diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbuio" "b/data_all_eng_slimpj/shuffled/split2/finalzzbuio" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbuio" @@ -0,0 +1,5 @@ +{"text":"\\section{}\n\n\\bibliographystyle{elsarticle-num} \n\n\n\\section*{Appendix A.}\n\\label{app:theorem}\n\n\n\nIn this appendix we prove the following theorem from\nSection~6.2:\n\n\\noindent\n{\\bf Theorem} {\\it Let $u,v,w$ be discrete variables such that $v, w$ do\nnot co-occur with $u$ (i.e., $u\\neq0\\;\\Rightarrow \\;v=w=0$ in a given\ndataset $\\dataset$). Let $N_{v0},N_{w0}$ be the number of data points for\nwhich $v=0, w=0$ respectively, and let $I_{uv},I_{uw}$ be the\nrespective empirical mutual information values based on the sample\n$\\dataset$. Then\n\\[\n\tN_{v0} \\;>\\; N_{w0}\\;\\;\\Rightarrow\\;\\;I_{uv} \\;\\leq\\;I_{uw}\n\\]\nwith equality only if $u$ is identically 0.} \\hfill\\BlackBox\n\n\\noindent\n{\\bf Proof}. We use the notation:\n\\[\nP_v(i) \\;=\\;\\frac{N_v^i}{N},\\;\\;\\;i \\neq 0;\\;\\;\\;\nP_{v0}\\;\\equiv\\;P_v(0)\\; = \\;1 - \\sum_{i\\neq 0}P_v(i).\n\\]\nThese values represent the (empirical) probabilities of $v$\ntaking value $i\\neq 0$ and 0 respectively. Entropies will be denoted\nby $H$. We aim to show that $\\fracpartial{I_{uv}}{P_{v0}} < 0$....\\\\\n\n\\section{Benchmark}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[Training Process in 150 epochs.]{\n\\includegraphics[width=0.45\\columnwidth]{figures\/mnist_top1_acc.pdf}\n}~~~~\n\\subfloat[Compression Ratio.]{\n\\includegraphics[width=0.45\\columnwidth]{figures\/mnist_cr.pdf}\n}\n\\caption{Experiments on UCF11. CR is short for Compression Ratio.}\n\\label{fig:ucf11}\n\\end{figure}\n\nUntil now, TDNs are mostly applied in computer vision field. Thus, aiming to validate performance of TedNet, we consider to conduct experiments on two datasets:\n\\begin{itemize}\n \\item The UCF11 Dataset contains 1,600 video clips of a resolution $320 \\times 240$ and is divided into 11 action categories. Each category consists of 25 groups of videos, within more than 4 clips in one group. \n \\item The Cifar10\/100 consists of 50,000 train images and 10,000 test images with size as $32 \\times 32 \\times 3$. CIFAR10 has 10 object classes and CIFAR100 has 100 categories.\n\\end{itemize}\n\n\n\n For the video classification task on UCF11, we adopt the same setting as described in literature \\cite{DBLP:conf\/aaai\/PanXWYWBX19}, where we extract feature of dimension 2048 from each frame of a video by Inception-V3~\\cite{DBLP:conf\/cvpr\/SzegedyVISW16}. Then throw these features as step inputs into TD-LSTMs. Results are shown in Figure~\\ref{fig:ucf11}. Almost every tensor decomposition model can achieve better accuracy except Tucker-2.\n\n\n\n\n For the image classification task on Cifar10\/100, we employ ResNet-32 as the backbone network. We show the results of corresponding TD-ResNet-32 implementations with various tensor decomposition in Table~\\ref{tbl:cifar}.\n\n Note that the results shown in Table~\\ref{tbl:cifar} and Figure~\\ref{fig:ucf11} are obtained without fine tuning parameters, and are just used for verifying the correctness of these algorithms. Thus the classification results does not mean the performance of these algorithms with the best parameter settings.\n\n\n\n\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{l|c|c|c|c|c|c|c} \n\\hline\n\\multicolumn{2}{l|}{} & \\multicolumn{3}{c|}{Cifar10} & \\multicolumn{3}{c}{Cifar100} \\\\ \n\\hline\n\\hline\nModel & Rank & Params & CR & Accuracy & Params & CR & Accuracy \\\\ \n\\hline\nResNet-32 & - & 0.46M & 1$\\times$ & 0.9228 & 0.47M & 1$\\times$ & 0.6804 \\\\ \n\\hline\nBTT-ResNet-32 & 4 & 0.08M & 6$\\times$ & 0.8955 & 0.08M & 6$\\times$ & 0.5661 \\\\ \n\\hline\nCP-ResNet-32 & 10 & 0.03M & 18$\\times$ & 0.8802 & 0.03M & 18$\\times$ & 0.4445 \\\\ \n\\hline\nTK2-ResNet-32 & 10 & 0.05M & 9$\\times$ & 0.8915 & 0.06M & 9$\\times$ & 0.5398 \\\\ \n\\hline\nTR-ResNet-32 & 10 & 0.09M & 5$\\times$ & 0.9076 & 0.09M & 5$\\times$ & 0.653 \\\\ \n\\hline\nTT-ResNet-32 & 10 & 0.09M & 5$\\times$ & 0.9020 & 0.10M & 5$\\times$ & 0.6386 \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Experiments on Cifar10\/100. Params denotes the number of parameters. CR means compression rate. The number of block-terms is set to 5 in BTT-ResNet-32.}\n\\label{tbl:cifar}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we present a package named TedNet that is specially designed for TDNs. TedNet is completely open-source and distributed under the MIT license. Compared with other related python packages, TedNet contains the most kinds of tensor decomposition.\n\n\\section*{Acknowledgements}\n\nThis paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204), and a key program of fundamental research from Shenzhen Science and Technology Innovation Commission (No. JCYJ20200109113403826).\n\n\n\\section{TedNet Details}\n\nTedNet is designed with the goal of building TDNs by calling corresponding APIs, which can extremely simplify the process of constructing TDNs. As shown in Figure~\\ref{fig:frame}, TedNet adopts Pytorch as the training framework because of its auto differential function and convenience to build DNN models. In addition, TedNet also uses NumPy~\\cite{DBLP:journals\/cse\/WaltCV11} to assist in tensor operations. The fundamental module of TedNet is \\textbf{\\_TNBase}, which is an abstracted class and inherits from \\textbf{torch.nn.Module}. Thus, TedNet models can be amicably combined with other Pytorch models. As an abstracted class, \\textbf{\\_TNBase} requires sub-classes to implement 4 functions~\\footnote{https:\/\/github.com\/tnbar\/tednet\/blob\/main\/tednet\/tnn\/tn\\_module.py}. On the right side of Figure~\\ref{fig:frame}, we show two main deep architectures of TedNet, namely TD ResNet and TD LSTM, which are probably the most frequently used backbone in convolutional neural networks and recurrent neural networks, respectively.\n\n\n\n\n\nUsually, DNNs are constructed with CNNs and Linears. The weight of a CNN is a 4-mode tensor $\\ca{C} \\in \\mathbb{R}^{K\\times K \\times C_{in} \\times C_{out}}$, where $K$ means the convolutional window, $C_{in}$ denotes the input channel and $C_{out}$ represents the counterpart output channel. And a Linear is a matrix $\\mathbf{W} \\in \\mathbb{R}^{I \\times O}$, where $I$ and $O$ are length of input and output feature respectively. Similar to DNNs, TDNs consist of TD-CNNs and TD-Linears(For simplification, TD- denotes the corresponding tensor decomposition model), whose weights $\\ca{C}$ and $\\textbf{W}$ are factorized with tensor decomposition. Following this pattern, \nthere are 5 frequently-used tensor decomposition (i.e. CP, Tucker-2, Block-Term Tucker, Tensor Train and Tensor Ring) in TedNet, which satisfies most of common situations. Notably, TedNet is an open-source package which supports Tensor Ring Decomposition. Besides, based on TD-CNNs and TD-Linears, TedNet has built some tensor decomposition based Deep Neural Networks, e.g. TD-ResNets, TD-RNNs.\n\n\n\\section{Installation and Illustrative Examples}\n\nThere are two ways to install TedNet. For the sake that the source code of TedNet is submitted to GitHub, it is feasible to install from the downloaded code by command \\textbf{python setup.py install}. Compared with aforementioned fussy way, another one, the recommended way is to install TedNet trough PyPI~\\footnote{https:\/\/pypi.org\/project\/tednet} by command \\textbf{pip install tednet}. After installation, all tensor decomposition models of TedNet can be used.\n\n A simple MNIST~\\cite{lecunGradientbasedLearningApplied1998a} classifier based on tensor ring is shown in Listing~\\ref{ll:tr}. The tensor ring module can be used by importing \\textbf{tednet.tnn.tensor\\_ring}. We utilize two fundamental tensor ring layers (i.e., TRConv2D, TRLinear) to build the sample classifier. In addition, it is very convenient to build a whole tensor ring network with only one line of code, e.g., TR-LeNet5~\\footnote{https:\/\/tednet.readthedocs.io\/en\/latest\/quick\\_start.html}. The usage of other decomposition is the same and more details can be found in the Document~\\footnote{https:\/\/tednet.readthedocs.io}.\n\n\n\n\n\n\n\\section{Introduction}\n\nTensor Decomposition Networks (TDNs) are constructed by decomposing deep neural layers with tensor formats. For the reason that the original tensor of a layer can be recovered from tensor decomposition cores, TDNs are often regarded as a compression method for the corresponding networks. Compared with traditional networks like Convolution Neural Networks (CNNs) and Recurrent Neural Networks (RNNs), TDNs can be much smaller and occupy a little memory. For example, TT-LSTM~\\cite{DBLP:conf\/icml\/YangKT17}, BTT-LSTM~\\cite{DBLP:conf\/cvpr\/YeWLCZCX18,DBLP:journals\/nn\/YeLCYZX20}, TR-LSTM~\\cite{DBLP:conf\/aaai\/PanXWYWBX19,li2021heuristic} are able to reduce 17,554, 17,414 and 34,192 times parameters with a higher accuracy than the original models. With light-weight architectures and good performance, TDNs are promising to be used in kinds of source-restricted scenes including mobile equipment and microcomputers. Due to these advantages, TDNs can often achieve comparably high accuracy with huge parameter reduction in a number of tasks, such as action recognition~\\cite{DBLP:journals\/pieee\/PanagakisKCONAZ21,sun2020human}. TDNs have also been implemented in FPGA for fast inference with ultra memory reduction~\\cite{zhang2021fpga} and multi-task learning to improve the representing ability~\\cite{wang2020concatenated}. Under this background, we design TedNet package for providing convenience for researchers to explore on TDNs.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figures\/frame.pdf}\n\\caption{The framework of TedNet. TedNet is based on Pytorch and adopts NumPy to process numerical calculations. Tensor decomposition (TD) can be applied to convolutional layers or linear layers. We implemented 5 variants of tensor decomposition methods, namely CP, Tucker, Tensor Ring, Tensor Train, and Block-term Tucker. Tensor decomposition can be fulfilled in convolution neural networks. An illustration of two tensorial classical neural blocks(i.e., ResNet and LSTM) that are built on the Tensor Decomposition Layer is shown in the right of the figure.}\n\\label{fig:frame}\n\\end{figure}\n\nThere are several related packages, such as T3F~\\cite{DBLP:journals\/jmlr\/NovikovIKFO20}, Tensorly~\\cite{DBLP:journals\/jmlr\/KossaifiPAP19}, TensorD~\\cite{DBLP:journals\/ijon\/HaoLYX18}, TensorNetwork~\\cite{roberts2019tensornetwork}, tntorch~\\cite{tntorch} , OSTD~\\cite{DBLP:conf\/iccvw\/SobralJJBZ15} and TensorTools~\\cite{williams2018unsupervised}. OSTD is constructed for low-rank decomposition and implemented with MATLAB. TensorTools based on NumPy~\\cite{DBLP:journals\/cse\/WaltCV11} implements CP decomposition only, while T3F is explicitly designed for Tensor Train Decomposition on Tensorflow~\\cite{DBLP:conf\/osdi\/AbadiBCCDDDGIIK16}. Similarly based on Tensorflow, TensorD supports CP and Tucker decomposition. By contrast, TedNet implements five kinds of tensor decomposition with backend Pytorch~\\cite{NEURIPS2019_9015}. TensorNetwork is built on Tensorflow and incorporates abundant tensor calculation tools. Nevertheless, TensorNetwork serves for tensor decomposition algorithms rather than TDNs. Tensorly supports with a variety of backends including CuPy, Pytorch, Tensorflow and MXNet~\\cite{chen2015mxnet}. Unfortunately, although Tensorly is powerful to process tensor algebra, tensor decomposition and tensor regressions, it still lacks support to Application Programming Interface (API) to build tensorial neural networks directly. Interestingly, Tensorly can assist to initialize TedNet network modules with its tensor decomposition operation. Compared with them, TedNet can \nset up a TDN layer quickly by calling API directly. In addition, we also provide three kinds of deep TDNs that are popular for researchers now. Due to the Dynamic Graph Mechanism of Pytorch, TedNet is also flexible to DEBUG for programmers.\n\n\n\n\n\n\n\n\n\n\\section*{Required Metadata}\n\n\n\n\n\\section*{Current code version}\n\\label{}\n\nAncillary data table required for subversion of the codebase. Kindly replace examples in right column with the correct information about your current code, and leave the left column as it is.\n\n\\begin{table}[!h]\n\\begin{tabular}{|l|p{6.5cm}|p{8.2cm}|}\n\\hline\n\\textbf{Nr.} & \\textbf{Code metadata description} & \\textbf{Please fill in this column} \\\\\n\\hline\nC1 & Current code version & 0.1.3 \\\\\n\\hline\nC2 & Permanent link to code\/repository used of this code version & $https:\/\/github.com\/tnbar\/tednet\/releases\/tag\/0.1.3$ \\\\\n\\hline\nC3 & Legal Code License & MIT License \\\\\n\\hline\nC4 & Code versioning system used & git \\\\\n\\hline\nC5 & Software code languages, tools, and services used & Python, Pytorch \\\\\n\\hline\nC6 & Compilation requirements, operating environments \\& dependencies & Python3.X, NumPy\\\\\n\\hline\nC7 & If available Link to developer documentation\/manual & $https:\/\/tednet.readthedocs.io\/en\/latest\/index.html$ \\\\\n\\hline\nC8 & Support email for questions & iperryuu@gmail.com\\\\\n\\hline\n\\end{tabular}\n\\caption{Code metadata (mandatory)}\n\\end{table}\n\\section{Installation and Usage}\n\nThere are two ways to install TedNet. For the sake that the source code of TedNet is submitted to GitHub, it is feasible to install from the downloaded code by command \\textbf{python setup.py install}. Compared with aforementioned fussy way, another one, the recommended way is to install TedNet trough PyPI~\\footnote{https:\/\/pypi.org\/project\/tednet} by command \\textbf{pip install tednet}. After installation, all tensor decomposition models of TedNet can be used.\n\nAn example of Tensor Ring is shown in Listing~\\ref{ll:tr}. Tensor ring decomposition can be used by import module \\textbf{tednet.tnn.tensor\\_ring}. The usage of other decomposition is the same and more details can be found in the Document~\\footnote{https:\/\/tednet.readthedocs.io}.\n\n\n\n\\begin{lstlisting}[language={Python}, label={ll:tr}, caption={\\bf The Usage of Tensor Ring Models}]\n# import tensor ring module\nimport tednet.tnn.tensor_ring as tr\n\n# TR-LSRM\nmodel = tr.TRLSTM([16, 16], [32, 32], [5, 5, 5, 5])\n\n# TR-Lenet-5\nmodel = tr.TRLeNet5(10, [6, 6, 6, 6])\n\n# TR-ResNet-20\nmodel = tr.TRResNet20([7, 7, 7, 7, 7, 7, 7], 10)\n\\end{lstlisting}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGauge-Higgs unification \\cite{Manton, Fairlie, Hosotani} is one of the attractive scenarios \nsolving the hierarchy problem without invoking supersymmetry. \nIn this scenario, \nHiggs doublet in the Standard Model (SM) is identified with \nthe extra spatial components of the higher dimensional gauge fields. \nRemarkable feature is that the quantum correction to Higgs mass is insensitive \nto the cutoff scale of the theory and calculable \nregardless of the non-renormalizability of higher dimensional gauge theory. \nThe reason is that the Higgs mass term as a local operator is forbidden \nby the higher dimensional gauge invariance. \nThe finite mass term is generated radiatively and expressed by the Wilson line phase as a non-local operator. \nThis fact has opened up a new avenue to the solution of the hierarchy problem \\cite{HIL}. \nSince then, much attention has been paid to the gauge-Higgs unification and \nmany interesting works have been done from various points of view \n\\cite{KLY}-\\cite{LHC}. \n\nThe finiteness of Higgs mass has been studied and verified in various models \nand types of compactification at one-loop level\\footnote{For the case of gravity-gauge-Higgs unification, \nsee \\cite{HLM}} \\cite{ABQ}-\\cite{LMH} and even at two loop level \\cite{MY, HMTY}. \nIt is natural to ask whether any other finite physical observables exist in the gauge-Higgs unification. \nThe naive guess is that such observables are in the gauge-Higgs sector of the theory if they ever exist. \nTwo of the present authors (C.S.L. and N.M.) studied the structure of divergences for S and T parameters \nin the gauge-Higgs unification since such parameters are described \nby higher dimensional gauge invariant operators with respect to gauge and Higgs fields, \nand are expected to be finite by virtue of the higher dimensional gauge symmetry. \nThe result is that both parameters are divergent (convergent) more than (in) five dimensions \nas expected from the naive power counting argument. \nHowever, a nontrivial prediction we have found, specific to the gauge-Higgs unification, is \nthat some linear combination of S and T parameters is finite even in six dimensions \\cite{LM}. \n\nIn our previous paper \\cite{ALM} we have found a more striking fact: \nwe have shown that the magnetic moment of fermion \nin the $(D+1)$ dimensional QED gauge-Higgs unification model \ncompactified on $S^1$ becomes finite for an arbitrary space-time dimension, \nregardless of the nonrenormalizability of the theory. \nActually, the reason is very simple. \nIn four dimensional space-time, \na dimension six gauge invariant local operator describes the magnetic moment: \n\\begin{eqnarray}\ni \\bar{\\psi}_L \\sigma^{\\mu\\nu} \\psi_R F_{\\mu\\nu} \\langle H \\rangle. \n\\label{MMM4}\n\\end{eqnarray}\nHowever, when included into the scheme of gauge-Higgs unification, \nthe Higgs doublet should be replaced by an extra space component of the higher dimensional gauge field $A_y$. \nThen the operator is forbidden by the higher dimensional gauge invariance, \nsince $A_y$ transforms inhomogeneously under the gauge transformation. \nThen, to preserve the gauge symmetry, \n$A_y$ should be further replaced by gauge covariant derivative $D_y$, \nand the relevant gauge invariant operator becomes \n\\begin{eqnarray}\ni \\bar{\\Psi} \\Gamma^{MN} D_L \\Gamma^L \\Psi F_{MN} \n\\label{MMMD}\n\\end{eqnarray}\nwhere $L, M$ and $N$ denote $D+1$ dimensional Lorentz indices. \nThe key observation of our argument is that the operator (\\ref{MMMD}), \nwhen $D_L$ is replaced by $\\langle D_L \\rangle$ with the gauge field $A_L$ replaced by its VEV, \nvanishes because of the on-shell condition $i \\langle D_L \\rangle \\Gamma^L \\Psi = 0$. \nFrom this fact, we can expect that the magnetic moment is finite \nand have shown that it is indeed the case by explicit diagrammatical calculations \\cite{ALM}. \nThis is the specific prediction of the gauge-Higgs unification to be contrasted with \nthe case of Randall-Sundrum model \\cite{DHR} or the universal extra dimension scenario \\cite{ADW, AD}, \nin which the magnetic moment of fermion diverges in the models with more than five space-time dimensions. \n\nAlthough this result was quite remarkable, the above model is too simple and not realistic. \nIn particular, \nthe gauge group $U(1)$ is too small to incorporate the standard model. \nIn this paper, we study more about the cancellation mechanism of ultraviolet (UV) divergences \nin a realistic gauge-Higgs unification model. \nWe consider $(D+1)$ dimensional $SU(3)$ gauge-Higgs unification model compactified \non an orbifold $S^1\/Z_2$ with a massive bulk fermion in a fundamental representation. \nThe orbifolding is indispensable to obtain the SM Higgs $SU(2)_L$ doublet \nsince Higgs originally behaves as an adjoint representation of the gauge group in the gauge-Higgs unification. \nWe consider here a simple orbifold $S^1\/Z_2$. \nIn the case of $S^1\/Z_2$, the bulk mass parameter of fermion must have odd $Z_2$ parity \nsince the fermion bulk mass term connects fermions with different chiralities and opposite $Z_2$ parities. \nIt is well known that the zero mode wave functions take an exponential profile along a compactified space coordinate \nand $D$-dimensional effective Yukawa couplings obtained by an overlap integral of zero mode wave functions \nare exponentially suppressed. \nIn this way, we can freely obtain the light fermion masses, which are otherwise of ${\\cal O}(M_W)$, \nby tuning the bulk mass parameter. \nOne might worry if our argument for the finiteness in the previous paper \nstill holds in the present orbifold model since the on-shell condition for the fermion is changed to \n$i \\Gamma^M \\langle D_M \\rangle \\Psi = M \\varepsilon(y) \\Psi$ \n($\\varepsilon(y):$ the sign function of $y$, the extra space coordinate) and also the brane localized operator \n\\begin{eqnarray}\ni \\bar{\\psi}_L \\Gamma^{\\mu\\nu} A_y \\Gamma^y \\psi_R F_{\\mu\\nu} \n\\label{braneAMM}\n\\end{eqnarray}\nseems to be allowed. \nHowever, these two worries are not necessary. \nAs for the first one, we note that the fermion $\\Psi$ in the operator (\\ref{MMMD}) \nshould be understood as the zero mode fermion. \nThough the operator (\\ref{MMMD}) does not vanish even after imposing the on-shell condition, \nthe remaining operator $M \\varepsilon(y) \\bar{\\Psi} \\Gamma^{MN} \\Psi F_{MN}$ has no correspondence \nin the standard model (in the standard model $\\bar{\\Psi}_L \\gamma^{MN} \\Psi_R$ is not gauge invariant), \nand therefore vanishes automatically for the zero-mode fermion $\\Psi$. \nAs for the second one, note that the shift symmetry \n$A_y \\to A_y + {\\rm const}$ is operative as a remnant of higher dimensional gauge symmetry\neven at the branes \\cite{GIQ}. \nTherefore, the brane localized operator (\\ref{braneAMM}) is forbidden by the shift symmetry. \nFurthermore, the UV finiteness is independent of how we compactify the extra space, \nbecause the information about the compactification is an infrared property of the theory. \nFrom these observations, we can expect the magnetic moment still to be finite \neven for the orbifold compactification and the presence of bulk mass term. \nThe primary purpose of this paper is to clarify the cancellation mechanism of UV divergences \nin the calculation of the anomalous magnetic moment in the framework of the present orbifold model. \n\n\nThis paper is organized as follows. \nIn the next section, we introduce our model and discuss the mass eigenvalues and mode functions of \nfermions and gauge bosons. \nIn section 3, we derive various interaction vertices, which are needed in the calculation of \nthe anomalous magnetic moment and generally valid without any approximation. \nIn section 4, we provide the general formulae for the contributions of $A_\\mu$ ($D$-dimensional gauge field) \nand $A_y$ ($D$-dimensional scalar) exchange diagrams to the anomalous magnetic moment. \nThe mechanism of cancellation of divergence is clarified in the case of small bulk mass in section 5. \nOur conclusions are given in section 6. \nThe detailed derivation of some useful properties \nconcerning the vertex functions are summarized in Appendices A and B. \n\n\n\\section{The Model}\n\nWe consider a $(D+1)$ dimensional $SU(3)$ gauge-Higgs unification model compactified on an orbifold $S^1\/Z_2$ \n($S^1$: a circle of radius $R$) with a massive bulk fermion in the fundamental representation of $SU(3)$ gauge group. \nThe Lagrangian is given by\n\\begin{eqnarray}\n{\\cal L} &=& -\\frac{1}{2} {\\rm Tr} (F_{MN} F^{MN}) + \\bar{\\Psi} (iD\\!