n(we used $G_e\\cap N=1=G_v\\cap N$\nfor all $v\\in V\\,-\\,V_1$ and $|G_v\\cap N|=p$ for all $v\\in V_1$\nto obtain the last equality).\nInserting this expression and the expression for $|I|$\nfrom Eq.~(\\ref{e-AinLemma}) into Eq.~(\\ref{e-rankF})\nyields the claimed formula for ${\\rm rank}\\of F$.\n\n\\medskip\n\\noindent {\\it Case 2.} $N\\cap G_e \\not= 1$.\n\nSince for all $v\\in V$ the edge group $G_e$ is contained in $G_v$\nby the hypothesis, $G_v$ centralizes $N\\cap G_e$.\nBut $G=\\gp{G_{v} \\mid v\\in V}$\nso $N\\cap G_e$ is  a central subgroup of $G$ of order $p$.\nThen, using ``bar'' to pass to the quotient modulo $N\\cap G_e$\nand the inductive hypothesis, for $\\ol G$ we have\n\\begin{eqnarray*}\n{\\rm rank}{(F)}-1&=& {\\rm rank}{(\\ol F)}-1 \\\\\n&=&[\\ol G:\\ol F]\\left(\\frac{|V|-1}{|\\ol G_e|}-\\sum_{v\\in V} \\frac{1}{|\\ol G_v|}\\right) \\\\\n&=&\\frac{1}{p}[G:F]\\left(\\frac{p(|V|-1)}{|G_e|}-\\sum_{v\\in V} \\frac{p}{|G_v|}\\right) \\\\\n&=&[G:F]\\left(\\frac{|V|-1}{|G_e|}-\\sum_{v\\in V} \\frac{1}{|G_v|}\\right)\n\\end{eqnarray*}\nas needed.\n\\end{proof}\n\n\nRecall that a pro-$p$ group $G$ acts\n{\\it faithfully} on a pro-$p$ tree $T$ if the kernel of the action is trivial;\nand $G$ acts {\\it irreducibly} on $T$\nif $T$ does not contain a proper $G$-invariant pro-$p$ subtree.\n\n\\begin{lemma}\\label{l-exchange1\nLet a pro-$p$ group $G$ act faithfully and irreducibly\non a pro-$p$ tree $T$.\nSuppose that $G_e$ is a minimal edge stabilizer and\nthe set of edges $E(T^{G_e})G$ is open in $T$.\nThen $T'\\!:=T\\,-\\, E(T^{G_e})G$ is a subgraph having each connected component a pro-$p$ tree.\n\nLet $\\ol T$ be the quotient graph obtained by collapsing distinct\nconnected components of $T'$ to distinct points.\nThen $\\ol T$ is a pro-$p$ tree on which\n$G$ acts faithfully and irreducibly, and $\\ol T=\\ol T^{G_e}G$.\n\\end{lemma}\n\n\\begin{proof}\nSince  $T'$ is closed and contains $V(T)$, it is a subgraph of $T$;\nhence its connected components \nare pro-$p$ trees.\nMoreover, $\\ol T$ is a $G$-graph, and by\n\\cite[Proposition p. 486]{Zalesskii:89},\nit is simply connected and hence a pro-$p$ tree.\n\nNow, we have $m\\in T'$ if and only if there exists a\nsubgroup $L\\le G_m$, an edge stabilizer, so that $L^g$ is not\ncontained in $G_e$ for every $g\\in G$.\nTherefore, since $G_e$ is a minimal edge stabilizer,\nwe conclude that all edge stabilizers\nof edges in $T^{G_e}G$ are conjugates of $G_e$.\n\nLet us show that $G_{\\ol e}$ is a conjugate of $G_e$ for every\n$\\ol e\\in E(\\ol T)$.\nLet $f\\in E(T)$ and $u,v$ be its end points which, \nby construction, belong to $T'$.\nFix $g\\in G_{\\ol e}=G_{\\ol u}\\cap G_{\\ol v}$.\nThen $ug$ and $vg$ belong\nrespectively to the same connected components as $u$ and $v$.\nThe collapsing procedure induces a\ncanonical epimorphism which is injective when restricted to\n$E(T^{G_e})G$.\nSince $\\ol T$ is a pro-$p$ tree we find that\nafter collapsing $e$ and $eg$ both map to $\\ol e$, and\nas edges of $eG$ under the collapsing procedure\nare not identified, $e=eg$ must follow.\nHence $G_{\\ol e}$ is a conjugate of $G_e$ indeed.\n\nSuppose that $G$ does not act irreducibly on $\\ol T$.\nSince $\\ol T$ is obtained by collapsing pro-$p$ subtrees,\nthe preimage of a proper $G$-invariant pro-$p$ subtree of $\\ol T$\nis a proper $G$-invariant pro-$p$ subtree of $T$; a contradiction.\n\nSuppose that $g\\in G$ acts trivially upon all of $\\ol T$.\nThen, in particular, $\\ol eg=\\ol e$ and,\nas edges of $eG$ under the collapsing procedure are not identified,\nwe must have $eg=e$, {\\it i.e.}, $g\\in G_e$.\nTherefore the kernel of the action of $G$ upon\n$\\ol T$ is contained in $G_e$ and so $G_e$ contains a normal\nsubgroup of $G$ which, by\n\\cite[Thm. 3.12]{RZ:00a}\nmust act trivially on $T$.\nHence $G$ acts faithfully on $\\ol T$.\n\\end{proof}\n\n\nRecall from the introduction that $G$ acts virtually freely\non a space $X$ if some open subgroup $H$ of $G$ acts freely on $X$.\n\n\\begin{lemma}\\label{l-treetech\nLet $G$ be a finitely generated pro-$p$ group acting\nfaithfully, irreducibly and virtually freely on a pro-$p$ tree $T$.\nThen there are a pro-$p$ tree $D$, an edge $e\\in E(D)$,\na finite subset $V\\subseteq D^{G_e}$ and\na finite subset $X\\subseteq G$ such that\n\\begin{itemize}\n\\item[{\\rm (a)}]\n$G$ acts faithfully upon $D$.\n\\item[{\\rm (b)}]\nAll edge stabilizers are pairwise conjugate;\nin particular, $D=D^{G_e}G$.\n\\item[{\\rm (c)}]\nfor every $v_1,v_2\\in V$, $v_1G=v_2G$ implies $v_1=v_2$.\n\\item[{\\rm (d)}]\n$X$ freely generates a free pro-$p$ subgroup $H$\nsuch that for $G_0\\!:=\\gp{G_v\\mid v\\in V}$\nwe have $G=\\gp{G_0, H}$ and $H\\cap G_0^G=1$.\n\\item[{\\rm (e)}] For each $x\\in X$, we have\n\\begin{equation*}\nG_e^x\\subseteq\\bigcup_{v\\in V} G_v \\, .\n\\end{equation*}\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof}\nLet $e\\in T$ be an edge with the stabilizer $G_e$ \nof minimal order.\nLet $\\Sigma$ denote the set of all non-trivial finite subgroups\n$L$ of $G$ that are not conjugate to a subgroup of $G_e$.\nSince $G$ is finitely generated,\nCorollary \\ref{c:finiteconjugacy}\nsays that there exist up to conjugation\nonly finitely many finite subgroups in $G$;\nin particular there is a finite subset $S$ of\n$\\Sigma$ such that $\\Sigma=\\{L^g\\mid L\\in S, \\ g\\in G\\}$.\nTherefore the subset\n$T_\\Sigma\\!:=\\{m\\in T\\mid \\exists L\\in\\Sigma, m\\in T^L\\}$,\nwhich is the union of all subtrees of fixed points $T^L$ for\nsubgroups $L\\in\\Sigma$ can be represented in the form\n$T_\\Sigma=\\bigcup_{L\\in S}T^LG$ and is hence a closed\n$G$-invariant subgraph of $T$.\nTherefore\n$E(T^{G_e})G=T\\,-\\, T_\\Sigma$ is open and we can apply\nLemma \\ref{l-exchange1}\nto obtain, by collapsing the connected components of $T_\\Sigma$,\na pro-$p$ tree $D$ on which $G$ acts irreducibly and faithfully with\nall edge stabilizers conjugate to $G_e$.\nThus $D$ satisfies (a) and (b).\n\nWe come to proving (c),(d) and (e).\nSet $N\\!:=\\gp{G_v\\mid v\\in V(D)}$.\nBy Lemma \\ref{l-action}\nthere is finite minimal subset $X$ of generators of\na retract $H$ in $G$ of $G\/N$ such that\n$D^{G_e}\\cap D^{G_e}x\\neq\\emptyset$ for every $x$ in $X$;\nin fact, as $G\/N$ is free pro-$p$ by\nTheorem \\ref{t:trees1}(b),\n$X$ freely generates $H$.\nMoreover, by the construction of $D$, there is only a finite subset\n$V$ of vertices up to translation with stabilizers that\nare not conjugates of $G_e$;\nto see this we observe that if vertices $v,w$ are both\nstabilized by $L\\in\\Sigma$, then $L$ stabilizes the geodesic $[v,w]$\n(see Theorem \\ref{t:trees1}(c))\nand so $v,w$ belong to the same connected component of $T_\\Sigma$.\n\nIt follows that $G=\\gp{G_v, H\\mid v\\in V}$ and\n$H\\cap \\gp{G_v\\mid v\\in V}=1$.\nMoreover, since $G$ is pro-$p$ we can reduce $V$ such that\nno two distinct vertices of it lie in the same orbit.\n\nBy construction, for every group element\n$x\\in X$ there is\na vertex $v_x\\in D^{G_e}$ with $v_xx\\inv\\in D^{G_e}$.\nWhen $f$ is any edge in the geodesic $[v_x,v_xx\\inv]$ then\n$G_e=G_f=G_{v_x}\\cap G_{v_x}^{x\\inv}$\n(see Theorem \\ref{t:trees1}(c)),\nso that, in particular, $G_e^{x}\\le G_{v_x}$.\nFinally modify $V$ by replacing for every $x\\in X$\na vertex $v\\in V$ by the vertex $v_x$ whenever $vG=v_xG$.\nThen we see that (c), (d), and (e) all hold.\n\\end{proof}\n\nIt is now convenient to introduce a notion of\n{\\em pro-$p$ {\\rm HNN}-extension} as a\ngeneralization of the construction described in\n\\cite[Sec. 4, p. 97]{RZ:00a}.\n\n\\begin{definition}\\label{d:HNN-ext}\n\\rm Suppose that $G$ is a pro-$p$ group, and for a finite set\n$X$, there are given monomorphisms $\\varphi_x:A_x\\to G$ for subgroups\n$A_x$ of $G$.\nThe {\\em {\\rm HNN}-extension}\n$\\tilde G\\!:={\\rm HNN}(G,A_x,\\varphi_x,x\\in X)$\nis defined to be the quotient of $G\\amalg F(X)$ modulo the\nequations $\\varphi_x(a_x)=a_x^x$ for all $x\\in X$.\nWe call $\\tilde G$ an {\\em {\\rm HNN}-extension} and\nterm $G$ the {\\em base group}, $X$ the set of {\\em stable letters},\nand the subgroups $A_x$ and $B_x\\!:=\\varphi_x(A_x)$\n{\\em associated}.\n\\end{definition}\n\nOne can see that every {\\rm HNN}-extension in the sense of the\npresent definition can be obtained\nby successively forming {\\rm HNN}-extensions, as defined\nin \\cite{RZ:00a}, each time defining the base group\nto be the just constructed group and then\nselecting a pair of associated subgroups\nand adding a new stable letter.\n\nThe {\\rm HNN}-extension $\\tilde G\\!:={\\rm HNN}(G,A_x,\\varphi_x,X)$\ncan also be defined by a {\\em universal property} as follows.\nThere are canonical maps\n$\\tilde f:G\\to\\tilde G$, $\\tilde f_x:A_x\\to\\tilde G$, $\\tilde g:X\\to \\tilde G$,\nwith $\\tilde f_x(a_x)^{\\tilde g(x)}=\\tilde f\\varphi_x(a_x)$\nfor all $a_x\\in A_x$,\nso that, given any pro-$p$ group $H$, any homomorphisms $f:G\\to H$, $f_x:A_x\\to H$ and a map $g:X\\to H$\nsuch that for all $x\\in X$ and all $a_x\\in A_x$\nwe have $f(\\varphi_i(a_x))=f_x(a_x)^{g(x)}$ then there is a unique\nhomomorphism $\\omega:\\tilde G\\to H$ with $f=\\omega\\tilde f$, $g=\\omega\\tilde g$, and, for all $x\\in X$,\n$f_x=\\omega\\tilde f_x$.\n\n\n\n\\begin{thm}\\label{t-treeacting\n Let $G$ be a finitely generated infinite pro-$p$ group acting\n virtually freely on a pro-$p$ tree $T$.\n Then $G$ splits either as an amalgamated free pro-$p$\n product or as a proper pro-$p$ ${\\rm HNN}$-extension\n over some edge stabilizer.\n\\end{thm}\n\n\\begin{proof}\\setcounter{claims}0\nWe consider $G$ to be a counterexample to the theorem\nwith minimal index $[G:F]$\nwhere $F$ is an open subgroup of $G$ acting freely on $T$.\n\n\\bcl\n$G$ does not have a non-trivial finite normal subgroup.\nIn particular, we can assume that\n$G$ acts on $T$ faithfully and irreducibly.\n\\ecl\n\nBy \\cite[Lemma 3.11]{RZ:00a} there\nexists a unique minimal $G$-invariant subtree in $T$.\nReplacing $T$ by this subtree we may assume that\nthe action of $G$ is irreducible.\n\nNow, if $G$ contains a non-trivial finite normal subgroup,\nit contains a central subgroup $C$ of order $p$.\nBy the minimality assumption on $[G:F]$\nand as $[{G\/C}:{FC\/C}]<[G:F]$\nthe quotient group $\\ol G\\!:=G\/C$ satisfies\nthe conclusion of the theorem, {\\it i.e.} $\\ol G$ is either\nan amalgamated free pro-$p$ product\n$\\ol G=\\ol G_1\\amalg_{\\ol H} \\ol G_2$ with finite\namalgamating subgroup or it is\nan ${\\rm HNN}$-extension $\\ol G={\\rm HNN}(\\ol G_1, \\ol H, t)$\nwith finite associated subgroups.\nThen $G$ is either a non-trivial amalgamated free pro-$p$ product\n$G=G_1\\amalg_H G_2$ or ${\\rm HNN}(G_1,  H, t)$\nwith $G_1,G_2, H$ being preimages of $\\ol G_1,\\ol G_2, \\ol H$ in $G$,\nrespectively, as needed.\nHence $G$ does not possess a non-trivial\nfinite normal subgroup.\nSince the vertex stabilizers are finite, the kernel of the\naction of $G$ upon $T$ is finite, hence it is trivial.\n\n\\medskip\n{\\em Thus, there exist $D$, $e$, $V$, $X$ and $G_0$\nhaving the properties {\\rm (a)}-{\\rm (e)} of\nLemma {\\rm \\ref{l-treetech}}.\nNote that the stabilizers of vertices in $D$ may well be infinite.}\n\n\\bcl\nThe pro-$p$ subgroup $H$ of $G$\nfreely generated by $X$ must be trivial.\n\\ecl\n\nSuppose that $H\\neq1$.\nLet $\\tilde G={\\rm HNN}(G_0, G_e, X)$ and\n$\\lambda:\\tilde G\\longrightarrow G$ be the epimorphism\ngiven by the universal property.\nBy induction on ${\\rm rank}\\of F$ we show that\n$\\lambda$ is an isomorphism.\n\nIt suffices to show that the rank of $F$ equals the rank of\n$\\tilde F\\!:=\\lambda^{-1}(F)$.\nIf $F_0\\!:=G_0\\cap F\\neq 1$ we can factor out\nthe normal closure of $F_0$ in $G$\n(and, if necessary, the kernel of the action as well)\nin order to obtain the quotient group $\\ol G$\nwhich acts on $D\/F_0^G$ and satisfies\n${\\rm rank}\\of{\\ol F}<{\\rm rank}\\of F$.\nTherefore the induced epimorphism\n$\\ol\\lambda:{\\rm HNN}(\\ol G_0,\\ol G_{\\ol e},X)\\to\\ol G$\nis an isomorphism, and it is not hard to see that\n${\\rm HNN}(\\ol G_0,\\ol G_{\\ol e},X)$ is isomorphic to\n$\\tilde G\/\\tilde F_0^{\\tilde G}$,\nwhere $\\tilde F_0\\!:=G_0\\cap \\tilde F$.\nThis shows that the image $\\ol F$ of $F$ in $\\ol G$ is\nisomorphic to  $\\tilde F\/\\tilde F_0^{\\tilde G}$.\nBy Proposition \\ref{p-freeedgeaction}\n$F$ is a free pro-$p$ product $F\\cong F_0\\amalg \\ol F$ and\n$\\tilde F\\cong  \\tilde F_0\\amalg \\tilde F\/\\tilde F_0^{\\tilde G}$, so $F\\cong \\tilde F$ and we are done.\nThus we may assume that $G_0$ is finite.\nNow applying\nLemma \\ref{l-rankformula}\nto $\\tilde G$ and $G$ we deduce that\n${\\rm rank}(\\tilde F)={\\rm rank}(F)$, so\n$\\lambda\\restr{\\tilde F}$ turns out to be an isomorphism,\ncontradicting $G$ being a counterexample.\nThis finishes the proof of the claim.\n\n\\bcl\nThe natural epimorphism\n\\begin{equation*}\n\\lambda:{\\coprod_{v\\in V}}\\raisebox{-1.2ex}{\\small $G_e$}\nG_{v}\\longrightarrow G\n\\end{equation*}\nfrom the free pro-$p$ product of vertex stabilizers $G_v$\namalgamated along the single edge group $G_e$\nonto $G$ is an isomorphism.\n\\ecl\n\nLet us use induction on ${\\rm rank}\\of F$.\nSince $F$ acts freely on $E(D)$\nby Proposition \\ref{p-freeedgeaction},\nfor each $v\\in V$ the intersection $F\\cap G_{v}$\nis a free factor of $F$, so similarly as in the proof of Claim 2\nwe can use the induction hypothesis,\nin order to achieve all $G_{v}$ to be finite.\n\nPut $\\tilde F=\\lambda^{-1}(F)$.\nSince $\\lambda$ restricted to all $G_v$ is injective,\nit suffices to prove that $\\lambda\\restr{\\tilde F}$ is an isomorphism.\nBut by applying\nLemma \\ref{l-rankformula} to $\\tilde G$ and $G$\nwe get that $F$ and $\\tilde F$ have the same rank\nand therefore $\\lambda$ is an isomorphism.\nThe result follows.\n\n\\medskip\n\nClaim 3 shows that $G$ is not a counterexample, a final contradiction.\n\\end{proof}\n\n\n\\begin{thm}\\label{t-fund\nA finitely generated pro-$p$ group $G$ acting\nvirtually freely on a pro-$p$ tree $T$\nis isomorphic to the\nfundamental pro-$p$ group $\\Pi_1(\\cG,\\Gamma)$ of a finite graph of\nfinite $p$-groups whose edge and vertex groups are isomorphic to\nthe stabilizers of some edges and vertices of $T$.\n\\end{thm}\n\n\\begin{proof}\nBy induction on the rank of a maximal normal free\npro-$p$ subgroup $F$ of $G$.\nIf ${\\rm rank}(F)=0$, that is $G$ is finite,\ntake as graph of groups the single vertex $G$.\nIn the general case, we apply\nTheorem \\ref{t-treeacting}\nto split $G$ as an amalgamated free pro-$p$ product\n$G=G_1\\amalg_K G_2$ or as a pro-$p$ {\\rm HNN}-extension\n$G={\\rm HNN}(G_1,K,t)$ over a finite subgroup $K$.\nMoreover, we are free to choose $K$ up to\nconjugation in $G_1$.\nThen every free factor, or the base group,\nsatisfies the induction hypothesis and\nso exists the fundamental group of a finite graph of finite $p$-groups.\nBy \\cite[Thm. 3.10]{ZM:89}\n$K$ is conjugate to some vertex group of $G_1$ and so\nwe may assume that\n$K$ is contained in a vertex group of $G_1$.\nNow in the case of an amalgamated product\nthere is $g_2\\in G_2$ such that $K^{g_2}$ is contained in\na vertex group of $G_2$, so $G$ admits a decomposition\n$G=G_1^{g_2}\\amalg_{K^{g_2}} G_2$.\nThus in both cases $G$ becomes\nthe fundamental group of a finite graph of finite $p$-groups.\n\\end{proof}\n\n\\begin{thm}\\label{t:subgrouptheorem\nLet $H$ be a finitely generated subgroup of the\nfundamental pro-$p$ group $G$ of a finite graph of finite $p$-groups.\nThen $H$ is the fundamental pro-$p$ group of\na finite graph of finite $p$-groups which are\nintersections of $H$ with some conjugates of vertex and edge\ngroups of $G$.\n\\end{thm}\n\n\\begin{proof}\nThe fundamental pro-$p$ group $G=\\Pi_1(\\cG,\\Gamma)$\nacts naturally on the standard pro-$p$ tree $T$\n({\\it cf.} \\cite[Sec. 3]{ZM:89})\nand therefore so does $H$.\nMoreover, since the graph $\\Gamma$ is finite,\nthere exists an open normal subgroup $U$ of $G$\nthat intersects all vertex groups trivially and so acts freely on $T$.\nThus Theorem \\ref{t-fund} can be applied.\n\\end{proof}\n\n\\begin{example}\\label{ex-4.3}\\rm\nLet $A$ and $B$ be groups of order $2$ and\n$G_0=\\gp{ A\\times B, t\\mid A^{t}=B}$ be a\npro-$2$ {\\rm HNN}-extension of $A\\times B$ with\nassociated subgroups $A$ and $B$.\nNote that $G_0$ admits an automorphism of order $2$ that\nswaps $A$ and $B$ and inverts $t$.\nLet $G=G_0\\rtimes C$ be the holomorph.\nSet $H_0=\\torgp{G_0}$ and $H=H_0\\rtimes C$.\nSince $G_0$ acts on its standard pro-$2$ tree\nwith finite vertex stabilizers, so does $H$.\nThe main result in\n\\cite{HZ:10}\nshows that $H$ does not decompose as the\nfundamental group of a profinite graph of finite $2$-groups.\nIts proof also shows that $H$ does not decompose as\nan amalgamated free pro-$p$ product or\nas a pro-$p$ {\\rm HNN}-extension over a finite group.