\\!\\!\\!\/ - M \\varepsilon(y)) \\Psi\n\\label{Lagrangian1}\n\\end{eqnarray}\nwhere the indices $M,N = 0,1,2,3 \\cdots, D$, \nthe $(D+1)$ dimensional gamma matrices are $\\Gamma^M = (\\gamma^\\mu, i \\gamma^{D+1})~(\\mu = 0,1,2,3, \\cdots, D-1)$, \n\\begin{eqnarray}\n&&F_{MN} = \\partial_M A_N - \\partial_N A_M - i g [A_M, A_N], \\\\\n&&D\\!\\!\\!\\!\/ = \\Gamma^M (\\partial_M -i g A_M), \\\\\n&&\\Psi = (\\psi_1, \\psi_2, \\psi_3)^T. \n\\end{eqnarray}\n$g$ denotes a gauge coupling constant in $(D+1)$ dimensional gauge theory. \n$M$ is a bulk mass of the fermion. \nHere we note that the bulk fermion mass must have an odd $Z_2$ parity \nto be consistent with an orbifold projection: we thus introduce here the bulk mass \nproportional to the sign function $\\varepsilon(y)$ of compactified extra space coordinate $y$. \n\nThe periodic boundary condition is imposed along $S^1$ \nand $Z_2$ parity assignments are taken as\n\\begin{eqnarray}\nA_\\mu = \n\\left(\n\\begin{array}{ccc}\n(+,+) & (+,+) & (-,-) \\\\\n(+,+) & (+,+) & (-,-) \\\\\n(-,-) & (-,-) & (+,+) \\\\\n\\end{array}\n\\right), \\quad \nA_y = \n\\left(\n\\begin{array}{ccc}\n(-,-) & (-,-) & (+,+) \\\\\n(-,-) & (-,-) & (+,+) \\\\\n(+,+) & (+,+) & (-,-) \\\\\n\\end{array}\n\\right), \n\\end{eqnarray}\n\\begin{eqnarray}\n\\Psi = \n\\left(\n\\begin{array}{c}\n\\psi_{1L}(+,+) + \\psi_{1R}(-,-) \\\\\n\\psi_{2L}(+,+) + \\psi_{2R}(-,-) \\\\\n\\psi_{3L}(-,-) + \\psi_{3R}(+,+) \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhere $\\mu = 0,1,2,3, \\cdots, D-1$ \nand $(+,+)$ means that $Z_2$ parities are even at the fixed points $y=0$ and $y=\\pi R$, for instance.\n$L, R$ on fermion denotes the chiral projection operator (for even $D$) defined as \n$L = \\frac{1+\\gamma^{D+1}}{2}, R = \\frac{1-\\gamma^{D+1}}{2}$. \nAs can be seen from the KK mode expansion consistent with the boundary conditions \nonly the fields with $(+,+)$ parities have massless modes as\n\\begin{eqnarray}\nA_\\mu^{(0)} = \\frac{1}{2}\n\\left(\n\\begin{array}{ccc}\nW_\\mu^3 +\\frac{B_\\mu}{\\sqrt{3}} & \\sqrt{2} W_\\mu^+ & 0 \\\\\n\\sqrt{2} W_\\mu^- & -W_\\mu^3 + \\frac{B_\\mu}{\\sqrt{3}} & 0 \\\\\n0 & 0 & -\\frac{2}{\\sqrt{3}}B_\\mu \\\\\n\\end{array}\n\\right), \\quad \nA_y^{(0)} = \\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{ccc}\n0 & 0 & \\phi^+ \\\\\n0 & 0 & \\frac{h -i \\phi^0}{\\sqrt{2}} \\\\\n\\phi^- & \\frac{h +i \\phi^0}{\\sqrt{2}} & 0 \\\\\n\\end{array}\n\\right), \n\\label{0modegauge}\n\\end{eqnarray}\nwhere $W_\\mu^3, W_\\mu^\\pm$ and $B_\\mu$ are the $SU(2)_L$ and $U(1)_Y$ gauge fields, respectively. \nFrom this expression, we see that the gauge symmetry $SU(3)$ is broken to $SU(2)_L \\times U(1)_Y$ \nby the boundary conditions. \nFurthermore, the SM Higgs doublet is just embedded into the off-diagonal elements of $A_y^{(0)}$. \nAs for the fermion, we obtain massless modes \n\\begin{eqnarray}\n\\Psi^{(0)} = \n\\left(\n\\begin{array}{c}\n\\psi_{1L}^{(0)} \\\\\n\\psi_{2L}^{(0)} \\\\\n\\psi_{3R}^{(0)} \\\\\n\\end{array}\n\\right) \n= \n\\left(\n\\begin{array}{c}\nu_L \\\\\nd_L \\\\\nd_R \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhich shows that chiral fermions are realized by orbifold projection. \n\n\nSome comments on this model are in order. \nFirst, the predicted Weinberg angle of this model is not realistic, $\\sin^2 \\theta_W = 3\/4$. \nAs was also discussed in \\cite{GW}, \nthe present $SU(3)$ model in five dimension ($D=4$) is inconsistent with \nthe experimental requirements $\\sin^2 \\theta_W \\simeq 1\/4$ \nand $\\rho = m_W^2\/(m_Z^2\\cos^2 \\theta_W) =1$. \nPossible way to cure the problem is to introduce an extra U(1) or \nthe brane localized gauge kinetic term \\cite{SSS}. \nSecond, the up quark remains massless and we have no up-type Yukawa coupling. \nA possible way out of this situation is to introduce second-rank symmetric tensors of $SU(3)$ \n(${\\bf 6}$ dimensional representation) \\cite{CCP}. \nThird point is that the fermion in the fundamental representation of $SU(3)$ has no lepton. \nIn order to incorporate the leptons, a third-rank symmetric tensor \n(${\\bf 10}$ dimensional representation) must be introduced. \nWhen such higher dimensional representations are added to the theory, there appear some massless exotic fermions and they should be removed from low energy sector \nby adding the brane localized fields to form brane localized mass terms with the exotic states. \n\nSince our primary purpose in this paper is to clarify \nthe cancellation mechanism of UV divergence for the magnetic moment, \nin this paper we calculate the $g-2$ of down quark belonging to the triplet, as an example of the anomalous magnetic moment of fermion. This makes our calculation greatly simplified since the fundamental representation has no massless exotic fermions and we does not need introducing additional brane localized fermions and mass terms. We hope that our results for UV finiteness remains unchanged even for the case of well-discussed muon $g-2$, since the operator analysis given in the introduction is independent of the representation of fermion. We also hope that our calculation of $g-2$ of $d$ quark will be relevant for, $e.g.,$ the electric dipole moment of neutron. \n\nThroughout this paper, what we mean by ``realistic\" is in it's restricted sence, i.e. we mean that the gauge-Higgs model we consider contains the gauge group of the Standard Model and reproduces correct order of small Yukawa coupling relevant for the calculation of the magnetic moment. \n\n\n\\subsection{The mass eigenvalues and mode functions of fermions}\n\nIn order to derive $D$-dimensional effective Lagrangian and Feynman rules \nnecessary for the calculations of the magnetic moment, \nfirst we have to obtain the $D$-dimensional mass eigenvalues and corresponding mode functions of fermions. \n\nWe first focus on the down quark sector $\\psi_d \\equiv (\\psi_2, \\psi_3)^t$ of the triplet fermion \n$(\\psi_1, \\psi_2, \\psi_3)^t$. \nThe $D$-dimensional mass term reads as \n\\begin{eqnarray}\n{\\cal L}_{{\\rm mass}} &=& \\bar{\\psi}_d [\\Gamma^y (i \\partial_y + g \\langle A_y \\rangle) -M \\varepsilon(y) ] \\psi_d, \n\\label{massfermion1} \\\\\ng \\langle A_y \\rangle &\\equiv& g_D v \n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n1 & 0 \\\\\n\\end{array}\n\\right) \\quad \n\\left( g_D \\equiv \\frac{g}{\\sqrt{2\\pi R}} \\right).\n\\label{massfermion2}\n\\end{eqnarray}\nWe diagonalize the matrix $g \\langle A_y \\rangle$ by an orthogonal transformation, \n\\begin{eqnarray}\n\\tilde{\\psi}_d = \n\\left(\n\\begin{array}{c}\n\\tilde{\\psi}_2 \\\\\n\\tilde{\\psi}_3 \\\\\n\\end{array}\n\\right) \n\\equiv \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n-\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right) \\psi_d \n= \\frac{1}{\\sqrt{2}} \\left(\n\\begin{array}{c}\n\\psi_2+\\psi_3 \\\\\n-\\psi_2+\\psi_3 \\\\\n\\end{array}\n\\right). \n\\end{eqnarray}\nIn terms of $\\tilde{\\psi}_{2,3}$, (\\ref{massfermion1}) can be rewritten as\n\\begin{eqnarray}\n\\bar{\\tilde{\\psi}_2} [\\Gamma^y (i \\partial_y + g_D v) - M \\varepsilon(y) ] \\tilde{\\psi}_2 \n+ \\bar{\\tilde{\\psi}_3} [\\Gamma^y (i \\partial_y - g_D v) - M \\varepsilon(y) ] \\tilde{\\psi}_3. \n\\label{massfermion3}\n\\end{eqnarray}\nNow, we try to find mass eigenvalues and mode functions of $\\tilde{\\psi}_2$. \nExpanding the $\\tilde{\\psi}_2(x,y)$ as \n$\\tilde{\\psi}_2(x,y) = \\sum_n \\frac{1}{\\sqrt{2}} [ d_{L}^{(n)}(x) f_{d_L}^{(n)}(y) + d_{R}^{(n)}(x) f_{d_R}^{(n)}(y) ]$, \nwe found the equations of motion \n\\begin{eqnarray}\n&&[ (i \\partial_y + g_D v)^2 + M^2 -2M (\\delta(y) - \\delta (y-\\pi R) ) ] f_{d_L}^{(n)}(y) = m_n^2 f_L^{(n)}(y), \n\\label{modeeqL}\\\\\n&&[ (i \\partial_y + g_D v)^2 + M^2 +2M (\\delta(y) - \\delta (y-\\pi R) ) ] f_{d_R}^{(n)}(y) = m_n^2 f_R^{(n)}(y)\n\\label{modeeqR}\n\\end{eqnarray} \nwhere $m_n$ is the mass eigenvalue of $n$-th KK mode with $n$ being an arbitrary integer. \n\nLet us solve (\\ref{modeeqL}). \nBy use of the boundary condition that $f_{d_L}^{(n)}$ is continuous \nand $\\partial_y f_{d_L}^{(n)}$ has a discontinuity $-2M f_{d_L}^{(n)}$ \nat the fixed point $y=0$, \nthe solution is known to take a form \n\\begin{eqnarray}\nf_{d_L}^{(n)}(y) &=& e^{ig_D v y} \\left[ \nC_1^{(n)} \\left\\{ \\cos(\\sqrt{m_n^2-M^2}y) \n- \\frac{M}{\\sqrt{m_n^2-M^2}} \\varepsilon(y) \\sin(\\sqrt{m_n^2-M^2}y) \\right\\} \\right. \\nonumber \\\\\n&& \\left. + C_2^{(n)} \\sin(\\sqrt{m_n^2-M^2}y) \\right] \n\\label{generalsolL}\n\\end{eqnarray}\nwhere $C_{1,2}^{(n)}$ are integration constants. \n\nThen the similar boundary conditions \nat the fixed point $y=\\pi R$ can be written as\n\\begin{eqnarray}\n&&0 = (w-1) \\left(\\cos \\varphi_n -\\frac{M}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n \\right) C_1^{(n)} \n+ (w+1) \\sin \\varphi_n C_2^{(n)}, \n\\label{bc1} \\\\\n&&0 = -(w+1) \\frac{m_n^2}{m_n^2-M^2} \\sin \\varphi_n C_1^{(n)} \n+ (w-1) \\left(\\cos \\varphi_n + \\frac{M}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n \\right) C_2^{(n)} \\nonumber \n\\label{bc2} \\\\\n\\end{eqnarray}\nwhere $w$ is a Wilson loop $w \\equiv e^{2\\pi i g_D Rv}$ and $\\varphi_n \\equiv \\sqrt{m_n^2-M^2} \\pi R$. \nThese boundary conditions determine the mass eigenvalues $m_n$ through the condition\n\\begin{eqnarray}\n\\frac{m_n^2}{m_n^2-M^2} \\sin^2(\\sqrt{m_n^2-M^2}\\pi R) = \\sin^2(g_D v \\pi R), \n\\label{masscond}\n\\end{eqnarray}\nwhich cannot be solved analytically for $m_n$, in general. \nAs a check, \nif we consider the case of $v=0$, we obtain\n\\begin{eqnarray}\nm_0 = 0, \\quad m_n^2 = \\left( \\frac{n}{R} \\right)^2 + M^2 \n\\end{eqnarray}\nwhich is a well known result. \nWe take a sign convention such that \n\\begin{eqnarray}\nm_n = \\frac{n}{R} + g_D v \n\\end{eqnarray}\nin the case of $M=0$. \nFor $m_n$ satisfying (\\ref{masscond}), the ratio of $C_1^{(n)}$ and $C_2^{(n)}$ is known to be fixed as\n\\begin{eqnarray}\nC_1^{(n)} \/ C_2^{(n)} = \\sqrt{\\cos(\\varphi_n-\\alpha_n)} \/ -i \\frac{m_n}{\\sqrt{m_n^2 - M^2}} \\sqrt{\\cos(\\varphi_n + \\alpha_n)}\n\\end{eqnarray}\nwhere $\\tan \\alpha_n \\equiv \\frac{M}{\\sqrt{m_n^2-M^2}}$. \nThus, we have obtained the mode functions\n\\begin{eqnarray}\n&&f_{d_L}^{(n)}(y) = F_{M, M_W}^{(n)}(y), \\quad f_{d_R}^{(n)}(y) = F_{-M, M_W}^{(n)}(y), \\\\\n&&F_{M,M_W}^{(n)}(y) = e^{ig_D vy}C^{(n)} \n\\left[\n\\sqrt{\\cos(\\varphi_n -\\alpha_n)} \\cos(\\sqrt{m_n^2-M^2}|y| + \\alpha_n) \\right. \\nonumber \\\\\n&& \\left. \\hspace*{4cm} - \\varepsilon(n) i \\sqrt{\\cos(\\varphi_n + \\alpha_n)} \\sin(\\sqrt{m_n^2-M^2}y) \n\\right], \\label{modeL}\n\\label{modeR}\n\\end{eqnarray}\nwhere the normalization constant $C^{(n)}$ is given by\n\\begin{eqnarray}\nC^{(n)} &=& \\left[\n2\\pi R \\cos \\varphi_n \\cos \\alpha_n \n- \\frac{2}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n \\cos \\alpha_n \\sin^2 \\alpha_n \n\\right]^{-1\/2}. \n\\label{normalization}\n\\end{eqnarray}\nThe ``sign function\" $\\varepsilon(n)$ is defined as $1$ for $n \\ge 0$ and $-1$ for $n < 0$. \n\nAs a matter of fact when the mode functions are substituted in (\\ref{massfermion3}) \nwe get a mass $-i m_n$ for the Dirac fermion $d^{(n)} = d_L^{(n)} + d_R^{(n)}$. \nThus we perform a chiral transformation\n\\begin{eqnarray}\n\\hat{\\psi}_d \\equiv \n\\left(\n\\begin{array}{c}\n\\hat{\\psi}_2 \\\\\n\\hat{\\psi}_3 \\\\\n\\end{array}\n\\right)\n= e^{-i\\frac{\\pi}{4}\\gamma^{D+1}} \\tilde{\\psi}_2,\n\\end{eqnarray}\nso that $d^{(n)}$ has a mass $m_n$, with $\\hat{\\psi}_2$ still being mode-expanded as \n\\begin{eqnarray}\n\\hat{\\psi}_2(x,y) = \\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} \n\\left(\nf_{d_L}^{(n)} d_L^{(n)} + f_{d_R}^{(n)} d_R^{(n)}\n\\right). \n\\label{modeexpsi2}\n\\end{eqnarray}\n\nBecause of the orbifolding, \n$\\hat{\\psi}_3$ is not independent of $\\hat{\\psi}_2$. \nIn fact, the $Z_2$ parity assignment $\\psi_2(x,-y) = \\gamma^{D+1} \\psi_2(x,y)$ and \n$\\psi_3(x,-y) = -\\gamma^{D+1} \\psi_3(x,y)$ tells us\n\\begin{eqnarray}\n\\hat{\\psi}_3 (x,y) = -\\gamma^{D+1} \\hat{\\psi}_2(x,-y). \n\\end{eqnarray}\nSince $\\psi_1$ does not get a mass due to the VEV $v$ and the $Z_2$ parity assignment is \n$\\psi_1(x,-y) = \\gamma^{D+1} \\psi_1(x,y)$, $\\hat{\\psi}_1 \\equiv e^{-i\\frac{\\pi}{4}\\gamma^{D+1}} \\psi_1$ \nis mode-expanded as\n\\begin{eqnarray}\n\\hat{\\psi}_1(x,y) &=& \\sum_{n=1}^\\infty \n\\left\\{\n\\frac{1}{\\sqrt{2}} (f_{u_L}^{(n)}(y) + f_{u_L}^{(n)}(-y)) u_L^{(n)}(x) \n+ \\frac{1}{\\sqrt{2}} (f_{u_R}^{(n)}(y) - f_{u_R}^{(n)}(-y)) u_R^{(n)}(x)\n\\right\\} \\nonumber \\\\\n&& + f_{u_L}^{(0)}(y) u_L^{(0)}(x), \n\\end{eqnarray}\nwhere the mode functions take relatively simple forms\n\\begin{eqnarray}\nf_{u_L}^{(n)}(y) &=& F_{M, 0}^{(n)}(y) = \\frac{1}{\\sqrt{2\\pi R}} \n\\left[\n\\cos \\left(\\frac{n}{R} |y| + \\alpha_n^{(0)} \\right) -i \\sin \\left(\\frac{n}{R} y \\right)\n\\right], \\\\\nf_{u_L}^{(0)}(y) &=& F_{M, 0}^{(0)}(y) = \\sqrt{\\frac{M}{1-e^{-2\\pi MR}}} e^{-M|y|}, \\\\ \nf_{u_R}^{(n)}(y) &=& F_{-M, 0}^{(n)}(y) = \\frac{1}{\\sqrt{2\\pi R}} \n\\left[\n\\cos \\left(\\frac{n}{R} |y| - \\alpha_n^{(0)} \\right) -i \\sin \\left(\\frac{n}{R} y \\right)\n\\right], \n\\end{eqnarray}\nwhere $\\alpha_n^{(0)}$ is defined by\n\\begin{eqnarray}\n\\tan \\alpha_n^{(0)} \\equiv \\frac{M}{n\/R}~(n \\ge 1). \n\\end{eqnarray}\nDirac fermion $u^{(n)}(x) \\equiv u_L^{(n)}(x) + u_R^{(n)}(x)$ has a mass\n\\begin{eqnarray}\n\\tilde{m}_n = \\sqrt{\\left(\\frac{n}{R} \\right)^2 + M^2}~(n \\ge 1), \n\\end{eqnarray}\nwhile $u_L^{(0)}(x)$ remains as a massless state: $\\tilde{m}_0 = 0$. \n\nWe thus have obtained the mass eigenstates of the fermion, \n\\begin{eqnarray}\n\\hat{\\Psi} &\\equiv& \n\\left(\n\\begin{array}{c}\n\\hat{\\psi}_1 \\\\\n\\hat{\\psi}_2 \\\\\n\\hat{\\psi}_3 \\\\\n\\end{array}\n\\right) \n= e^{-i \\frac{\\pi}{4}\\gamma_y} {\\cal O} \\Psi \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{c}\n\\sum_{n=1}^\\infty \\frac{1}{\\sqrt{2}} \n\\left\\{\n(f_{u_L}^{(n)}(y) + f_{u_L}^{(n)}(-y)) u_L^{(n)}(x) \n+ (f_{u_R}^{(n)}(y) - f_{u_R}^{(n)}(-y)) u_R^{(n)}(x)\n\\right\\}\n+ f_{u_L}^{(0)}(y) u_L^{(0)}(x) \\\\\n\\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} (f_{d_L}^{(n)}(y) d_L^{(n)}(x) + f_{d_R}^{(n)}(y) d_R^{(n)}(x)) \\\\\n\\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} (-f_{d_L}^{(n)}(-y) d_L^{(n)}(x) + f_{d_R}^{(n)}(-y) d_R^{(n)}(x)) \\\\\n\\end{array}\n\\right), \\nonumber \\\\\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\cal O} &\\equiv& \n\\left(\n\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n0 & -\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right). \n\\label{rotation}\n\\end{eqnarray}\nLet us comment on the quantum mechanical supersymmetry (QMS) hidden in the mode functions of fermions. \nIt has been demonstrated that the mode functions of $A_\\mu$ and $A_y$ form a supermultiplet \nfor each non-zero KK modes, which reflects the Higgs-like mechanism to form massive gauge bosons \n$A_\\mu^{(n)}(n \\ne 0)$ \\cite{LNSS}. \nA similar thing is expected to take place in the fermion sector; \n$f_L^{(n)}$ and $f_R^{(n)}$ are expected to form a supermultiplet, \nas they are ``partner\" to form a massive Dirac fermion. \nIn fact, they are known to be related by a supercharge $Q$: \n\\begin{eqnarray}\n&&Q\n\\left(\n\\begin{array}{c}\nF_{M, M_W}^{(n)} \\\\\nF_{-M, M_W}^{(n)} \\\\\n\\end{array}\n\\right)\n= m_n \n\\left(\n\\begin{array}{c}\nF_{M, M_W}^{(n)} \\\\\nF_{-M, M_W}^{(n)} \\\\\n\\end{array}\n\\right), \n\\label{2.35}\\\\\n&&Q = \n\\left(\n\\begin{array}{cc}\n0 & i \\partial_y + M_W -i M \\varepsilon(y) \\\\\ni \\partial_y + M_W +i M \\varepsilon(y) & 0 \\\\\n\\end{array}\n\\right)\n\\label{2.36}\n\\end{eqnarray}\nwith $Q^2$ giving the differential operators in (\\ref{modeeqL}) and (\\ref{modeeqR}), \nnamely the Hamiltonian in QMS. \n\n\\subsection{The mass eigenvalues and mode functions of gauge bosons}\nNext, we turn to the mass eigenvalues and mode functions of $D$-dimensional gauge bosons and Higgs scalars, \n$i.e.~A_\\mu$ and $A_y$. \nFirst we explicitly write the gauge bosons and Higgs scalar fields as follows, \n\\begin{eqnarray}\nA_\\mu(x, y) &=& \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_\\mu}{\\sqrt{3}} & \\frac{W^+_\\mu}{\\sqrt{2}} & \\frac{\\phi^+_\\mu}{\\sqrt{2}} \\\\\n\\frac{W^-_\\mu}{\\sqrt{2}} & -\\frac{\\gamma_\\mu}{2\\sqrt{3}} -\\frac{Z_\\mu}{2} & \\frac{h_\\mu-i\\phi^0_\\mu}{2} \\\\\n\\frac{\\phi^-_\\mu}{\\sqrt{2}} & \\frac{h_\\mu +i \\phi^0_\\mu}{2} & -\\frac{\\gamma_\\mu}{2\\sqrt{3}} + \\frac{Z_\\mu}{2} \\\\\n\\end{array}\n\\right), \n\\label{gaugematrix} \\\\\nA_y(x, y) &=& \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_y}{\\sqrt{3}} & \\frac{W^+_y}{\\sqrt{2}} & \\frac{\\phi^+}{\\sqrt{2}} \\\\\n\\frac{W^-_y}{\\sqrt{2}} & -\\frac{\\gamma_y}{2\\sqrt{3}} -\\frac{Z_y}{2} & \\frac{h-i\\phi^0}{2} \\\\\n\\frac{\\phi^-}{\\sqrt{2}} & \\frac{h +i \\phi^0}{2} & -\\frac{\\gamma_y}{2\\sqrt{3}} + \\frac{Z_y}{2} \\\\\n\\end{array}\n\\right). \n\\label{Aymatrix}\n\\end{eqnarray}\nEach field has a mode expansion depending on its $Z_2$ parity, \n\\begin{eqnarray}\nA_{\\mu,y}(x,y) &=& \\frac{1}{\\sqrt{2\\pi R}} A_{\\mu,y}^{(0)}(x) \n+ \\frac{1}{\\sqrt{\\pi R}} \\sum_{n=1}^\\infty A_{\\mu,y}^{(n)}(x) \\cos \\left(\\frac{n}{R}y \\right)~({\\rm even}), \\\\\nA_{\\mu,y}(x,y) &=& \n\\frac{1}{\\sqrt{\\pi R}} \\sum_{n=1}^\\infty A_{\\mu,y}^{(n)}(x) \\sin \\left(\\frac{n}{R}y \\right)~({\\rm odd})\n\\end{eqnarray}\nPutting these mode functions into the term ${\\rm Tr}(F_{\\mu y})^2$ and integrating over $y$ coordinate lead to \nthe necessary quadratic terms concerning nonzero KK modes;\n\\begin{eqnarray}\n{\\cal L}_{{\\rm quadratic}} &=& \\sum_{n=1}^\\infty \\left[ \n\\frac{1}{2} \\left(\\partial_\\mu \\gamma_y^{(n)} + \\frac{n}{R} \\gamma_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2} \\left(\\partial_\\mu h^{(n)} - \\frac{n}{R} h_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2}[(\\partial_\\mu Z_y^{(n)})^2 + (\\partial_\\mu \\phi^{0(n)})^2] \\right. \\nonumber \\\\\n&& \\left. + \\frac{1}{2}\\left[ \\left( \\frac{n}{R} \\right)^2 + (2g_D v)^2 \\right] [ (Z_\\mu^{(n)})^2 + (\\phi_\\mu^{0(n)})^2 ]\n+ 4\\frac{n}{R} g_D v Z_\\mu^{(n)} \\phi^{\\mu0(n)} \\right. \\nonumber \\\\\n&& \\left. + (\\partial^\\mu Z_y^{(n)}) \\left(\\frac{n}{R} Z_\\mu^{(n)} +2g_D v \\phi_\\mu^{0(n)} \\right) \n- (\\partial^\\mu \\phi^{0(n)}) \\left(\\frac{n}{R} \\phi_\\mu^{0(n)} +2g_D v Z_\\mu^{(n)} \\right) \\right. \\nonumber \\\\\n&& \\left. +|\\partial_\\mu W_y^{+(n)}|^2 + |\\partial_\\mu \\phi^{+(n)}|^2 \n+ \\left[\\left( \\frac{n}{R} \\right)^2 + (g_D v)^2 \\right](|W_\\mu^{+(n)}|^2 + |\\phi_\\mu^{+(n)}|^2) \\right. \\nonumber \\\\\n&& \\left. + \\left\\{\ni\\frac{n}{R} 2g_D v W_\\mu^{+(n)} \\phi^{-(n)\\mu} \n+(\\partial^\\mu W_y^{+(n)}) \\left(\\frac{n}{R} W_\\mu^{-(n)} +ig_D v \\phi_\\mu^{-(n)} \\right) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. \n+(\\partial^\\mu \\phi^{+(n)}) \\left(-\\frac{n}{R} \\phi_\\mu^{-(n)} +ig_D v W_\\mu^{-(n)} \\right) + {\\rm h.c.}\n\\right\\} \\right]\n\\end{eqnarray}\nThe mixings between $Z$ and $\\phi^0$, $W^\\pm$ and $\\phi^\\pm$ due to the VEV $v$ \nnecessitates the following diagonalization,\n\\begin{eqnarray}\n&&\\left(\n\\begin{array}{c}\nZ_{1\\mu}^{(n)} \\\\\nZ_{2\\mu}^{(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nZ_{\\mu}^{(n)} \\\\\n\\phi_{\\mu}^{0(n)} \\\\\n\\end{array}\n\\right), \\quad \n\\left(\n\\begin{array}{c}\nZ_{1y}^{(n)} \\\\\nZ_{2y}^{(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nZ_y^{(n)} \\\\\n\\phi^{0(n)} \\\\\n\\end{array}\n\\right), \n\\label{rotationZ}\\\\\n&&\\left(\n\\begin{array}{c}\nW_{1\\mu}^{+(n)} \\\\\nW_{2\\mu}^{+(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nW_{\\mu}^{+(n)} \\\\\ni \\phi_{\\mu}^{+(n)} \\\\\n\\end{array}\n\\right), \\quad \n\\left(\n\\begin{array}{c}\nW_{1y}^{+(n)} \\\\\nW_{2y}^{+(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nW_y^{+(n)} \\\\\ni \\phi^{+(n)} \\\\\n\\end{array}\n\\right). \\nonumber \n\\label{rotationW}\\\\\n\\end{eqnarray}\nThus, the quadratic terms including the zero mode sector read as\n\\begin{eqnarray}\n{\\cal L}_{{\\rm quadratic}} &=& \\sum_{n=1}^\\infty \\left[ \n\\frac{1}{2} \\left(\\partial_\\mu \\gamma_y^{(n)} + \\frac{n}{R} \\gamma_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2} \\left(\\partial_\\mu h^{(n)} - \\frac{n}{R} h_\\mu^{(n)} \\right)^2 \\right. \\nonumber \\\\\n&& \\left. \n+ \\frac{1}{2} \\left[ \\partial_\\mu Z_{1y}^{(n)} + \\left(\\frac{n}{R} + M_Z \\right) Z_{1\\mu}^{(n)} \\right]^2 \n+ \\frac{1}{2} \\left[ \\partial_\\mu Z_{2y}^{(n)} + \\left(\\frac{n}{R} - M_Z \\right) Z_{2\\mu}^{(n)} \\right]^2 \\right. \\nonumber \\\\\n&& \\left. + \\left|\\partial_\\mu W_{1y}^{+(n)} + \\left(\\frac{n}{R} + M_W \\right) W_{1\\mu}^{+(n)} \\right|^2 \n+ \\left|\\partial_\\mu W_{2y}^{+(n)} + \\left(\\frac{n}{R} - M_W \\right) W_{2\\mu}^{+(n)} \\right|^2 \\right] \\nonumber \\\\\n&& \n+ \\frac{1}{2}(\\partial_\\mu h^{(0)})^2 + \\frac{1}{2}(\\partial_\\mu \\phi^{0(0)} + M_Z Z_\\mu^{(0)})^2 \n+|\\partial_\\mu (i\\phi^{+(0)}) +M_W W_\\mu^{+(0)}|^2\n\\label{gaugemass}\n\\end{eqnarray}\nwhere $M_Z = 2g_D v =2M_W$ (recall that $\\sin \\theta_W = \\frac{\\sqrt{3}}{2}$). \n\nWe note that the Higgs-like mechanism works between the partner of the pairs $(Z_{1,2\\mu}^{(n)}, Z_{1,2y}^{(n)})$ \nand $(W_{1,2\\mu}^{+(n)}, W_{1,2y}^{+(n)})$. \nAs we will see later, this structure is crucial for the cancellation of UV divergences \nin the calculation of the anomalous moment. \nThe mass spectrum for degenerate pairs of the gauge bosons and would-be N-G bosons \n(in the 't Hooft-Feynman gauge) are summarized as\n\\begin{eqnarray}\n(Z_{1\\mu}^{(n)}, Z_{1y}^{(n)}) &:& \\frac{n}{R} + M_Z, \\\\\n(Z_{2\\mu}^{(n)}, Z_{2y}^{(n)}) &:& \\frac{n}{R} - M_Z, \\\\\n(W_{1\\mu}^{+(n)}, W_{1y}^{+(n)}) &:& \\frac{n}{R} + M_W, \\\\\n(W_{2\\mu}^{+(n)}, W_{2y}^{+(n)}) &:& \\frac{n}{R} - M_W. \n\\end{eqnarray}\n\nWe thus find that just as in the sector of fermions, the mass eigenstates are obtained \nby the orthogonal transformation due to ${\\cal O}$ in (\\ref{rotation}): \n\\begin{eqnarray}\n\\tilde{A}_\\mu &\\equiv& {\\cal O} A_\\mu {\\cal O}^t \n= \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_\\mu}{\\sqrt{3}} & \\frac{W_\\mu^+ + \\phi_\\mu^+}{2} & \\frac{-W_\\mu^+ + \\phi_\\mu^+}{2} \\\\\n\\frac{W_\\mu^- + \\phi_\\mu^-}{2} & -\\frac{\\gamma_\\mu}{2 \\sqrt{3}} + \\frac{h_\\mu}{2} & \\frac{Z_\\mu -i \\phi_\\mu^0}{2} \\\\\n\\frac{-W_\\mu^- + \\phi_\\mu^-}{2} & \\frac{Z_\\mu +i \\phi_\\mu^0}{2} & -\\frac{\\gamma_\\mu}{2 \\sqrt{3}} - \\frac{h_\\mu}{2} \\\\\n\\end{array}\n\\right) \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) & \n\\frac{1}{2} \\hat{W}_\\mu^{+(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2} \\hat{W}_\\mu^{-(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \n-\\frac{1}{2 \\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) + \\frac{1}{2}h_\\mu^{(n)}S_n \\\\\n-\\frac{1}{2} \\hat{W}_\\mu^{-(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \\frac{1}{2}\\hat{Z}_\\mu^{(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n&& \\left.\n\\begin{array}{c}\n-\\frac{1}{2} \\hat{W}_\\mu^{+(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2}\\hat{Z}_\\mu^{(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{1}{2 \\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) - \\frac{1}{2}h_\\mu^{(n)} S_n \\\\\n\\end{array}\n\\right), \n\\label{gaugefield1} \\\\\n\\tilde{A}_y &\\equiv& {\\cal O} A_y {\\cal O}^t \n= \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_y}{\\sqrt{3}} & \\frac{W_y^+ + \\phi^+}{2} & \\frac{-W_y^+ + \\phi^+}{2} \\\\\n\\frac{W_y^- + \\phi^-}{2} & -\\frac{\\gamma_y}{2 \\sqrt{3}} + \\frac{h}{2} & \\frac{Z_y -i \\phi^0}{2} \\\\\n\\frac{-W_y^- + \\phi^-}{2} & \\frac{Z_y +i \\phi^0}{2} & -\\frac{\\gamma_y}{2 \\sqrt{3}} - \\frac{h}{2} \\\\\n\\end{array}\n\\right) \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} \\gamma^{(n)}_y S_n & \n\\frac{1}{2} \\hat{\\phi}^{+(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2} \\hat{\\phi}^{-(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \n-\\frac{1}{2 \\sqrt{3}} \\gamma^{(n)}_y S_n + \\frac{1}{2} \\left( h^{(n)} C_n + \\frac{h^{(0)}}{\\sqrt{2\\pi R}} \\right) \\\\\n\\frac{1}{2} \\hat{\\phi}^{-(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \\frac{i}{2} \\hat{\\phi}^{0(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n&& \\left.\n\\begin{array}{c}\n\\frac{1}{2} \\hat{\\phi}^{+(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{i}{2}\\hat{\\phi}^{0(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{1}{2 \\sqrt{3}} \\gamma^{(n)}_y S_n -\\frac{1}{2} \\left( h^{(n)} C_n + \\frac{h^{(0)}}{\\sqrt{2\\pi R}} \\right) \\\\\n\\end{array}\n\\right), \n\\end{eqnarray}\nwhere $C_n \\equiv \\frac{\\cos \\left(\\frac{n}{R}y \\right)}{\\sqrt{\\pi R}}, S_n \\equiv \\frac{\\sin \\left(\\frac{n}{R}y \\right)}{\\sqrt{\\pi R}}$, \nand the mode sum is for $n \\ge 1$ in the case of $C_n$ and $S_n$, while the sum is for all integer \nin the case of $e^{\\pm i \\frac{n}{R} y}\/\\sqrt{2\\pi R}$. \nWe have used the following notation, combining $Z_{1\\mu,y}^{(n)}, W_{1\\mu,y}^{+(n)}$ with $Z_{2\\mu,y}^{(n)}, W_{2\\mu,y}^{+(n)}$, \nrespectively: \n\\begin{eqnarray}\n&& \\hat{Z}_\\mu^{(n)} = Z_{1\\mu}^{(n)}, \\hat{Z}_\\mu^{(-n)} = Z_{2\\mu}^{(n)}~(n \\ge 1), \\hat{Z}_\\mu^{(0)} = Z_\\mu^{(0)} \\\\\n&& \\hat{\\phi}^{0(n)} = -Z_{1y}^{(n)}, \\hat{\\phi}^{0(-n)} = Z_{2y}^{(n)}~(n \\ge 1), \\hat{\\phi}^{0(0)} = \\phi^{0(0)} \\\\\n&& \\hat{W}_\\mu^{+(n)} = W_{1\\mu}^{+(n)}, \\hat{W}_\\mu^{+(-n)} = W_{2\\mu}^{+(n)}~(n \\ge 1), \\hat{W}_\\mu^{+(0)} = W_\\mu^{+(0)} \\\\\n&& \\hat{\\phi}^{+(n)} = -i W_{1y}^{+(n)}, \\hat{\\phi}^{+(-n)} = i W_{2y}^{+(n)}~(n \\ge 1), \\hat{\\phi}^{+(0)} = \\phi^{+(0)},\n\\end{eqnarray}\nso that the mass squared of $(\\hat{Z}_\\mu^{(n)}, \\hat{\\phi}^{0(n)})$ and $(\\hat{W}_\\mu^{+(n)}, \\hat{\\phi}^{+(n)})$ \nare $\\left(\\frac{n}{R} + M_Z \\right)^2$ and $\\left(\\frac{n}{R} + M_W \\right)^2$, respectively. \n\n\\section{Interaction vertices}\nHaving obtained the mode functions of the fermion $\\hat{\\Psi}$ and the ``gauge-Higgs\" sector $\\tilde{A}_\\mu, \\tilde{A}_y$, \nwe are ready to derive $D$-dimensional gauge and Yukawa interaction vertices of $\\hat{\\Psi}$ \nby the overlap integral of the relevant mode functions with respect to $y$, \nin the relevant part of the Lagrangian, \n\\begin{eqnarray}\ng \\bar{\\hat{\\Psi}} [\\tilde{A}_\\mu \\gamma^\\mu - \\tilde{A}_y]\\hat{\\Psi}. \n\\end{eqnarray}\n\nLet us start with the interaction vertex of $\\gamma_\\mu$, with $d$ quark, \nwhich is obtained by an integral\n\\begin{eqnarray}\n&&-\\frac{g}{2\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_\\mu(x,y) \n\\left[\n\\bar{\\hat{\\psi}}_2(x,y) \\gamma^\\mu \\hat{\\psi}_2(x,y) + \\bar{\\hat{\\psi}}_3(x,y) \\gamma^\\mu \\hat{\\psi}_3(x,y) \n\\right] \\nonumber \\\\\n&=& -\\frac{g}{\\sqrt{3}} \\int_{-\\pi R}^{\\pi R}dy \\gamma_\\mu(x,y) \\bar{\\hat{\\psi}}_2(x,y) \\gamma^\\mu \\hat{\\psi}_2(x,y) \n\\label{Intvtx}\n\\end{eqnarray}\nwhere $\\gamma_\\mu(x,y)$ denotes the photon field and \nthe properties $\\hat{\\psi}_3(x,y) = -\\gamma^{D+1} \\hat{\\psi}_2(x,-y)$ \nand $\\gamma_\\mu(x,-y) = \\gamma_\\mu(x,y)$ are used. \nSubstituting (\\ref{modeexpsi2}) and the $\\gamma_\\mu$ part of (\\ref{gaugefield1}) in (\\ref{Intvtx}), \nwe get the $D$-dimensional vertex function necessary for the calculation of $g-2$ of the zero-mode $d^{(0)}$, \n\\begin{eqnarray}\n-\\frac{g}{2\\sqrt{3}} \\gamma_\\mu^{(m)} \\bar{d}^{(n)} \\gamma^\\mu V_{\\gamma_\\mu}^{nm} d^{(0)},\n\\label{3.3}\n\\end{eqnarray}\nwhere the vertex function $V_{\\gamma_\\mu}^{nm}$ is defined as\n\\begin{eqnarray}\nV_{\\gamma_\\mu}^{nm} &=& I_c^{nm} (M, M_W,; M, M_W)L + I_c^{nm} (-M, M_W,; -M, M_W)R \\nonumber \\\\\n&=& (L + (-1)^{n+m}R) I_c^{nm}(M, M_W; M, M_W)~(m \\ge 1), \\label{3.4} \\\\\nV_{\\gamma_\\mu}^{n0} &=& (L + (-1)^{n}R) I^{n0}(M, M_W; M, M_W)\n\\end{eqnarray}\nin terms of functions\n\\begin{eqnarray}\n&&I_c^{nm}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \n\\cos \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y)~(m \\ge 1), \n\\label{3.6}\\\\\n&&I^{n0}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{2\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \nF_{M_1, M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y). \n\\label{3.7}\n\\end{eqnarray}\nHere we have used the fact that the vertices for the left-handed and right-handed $d$ quark are related \nwith the factor $(-1)^{m+n}$, as is shown in the Appendix A. \nFor the case of $M_1=M_3=M$ and $M_2=M_4=M_W$, the explicit forms of these functions are\n\\begin{eqnarray}\n&&I_c^{nm}(M, M_W; M, M_W) \n= \\frac{1}{\\sqrt{\\pi R}} C^{(n)}(\\varphi_n, \\alpha_n)^* C^{(0)}(\\varphi_0, \\alpha_0) \n\\left\\{\n\\sqrt{\\cos(\\varphi_n-\\alpha_n)} \\sqrt{\\cos(\\varphi_0 -\\alpha_0)} \\right. \\nonumber \\\\\n&& \\left. \\left[\n-\\frac{\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\alpha_n + \\varphi_0 + \\alpha_0) - \\sin(\\alpha_n + \\alpha_0)\n\\right) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. \n-\\frac{\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\alpha_n - \\varphi_0 - \\alpha_0) - \\sin(\\alpha_n - \\alpha_0)\n\\right)\n\\right] \\right. \\nonumber \\\\\n&& \\left. + \\varepsilon(n) \n\\sqrt{\\cos(\\varphi_n+\\alpha_n)} \\sqrt{\\cos(\\varphi_0 + \\alpha_0)} \\right. \\nonumber \\\\\n&& \\left. \\left[\n\\frac{\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\varphi_0) \n\\right) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. \n-\\frac{\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n - \\varphi_0) \n\\right)\n\\right]\n\\right\\}, \\label{Icnm}\\\\\n&& I^{n0}(M, M_W; M, M_W) = \\delta_{n0} \\frac{1}{\\sqrt{2\\pi R}}. \n\\label{In0} \n\\end{eqnarray}\nNote that (\\ref{In0}) is nothing but the orthonormality of the mode functions. \nWe therefore find that the $\\gamma_\\mu^{(0)}$ coupling is of ordinary form: \n\\begin{eqnarray}\n-\\frac{g}{2\\sqrt{3}} \\frac{1}{\\sqrt{2\\pi R}} \\gamma_\\mu^{(0)} \\bar{d}^{(0)} \\gamma^\\mu d^{(0)} \n= -\\frac{e_D}{3} \\gamma_\\mu^{(0)} \\bar{d}^{(0)} \\gamma^\\mu d^{(0)}, \n\\end{eqnarray}\nwhere $\\frac{g}{\\sqrt{2\\pi R}} =g_D, g_D \\sin \\theta_W = \\frac{\\sqrt{3}}{2} g_D = e_D$ are used. \n\nThe interaction vertex of $\\gamma_y$ with $d$ quark is derived by a similar step \nas in the case of $\\gamma_\\mu$ vertex: \n\\begin{eqnarray}\n&&\\frac{g}{2\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_y(x,y) \n\\left[\n\\bar{\\hat{\\psi}}_2(x,y) \\hat{\\psi}_2(x,y) + \\bar{\\hat{\\psi}}_3(x,y) \\hat{\\psi}_3(x,y)\n\\right] \\nonumber \\\\\n&=& \\frac{g}{\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_y(x,y) \\bar{\\hat{\\psi}}_2(x,y) \\hat{\\psi}_2(x,y) \n\\to \\frac{g}{2\\sqrt{3}} \\gamma^{(m)}_y \\bar{d}^{(n)} V_{\\gamma_y}^{nm} d^{(0)}~(m \\ge 1), \n\\end{eqnarray}\nwhere the vertex function $V_{\\gamma_y}^{nm}$ is defined as \n\\begin{eqnarray}\nV_{\\gamma_y}^{nm} = (L + (-1)^{n+m}R) I_s^{nm}(-M, M_W; M, M_W)\n\\end{eqnarray}\nin terms of a function\n\\begin{eqnarray}\nI_s^{nm}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \n\\sin \\left(\\frac{m}{R}y \\right) F_{M_1,M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y)~(m \\ge 1). \n\\label{3.13}\n\\end{eqnarray}\n\nAs the matter of fact, \n$I_s^{nm}$ is not an independent function, and is related to $I_c^{nm}$. \nThis is because in the non-zero KK mode sector $\\gamma_y^{(m)}$ behaves as a would-be N-G boson \nto be ``eaten\" by $\\gamma_\\mu^{(m)}$; Higgs-like mechanism is operative and \nthe coupling of $\\gamma_y^{(m)}$ should be equivalent to that of the longitudinal component \nof $\\gamma_\\mu^{(m)}$ (``equivalence theorem\"). \nOr such relation may be attributed to the quantum mechanical supersymmetry, as is shown in the Appendix B. \nAnyway, we get a relation between $I_s^{nm}$ and $I_c^{nm}$, which means\n\\begin{eqnarray}\nV_{\\gamma_y}^{nm} = i \\frac{m_n-(-1)^{m+n}m_d}{\\frac{m}{R}} V_{\\gamma_\\mu}^{nm}~(m \\ge 1). \n\\end{eqnarray}\n\nThe interaction terms of $h_\\mu$ and $h$ are derived in a similar way, \nand we just give the result: \n\\begin{eqnarray}\n&&\\frac{g}{2} h_\\mu^{(n)} \\bar{d}^{(n)} \\gamma^\\mu V_{h_\\mu}^{nm} d^{(0)}, \\\\\n&&V_{h_\\mu}^{nm} = (L + (-1)^{m+n}R) I_s(M, M_W; M, M_W)~(m \\ge 1), \\\\\n&&-\\frac{g}{2} h^{(n)} \\bar{d}^{(n)} V_h^{nm} d^{(0)}, \\\\\n&&V_h^{nm} = \\left\\{\n\\begin{array}{c}\n(L + (-1)^{m+n} R) I_c^{nm}(-M, M_W; M, M_W)~(m \\ge 1) \\\\\n(L + (-1)^{n} R) I^{n0}(-M, M_W; M, M_W)~(m=0). \\\\\n\\end{array}\n\\right.\n\\end{eqnarray}\nAgain a relation holds:\n\\begin{eqnarray}\nV_h^{nm} = -i \\frac{m_n - (-1)^{m+n}m_d}{\\frac{m}{R}} V_{h_\\mu}^{nm}~(m \\ge 1). \n\\end{eqnarray}\n\n\nIn the case of interaction terms of $\\hat{Z}_\\mu$ and $\\hat{\\phi}^0$, \nthere appears a transition between $\\hat{\\psi}_2$ and $\\hat{\\psi}_3$ and \nthe vertex functions are described by new types of functions, $\\tilde{I}_c$ and $\\tilde{I}_s$.\nNamely, the interaction term of $\\hat{Z}_\\mu$ is (for an arbitrary integer $m$)\n\\begin{eqnarray}\n&&-\\frac{g}{2} \\hat{Z}_\\mu^{(m)} \\bar{d}^{(n)} \\gamma^\\mu V_{Z_\\mu}^{nm} d^{(0)}, \\\\\n&&V_{Z_\\mu}^{nm} = (L - (-1)^{m+n}R)\\frac{1}{\\sqrt{2}} \n\\left( \\tilde{I}_c^{nm}(M, M_W; M, M_W) + i \\tilde{I}_s^{nm}(M, M_W; M, M_W) \\right), \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\tilde{I}_c^{nm}(M_1, M_2; M_3, M_4) &\\equiv& \\frac{1}{\\sqrt{\\pi R}} \n\\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(-y)^* F_{M_3, M_4}^{(0)}(y), \\\\\n\\tilde{I}_s^{nm}(M_1, M_2; M_3, M_4) &\\equiv& \\frac{1}{\\sqrt{\\pi R}} \n\\int_{-\\pi R}^{\\pi R} dy \\sin \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(-y)^* F_{M_3, M_4}^{(0)}(y). \n\\end{eqnarray}\nThe interaction term of $\\hat{\\phi}^0$ reads as\n\\begin{eqnarray}\n&&-\\frac{g}{2} \\hat{\\phi}^{0(m)} \\bar{d}^{(n)} V_{\\phi^0}^{nm} d^{(0)}, \\nonumber \\\\\n&&V_{\\phi^0}^{nm} = (L - (-1)^{m+n}R)\\frac{i}{\\sqrt{2}} \n\\left( \\tilde{I}_c^{nm}(-M, M_W; M, M_W) + i \\tilde{I}_s^{nm}(-M, M_W; M, M_W) \\right). \\nonumber \\\\\n\\end{eqnarray}\nSome relations hold between $\\tilde{I}_s^{nm}$ and $\\tilde{I}_c^{nm}$ which mean, \n\\begin{eqnarray}\nV_{\\phi^0}^{nm} = i\\frac{m_n + (-1)^{m+n}m_d}{\\frac{m}{R} + M_Z}V_{Z_\\mu}^{nm}. \n\\end{eqnarray}\n\nFinally, in the case of interaction terms of $\\hat{W}_\\mu^+$ and $\\hat{\\phi}^+$ \nthe transition between $\\hat{\\psi}_1$ and $\\hat{\\psi}_2$ or $\\hat{\\psi}_3$ appears, \nwhich makes the vertex function a little complicated: \nfor $n \\ge 0$ and an arbitrary integer $m$, \n\\begin{eqnarray}\n&&\\frac{g}{2} \\hat{W}_\\mu^{+(m)} \\bar{u}^{(n)} \\gamma^\\mu V_{W_\\mu^+}^{nm} d^{(0)}, \\\\\n&&V_{W_\\mu^+}^{nm} = \n\\left\\{\n\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\left( I_c^{nm}(M, 0; M, M_W) +i I_s^{nm}(M, 0; M, M_W) \\right) (L + (-1)^{n+m}R) \\\\\n+ \\frac{1}{\\sqrt{2}} \\left( \\tilde{I}_c^{nm}(M, 0; M, M_W) +i \\tilde{I}_s^{nm}(M, 0; M, M_W) \\right) (L - (-1)^{n+m}R)~(n \\ge 1), \\\\\n\\left[I_c^{0m}(M, 0; M, M_W) +i I_s^{0m}(M, 0; M, M_W) \\right] L~(n=0), \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n\\\\\n&&-\\frac{g}{2} \\hat{\\phi}^{+(m)} \\bar{u}^{(n)} V_{\\phi^+}^{nm} d^{(0)}, \\\\\n&&V_{\\phi^+}^{nm} = \n\\left\\{\n\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\left( I_c^{nm}(-M, 0; M, M_W) +i I_s^{nm}(-M, 0; M, M_W) \\right) (L + (-1)^{n+m}R) \\\\\n-\\frac{1}{\\sqrt{2}} \\left( \\tilde{I}_c^{nm}(-M, 0; M, M_W) +i \\tilde{I}_s^{nm}(-M, 0; M, M_W) \\right) (L - (-1)^{n+m}R)~(n \\ge 1), \\\\\n\\left[I_c^{0m}(-M, 0; M, M_W) +i I_s^{0m}(-M, 0; M, M_W) \\right] (-1)^m R~(n=0). \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n\\end{eqnarray}\nWriting $V_{W_\\mu^+}^{nm}$ as\n\\begin{eqnarray}\nV_{W_\\mu^+}^{nm} = \\alpha^{nm} L + \\beta^{nm} R, \n\\end{eqnarray}\n$V_{\\phi^+}^{nm}$ is written as (noting $\\tilde{m}_0=0$), \n\\begin{eqnarray}\nV_{\\phi^+}^{nm} = -\\frac{\\tilde{m}_n \\alpha^{nm} - m_d \\beta^{nm}}{\\frac{m}{R} +M_W} L \n-\\frac{\\tilde{m}_n \\beta^{nm} - m_d \\alpha^{nm}}{\\frac{m}{R} +M_W} R. \n\\end{eqnarray}\n\nThe Feynman rules for the $D$-dimensional gauge and Yukawa interactions of $\\tilde{A}_\\mu, \\tilde{A}_y$ \nand $\\hat{\\psi}$ are readily read off from the results obtained above. \nFor instance, the Feynman rule for the $\\gamma_\\mu^{(n)}~(n \\ge 1)$ vertex is given by Fig. \\ref{photongaugeint}. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=8cm]{fig01.ps}\n \\end{center}\n\\caption{\nThe KK photon vertex between the down quark and KK fermion. \n} \n\\label{photongaugeint}\n\\end{figure}\n\nWe also need the Feynman rule for the three point self-interactions of the photon $\\gamma_\\mu^{(0)}$ \nwith charged gauge and Higgs bosons $\\hat{W}_\\mu^{\\pm(n)}, \\hat{\\phi}^{\\pm(n)}$ \nfor the calculation of $g-2$. \nWe skip all the detail of the derivation and just display the result: \nthe relevant terms of $D$-dimensional lagrangian are\n\\begin{eqnarray}\n&&i e_D \\left\\{\n(\\partial_\\mu \\gamma_\\nu^{(0)} - \\partial_\\nu \\gamma^{(0)}_\\mu ) \\hat{W}^{+(n)\\mu} \\hat{W}^{-(n)\\nu} \n- (\\partial_\\mu \\hat{W}_\\nu^{+(n)} - \\partial_\\nu \\hat{W}^{+(n)}_\\mu ) \\gamma^{(0)\\mu} \\hat{W}^{-(n)\\nu} \n\\right. \\nonumber \\\\\n&& \\left. \n+ (\\partial_\\mu \\hat{W}_\\nu^{-(n)} - \\partial_\\nu \\hat{W}^{-(n)}_\\mu ) \\gamma^{(0)\\mu} \\hat{W}^{+(n)\\nu} \n\\right\\} \\nonumber \\\\\n&&+ e_D \\gamma^{(0)\\mu} \n\\left\\{\n\\left( \\frac{n}{R} + M_W \\right) (\\hat{W}_\\mu^{+(n)} \\hat{\\phi}^{-(n)} + \\hat{W}_\\mu^{-(n)} \\hat{\\phi}^{+(n)}) \n+ i [(\\partial_\\mu \\hat{\\phi}^{+(n)}) \\hat{\\phi}^{-(n)} - (\\partial_\\mu \\hat{\\phi}^{-(n)}) \\hat{\\phi}^{+(n)} ]\n\\right\\}, \\nonumber \\\\\n\\end{eqnarray}\nwhere $n$ can be an arbitrary integer. \nA nice thing here is that if we regard $\\hat{W}_\\mu^{+(n)}$ as $W_\\mu^{+(0)}$, \nthe lagrangian is just the same as that in the standard model, \nexcept that now $M_W$ is replaced by $\\frac{n}{R} + M_W$. \nThus to obtain the Feynman rule is straightforward. \nWe get, {\\it e.g.}, \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=13cm]{fig_null.ps}\n \\end{center}\n\\end{figure}\n\n\n\\section{General formulae for the anomalous magnetic moment}\nIn this section, we derive general formulae for $A_\\mu$ and $A_y$-exchange diagrams \ncontributing to the anomalous magnetic moment, \nwhere $A_\\mu, A_y$ denote generic $D$-dimensional gauge and scalar bosons, respectively. \n\n\\subsection{$A_\\mu$-exchange diagram}\n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=7cm]{fig02.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA generic gauge interaction vertex of the $D$-dimensional gauge boson $A_\\mu^{(m)}$. \n} \n\\label{gaugeint}\n\\end{figure}\nWe first derive a general formula for the $A_\\mu^{(m)}$-exchange diagram due to the vertex in Fig. \\ref{gaugeint}, \nwith $A_\\mu^{(m)}$ and $\\psi^{(n)}$ being generic mass eigenstates of $A_\\mu$ and fermion ($d$ or $u$), respectively, \nwith masses $M_m$ and $m_n$. \nFor simplicity, hereafter, we indicate $d^{(0)}$ just as $d$. \n\n\nThe diagram contributing to the anomalous magnetic moment is shown in Fig. \\ref{gaugeex}. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=5cm]{fig03.ps} \n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe contribution of $A_\\mu^{(m)}$-exchange to the anomalous magnetic moment. \n$p, p'$\nare external momenta of $d$ quark. \n} \n\\label{gaugeex}\n\\end{figure}\nThe $\\gamma_\\mu^{(0)}$ coupling at the tree level is modified \ndue to the quantum correction into\n\\begin{eqnarray}\n-\\frac{e_D}{3} \\bar{d}(\\gamma^\\mu + \\Gamma^\\mu) d. \n\\end{eqnarray}\nAmong a few term in $\\bar{d} \\Gamma^\\mu d$, we are interested in the term \nproportional to $p^\\mu + p'^\\mu$ with a form factor $F_2(0)$.\n\\begin{eqnarray}\n\\bar{d} \\Gamma^\\mu d \\to \\bar{d} \\left[-\\frac{1}{2m_d} (p^\\mu + p'^\\mu) F_2(0) \\right] d. \n\\label{gaugeex1}\n\\end{eqnarray}\nIt is the form factor $F_2(0)$ that gives the anomalous magnetic moment: \n$a=\\frac{g-2}{2} = F_2(0)$. \n\nWe obtain the contribution of Fig. \\ref{gaugeex} to the form factor $F_2(0)$ as \n\\begin{eqnarray}\nF_2^{A_\\mu^{(m)}}(0) &=& -4i \\frac{Q(\\psi)}{Q(d)}\n\\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\times \\nonumber \\\\ \n&& \\frac{-\\frac{a^*b+ab^*}{2} [4-DX] m_d m_n \n+ \\frac{|a|^2+|b|^2}{2}(1 - X)[4-(D-2)X] m_d^2}{[l^2 + X(1-X)m_d^2 - X m_n^2 - (1 - X) M_m^2]^3} \n\\label{g-2gauge}\n\\end{eqnarray}\nwhere $Q(\\psi)$ and $Q(d)$ are electric charges of $\\psi^{(n)}$ and $d$ quark, respectively. \n\n\n\n\\subsection{$A_y$-exchange diagram}\n\n\nLet us move to the calculation of $A_y^{(m)}$-exchange diagram due to the vertex shown in Fig. \\ref{Ayint} \nwhere $A_y^{(m)}$ is a generic mass eigenstate of $A_y$ with a mass $M_m$. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=6cm]{fig04.ps} \n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA generic interaction vertex of the $D$-dimensional scalar $A_y^{(m)}$. \n} \n\\label{Ayint}\n\\end{figure}\nThe diagram to evaluate is displayed in Fig. \\ref{Ayex}. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=5cm]{fig05.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\n$A_y^{(m)}$-exchange diagram contributing to the anomalous magnetic moment. \nThe external and internal momenta are defined in a similar way as in the $A_\\mu^{(m)}$-exchange diagram. \n} \n\\label{Ayex}\n\\end{figure}\nSimilarly to the case of the $A_\\mu^{(m)}$-exchange diagram, \nthe one-loop correction is given by\n\\begin{eqnarray}\nF_2^{A_y^{(m)}}(0) = 4i \\frac{Q(\\psi)}{Q(d)}\n\\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \n\\frac{\\frac{a'^*b'+a'b'^*}{2} m_d m_n + \\frac{|a'|^2+|b'|^2}{2} (1 - X) m_d^2}{[l^2 +X(1-X) m_d^2 - X m_n^2 - (1 - X) M_m^2]^3}. \n\\label{g-2Ay}\n\\end{eqnarray}\n\\subsection{Diagrams due to the three point self-interaction}\nWe also need a general formula for the diagrams due to the three point self-interactions of \n$\\gamma_\\mu^{(0)}$ with $A_\\mu^{(m)}$ and\/or $A_y^{(m)}$. \nWe first consider the contribution of a diagram (Fig. \\ref{3ptgauge})\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=5cm]{fig06.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA diagram contributing to the anomalous magnetic moment due to a triple gauge boson vertex. \n} \n\\label{3ptgauge}\n\\end{figure}\ndue to the vertex shown in Fig. \\ref{gaugeint} and Fig. \\ref{3ptvtx}\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=10cm]{fig07.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA triple gauge boson vertex. \n} \n\\label{3ptvtx}\n\\end{figure}\nwhere the factor $e_D$ is assigned assuming $A_\\mu^{(m)}$ is $\\hat{W}_\\mu^{-(m)}$, \nas is really the case. \nThe contribution of Fig. \\ref{3ptgauge} to $F_2(0)$ is given as\n\\begin{eqnarray}\nF_2(0)^{A_\\mu^{(m)} A_\\nu^{(m)}} &=& \n12i \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\frac{-\\frac{a^*b + ab^*}{2}[4-D+(D-1)X]m_d m_n \n+ \\frac{|a|^2+|b|^2}{2} X [5-D+(D-2)X] m_d^2}{[l^2 + X(1-X)m_d^2 -X M_m^2 -(1-X)m_n^2]^3}. \n\\nonumber \\\\\n\\end{eqnarray}\n\nNext we consider the contribution of combined two diagrams shown in Fig. \\ref{3ptmix}\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=10cm]{fig08.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nDiagrams contributing to the anomalous magnetic moment \ndue to the $\\gamma_\\mu$-$A^\\mu$-$A_y$ coupling. \n} \n\\label{3ptmix}\n\\end{figure}\ndue to the vertices in Fig. \\ref{gaugeint}, Fig. \\ref{Ayint} and additional one, Fig. \\ref{mixed}. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=6cm]{fig09.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe 3-point vertex relevant for Fig. \\ref{3ptmix}.\n} \n\\label{mixed}\n\\end{figure}\nThe contribution to $F_2(0)$ from Fig. \\ref{3ptmix} is\n\\begin{eqnarray}\nF_2^{A_\\mu^{(m)}A_y^{(m)}}(0) = -12i M_m \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\frac{\\frac{1}{4}(b'^* a + a'^* b + b' a^* + a'b^*)X m_d}{[l^2 + X(1-X) m_d^2 - X M_m^2 - (1-X) m_n^2]^3}. \n\\nonumber \\\\\n\\end{eqnarray}\n\nFinally, we consider the contribution of Fig. \\ref{gyy}\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=5.5cm]{fig10.