\n\\end{example}\n\n\n\n\\section{\\texorpdfstring{$2$}{2}-generated subgroups}\n\\label{s:2-generated}\n\nThe final section is devoted to the proof of\nTheorem \\ref{t:freeorabelian_intro}.\nSo, henceforth, $G:=A\\amalg_{C} B$ is a free pro-$p$ product of\n$A$ and $B$ with procyclic amalgamating subgroup $C$\nsatisfying the following assumptions:\n\\begin{itemize}\n \\item[{\\rm (i)}]\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n\n \\item[{\\rm (ii)}]\n  each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group.\n\\end{itemize}\n\n\\begin{lemma}\\label{l:cent}\nFor every subgroup $D\\le C$ we have $N_G(D)=C_G(D)$.\n\\end{lemma}\n\n\\begin{proof}\nBy the pro-$p$ version of \\cite[Cor. 2.7(ii)]{RZ:96},\n\\begin{equation*}\nN_{G}(D)=N_{A}(D)\\amalg_{C}N_{B}(D)\\, .\n\\end{equation*}\nSince solvable $2$-generated subgroups of $A$ and $B$ are abelian,\n$N_{A}(D)=C_{A}(D)$ and $N_{B}(D)=C_{B}(D)$;\nhence $N_G(D)=\\gp{C_{A}(D),C_{B}(D)}\\subseteq C_G(D)$,\nas needed.\n\\end{proof}\n\n\\begin{thm}\\label{t:freeorabelian}\n  Let $G=A\\amalg_{C} B$ be a free pro-$p$ product of\n  $A$ and $B$ with procyclic amalgamating subgroup $C$.\n  Suppose that\n  the centralizer in $G$ of each non-trivial closed subgroup of $C$ is\n  a free abelian pro-$p$ group and contains $C$ as a direct factor.\n  If each $2$-generated pro-$p$ subgroup of $A$\n  and each $2$-generated pro-$p$ subgroup of $B$\n  is either a free pro-$p$ group or a free abelian pro-$p$ group\n  then so is each $2$-generated pro-$p$ subgroup of $G$.\n\\end{thm}\n\n\\begin{proof}\nLet $T$ be the standard pro-$p$ tree on which $G$ acts\n({\\it cf.} \\cite[Sec. 4]{RZ:00a})\nand let $L$ be a $2$-generated pro-$p$ subgroup of $G$.\nIt follows from the definition of $T$ that if\n$L$ stabilizes a vertex of $T$, then $L$ is up to conjugation\nin one of the free factors of $G$; hence $L$ is either free pro-$p$ or\nfree abelian pro-$p$, by hypothesis (ii).\n\nLet us assume that $L$ fixes no vertex of $T$.\nSince $L$ is finitely generated, we have \n$L\\cong \\varprojlim L\/U_n$ where\n$\\{U_n\\mid n\\in{\\mathbb N}\\}$ is a set of open normal subgroups of\n$L$ with $\\bigcap U_n =1$.\nRecall our notation $\\widetilde{U_n}$\nfor the closed subgroup of $U_n$ generated by all vertex\nstabilizers with respect to the action of $U_n$ on $T$.\nWe consider the infinite set $I$ of integers $n$ such that\n$U_n\/\\widetilde{U_n}$ is an infinite free pro-$p$ group\n({\\it cf.} Theorem \\ref{t:trees1}(b)).\nSo, defining $L_n\\!:=L\/\\widetilde{U_n}$ we see that\n$L_n$ acts virtually freely on a pro-$p$ tree $T\/\\widetilde{U_n}$\n({\\it cf.} Theorem \\ref{t:trees1}(a))\nand so we are in position to apply\nTheorem \\ref{t-fund}\nto each of them.\nThus $L_n=\\Pi_1(\\cL_n,\\Gamma_n)$ is the fundamental pro-$p$ group of\na finite graph of finite $p$-groups whose edge and vertex groups\nare stabilizers of certain edges and vertices of $T\/\\tilde U_n$.\nClearly we have $L\\cong \\varprojlim \\{L_n, \\varphi_{nm}, I\\}$\nwhere each $\\varphi_{nm}$ is the canonical map.\n\nNow, since $L\/\\widetilde{L}$ is a free pro-$p$ group\nof rank at most $2$,\nwe need to consider only the two cases\n$L=\\widetilde{L}$ and $L\/\\widetilde{L}\\cong {\\mathbb Z}_p$;\nin the remaining case, when $d(L\/\\widetilde L)=2$,\n$L$ is itself free pro-$p$ of rank 2 -- by the Hopfian property.\nWe can assume that $\\widetilde{L}\\neq 1$, otherwise\nthere is nothing to prove.\n\n\\medskip\n\\noindent {\\it Case 1}.  $L=\\widetilde{L}$.\n\n\\medskip\n\nWe claim that $\\Gamma_n$ is a tree.\nIf not then there is an edge $e\\in \\Gamma_n$\nso that $L_n={\\rm HNN}(P_n,G(e),t)$ for $G(e)$ finite.\nBut then there is a homomorphism from $L_n$ onto ${\\mathbb Z}_p$\ncontradicting $L_n=\\torgp{L_n}$.\n\nThen in light of\nLemma \\ref{l:2gen}(c) and of Proposition \\ref{p:treeprod},\nwe have inverse systems of conjugates of $K_n$ and $D_n$; \nfollowing the notation of the referred Proposition, we define\ntwo procyclic groups $K\\!:=\\varprojlim K_n'$ and $D \\!:=\n\\varprojlim D_n''$.\n\nWe claim that $D=1$.\nNote that since each $D_n$ is an edge stabilizer\nwith respect to the $L_n$-action, we have\n$D=L\\cap C^g$, for some $g\\in G$. \nSince $C_L(D) = L\\cap C_G(D)$, it follows from (i) that \n$C^g$ is a direct factor of $C_G(D)$,\nhence $D$ is a direct factor of $C_L(D)$.\nSuppose on the contrary that $D\\neq 1$. \nSince the procyclic group $K$ contains $D$,\nit follows that $D=K$. \nNow, the projection $K\\to K_{n_0}'$ is\nsurjective for some sufficiently large $n_0$, by\nLemma \\ref{l:invsys}(a).\nHence $D_{n_0}=K_{n_0}$; a contradiction to the\nnon-fictitious decomposition of $L_{n_0}$.\n\nThus $D=1$, and so\n$\\varprojlim {D_n''}^{L_n} =1$, by\nLemma \\ref{l:invsys}(b).\nThen, writing $L_n\\cong K_n'\\amalg_{D_n''} R_n'$\nwe have\n$L\\cong \\varprojlim L_{n}\/{D_n''}^{L_n}\n\\cong \\varprojlim (K_n'\/D_n'' \\amalg R_n'\/{D_n''}^{R_n'})$.\nNow, if $d(L_n\/D_n''^{L_n})=1$ for every $n$, then\n$L$ is procyclic; thus\nwithout loss of generality we may and do assume that each\n$L_{n}\/{D_n''}^{L_n}$ is $2$-generated.\nSince $K_n'\/D_n''$ is $1$-generated,\nso is $R_n'\/{D_n''}^{R_n'}$.\nTherefore $L\\cong \\mathbb{Z}_p\\amalg \\mathbb{Z}_p\\,$,\nby Proposition \\ref{p:decompfp}.\nOur proof is finished for {\\it Case 1}.\n\n\n\\medskip\n\\noindent {\\it Case 2}. $L\/\\widetilde{L}\\cong {\\mathbb Z}_p$.\n\n\\medskip\n\nFor $n\\in \\mathbb N$ we have ${\\mathbb Z}_p\\cong\nL\/\\widetilde L\\cong L_n\/(\\widetilde L\/\\widetilde U_n)$\nand therefore $\\Gamma_n$ cannot be a tree.\nThen we can select a suitable edge $e_n$ of $\\Gamma_n$,\nset $\\Delta_n:=\\Gamma_n\\,-\\,\\{e_n\\}$, and present\n$L_n={\\rm HNN}(K_n,D_n,t_n)$\nwith cyclic edge group $D_n$ of $e_n$ and\n$K_n=\\Pi_1({{\\mathcal G}_n}\\restr{\\Delta_n},\\Delta_n)$.\n\nSince $\\widetilde{L}\/\\widetilde{U_n}$ is generated by torsion,\nas a consequence of\nTheorem \\ref{t:hnnproperties}(a), it is contained in ${K_n}^{L_n}$;\nso, ${\\langle {\\rm tor}(L_n) \\rangle}={K_n}^{L_n}$.\nBy \\cite[Prop. 1.7(ii)]{Zalesskii:04},\n$K_n\/{\\langle {\\rm tor}(K_n)\\rangle}$ is a free pro-$p$ group,\nwhence ${\\langle {\\rm tor}(L_n) \\rangle}$ has trivial image\nin the quotient ${\\rm HNN}(K_n\/{\\langle {\\rm tor}(K_n)\\rangle},1, t_n)$\nof $L_n$.\nThus $K_n={\\langle {\\rm tor}(K_n)\\rangle}$.\nSince $K_n$ acts on the pro-$p$ tree $T\/\\widetilde{U_n}$\nwe have $K_n=\\widetilde{K_n}$\n({\\it cf.} Theorem \\ref{t:trees1}(d)),\nso in particular, $\\Delta_n$ must be a tree.\nPassing now to a cofinal subset of $\\mathbb N$,\nif necessary, we may assume that for all $n$ either\n$\\Delta_n$ is a single vertex or $\\Delta_n$ contains an edge.\nWe discuss the two subcases.\n\n\\medskip\n\\noindent {\\it Subcase 2($\\alpha$).\nFor each $n\\in\\mathbb N$ the tree $\\Delta_n$ is a single vertex.}\n\n\\medskip\n\nThen $K_n$ is finite.\nPassing again to a cofinal subset of $\\mathbb N$, if necessary,\nwe can, making use of\nTheorem \\ref{t:hnnproperties}(a)\nand a projective limit argument, arrange that\n$\\varphi_{n+1n}(K_{n+1})\\le K_n$ holds for all $n$.\nPassing again to a cofinal subset of $\\mathbb N$, if necessary,\nand making use of\nLemma \\ref{l:2gen}(b)\nwe can arrange that for all $n$ either\n$K_n$ is cyclic or that $d(K_n)=2$.\nWe shall discuss the situations when $K\\!:=\\varprojlim K_n$\nis procyclic and when $d(K)=2$.\n\nIf $K$ is procyclic, then for every $m$ there exists $n>m$\nsuch that the $\\varphi_{nm}(K_n)$ is cyclic and so\n$\\varphi_{nm}(D_n)=\\varphi_{nm}(D_n^{t_n})$.\nHence $\\varphi_{nm}(t_n)$ normalizes $\\varphi_{nm}(D_n)$\nand so $L_m=N_{L_m}(\\varphi_{nm}(D_n))$.\nSince $L=\\varprojlim L_m$ it follows that\n$D\\!:=\\varprojlim D_m$ is normal in $L$.\nSince $E(T)$ is a compact $L$-space, setting in\nLemma \\ref{l:fp}\n$X\\!:=E(T)$, $G\\!:=L$, and $S_n\\!:=D_n$,\nwe find $e\\in E(T)$ with $D\\le G_e$.\nTherefore $D^g\\le C$ for some $g\\in G$ and,\nif $D\\neq 1$, making use of \nLemma \\ref{l:cent},\nwe find that $L\\cong {\\mathbb Z}_p\\times {\\mathbb Z}_p$\nby hypothesis (i), as needed.\n\nNext assume that $D=1$.\nIt follows from\nLemma \\ref{l:invsys}(b)\nthat $\\varprojlim D_m^{L_m}=1$ and\nso $L=\\varprojlim L_m\/D_m^{L_m}$.\nObserving that\n$L_m\/D_m^{L_m}\n=(K_m\/K_m\\cap D_m^{L_m})\\amalg \\langle t_m\\rangle$\nProposition \\ref{p:decompfp}\nimplies that $L\\cong {\\mathbb Z}_p\\amalg {\\mathbb Z}_p$,\nwhence the result when $K$ is procyclic.\n\nFor finishing Subcase 2($\\alpha)$\nwe can now assume that $d(K)=2$.\nThen\nLemma \\ref{l:invsys}(a)\nin conjunction with a projective limit argument implies that\n$\\varphi_{n+1 n}(K_{n+1})\\cong K_n$ for every $n$.\nBy virtue of\nProposition \\ref{p:decomphnn},\nwe have inverse systems of conjugates $K_n'$ and $D_n''$\nof the finite $p$-groups $K_n$ and\n$D_n$, and $L ={\\rm HNN}(K,D,t)$ where $K\\!:=\\varprojlim K_n'$ and\n$D\\!:=\\varprojlim D_n''$ is procyclic.\nWe must have $D\\neq 1$,\nelse $L\\cong K\\amalg \\gp t$, and so $2=d(K)=d(L)-1=1$;\na contradiction.\n\nAn application of Lemma \\ref{l:fp} shows that\n$K$ stabilizes a vertex in $T$; it is therefore, up to conjugation,\ncontained in either $A$ or $B$ and so by hypothesis (ii) is either\nfree pro-$p$ or free abelian pro-$p$.\nIn the first case we observe that\nLemma \\ref{l:2gen}(b)\nimplies that $K=D\\amalg D^t$ and so\n$L=D\\amalg \\langle t\\rangle$ is a free pro-$p$ group.\n\nSo assume in the sequel that\n$K$ is a free pro-$p$ abelian group.\nNote that $L={\\rm HNN}(K,D,t)$ contains\n$H\\!:=K\\amalg_DK^t$ which is not abelian.\nOn the other hand since $E(T)$ is a compact $L$-space, setting in\nLemma \\ref{l:fp}\n$X\\!:=E(T)$, $G\\!:=L$, and $S_n\\!:=D_n$\nwe find $e\\in E(T)$ with $D\\le G_e$.\nHence $D\\le C^g$ for suitable $g\\in G$.\nSince $D\\le C^g$, by hypothesis (i) $C_G(D)$ is abelian,\nand it contains $H$; a contradiction.\nHence we are done with Subcase 2($\\alpha$).\n\n\n\\medskip\n\n\\noindent {\\it Subcase 2($\\beta$).\nFor each $n\\in\\mathbb N$ the tree $\\Delta_n$ contains an edge.}\n\n\\medskip\n\nLemma \\ref{l:2gen}(c)  and Proposition \\ref{p:treeprod}\nimply that $K_n$ can be written as\n$K_n=X_n\\amalg_{Z_n}W_n$, with cyclic $p$-groups $X_n$,\nand there are inverse systems $\\{X'_n\\}$ and $\\{Z''_n\\}$\nwith $Z''_n\\le X'_n$ of conjugates of $X_n$ and $Z_n$\nrespectively.\nDefine procyclic groups $X=\\varprojlim X_n'$ and\n$Z=\\varprojlim Z_n''$.\nWe must have $Z\\neq X$ else by\nLemma \\ref{l:invsys}(a) we could find $n$ with $Z_n=X_n$\nand so the decomposition $K_n=X_n\\amalg_{Z_n}W_n$\nwould be fictitious; a contradiction.\nSetting in\nLemma \\ref{l:fp}\n$X:=E(T)$, $G:=L$, and $S_n:=Z_n''$\nwe find $e\\in E(T)$ with $Z\\le G_e$.\nHence there is $g\\in G$ with $Z\\le C^g$.\nNow, since $Z\\neq X$, \nhypothesis (i) implies $Z=1$.\nLet $\\bar K_n=K_n\/{Z_n}^{K_n}$\nand let $\\bar D_n$ be the canonical image of $D_n$ in $\\bar K_n$.\nThen, we consider\n\\begin{equation*}\n\\bar L_n \\!=\\! L_n\/{Z_n}^{L_n} \\!=\\!\n\\text{\\rm HNN}(\\bar K_n, \\bar D_n, \\bar t_n) \\!=\\! \\text{\\rm HNN}\n(X_n\/{Z_n}^{X_n}\\amalg W_n\/{Z_n}^{W_n}, \\bar D_n, \\bar t_n)\n\\, .\n\\end{equation*}\nBy Lemma \\ref{l:2gen}(b),\neach pro-$p$ group $K_n$ is at most $2$-generated,\nhence considering $\\bar L_n$ modulo its Frattini subgroup,\nwe can conclude that $d(W_n\/{Z_n}^{W_n})=1$.\nSo, taking into account\nLemma \\ref{l:2gen}(b) we conclude that\n$X_n\/{Z_n}^{X_n}$ and $W_n\/{Z_n}^{W_n}$ are isomorphic\ncyclic $p$-groups.\nThus $\\bar L_n\\cong X_n\/{Z_n}^{X_n} \\amalg\n\\langle\\bar t_n\\rangle$.\nBy Lemma \\ref{l:invsys}(b) and Proposition \\ref{p:decompfp}\nwe obtain that $L\\cong \\varprojlim\n\\bar L_n \\cong {\\mathbb Z}_p\\amalg {\\mathbb Z}_p$.\nThis concludes the proof of the theorem.\n\\end{proof}\n\n\\begin{cor}\\label{c:2-free}\nSuppose that neither $A$ nor $B$ contains a\n$2$-generated non-procyclic abelian subgroup.\nThen any $2$-generated subgroup $L$ of $G$\nis a free pro-$p$ group.\n\\end{cor}\n\n\\begin{proof}\nSuppose that $L$ is a free abelian pro-$p$ group of rank $2$.\n\nLet $T$ be the standard pro-$p$ tree on which $G$ acts.\nThen by\n\\cite[Thm. 3.18]{RZ:00a}\neither $L$ stabilizes a vertex or there\nis an edge $e$ of $T$ such that $L\/L_e\\cong{\\mathbb Z}_p$.\nBut $L$ cannot stabilize a vertex; else it is conjugate to \na subgroup of one of the free factors of $G$,\ncontradicting the supposition.\n\nTherefore $L\/L_e\\cong {\\mathbb Z}_p$ for some edge $e$.\nSince $d(L)=2$ we must have $L_e \\not= 1$.\nConjugating $L$ by some element of $G$\nwe may assume that $L_e$ is contained in $C$.\nThen, $L=N_G(L_e)=C_G(L_e)$, by\nLemma \\ref{l:cent}(a), and,\nby the centralizer condition of the theorem,\n$L=C\\cong {\\mathbb Z}_p$, a contradiction.\nThus, by\nTheorem \\ref{t:freeorabelian},\n$L$ must be free pro-$p$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe central prediction of the Weakly Interacting Massive Particles (WIMP)  paradigm is that Dark Matter (DM) particles should have a thermally averaged \nannihilation cross section of $\\langle \\sigma v \\rangle \\sim 10^{-26} \\, \\mathrm{cm^3 \\, s^{-1}}$ during freeze-out. \nIn many DM models, the present-day annihilation cross section in astrophysical systems is predicted to be of a similar magnitude, \nproviding a clear target for indirect detection experiments searching for the products of DM annihilation processes.\n\nWhile the most robust constraints on the DM annihilation cross section stem from observations of the {CMB~\\cite{Aghanim:2018eyx} and of the} $\\gamma$-ray sky, \nin particular from Fermi-LAT~\\cite{Ackermann:2013yva,Ackermann:2015zua,Ackermann:2015lka}, highly complementary information\ncan be obtained by precisely measuring the flux \nof charged anti-particles arriving on Earth. Very recently, AMS-02 has released results \nfrom the first seven years of data taking~\\cite{Aguilar:2021tos}, which include in particular the flux of antiprotons with unprecedented precision. \nTheoretical predictions for this flux however require detailed modelling of the production and propagation of \ncharged cosmic rays (CRs) in the Galaxy, which are subject to significant uncertainties {and are currently constrained using CR data (see e.g.\\ Ref.~\\cite{Korsmeier:2021brc}), as well as their non-thermal emissions (see e.g.\\ Ref.~\\cite{Orlando:2017mvd}).} \n\nWhile various numerical codes, such as \\textsc{Galprop}~\\cite{Strong:1998fr} and \\textsc{Dragon}~\\cite{Evoli:2008dv}, exist to address this challenge and simulate the propagation of CRs, they require as input a large number of parameters that need to be varied to assess their impact on the predictions. \nAs a result these simulations are typically computationally so expensive that they become prohibitive in the context \nof a global analysis of DM models, where also the fundamental model parameters need to be varied~\\cite{Ambrogi:2018jqj}. Recent analyses of the AMS-02 antiproton data have therefore typically focused on simplified DM models with only a single annihilation channel, see e.g.\\ Ref.~\\cite{Cuoco:2016eej, Cui:2016ppb, Reinert:2017aga,Cholis:2019ejx}. \n\n\nIn the present work we explore the potential of artificial neural networks (ANNs) to solve this issue and substantially speed up \nthe calculation of predictions for the primary antiproton flux for a very broad range of DM models.\\footnote{For other recent works on the use of machine learning for cosmic ray propagation in the context of DM we refer to Refs.~\\cite{Lin:2019ljc,Tsai:2020vcx}.} Specifically, we employ recurrent neural networks (RNNs), which are particularly well suited for the prediction of continuous spectra. \nThe network is trained on a large sample of antiproton fluxes based on propagation parameters that are chosen to broadly agree with recent AMS-02 data, and a general parametrisation of the properties of the DM particle in terms of its mass and the branching fractions for various different final states. Using the same approach we have also developed and trained ANNs to accurately predict further CR species, like secondary antiprotons, protons or helium.\n\nThe predictions of the network can then be used to calculate the likelihood of the AMS-02 data for a given DM model and varying propagation parameters in order to calculate exclusion limits using a range of frequentist or Bayesian methods.  However, it is important to ensure that the resulting constraints are not biased by regions of parameter space for which the ANN has not been sufficiently trained. In the Bayesian approach this potential pitfall is avoided by evaluating the likelihood for a fixed sample of propagation parameter points drawn from the posterior probability distribution in the absence of a DM signal. The marginalisation over propagation uncertainties can then be performed via importance sampling, i.e.