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA diagram contributing to the anomalous magnetic moment \ndue to the $\\gamma_\\mu$-$A_y$-$A_y$ coupling. \n} \n\\label{gyy}\n\\end{figure}\ndue to another additional vertex of Fig. \\ref{3ptgyy}. \n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=8cm]{fig11.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe 3-point vertex relevant for Fig. \\ref{gyy}. \n} \n\\label{3ptgyy}\n\\end{figure}\nThe contribution to $F_2(0)$ is\n\\begin{eqnarray}\nF_2^{A_y^{(m)}A_y^{(m)}}(0) &=& -12i \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\frac{\n\\frac{1}{2}(a'^* b' + a'b'^{*})(1-X) m_d m_n \n+ \\frac{1}{2}(|a'|^2 + |b'|^2) X(1-X) m_d^2}{[l^2 + X(1-X) m_d^2 - X M_m^2 - (1-X) m_n^2]^3}. \n\\end{eqnarray}\n\nHaving the general formulae, \nthe contribution of each type of $A_\\mu^{(m)}$ or $A_y^{(m)}$-exchange diagram is readily written down \nby use of the vertex functions derived in the previous section. \nFor instance, the contribution of $\\gamma_\\mu^{(m)}$-exchange diagram $(m \\ge 1)$ \nis obtained by setting (see (\\ref{3.3}), (\\ref{3.4}))\n\\begin{eqnarray}\na = (-1)^{m+n} b = -\\frac{g}{2\\sqrt{3}} I_c^{nm}(M, M_W; M, M_W)\n\\end{eqnarray}\nin (\\ref{g-2gauge}): \n\\begin{eqnarray}\nF_2^{\\gamma_\\mu^{(m \\ne 0)}}(0) &=& -\\frac{g^2}{3} i \\sum_{m=1}^\\infty \\sum_{n=-\\infty}^\\infty \n\\left[ I_c^{nm}(M, M_W; M, M_W) \\right]^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\frac{-(-1)^{n+m} [4-DX] m_d m_n +(1-X) [4-(D-2)X] m_d^2}{[l^2 + X(1-X) m_d^2 -X m_n^2 -(1-X)(\\frac{m}{R})^2]^3}. \n\\end{eqnarray}\nTo get the other contributions is also straightforward and we do not write them down explicitly here. \nEven though in this way we can get the exact formulae for the contributions to the magnetic moment, \nto get the final analytic results is a hard task, since the vertex functions like $I_c(M, M_W; M, M_W)$ \nare rather complicated and $m_n$ to describe the function cannot be given analytically, \nunless some approximation is applied or some extreme cases are considered. \n\n\n\\section{The cancellation of divergences}\n\nWe are now ready to focus on the main issue of this paper, \n{\\em i.e.} the cancellation mechanism of UV divergence in the contribution to $g-2$. \nAs discussed in the introduction, \nthe anomalous magnetic moment in the present model is expected to be finite \nrelying on the operator analysis, similarly to the case of toy model, \n$(D+1)$ dimensional QED \\cite{ALM}. \nIn Ref. \\cite{ALM}, the cancellation seems to take place \nbetween photon-exchange and Higgs-exchange diagrams. \nIn the present model, however, these contributions behave differently. \nFor instance, the Yukawa coupling of Higgs to $d$ quark is suppressed compared to the gauge coupling. \nWe will find that in the present realistic model, \nthe cancellation takes place between the contributions of the partners $(A_\\mu^{(m)}, A_y^{(m)})$ \nhaving the same quantum number, such as $(\\hat{W}_\\mu^{+(m)}, \\hat{\\phi}^{+(m)})$, \namong which the Higgs(-like) mechanism is at work. \nIt is interesting to realize that they also form ``super-partners\" of the quantum mechanical SUSY \npresent in the higher dimensional gauge theories \\cite{LNSS}.\n\n\nAs has been discussed in the end of previous section, \nthe exact formulae for the contributions of each diagram are not easy to handle. \nSince the cancellation should occur for the arbitrary bulk mass $M$, \nbeing supported by the operator analysis, \nand our main purpose is to confirm the cancellation mechanism, \nwe simplify the analysis by considering the case of small $M$ in this section. \nWe will retain only the term up to ${\\cal O}(M)$ in the contribution of each diagram. \nAs the matter of fact, \nit turns out that the term linear in $M$ in each contribution vanishes. \nThis comes from the fact that the functions $I_c^{nm}$, {\\em etc.}, \nappearing in the fermion vertices satisfy a relation, as is shown in Appendix A: \n\\begin{eqnarray}\nI_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W). \n\\end{eqnarray}\nSince such functions appear twice in each diagram, \nwe conclude the contribution of each diagram is an even function of $M$. \nHence the term linear in $M$ actually disappears. \nWe thus take the limit of $M \\to 0$. \n\nIn the limit $M \\to 0$, $m_n$ reduces to $\\frac{n}{R}+M_W$ and the functions $I_c^{nm}$ etc. \nare greatly simplified, reflecting the (partial) recovery of momentum conservation \nalong the extra dimension: \n\\begin{eqnarray}\n&&I_c^{nm}(0, M_2; 0, M_4) = \\frac{1}{2\\sqrt{\\pi R}} (\\delta_{n,m} + \\delta_{-n,m})~(m \\ne 0), \\\\\n&&I_c^{n0}(0, M_2; 0, M_4) = \\frac{1}{\\sqrt{2\\pi R}} \\delta_{n,0}, \\\\\n&&I_s^{nm}(0, M_2; 0, M_4) = \\frac{i}{2\\sqrt{\\pi R}} (\\delta_{n,m} - \\delta_{-n,m})~(m \\ne 0), \\\\\n&&\\tilde{I}_c^{nm}(0, M_2; 0, M_4) = \\frac{1}{2\\sqrt{\\pi R}} (\\delta_{n,m} + \\delta_{-n,m})~(m \\ne 0), \\\\\n&&\\tilde{I}_s^{nm}(0, M_2; 0, M_4) = -\\frac{i}{2\\sqrt{\\pi R}} (\\delta_{n,m} - \\delta_{-n,m})~(m \\ne 0).\n\\end{eqnarray}\n %\n\nThen the vertex functions such as $V_{\\gamma_\\mu}^{nm}$ in this limit \nalso take simple forms. \nFor instance, $V_{\\gamma_\\mu}^{nm}=\\frac{1}{2\\sqrt{\\pi R}}(\\delta_{n,m} + \\delta_{-n,m})~(m \\ge 1)$. \nHaving the Feynman rules for the interaction vertices of fermions, \nwe readily get the contribution to $g-2$, {\\em i.e.} $F_2(0)$, \nfrom each one-loop diagram, by use of the general formulae derived in section 4. \nWe display the results below, dividing them into the sectors of \ncharged current, neutral current, etc. \n\n\\subsection{The charged current sector}\n\nWe first display the contribution of $\\hat{W}_\\mu^+$-exchange \ndiagram (obtained with $A_\\mu^{(m)}$ being replaced by $\\hat{W}_\\mu^{+(\\pm n)}$ in Fig. \\ref{gaugeex}),\nwhich we denote by $F_2^{(W)}(0)$: \n\\begin{eqnarray}\nF_2^{(W)}(0) &=& -4i (-2) \\left( \\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1dX X \\nonumber \\\\\n&& \\times \\sum_{n=-\\infty}^\\infty \n\\frac{(4-DX)\\frac{n}{R}M_W +(1-X) [4-(D-2)X]M_W^2}{[l^2-(\\frac{n}{R}+(1-X)M_W)^2]^3}\n\\label{F2W}\n\\end{eqnarray}\nwhere the factor $(-2)$ comes from $Q(u)\/Q(d)$ and \nthe facts $m_d = M_W$, $\\tilde{m}_n =\\frac{n}{R}$, $M_{\\hat{W}_\\mu^{+(n)}} = \\frac{n}{R} + M_W$ \n(in the limit $M \\to 0$) have been used. \nA wisdom to treat the UV divergence is to invoke Poisson resummation \nand extract the ``zero-winding\" sector. \nHere, however, we just take the limit of ``de-compactification\", $R\\to \\infty$, \nas the zero-winding sector is easily known to correspond to the limit. \nIn this limit $\\frac{n}{R}$ may be replaced by the continuous extra space momentum $p_y$. \nThus, taking the limit $R \\to \\infty$, replacing $\\frac{n}{R}$ by $p_y$ \nand performing a shift of the momentum, $p_y +(1-X)M_W \\to p_y$, \nwe get the divergent part $F_2(0)_{{\\rm div.}}$ of the contribution, \n\\begin{eqnarray}\nF_2^{(W)}(0)_{{\\rm div.}} = i \\frac{2\\pi}{3} R g_D^2 \\int \\frac{d^Dl d p_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{[l^2-p_y^2]^3}. \n\\label{Wexdiv}\n\\end{eqnarray}\n\nWe next display the contribution of the exchange of the partner of $W_\\mu^+$, \n{\\em i.e.} $\\hat{\\phi}^+$, whose diagram is obtained with $A_y^{(m)}$ being replaced by $\\hat{\\phi}^{+(\\pm n)}$ in Fig. \\ref{Ayex}. \n\\begin{eqnarray}\nF_2^{(\\phi^+)}(0) = 4i (-2) \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\sum_{n=-\\infty}^\\infty \\frac{-X \\frac{n}{R} M_W + X(1-X)M_W^2}{[l^2-(\\frac{n}{R} + (1-X) M_W)^2]^3}. \n\\label{F2phi}\n\\end{eqnarray}\nTaking the same step as we took above, \nwe obtain the divergent part of the contribution, \n\\begin{eqnarray}\nF_2^{(\\phi^+)}(0)_{{\\rm div.}} =-i \\frac{2\\pi}{3} Rg_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\frac{M_W^2}{[l^2-p_y^2]^3}. \n\\label{phiexdiv}\n\\end{eqnarray}\nWe realize , as we anticipated, the divergence exactly cancel out \nbetween the contributions of ``partners\" $(\\hat{W}_\\nu^{+(n)}, \\hat{\\phi}^{+(n)})$ \nshown in (\\ref{Wexdiv}) and (\\ref{phiexdiv}), \nthough the original forms of (\\ref{F2W}) and (\\ref{F2phi}) are quite different. \nLet us note that the Higgs(-like) mechanism is operative \nboth in the non-zero KK mode sector $(n \\ne 0)$ and the zero-mode sector $(n=0)$ \nfor the charged gauge-Higgs bosons, being triggered by the KK mass $\\frac{n}{R}$ for $n \\ne 0$ \nand by the VEV of $A_y$ for $n=0$ ({\\em i.e.} the ordinary Higgs mechanism). \n\nIn addition to these diagrams, \nwe have to evaluate the contributions due to the three point self-interaction vertices, \nwhose diagrams are obtained with $A_\\mu^{(m)}$ and $A_y^{(m)}$ being replaced \nby $\\hat{W}_\\mu^{+(\\pm n)}$ and $\\hat{\\phi}^{+(\\pm n)}$ \nin Figs. \\ref{3ptgauge}, \\ref{3ptmix}, and \\ref{gyy}. \n %\n %\n %\n %\nThe contributions of each diagram are\n\\begin{eqnarray}\nF_2^{(WW)}(0) &=& 3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \n\\frac{-[D-4-(D-1)X] \\frac{n}{R} M_W + X[5-D+(D-2)X]M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}, \\\\\nF_2^{(W \\phi^+)}(0) &=& 3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\sum_{n=-\\infty}^\\infty \n\\frac{X \\frac{n}{R} M_W + X M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}, \\\\\nF_2^{(\\phi^+ \\phi^+)}(0) &=& -3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\sum_{n=-\\infty}^\\infty \n\\frac{-(1-X) \\frac{n}{R} M_W + X (1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}. \n\\end{eqnarray}\nIt is straightforward to show that \nthe divergent part just cancel out; \n\\begin{eqnarray}\nF_2^{(WW)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& 6\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \\\\\nF_2^{(W \\phi^+)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& 6\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \\\\\nF_2^{(\\phi^+ \\phi^+)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& -12\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\left( F_2^{(WW)}(0) + F_2^{(W \\phi^+)}(0) + F_2^{\\phi^+ \\phi^+}(0) \\right)_{{\\rm div.}} = 0. \n\\end{eqnarray}\n\n\\subsection{The neutral current sector}\nTaking a similar step to the case of the charged current sector, \nthe contribution due to $\\hat{Z}_\\mu$-exchange diagram \nis known to be given as\n\\begin{eqnarray}\nF_2^{(Z)}(0) &=& -4i \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \n\\sum_{n=-\\infty}^\\infty \\frac{(4-DX)(\\frac{n}{R}+M_W)M_W +(1-X) [4 - (D-2)X] M_W^2}{[l^2-(\\frac{n}{R}+(2-X)M_W)^2]^3},\n\\end{eqnarray}\nwhere $m_n = \\frac{n}{R}+M_W, M_{\\hat{Z}_\\mu^{(n)}} = \\frac{n}{R} + 2M_W$ has been used. \n\nThe $R\\to \\infty$ limit and the shift $p_y +(2-X) M_W \\to p_y$ gives\n\\begin{eqnarray}\nF_2^{(Z)}(0)_{{\\rm div.}} = -i \\frac{\\pi}{3}R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3}. \n\\end{eqnarray}\n\n\nThe contribution due to $\\hat{\\phi}^0$-exchange diagram \nis given as\n\\begin{eqnarray}\nF_2^{(\\phi^0)}(0) &=& 4i \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \n\\sum_{n=-\\infty}^\\infty \\frac{-X (\\frac{n}{R}+M_W) M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R}+(2-X)M_W)^2]^3}. \n\\end{eqnarray}\nThe divergent part reads as\n\\begin{eqnarray}\nF_2^{(\\phi^0)}(0)_{{\\rm div.}} = i \\frac{\\pi}{3}R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3}. \n\\end{eqnarray}\nWe thus confirm the cancellation of the divergence,\n\\begin{eqnarray}\nF_2^{(Z)}(0)_{{\\rm div.}} + F_2^{(\\phi^0)}(0)_{{\\rm div.}} = 0. \n\\end{eqnarray}\n\n\\subsection{The photon sector}\nIn this subsection, we discuss the contributions of photon ($\\gamma_\\mu^{(n)}$)-exchange \nand the exchange of $\\gamma_y^{(n)}$, the partner of $\\gamma_\\mu^{(n)}$ for $n \\ge 1$. \nIn clear contrast to the previous cases, \nthe Higgs mechanism does not exist for the zero mode sector, $n=0$, \nand also $\\gamma_y^{(0)}$ is absent due to the orbifolding. \nThis suggests that the contribution of $n=0$ sector differs from \nwhat we obtain by formally setting $n=0$ in the formula valid for $n \\ne 0$. \nIn fact, the contribution of the $\\gamma_\\mu^{(n)}$-exchange ($n \\ge 0$)\nis given by \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0) &=& -4i \\left(-\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{-(4-DX)(\\frac{n}{R}+M_W)M_W + (1-X)[4-(D-2)X] M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&-4i \\left(-\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dXX^2 \\frac{[(D-2)X-2]M_W^2}{(l^2 - X^2 M_W^2)^3}. \n\\label{ptng-2}\n\\end{eqnarray}\nThe second term is to adjust the discrepancy mentioned above. \nThe divergent part reads as \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0)_{{\\rm div.}} &=& -i \\frac{2\\pi}{27} R e_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& -i \\frac{2}{9} e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2] M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{ptndiv}\n\\end{eqnarray}\nSimilarly, the contribution of the $\\gamma_y^{(n)}$-exchange ($n \\ge 1$) \nis given by \n\\begin{eqnarray}\nF_2^{(\\gamma_y)}(0) &=& 4i \\left(\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{X(\\frac{n}{R}+M_W)M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&-4i \\left(\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dXX^2 \\frac{(2-X) M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{Ayg-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(\\gamma_y)}(0)_{{\\rm div.}} &=& i \\frac{2\\pi}{27} R e_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& -i \\frac{2}{9} e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{Aydiv}\n\\end{eqnarray}\nWe thus realize that although the cancellation of divergence is ``almost\" complete, \nthere remains a ``partial\" $D$-dimensional divergence originated from the $n=0$ sector, \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0)_{{\\rm div.}} + F_2^{(\\gamma_y)}(0)_{{\\rm div.}} \n= -i \\frac{2}{9} (D-3)e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^3 \\frac{M_W^2}{(l^2-X^2 M_W^2)^3}. \n\\label{zeromodediv}\n\\end{eqnarray}\n\n\\subsection{The Higgs sector}\n\nFinally, in this subsection we discuss the contribution of the exchange \nof the partner of the Higgs $h_\\mu^{(n)}$, and the contribution of the Higgs ($h^{(n)}$)-exchange. \nThe situation concerning the UV divergence is similar to that in the photon sector, \nand we just summarize the result below. \n\nThe contribution of the $h_\\mu^{(n)}$-exchange ($n \\ge 1$)\n\nis given by \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0) &=& -4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{-(4-DX)(\\frac{n}{R}+M_W)M_W + (1-X)[4-(D-2)X] M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&+4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2]M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{hmug-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0)_{{\\rm div.}} &=& -i \\frac{\\pi}{6} R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& +i \\frac{1}{2} g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2] M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{hmudiv}\n\\end{eqnarray}\n\nThe contribution of the $h^{(n)}$-exchange ($n \\ge 0$) \nis given by \n\\begin{eqnarray}\nF_2^{(h)}(0) &=& 4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{X(\\frac{n}{R}+M_W) M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&+4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{hg-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(h)}(0)_{{\\rm div.}} &=& i \\frac{\\pi}{6} R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& +i \\frac{1}{2} g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{Aydiv}\n\\end{eqnarray}\nAgain, the cancellation of UV divergence turns out to be not complete, \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0)_{{\\rm div.}} + F_2^{(h)}(0)_{{\\rm div.}} \n= i \\frac{1}{2} (D-3)g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^3 \\frac{M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{zeromodehdiv}\n\\end{eqnarray}\n\nTo summarize this section, \nwe have found the cancellation mechanism of UV divergence \nbetween the contributions of the pairs of $D$-dimensional vector bosons and scalars \n$(A_\\mu, A_y)$, which form ``partners\" of Higgs(-like) mechanism and at the same time \nthe partners of quantum mechanical SUSY. \nThe cancellation is complete for the charged current and neutral current sectors, \n$(\\hat{W}_\\mu^{+(n)}, \\hat{\\phi}^{+(n)})$ and $(\\hat{Z}_\\mu^{(n)}, \\hat{\\phi}^{0(n)})$, \nwhile it is incomplete for the photon and Higgs sectors, \n$(\\gamma_\\mu^{(n)}, \\gamma_y^{(n)})$ and $(h_\\mu^{(n)}, h^{(n)})$. \nThe remaining ``partial\" and lower dimensional ($D$-dimensional) divergence \n({\\rm i.e.} the sum of Eqs. (\\ref{zeromodediv}) and (\\ref{zeromodehdiv}), by use of $e_D^2 =\\frac{3}{4} g_D^2$) \n\\begin{eqnarray}\nF_2(0)_{{\\rm div.}} = i \\frac{1}{3}(D-3)g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X^3 \\frac{M_W^2}{(l^2 -X^2 M_W^2)^3}, \n\\label{remaindiv}\n\\end{eqnarray}\nseems to be attributed to the absence of Higgs mechanism for $n=0$ in the photon and Higgs sectors. \nAs the matter of fact, however, Eq.(\\ref{remaindiv}) is finite, so is the anomalous magnetic moment, \nfor 5 or 6 dimensional ($D=4~{\\rm or}~5$) space-time. \nLet us note the $g-2$ is divergent already at 6 dimensional space-time \nin other types of higher dimensional gauge theories, \n{\\em e.g.} in the scenario of universal extra dimension. \n\n\n\n\\section{Conclusions}\nIn this paper, \nwe have discussed the finiteness of the anomalous magnetic moment of fermion \nin a realistic model of gauge-Higgs unification. \nOur main purpose is to clarify the cancellation mechanism of UV divergences \nin various contributing Feynman diagrams. \nOur expectation that the anomalous moment should be finite and calculable, \nin spite of the fact that higher dimensional gauge theories are argued to be non-renormalizable, \nis based on an operator analysis: \nan operator corresponding to the dimension six gauge invariant operator \ndescribing the anomalous moment in the standard model \nis forbidden because of the higher dimensional gauge symmetry \npresent in the gauge-Higgs unification and the on-shell condition for the fermion, \nas was discussed in the introduction. \n\nIn our previous paper, \nwe adopted a toy model, i.e. higher dimensional QED compactified on $S^{1}$, \nwhere $D$-dimensional gauge and scalar fields $(A_{\\mu}, A_{y})$ were identified with photon and Higgs fields, respectively, \nand showed by explicit calculation that the anomalous moment is in fact finite \nfor arbitrary space-time dimensions \\cite{ALM}. \nAlthough the result is quite remarkable, \nthis toy model is not realistic and has a few drawbacks: \nthe gauge group is too small to incorporate the $SU(2)$ doublet Higgs \nand the masses of light fermions or their small Yukawa couplings cannot be taken into account. \nIt cannot reproduce the famous result of Schwinger on the magnetic moment \\cite{Schwinger}, \nagain due to the unrealistic Yukawa coupling. \n \nThese unsatisfactory points are overcomed in this paper \nby considering a realistic $D+1$ dimensional ($D = 4,5,$ etc.) $SU(3)$ gauge-Higgs unification model \ncompactified on an orbifold $S^{1}\/Z_{2}$, with matter fermions of $SU(3)$ triplet. \nThe small Yukawa coupling is achieved by introducing $Z_{2}$-odd bulk mass $M$ for the fermions.\n\nAfter deriving various general formulae which are valid for arbitrary $M$ \nand are used to obtain the anomalous moment, \nwe have discussed the cancellation mechanism of the UV divergences \nfor the simplified case of small $M$. \nSince our operator analysis concerning the finiteness strongly \ndepends on the higher dimensional gauge symmetry, \nit will be natural to expect that the cancellation of UV divergence is realized \nbetween the contributions of the pair of $D$-dimensional gauge and scalar bosons \n$(A_{\\mu}, A_{y})$ with the same quantum number. $(A_{\\mu}, A_{y})$, at least for non-zero KK modes, \nplay the roles as the ``partners\" in the Higgs-like mechanism \nto form massive gauge bosons (from $D$-dimensional point of view), \nwhich is the manifestation of the higher dimensional gauge symmetry. \nWe also would like to point out that the pair $(A_{\\mu}, A_{y})$ is known to behave \nas a multiplet of quantum mechanical SUSY \\cite{LNSS}, \nso the cancellation may be regarded as the consequence of the supersymmetry. \nLet us note that in this model photon and Higgs have different quantum numbers and \nthe divergence cancellation does not take place between the contributions of these particles, \nin clear contrast to the case of our previous paper \\cite{ALM}. \n\nWe have confirmed these expectations by explicit calculation of Feynman diagrams. \nConcerning the contributions of charged and neutral currents \ndue to the pairs $(\\hat{W}_{\\mu}^{\\pm (n)}, \\hat{\\phi}^{\\pm (n)})$ and $(\\hat{Z}_{\\mu}^{(n)}, \\hat{\\phi}^{0(n)})$, \nthe zero-mode sector also has ordinary Higgs mechanism. \nThus the UV divergence has been shown to be completely cancelled \nbetween the contributions of each partners. \nOn the other hand, \nconcerning the remaining contributions of photon and Higgs sector, \n$(\\gamma_{\\mu}^{(n)}, \\gamma_{y}^{(n)})$ and $(h_{\\mu}^{(n)}, h^{(n)})$, \nthe cancellation is not complete and there remains a UV divergence (but of lower degree) \nwhich originates from the zero modes. \nLet us recall that for these sectors the zero modes do not have ordinary Higgs mechanism and \none member of each partners is missing due to the orbifolding. \nSo far, we do not have any good reasoning why the cancellation mechanism does not perfectly work, \nwhile the operator analysis seems to be valid. \nThis issue remains to be settled. \nNevertheless, we would like to point out \nthat we have completely finite anomalous moments for 5 and 6 dimensional space-time, \nalthough the anomalous moment is divergent for the case of 6 dimensions \nin other higher dimensional gauge theories such as ``universal extra dimension\". \n\nIn our previous paper, \nwe could not reproduce the Schwinger's result in ordinary QED, \nsince the contribution of the Higgs-exchange was comparable to that of photon-exchange \nas the Yukawa coupling was of the order of the gauge coupling. \nThis drawback is overcomed in the present model \nby introducing the bulk mass which causes the localization of Weyl fermions \nwith different chiralities at two different fixed points. \nWe will report in the forthcoming paper \\cite{ALM2} \nthat the Schwinger's result is indeed reproduced. \nIn the paper, we will also discuss the constraint on the compactification scale \nby comparing our prediction on the anomalous magnetic moment \nincluding the contributions of non-zero KK modes \nwith the allowed deviation of the data from the prediction of the standard model. \n\n\n\\subsection*{Acknowledgment}\n\nThe work of the authors was supported \nin part by the Grant-in-Aid for Scientific Research \nof the Ministry of Education, Science and Culture, No.18204024 and No. 20025005. \n\n\\begin{appendix}\n\n\\setcounter{equation}{0}\n\\section{The derivation of the property \\\\$I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$}\nIn this appendix, the relation between the vertex functions \ndue to the fermions with different chiralities is discussed. \nWe take a typical example of $I_c^{nm}(M, M_W; M, M_W)$ and \n$I_c^{nm}(-M, M_W; -M, M_W)$ defined in (\\ref{3.6}) and (\\ref{3.7}) \nto show the property \\\\$I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$. \nSince the interchange of the chirality $L \\leftrightarrow R$ corresponds \nto the sign flip of the bulk mass $M \\to -M$, therefore the exchange of the fixed points \n$y \\leftrightarrow y-\\pi R$, we expect that we can obtain $f_{d_R}$ by shifting $f_{d_L}$ by $\\pi R$ \nwith possible phase change. \nAs the shift does not mix the real and imaginary parts, \nwe expect that such property holds in each part of even and odd functions of $y$. \n\nLet us start with left-handed mode function,\n\\begin{eqnarray}\nf_{d_L}^{(n)}(y) &=& F_{M, M_W}^{(n)}(y) \\nonumber \\\\\n&=& e^{iM_W y} C^{(n)} \n\\left[\n\\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right. \\nonumber \\\\\n&& \\left. -\\varepsilon(n) i \\sqrt{\\cos(\\varphi_{n} +\\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y)\n\\right] \\nonumber \\\\\n&=&\nC^{(n)} \n\\left[\n\\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right. \\nonumber \\\\\n&& \\left. + \\varepsilon(n) \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y) \n\\right. \\nonumber \\\\\n&& \\left. +i \\left(\n-\\varepsilon(n) \\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y) \n\\right. \\right. \\nonumber \\\\\n&& \\left. \\left. + \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right) \n\\right], \n\\end{eqnarray}\nwhere $\\varepsilon(n)=1$ for $n \\ge 0$, $-1$ for $n < 0$.