\\ by appropriately reweighing and combining the points in the sample. This approach is particularly well suited for the analysis of antiproton data, since the propagation parameters are rather \ntightly constrained by the proton flux and the secondary antiproton flux, so that the presence of a DM signal \ndoes not dramatically shift the relevant regions of parameter space. \n\nWe emphasise that, while the initial generation of a sample from the posterior is computationally expensive, it does not require specific assumptions on the properties of DM and therefore only needs to be carried out once in advance. Moreover, the same simulation step can be used to set up the training data for the ANN, ensuring that the network is trained specifically on the most interesting regions of parameter space. Once training is completed, the remaining steps are computationally cheap and can be performed for a large number of DM parameters. Indeed, the full marginalisation over propagation parameters can be performed in a similar amount of time as it would take to \nsimulate a single parameter point in the conventional approach.\n\nWe apply our fully trained ANN to a number of cases of particular interest. For the case of DM annihilations exclusively into bottom quarks we show that the most recent AMS-02 data leads to results that are compatible with previous studies. \nIn particular, we recover a notable excess for DM masses around 100 GeV in the case that no correlations in the AMS-02 data are considered. \nWe also present new constraints on the well-studied model of scalar singlet DM and find that antiproton data places competitive constraints on this model. \nHowever, we emphasise  that the ANN is not limited to these cases and can be applied to a wide variety of DM models. \nMoreover, the general approach that we present can be extended to consider different propagation models (provided a suitable simulator exists), systematic uncertainties (such as correlations in the AMS-02 data) or cross-section uncertainties, enabling the community to fully explore the wealth of the available CR data.\n\nThe remainder of this work is structured as follows. In section~\\ref{sec:cr} we briefly review the fundamental concepts of CR production and propagation and present the specific implementation that we adopt in the present work. We also carry out a first analysis of the most recent AMS-02 data and perform a parameter scan to identify the most interesting regions of parameter space. In section~\\ref{sec:ANN} we introduce our machine learning approach to simulating CRs and discuss how we train and validate our ANNs. Finally, in section~\\ref{sec:constraints} we apply the fully trained ANNs to constrain DM models. We present the relevant statistical methods and discuss the resulting exclusion limits.\n\n\\section{Cosmic-ray antiprotons in the Galaxy}\\label{sec:cr}\n\nFor the following discussion it is useful to distinguish between primary and secondary CRs. Primary CRs are directly accelerated and emitted by standard astrophysical sources \nlike supernova remnants or pulsars. But also more exotic scenarios such as the production of (anti)particles by DM annihilation or decay are considered as primary origin. \nProtons provide the dominant contribution to primary CRs (about 90\\%) while helium (He) makes up about 10\\%. Heavier nuclei only contribute at the percent level.\nOn the other hand, secondary CRs are produced during the propagation of primary CRs by fragmentation or decay.\nMore specifically, when the primary CRs interact with the gas in the Galactic disc, commonly called interstellar medium (ISM), secondary particles are produced.\nBecause of the different production mechanism, secondaries are suppressed with respect to primary CRs.\nIt is commonly believed that CR antiprotons do not have standard astrophysical sources%\n\\footnote{\n\tWe note that the possibility of primary antiprotons that are directly produced and accelerated at supernova remnants\n\t\\cite{Blasi:2009bd,Mertsch:2014poa,Mertsch:2020dcy, Kohri:2015mga} is also discussed in literature. \n}\nsuch that their dominant contribution comes from secondary production. As a consequence, antiprotons are \nsuppressed by 4--5 orders of magnitude with respect to protons, which makes them (together with other antimatter CRs, e.g.\\ antideuterons \\cite{Aramaki:2015pii,vonDoetinchem:2020vbj}) \na promising channel for constraining DM signals. \n\nIn this section we first discuss the production of antiprotons in the annihilation of dark matter particles in our Galaxy, followed by a discussion of backgrounds from secondary antiprotons. We then present the framework that we use to simulate CR propagation and the strategy to fit the resulting spectra to data. Finally, we perform a scan over the propagation parameters in order to create the training set for the machine learning approach introduced in section~\\ref{sec:ANN}.\n\n\\subsection{Antiprotons from dark matter annihilation}\\label{sec::antipdm}\n\nCR antiprotons are a long standing target used to search for signals of WIMP DM in our Galaxy\n\\cite{Bergstrom:1999jc,Donato:2003xg,Bringmann:2006im,Donato:2008jk,Fornengo:2013xda,Evoli:2011id,Bringmann:2014lpa,Pettorino:2014sua,Cirelli:2014lwa,Cembranos:2014wza,Hooper:2014ysa,Boudaud:2014qra,Giesen:2015ufa,Evoli:2015vaa,Johannesson:2016rlh,Luque:2021ddh,DiMauro:2021qcf}.\nMore recently, there has been a discussion of an antiproton excess at about 20~GeV, which could be fitted with a \nprimary DM source~\\cite{Cuoco:2016eej, Cui:2016ppb, Reinert:2017aga, Cuoco:2019kuu, Cholis:2019ejx}. \nHowever, the excess might also be accounted for by a combination of systematic effects~\\cite{Boudaud:2019efq, Heisig:2020nse, Heisig:2020jvs}.\nIf DM particles annihilate into standard model particle final states $f$ within the diffusion halo of our Galaxy as ${\\rm DM}\\!+\\!{\\rm DM} \\to f\\!+\\!\\bar{f}$, \nwe expect a corresponding flux contribution to antiprotons in CRs, coming from the subsequent decay of for example $q \\!+\\!\\bar{q}$ modes (see e.g.\\ \\cite{Cirelli:2010xx}). \nThe source term of this primary antiproton component,  $q_{\\bar{p}}^{(\\mathrm{DM})}$, is a function of the Galactic coordinates $\\bm{x}$ and the antiproton kinetic energy $E_\\mathrm{kin}$. For a generic combination of standard model final states $f$ it reads: \n\\begin{eqnarray}\n  \\label{eqn::pbar_DM_source_term}\n  q_{\\bar{p}}^{(\\mathrm{DM})}(\\bm{x}, E_\\mathrm{kin}) =\n      \\frac{1}{2} \\left( \\frac{\\rho(\\bm{x})}{m_\\mathrm{DM}}\\right)^2  \n      \\sum_f \\left\\langle \\sigma v \\right\\rangle_f \\frac{\\mathrm{d} N^f_{\\bar{p}}}{\\mathrm{d} E_\\mathrm{kin}} \\; .\n\\end{eqnarray}\nThe factor $1\/2$ in eq.~\\eqref{eqn::pbar_DM_source_term} corresponds to Majorana fermion DM.\nFurthermore, $m_\\mathrm{DM}$ is the DM mass, $\\rho(\\bm{x})$ the DM halo energy density profile, and $\\langle \\sigma v \\rangle_f$ is the thermally \naveraged annihilation cross section for the individual final states $f$. In the following, we fix $\\langle \\sigma v \\rangle$ independent of $f$ and account for this by assigning branching fractions into the relevant final states.\nFinally, $\\mathrm{d} N^f_{\\bar{p}}\/\\mathrm{d} E_\\mathrm{kin}$ denotes the energy spectrum of antiprotons for a single DM annihilation. \nThis quantity depends on the DM mass and the standard model final state. Here we implement the  widely used tabulated results for the antiproton energy spectrum \npresented in Ref.~\\cite{Cirelli:2010xx} which include electroweak corrections.%\n\\footnote{\n\tIf DM annihilates into a pair of $W$ or $Z$ bosons it is possible to produce one of them off-shell. This possibility is not taken \n\tinto account in the original tables. We extend the tables of $W$ and $Z$ bosons to lower DM masses using the tables from Ref.~\\cite{Cuoco:2017rxb}.\n}\n\n\nWe assume that the DM density in our Galaxy follows an NFW profile~\\cite{Navarro:1995iw}  \n$\\rho_{\\mathrm{NFW}}(r) = \\rho_h \\, r_h\/r\\, \\left( 1 + r\/r_h \\right)^{-2}$, \nwith a scale radius of $r_h=20\\;$kpc and a characteristic halo density, $\\rho_h$, which is normalised such that the local \nDM density at the solar position of $8.5\\;$kpc is fixed to $0.43\\;$GeV\/cm$^3$~\\cite{Salucci:2010qr}, compatible also with more recent estimates \\cite{deSalas:2020hbh}.\nWe note that the NFW profile is only one of many viable DM profiles currently investigated. Other profiles \ncan have a significantly different behavior towards the Galactic center, see e.g.\\ the discussion in Ref.~\\cite{Benito:2019ngh}.\nHowever, we stress that choosing a different DM density profile only has a small impact on the results presented in this paper  \nsince CR antiprotons from DM annihilation dominantly arrive from the local environment. \nTherefore they are mostly sensitive to the local DM density and the resulting flux depends only weakly on the shape of the DM density profile at \nthe Galactic center. \nMore specifically, the impact of changing the cuspy NFW profile to the cored Burkert profile~\\cite{Burkert:1995yz} has been quantified \nin Ref.~\\cite{Cuoco:2017iax}; it was found that a core radius of $5\\;$kpc only weakens DM limits by about 20\\%. \n\n\n\n\\subsection{Secondary antiprotons}\\label{sec::sec}\n\nThe ISM consists of roughly  90\\% hydrogen (H) and 10\\% He. Thus secondary antiprotons are mostly produced by the \ninteraction of $p$ and He CRs with the H and He components of the ISM. \nThe source term for the secondary antiprotons $q_{\\bar p}^{(\\mathrm{sec})}$ is thus given by the convolution of the primary CR fluxes $\\phi$ of isotope $i$, \nthe ISM density $n_{\\mathrm{ISM}}$ of component $j \\in \\lbrace \\mathrm{H}, \\mathrm{He} \\rbrace$, and the energy-differential \nproduction cross section $\\mathrm{d}\\sigma_{ij\\rightarrow\\bar p}\/\\mathrm{d}  E_{\\mathrm{kin},\\bar{p}}$:\n\\begin{eqnarray}\n\t\\label{eqn::pbar_sec_source_term}\n\tq_{\\bar p}^{(\\mathrm{sec})}({\\bm x},E_{\\mathrm{kin},\\bar{p}}) &=&  \n\t                                \\!\\!\\!\\!\\sum\\limits_{j \\in \\lbrace \\mathrm{H}, \\mathrm{He} \\rbrace} \\!\\!\\!\\! 4\\pi \\,n_{\\mathrm{ISM},j}({\\bm x}) \n\t                                \\sum\\limits_{i} \n\t                                \\int\n\t                                \\mathrm{d} E_{\\mathrm{kin},i} \\,\n                                    \\phi_i  ( E_{\\mathrm{kin},i}) \\, \n                                    \\frac{\\mathrm{d}\\sigma_{ij\\rightarrow\\bar p}}{\\mathrm{d}  E_{\\mathrm{kin},\\bar{p}} }(E_{\\mathrm{kin},i} ,  E_{\\mathrm{kin},\\bar{p}} )\\,.\n\\end{eqnarray}\nBy construction, secondaries are suppressed with respect to primary CRs. In the case of antiprotons, the experimentally \nobserved suppression compared to protons is {5 orders of magnitude at 1~GV and increases to about 4 orders of magnitude above 10~GV}.\nSince secondary CRs constitute the dominant contribution of the measured antiproton flux, \nconsidering standard astrophysical sources only already results in a good fit to the data \\cite{Korsmeier:2016kha, Cuoco:2016eej, Boudaud:2019efq}, \nsee also  discussion in section~\\ref{sec:fitams}.\n\nThe cross section of secondary antiproton production is a very important ingredient of eq.~\\eqref{eqn::pbar_sec_source_term}, which has been discussed \nby various groups recently~\\cite{diMauro:2014zea,Winkler:2017xor,Korsmeier:2018gcy,Kachelriess:2019ifk}. \nIn general there are two different strategies to determine this cross section. On the one hand, Monte Carlo generators, \nwhich are tuned to the relevant cross section data~\\cite{Kachelriess:2019ifk}, can be used to infer the relevant cross section.\nOn the other hand, a parametrisation of the Lorentz invariant cross section can be fitted to all available cross section data. \nThen the required energy-differential cross section is obtained by an angular integration~\\cite{diMauro:2014zea,Winkler:2017xor,Korsmeier:2018gcy}. \nWe follow the second approach and use the analytic cross section parametrisation from Ref.~\\cite{Winkler:2017xor} with the updated parameters from Ref.~\\cite{Korsmeier:2018gcy}. \nAn important advantage of the analytic cross section parametrisation is that it is explicitly tuned to cross-section data at low energies, and therefore more reliable \nbelow antiproton energies of $\\sim 10$~GeV as discussed in Ref.~\\cite{Donato:2017ywo}.\n\nFinally, we consider that secondary antiprotons may scatter inelastically with the ISM and lose energy. This antiproton contribution, commonly referred to as tertiary \\cite{Moskalenko:2001ya},\nis suppressed with respect to the secondaries.\n\n\n\\subsection{Propagation in the Galaxy and solar modulation}\\label{sec::prop}\nThe sources, acceleration and propagation of Galactic CRs are research topics by themselves \\cite{Amato:2017dbs, Gabici:2019jvz}. \nFast evolution and progresses has been driven in the last years by newly available and very precise data by \nAMS-02 \\cite{Aguilar:2021tos}, PAMELA \\cite{Adriani:2014xoa} and Voyager \\cite{2013Sci...341..150S}. \nSome recent developments include the studies of systematic uncertainties from solar modulation, correlated experimental data points, \nsecondary production\/fragmentation cross sections as well as detailed studies of propagation phenomena below a rigidity of 10 GV to disentangle \ndiffusion and reacceleration \\cite{Genolini:2019ewc,Evoli:2019wwu,Evoli:2019iih,Boschini:2018baj,Boschini:2019gow,Weinrich:2020cmw,Weinrich:2020ftb,Luque:2021nxb,Luque:2021joz,Schroer:2021ojh, Korsmeier:2021brc}, {where the rigidity $R$ of a CR particle \nis defined as its momentum divided by the absolute value of its charge}.\nHere we will not explore these exciting directions and instead focus on one standard setup of CR propagation, which was already studied in the context DM searches with antiprotons in Ref.~\\cite{Cuoco:2019kuu}. The machine learning approach and the statistical methods introduced below can however be readily applied also to alternative assumptions and more refined descriptions. We briefly summarise below the main ingredients  of this specific approach and refer to Ref.~\\cite{Cuoco:2019kuu} for a more detailed discussion.\n\n\\medskip\n\nCharged CRs propagate within a diffusion halo assumed to be cylindrically symmetric, which extends a few kpc above and below the Galactic plane. \nIn particular, it has a fixed radial extent of 20~kpc, while the \nhalf height of the diffusion halo is denoted by $z_\\mathrm{h}$ and typically enters CRs fits as a free parameters (see section~\\ref{sec:fitams}).\nWhen CRs cross the boundary of the diffusion halo they escape from the Galaxy, while the propagation within the halo is described by a chain of coupled diffusion equations.\n\nThe diffusion equation for the {CR number density per volume and absolute momentum} \n$\\psi_i (\\bm{x}, p, t)$ of CR species $i$ at position $\\bm{x}$ and momentum $p$  is given by~\\cite{StrongMoskalenko_CR_rewview_2007}:\n\\begin{eqnarray}\n  \\label{eqn::propagationEquation}\n  \\frac{\\partial \\psi_i (\\bm{x}, p, t)}{\\partial t} = \n    q_i(\\bm{x}, p) &+&  \n    \\bm{\\nabla} \\cdot \\left(  D_{xx} \\bm{\\nabla} \\psi_i - \\bm{V} \\psi_i \\right)  \\\\ \\nonumber\n     &+&  \\frac{\\partial}{\\partial p} p^2 D_{pp} \\frac{\\partial}{\\partial p} \\frac{1}{p^2} \\psi_i - \n    \\frac{\\partial}{\\partial p} \\left( \\frac{\\mathrm{d} p}{\\mathrm{d} t} \\psi_i  \n    - \\frac{p}{3} (\\bm{\\nabla \\cdot V}) \\psi_i \\right) -\n    \\frac{1}{\\tau_{f,i}} \\psi_i - \\frac{1}{\\tau_{r,i}} \\psi_i \\; .\n\\end{eqnarray}\nWe briefly describe each of the terms in eq.~\\eqref{eqn::propagationEquation} below. \nTo solve these equations numerically we employ \\textsc{Galprop}~56.0.2870~\\cite{Strong:1998fr,Strong:2015zva} and \\textsc{Galtoollibs}~855\\footnote{https:\/\/galprop.stanford.edu\/download.php} \n\twith a few custom modification as described in Ref.~\\cite{Cuoco:2019kuu}. Alternatively, solutions might be obtained analytically, utilizing various simplifying assumption \\cite{Putze:2010zn,Maurin:2018rmm}, or using other fully numerically codes like \\textsc{Dragon}~\\cite{Evoli:2008dv,Evoli:2017vim} or \\textsc{Picard}~\\cite{Kissmann:2014sia}.\n\\textsc{Galprop} assumes that CRs are in a steady state and solves the diffusion equations on a 3-dimensional grid.\nTwo dimensions describe the spatial distribution of CRs, the radial distance $r$ from the Galactic center and distance $z$ perpendicular to the plane, and \none dimension contains the CR's kinetic energy. The grid points of the spatial dimensions are spaced linearly with step size of $\\Delta r = 1$\\;kpc \nand $\\Delta z = 0.1$\\;kpc, respectively, while the grid is spaced logarithmically in kinetic energy with a ratio between successive grid points of 1.4.\n\nThe source term $q_i$ in eq.~(\\ref{eqn::propagationEquation}) depends on the CR species. For secondary antiprotons and antiprotons from DM annihilation it takes the form of \neq.