\n \nConcentrating on the real part of $f_{d_L}^{(n)}\/C^{(n)}$ ($C^{(n)}$ is invariant under $M \\to -M$), \nand making a shift $y \\to y-\\pi R$, we find\n\\begin{eqnarray}\n&&[f_{d_L}^{(n)}(y-\\pi R)\/C^{(n)}]_{{\\rm real}} \\nonumber \\\\\n&=& \n\\cos(M_W (y-\\pi R)) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \n\\cos(\\sqrt{m_{n}^2-M^2}(\\pi R -y) + \\alpha_{n}) \\nonumber \\\\\n&&+ \\varepsilon(n) \\sin(M_W (y-\\pi R)) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}(y-\\pi R)) \\nonumber \\\\\n&=& \\cos(M_W y) \n\\left[\n\\left(\n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos \\varphi_{n} \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. + \\varepsilon(n) \n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin (\\varphi_{n} - \\alpha_{n}) \n\\right) \\cos(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n}) \\right. \\nonumber \\\\\n&& \\left. + \\left(\n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin \\varphi_{n} \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. -\\varepsilon(n) \n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\cos (\\varphi_{n} - \\alpha_{n}) \n\\right) \\sin(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n})\n\\right] \\nonumber \\\\\n&& + \\sin(M_W y) \n\\left[\n\\left(\n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin (\\varphi_{n}+ \\alpha_{n}) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. +\\varepsilon(n) \n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\cos \\varphi_{n} \n\\right) \\sin(\\sqrt{m_{n}^2-M^2}y) \\right. \\nonumber \\\\\n&& \\left. + \\left(\n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos (\\varphi_{n} + \\alpha_{n}) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. -\\varepsilon(n) \n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin \\varphi_{n} \n\\right) \\cos(\\sqrt{m_{n\\pm}^2-M^2}y)\n\\right] \\nonumber \\\\\n&=& \n(-1)^n\n\\left[ \n\\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \n\\cos(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n}) \\right. \\nonumber \\\\\n&& \\left. \n+ \\varepsilon(n) \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y)\n\\right] \\nonumber \\\\\n&=& (-1)^n [F_{-M, M_W}^{(n)}(y)\/C^{(n)}]_{{\\rm real}} \\nonumber \\\\\n&=& (-1)^n [f_{d_R}^{(n)}(y)\/C^{(n)}]_{{\\rm real}}\n\\end{eqnarray}\nwhere \nwe used the relations\n\\begin{eqnarray}\n\\sin(M_W \\pi R) \\cos \\alpha_{n} &=& \\varepsilon(n) (-1)^n \\sin \\varphi_{n}, \\\\\n\\cos(M_W \\pi R) \\sin \\varphi_{n} &=& \n\\varepsilon(n) \\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} \\pm \\alpha_{n})} \n\\sqrt{\\cos(\\varphi_{n} \\mp \\alpha_{n})} \n\\end{eqnarray}\nwhich can be derived from\n\\begin{eqnarray}\n\\sin^2 \\varphi_n = \\sin^2(M_W \\pi R) \\cos^2 \\alpha_n \n= \\tan^2(M_W \\pi R) \\cos(\\varphi_n+\\alpha_n) \\cos(\\varphi_n-\\alpha_n). \n\\end{eqnarray}\nA similar relation for the imaginary part can be verified. \nThus, we obtain \n\\begin{eqnarray}\n&&\\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) f_{d_R}^{(n)}(y)^* f_{d_R}^{(0)}(y) \\nonumber \\\\\n&=& 2 \\int_0^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) \n\\left[\nf_{d_R}^{(n)}(y)_{{\\rm real}} f_{d_R}^{(0)}(y)_{{\\rm real}} + f_{d_R}^{(n)}(y)_{{\\rm imaginary}} f_{d_R}^{(0)}(y)_{{\\rm imaginary}} \n\\right] \\nonumber \\\\\n&=& (-1)^n 2\\int^{\\pi R}_{0} dy (-1)^m \\cos \\left(\\frac{m}{R}(y-\\pi R) \\right) \\times \\nonumber \\\\\n&& \\left[\nf_{d_L}^{(n)}(y-\\pi R)_{{\\rm real}} f_{d_L}^{(0)}(y-\\pi R)_{{\\rm real}} \n+ f_{d_L}^{(n)}(y-\\pi R)_{{\\rm imaginary}} f_{d_L}^{(0)}(y-\\pi R)_{{\\rm imaginary}} \n\\right] \\nonumber \\\\\n&=& (-1)^{n+m} \\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) f_{d_L}^{(n)}(y)^* f_{d_L}^{(0)}(y),\n\\end{eqnarray}\nwhich concludes the fact $I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$. \n\n\\section{The ``equivalence theorem\"}\n\nIn this appendix, \nwe discuss the relation between the vertex functions of $D$-dimensional gauge boson and scalar $(A_{\\mu}, A_{y})$, \nwhich are the partners of the Higgs-like mechanism in the sector of non-zero K-K modes \nor form a multiplet of quantum mechanical SUSY. \nThe relation is what we naturally expect from the fact \nthat the interaction of the longitudinal component of the massive gauge boson is equivalent to \nthat of the would-be Nambu-Goldstone boson. \nAs the typical example, we will derive a relation between \n$I_{c}^{(nm)}$ and $I_{s}^{(nm)}$ defined in (\\ref{3.6}) and (\\ref{3.13}). \n\nA key ingredient is the fact that right- and left-handed Weyl fermions form a multiplet of quantum mechanical SUSY, \nwhose transformation is given as (see (\\ref{2.35}) and (\\ref{2.36})) \n\\begin{eqnarray} \n&& i(\\partial_{y} -iM_{W} -M \\epsilon (y)) f_{dR}^{(n)} (y) = m_{n} f_{dL}^{(n)} (y), \\\\ \n&& i(\\partial_{y} -iM_{W} +M \\epsilon (y)) f_{dL}^{(n)} (y) = m_{n} f_{dR}^{(n)} (y). \n\\end{eqnarray}\nBy use of these relations we can verify (for $m \\geq 1$)\n\\begin{eqnarray} \n&& I_{s}^{(nm)}(-M, M_{W}; M, M_{W}) \n= \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\sin (\\frac{m}{R}y) f_{dR}^{(n)}(y)^{\\ast} \nf_{dL}^{(0)}(y) \\nonumber \\\\ \n&=& \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[(\\partial_{y} f_{dR}^{(n)}(y)^{\\ast})f_{dL}^{(0)}(y) + f_{dR}^{(n)}(y)^{\\ast}(\\partial_{y} f_{dL}^{(0)}(y))] \\nonumber \\\\ \n&=& \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[((\\partial_{y} +iM_{W} - M \\epsilon (y))f_{dR}^{(n)}(y)^{\\ast})f_{dL}^{(0)}(y) \\nonumber \\\\ \n&& \\hspace*{10mm} + f_{dR}^{(n)}(y)^{\\ast}((\\partial_{y} -iM_{W} + M \\epsilon (y))f_{dL}^{(0)}(y))] \\nonumber \\\\ \n&=& \\frac{i}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[m_{n} f_{dL}^{(n)}(y)^{\\ast}f_{dL}^{(0)}(y) - m_{d} f_{dR}^{(n)}(y)^{\\ast}f_{dR}^{(0)}(y)] \\nonumber \\\\ \n&=& i \\frac{1}{(\\frac{m}{R})} [m_{n} - (-1)^{m+n} m_{d}] I_{c}^{(nm)} (M, M_{W}; M, M_{W}), \n\\end{eqnarray}\nwhere the step to move to the second line is due to a partial integral. \nThis relation immediately leads to the relation between the vertex functions, \n\\begin{eqnarray} \nV_{\\gamma_y}^{nm} = i \\frac{m_{n} - (-1)^{m+n} m_{d}}{(\\frac{m}{R})} V_{\\gamma_{\\mu}}^{nm} \\ \\ (m \\geq 1). \n\\end{eqnarray}\n\n\n\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nIn this paper the authors explicitly describe in terms of generators and relations and three families of polynomials, the universal central extension of an algebra appearing in work of Date, Jimbo, Kashiwara and Miwa (see \\cite{MR701334,MR823315}), where they study integrable systems arising from Landau-Lifshitz differential equation. Two of these families of polynomials are described below in terms of elliptic integrals and the other family is a variant of certain ultraspherical polynomials. The authors Date, Jimbo, Kashiwara and Miwa solved the Landau-Lifshitz equation using methods developed in some of their previous work on affine Lie algebras. The hierarchy of this equation is written in terms of free fermions on an elliptic curve. The infinite-dimensional Lie algebra mentioned above is shown to act on solutions of the Landau-Lifshitz equation as infinitesimal B\\\"acklund transformations where they derive an $N$-soliton formula.These authors arrive at an algebra that is a one dimensional central extension of $\\mathfrak g\\otimes \\mathbb C[t,t^{-1},u|u^2=(t^2-b^2)(t^2-c^2)]$ where $b\\neq \\pm c$ are complex constants and $\\mathfrak g$ is a simple finite dimensional Lie algebra defined over the complex numbers. Below we explicitly describe its four dimensional universal central extension. Modulo the center this algebra is a particular example of a Krichever-Novikov current algebra (see (\\cite{MR902293}, \\cite{MR925072}, \\cite{MR998426}). A fair amount of interesting and fundamental work has be done by Krichever, Novikov, Schlichenmaier, and Sheinman on the representation theory of certain one dimensional central extensions of these latter current algebras and of analogues of the Virasoso algebra. In particular Wess-Zumino-Witten-Novikov theory and analogues of the Knizhnik-Zamolodchikov equations are developed for these algebras (see the survey article \\cite{MR2152962}, and for example \\cite{MR1706819}, \\cite{MR1706819},\\cite{MR2072650},\\cite{MR2058804},\\cite{MR1989644}, and \\cite{MR1666274}).\n\n\nM. Bremner on the other hand has explicitly described in terms of generators, relations and certain families of polynomials (ultraspherical and Pollacyk) the structure constants for the universal central extension of algebras of the form $\\mathfrak g\\otimes \\mathbb C[t,t^{-1},u|u^2=p(t)]$ where $p(t)=t^2-2bt+1$ and $p(t)=t^3-2bt^2+t$ (see \\cite{MR1249871,MR1303073}). He determined more generally the dimension of the universal central extension for affine Lie algebras of the form $\\mathfrak g\\otimes R$ where $R$\nis the ring of regular functions defined on an algebraic curve with any number of points removed. He obtained this using C. Kassel's result (\\cite{MR772062}) where one knows that the center is isomorphic as a vector space to $\\Omega_R^1\/dR$ (the {\\it space of K\\\"ahler differentials of $R$ modulo exact forms}). We will review this material below as needed. \n\nIn previous work of the authors (see \\cite{MR2373448,MR2541818}) we used Bremner's aforementioned description to obtain certain free field realizations of the four point and elliptic affine algebras depending on a parameter $r=0,1$ which correspond to two different normal orderings. These later realizations are analogues of Wakimoto type realizations which have been used by Schechtman and Varchenko and various other authors in the affine setting to pin down integral solutions to the Knizhnik-Zamolodchikov differential equations (see for example \\cite{MR1136712}, \\cite{MR1138049}, \\cite{MR2001b:32028}, \\cite{MR1077959}). Such realizations have also been used in the study of Drinfeld-Sokolov reduction in the setting of $W$-algebras and in E. Frenkel's and B. Feigin's description of the center of the completed enveloping algebra of an affine Lie algebra (see \\cite{MR1309540}, \\cite{MR2146349}, and \\cite{MR1187549}). In future work we plan to use results of this paper to describe free field realizations of the universal central extension of the algebras of Date, Jimbo, Kashiwara and Miwa (which, since this is a mouth full, we will call DJKM algebras). \n\n\n\\section{Universal Central Extensions of Current Algebras} Let $R$ be a commutative algebra defined over $\\mathbb C$.\nConsider the left $R$-module with action $f( g\\otimes h ) = f g\\otimes h$ for $f,g,h\\in R$ and let $K$ be the submodule generated by the elements $1\\otimes fg -f \\otimes g -g\\otimes f$. Then $\\Omega_R^1=F\/K$ is the module of K\\\"ahler differentials. The element $f\\otimes g+K$ is traditionally denoted by $fdg$. The canonical map $d:R\\to \\Omega_R^1$ by $df = 1\\otimes f + K$. The {\\it exact differentials} are the elements of the subspace $dR$. The coset of $fdg$ modulo $dR$ is denoted by $\\overline{fdg}$. As C. Kassel showed the universal central extension of the current algebra $\\mathfrak g\\otimes R$ where $\\mathfrak g$ is a simple finite dimensional Lie algebra defined over $\\mathbb C$, is the vector space $\\hat{\\mathfrak g}=(\\mathfrak g\\otimes R)\\oplus \\Omega_R^1\/dR$ with Lie bracket given by\n$$\n[x\\otimes f,Y\\otimes g]=[xy]\\otimes fg+(x,y)\\overline{fdg}, [x\\otimes f,\\omega]=0, [\\omega,\\omega']=0,\n$$\n where $x,y\\in\\mathfrak g$, and $\\omega,\\omega'\\in \\Omega_R^1\/dR$ and $(x,y)$ denotes the Killing form on $\\mathfrak g$. \n \n\nConsider the polynomial\n$$\np(t)=t^n+a_{n-1}t^{n-1}+\\cdots+a_0\n$$\nwhere $a_i\\in\\mathbb C$ and $a_n=1$. \nFundamental to the description of the universal central extension for $R=\\mathbb C[t,t^{-1},u|u^2=p(t)]$ is the following:\n\\begin{thm}[\\cite{MR1303073},Theorem 3.4] Let $R$ be as above. The set \n$$\n\\{\\overline{t^{-1}\\,dt},\\overline{t^{-1}u\\,dt},\\dots, \\overline{t^{-n}u\\,dt}\\}\n$$\n forms a basis of $\\Omega_R^1\/dR$ (omitting $\\overline{t^{-n}u\\ dt}$ if $a_0=0$). \n\\end{thm}\nSet $u^m=p(t)$. Then $u\\,d(u^m)=mu^mdu$ and\n$$\n\\sum_{j=1}^nja_jt^{j-1}u\\,dt-m\\left(\\sum_{j=0}^na_jt^j\\,du\\right)=0\n$$\nor \n$$\np'(t)udt-mp(t)du=0.\n$$\nMultiplying by $t^i$ we get \n\\begin{equation}\\label{lady1}\n\\sum_{j=1}^nja_jt^{i+j-1}u\\,dt-m\\left(\\sum_{j=0}^na_jt^{i+j}\\,du\\right)=0\n\\end{equation}\n\n\n\\begin{lem} If $u^m=p(t)$ and $R=\\mathbb C[t,t^{-1},u|u^m=p(t)]$, then in $\\Omega_R^1\/dR$, one has\n\\begin{equation}\\label{recursionreln}\n((m+1)n+im)t^{n+i-1}u\\,dt \\equiv - \\sum_{j=0}^{n-1}((m+1)j+mi)a_jt^{i+j-1}u\\,dt\\mod dR\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe have expanding $d(t^{i+j}u)$\n$$\n(i+j)t^{i+j-1}u\\,dt\\equiv-t^{i+j}\\,du\\mod dR.\n$$\nso that \\eqnref{lady1} implies \n\\begin{equation}\n\\sum_{j=0}^nja_jt^{i+j-1}u\\,dt+m\\left(\\sum_{j=0}^n(i+j)a_jt^{i+j-1}u\\,dt\\right)=0 \\mod dR\n\\end{equation}\nor\n\\begin{equation}\n\\sum_{j=0}^n((m+1)j+mi)a_jt^{i+j-1}u\\,dt\\equiv 0\\mod dR\n\\end{equation}\nThis gives \\eqnref{recursionreln}.\n\n\n\\end{proof}\n\n\n\n\\section{Description of the universal central extension of Date-Jimbo-Miwa-Kashiwara algebras}\nIn the Date-Jimbo-Miwa-Kashiwara setting one takes $m=2$ and $p(t)=(t^2-a^2)(t^2-b^2)=t^4-(a^2+b^2)t^2+(ab)^2$ with $a\\neq \\pm b$ and neither $a$ nor $b$ is zero. We fix from here onward $R=\\mathbb C[t,t^{-1},u\\,|\\,u^2= (t^2-a^2)(t^2-b^2)]$. As in this case $a_0=(ab)^2$, $a_1=0$, $a_2=-(a^2+b^2)$, $a_3=0$ and $a_4=1$, then letting $k=i-2$ the recursion relation in \\eqnref{recursionreln} looks like\n\\begin{align*}\n(6+2k)\\overline{t^{k}u\\,dt} \n&=-2(k-3)(ab)^2\\overline{t^{k-4}u\\,dt} +2k(a^2+b^2)\\overline{t^{k-2}u\\,dt}.\n\\end{align*}\nAfter a change of variables we may assume that $a^2b^2=1$. Then the recursion relation looks like\n\\begin{equation}\\label{recursionreln1}\n(6+2k)\\overline{t^{k}u\\,dt} \n=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt},\n\\end{equation}\nafter setting $c=(a^2+b^2)\/2$, so that $p(t)=t^4-2ct^2+1$. Let $P_k:=P_k(c)$ be the polynomial in $c$ satisfy the recursion relation \n$$\n(6+2k)P_k(c) \n=4k cP_{k-2}(c)-2(k-3)P_{k-4}(c)\n$$\nfor $k\\geq 0$.\nThen set\n$$\nP(c,z):=\\sum_{k\\geq -4}P_k(c)z^{k+4}=\\sum_{k\\geq 0}P_{k-4}(c)z^{k}.\n$$\nso that after some straightforward rearrangement of terms we have\n\\begin{align*}\n0&=\\sum_{k\\geq 0}(6+2k)P_k(c)z^k \n-4c\\sum_{k\\geq 0}kP_{k-2}(c)z^{k} +2\\sum_{k\\geq 0}(k-3)P_{k-4}(c)z^{k} \\\\\n&=(-2z^{-4} +8cz^{-2}-6)P(c,z) +(2z^{-3}-4cz^{-1}+2z)\\frac{d}{dz}P(c,z) \\\\\n&\\quad+(2z^{-4}-8cz^{-2})P_{-4}(c) -4cP_{-3}(c) z^{-1} -2P_{-2}(c)z^{-2} -4P_{-1}(c)z^{-1}.\n\\end{align*}\nWe then multiply the above through by $z^{4}$ to get \n\\begin{align*}\n0&=(-2+8cz^{2}-6z^4)P(c,z) +(2z-4cz^{3}+2z^5)\\frac{d}{dz}P(c,z) \\\\\n&\\quad+(2-8cz^{2})P_{-4}(c) -4cP_{-3}(c) z^{3} -2P_{-2}(c)z^{2} -4P_{-1}(c)z^{3}.\n\\end{align*}\n\nHence $P(c,z)$ must satisfy the differential equation\n\\begin{equation}\\label{funde}\n\\frac{d}{dz}P(c,z)-\\frac{3z^4-4c z^2+1}{z^5-2cz^3+z}P(c,z)=\\frac{2\\left(P_{-1}+cP_{-3} \\right)z^3 +P_{-2} z^2+(4cz^2-1)P_{-4} }{z^5-2cz^3+z}\n\\end{equation}\nThis has integrating factor\n\\begin{align*}\n\\mu(z)&\n=\\exp \\int\\left( \\frac{-2 \\left(z^3-cz\\right)}{1-2 c z^2+z^4 }-\\frac{1}{z}\\right)\\,dz \\\\\n&=\\exp(-\\frac{1}{2} \\ln(1-2 c z^2+z^4)-\\ln (z))=\\frac{1}{z\\sqrt{1-2 c z^2+z^4}}.\n\\end{align*}\n\n\\subsection{Elliptic Case 1}\nIf we take initial conditions $P_{-3}(c)=P_{-2}(c)=P_{-1}(c)=0$ and $P_{-4}(c)=1$ then we arrive at a generating function \n$$\nP_{-4}(c,z):=\\sum_{k\\geq -4}P_{-4,k}(c)z^{k+4}=\\sum_{k\\geq 0}P_{-4,k-4}(c)z^{k},\n$$\ndefined in terms of an elliptic integral\n\\begin{align*}\nP_{-4}(c,z)&=z\\sqrt{1-2 c z^2+z^4}\\int \\frac{4cz^2-1}{z^2(z^4-2c z^2+1)^{3\/2}}\\, dz.\n\\end{align*}\n One way to interpret the right hand integral is to expand $(z^4-2c z^2+1)^{-3\/2}$ as a Talyor series about $z=0$ and then formally integrate term by term and multiply the result by the Taylor series of $z\\sqrt{1-2 c z^2+z^4}$. More precisely one integrates formally with zero constant term\n $$\n \\int (4c-z^{-2})\\sum_{n=0}^\\infty Q_n^{(3\/2)}(c)z^{2n}\\,dz =\\sum_{n=0}^\\infty \\frac{4cQ_n^{(3\/2)}(c)}{2n+1}z^{2n+1} -\\sum_{n=0}^\\infty \\frac{Q_n^{(3\/2)}(c)}{2n-1}z^{2n-1}\n $$ \n where $Q_n^{(\\lambda)}(c)$ is the $n$-th Gegenbauer polynomial.\nAfter multiplying this by \n$$\nz\\sqrt{1-2cz^2+z^4}=\\sum_{n=0}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\n$$\none arrives at the series $P_{-4}(c,z)$.\n\n\n\\subsection{Elliptic Case 2}\nIf we take initial conditions $P_{-4}(c)=P_{-3}(c)=P_{-1}(c)=0$ and $P_{-2}(c)=1$ then we arrive at a generating function defined in terms of another elliptic integral:\n\\begin{align*}\nP_{-2}(c,z)&=z\\sqrt{1-2 c z^2+z^4}\\int \\frac{1}{ (z^4-2c z^2+1)^{3\/2}}\\, dz.\n\\end{align*}\n\n\n\n\\subsection{Gegenbauer Case 3}\nIf we take $P_{-1}(c)=1$, and $P_{-2}(c)=P_{-3}(c)=P_{-4}(c)=0$ and set \n$$\nP_{-1}(c,z)=\\sum_{n\\geq 0}P_{-1,n-4}z^n,\n$$\nthen we get a solution which after solving for the integration constant can be turned into a power series solution \n\\begin{align*}\nP_{-1}(c,z)&=(z\\sqrt{1-2 c z^2+z^4})\\left(\\int \\frac{2cz^3}{t\\sqrt{1-2 c z^2+z^4}(z^5-2c z^3+z)}\\, dt+C\\right) \\\\\n&=\\frac{ z(c-z^3)}{c^2-1}-\\frac{c}{c^2-1}z\\sqrt{z^4-2cz^2+1} \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-cz\\sqrt{z^4-2c z^2+1}\\right) \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-\\sum_{k=0}^\\infty c Q_n^{(-1\/2)}(c)z^{2n+1}\\right) \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-cz+c^2z^3-\\sum_{k=2}^\\infty c Q_n^{(-1\/2)}(c)z^{2n+1}\\right) \n\\end{align*}\nwhere $Q^{(-1\/2)}_n(c)$ is the $n$-th Gegenbauer polynomial. Hence\n\\begin{align*}\nP_{-1,-4}(c)&=P_{-1,-3}(c)=P_{-1,-2}(c) =P_{-1,2m}(c)=0, \\\\\nP_{-1,-1}(c)&=1, \\\\\nP_{-1,2n-3}(c)&=\\frac{-cQ_{n}(c)}{c^2-1},\n\\end{align*}\nfor $m\\geq 0$ and $n\\geq 2$ .\nThe $Q^{(-1\/2)}_n(c)$ are known to satisfy the second order differential equation:\n\\begin{align*}\n(1-c^2)\\frac{d^2}{d^2 c}Q^{(-1\/2)}_n(c)+n(n-1)Q^{(-1\/2)}_{n}(c)=0\n\\end{align*}\nso that the $P_{-1,k}:=P_{-1,k}(c)$ satisfy the second order differential equation\n\\begin{align*}\n(c^4-c^2)\\frac{d^2}{d^2 c}P_{-1,2n-3}+2c(c^2+1)\\frac{d}{d c}P_{-1,2n-3}+(-c^2n(n-1)-2)P_{-1,2n-3}=0\n\\end{align*}\nfor $n\\geq 2$.\n\n\\subsection{Gegenbauer Case 4}\nNext we consider the initial conditions $P_{-1}(c)=0=P_{-2}(c)=P_{-4}(c)=0$ with $P_{-3}(c)=1$ and set \n$$\nP_{-3}(c,z)=\\sum_{n\\geq 0}P_{-3,n-4}(c)z^n,\n$$\nthen we get a power series solution\n\\begin{align*}\nP_{-3}(c,z)&=(z\\sqrt{1-2 c z^2+z^4})\\left(\\int \\frac{2cz^3}{z\\sqrt{1-2 c z^2+z^4}(z^5-2c z^3+z)}\\, dz+C\\right) \\\\\n&=\\frac{ cz(c-z^3)}{c^2-1}-\\frac{1}{c^2-1}z\\sqrt{z^4-2cz^2+1} \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-z\\sqrt{z^4-2c z^2+1}\\right) \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-\\sum_{k=0}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\\right) \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-z+cz^3-\\sum_{k=2}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\\right) \\\\\n\\end{align*}\nwhere $Q^{(-1\/2)}_n(c)$ is the $n$-th Gegenbauer polynomial. Hence\n\\begin{align*}\nP_{-3,-4}(c)&=P_{-3,-2}(c)=P_{-3,-1}(c) =P_{-1,2m}(c)=0, \\\\\nP_{-3,-3}(c)&=1, \\\\\nP_{-3,2n-3}(c)&=\\frac{-Q_{n}(c)}{c^2-1},\n\\end{align*}\nfor $m\\geq 0$ and $n\\geq 2$ and hence \\begin{align*}\n(c^2-1)\\frac{d^2}{d^2 c}P_{-3,2n-3}+4c \\frac{d}{d c}P_{-3,2n-3} -(n+1)(n-2)P_{-3,2n-3} =0\n\\end{align*}\nfor $n\\geq 2$ and $P_{-1,2n-3}=cP_{-3,2n-3}$ for $n\\geq 2$.