~\\eqref{eqn::pbar_sec_source_term} and eq.~\\eqref{eqn::pbar_DM_source_term}, respectively. \nFor primary CRs the source term factorizes into a spatial and a rigidity-dependent term. \nThe spatial term traces the distribution of supernova remnants.%\n\\footnote{\n    We use the default prescription of \\textsc{Galprop} where the parameters of the\n    source term distribution are fixed to $\\alpha = 0.5$, $\\beta=2.2$, $r_s=8.5$~kpc, and $z_0=0.2$~kpc.\n    This is slightly different from recent values in the literature \\cite{Green:2015isa}.\n    We note, however, that nuclei are only very weakly sensitive to the chosen distribution as discussed in Ref.~\\cite{Korsmeier:2016kha}.\n} \nOn the other hand, the rigidity dependence is modeled as a smoothly broken power-law:\n\\begin{eqnarray}\\label{eq::psp}\n    \\label{eqn::SourceTerm_2}\n    q_R(R)     &=&   \\left( \\frac{R}{R_0} \\right)^{-\\gamma_1}\n                     \\left( \\frac{R_0^{1\/s}+R^{1\/s} }\n                                 {2\\,R_0^{1\/s}      } \\right)^{-s (\\gamma_2-\\gamma_1)},\n\\end{eqnarray}\nwhere $R_0$ is the break position and  $\\gamma_{1,i}$ and $\\gamma_{2,i}$ are the\nspectral indices above and below the break for the CR species $i$, respectively.\nThe parameter $s$ regulates the amount of smoothing at the break.\n{\n     In the following analysis we will assume that all primary nuclei except for protons have a \n     universal injection spectrum such that we adopt $\\gamma_{1,i}=\\gamma_{1}$ and $\\gamma_{2,i}=\\gamma_{2}$. \n     For protons we allow different spectral behaviour and keep the subscript $i=p$.}\nThe broken power-law spectrum in eq.~(\\ref{eq::psp}) is a widely used  phenomenological approximation which describes well the data in the considered rigidity range. \nAll CR species are affected by several processes that contribute to CR propagation, which are diffusion, reacceleration, convection, and energy losses.\nWe assume that diffusion is spatially homogeneous and isotropic. In this case, the diffusion coefficient, $D_{xx}$, \ncan be modeled as a broken power-law in rigidity\n\\begin{eqnarray}\n    \\label{eqn::diffusionConstant}\n    D_{xx} &=&\n        \\begin{cases} \n       \t    \\beta D_{0} \\left( \\frac{R}{4 \\, \\mathrm{GV}} \\right)^{\\delta} &\\text{if}\\; R<R_1\\\\\n       \t    \\beta D_{0} \\left( \\frac{R_1}{4 \\, \\mathrm{GV}} \\right)^{\\delta} \\left( \\frac{R}{R_1} \\right)^{\\delta_h} &\\text{otherwise}\\,,\n        \\end{cases}\n\\end{eqnarray}\nwhere $D_0$ is an overall normalisation and $\\delta$ and $\\delta_h$ are the power-law indices below and above the break at position $R_1$.\nAt low energies the diffusion coefficient is proportional to the velocity $\\beta=v\/c$ of the CRs.\nWe allow for a diffusive reacceleration of CRs by scattering off Alfv\\`en magnetic waves. The amount of reacceleration \nis then determined by the velocity $v_\\mathrm{Alfven}$  of the waves  \\cite{Ginzburg:1990sk,1994ApJ...431..705S}:\n\\begin{eqnarray}\n    \\label{eqn::DiffusivReaccelerationConstant}\n    D_{pp} = \\frac{4 \\left(p \\, v_\\mathrm{Alfven} \\right)^2 }{3(2-\\delta)(2+\\delta)(4-\\delta)\\, \\delta \\, D_{xx}} \\; .\n\\end{eqnarray}\n\nThe terms proportional $\\bm{V}(\\bm{x})$ in eq.~(\\ref{eqn::propagationEquation}) describe  convective winds which drive the CRs away from the Galactic plane. They are taken constant and orthogonal \nto the Galactic plane, such that $\\bm{V}(\\bm{x})= {\\rm sign}(z)\\, v_{0,{\\rm c}}\\,{\\bm e}_z$.\nThe remaining terms describe different contributions of energy losses, for which we adopt the \ndefault \\textsc{Galprop} implementation. In particular, continuous energy losses like ionisation or bremsstrahlung are \nincluded in the term $\\mathrm{d} p\/\\mathrm{d} t$, while catastrophic energy losses by fragmentation or decay \nare modeled by the last two terms. The parameters $\\tau_f$ and $\\tau_r$ are the corresponding lifetimes. \n\nWe emphasise again that the setup of CR propagation described above reflects one specific choice. The available measurements of \nCR nuclei are also described well by other setups. In particular, a model without diffusive re-acceleration,\nbut with an additional break in the diffusion coefficient between 5 and 10\\;GeV \nis currently discussed in the literature \\cite{Genolini:2019ewc,Weinrich:2020cmw,Korsmeier:2021brc}. \n\n\\medskip\n\nCRs measured at the top of the atmosphere  have to traverse a significant part of the heliosphere where they\nare deflected and decelerated by solar winds. The strength of this effect varies in a 22-year cycle and is commonly \nknown as solar modulation. It mostly affects low-energetic CRs; in practice the impact on the spectra above a few tens of \nGV is negligible. We use the common force-field approximation~\\cite{Fisk:1976aw} to model the impact on the CR spectra:\n\\begin{eqnarray}\n    \\label{eqn::solarModulation}\n    \\phi_{\\oplus,i}(E_{\\oplus,i}) &=& \\frac{E_{\\oplus,i}^2 - m_i^2}{E_{\\text{LIS}, i}^2 - m_i^2} \\phi_{\\text{LIS}, i}(E_{\\text{LIS}, i}) \\,,\\\\\n    E_{\\oplus,i} &=& E_{\\text{LIS}, i} - e|Z_i|\\varphi_{i}\\,.\n\\end{eqnarray} \n{Here $\\phi$ and $E$ label the energy-differential flux and the kinetic energy, respectively. \nThe subscripts on the energy or flux denote the position which can either be local interstellar (LIS) or top of the atmosphere ($\\oplus$). \nFurthermore, }$Z_i$ is the charge number, $e$ is the elementary charge, and $\\varphi_{i}$ is the solar modulation potential. \nThe potential is known to be time and charge-sign dependent.\nWe note that the force-field approximation is probably an oversimplified treatment of solar modulation.%\n\\footnote{\n    In more sophisticated models solar modulation is described by propagation equations similar to eq.~\\eqref{eqn::propagationEquation} but tuned to the \n    environment of the heliosphere. These are typically solved numerically~\\cite{Kappl:2015hxv,Vittino:2017fuh,Boschini:2017gic,Fiandrini:2020puf,Ngobeni:2020quz}.\n}\nTo minimise systematic impacts from solar modulation on our results we will exclude data below 5~GV from our analysis. Furthermore, \nwe allow a different solar modulation potential for antiprotons  to account for a possible charge-sign dependence.  \n\n\\subsection{Fit to AMS-02 data}\\label{sec:fitams}\n\nIn the following we summarise very briefly the considered data sets and the fit strategies, where the latter are directly adopted from Ref.~\\cite{Cuoco:2019kuu}. \nThe most precise measurement of CR antiprotons above 1~GV is currently provided by the AMS-02 experiment \\cite{Aguilar:2021tos}. \nWe consider the data sets of proton, helium, and {the antiproton-to-proton ratio} from AMS-02~\\cite{Aguilar:2021tos} collected over 7 years from 2011 to \n2018 and complement with low-energy data for protons and helium from Voyager~\\cite{Cummings:2016pdr}.\nWhen fitting the CR data with the model outlined below, the CR likelihood is defined by \n\\begin{eqnarray}\n  \t\\label{eqn::likelihood_CR}\n\t-2\\,\\log{{\\cal L}_{{\\rm CR}} }(\\bm\\theta) =  \\chi^2_{\\rm CR}(\\bm\\theta) =  \\sum\\limits_{e,s,i} \t\n\t\t\t\t\t\t\t\\left(\\frac{\\phi^{(e)}_{{ s},i}- \\phi^{(\\text{m})}_{s,i,e}(\\bm \\theta)}\n\t\t\t\t\t\t\t           {\\sigma^{\\left(e\\right)}_{s,i}}\\right)^2 \\,,\n\\end{eqnarray}\nwhere $\\phi^{(e)}_{{s},i}$ denotes the flux of the CR species $s$ that was measured by the experiment $e$\nat the rigidity $R_i$ or energy $E_{\\mathrm{kin},i}$, \nwhile $\\phi^{(\\text{m})}_{s,i,e}(\\theta)$ is the flux computed with \\textsc{Galprop}  for the corresponding species and energy. \nFinally, $\\sigma^{\\left(e\\right)}_{s,i}$ is the experimental uncertainty of the flux measurement.\nThe AMS-02 experiment provides separate statistical and systematic uncertainties. Here we assume that \nthe systematic uncertainties are uncorrelated and add the two contribution in quadrature. This \nis certainly a simplified treatment.\nIn particular, it was shown that the significance of a potential excess can depend critically on the assumptions made for  the correlations of this\nuncertainty~\\cite{Boudaud:2019efq,Heisig:2020nse}. \nHowever, we expect that the impact on DM limits is less severe,\n{\n    because of two opposing effects: The covariance matrices as modeled in Refs.~\\cite{Boudaud:2019efq,Heisig:2020nse}\n    contain contributions with both large and small correlation lengths. A large correlation length corresponds \n    to a change of the overall normalisation which looks very different from a peaked signature \n    such as expected from DM annihilation. This potentially leads to \n    a stronger DM limit. On the other hand, a small correlation length allows for signatures similar to those from DM and, therefore, potentially weakens the limit.\n    Overall, we expect these two effects to partly cancel each other. \n}\nWe leave the study of different systematics and correlations within the new methods introduced in this paper to future investigation.\n\n\nWithin the phenomenological description of CR injection and propagation outlined in section~\\ref{sec::prop}, the parameters of \neq.~\\eqref{eqn::propagationEquation} are largely unconstrained a priori and are directly inferred from CR data. \nWe allow for a total of 15 parameters to describe CR injection and propagation. To sample this large parameter space efficiently we use the Nested Sampling algorithm implemented in the \\textsc{MultiNest} code~\\cite{Feroz:2008xx}. The computing efficiency is increased even further by exploiting a hybrid strategy where only a subset of parameters \tis sampled by \\textsc{MultiNest} (``slow parameters'') and the remaining parameters are profiled on-the-fly (``fast parameters''). \tThe slow parameters are the ones that are needed as input for \\textsc{Galprop} and thus changing them is time consuming. More specifically, these are the following eleven parameters: the slopes of the primary injection spectra $\\gamma_{1,p}$, $\\gamma_{1}$, $\\gamma_{2,p}$, \nand $\\gamma_{2}$, the break position $R_0$ and its smoothing $s$, the normalisation $D_0$ and slope\n$\\delta$ of the diffusion coefficient, the half-height of the diffusion halo $z_\\mathrm{h}$, and the velocities of\nconvection $v_{0c}$ and Alfv{\\`e}n magnetic waves  $v_\\text{Alfv{\\`e}n}$.\nThe scan ranges for all of these parameters are summarised in table~\\ref{tab:param_ranges_v2}.\n{In the following we will give results in the frequentist and the Bayesian interpretation. \nFor the Bayesian interpretation we assume flat priors in the scan ranges.}\n\nThe four remaining parameters describe the normalisation of the proton ($A_p$) and helium ($A_\\mathrm{He}$) fluxes and the solar modulation potentials ($\\varphi_{\\text{AMS-02},p,{\\rm He}}$ for $p$ and He and $\\varphi_{\\text{AMS-02},{\\bar{p}}}$ for  ${\\bar{p}}$). These are the fast parameters, which are treated in a simplified way in our analysis and therefore can be varied much more easily.\nInstead of explicitly including them in the \\textsc{MultiNest} parameter scans, we profile over them on-the-fly at each likelihood evaluation of \\textsc{MultiNest}, i.e.\\ we maximise the likelihood over the fast parameters using \\textsc{Minuit}~\\cite{James:1975dr}. A very weak Gaussian prior is applied to $\\varphi_{\\text{AMS-02},{\\bar{p}}}$ by adding to the main likelihood the term\n$-2\\,\\log({{\\cal L}_{{\\rm SM}} }) = (\\varphi_{\\text{AMS-02},p,{\\rm He}}-\\varphi_{\\text{AMS-02},{\\bar{p}}})^2 \/ \\sigma_\\varphi^2$\nwhere $\\sigma_\\varphi = 100$ MV,%\n\\footnote{\n\t {The prior expresses that the solar modulation potential of antiprotons \n     and the one of protons and helium are related, even if they are not forced to be the same.\n     In Ref.~\\cite{Vittino:2019yme} the average difference between the potential of electrons and positrons \n     was found to be around 100 MV.}\n}\nwhile no priors are applied on $\\varphi_{\\text{AMS-02},p,{\\rm He}}$. \n\nWe truncate the rigidity range of the fit to the range between 5 to 300\\;GV. \nAs mentioned above, data below 5\\;GV is \nexcluded to avoid a strong bias from our modeling of solar modulation.%\n\\footnote{\n    It was shown in Ref.~\\cite{Cuoco:2019kuu} that the cut at $R=5$\\;GV does not artificially enhance the significance of a potential DM signal.  \n}\nAt high energies, the spectra of CR nuclei show a break at $R\\sim300$\\;GV, which is more pronounced \nin secondaries with respect to primaries~\\cite{Aguilar:2018njt,Aguilar:2017hno}.\nWhile in general it would be possible to introduce spectral breaks in the injection spectrum or in the diffusion coefficient, \nonly the latter naturally explains the different behavior of the primaries and secondaries~\\cite{Genolini:2017dfb}. We therefore \nfix the parameters of eq.~\\eqref{eqn::diffusionConstant} to $R_1=300$ GV and $\\delta_h=\\delta - 0.12$. The proton and helium data \nof AMS-02 is described well by this choice. Truncating our fit at $R\\sim300$\\;GV avoids unnecessary bias.%\n\\footnote{\n    As an alternative, $R_1$ and $\\delta_h-\\delta$ could be treated at free parameters in the fit which would, however, increase the \n    complexity of the already high-dimensional parameter fit. \n}\n\n\\begin{table}[t]\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\caption{\n        Results of the CR fits to AMS-02 and Voyager data of protons, helium, and antiprotons. The parameter ranges for the \\textsc{MultiNest} scan are stated in column 2. \n        In the remaining columns we state the best-fit parameter values and their uncertainty at the 68\\% C.L. for a fit with and without a DM signal. \n        Results are given both in the frequentist and the Bayesian interpretation. \n        \\label{tab:param_ranges_v2}\n\t }\n\t\\begin{tabular}{lccccc}\n\t\t\\hline\n\t\t\\hline\n\t\t                                                                    &               &  \\multicolumn{2}{c}{Frequentist}                                                        & \\multicolumn{2}{c}{Bayesian}                                                            \\\\\n\t\tParameter                                                           &  Scan ranges  & w\/o DM                                     & w\/ DM                                      & w\/o DM                                     & w\/ DM                                      \\\\\n\t\t\\hline\n\t\t$\\gamma_{1,p}$                                                      & $[1.2, 2]   $ & $                  {1.80}^{+0.04}_{-0.03}$ & $                  {1.79}^{+0.07}_{-0.06}$ & $                  {1.77}^{+0.07}_{-0.04}$ & $                  {1.68}^{+0.14}_{-0.07}$ \\\\\n\t\t$\\gamma_{1}$                                                        & $[1.2, 2]   $ & $                  {1.79}^{+0.04}_{-0.04}$ & $                  {1.74}^{+0.08}_{-0.06}$ & $                  {1.75}^{+0.07}_{-0.04}$ & $                  {1.63}^{+0.15}_{-0.07}$ \\\\\n\t\t$\\gamma_{2,p}$                                                      & $[2.3, 2.6] $ & $               {2.405}^{+0.013}_{-0.007}$ & $                  {2.48}^{+0.02}_{-0.03}$ & $                  {2.41}^{+0.01}_{-0.01}$ & $                  {2.48}^{+0.02}_{-0.03}$ \\\\\n\t\t$\\gamma_{2}$                                                        & $[2.3, 2.6] $ & $               {2.357}^{+0.014}_{-0.005}$ & $                  {2.42}^{+0.02}_{-0.03}$ & $               {2.366}^{+0.009}_{-0.012}$ & $                  {2.42}^{+0.02}_{-0.02}$ \\\\\n\t\t$R_{0}\\,\\mathrm{[ 10^{3}\\;MV]}$                                     & $[1, 20]    $ & $                  {7.92}^{+0.82}_{-0.80}$ & $                  {7.32}^{+1.16}_{-0.83}$ & $                  {7.06}^{+0.93}_{-1.04}$ & $                  {6.42}^{+0.97}_{-1.13}$ \\\\\n\t\t$s_{}$                                                              & $[0.1, 0.9] $ & $                  {0.37}^{+0.03}_{-0.03}$ & $                  {0.40}^{+0.03}_{-0.04}$ & $                  {0.38}^{+0.04}_{-0.04}$ & $                  {0.44}^{+0.04}_{-0.06}$ \\\\\n\t\t$D_{0}\\,\\mathrm{[ 10^{28}\\;cm^2\/s]}$                                & $[0.5, 10]  $ & $                  {2.05}^{+1.48}_{-0.39}$ & $                  {2.92}^{+2.09}_{-0.96}$ & $                  {3.58}^{+1.30}_{-0.73}$ & $                  {5.37}^{+1.52}_{-1.78}$ \\\\\n\t\t$\\delta$                                                      & $[0.2, 0.6] $ & $               {0.419}^{+0.009}_{-0.012}$ & $                  {0.35}^{+0.03}_{-0.02}$ & $                  {0.42}^{+0.01}_{-0.01}$ & $                  {0.33}^{+0.03}_{-0.03}$ \\\\\n\t\t$v_\\mathrm{Alfven}\\,\\mathrm{[km\/s]}$                                & $[0, 30]    $ & $                  {8.84}^{+1.45}_{-2.58}$ & $                 {10.25}^{+2.12}_{-2.06}$ & $                  {6.02}^{+3.57}_{-2.51}$ & $                  {7.70}^{+4.15}_{-3.10}$ \\\\\n\t\t$v_{0,\\mathrm{c}}\\,\\mathrm{[km\/s]}$                                 & $[0, 60]    $ & $                  {0.09}^{+1.08}_{-0.08}$ & $                  {0.90}^{+6.77}_{-0.78}$ & $                  {2.48}^{+0.32}_{-2.48}$ & $                {13.36}^{+2.44}_{-13.36}$ \\\\\n\t\t$z_\\mathrm{h}\\,\\mathrm{[kpc]}$                                      & $[2, 7]     $ & $                  {2.60}^{+2.25}_{-0.48}$ & $                  {2.79}^{+2.87}_{-0.75}$ & $                  {4.70}^{+2.30}_{-0.86}$ & $                  {4.84}^{+2.13}_{-0.75}$ \\\\\n\t\t$\\log_{10}(m_\\mathrm{DM}\/\\mathrm{MeV})$                          & $[4, 7]     $ & -                                          & $                  {5.07}^{+0.03}_{-0.05}$ & -                                          & $                  {5.08}^{+0.04}_{-0.05}$ \\\\\n\t\t$\\log_{10}( {\\langle} \\sigma v {\\rangle} \\mathrm{s\/cm^{3}})$     & $[-27,-22]  $ & -                                          & $                {-25.42}^{+0.22}_{-0.48}$ & -                                          & $                {-25.76}^{+0.13}_{-0.26}$ \\\\\n\t\t$\\varphi_{\\mathrm{AMS-02,pHe}}\\,\\mathrm{[GV]}$                      &               & $                  {0.26}^{+0.04}_{-0.03}$ & $                  {0.25}^{+0.05}_{-0.03}$ & $                  {0.30}^{+0.04}_{-0.05}$ & $                  {0.28}^{+0.04}_{-0.06}$ \\\\\n\t\t$(\\varphi_{\\bar p} - \\varphi_{p})_\\mathrm{AMS-02} \\,\\mathrm{[GV]}$  &               & $               {0.200}^{+0.000}_{-0.036}$ & $                  {0.13}^{+0.07}_{-0.12}$ & $               {0.177}^{+0.023}_{-0.001}$ & $                  {0.09}^{+0.11}_{-0.03}$ \\\\\n\t\t$A_{\\mathrm{p,AMS-02}}$                                             &               & $               {1.173}^{+0.004}_{-0.003}$ & $               {1.173}^{+0.003}_{-0.004}$ & $               {1.178}^{+0.004}_{-0.004}$ & $               {1.177}^{+0.004}_{-0.004}$ \\\\\n\t\t$A_{\\mathrm{He,AMS-02}}$                                            &               & $               {1.257}^{+0.006}_{-0.014}$ & $                  {1.20}^{+0.02}_{-0.01}$ & $               {1.253}^{+0.010}_{-0.010}$ & $                  {1.20}^{+0.02}_{-0.02}$ \\\\\n\t\t$\\chi^2_{\\mathrm{p,AMS-02}}$                                        &               & $                                     7.2$ & $                                     6.2$ \\\\\n\t\t$\\chi^2_{\\mathrm{He,AMS-02}}$                                       &               & $                                     3.2$ & $                                     2.1$ \\\\\n\t\t$\\chi^2_{\\mathrm{pbar\/p,AMS-02}}$                                   &               & $                                    35.0$ & $                                    21.5$ \\\\\n\t\t$\\chi^2_{\\mathrm{p,Voyager}}$                                       &               & $                                     7.9$ & $                                     4.1$ \\\\\n\t\t$\\chi^2_{\\mathrm{He,Voyager}}$                                      &               & $                                     3.9$ & $                                     3.2$ \\\\\n\t\t$\\chi^2$                                                            &               & $                                    57.2$ & $                                    37.1$ \\\\\n\t\t\\hline \\hline\n\t\\end{tabular}\n\t\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\nTo gain a first understanding of the allowed regions of parameter space, we perform a CR fit as detailed above. The scan is conducted using 1000 live points, a stopping criterion of \\textsc{tol=0.1} and an enlargement factor \\textsc{efr=0.7}. \nThe final efficiency of the scan is found to be around 9\\%, \nwith about 350\\,000\nlikelihood evaluations in total.\n     The fit is heavily parallelised using 96 cores at the same time. The \\textsc{Galprop} code is parallelised using \\textsc{openMP} while the nested sampling algorithm of\n     \\textsc{MultiNest} can be expanded to multiple \\textsc{MPI} tasks. We follow a hybrid strategy with 24 \\textsc{MPI} tasks using 4 cores for each task. \n     We have verified that the parallelisation efficiency lies above 70\\%. \n     In total the fit requires about 5.5 days to converge and consumes 12500 cpu hours, which means that a single likelihood evaluation requires on average about 130 cpu seconds. \nTo perform this fit, \\textsc{MultiNest} starts by broadly sampling the entire parameter space and then continuously shrinks to the allowed parameters. The result is an ensemble of parameter points which is denser in the most interesting parameter region. We will make use of this property in the following section, where our goal is to train the ANNs in such a way that they perform particularly well in the parameter range preferred by data. Thus, we save all the sample points during the fit and then use them as a starting point for the training in section~\\ref{sec:ANN}.\n\nIn table~\\ref{tab:param_ranges_v2} we summarise the best-fit (most probable) values of the various parameters based on a frequentist (Bayesian) interpretation as well as their 68\\% confidence intervals (credible intervals).\nThe best-fit point corresponds to $\\chi^2 = 57.2$ for the AMS-02 data. \nThese results broadly agree with the ones from Ref.~\\cite{Cuoco:2019kuu} even though we use the more recent 7-year AMS-02 data for $p$, He, and $\\bar p$.\nAs expected, for the parameters that are well-constrained by data, there is good agreement between the frequentist and the Bayesian approach. For less constrained parameters (such as for example $z_\\mathrm{h}$) there can be sizeable differences between the best-fit point (obtained by maximizing the profile likelihood) and the most probable point (obtained by maximizing the marginalised likelihood). We will return to this issue in section~\\ref{sec:constraints}.\n\nPrevious analysis have discussed a potential DM signal that could be accommodated at antiproton energies between 10 and 20 GeV where the antiproton flux shows a small anomaly at the level of a few percent. This potential signal corresponds, for example, to DM particles with a mass of about 80~GeV that self-annihilate into $b\\bar{b}$ final states at a thermal cross section. However, the significance of this potential signal has been discussed controversially in the literature. The most recent works suggest that the anomaly is well explained by the combination of several systematic uncertainties, namely uncertainties in the secondary antiproton production cross section, correlated systematics in the AMS-02 data, and some additional \nfreedom in the CR propagation model \\cite{Boudaud:2019efq, Heisig:2020nse, Heisig:2020jvs}, which we do not include here. \n\nThe focus of this work lies instead on developing new methods for exploiting ANNs and importance sampling to derive DM limits. In contrast to a DM signal, we expect the limits to be only weakly dependent on those systematics and leave their investigation to future studies. Nevertheless, for comparison we also perform one fit where antiprotons from DM annihilation are included. We choose DM annihilation into a pair of $b\\bar b$ quarks as our benchmark. In this case,\ntwo further parameters are considered, the mass of the DM particle $m_{\\rm DM}$ and the thermally averaged annihilation cross section $\\langle \\sigma v \\rangle$ (see eq.~\\eqref{eqn::pbar_sec_source_term}). We explore values of $m_{\\rm DM}$ from 10\\;GeV to 10\\;TeV and values of $\\langle \\sigma v \\rangle$ between $10^{-22}$ and $10^{-27}\\;\\mathrm{cm^3\/s}$, with our results being independent of the precise choice of these ranges.\nTheses two additional parameters are sampled with \\textsc{MultiNest}. \n\nThe results of the additional fit are also shown in table~\\ref{tab:param_ranges_v2}.\nIncluding a DM signal formally improves the $\\chi^2$ by 20.1 which, however, \ngiven the discussion above should not be interpreted as significant. Nonetheless, we can take this value as a point of \ncomparison for the performance of the ANN in section~\\ref{sec:constraints}. \n{We furthermore observe that, while the additional DM signal affects most CR propagation parameters only marginally, there is a sizeable shift of the preferred parameter regions for $\\gamma_{2}$, $\\gamma_{2,p}$ and $\\delta$. While this shift is likely overestimated in our analysis for the reasons mentioned earlier, it highlights the challenges for the training of the ANN (see section~\\ref{sec:ANN}) and for the statistical inference via importance sampling (see section~\\ref{sec:constraints}).}\n\nIn the previous paragraph, we focused on a specific case of DM annihilation into a pair of bottom quarks which serves as an example and a point of comparison. In general,\nmuch more complex scenarios with a range of different final states and combinations at different branching fractions are possible. The naive approach to obtain\nresults would be to perform an entirely new parameter scan for each case of interest which obviously requires a substantial amount of computational resources. \nInstead, in the following we will discuss methods to speed up the calculation of CR spectra in a model-independent fashion to quickly obtain constraints \nfor any given DM model. \n\n\n\\section{Deep neural network setup and training}\n\\label{sec:ANN}\n\nOur aim is to predict the output of \\textsc{Galprop} for a wide range of input parameters representing both uncertainties in the propagation model and the unknown properties of DM. This output can then be used to calculate experimental likelihoods as described in section~\\ref{sec:fitams} without computationally expensive simulations. To achieve this goal, we build and train suitable ANNs and validate their performance. Considering the two different contributions to the antiproton flux (i.e.\\ primary and secondary CRs), we construct two separate ANNs to provide fast predictions of each component based on the relevant physical parameters. \nWe will refer to the networks for the DM component and the secondary component as DMNet and sNet, respectively.\nAs the underlying method in the development of the neural networks is the same, both ANNs will be presented in parallel in this section. \n\n\\subsection{Training Set}\n\\label{sec:training_set}\n\nThe information that a neural network should be able to learn, in general, has to be represented in the data that is used to train the network. This allows for the interpolation of data within the parameter space that would, in a conventional approach, require new simulations. To remain impartial on the specific parameters of the DM model, we consider a wide range in the mass of the DM particle from 5 GeV to 5 TeV and randomly sample from a logarithmic distribution in this range. A similar approach is taken for the branching fractions, where we consider all SM final states that give a non-negligible contribution to a CR antiproton flux~\\cite{Cirelli:2010xx}: $q\\bar{q}$, $c\\bar{c}$, $b\\bar{b}$, $t\\bar{t}$, $W^+W^-$, $ZZ$, $hh$ and $gg$. We logarithmically sample each branching fractions in the range $[10^{-5}, 1]$ and then normalise the result in such a way that the sum of all  branching fractions equals one. The DM annihilation cross section is fixed to $\\langle \\sigma v \\rangle = 3 \\times 10^{-26} \\, \\text{cm}^3 \\, \\text{s}^{-1}$ in the complete training set, as variations in this parameter can be included at a later stage by an appropriate rescaling of the flux. \nThese DM parameters, which we will collectively denote by $\\mathbf{x}_\\text{DM}$, are only relevant to the DM component of the antiproton flux and the corresponding neural network, while the secondary flux is independent of $\\mathbf{x}_\\text{DM}$ and hence these parameters will not be used as inputs to the sNet. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = \\textwidth,clip,trim=5 0 5 0]{.\/figures\/2dhist_training_data_v2.png}\n\t\\caption{\\textit{Triangle plot:} One and two dimensional histograms {showing the frequency} of propagation parameters used in the training set, constructed in such a way that the highest density is achieved in the regions most favoured by the combination of AMS-02 proton, antiproton and helium data without DM signal. \\textit{Top right:} One dimensional histogram of the training set for each of the branching fractions {$\\chi \\chi \\rightarrow$ SM SM}.}\n\t\\label{img:training_data_distibution}\n\\end{figure}\n\nFor the propagation parameters we face the significant challenge that the parameter space introduced in section~\\ref{sec:cr} is 11-dimensional and that only a very small volume of this parameter space gives an acceptable fit to AMS-02 data. If we were to simply perform a grid scan or draw random samples from this parameter space, we would include large regions of parameter space for which accurate predictions are unnecessary, as they will anyways be strongly excluded. Conversely, in the preferred regions of parameter space, we want to achieve an accuracy that is significantly better than the typical relative errors of about 5\\% in AMS-02 data, which requires large amounts of training data.\n\nTo obtain sufficiently accurate network predictions in the most interesting regions of parameter space without spending unnecessary time on the simulation and training of less interesting regions, we want to make use of the AMS-02 data already for the creation of the training set. Indeed, we can directly use the \\textsc{MultiNest} scan described in section~\\ref{sec:fitams} to obtain a sample of propagation parameters (denoted by $\\bm{\\theta}_\\text{prop}$ in the following) that is focused on the regions of parameter space with the highest likelihood (see also Ref.~\\cite{Coccaro:2019lgs}). {Since in the following we will be most interested in the calculation of exclusion limits, we will base our training on the \\textsc{MultiNest} scan without DM signal. For a detailed investigation of the excess the same procedure outlined below could be applied to the sample of propagation parameters from the \\textsc{MultiNest} scan with DM signal.}\n\nFor the creation of the training set we exclude any parameter point in the \\textsc{MultiNest} sample that gives a poor fit to AMS-02 data, specifically with $\\Delta \\chi^2 \\ge 30$ compared to the best-fit point. This results in a total of 117335 remaining parameter points which we show in figure~\\ref{img:training_data_distibution}. We emphasise that for each parameter the training data extends well beyond the 68\\% confidence\/credible intervals {without DM annihilations quoted in table~\\ref{tab:param_ranges_v2}}. To ensure a sufficiently good coverage also of the DM parameter space, we sample 8 combinations of DM parameters for each propagation parameter point, leading to a very large simulation set of $\\mathcal{O} (10^6)$ parameter points. \n\n\\subsection{Neural Network Architectures}\n\\label{sec:architectures}\n\nAlthough the two networks that we use to predict the two components of the antiproton flux can be set up and trained in a similar way, we face distinct challenges in each component. For the DMNet the key challenge is the very large number of input parameters, namely the DM mass plus 8 branching fractions in $\\mathbf{x}_\\text{DM}$ and a total of 11 propagation parameters in $\\bm{\\theta}_\\text{prop}$, each with a different physical effect on the output, i.e.\\ the antiproton flux. As we want to have accurate predictions for variations in each of the parameters, we treat the DM mass, the branching fractions, and the propagation parameters as three distinct sets of inputs, which are first processed by three independent dense networks before combining the outputs (see below).\n\nFor the sNet the key challenge is to achieve sufficient accuracy in the prediction of the secondary antiprotons flux, which is tightly constrained by AMS-02 data. Given these constraints, the secondary antiproton flux only exhibits relatively small variations across the training set, which nevertheless need to be accurately captured using the sNet. To achieve the desired accuracy, we provide an increased number of trainable parameters that define the network.\nAs we will show within the following sections, the training duration consequently increases with respect to the DMNet but a very good accuracy is achieved.\n\nRather than directly feeding the physical parameters as inputs to the network, we map the logarithm of $\\mathbf{x}_\\text{DM}$ to values in the range $\\left[0, 1\\right]$ and the remaining parameters $\\bm{\\theta}_\\text{prop}$ to a distribution with a mean of 0 and a standard deviation of 1. Each of the networks is then trained in a supervised approach. The simulated fluxes serve as training labels or `true' fluxes to which the network output can be compared.\n\nGiven the large variations in the CR fluxes that are desired as the output of the ANNs, here we choose a natural scaling of the original (simulated) flux $\\Phi ( E )$ for the sNet outputs, \n\\begin{equation}\n\t\\tilde{\\Phi}_\\text{s} (E) = \\log_{10} \\left( \\Phi(E) \\, E^{2.7} \\right) \\, .\n\t\\label{eq:trafo}\n\\end{equation}\nThe $\\log_{10}$ further decreases the variations in the flux values, which would otherwise cover several orders of magnitude.\nThe energies and their respective fluxes are binned values, identical to the output from the simulations, which extend over the energy range of the AMS-02 antiproton measurement. Consequently, we have sequences of distinct values in the scaled flux as training labels. The transformation in eq.~(\\ref{eq:trafo}) is easily invertible and thus allows for direct comparison of the network output to the simulated spectra.\n\nAs the DM component of the flux predominantly scales with the DM mass, we choose a different scaling for that flux component, \n\\begin{equation}\n\t\\tilde{\\Phi}_\\text{DM} (x) = \\log_{10} \\left( m_\\text{DM}^3 \\, x \\, \\Phi(E) \\right) \\, ,\n\t\\label{eq:trafo_DM}\n\\end{equation}\nwhere $x = E\/m_\\text{DM}$ is a dimensionless quantity. We use a grid in $x$ with 40 points logarithmically spaced in the interval $[10^{-3.7},1]$, on which we evaluate the training labels and DMNet output. The advantage of this scaling compared to eq.~\\eqref{eq:trafo} is that it substantially reduces the impact of changing the DM mass and therefore leads to much less variation across the training set.\\footnote{To first approximation the DM component of the antiproton flux follows the source term $\n\\Phi_{\\overline{p}, \\text{DM}} \\left(E \\right) \\propto q_\\text{DM} \\propto m_\\text{DM}^{-2} \\, \\mathrm{d} N\/\\mathrm{d} E \\propto m_\\text{DM}^{-3} x^{-1} \\, \\mathrm{d} N\/ \\mathrm{d} \\log_{10} x$, where $\\mathrm{d} N \/ \\mathrm{d} \\log_{10} x$ depends only very mildly on $m_\\text{DM}$.\n}\nThis is illustrated in figure~\\ref{img:training_outputs}, which shows the resulting DM antiproton fluxes $\\tilde{\\Phi}_\\text{DM}$ as a function of $x$ for a representative set of final state combinations and DM masses in the training set. We find that for each combination of input parameters we obtain a slowly-varying function of $x$ that reaches a maximum and then drops towards $x \\to 1$. The general trend is similar across the entire range of DM masses that we consider, but some information on the DM mass is retained. We find that this approach significantly improves the training of the DMNet compared to the scaling in eq.~\\eqref{eq:trafo}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.7\\textwidth]{.\/figures\/Output_trafo.png}\n\t\\caption{Transformed DM antiproton fluxes following eq.~\\eqref{eq:trafo_DM} for our training set, which varies propagation parameters, branching fractions and DM masses as discussed in sec.~\\ref{sec:training_set}. The modest amount of variation across different parameter points results in a more easily processable version of the input \\textsc{GALPROP} simulated flux for the DMNet.} \n\t\\label{img:training_outputs}\n\\end{figure}\n\nSubsequent to the pre-processing of the input, the ANNs contain densely connected ('dense') layers, that process the information from the inputs. To address the individual challenges for the networks we set up the architecture as depicted in figure~\\ref{img:architecture}. We provide dense layers for each of the different inputs in the DMNet which are concatenated in the next step and followed by large dense layers. In the sNet the pre-processed input is fed through a more intricate set of dense layer, specifically (56, 28, 14, 28, 56) nodes in the set of layers. We use ReLU activations and add a small dropout probability of 0.1\\% between the layers. The precise values of these hyperparameters do not significantly affect the training performance.\n\nThe main feature of each of the networks is a recurrent layer. The choice to work with a recurrent setup instead of other network architecture types has lead to significant improvements in the architecture development process. Even though the typical application for RNNs is time-series data, we find our spectra as functions of energy to be handled just as well by this network type. In particular, it can be reasoned that the information on the flux that is contained in a specific energy bin is highly correlated with the prior and subsequent energy bins and a network architecture that is able to propagate the information of neighbouring units is very beneficial for the task at hand.\nWe chose a GRU layer as proposed in Ref.~\\cite{DNN:gru} in the DMNet and a LSTM layer following \\cite{DNN:lstm} in the sNet. Each of these layer types is useful for long data sequences and far-reaching information propagation without leading to vanishing or exploding gradients during training. While both methods can in principle be used for either networks, the final implementations that achieved the best results was based on different layer types.\nAs network output a final dense layer is set up. We build the networks using the deep learning API \\textsc{Keras} \\cite{DNN:keras} which uses \\textsc{Tensorflow} \\cite{DNN:tensorflow} as backend. \n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 1\\textwidth]{.\/figures\/Archtecture_sketch_both_simple.pdf}\n\t\t\\begin{tabular}{ll}\n            \\hline \\hline\n\t\t\t\\multicolumn{2}{l}{\\textbf{Hyperparameters}} \\\\\n\t\t\t\\hline\n\t\t\tActivation & ReLU \\\\ \n\t\t\tDropout fraction & 0.1 \\% \\\\ \n\t\t\tOptimizer & Adam, learning rate scheduling $l \\in [10^{-2}, 10^{-5}]$, patience 10 epochs \\\\ \n\t\t\tLoss & Mean squared error (MSE) \\\\\n\t\t\tBatch size & 500 \\\\ \n\t\t\tValidation split & 20~\\% \\\\\n\t\t\tEarly stopping & Monitor val. loss, patience = 40 \\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\caption{Schematic of the network structure. \\textit{Top left:} Architecture set up for handling the complete set of inputs. This network type can be used to be trained on the DM component of the $\\overline{p}$ flux (DMNet). \\textit{Top right:} Simplified architecture for networks that require only the CR propagation parameters as input. This network architecture is designed for learning the $\\overline{p}_\\text{secondary}$ fluxes (sNet) and can be employed to train on proton and helium spectra as well (see appendix~\\ref{app:p_and_He}). \\textit{Bottom:} The hyperparameters used during the training process for each of these networks.}\n\t\\label{img:architecture}\n\\end{figure}\n\n\\subsection{Training process}\n\\label{sec:train_process}\n\nWe use approximately $75$\\% of the previously described simulation set for the network training.\\footnote{Note that the sNet has a smaller training set compared to the DMNet, as here we have fewer unique spectra following from our parameter sampling for simulating the training set.} The remainder is used as a test set on which network performance evaluations can be conducted. Within the training set, a validation split of $20$\\% is used during training to monitor the generalisation capabilities of the network. Unlike the training loss, the loss calculated on the validation set is not used to update the model parameters during the optimisation process. \n\nThe network training was conducted using the ADAM optimizer~\\cite{adam}  and a mean squared error (MSE) loss. The initial learning rate of $10^{-2}$ is decreased during the training process, based on the behaviour of the validation loss, for an optimal convergence to a minimal loss. After the learning rate reaches its predefined minimum (lr = $10^{-5}$) the training process is terminated after 40 epochs without improvement of the validation loss, using an early stopping mechanism. This process helps ensure the convergence of the network optimisation. The MSE loss for both the training and validation loss over the training epochs is shown in figure \\ref{img:loss_curves} for both ANNs. \n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/loss_full_DM.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/loss_secondary.pdf}\n\t\\end{minipage}\n\t\\caption{Evolution of the MSE loss for the DMNet \\textit{(left)} and sNet \\textit{(right)} over the training epochs.}\n\t\\label{img:loss_curves}\n\\end{figure}\n\nWe performed the training on a V100-SXM2 GPU. Given the depth of the individual networks, this resulted in training durations of about $4$~minutes per epoch of the DMNet and about $12$~minutes per epoch for the sNet. \n\n\\subsection{Validation of the Network Performance}\n\nTraining performance measures, such as the loss based on the training set, can be helpful while adjusting the architecture and hyperparameters of the networks. The usage of the networks however, requires an evaluation of their ability to replace the simulations.\nUsing the fully trained networks we can compare the simulated spectra from \\textsc{Galprop} within the test set to the network predictions based on the same parameter point. An example for such a comparison is shown in figure~\\ref{img:example_fluxes}. We show the simulated spectra and the output of the respective ANNs for both the secondary and DM component of the antiproton flux (as well as for their sum). In the top panel we depict the fluxes in physical space alongside the AMS-02 antiproton data, demonstrating that the network provides fluxes that are extremely similar to the corresponding simulations. This is illustrated even more clearly in the bottom panel, which shows the relative differences between the ANN and the simulation with respect to the simulated total antiproton flux, compared to the relative uncertainties of the AMS-02 antiproton data. Prior to plotting each CR flux we infer the solar modulation and overall normalization by maximizing the likelihood for the AMS-02 data, as outlined in section~\\ref{sec:fitams}. This enables a fit to the data measured within the heliosphere and is automatically applied to each CR flux evaluated in the following. As this is not computationally expensive, it is not necessary to already include this step in the training process for the ANN. The parameters inferred for the \\textsc{Galprop} and ANN fluxes respectively are in agreement with each other.\nThe parameter point for the specific example presented in figure~\\ref{img:example_fluxes} was randomly selected from the extensive test set.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Antiprotons.pdf}\n\t\\caption{Exemplary comparison of the ANN versus \\textsc{Galprop} antiprotons flux of only the DM component and the combination of secondary and DM component where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. The lower panel depicts the relative difference between the \\textsc{Galprop} (`true') and ANN (`predicted') fluxes with respect to the \\textsc{Galprop} flux compared to the relative AMS-02 uncertainty. The listed solar modulation potential and overall normalization were inferred based on the AMS-02 data for each combined antiproton flux as described in section~\\ref{sec:fitams}.}\n\t\\label{img:example_fluxes}\n\\end{figure}\n\nWe conclude that the accuracy of the sNet is fully sufficient: the relative difference between the fluxes predicted by the ANN and the simulated \\textsc{Galprop} fluxes are always well below the relative uncertainty of the AMS-02 measurements. The architecture and training process used for the sNet can analogously be applied to train an ANN on proton and Helium spectra based on the same \\textsc{Galprop} simulation set, achieving a comparable accuracy. We provide additional details on these networks in appendix~\\ref{app:p_and_He}.\n\nGiven that the DMNet is trained on a parameter space of much higher dimensionality, it is unsurprising that its predictions are on average less accurate than the ones of the sNet. Indeed when calculating the relative differences between simulations and network predictions for the DM component only, we find that only 72\\% of samples lie on average within the AMS-02 relative uncertainties. However, it is essential to realise that in any realistic setting the DM component will only constitute a subdominant contribution to the antiproton flux. Indeed, if the DM contribution in a given bin significantly exceeds the uncertainty of the AMS-02 data (which is typically at the level of 5\\%) the model is expected to be excluded. \n\nIn order to provide more realistic estimates of the general accuracy and stability of the DMNet performance within the test set, we therefore focus on DM signals that contribute 5\\% to the total antiproton flux in the bin where the relative DM contribution is largest. We then calculate the differences between simulations and network output relative to the total antiproton flux. This approach shows that even if the DMNet itself is only accurate at the level of 10\\%, the total antiproton flux can still be predicted with an accuracy at the sub-percent level for allowed DM models.\n\nIn figure~\\ref{img:residual_bands}, we show this accuracy estimate for a total of 3000 DM component samples from the test set (1000 samples each for three different mass bins corresponding to the three different rows). Here we compute the deviation between DMNet prediction and \\textsc{Galprop} simulation to the corresponding total antiproton flux, as in the lower panel of figure~\\ref{img:example_fluxes}. Since the deviations are found to be miniscule when compared to the AMS-02 relative uncertainties, we provide in the right column of figure~\\ref{img:residual_bands} a zoomed-in version. It can be seen that the uncertainty bands (containing the central 68\\% of the network predictions) are typically at the level of 0.1\\% and do not exhibit any systematic shifts nor any significant dependence on the DM mass.  \n\nIn the following we will be interested in comparing the total antiproton flux to data in order to determine which DM signals are allowed by observations. The comparison between the network accuracy and the AMS-02 uncertainties in figure~\\ref{img:residual_bands} clearly shows that it is fully sufficient for this purpose to use the ANNs instead of running \\textsc{Galprop}. Indeed, we will show explicitly in the next section that both approaches lead to very similar values for the $\\chi^2$ statistic described in section~\\ref{sec:fitams}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/residual_bands_tot.png}\n\t\\caption{Relative deviations between predicted and simulated fluxes for the DM component binned into three mass bins and their 68 percentile. Each of the panel shows 1000 samples from the test set. As in the lower panels in figure~\\ref{img:example_fluxes}, in the left panel we again compare this to the benchmark of the relative AMS-02 uncertainties. The right panel shows a zoomed-in version of the left panel in the interval [-0.010, 0.010].}\n\t\\label{img:residual_bands}\n\\end{figure}\n\nThe fully trained networks, as described in this section and appendix~\\ref{app:p_and_He}, are publicly available as \\textsc{DarkRayNet} at \\url{https:\/\/github.com\/kathrinnp\/DarkRayNet}. In this repository, we provide an interface to easily access flux predictions for the corresponding CR species. This tool can for example be used for indirect DM searches as we outline in our analysis in the subsequent section.  \n\n\\section{Constraining the dark matter annihilation cross section}\n\\label{sec:constraints}\n\n\\subsection{Statistical method}\n\\label{sec:marg_importance}\n\nThe ANNs described in the previous section enable us to obtain predictions of the primary and secondary antiproton flux as a function of the DM parameters $\\mathbf{x}_\\text{DM}$ and the propagation parameters $\\bm{\\theta}_\\text{prop}$. Given data from observations we can then construct a likelihood function $\\mathcal{L}(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop})$. We emphasise that, given suitable predictions for the CR fluxes, this likelihood is quick to evaluate and therefore does not need to be predicted by the ANN. This has the significant advantage that the ANN does not need to learn the various fluctuations that may be present in the data.