\n\\section{Main result}\nFirst we give an explicit description of the cocyles contributing to the {\\it even} part of the DJKM algebra. \n\n\\begin{prop}[cf. \\cite{MR1303073}, Prop. 4.2] \\label{cocyclecalc} Set $\\omega_0=\\overline{t^{-1}\\,dt}$. \n For $i,j\\in\\mathbb Z$ one has\n\\begin{equation}\nt^i\\,d(t^j)= j \\delta_{i+j,0}\\omega_0\n\\end{equation}\nand \n\\begin{equation}\nt^{i-1}u\\,d(t^{j-1}u)=\\left(\\delta_{i+j,-2}(j+1) -2cj\\delta_{i+j,0} +(j-1)\\delta_{i+j,2}\\right)\\omega_0.\n\\end{equation}\n\n\\end{prop}\n\\begin{proof} First observe that \n$2u\\,du=d(u^2)=(4t^3-4c t)\\,dt$.\n The second congruence then follows from\n\\begin{align*}\nt^{i-1}u\\,d(t^{j-1}u)&=(j-1)t^{i+j-3}u^2\\,dt+t^{i+j-2}u\\,du \\\\\n&=(j-1)t^{i+j-3}(t^4-2ct^2+1)\\,dt+2t^{i+j-2}(t^3-c t)\\,dt \\\\\n&=(j-1)(t^{i+j+1}-2ct^{i+j-1}+t^{i+j-3})\\,dt+2(t^{i+j+1}-c t^{i+j-1})\\,dt \\\\\n&=(j+1)t^{i+j+1}\\,dt -2cjt^{i+j-1}\\,dt +(j-1)t^{i+j-3}\\,dt. \n\\end{align*}\n\\end{proof}\n\nThe map $\\sigma:R\\to R$ given by $\\sigma(t)=t^{-1}$, $\\sigma(u)=t^{-2}u$ is an algebra automorphism as \n$\\sigma(u^2)=t^{-4}u^2=1-2ct^{-2}+t^{-4}=\\sigma(1-2ct^{2}+t^{4})$. This descends to a linear map $\\sigma:\\Omega_R^1\/dR$ where\n\\begin{align*}\n\\sigma(\\overline{t^{-1}\\,dt})&=-\\overline{t^{-1}\\,dt},\\\\\n\\sigma(\\overline{t^{-1}u\\,dt)}&=\\overline{t(t^{-2}u)d(t^{-1})}=-\\overline{t^{-3}u\\,dt}, \\\\\n\\sigma(\\overline{t^{-2}u\\,dt)}&=\\overline{t^2(t^{-2}u)d(t^{-1})}=-\\overline{t^{-2}u\\,dt}, \\\\\n\\sigma(\\overline{t^{-3}u\\,dt)}&=-\\overline{t^{-1}u\\,dt}, \\\\ \n\\sigma(\\overline{t^{-4}u\\,dt)}&=\\overline{t^4(t^{-2}u)d(t^{-1})}=-\\overline{u\\,dt}=-\\overline{t^{-4}u\\,dt },\n\\end{align*} \nwhereby the last identity follows from the recursion relation \\eqnref{recursionreln1} with $k=0$. Setting $\\omega_{-k}=\\overline{t^{-k}u\\,dt}$, $k=1,2,3,4$, then $\\sigma(\\omega_{-1})=-\\omega_{-3}$, and $\\sigma(\\omega_{-l})=-\\omega_{-l}$ for $l=2,4$.\n\n\\begin{thm} Let $\\mathfrak g$ be a simple finite dimensional Lie algebra over the complex numbers with the Killing form $(\\,|\\,)$ and define $\\psi_{ij}(c)\\in\\Omega_R^1\/dR$ by\n\\begin{equation}\n\\psi_{ij}(c)=\\begin{cases} \n\\omega_{i+j}&\\quad \\text{ for }\\quad i+j=1,0,-1,-2 \\\\\nP_{-3,i+j-2}(c) (\\omega_{-3}+c\\omega_{-1})&\\quad \\text{for} \\quad i+j =2n-1\\geq 3,\\enspace n\\in\\mathbb Z, \\\\\nP_{-3,i+j-2}(c) (c\\omega_{-3}+\\omega_{-1})&\\quad \\text{for} \\quad i+j =-2n+1\\leq - 3, n\\in\\mathbb Z, \\\\\nP_{-4,|i+j|-2}(c) \\omega_{-4} +P_{-2,|i+j|-2}(c)\\omega_{-2}&\\quad\\text{for}\\quad |i+j| =2n \\geq 2, n\\in\\mathbb Z. \\\\\n\\end{cases}\n\\end{equation}\nThe universal central extension of the Date-Jimbo-Kashiwara-Miwa algebra is the $\\mathbb Z_2$-graded Lie algebra \n$$\n\\widehat{\\mathfrak g}=\\widehat{\\mathfrak g}^0\\oplus \\widehat{\\mathfrak g}^1,\n$$\nwhere\n$$\n\\widehat{\\mathfrak g}^0=\\left(\\mathfrak g\\otimes \\mathbb C[t,t^{-1}]\\right)\\oplus \\mathbb C\\omega_{0},\\qquad \\widehat{\\mathfrak g}^1=\\left(\\mathfrak g\\otimes \\mathbb C[t,t^{-1}]u\\right)\\oplus \\mathbb C\\omega_{-4}\\oplus \\mathbb C\\omega_{-3}\\oplus \\mathbb C\\omega_{-2}\\oplus \\mathbb C\\omega_{-1}\n$$\nwith bracket\n\\begin{align*}\n[x\\otimes t^i,y\\otimes t^j]&=[x,y]\\otimes t^{i+j}+\\delta_{i+j,0}j(x,y)\\omega_0, \\\\ \\\\\n[x\\otimes t^{i-1}u,y\\otimes t^{j-1}u]&=[x,y]\\otimes (t^{i+j+2}-2ct^{i+j}+t^{i+j-2}) \\\\\n &\\hskip 40pt+\\left(\\delta_{i+j,-2}(j+1) -2cj\\delta_{i+j,0} +(j-1)\\delta_{i+j,2}\\right)(x,y)\\omega_0, \\\\ \\\\\n[x\\otimes t^{i-1}u,y\\otimes t^{j}]&=[x,y]u\\otimes t^{i+j-1}+ j(x,y)\\psi_{ij}(c).\n\\end{align*}\n\n\\end{thm}\n\n\n\\begin{proof}\nThe first two equalities follow from \\propref{cocyclecalc}. For the last one we first observe that for $k=i+j-2\\neq -3$,\n\\begin{align*}\nj\\omega_{ij}(c)=\\overline{ t^{i-1}u\\,d( t^{j})}&=j\\overline{t^{i+j-2}u\\,dt} \\\\\n&=j\\left(\\frac{-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}}{6+2k}\\right),\n\\end{align*}\nwhere the last equality is derived from \\eqnref{recursionreln1}.\nThen by setting $k=0,1,2,3,4,5$ in \\eqnref{recursionreln1}\n\\begin{equation*}\n(6+2k)\\overline{t^{k}u\\,dt} \n=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}.\n\\end{equation*}\ngives us \n\\begin{align*}\n6\\overline{u\\,dt} &=6\\overline{t^{-4}u\\,dt} , \\\\\n 8\\overline{tu\\,dt} &=4\\overline{t^{-3}u\\,dt} +4c\\overline{t^{-1}u\\,dt}, \\\\\n 10\\overline{t^{2}u\\,dt} \n&=2\\overline{t^{-2}u\\,dt} +8c\\overline{ u\\,dt}, \\\\ \n12\\overline{t^{3}u\\,dt} \n&=12c\\overline{tu\\,dt} ,\\\\\n14\\overline{t^{4}u\\,dt} \n&=-2\\overline{u\\,dt} +16c\\overline{t^{2}u\\,dt}, \\\\\n16\\overline{t^{5}u\\,dt} \n&=-4\\overline{tu\\,dt} +20c\\overline{t^{3}u\\,dt}, \\\\\n(6+2k)\\overline{t^{k}u\\,dt} \n&=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt} .\n\\end{align*}\nHence when $i+j-2=k=0,1,2,3,4,5$\n\\begin{align*}\n\\overline{u\\,dt} &=\\omega_{-4} , \\\\\n\\overline{tu\\,dt} &=\\frac{1}{2}\\left(\\omega_{-3} +c\\omega_{-1}\\right), \\\\\n \\overline{t^{2}u\\,dt} \n&=\\frac{1}{5} \\omega_{-2} +\\frac{4c}{5}\\omega_{-4}, \\\\ \n\\overline{t^{3}u\\,dt} \n&=\\frac{c}{2}\\left(\\omega_{-3} +c\\omega_{-1}\\right), \\\\\n \\overline{t^{4}u\\,dt} \n&=-\\frac{1}{7}\\overline{u\\,dt} +\\frac{8}{7}c\\overline{t^{2}u\\,dt}=-\\frac{1}{7}\\omega_{-4} +\\frac{8}{7}c\\left(\\frac{1}{5} \\omega_{-2} +\\frac{4c}{5}\\omega_{-4}\\right)\\\\\n&=\\left(\\frac{32c^2-5}{35}\\right)\\omega_{-4} +\\frac{8}{35}c \\omega_{-2} , \\\\ \n \\overline{t^{5}u\\,dt} \n&=-\\frac{1}{8} \\left(\\omega_{-3} +c\\omega_{-1}\\right)+\\frac{5c^2}{8}\\left(\\omega_{-3} +c\\omega_{-1}\\right) \\\\\n&=\\frac{5c^2-1}{8} \\left(\\omega_{-3} +c\\omega_{-1}\\right),\n\\\\\n\\overline{t^{k}u\\,dt} \n&=\\frac{-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}}{6+2k} .\n\\end{align*}\nThus by induction using the last equation above for $i+j-2=k=2n-3\\geq 1$, $n\\in\\mathbb Z$, we have \n\\begin{align}\\label{oddcase}\n\\omega_{ij}(c)\n&=P_{-3,i+j-2}(c)\\left(\\omega_{-3} +c\\omega_{-1}\\right),\n\\end{align}\nand for $i+j-2=k=2n-2\\geq 0$, $n\\in\\mathbb Z$, we have \n\\begin{align}\\label{evencase}\n\\omega_{ij}(c)\n&=P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}.\n\\end{align}\nApplying $\\sigma$ to \\eqnref{oddcase} for $i+j-2=k=2n-3\\geq 1$ to obtain \n\\begin{align*}\nj\\sigma(\\omega_{ij}(c))=\\overline{ t^{-i+1}u\\,d( t^{-j})}&=-j\\overline{t^{-i-j-2}u\\,dt} \\\\\n&=j\\sigma\\left(P_{-3,i+j-2}(c)\\left(\\omega_{-3} +c\\omega_{-1}\\right) \\right) \\\\ \n&=-jP_{-3,i+j-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\nHence for $i+j-2=2n-3\\geq 1$\n\\begin{align*}\n\\omega_{-i,-j}(c)=\\overline{t^{-i-j-2}u\\,dt} \n&=P_{-3,i+j-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\nSetting $i'=-i$ and $j'=-j$ we get for $i'+j'-2=-k-4=-2n+3\\leq -5$\n\\begin{align*}\n\\omega_{i'j'}(c)=\\overline{t^{i'+j'-2}u\\,dt} \n&=P_{-3,|i'+j'|-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\n\nSimilarly if we apply $\\sigma$ to \\eqnref{evencase} for $i+j=2n\\geq 2$, $n\\in\\mathbb Z$, we obtain \n\\begin{align*}\nj\\sigma(\\omega_{ij}(c))=\\overline{ t^{-i+1}u\\,d( t^{-j})}&=-j\\overline{t^{-i-j-2}u\\,dt} \\\\\n&=j\\sigma\\left(P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2} \\right) \\\\ \n&=-j\\left(P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}\\right)\n\\end{align*}\nHence for $i+j=2n\\geq 2$\n\\begin{align*}\n\\omega_{-i,-j}(c)=\\overline{t^{-i-j-2}u\\,dt} \n&=P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}.\n\\end{align*}\nSetting $i'=-i$ and $j'=-j$ we get for $i'+j'=-2n\\leq -2$\n\\begin{align*}\n\\omega_{i'j'}(c)=\\overline{t^{i'+j'-2}u\\,dt} \n&=P_{-4,|i'+j'|-2}(c) \\omega_{-4} +P_{-2,|i'+j'|-2}(c)\\omega_{-2}.\n\\end{align*}\n\\end{proof}\nOne might want to compare the above theorem with the results that M. Bremner obtained for the elliptic and four point affine Lie algebra cases (\\cite[Theorem 4.6]{MR1303073} and \\cite[Theorem 3.6]{MR1249871} respectively).\n\n\n\n\n\\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently, an interesting effect was discovered by Ba\\~{n}ados, Silk\nand West \\cite{ban}, called usually the BSW effect after the names of the\nauthors: if two particles collide near the black hole horizon, the energy $%\nE_{c.m.}$ in their centre of mass frame can grow indefinitely large,\nprovided the parameters of one of the particles are fine-tuned. Immediately\nafter this observation, several considerations of theoretical nature were\nbrought forward suggesting that there must be restrictions that would\nprevent the realization of this effect. One of the basic objections is\nconnected with the force of gravitational radiation acting on particles. It\nwas pushed forward in \\cite{berti} and is mentioned from time to time in\nconsequent works starting from \\cite{ted}. There are also other similar effects which seem to\nrestrict the divergence of $E_{c.m.}$ -- say, synchrotron radiation by\ncharged particles near black holes \\cite{fr}.\n\nMeanwhile, the influence of the force of gravitational radiation (or any\nother force) on the BSW effect is not so obvious. First of all, the BSW\neffect is prepared from two main ingredients -- the presence of the horizon\nand the presence of special \"critical\" trajectories, (see below). It was\nshown in \\cite{gc}, with minimal assumptions, that even for neutral\nparticles and nongeodesic motion, such trajectories do exist. Therefore, the\nquestion is whether or not the force destroys these trajectories. If this\nhappens, the BSW effect is restricted. However, for a weak force, one can expect a large bound on $E_{c.m.}$. For instance, the analysis of\nparticle's motion on the innermost stable orbit near the Kerr black hole\nwith gravitational radiation taken into account showed that $E_{c.m.}$ can\nbe far beyond the Planck energy for collision of dark matter particles near\na stellar mass near-extremal black hole \\cite{insp}. The analysis suggested in \\cite{insp}, however, concerns special (although important for astrophysics) cases: it applies to\nnear-extremal Kerr black holes when fine-tuning required for the BSW effect\nis realized on circular orbits. It also remains incomplete since not all\nfactors responsible for the self-force are taken into account. Meanwhile, it\nis of interest to elucidate the issue under discussion in a\nmodel-independent way.\n\nIn this paper we develop such a general approach and analyze the BSW effect under the influence of a generic force near the horizon of a generic axially symmetric stationary ``dirty'' black hole (i.e. a black hole that is surrounded by matter, so its metric may deviate from the Kerr one). Here we only consider the case of an extremal horizon of a maximally rotating black hole. The approach used is applicable, with minimal modifications, to static or charged black holes, as shown explicitly for the case of Reissner-Nordstr\\\"{o}m metric.\n\nWe consider the conditions the force should satisfy for the effect to be either preserved in some form or not. The analysis is made in terms of tetrad components of the corresponding quantities in the frames attached both to an observer orbiting the black hole, and the one crossing the horizon. The nature of the force itself is not specified, we only assume that its tetrad components in the particle's proper frame are finite and restrict our consideration to equatorial motion. \nWe show that the BSW effect survives any force that satisfies the following assumptions: (i) it remains finite near the horizon, and (ii) its azimuthal component tends to zero fast enough (more detailed definition is given below). In case the above condition is not satisfied, e.g. the azimuthal force does not vanish in the horizon limit, the weaker version of the effect is realized whenever the acceleration's amplitude is small enough (as should be for e.g. radiation reaction). For the latter case, we find generic bounds on $E_{c.m.}$.\n\nIt is worth stressing that the BSW effect reveals itself not only for extremal black holes, but also for nonextremal ones. The mechanism in the latter case, however, is generally different, as it requires multiple scattering, which for extremal black holes is not necessary \\cite{gp} (see also \\cite{prd}). Correspondingly, we postpone consideration of the BSW effect with a force near nonextremal horizons and, in the present paper, restrict ourselves to the extremal case. The effect for near-extremal horizons, considered in \\cite{insp}, occupies an intermediate position between the two. This problem contains some subtleties on its own related to the properties of near-circular orbits and in the general setting also needs separate treatment.\n\nThere are two aspects of the BSW effect --- the behavior of $E_{c.m.}$ near the horizon and the properties of energies of the collision outcome measured at infinity. The typical energies at infinity are quite modest even in the absence of force \\cite{inf1,inf2,inf3}, so taking the force into account can only change them slightly. It is the first aspect which is nontrivial and is being discussed in the present paper. \n\nThe paper is organized as follows. In Sec. II, we consider classification of\nparticles relevant for the BSW effect and discuss novel features that the\nforce brings into the system. In Sec. III, we consider behavior of\nacceleration near the horizon in different frames (attached to an observer orbiting the black hole or to one crossing the horizon). In Sec. IV, we illustrate\ngeneral relationships using the Reissner-Nordstr\\\"{o}m metric as an example.\nIn Sec. V, we consider generic motion in the equatorial plane under the action of finite forces and derive conditions on the force that allow or forbid critical trajectories.\nIn Sec. VI, we estimate the bounds on $E_{c.m.}$ for the case when the force is least favourable for the effect but small. In Sec. VII, we discuss pure kinematic restrictions on particle's trajectories (valid even in the absence of force) which can influence the properties of the BSW effect. Sec. VIII is devoted to conclusion.\n\n\\section{Particles' kinematics near extremal horizons}\n\n\\subsection{A particle in axially symmetric metric}\n\nWe consider the axially symmetric stationary metric written (at least in the\nvicinity of the horizon) in coordinates which are obtained from the Gaussian\nnormal ones by replacing the distance to the horizon $n$ with the\nradial coordinate $r$, defined so that\\footnote{Such as the quasiglobal coordinate of \\cite{BrRub}, ch.3.} $A(r)\\sim N^{2}$ in the\nhorizon limit, where $N^{2}\\rightarrow 0$ (hereafter $c=1$): \n\\begin{equation}\nds^{2}=-N^{2}dt^{2}+g_{\\phi }(d\\phi -\\omega dt)^{2}+\\frac{dr^{2}}{A}%\n+g_{z}dz^{2}.\n\\end{equation}\n\nLet there be some arbitrary, not necessarily geodesic, particle of mass $m$,\nfour-velocity $u^{\\mu }$ and four-momentum \n\\begin{equation*}\np^{\\mu }=mu^{\\mu }.\n\\end{equation*}%\nIt is convenient to represent the four-velocity, both with upper and lower\nindices, by the components of its four-momentum in the following way: \n\\begin{align}\nu^{\\mu }& =\\frac{1}{m}\\Big(\\frac{X}{N^{2}},\\frac{L}{g_{\\phi }}+\\frac{\\omega X%\n}{N^{2}},p^{r},p^{z}\\Big); \\label{Uu} \\\\\nu_{\\mu }& =\\frac{1}{m}\\Big(-E,L,\\frac{1}{A}p^{r},g_{z}p^{z}\\Big), \\label{Ud}\n\\end{align}%\nwhere $E=-mu_{0}$ is energy, $L=mu_{\\phi }$ angular momentum and \n\\begin{equation}\nX=E-\\omega L. \\label{x}\n\\end{equation}\nDue to forward in time condition, $X$ is always positive.\n\nFor a free particle on a geodesic trajectory the energy $E$ and angular momentum \n$L$ are conserved; eqs. (\\ref{Uu}), (\\ref{Ud}) are nothing but the equations\nof motion with given fixed values of $E$ and $L$. In the general case, $E$\nand $L$ are not conserved and together with $u^{z}$ should be treated as\nfunctions of the particle's proper time. Nonetheless, we still write the\ncomponents of the four-velocity in the same form (\\ref{Uu}), (\\ref{Ud})\nwhich can be considered simply as useful parametrization.\n\nThe normalization condition $u^\\mu u_\\mu =-1$ can be written as \n\\begin{equation}\n\\frac{1}{A}(p^r)^2 +g_{z}(p^z)^2 =\\frac{X^2}{N^2}-\\frac{L^2}{g_\\phi}-m^2 . \\label{norm}\n\\end{equation}\n\nThen $p^{r}$ is expressed through the three independent parameters $E$, $L$\nand $u^{z}$: \n\\begin{equation}\np^{r}=\\pm \\frac{\\sqrt{A}}{N}\\;Z, \\label{pr}\n\\end{equation}\nwhere \n\\begin{align}\nZ^{2} \n& =X^{2}-N^{2}\\Big[\\frac{L^{2}}{g_{\\phi }}+g_{z}(p^{z})^{2}+m^{2}\\Big]. \\label{z}\n\\end{align}\n\nThe formulas in this section are applicable also to massless particles, with\nthe only difference that one has to set $m=0$. The four-momentum then, in\nthe appropriate parametrization of the worldline, is related to the wave\nvector $k^{\\mu }$ as $p^{\\mu }=\\hbar k^{\\mu }$.\n\n\\subsection{Two particles' collision near horizon}\n\nThe energy $E_{i\\;c.m.}$ of a particle $i$ with four-momentum $p_{i}^{\\mu }$\nin its center of mass (c.m.) frame is simply its rest mass, i.e. the norm of\nits four-momentum: \n\\begin{equation}\nE_{i\\;c.m.}^{2}=m_{i}^{2}=-p_i^{\\mu }p_{i\\,\\mu }.\n\\end{equation}%\nLikewise, for two particles with masses $m_{1}$ and $m_{2}$ and\nfour-velocities $u_{1}^{\\mu }$ and $u_{2}^{\\mu }$ the center of mass energy $%\nE_{c.m.}$ at the collision event is the norm of their total four-momentum \n\\begin{equation}\nE_{c.m.}^{2}=-(p_{1}^{\\mu }+p_{2}^{\\mu })(p_{1\\mu }+p_{2\\mu\n})=m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\\gamma _{c.m.}, \\label{cm}\n\\end{equation}%\nwhere \n\\begin{equation}\n\\gamma _{c.m.}=-u_{1\\mu }u_{2}^{\\mu }\n\\end{equation}%\nis the relative Lorentz factor.\n\nThe contraction can be written as \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{1}X_{2}-Z_{1}Z_{2}}{N^{2}}-\\frac{L_{1}L_{2}%\n}{g_{\\phi }}-g_{z}p_{1}^{z}p_{2}^{z}. \\label{ga}\n\\end{equation}\n\nFor a collision of a massive particle of mass $m$ and a photon one obtains\nthat \n\\begin{equation}\nE_{c.m.}^{2}=m^{2}+2m\\hbar \\omega _{det},\n\\end{equation}%\nwhere $\\omega _{det}$ $=-k_{\\mu }u^{\\mu }$ is the photon's frequency as\ndetected in the frame of this massive particle.\n\n\\subsection{Usual and critical particles near extremal horizons}\n\nConsider a particle in the vicinity of a regular extremal horizon, for which \n\\cite{reg}\n\\begin{align}\nN^{2}(r)&\\sim (r-r_{H})^{2}, \\label{Extr}\\\\\n\\omega (r)&=\\omega _{H}-\\omega _{1}(z)N+O(N^{2}),\\qquad \\omega _{H}=const,\n\t\\label{OmExpansion}\n\\end{align}\nwhere $r=r_{H}$ is the horizon. The regularity of the horizon implies \\cite{reg} that,\nin particular, $\\omega_H$ is a constant, and that other metric functions can\nalso be expanded into series by $(r-r_{H})$ with positive powers. Note also\nthe sign by $\\omega_1$, defined so for consistency with earlier works.\n\nFrom the normalization condition (\\ref{norm}) then a particle's\nfour-velocity components can always be presented as series by $N$ (though\nthey can diverge at the horizon). Assuming $L$ and $E$ are finite, \n\\begin{equation}\nE=E_{H}+E_{1}N+O(N^{2}),\\quad L=L_{H}+L_{1}N+O(N^{2}), \\label{el}\n\\end{equation}%\nand then \n\\begin{align}\nX=& X_{H}+O(N), \\\\\n& X_{H}=E_{H}-\\omega _{H}L_{H}.\n\\end{align}\n\nFor a usual (generic) particle $X_{H}\\neq 0$. The normalization (\\ref{norm})\nthen implies that \n\\begin{equation}\nu^{r}=O(1), \\label{ru}\n\\end{equation}%\n$u^{z}=O(1\/N)$, so the particle reaches the horizon in finite proper time $%\n\\tau \\sim \\int dr<\\infty $.\n\nHowever, there are also worldlines of particles with angular momentum\nfine-tuned to energy in such a way that $X_{H}=0$, so that \n\\begin{equation}\nX=O(N). \\label{xc}\n\\end{equation}%\nSuch particles are called critical.\n\nFor critical particles the right hand side of normalization condition(\\ref{norm}) is bounded, and as the left hand side there is a sum of squares, we obtain \n\\begin{equation}\nu^r =O(N),\\quad u^z =O(1). \\label{Ubounded}\n\\end{equation}\n\nThen the equation of radial motion in the main order by $N$ is \n\\begin{equation}\n\\frac{dr}{d\\tau }=-\\frac{r-r_{H}}{\\tau _{0}}, \\label{rt}\n\\end{equation}%\nwhere $\\tau _{0}$ is a constant for motion in equatorial plane\\footnote{In general, the coordinate $z$ can oscillate between some limiting values, see \\cite{ne} for the Kerr metric and \\cite{jh} for discussion of a more general case.}; its solution is \n\\begin{equation}\nr-r_{H}=r_{0}e^{-\\tau \/\\tau _{0}} \\label{ur}\n\\end{equation}%\nand the proper time of reaching the horizon diverges as $\\ln (r-r_H )$. In case $u^{r}$ is of higher order that $N$, the divergence is stronger (i.e. if $u^{r}\\sim (r-r_{H})^2$,\nthen $\\tau $ diverges as $(r-r_{H})^{-1}$).\n\n\\subsubsection{Example: the Kerr metric}\n\nIt is instructive to look at the critical trajectory for the Kerr metric.\nLet us restrict ourselves to equatorial motion $\\theta =\\frac{\\pi }{2}$.\nThen, the metric coefficients near the horizon of the extremal Kerr black\nhole read \n\\begin{align}\n&N\\approx \\frac{r-r_{H}}{2r_{H}},\\qquad \\sqrt{A}\\approx \\frac{r-r_{H}}{r_{H}}%\n,\\qquad (g_{\\phi })_{H}=4r_{H}^{2}, \\\\\n&\\omega _{H}=\\frac{1}{2r_{H}},\\qquad \\omega -\\omega _{H}\\approx -\\frac{%\nr-r_{H}}{2r_H^2}.\n\\end{align}\nThen, it follows from eqs. (\\ref{pr}), (\\ref{z}) that the trajectory of the\nparticle with $E=\\omega _{H}L$ has exactly the form (\\ref{rt}) with \n\\begin{equation}\n\\tau _{0}=\\frac{r_{H}}{\\sqrt{3\\frac{E^{2}}{m^{2}}-1}}.\n\\end{equation}\n\n\\subsection{The BSW effect}\n\nConsider the collision of two particles. For a usual (generic) particle,\nassuming $E$, $L$ and $u^z$ are finite\\footnote{Those are natural assumptions, but for justification see the section on\ndynamics below.}, \n\\begin{equation}\nX=X_H +O(N),\\quad Z=X+O(N^2).\n\\end{equation}\n\nThen the relative Lorentz factor at the collision event of two usual\nparticles is \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{1}X_{2}-Z_{1}Z_{2}}{N^{2}}+O(1)=O(1).\n\\label{gacm}\n\\end{equation}\n\nHowever, for a critical particle \n\\begin{equation}\nX=X_{N}N+O(N^{2}),\\quad Z=Z_{N}N+O(N^{2}).\n\\end{equation}%\nThen for two critical particles $\\gamma _{c.m.}$ is also bounded, but the\nrelative Lorentz factor at the collision event of a critical (1) particle\nand a usual (2) particle is \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{H}^{(2)}(X_{N}^{(1)}-Z_{N}^{(1)})}{N}%\n+O(1)\\rightarrow \\infty . \\label{div}\n\\end{equation}\n\nSo, the BSW effects occurs whenever one usual and one critical particle collide near the horizon. Geodesic particles can be\ncritical just due to the choice of initial conditions which fix $E$ and $L$,\nso one can always achieve $X_{H}=0.