\n\nThe likelihood function can then be used to constrain both $\\mathbf{x}_\\text{DM}$ and $\\bm{\\theta}_\\text{prop}$. In the present work we primarily focus on the constraints on the DM parameter space, meaning that we will treat the propagation parameters simply as nuisance parameters that need to be varied in order to draw robust conclusions. The two main ways to achieve this is to either calculate the profile likelihood\n\\begin{equation}\n \\hat{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) = \\mathcal{L}(\\mathbf{x}_\\text{DM},\\hat{\\bm{\\theta}}_\\text{prop}(\\mathbf{x}_\\text{DM})) \\; ,\n\\end{equation}\nwhere $\\hat{\\bm{\\theta}}_\\text{prop}(\\mathbf{x}_\\text{DM})$ denote the propagation parameters that maximise the likelihood for given DM parameters $\\mathbf{x}_\\text{DM}$, or to calculate the marginalised likelihood\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) = \\int \\mathcal{L}(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop}) p(\\bm{\\theta}_\\text{prop}) \\mathrm{d}\\bm{\\theta}_\\text{prop} \\; ,\n\\end{equation}\nwhere $p(\\bm{\\theta}_\\text{prop})$ denotes the prior probability for the propagation parameters. Given sufficiently constraining data, the profile likelihood and the marginalised likelihood are expected to be similar and the dependence of the result on the chosen priors should be small. We find that this is largely true for the case considered here, with some notable exceptions to be discussed below.\n\nFrom the point of view of our machine learning approach, however, the two ways of varying the nuisance parameters are very different. The profile likelihood depends only on the antiproton flux for a single value of $\\bm{\\theta}_\\text{prop}$, meaning that highly accurate predictions are needed close to the maximum of the likelihood. For extreme choices of the DM parameters this maximum may be pushed to corners of parameter space where the network has not been sufficiently trained. A single outlier in the prediction will then completely bias the result and lead to numerical instabilities {when sampling the parameter space}. This makes accurate calculations of the profile likelihood a highly challenging task.\n\nThe marginalised likelihood, on the other hand, depends on the likelihood across a range of propagation parameters, which should have substantial overlap with the parameter regions seen during training. The impact of individual outliers in the predictions is also reduced significantly compared to the case of the profile likelihood, making the calculation of marginalised likelihoods based on ANN predictions more robust. Nevertheless, the challenge remains to ensure that results are not biased by regions of parameter space where only little training has been performed. In the present work we address this challenge using the technique of importance sampling~\\cite{mcbook}, which we describe in the following.\\footnote{For a different approach to Bayesian analyses of cosmic ray propagation with the help of neural networks we refer to Ref.~\\cite{Johannesson:2016rlh}.}\n\nFirst of all, we note that an approximate marginalisation can be performed by drawing a random sample of parameter points $\\bm{\\theta}_i$ from the prior probability $p(\\bm{\\theta}_\\text{prop})$ and calculating the sum\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i) \\; .\n\\end{equation}\nIn fact, the same can be done by drawing a random sample from any probability distribution function $q(\\bm{\\theta}_\\text{prop})$ provided the individual points are reweighted accordingly (so-called importance sampling):\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{\\sum_{i=1}^N \\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i) \\frac{p(\\bm{\\theta}_i)}{q(\\bm{\\theta}_i)}}{\\sum_{i=1}^N \\frac{p(\\bm{\\theta}_i)}{q(\\bm{\\theta}_i)}} \\; .\n \\label{eq:sample}\n\\end{equation}\nA particularly interesting case is that $q(\\bm{\\theta}_\\text{prop})$ is taken to be the posterior probability for the propagation parameters in the absence of a DM signal, i.e.\n\\begin{equation}\n\tq(\\bm{\\theta}_\\text{prop}) \\propto \\mathcal{L}(\\mathbf{x}_\\text{DM} = 0, \\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\equiv \\mathcal{L}_0(\\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\, .\n\\end{equation}\nIn this case $p(\\bm{\\theta}_i) \/ q(\\bm{\\theta}_i) \\propto 1\/\\mathcal{L}_0(\\theta_i)$ and hence\n\\begin{equation}\n \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM}) \\approx \\frac{\\sum_{i=1}^N  \\frac{\\mathcal{L}(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_i)}{\\mathcal{L}_0(\\bm{\\theta}_i)}}{\\sum_{i=1}^N \\frac{1}{\\mathcal{L}_0(\\bm{\\theta}_i)}} \\; .\n\\end{equation}\n\nThe great advantage of this approach is that the likelihood is only evaluated for plausible values of the propagation parameters, meaning for values that give a large posterior probability in the absence of a DM signal. These are exactly the same parameter regions on which we have focused for the training of the ANNs described above. Indeed, it is possible to generate the training data and sample the posterior probability using the same prior probabilities and the same \\textsc{MultiNest} runs such that a large overlap between the two is ensured.\\footnote{We emphasise that the posterior sample is not part of the training data, i.e.\\ the ANN is never evaluated on the exact same values seen during training.} Another significant advantage is that it is straight-forward to include additional constraints on the propagation parameters that are independent of the DM parameters and therefore not part of the ANN training. For example, to also include likelihoods for proton data $\\mathcal{L}_p$ and He data $\\mathcal{L}_\\text{He}$, it is sufficient to draw a sample from the joint posterior\n\\begin{equation}\n q(\\bm{\\theta}_\\text{prop}) \\propto \\mathcal{L}_0(\\bm{\\theta}_\\text{prop}) \\, \\mathcal{L}_p(\\bm{\\theta}_\\text{prop}) \\, \\mathcal{L}_\\text{He}(\\bm{\\theta}_\\text{prop}) \\, p(\\bm{\\theta}_\\text{prop}) \\, .\n\\end{equation}\n\nTo conclude this discussion, we note that in the case that the likelihood can be written in terms of a $\\chi^2$ function, $\\mathcal{L} \\propto e^{- \\chi^2\/2}$, we can define a marginalised $\\chi^2$ function as $\\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) \\equiv -2 \\log \\bar{\\mathcal{L}}(\\mathbf{x}_\\text{DM})$. Importance sampling then yields\n\\begin{equation}\n\t\\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) = - 2 \\log {  \\frac{\\sum_{i=1}^N  \\exp{ \\left(- \\frac{\\Delta\\chi^2(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_\\text{prop})}{2} \\right)}} {\\sum_{i=1}^N  \\exp{ \\left(\\frac{\\chi^2_0(\\bm{\\theta}_\\text{prop})}{2} \\right)} }} \\, ,\n\t\\label{eq:marg_chi}\n\\end{equation}\nwhere $\\chi^2_0(\\bm{\\theta}_\\text{prop}) = \\chi^2(\\mathbf{x}_\\text{DM} = 0, \\bm{\\theta}_\\text{prop})$ and $\\Delta\\chi^2(\\mathbf{x}_\\text{DM}, \\bm{\\theta}_\\text{prop}) = \\chi^2(\\mathbf{x}_\\text{DM},\\bm{\\theta}_\\text{prop}) - \\chi^2_0(\\bm{\\theta}_\\text{prop})$.\nTo calculate confidence intervals and exclusion limits for the DM parameters, we then define\n\\begin{equation}\n\t\\Delta \\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) = \\bar{\\chi}^2(\\mathbf{x}_\\text{DM}) - \\bar{\\chi}^2_0\\, .\n\t\\label{eq:upper_bound}\n\\end{equation}\nHence, $\\Delta \\bar{\\chi}^2 < 0$ corresponds to a preference for a DM signal, while parameter points with $\\Delta \\bar{\\chi}^2 > 3.84$ can be excluded at 95\\% confidence level.\\footnote{Note that although our treatment of nuisance parameters is motivated by Bayesian statistics, we still interpret the resulting marginalised likelihood using frequentist methods, such that there is no need to choose priors for the DM parameters.}\n\n\\subsection{Example A: Single Dark Matter Annihilation Channel}\n\\label{sec:example_A}\n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/MN_sample_galprop_vs_NN_pbar_delta_100GeV.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/MN_sample_galprop_vs_NN_pbar_delta_1TeV.pdf}\n\t\\end{minipage}\n\t\\caption{One and two dimensional histograms of $\\Delta \\chi^2$ for the AMS-02 antiproton measurement based on the antiproton fluxes provided by the Neural Network and \\textsc{Galprop} for different combinations of propagation parameters. \n\tWe consider the annihilations of DM particles with $m_\\text{DM} = 100$ GeV \\textit{(left)} and 1 TeV \\textit{(right)} into $b \\overline{b}$ with a cross section of $\\langle \\sigma v \\rangle = 10^{-26}$ cm$^3$ s$^{-1}$. The values for $\\Delta \\hat{\\chi}^2$ indicated by the black dashed lines represent the marginalised values obtained by the importance sampling technique described in section~\\ref{sec:marg_importance}. } \n\t\\label{img:chi_comp}\n\\end{figure}\n\nLet us first consider a frequently-used benchmark scenario and assume that the DM particles annihilate exclusively into pairs of bottom quarks, such that the injection spectrum is fully characterised by the (velocity-independent) annihilation cross section $\\langle \\sigma v \\rangle$ and the DM mass $m_\\text{DM}$. As a first step, we can then calculate $\\Delta \\chi^2(m_\\text{DM}, \\langle \\sigma v \\rangle, \\bm{\\theta}_\\text{prop})$ for different values of the propagation parameters. Figure~\\ref{img:chi_comp} compares the results that we obtain when using the ANN predictions of the antiproton flux and when employing \\textsc{Galprop}. The two panels correspond to different values of the DM mass and use the same 10122 sets of propagation parameters drawn randomly from the posterior distribution $q(\\bm{\\theta}_\\text{prop})$ as discussed above. In both cases we find a very strong correlation between the two ways of calculating $\\Delta \\chi^2$ ($r > 0.98$). Indeed, for 95\\% of parameter points the absolute difference in $\\Delta \\chi^2$ is smaller than $2.1$ ($0.9$) for $m_\\text{DM} = 100\\,\\mathrm{GeV}$ ($m_\\text{DM} = 1\\,\\mathrm{TeV}$), confirming the excellent performance of our ANN.\n\nIn each case we use a dashed line to indicate $\\Delta \\bar{\\chi}^2$ as defined in eq.~(\\ref{eq:marg_chi}). We emphasise that, since we average over $\\exp(-\\Delta \\chi^2 \/ 2)$, the final result is dominated by the points with the smallest $\\Delta \\chi^2$. Again, we find very good agreement between the marginalised $\\Delta \\chi^2$ obtained from the ANN and from \\textsc{Galprop}. The values obtained in the left panel correspond to a substantial preference for a DM signal, while the parameter point considered in the right panel is slightly disfavoured by data. Although the value $\\Delta \\bar{\\chi}^2 = -31.5$ ($-32.7$) that we obtain for $m_\\text{DM} = 100 \\, \\mathrm {GeV}$ from the ANN (\\textsc{Galprop}) would at face value correspond to quite a significant excess, we {emphasize that our set-up is not designed to provide an accurate characterisation of this excess. In particular we} caution the reader that due to our simplified implementation of AMS-02 data (in particular neglecting correlations) this number should be interpreted with care. We expect that a more detailed analysis of AMS-02 data would lead to a much lower significance.\n\nComparing the evaluations of the marginalised $\\Delta \\chi^2$ with the ANN and \\textsc{Galprop} respectively, the reduction of the computational cost achieved with our neural network method becomes apparent. For the ANN the prediction of the set of CR fluxes for each of the specific DM parameter points only takes $\\mathcal{O}(1)$ cpu second in total for the 10122 parameter points, but the calculation of the respective $\\chi^2$ while inferring the solar modulation potential takes up the majority of the computation time ($\\mathcal{O}(10)$ cpu seconds in total). This time is however negligible compared to the \\textsc{Galprop} simulations which take $\\mathcal{O}(10)$ cpu hours to obtain the same number of CR fluxes.\n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/chi_dist_zoom.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.5\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/chi_dist_marg_zoom.pdf}\n\t\\end{minipage}\n\t\\caption{$\\Delta \\chi^2$ for the AMS-02 antiproton measurement based on the antiproton fluxes provided by the Neural Network and \\textsc{Galprop} as a function of $\\langle \\sigma v \\rangle$ and for different values of $m_\\text{DM}$. We assume a dominant DM DM $\\rightarrow \\, b \\overline{b}$ annihilation in each case. \\textit{Left:} Propagation parameters are fixed to the best-fit values in a frequentist setup when only secondary antiprotons are considered (see table~\\ref{tab:param_ranges_v2}). \\textit{Right:} Propagation parameters are marginalised over using importance sampling. We also include the 95 \\% upper bound values of the annihilation cross section following eq.~(\\ref{eq:upper_bound}).}\n\t\\label{img:chi_dist_zoom}\n\\end{figure}\n\nA complementary perspective to the results in figure~\\ref{img:chi_comp} is provided in figure~\\ref{img:chi_dist_zoom}, which shows $\\Delta \\chi^2$ as a function of $\\langle \\sigma v \\rangle$ for different values of the DM mass. In the left panel we fix the propagation parameters to their best-fit values in the absence of a DM signal (see table~\\ref{tab:param_ranges_v2}), while in the right panel we marginalise over all propagation parameters using importance sampling. Solid (dotted) curves correspond to the ANN (\\textsc{Galprop}) predictions and again show excellent agreement. The horizontal dashed lines indicate the 95\\% confidence level upper bound on $\\langle \\sigma v \\rangle$ obtained following eq.~(\\ref{eq:upper_bound}).\n\nAs expected, allowing variations in the propagation parameters generally leads to smaller values of $\\Delta \\chi^2$ and hence relaxes the upper bounds on the annihilation cross section. This effect is most dramatic for the case $m_\\text{DM} = 100 \\, \\mathrm{GeV}$ (blue line), where there is a preference for a DM signal in the data and hence the exclusion limit is relaxed by about an order of magnitude. The small bumps in the blue curve in the right panel are a result of the finite size of the sample of propagation parameters used for the marginalisation and result from the approximation made in eq.~(\\ref{eq:sample}).\n\nRepeating this procedure for different values of the DM mass, we can obtain exclusion limits on $\\langle \\sigma v \\rangle$ as a function of $m_\\text{DM}$. These are shown in figure~\\ref{img:bounds_bb} for the case of fixed propagation parameters (left) and when marginalising over propagation parameters (right). The colour shading indicates parameter regions where $\\Delta \\chi^2 > 0$, such that a DM signal is disfavoured, while greyscale is used to indicate parameter regions where $\\Delta \\chi^2 < 0$ such that a DM signal is preferred. We find that this is the case for DM masses in the range $50\\text{--}250\\,\\mathrm{GeV}$. Again, marginalisation leads to relaxed exclusion bounds  and an increased preference for a DM signal. We reiterate however that the magnitude of this preference is likely overestimated in our analysis.\n\n\\begin{figure}[t]\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/bb_both.pdf}\n\t\\caption{$\\Delta \\chi^2$ for the AMS-02 antiproton measurement as a function of $\\langle \\sigma v \\rangle$ and $m_\\text{DM}$ using {the fixed propagation parameters specified in table~\\ref{tab:param_ranges_v2}} (\\emph{left}) and performing the marginalisation via importance sampling (\\emph{right}). The dashed lines represent the 95~\\% CL upper bounds on the annihilation cross section. The white regions in the upper part of each panel correspond to $\\Delta \\chi^2 > 1000$ and are excluded to improve numerical stability.}\n\t\\label{img:bounds_bb}\n\\end{figure}\n\nTo assess the impact of marginalisation let us finally compare our results with those obtained using a profile likelihood. As discussed in section~\\ref{sec:marg_importance}, special care needs to be taken when using the ANN predictions to calculate a profile likelihood in order to ensure that the result is not dominated by regions of parameter space with insufficient training data. We achieve this goal by restricting the allowed parameter regions as follows: $0.1 < s < 0.6$, $1 \\, \\mathrm{GV} < R_0 < 10 \\, \\mathrm{GV}$, $0.35 < \\delta < 0.6$ and $2.3 < \\gamma_{2,(p)} < 2.5$. We then use \\textsc{MultiNest} to explore the remaining parameter space for fixed values of the DM mass and varying annihilation cross section in order to find the largest value of $\\langle \\sigma v \\rangle$ such that $\\Delta \\hat{\\chi}^2(m_\\text{DM}, \\langle \\sigma v \\rangle) \\equiv  -2 \\Delta \\log \\hat{\\mathcal{L}}((m_\\text{DM}, \\langle \\sigma v \\rangle)) < 3.84$. Repeating this procedure for different values of $m_\\text{DM}$ then yields the exclusion limit.