$ The question is how resilient is the\ncriticality attribute with respect to acceleration: whether a particle can\nremain critical under the action of finite forces, such as radiation\nreaction.\n\n\\subsection{Generalization: usual, critical and sub-critical particles}\n\nIn the absence of external forces acting on a particle, in the vicinity of a\nregular horizon, where all metric functions can be expanded into series by\nthe radial coordinate $r$, the geodesic equation induces the same\ntype of expansions for the parameters of a particle, such as $X$ and $E$.\nTherefore there are only two principally different types of particles: usual\nand critical ones. If we want to take into account forces acting on a\nparticle, however, we have to allow for more general setting. In particular,\nwe assume that acceleration components in the proper frame of a particle and \n$X$ can behave as $\\xi^{q}$ and $\\xi^{p}$ respectively, with some real $q$ and $p$, where\n\\begin{equation}\n \\xi \\equiv r-r_{H}. \\label{xi}\n\\end{equation}\nHereafter we consider this reasonably general while still relatively simple model.\n\nIf a particle reaches the horizon, $X$ must tend to zero more slowly than $N$, so that $Z^2$ remains positive. This is only possible for $p\\leq 1$. On the other hand, it is reasonable to\nrestrict our consideration to finite $E$ and $L$, and thus $X$, so $p\\geq 0$. Then there are three possible particle types, distinguished by $p$ in \n\\begin{equation}\nX\\sim \\xi^ p .\n\\end{equation}\n\n\\begin{enumerate}\n\\item $p=0$: \\textbf{usual particles}. \n\\begin{equation}\nX_u=x_H + x_1 \\xi +\\ldots, \\qquad Z_u=X+O(\\xi^2).\n\\end{equation}\n\n\\item $p=1$: \\textbf{critical particles}. \n\\begin{equation}\nX_{cr}=x_1 \\xi + x_2 \\xi^2 +\\ldots,\\qquad Z_{cr}=O(\\xi) .\n\\end{equation}\n\n\\item $p\\in (0,1)$: the intermediate case, which will be called \\textbf{sub-critical} particles hereafter: \n\\begin{equation}\nX_{sc}=\\alpha \\xi^{p}(1+x_1 \\xi+\\ldots),\\qquad Z_{sc}=X_{sc}+O(\\xi^{2-p}).\n\\end{equation}\nTheir proper time of reaching the horizon $\\sim \\int d\\xi \/Z_{sc}$ is finite.\n\\end{enumerate}\n\nAs shown above (\\ref{ga}), for collision of two particles the relative Lorentz factor\nis \n\\begin{equation}\n\\gamma_{c.m.} =\\frac{X_1 X_2 -Z_1 Z_2}{m_1 m_2 N^2}+O(1). \\label{RelGamma}\n\\end{equation}\nFor collision of two usual or two critical particles near extremal horizon, for which $N^2 \\sim\n\\xi^2$ (\\ref{Extr}), we have $\\gamma_{c.m.} =O(1)$; for usual and critical $\\gamma_{c.m.} \\sim 1\/\\xi$.\nLikewise for usual and sub-critical one obtains \n\\begin{equation}\n\\gamma_{c.m.}\\sim \\xi^{-p} \\to \\infty ;\n\\end{equation}\nfor critical and sub-critical \n\\begin{equation}\n\\gamma_{c.m.} \\sim \\xi^{p-1}\\to \\infty . \\label{SubGamma}\n\\end{equation}\nSo, the corresponding particles behave as\ncritical in collisions with usual ones and as usual in collisions with\ncritical ones.\n\nThis result can be derived in the general setting. First of all, let there\nbe a particle with \n\\begin{align}\n&X=\\alpha \\xi^p (1+O(\\xi)),\\qquad p\\in [0,1]; \\label{alpha} \\\\\n& \\frac{L^2}{g_{\\phi}}+m^2 =\\beta^2+O(\\xi),\\qquad \\alpha,\\beta\\sim 1. \\label{beta}\n\\end{align}\n\nThen \n\\begin{equation}\nZ-X =-C\\xi^{2-p}(1+O(\\xi)), \\label{Z-X}\n\\end{equation}\nwhere \n\\begin{equation}\nC=\\left\\{\\begin{array}{ll}\n\t\\beta^2 \/ 2\\alpha & \\quad \\text{for}\\quad p< 1; \\\\ \n\t\\alpha-\\sqrt{\\alpha^2 -\\beta^2} & \\quad \\text{for}\\quad p=1 .\n\\end{array}\\right.\n\\end{equation}\nAs due to forward in time condition $\\alpha>0$ and $\\beta^2$ is also positive, $C$ is strictly positive as well.\n\nNow suppose we have two such particles, with $p_{1}$ and $p_{2}$, colliding\nnear the horizon. Then using (\\ref{RelGamma}) and (\\ref{Z-X}), the relative\nLorentz factor is reduced to \n\\begin{align}\nm_{1}m_{2}\\,\\gamma _{c.m.}& \n\t=O(1)+\\big(1+O(\\xi )\\big)\\Big[C_{1}\\alpha _{2}\\xi^{p_{2}-p_{1}}\n\t+C_{2}\\alpha _{1}\\xi ^{p_{1}-p_{2}}\\Big] \\label{gamma12-1} \\\\\n& \\sim \\xi ^{-|p_{1}-p_{2}|}\\big[1+O(\\xi )+O(\\xi ^{2|p_{1}-p_{2}|})\\big],\n\\label{gamma12-2}\n\\end{align}%\nand thus \n\\begin{equation}\n\\gamma_{c.m.}\\sim \\xi ^{-|p_{1}-p_{2}|}. \\label{p1-p2}\n\\end{equation}%\nHere, gamma becomes finite only if $p_{1}=p_{2}$. We see that consideration\nof sub-critical particles is convenient, as it allows to describe usual and\ncritical particles in a more coherent and unified way, while at the same\ntime providing greater generality, necessary when dealing with non-geodesic\nmotions.\n\n\\section{Dynamics}\n\n\\subsection{OZAMO and FZAMO frames}\n\nThere are two main qualitatively different frames of reference in the\nvicinity of a black hole horizon. The tetrad vectors and tetrad components\nof different quantities will be denoted by superscripts in parenthesis,\nwhile low case ``o'' or ``f'' in the subscript will denote which frame is used, i.e. $%\na_{o}^{(t)}$ is the $t$-component of acceleration in the OZAMO frame (see\nbelow).\n\n\\paragraph{OZAMO.}\n\nThe first kind of frame is attached to an observer who is orbiting the black hole with\nconstant $r$, having constant energy and zero angular momentum. We will call\nit OZAMO for orbital zero angular momentum observer\\footnote{This observer is usually called just ZAMO in textbooks, but we need to be more specific.}. It is the analogue of\nthe static observer in a static spacetime, and it becomes lightlike in the\nhorizon limit \\cite{72}.\n\nThe tetrad 1-forms of the OZAMO frame, denoted by small ``o'' subscripts,\nread \n\\begin{align}\ne^{(t)}_o& =-Ndt; \\label{zamo} \\\\\ne^{(\\phi )}_o& =g_{\\phi }^{1\/2}\\;(d\\phi-\\omega dt); \\\\\ne^{(r)}_o& =A^{-1\/2}dr; \\\\\ne^{(z)}_o& =g_{z}^{1\/2}dz. \\label{zamo-z}\n\\end{align}\n\nIf another particle's four-velocity is $u^{\\mu }$, then its Lorentz factor\nin this frame is \n\\begin{equation}\n\\gamma=-u^{\\mu }(e_{o }^{(t)})_\\mu=\\frac{X}{mN}. \\label{gx}\n\\end{equation}\nThus for a particle with $X\\sim \\xi^p$\n\\begin{equation}\n\\gamma \\sim \\xi^{p-1}; \\label{gx1}\n\\end{equation}\nfor a usual particle it diverges in the horizon limit, while for a critical\none it stays finite.\n\nA particle's acceleration is \n\\begin{equation}\na^{\\mu }\\equiv u^{\\nu }\\nabla _{\\nu }u^{\\mu }.\n\\end{equation}\nIts tetrad components in the OZAMO frame \n\\begin{equation*}\na^{(i)}_o=a^\\mu (e^{(i)}_o)_{\\mu},\\qquad i=t,\\phi,r,z,\n\\end{equation*}\nare equal to \n\\begin{align}\na^{(t)}_o& =Na^{t}; \\label{a} \\\\\na^{(\\phi) }_o& =\\sqrt{g_{\\phi }}\\;(a^{\\phi }-\\omega a^{t}); \\label{a-phi} \\\\\na^{(r)}_o& =\\frac{1}{\\sqrt{A}}\\;a^{r}; \\label{a-r}\\\\\na^{(z)}_o& =\\sqrt{g_{z}}a^{z}; \\label{a-z}\n\\end{align}%\nthe acceleration scalar then can be presented as \n\\begin{equation}\na^{2}\\equiv a^{\\mu }a_{\\mu } =-(a^{(t)}_o)^{2} +(a^{(\\phi)}_o)^{2}\n+(a^{(r)}_o)^{2}+(a^{(z)}_o)^{2}.\n\\end{equation}\n\nThe OZAMO orbits the horizon at constant $r$ and does not cross it,\ntherefore it is not classified as either usual or critical particle, which\ndoes cross or approach the horizon in infinite proper time respectively. However, it\nis useful to note, that, as its Lorentz factor is finite with respect to a\ncritical particle, and vice versa, in the discussion that follows, OZAMO and\ncritical particles behave similarly.\n\nIt is well-known, that an OZAMO frame breaks down at the horizon, thus strictly speaking at the horizon it is not a valid frame and OZAMO is not an observer in the traditional sense. Hereafter, what we refer to as the values of some quantities measured in the OZAMO frame in the horizon limit are the limits of the corresponding quantities measured in successive different OZAMO frames, with different $r_{ZAMO}$, when $r_{ZAMO}\\to r_H$.\n\n\\paragraph{FZAMO.}\n\nThe other important frame of reference is realized by one of the usual\nparticles crossing the horizon. For simplicity, it is convenient to take for\nsuch an observer $L=0$ similarly to OZAMO and, additionally, $E=m$. Thus we\nwill call the corresponding observer FZAMO for falling zero angular\nmomentum observer. Its frame $\\{e_{f}^{(i)}\\}$, with $i=t,\\phi ,r,z$, is\nconstructed by making a local Lorentz transformation from the OZAMO in the\ndirection towards the horizon\\footnote{Note that this is the transformation for one-forms; vectors are\ntransformed by the inverse matrix, which differs by the sign of $v_{f}$.}: \n\\begin{align}\n\\begin{pmatrix}\n(e^{(t)}_{f})_{\\mu} \\\\ \n(e^{(r)}_{f})_{\\mu}%\n\\end{pmatrix}\n&=\\gamma_f\n\\begin{pmatrix}\n1 & v_f \\\\ \nv_f & 1%\n\\end{pmatrix}\n\\begin{pmatrix}\n(e^{(t)}_o)_{\\mu} \\\\ \n(e^{(r)}_o)_{\\mu}\n\\end{pmatrix}\n\\label{rrt}\n\\end{align}%\nThe FZAMO's Lorentz factor in the OZAMO frame is $\\gamma _{f}=-\\left( u^{\\mu\n}\\right) (e_{o}^{(t)})_{\\mu }$, where $u^{\\mu }$ is given by (\\ref{Uu}) with $L=0$ and $E=m$, and $v_{f}=\\sqrt{1-\\gamma_{f}^{-2}}$.\n\nThen, \n\\begin{equation}\n\\gamma _{f}=\\frac{1}{N},\\qquad v_{f}=\\sqrt{1-N^{2}}. \\label{va}\n\\end{equation}\n\nThe corresponding tetrad components of acceleration \n\\begin{equation}\na_{f}^{(i)}=a^{\\mu }(e_{f}^{(i)})_{\\mu },\\qquad i=t,\\phi ,r,z \\label{aa}\n\\end{equation}%\nare related to $a_{o}^{(i)}$ by the respective Lorentz transformation\nwhich becomes singular on the horizon, where $N\\rightarrow 0$, $\\gamma\n_{f}\\rightarrow \\infty $.\n\n\\paragraph{Proper frame.}\n\nFor non-critical particles with $p<1$ the Lorentz factor relative to the\nOZAMO frame diverges as $\\gamma \\sim \\xi ^{p-1}$ (\\ref{SubGamma}). Thus the\ncorrect reference frame for it will have the same behaviour of Lorentz\nfactor. We will construct it, analogously to FZAMO, by making the\ncorresponding boost in the radial direction, and call it for simplicity the\nproper frame for a particle, although it may not be exactly proper. What is\nimportant is that, in contrast to the OZAMO, the particle's velocity in it\nstays finite (does not tend to $c$).\n\nThus, given a particle's Lorentz factor in the OZAMO frame, $\\gamma$\n(without subscripts), the tetrad components of acceleration in the proper\nframe are \n\\begin{equation}\n\\begin{pmatrix}\na^{(t)}_{pr} \\\\ \na^{(r)}_{pr}%\n\\end{pmatrix}\n=\\gamma \n\\begin{pmatrix}\n1 & v \\\\ \nv & 1%\n\\end{pmatrix}\n\\begin{pmatrix}\na^{(t)}_o \\\\ \na^{(r)}_o%\n\\end{pmatrix}%\n. \\label{AProper}\n\\end{equation}\nFor a usual or critical particle this reduces to the already considered\nOZAMO and FZAMO frames respectively, while for sub-critical particles the proper frame\ndoes not coincide with either one of those.\n\n\\subsection{Acceleration in different frames}\n\nWhen describing particles' motion near the horizon, we must restrict\nourselves to particles with finite acceleration. This necessarily means that\nthe acceleration scalar $a^{2}$ should be finite. It would seem that it is\nnatural to demand that tetrad components of acceleration are finite as well.\nHowever, as shown above, when we describe a particles' motion near the\nhorizon, we have different frames of reference, which are related to each\nother by singular Lorentz transformations. This means that finite tetrad\ncomponents of acceleration in one of the frames may correspond to diverging\ntetrad components in the other or vice versa.\n\nThe frame in which tetrad components of a particle's acceleration should be\nfinite is the instantly comoving frame, or equivalently, any frame which\nmoves with finite Lorentz factor with respect to that. For example, recall\nthe reasonably realistic problem of a charged particle in a uniform electric\nfield in Special Relativity. The tetrad components of acceleration in the\nlaboratory frame (with Minkowski metric) diverge proportionally to the\nLorentz factor, while those in the instantly comoving frame (with the\nRindler metric and the horizon) are constant (see, e.g. p. 403 of \\cite{car}%\n).\n\nFor a critical particle then acceleration is adequately and most easily\nmeasured in the OZAMO frame. For a usual particle we would have to attach\nthe tetrad also to one of the usual particles, for example to FZAMO. Due to\nnormalization of four-velocity $u^\\mu u_\\mu =-1$, which implies $a^\\mu u_\\mu\n=0$, in each case it is sufficient to show that three of the four\ntetrad components are finite.\n\n\\subsection{Energy and angular momentum}\n\nIf $\\xi ^{\\mu }$ is a Killing vector field, then \n\\begin{equation}\n\\frac{d}{d\\tau }(\\xi ^{\\mu }u_{\\mu })=\\xi ^{\\mu }a_{\\mu }.\n\\end{equation}%\nIn a stationary axisymmetric metric we have two Killing vectors $\\xi\n_{t}^{\\mu }=\\delta _{t}^{\\mu }$ and $\\xi _{\\phi }^{\\mu }=\\delta _{\\phi\n}^{\\mu }$, which give \n\\begin{align}\n\\frac{1}{m}\\frac{dE}{d\\tau }& =(N^{2}-\\omega ^{2}g_{\\phi })a^{t}+\\omega\ng_{\\phi }a^{\\phi }; \\\\\n\\frac{1}{m}\\frac{dL}{d\\tau }& =-\\omega g_{\\phi }a^{t}+g_{\\phi }a^{\\phi },\n\\end{align}%\nor through the tetrad components in the OZAMO frame (\\ref{a}, \\ref{a-phi})\n\\begin{align}\n\\frac{1}{m}\\frac{dE}{d\\tau }& =Na^{(t)}_o +\\omega \\sqrt{g_{\\phi }}%\n\\;a^{(\\phi)}_o; \\label{dEdt} \\\\\n\\frac{1}{m}\\frac{dL}{d\\tau }& =\\sqrt{g_{\\phi }}\\;a^{(\\phi)}_o. \\label{dLdt}\n\\end{align}%\n\nIt is clear, that if the proper time of crossing the horizon is finite, as\nis the case for the usual particles, then the finiteness of $Na^{(t)}_o$ and \n$a^{(\\phi)}_o$ implies that $E$ and $L$ are also bounded. However, this does\nnot seem to be necessarily so for critical particles, for which the proper\ntime of reaching the horizon diverges.\n\n\\subsection{Dynamic restrictions on a particle's velocity}\n\nLet us enumerate and classify all the possible variants of particle's type\nof asymptotic motion in the vicinity of the horizon, now in more detail than\nin the section on kinematics, so as to focus below only on those that are\nnot explicitly non-physical. \n\nFirst of all, diverging $L$, as seen from (\\ref{dLdt}), would correspond to\ncontinuous acceleration in the $\\phi$ direction, which would cost formally\ninfinitely large amounts of fuel per a unit mass particle. If one has the\nresources to make such experiments, he would not need the BSW effect in\norder to observe (formally) infinite energy in the center of mass frame. So\nthis variant is of not much interest.\n\nSecondly, one could imagine divergent $u^z$. Such a particle would have\nvelocity tending to $c$ and directed along the $z$ axis (or at finite angle\nwith respect to it) both in the OZAMO and FZAMO frames. This would mean that the\nparticle is ``accelerated'' (in the sense that its velocity increases) not\nonly in radial direction, but also along the horizon surface. This would be\nvery strange behaviour, and in the Kerr metric such particles are naturally\nabsent \\cite{ne}. We will not consider this variant here.\n\nGiven these two natural assumptions, from the normalizing condition \n\\begin{equation}\nZ^{2}=X^{2}-N^{2}\\tilde{\\beta}^{2},\n\\end{equation}%\nwhere $\\tilde{\\beta}$ in the horizon limit tends to a positive real number,\nfinite and separated from zero. Consequently, for a particle reaching the\nhorizon, where $Z^{2}$ must remain positive, and at the same time finite $X$ and $E$ (see discussion after Eq. (\\ref{xi})), we have\n\\begin{equation}\nX\\sim \\xi ^{p},\\quad \\text{with}\\quad p\\in[0,1],\n\\end{equation}%\nwhich corresponds to usual, sub-critical and critical particles as discussed\nabove.\n\n\\subsection{Usual particles}\n\nFor a usual particle $X_{H}\\neq 0$ by definition. As discussed above, the\ntetrad components of its acceleration in the FZAMO frame $a^{(i)}_f$ must be\nfinite. Then the components in the OZAMO frame $a^{(i)}_o$, related to them\nvia the singular Lorentz transform (\\ref{va}), with $\\gamma_f\n=1\/N$, can diverge as $1\/N$. Writing out explicitly the asymptotics for the $%\nt$ and $r$ components in both frames, we get \n\\begin{align}\na^{(t)}_f &= (a^{(t)}_f)_{0}+(a^{(t)}_f)_1 N+O(N^2); \\label{af-t} \\\\\na^{(r)}_f &= (a^{(r)}_f)_{0}+(a^{(r)}_f)_1 N+O(N^2); \\label{af-r} \\\\\na^{(t)}_o &=+\\frac{ (a^{(t)}_f)_{0}- (a^{(r)}_f)_{0}}{N} +\\big[ %\n(a^{(t)}_f)_{1}-(a^{(r)}_f)_{1}\\big]+O(N); \\label{ao-t} \\\\\na^{(r)}_o &=-\\frac{ (a^{(t)}_f)_{0}- (a^{(r)}_f)_{0}}{N} -\\big[ %\n(a^{(t)}_f)_{1}-(a^{(r)}_f)_{1}\\big]+O(N), \\label{ao-r}\n\\end{align}\n\nThe $\\phi $ and $z$ components are the same in the two frames and must be\nbounded: \n\\begin{align}\na^{(\\phi)}_f &=a^{(\\phi)}_o =O(1); \\\\\na^{(z)}_f &=a^{(z)}_o =O(1).\n\\end{align}\nThen, we see that if $a^{(i)}_f=O(1)$, the right hand side of (\\ref{dEdt}) is\nfinite. The left hand side is also finite, as for a usual particle, given $%\nu^{r}\\sim 1$ (\\ref{ru}), $dr\\sim d\\tau \\sim dN$.\n\nThe explicit expressions for $a_{o}^{(r)}$ and $a_o^{(z)}$ are \n\\begin{align}\na_o^{(r)}=& \\frac{1}{\\sqrt{A}}\\Big\\{(u^{r}\\partial _{r}+u^{z}\\partial\n_{z})u^{r} -\\frac{A^{\\prime }}{2A}(u^{r})^{2} -\\frac{A}{2}\\partial\n_{r}g_{z}(u^{z})^{2}+ \\notag \\\\\n& -\\frac{A}{2}\\Big[X^2 \\partial_r N^{-2}-L^2 \\partial_r g_\\phi^{-1}\n\t-2\\frac{XL}{N^2}\\partial_r \\omega\\Big]\\Big\\}; \\label{aR}\n\\\\\na_o^{z}=& \\sqrt{g_z}\\Big\\{ (u^{r}\\partial _{r}+u^{z}\\partial _{z})u^{z} +%\n\\frac{\\partial _{z}g_{z}}{2g_{z}}(u^{z})^{2} \n\t+\\frac{\\partial _{r}g_{z}}{g_{z}}u^{r}u^{z}+ \\notag \\\\\n& -\\frac{1}{2 g_z}\\Big[X^2 \\partial_z N^{-2}-L^2 \\partial_z g_\\phi^{-1}\n\t-2\\frac{XL}{N^2}\\partial_z \\omega\\Big]\\Big\\}.\n\\label{aZ}\n\\end{align}\n\nThe conditions $a^{(i)}_f=O(1)$ can be reformulated in the form of\nrestrictions on the coefficients $\\alpha_k$ and $\\beta_k$ in the expansions \n\\begin{align}\nu^{r}&=\\alpha _{0}(z)+\\alpha _{1}(z)(r-r_{H})+O((r-r_{H})^{2}), \\label{u} \\\\\nu^{z}&=\\beta _{0}(z)+\\beta _{1}(z)(r-r_{H})+O((r-r_{H})^{2}).\n\\end{align}\n\n\\subsection{Critical particles}\n\nSuch particles approach the horizon but, in contrast to usual ones, the\nprocess takes infinite proper time. On the other hand, as seen from (\\ref{xc}) and (\\ref{gx}), their Lorentz factor in the OZAMO frame $\\gamma $ is\nfinite, and the velocity is $v<1$, so the tetrad components of acceleration\nin the OZAMO frame must be finite. As mentioned above, we consider only\nmotion with $E$ and $L$ bounded in the horizon limit $\\tau \\rightarrow\n\\infty $. This means that $a_{o}^{(\\phi )}$ should be not only bounded, but\nintegrable (\\ref{dLdt}): $\\int d\\tau a_{o}^{(\\phi )}<\\infty $. If we assume\nthat $a_{o}^{(\\phi )}$ is expandable in power series by $r$ with integer powers, this means \n\\begin{equation}\na_{o}^{(\\phi )}=O(N).\n\\end{equation}%\nWith this condition satisfied, and Eq. (\\ref{ao-t}) taken into account, the boundedness of $E$ from (\\ref{dEdt}) does\nnot give any more restrictions on $a_{o}^{(t)}$. Then, using that $E$, $L$,\nand $u^{z}$ are bounded (\\ref{el}), while $u^{r}$ and $X$ are $O(N)$, and $\\omega\n_{H}=const$ (which follows from regularity \\cite{reg}), it is easy to see\nthat all the terms in (\\ref{aR}) and (\\ref{aZ}) are automatically finite, so \n\\begin{equation}\na_{o}^{(r)},a_{o}^{(z)}=O(1).\n\\end{equation}\nThus all components of acceleration of a critical particle in the OZAMO\nframe, and therefore in the instantly comoving proper frame, are finite\nunconditionally. This is in contrast to usual particles, for which the\nconditions $a_{f}^{(i)}=O(1)$ impose some additional constraints on $\\alpha\n_{k}$ and $\\beta _{k}$ in (\\ref{u}).\n\nIn the FZAMO frame, and the frame of any usual particle, the picture looks\ndifferent, as the relative Lorentz factor of a usual and critical particle\ndiverges as $1\/N$. Using the Lorentz transformation (\\ref{rrt})\nbetween the OZAMO and FZAMO frames, with $\\gamma_f \\sim 1\/N$, we see that $%\na^{(t)}_f$ and $a^{(r)}_f$ can diverge as $1\/N$. Using (\\ref{rrt}), the asymptotics of these components of acceleration in the two\nframes can be brought to the form \n\\begin{align}\na^{(t)}_o &= (a^{(t)}_o)_{0}+(a^{(t)}_o)_1 N+O(N^2); \\\\\na^{(r)}_o &= (a^{(r)}_o)_{0}+(a^{(r)}_o)_1 N+O(N^2); \\\\\na^{(t)}_f &=\\frac{ (a^{(t)}_o)_{0}+ (a^{(r)}_o)_{0}}{N} +\\big[ %\n(a^{(t)}_o)_{1}+(a^{(r)}_o)_{1}\\big]+O(N); \\\\\na^{(r)}_f &=\\frac{ (a^{(t)}_o)_{0}+ (a^{(r)}_o)_{0}}{N} +\\big[ %\n(a^{(t)}_o)_{1}+(a^{(r)}_o)_{1}\\big]+O(N).\n\\end{align}\nThe $\\phi$ and $z$ components in the two frames are the same and therefore,\nas shown above, satisfy \n\\begin{align}\na^{(\\phi)}_o &=a^{(\\phi)}_f =O(N); \\\\\na^{(z)}_o &=a^{(z)}_f =O(1).\n\\end{align}\n\nThus we have two mutually complimentary cases. In the OZAMO frame $r$ and $t$\ncomponents of acceleration diverge for usual particles and stay finite for\nthe critical ones. In the FZAMO frame, the situation is opposite: $r$ and $t$\ncomponents of acceleration are finite for usual particles and diverge for\nthe critical ones. The $\\phi$ and $z$ components are the same in the two\nframes and are finite. For critical particles, additionally $%\na_{o}^{(\\phi)}=O(N)$ near the horizon for energy and angular momentum to\nremain bounded.\n\n\\section{Example: the Reissner-Nordstr\\\"{o}m metric}\n\nThe approach and results of the present paper are also valid in the case of\nthe electromagnetic interaction with minimal changes: in eq. (\\ref{x}) one\nshould make the replacement $X\\rightarrow X-q\\varphi $, where $\\varphi $ is\nthe electrostatic potential, and $q$ is the particle's charge. In order to\ndemonstrate this, it is instructive to consider as an example the extremal\nReissner-Nordstr\\\"{o}m metric. In this case the metric functions are \n\\begin{equation}\nN=\\sqrt{A}=1-\\frac{r}{r_{H}},\\qquad \\omega =0,\\qquad g_{\\phi }=r^{2},\n\\end{equation}%\nand the electromagnetic field potential is \n\\begin{equation}\nA_{\\mu }=-\\varphi \\delta _{\\mu }^{t},\\qquad \\varphi =\\frac{Q}{r},\n\\end{equation}%\nwhere \n\\begin{equation*}\nQ=r_{H}\n\\end{equation*}%\nis the extremal black hole's charge, so that the only nonvanishing\ncomponents of the electromagnetic field tensor are \n\\begin{equation*}\nF_{rt}=-F_{tr}=\\frac{Q}{r^{2}}.