\n\nThe results are shown in figure~\\ref{img:bounds_bb_compare} together with the exclusion limits obtained for fixed propagation parameters and when marginalising over propagation parameters as shown in figure~\\ref{img:bounds_bb}. We find that in most regions of parameter space the profile likelihood approach yields somewhat weaker exclusion limits than the marginalisation. Such a difference is to be expected whenever substantial tuning in the propagation parameters is required in order to accommodate a DM signal. For example, for $m_\\text{DM} = 1 \\, \\mathrm{TeV}$ and $\\langle \\sigma v \\rangle = 5 \\times 10^{-26} \\, \\mathrm{cm^3 \\, s^{-1}}$ we find that $\\Delta \\hat{\\chi}^2 < 3.84$ can be achieved only if $D_0$, $v_{0,c}$ and $z_\\mathrm{h}$ all take values close to their lower bounds. Such a tuning is not penalised in the profile likelihood, but the contribution of these solutions to the marginalised likelihood will be suppressed according to the small volume of the viable parameter space. The same conclusion can be reached from the right panel of figure~\\ref{img:chi_comp}: Although there are sets of propagation parameters that yield $\\Delta \\chi^2 \\approx 0$, most parameter combinations give significantly larger $\\Delta \\chi^2$, such that marginalisation leads to $\\Delta \\hat{\\chi}^2 \\approx 2.6$, close to the 95\\% confidence level upper bound. In other words, the difference between the two approaches is a direct consequence of the different statistical methods and not an artefact of the ANN predictions.\n\nIn general the dependence of the DM limit on the chosen value for the halo height is very well known. To first order the normalisation of the \nDM flux is proportional to $z_\\mathrm{h}$ and thus the DM limit is anti-proportional to $z_\\mathrm{h}$ as again nicely demonstrated in \na very recent analysis \\cite{Genolini:2021doh}.\nThe CR fit conducted in section \\ref{sec:cr} varies $z_\\mathrm{h}$ between 2 and 7 kpc. Because of the well-known $z_\\mathrm{h}$-$D_0$ \ndegeneracy the resulting posterior of $z_\\mathrm{h}$ is almost flat in the entire fit range. \nThe DM limit derived from the marginalisation of the $\\Delta \\hat{\\chi}^2$ should be understood to refer to 4.8 kpc, \nnamely the average value of $z_\\mathrm{h}$ in the posterior. This is in perfect agreement with recent analyses \nof secondary fluxes by AMS-02~\\cite{Evoli:2019iih,Weinrich:2020ftb,Luque:2021joz,Korsmeier:2021brc}.\nOn the other hand, when limits are derived in a frequentist approach and in the absence of a DM preference, \n$z_\\mathrm{h}$ values are pushed towards the lower boundary of the fit range at 2 kpc. \nThis again explains the difference between the marginalised and profiled limit in the figure~\\ref{img:bounds_bb_compare}.\nOne possible way to study the $z_\\mathrm{h}$ dependence explicitly in the marginalisation framework is to \nfurther restrict the range of $z_\\mathrm{h}$. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.7\\textwidth]{.\/figures\/Compare_bounds_part.pdf}\n\t\\caption{A comparison of the 95~\\% CL exclusion bounds in figure~\\ref{img:bounds_bb} (blue and light blue) with the bounds obtained when profiling over the propagation parameters using the CR spectra provided by our ANNs (green). The black dashed line indicates the thermal annihilation cross section for WIMPs from \\cite{Steigman2012}.}\n\t\\label{img:bounds_bb_compare}\n\\end{figure}\n\nThe differences between marginalised and profiled limits are particularly relevant given how they affect the conclusions drawn from figure~\\ref{img:bounds_bb_compare}. When using the marginalised likelihood we find that the thermal cross section (indicated by the black dashed line) can be excluded for DM masses in the range $300\\text{--}2000\\,\\mathrm{GeV}$, implying that WIMP models in this mass range can only be viable if the injection of antiprotons are suppressed. When using the profile likelihood, on the other hand, almost the entire mass range above $70 \\, \\mathrm{GeV}$ is found to be viable. \tWe note that the agreement between the frequentist and Bayesian approach will improve with a better determination of $z_\\mathrm{h}$ as expected from the analysis of the forthcoming Be isotope measurements by AMS-02 \\cite{Derome:ICRC2021}.\n\nIn addition to the reduction in computing time achieved when using the ANN instead of \\textsc{Galprop}, we find that the use of importance sampling leads to another improvement compared to the more conventional profiling approach.\nCrucially, our marginalisation using importance sampling is based on a fixed set of 10122 data points in the propagation model, which can be evaluated in parallel. The ANN therefore gives a negligible contribution to the time needed to calculate the upper bound on the annihilation cross section for each of the 100 mass bins shown in figures~\\ref{img:bounds_bb} and \\ref{img:bounds_bb_compare}. \nFor the profiling approach on the other hand the evaluation of the data points cannot be performed in parallel by the ANN due to their sampling. This leads to an increase in computation time, such that  the speed-up of the runtime when using the ANN instead of \\textsc{Galprop} is reduced to two orders of magnitude (rather than three orders of magnitude for importance sampling).\n\n\\subsection{Example B: Scalar Singlet Dark Matter}\n\nWe now illustrate the use of the ANN for the analysis of a specific model of DM with a singlet scalar field $S$.  Imposing a $Z_2$ symmetry, $S \\to -S$, the scalar particle is stable and thus a DM candidate. The Lagrangian of this scalar singlet DM (SSDM) model reads~\\cite{Silveira:1985rk,McDonald:1993ex,Burgess:2000yq} \n\\begin{equation}\n{\\cal L} = {\\cal L}_\\text{SM} + \\frac 12 \\partial_\\mu  S  \\partial^\\mu  S  - \\frac12 m_{S,0}^2S^2- \\frac 14 \\lambda_S  S^4- \\frac 12 \\lambda_{H\\!S}\\, S^2 H^\\dagger H\\,, \n\\label{eq:lagr}\n\\end{equation}\nwhere ${\\cal L}_\\text{SM}$ is the Standard Model Lagrangian and $H$ is the Standard Model Higgs field. After electroweak symmetry breaking, the last three terms of the Lagrangian become\n\\begin{equation}\n{\\cal L} \\supset  - \\frac12 m_{S}^2\\, S^2- \\frac 14 \\lambda_S\\,  S^4 - \\frac 14 \\lambda_{H\\!S}\\, h^2 S^2 - \\frac {1}{2} \\lambda_{H\\!S}\\, v h S^2\\,,\n\\label{eq:ewbr}\n\\end{equation}\nwith $H = (h+v, 0)\/\\sqrt{2}\\,$, $v = 246\\,$GeV, and where we introduced the physical mass of the singlet \nfield, $m_S^2  = m_{S,0}^2 + \\lambda_{H\\!S} \\,v^2 \/ 2$. The DM phenomenology of the SSDM has been extensively studied in the literature, see e.g.\\ \\cite{Cline:2013gha,Beniwal:2015sdl,Cuoco:2016jqt,Cuoco:2017rxb,GAMBIT:2017gge,Athron:2018ipf} and references therein. \n\nThe DM phenomenology of the SSDM is fully specified by the mass of the DM particle, $m_S=m_{\\rm DM}$, and the strength of the coupling between the DM and Higgs particle, $\\lambda_{H\\!S}$. Below the Higgs-pair threshold, $m_S < m_h$, DM annihilation proceeds through $s$-channel Higgs exchange only, and the relative weight of the different SM final states is determined by the SM Higgs branching ratios, independent of the Higgs-scalar coupling $\\lambda_{H\\!S}$. Above the Higgs-pair threshold, $m_S \\ge m_h$, the $hh$ final state opens up. The strength of the annihilation into Higgs pairs, as compared to $W$, $Z$ or top-quark pairs, depends on the size of the Higgs-scalar coupling. For our specific analysis we require that the SSDM provide the correct DM relic density, $\\Omega h^2 = 0.1198\\pm 0.0015$~\\cite{Ade:2015xua}, which in turn determines the size of $\\lambda_{H\\!S}$ for any given DM mass $m_S$. The corresponding branching fractions for DM annihilation within the SSDM are shown in figure~\\ref{img:bounds_SSDM} (left panel) as a function of the DM mass. \n\nUsing the ANN we analyse the $\\Delta\\chi^2$ distribution of the model, marginalising over propagation uncertainties as described in section~\\ref{sec:marg_importance}. The result is shown in figure~\\ref{img:bounds_SSDM} (right panel). Comparing figure~\\ref{img:bounds_SSDM} with the analogous result for the single annihilation channel into $b\\bar{b}$, figure~\\ref{img:bounds_bb} (right panel), we observe a similar overall shape of the $\\Delta\\chi^2$ distribution. \n\nFor light DM the SSDM annihilates dominantly into bottom final states, so one expects results that are very similar to the case of the single $b\\bar{b}$ channel. However, for the smallest DM masses that we consider ($m_\\chi \\approx 10 \\, \\mathrm{GeV}$) we find that the constraints become considerably stronger when including even a sub-dominant contribution from $c\\bar{c}$. The reason is that in this mass range, most antiprotons resulting from annihilation into bottom quarks have energies below $5\\,\\mathrm{GeV}$ and do therefore not give a contribution in our fits. Annihilation into charm quarks, on the other hand, can give rise to more energetic antiprotons, leading to stronger constraints. For DM masses above about $50 \\, \\mathrm{GeV}$, a variety of SM final states contributes in the SSDM, including in particular $WW$, $hh$ and $ZZ$.  However, as shown in Ref.~\\cite{Cuoco:2017iax}, the limits for heavy DM are similar for these final states and for annihilation into bottom quarks, so that the overall constraints for the SSDM are comparable to those for annihilation into bottom quarks only. \n\n\\begin{figure}[t]\n\t\\begin{minipage}{0.435\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/SSDM_fractions.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{0.565\\textwidth}\n\t\t\\includegraphics[width = 1\\textwidth]{.\/figures\/SSDM_marginalized.pdf}\n\t\\end{minipage}\n\t\\caption{\\textit{Left:} Mass dependence of branching fractions of $S S \\rightarrow \\text{SM} \\text{ SM}$ in the SSDM model for $\\lambda_\\mathrm{HS}$ fixed by the relic density requirement. \\textit{Right:} Marginal $\\chi^2$ distribution of the $\\langle \\sigma v \\rangle - m_\\text{DM}$ parameter space in the SSDM model.}\n\t\\label{img:bounds_SSDM}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThe analysis of cosmic ray (CR) antiprotons is a powerful method for the indirect detection of dark matter (DM). The accurate experimental measurements, in particular from AMS-02, allow to probe DM annihilation cross sections close to the value predicted by thermal freeze-out for a wide range of DM masses. However, a precise description of CR propagation through the Galaxy is required to exploit the potential of the experimental data. The propagation models depend on a large number of parameters, and the  standard numerical simulation tools, such as \\textsc{Galprop}, are computationally expensive. Therefore, global analyses of generic models of DM can only be carried out with an immense computational effort, if at all. \n\nIn this work we have developed an artificial neural network (ANN) that allows extremely fast and accurate predictions of the cosmic ray flux for generic DM models. Specifically, we have employed recurrent neural networks (RNNs) to predict the CR energy spectrum. RNNs are particularly well suited to learn the correlations between the fluxes contained in neighbouring energy bins. Additional improvements in performance are achieved by grouping input parameters that have similar physical origin and by performing a suitable rescaling of the output spectra. \n\nWe have trained the ANN with a large set of antiproton fluxes simulated with \\textsc{Galprop}, where the propagation parameters have been chosen to be broadly compatible with the most recent AMS-02 data, and a generic parametrisation of the dark matter model in terms of the DM mass and the branching fractions for the annihilation into various \nStandard Model final states. We emphasise that the contribution of different DM models to the antiproton flux only has a marginal impact on the preferred range of the propagation parameters. It is therefore possible to focus the training of the ANN on the relevant range of propagation parameters without specifying the details of the DM model in advance. We have validated the performance and accuracy of the network by comparing both the predicted antiproton fluxes and the resulting AMS-02 likelihoods to the ones obtained from explicit \\textsc{Galprop} simulations for a range of different propagation and DM model parameters.\n\nWe have then used the neural network predictions to test specific DM models against current AMS-02 data. We have focused on the DM parameter space and treated the propagation parameters as nuisance parameters by calculating both the corresponding profile and marginalised likelihoods. While the former approach requires an explicit restriction of the parameter space to the regions where the ANN has been sufficiently trained, this requirement can be automatically fulfilled in the latter case by employing importance sampling. Comparing the ANN to \\textsc{Galprop} we find a speed-up in runtime of about two (three) orders of magnitude when using profiling (importance sampling).\n\nFor DM annihilation into bottom quarks we have obtained results that are consistent with previous studies based on simulations and a profile likelihood approach. We find more stringent bounds on the DM parameter space when using the marginalised likelihood; here a thermal cross section can be excluded for DM annihilating fully into bottom quarks for DM  masses in the range between approximately 300~GeV and 2~TeV. \nTo illustrate the flexibility of our approach, we have also used the ANN to derive constraints on scalar singlet DM, for which DM annihilation results in a variety of Standard Model final states with branching fractions that depend strongly on the DM mass. \n\nThe ANN developed in this work, and the corresponding method for efficient training, can {also be used to study more closely the potential DM interpretation of the antiproton excess around 20 GeV, for example regarding the impact of correlations in AMS-02 data. Moreover, it can} be easily extended to alternative propagation models and can be applied to a wide class of DM scenarios.\nIt will thus be possible to fully exploit the potential of current and future cosmic-ray data in global analyses of general DM models. {In future work a transformation of the ANNs into Bayesian neural networks can be incorporated in the analysis. With this step, additional more in-depth studies of the uncertainties of the network predictions will be possible.} The fully trained networks together with a suitable user interface are publicly available as \\textsc{DarkRayNet} at \\url{https:\/\/github.com\/kathrinnp\/DarkRayNet}.\n\n\\acknowledgments\n\nWe thank Thorben Finke and Christoph Weniger for discussions, Alessandro Cuoco and Jan Heisig for helpful comments on the manuscript and Sven Guenther for testing \\textsc{DarkRayNet}. F.K.\\ is  supported by  the  Deutsche Forschungsgemeinschaft (DFG) through the Emmy Noether Grant No.\\ KA 4662\/1-1.\nM.Ko.\\ is partially supported by the Swedish National Space Agency under contract 117\/19 and the European Research Council under grant 742104.\nSimulations and ANN training were performed with computing resources granted by RWTH Aachen University under project jara0184 and rwth0754.\n\n\n\\begin{appendix}\n\n\n\\section{Predicting proton and helium spectra}\n\\label{app:p_and_He}\n\nWhen simulating the antiproton fluxes as described in section~\\ref{sec:training_set} we can  also obtain the CR spectra of protons, deuterium, and helium ($^3$He and $^4$He) without significant additional computation costs due to the setup of \\textsc{Galprop}. The task of modelling these spectra using an ANN is very comparable with the task fulfilled by the sNet. We have thus examined the ability of the sNet architecture (as described in sec.~\\ref{sec:architectures}) to also accurately predict proton and helium spectra. The inputs of the sNet remain the same, but we have extended the length of the final output layer, to accommodate a wider energy range, appropriate for the proton and Helium AMS-02 and Voyager data. Using also the same training process (see sec.~\\ref{sec:train_process}) we achieve a similar accuracy as for the secondary antiprotons, as each of the predictions deviates from the simulations only marginally with respect to the experimental uncertainties. In figures~\\ref{img:example_fluxes_p} and \\ref{img:example_fluxes_He} we show exemplary results for protons, resp. helium, and their individual components analogous to figure~\\ref{img:example_fluxes}. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Protons.pdf}\n\t\\caption{Exemplary comparison of the simulated versus predicted protons flux of the individual components protons and Deuterium and the combination of both where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. Lower panel as figure~\\ref{img:example_fluxes}.}\n\t\\label{img:example_fluxes_p}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.8\\textwidth]{.\/figures\/Comp_Helium.pdf}\n\t\\caption{Exemplary comparison of the simulated versus predicted He flux of the individual components $^3$He and $^4$He and the combination of both where the listed parameters and simulated fluxes are randomly sampled from the test set. Each component of the neural network flux is predicted by the individual networks. Lower panel as figure~\\ref{img:example_fluxes}.}\n\t\\label{img:example_fluxes_He}\n\\end{figure}\n\n\\end{appendix}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}