\n\\end{equation*}\n\nFor a particle of charge $q$ moving radially towards the horizon the\nfour-momentum can be parametrized as \n\\begin{equation}\np_\\mu=m u_\\mu=-(X,0,Z\/N^2 ,0) ,\n\\end{equation}\nthen the normalization condition implies \n\\begin{equation} \\label{zz}\nZ=\\sqrt{X^2 -m^2 N^2}.\n\\end{equation}\nThe equation of motion \n\\begin{equation}\nma^{\\mu }=qF^{\\mu \\nu }u_{\\nu },\n\\end{equation}\nhas the integral of motion \n\\begin{equation} \\label{RN-energy}\nE=X+q\\varphi =const.\n\\end{equation}\n\nFor usual particles, with $X_H \\neq 0$, \n\\begin{equation}\np^{r}=-X_{H}-\\frac{q}{r_H}(r-r_{H})+O((r-r_{H})^{2}),\n\\end{equation}%\nin agreement with (\\ref{u}).\n\nFor a critical particle\n\\begin{equation}\nE=q,\\quad X=qN,\\quad Z=N\\sqrt{q^{2}-m^{2}}.\\label{RNcrit}\n\\end{equation}\nThen, integrating the equation for radial motion \n\\begin{equation*}\n\\frac{dr}{d\\tau}=-\\frac{Z}{m} =-N(r)\\sqrt{q^2 \/m^2 -1},\n\\end{equation*}\nit is easy to obtain that in the horizon limit the same asymptotic as in eq.\n(\\ref{ur}) holds, with the characteristic time \n\\begin{equation}\n\\tau _{0}=r_H \\Big(\\frac{q^2}{m^2}-1\\Big)^{-1\/2}.\n\\end{equation}\n\nNow, we will consider the acceleration measured by the two types of\nobservers.\n\n\\subsection{Static observers}\n\nThe tetrad (\\ref{zamo})--(\\ref{zamo-z}) in this case turns into the tetrad\nof a static observer. Then, using (\\ref{zamo}) and (\\ref{a}), we obtain \n\\begin{align}\nm a^{(t)}_o&=-\\frac{qQ}{r^{2}}\\,\\frac{Z}{mN}, \\\\\nm a^{(r)}_o&=+\\frac{qQ}{r^{2}}\\,\\frac{X}{mN}, \\\\\n&m^2 a^{2}=\\Big(\\frac{qQ}{r^2}\\Big)^2 .\n\\end{align}\n\nFor the critical particle (\\ref{RNcrit}) both components of acceleration \n\\begin{align}\nm a^{(t)}_o&=-\\frac{qQ}{r^{2}}\\sqrt{q^{2}\/m^{2}-1}, \\\\\nm a^{(r)}_o&=\\frac{qQ}{r^{2}}\\frac{E}{m}\n\\end{align}\nare finite on the horizon, and can be expanded into a series by $(r-r_{H})$\nor $N$.\n\nHowever, for a usual particle, with $X_{H}\\neq 0$, near the horizon \n\\begin{equation*}\nX=X_{H}+O(N),\\quad Z=X+O(N^2),\n\\end{equation*}\nso \n\\begin{align}\n&a^{(r)}_o \\approx -a^{(t)}_o= \\frac{a_{-1}}{N}+O(1),\n\\end{align}\nwhere \n\\begin{equation}\na_{-1}=\\frac{q}{Q}\\frac{X_{H}}{m^{2}}.\n\\end{equation}\nThus both components diverge near the horizon, in accordance with (\\ref{ao-r}%\n,\\ref{ao-t}), while satisfying \n\\begin{equation}\na^{(r)}_o+a^{(t)}_o=O(N). \\label{rt-sum}\n\\end{equation}\n\n\\subsection{Falling observers}\n\nThe falling frame $e^{(i)}_f$ is attached to a particle falling into the\nblack hole according to (\\ref{rrt})--(\\ref{va}), with the Lorentz factor $%\n\\gamma_{f}=1\/N$ and velocity $v_f =1-O(N^2)$ in the static frame.\n\nIn this frame the tetrad components of acceleration are equal to \n\\begin{align}\nm a^{(t)}_f&=-\\gamma_F \\;\\frac{qQ}{r^{2}}\\,\\frac{Z-v_F X}{mN}; \\\\\nm a^{(r)}_f&=+\\gamma_F \\;\\frac{qQ}{r^{2}}\\,\\frac{X-v_F Z}{mN}.\n\\end{align}\n\nFor usual particles, in the horizon limit $N\\rightarrow 0$, $X_{H}\\neq 0$.\nThen (\\ref{va}) and (\\ref{zz}) imply that $Z-X=O(N^{2})$, so $a^{(t)}_f$ and $a^{(r)}_f$\nare finite.\n\nIf the particle under consideration is critical, then $Z\\sim X\\sim N$, and\nboth components of acceleration diverge: \n\\begin{align}\na^{(t)}_f=&\\frac{\\tilde{a}_{-1}}{N}+O(N), \\\\\na^{(r)}_f=&\\frac{\\tilde{a}_{-1}}{N}+O(N), \\\\\n&\\tilde{a}_{-1} =r_H^{-1}\\,\\frac{q}{m} \\Big[\\frac{q}{m}-\\sqrt{%\n\\frac{q^2}{m^2}-1}\\Big].\n\\end{align}\nThus we see that, indeed, all the general properties (\\ref{af-t}), (\\ref{af-r}), (\\ref{ao-t}), (\\ref{ao-r}), described in the preceding section, are\nexplicitly verified in this exactly solvable case.\n\n\\section{BSW effect under finite forces: equatorial motion}\n\n\\subsection{Motion in equatorial plane}\n\nLet $m=1$, and let us consider motion in the equatorial plane so that $%\nu^{z}=0$ and $a_{o}^{(z)}=0$. Then for arbitrary motion we have 1) the\nnormalization condition for velocity \n\\begin{equation}\nu^{r}=-\\frac{\\sqrt{A}}{N}Z,\\qquad Z=\\sqrt{X^{2}-N^{2}\\Big[\\frac{L^{2}}{%\ng_{\\phi }}+1\\Big]}, \\label{zeq}\n\\end{equation}%\nand 2) orthogonality condition for acceleration, which can be written in terms of (\\ref{a}--\\ref{a-z}) as \n\\begin{align}\n0=u_{\\mu }a^{\\mu }& =+u_{t}a^{t}+u_{\\phi }a^{\\phi }+u_{r}a^{r} \\\\\n& =-Ea^{t}+La^{\\phi }+A^{-1}u^{r}a^{r} \\\\\n& =-\\frac{X}{N}a_{o}^{(t)}+\\frac{L}{\\sqrt{g_{\\phi }}}a_{o}^{(\\phi )}\n\t+\\frac{u^{r}}{\\sqrt{A}}a_{o}^{(r)}. \\label{Orth3a}\n\\end{align}%\nGenerically, at least two of the three components of $a^{(i)}$ have to be\nnon-zero if there is acceleration. Also for simplicity we will assume\\footnote{This assumption is purely technical. In general, one should write $A=N^2 B$, where $B$ is some bounded function which does not vanish at the horizon. Its form does not affect the results qualitatively but leads to \nmore cumbersome expressions. Thus we put for simplicity $B=1$, which also\nfixes the time scale.} that $A=N^{2}$, so that $u^{r}=-Z$ and orthogonality condition takes form\n\\begin{equation}\n\\frac{X}{N}a_{o}^{(t)}-\\frac{L}{\\sqrt{g_{\\phi }}}a_{o}^{(\\phi )}\n\t+\\frac{Z}{N}a_{o}^{(r)}=0.\\label{Orth3}\n\\end{equation}\n\nOf the four components of the equation of motion \n\\begin{equation}\n(u^\\mu \\nabla_\\mu) u^\\nu=a^\\nu\n\\end{equation}\none is trivial\\footnote{In the equatorial plane derivatives of metric functions by $z$ in (\\ref{aZ}) must vanish due to symmetry.} $a^{z}=0$, and the other three are related through the\northogonality condition, so it is always sufficient to consider only two\ncomponents, for example (\\ref{dEdt}) and (\\ref{dLdt}), which can be written\nin terms of $X$ and $L$ as \n\\begin{align}\n\\frac{dX}{d\\tau}&=N a_o^{(t)}-L\\frac{d\\omega}{d\\tau}; \\label{Xeq} \\\\\n\\frac{dL}{d\\tau}&=\\sqrt{g_\\phi}\\; a_o^{(\\phi)} . \\label{Leq}\n\\end{align}\nAs $dr\/d\\tau =u^r =-Z$, in terms of $X$ and derivatives by $\\xi\\equiv (r-r_H\n)$, which are denoted by primes, this can be written as \n\\begin{align}\n&X^{\\prime}+L\\omega^{\\prime}=-\\frac{N}{Z} a_o^{(t)}; \\label{Xeq2} \\\\\n&L^{\\prime}=-\\frac{\\sqrt{g_\\phi}}{Z}\\; a_o^{(\\phi)} . \\label{Leq2}\n\\end{align}\nIt can be checked that, indeed, in case $u^z =0$, equations (\\ref{Xeq}) and (%\n\\ref{Leq}) together with (\\ref{Orth3}) give (\\ref{aR}).\n\n\n\n\\subsection{Acceleration in proper frame}\nExpressing acceleration components in the OZAMO frame through the particle's parameters $E$ and $L$ from (\\ref{Xeq2},\\ref{Leq2}) and the orthogonality condition (\\ref{Orth3}), one obtains\n\\begin{align}\na_o^{(\\phi)}&=-\\frac{Z}{\\sqrt{g_\\phi }}L' ;\\label{EQao-phi}\\\\\na_o^{(t)}&=-\\frac{Z}{N}(X'+L\\omega');\\label{EQao-t}\\\\\na_o^{(r)}&=-\\frac{X}{Z}a_o^{(t)}-N\\frac{LL'}{g_\\phi}.\\label{EQao-r}\n\\end{align}\nFor critical particles the OZAMO frame is the proper frame. For other types of particles the $r$ and $t$ components of acceleration in the proper frame are given by (\\ref{AProper}) with Lorentz factor (\\ref{gx}) \n\\begin{equation}\n\\gamma =\\frac{X}{N}, \\label{gx2}\n\\end{equation}\nwhile $a_{pr}^{(\\phi)}=a_{o}^{(\\phi)}$ for any type. Using (\\ref{EQao-phi}--\\ref{EQao-r}), this gives\n\\begin{equation}\n\\begin{pmatrix}\na_{pr}^{(t)}\\\\a_{pr}^{(r)}\n\\end{pmatrix}=\n\\frac{X}{N}\\left\\{\n\t\\frac{a_o^{(t)}}{Z}\n\t\\begin{pmatrix}\n\t\tZ-vX \\\\ Zv-X\n\t\\end{pmatrix}\n\t-N\\begin{pmatrix}\n\t\tv\\\\1\n\t\\end{pmatrix}\n\t\t\\frac{LL'}{g_\\phi}\\right\\}. \\label{ACCproper}\n\\end{equation}\n\nSuppose we have a particle with\n\\begin{align}\n\tX&=\\alpha \\xi^p (1+O(\\xi)),\\qquad \t\\gamma=\\frac{X}{\\xi},\\\\\n\tv&=\\sqrt{1-\\gamma^{-2}}=1-\\frac{1}{2\\alpha}\\xi^{2(1-p)}(1+O(\\xi)),\n\\end{align}\nwhere $p<1$. Using (\\ref{Z-X}), we get\n\\begin{align}\nZ-Xv&=(\\tfrac12 -C)\\xi^{2-p}(1+O(\\xi)), \\label{Z-Xv}\\\\\nX-Zv&=(\\tfrac12 +C)\\xi^{2-p}(1+O(\\xi)).\\label{X-Zv}\n\\end{align}\n\n\\subsubsection{Usual particles}\nFor a usual particle $p=0$, so assuming $L'$ is bounded, $a_o^{(\\phi,r,t)}=O(1)$, while $\\gamma \\sim 1\/\\xi$, and one can easily check term by term that \nacceleration in the proper frame (\\ref{ACCproper}) is always bounded: as expected, \\textbf{for usual particles there are no additional requirements}.\n\n\\subsubsection{Sub-critical particles}\nFor a sub-critical particle $p\\in(0,1)$. As $X\\sim Z$, the derivative $X^{\\prime}\\sim \\xi^{p-1}$ in (\\ref{EQao-t}) diverges, while $L\\omega^{\\prime}=O(1)$, so \n\\begin{equation}\n\\frac{a_o^{(t)}}{Z}\\sim \\xi^{p-2}.\n\\end{equation}\nThen taking into account (\\ref{Z-Xv}--\\ref{X-Zv}), the first term in the braces of (\\ref{ACCproper}) is $O(1)$, and different in the two rows, thus separated from zero. \n\nThe second term could only compensate the first one (in one of the two rows), if $L' \\sim \\xi^{-1}$, which would imply divergent $L\\sim \\ln \\xi$. Therefore the quantity in the braces is finite and separated from zero, so the proper acceleration diverges as (\\ref{gx2})\n\\begin{equation}\n\\gamma =\\frac{X}{N}\\sim \\xi^{p-1}\\to \\infty .\n\\end{equation}\nThis means that there are \\textbf{no sub-critical particles} with finite acceleration for motion in the equatorial plane.\n\n\\subsubsection{Critical particles}\nThe only remaining case to be considered is critical particles. Although (\\ref{ACCproper}) for them is unnecessary, one restores the acceleration in the OZAMO frame from it by setting $\\gamma=1$ and $v=0$. We see that\\footnote{Remember that components $a_o^{(i)}$ are related through the orthogonality condition (\\ref{Orth3}); if two of them are finite, then the third is bounded as well.}\n\\begin{align}\n\ta_o^{(\\phi)}&\\sim \\xi L' ;\\\\\n\ta_o^{(r,t)}&\\sim (X'+L\\omega '),\n\\end{align}\nso in order for such trajectory to be realized we need the azimuthal force to tend to zero fast enough:\n\\begin{equation}\na_o^{(\\phi)}=O(\\xi).\n\\end{equation}\nThere is no restriction on the radial component: it can be of the order of unity, as it will still be possible to fine-tune a critical particle by the appropriate choice of initial condition (this will be shown in more detail in the next Section). Thus the radial component does not affect or hinder the existence of critical trajectories and consequently the BSW effect. This is in agreement with the already established fact that the radial force itself is the reason for the BSW effect near charged nonrotating black holes \\cite{jl}.\n\n\\subsection{Example: azimuthal dissipative force}\nLet us consider the particular case when the radial force, which does not hinder critical particles anyway, is absent:\n\\begin{equation}\na_{o}^{(r)}= 0,\\qquad a_{o}^{(t)}, a_{o}^{(\\phi)}\\neq 0 .\n\\end{equation}\nUsing orthogonality (\\ref{Orth3}), \n\\begin{equation}\na_o^{(t)}=\\frac{N}{X}\\frac{L}{\\sqrt{g_\\phi}} a_o^{(\\phi)} , \\label{OrthAD}\n\\end{equation}\nso in terms of derivatives with respect to $\\xi$ Eqs. (\\ref{Xeq2}) and (\\ref{Leq2}) can be rewritten as \n\\begin{align}\n&g_\\phi X(X^{\\prime}+L\\omega ^{\\prime})=N^2 LL^{\\prime}; \\label{AD-eq1} \\\\\n&a_o^{(t)}=-N \\frac{Z}{X}\\frac{LL^{\\prime}}{g_\\phi} ; \\label{AD-eq2}\\\\\n&a_o^{(\\phi)}=-Z\\frac{L^{\\prime}}{\\sqrt{g_\\phi}}.\\label{AD-eq3}\n\\end{align}\n\n\\subsubsection{Tuning a critical particle}\nIn this section we show in more detail how one would tune the particle to be\ncritical $X\\sim \\xi$.\n\nAssuming expansions \n\\begin{align}\nN^2 &=\\nu_2 \\xi^2 +\\nu_3 \\xi^3 +\\ldots , \\\\\n\\omega&=\\omega_H -\\omega_1 \\xi +\\omega_2 \\xi^2 +\\ldots , \\\\\ng_\\phi &=g_H +g_1 \\xi +g_2 \\xi^2 +\\ldots , \\\\\nX&=x_1 \\xi +x_2 \\xi^2 +\\ldots , \\\\\nL&=l_H + l_1 \\xi +l_2 \\xi^2 +\\ldots ,\n\\end{align}\nfrom (\\ref{AD-eq1}) we obtain in consecutive orders \n\\begin{align}\nl_H &=\\frac{x_1}{\\omega_1}; \\\\\nl_{1}&=2\\frac{x_1 \\omega_2 +x_2 \\omega_1}{\\omega_1^2 +\\nu_2 \/g_H}; \\\\\nl_{2}&=l_{2} (x_1, x_2, x_3),\\quad \\ldots\n\\end{align}\n\nThen \n\\begin{equation}\n\\frac{Z^{2}}{\\xi^2}\n\t\\approx x_1^2 \\Big[1-\\frac{\\nu_2}{g_H}\\omega_1^{-2}\\Big]-\\nu_2 .\n\\end{equation}\nThere is a critical particle for \n\\begin{equation}\n|x_1| >x_{1\\min},\n\\end{equation}\nand there is a solution \n\\begin{equation}\nx_{1\\min}=\\frac{\\nu_2}{1-\\frac{\\nu_2}{g_H}\\omega_1^{-2}}\n\\end{equation}\nas long as \n\\begin{equation}\n\\omega_1^2 > \\frac{\\nu_2}{g_H}. \\label{OmegaCond}\n\\end{equation}\n\nFrom (\\ref{AD-eq2}) we get \n\\begin{equation}\na_o^{(t)}=-\\frac{Z}{N}(X^{\\prime}+L\\omega ^{\\prime})\n\\end{equation}\nand after substitution of expansions for $X,L$ and $\\omega$, \n\\begin{equation}\na_o^{(t)}\\approx -\\frac{2Z}{\\omega_1 \\sqrt{\\nu_2}}\\frac{x_1 \\omega_2 +x_2\n\\omega_1}{1+g_H \\omega_1^2 \/\\nu_2}\\sim Z \\sim \\xi .\n\\end{equation}\nFurther terms are obtained straightforwardly but they are quite cumbersome.\n\nSo, if acceleration is expanded in a series by $\\xi$ \n\\begin{equation}\na_o^{(t)}=a_1 \\xi +a_2 \\xi^2 +\\ldots,\n\\end{equation}\nin the first order we obtain $a_{1}(x_1 ,x_2)$. As long as $a =O(\\xi)$, and the\nmetric coefficients satisfy (\\ref{OmegaCond}), we can take arbitrary $x_{1}$\nsuch that $|x_1|>x_{1\\min}$ (or equivalently $l_{H}=x_{1}\/\\omega_{1}$). Then\nfor the given $a_1$ in the first order we obtain $x_{2}(a_1)$, in the next\norder $x_3 (a_1, a_2)$ and so on. The set of critical trajectories\\footnote{For large enough $|x_1|$ the turning point, given by $Z=0$, will be at finite values of $(r-r_H )$ from the horizon. Thus it will be at the coordinate distance that does not have to be small in order to gain arbitrarily large $E_{c.m.}$ at the collision event near the horizon. This is in contrast to the case discussed in \\cite{BSW2}, which is realized near the turning point of a usual particle with small $X_H$, and the turning point itself must be close to the horizon: the nearer it is, the larger $E_{c.m.}$ is achieved.} is parametrized by one free parameter $x_1$ (or $l_H$).\n\n\\subsubsection{Other realizations of critical trajectories}\nSuppose now the azimuthal force tends to zero as $\\xi^s$ with some integer $s>1$. Then from (\\ref{AD-eq3}) we see that $L' \\sim \\xi^{s-1}$, and therefore expansion (\\ref{el}) for $L$ near the horizon takes the form\n\\begin{equation}\nL=L_H +L_{s}\\xi^{s}(1+o(1)).\\label{LS}\n\\end{equation}\nIt is perfectly consistent with the particle being critical, so that $X\\sim \\xi$: (\\ref{AD-eq1}) can be satisfied for any integer $s>1$ and solved for $X(L)$ (or, equivalently, $E(L)$) in each consecutive order by $\\xi$. Let us consider, for example, the case $s=2$. Assuming\n\\begin{equation}\nX=x_1 \\xi +x_2 \\xi^2 +x_3 \\xi^3 +O(\\xi^4),\n\\end{equation}\nfrom (\\ref{AD-eq1}) in consecutive orders one obtains\n\\begin{align}\nx_{1}&=L_{H}\\omega_{1};\\\\\nx_{2}&=-2\\omega_{2}L_{H};\\\\\nx_{3}&=-\\omega_3 L_H +\\frac{\\omega_1}{3}L_2 +\\frac{2}{3g_{\\phi H}}L_H L_2 ;\\\\\n\\ldots \\notag\n\\end{align}\nThis can be turned around to give $L_H (x_1)$ and $L_2 (x_3)$, but $x_2 \/ x_1$ is fixed to metric function coefficients; in terms of $E(L)$ this is\n\\begin{equation}\nE=\\omega_H L_H +(L_2 \\omega_H -L_H \\omega_2)\\xi^2 +\\ldots .\n\\end{equation}\nFor other integer $s$ the procedure is analogous. \n\n\n\\section{Energy bounds in collisions with near-critical particles}\nWe have seen in the previous section that as long as the azimuthal force is weak enough, critical particles exist and can be tuned via initial conditions. Then the BSW effect in its primary version \\cite{ban} manifests itself. However, what if this is not the case and azimuthal force is e.g. separated from zero on the horizon? The condition for critical particles is that\n\\begin{equation}\na^{(\\phi)} =O(x) \\label{Need},\n\\end{equation}\nwhere\n\\begin{equation}\nx=\\frac{\\xi}{r_H} \\label{Xdimensionless}\n\\end{equation}\nis the dimensionless radial coordinate. Suppose that instead\n\\begin{equation}\na^{(\\phi)}(x)\\approx \\frac{a_{0}x ^{\\lambda }}{r_H},\\qquad \\lambda <1 , \\label{Asmall}\n\\end{equation}\nso that (\\ref{Need}) is violated, and we factored out the dimensional quantity $r_H^{-1}$, so that $a_0$ is dimensionless.\n\nDoes it mean that the BSW effect necessarily breaks down? Under no\nadditional assumptions -- yes. However, what if $a_0$ is small? Radiation\nreaction forces are usually considered very small (see e.g. \\cite{ne}, \\cite{Kesden}). In that case, the question is how high $E_{c.m.}$ can be\nachieved for the given small $a_{0}$ ? \n\nLet us reformulate the condition that \\emph{is} satisfied (\\ref{Asmall}) via\nanother small parameter: \n\\begin{equation}\na^{(\\phi)} (x)\\approx r_H^{-1} \\; x^{\\lambda}x_{m}^{1-\\lambda},\n\\end{equation}\nwhere \n\\begin{equation}\nx_{m}=a_0^{\\frac{1}{1-\\lambda}}\\ll 1 .\n\\end{equation}\nThen \n\\begin{equation}\na^{(\\phi)}(x_m) \\approx r_H^{-1} x_m\n\\end{equation}\nand for all $x\\gtrsim x_m$ the necessary condition for acceleration (\\ref{Need}) is\neffectively obeyed.\n\nBut then for $\\xi\\gtrsim r_H x_m$ the trajectory of a particle can be\neffectively tuned to be critical, (or sub-critical, for the chosen $p$), at\nwill. At the near-horizon end of this region, i.e. at $\\xi_{m}\\sim r_H x_m$,\nthe Lorentz factor with a usual particle with $p_2 =0$ will behave as\ndescribed (\\ref{p1-p2}) and can grow very large. Thus, for $p=1$ (the particle is tuned to be critical) from (\\ref{gamma12-1}) we get \n\\begin{equation}\n\\gamma_{12}^{(max)} \\approx \n\t\\gamma_{12}(x_m) \\approx \n\t\t\\frac{C_1 \\alpha_2}{r_H} \\cdot a_0^{-\\frac{1}{1-\\lambda}}.\n\\end{equation}\n\nWe see that as long as the amplitude $a_0$ of the azimuthal force acting on the particle is small enough, the BSW effect survives almost any\nkind of perturbation: one only has to calculate accurately the corresponding tuning parameters for the effectively critical trajectories.\n\n\\section{Kinematic restrictions on critical particles and two types of the\nBSW effect}\n\nIn the preceding Section, it was assumed that collision occurred not exactly\non the horizon but at some coordinate distance from it, its scale being tied\nto the amplitude of azimuthal acceleration, which is supposed to be small. The force, being too large, prevents\nthe critical particle from approaching the horizon. In this sense, the\nreason of it is dynamic. Meanwhile, even if the external force is small enough or absent at all, pure\nkinematic factors can also create an obstacle for reaching the horizon.\n\nLet us remind the situation with geodesic particles. If the horizon is\nnonextremal, the critical particle cannot reach the horizon at all.\nNonetheless, it was demonstrated in \\cite{gp} for the Kerr metric and in \n\\cite{prd} for generic dirty axially symmetric black holes, that $E_{c.m.}$\ncan be made as large as one likes provided (i) the critical particle is\nreplaced with a slightly noncritical one, (ii) the coordinate distance\nbetween the point of collision and horizon is adjusted to the small\ndeviation of the particle's parameters from the values corresponding to the\ncritical case.\n\nNow, we are dealing with an extremal horizon but there is a special situation \nwhen $X\\sim \\xi^p$ with $p>1$ (``supercritical'' particle). Such a particle cannot reach the horizon (in this sense it is similar to the case of the nonextremal horizon).\n\nThen, instead of taking a critical particle, we can choose a usual one with sufficiently\nsmall $X_{H}$. More precisely, let us consider expansion for $X$ of the form \n\\begin{equation}\nX=X_{H}+\\alpha_{s}\\xi ^{s}(1+x_{1}\\xi +\\ldots ),\\qquad p>1. \\label{xpp}\n\\end{equation}\n\nThen, we look for the region in which both terms in $Z$ (\\ref{z}) are of the\nsame order of magnitude. This is achieved at $\\xi\\approx \\xi_c \\sim r_H X_H$. Then we can neglect the correction in (\\ref{xpp}), so that \n\\begin{equation}\nZ(\\xi_c)\\sim X_{H}\\sim \\xi_c \\sim N(\\xi_c ),\n\\end{equation}%\nand therefore (\\ref{gacm}) implies \n\\begin{equation}\n\\gamma _{c.m.}^{(max)}\n\t\\approx \\gamma_{c.m.}(\\xi_c)\n\t\t\\sim N^{-1}(\\xi_c) \n\t\t\\sim \\xi_{c} ^{-1} . \\label{bsw}\n\\end{equation}\n\nThus one can distinguish between two main types of the BSW effect: BSW 1, in\nwhich the critical particle can approach the horizon, so that the\nhorizon limit can be taken, and BSW 2, for which the critical\nparticle does not reach the horizon. We see that, in general, the presence\nof the external force is compatible with both types of the BSW effect\\footnote{Another, more detailed classification of trajectories and corresponding\ntypes of the BSW effect can be found in Sec. IV of \\cite{ne} for the Kerr\nmetric and in \\cite{jh} for general dirty rotating axially symmetric black\nholes.}.\n\nIt is worth noting that even in the absence of force, the expansion for $X$\ncan take the form (\\ref{xpp}), if the linear terms cancel each other. Say,\nthis happens for solutions near the so-called ultraextremal horizon ($%\nN^{2}\\sim (r-r_{H})^{3}$) in special \"exotic\" metrics described in\nSec.\\,IV\\,B.\\,5 of \\cite{reg}, for which $\\partial_r \\omega \\big|_H\n=\\partial^2_r \\omega \\big|_H =0$, and thus $s$ can be equal to $2$ or $3$.\nCorrespondingly, the BSW-2 effect can be realized near such horizons.\n\n\\section{Conclusion}\n\nIn general, three main circumstances were considered as the factors which\nwere expected to restrict the indefinite growth of $E_{c.m.}$ and thus\ncreate obstacles to the manifestation of the BSW effect. These are (i)\nself-gravitation, (ii) deviation of a black hole from extremality \\cite{ted}%\n, and (iii) the force due to backreaction of gravitational or\nelectromagnetic radiation. As far as self-gravitation is concerned, it was\nshown in \\cite{shell} that for collisions of massive shells, either the BSW\neffect does not occur or it occurs but in the region inaccessible by a\nremote observer. However, in this case the shell does not approach the\nhorizon from the viewpoint of an external observer. As there is no horizon,\nthere is no BSW effect. Factor (ii) was analyzed in \\cite{gp} for the Kerr\nmetric where it was shown that for nonextremal black holes the BSW effect\ndoes exist (this conclusion was generalized in \\cite{prd} to generic dirty\naxially symmetric black holes). \n\nAnd, in the present work, we showed for extremal horizons that the BSW effect is compatible with a nonzero force under rather general assumptions: the radial force should be finite and the azimuthal force should tend to zero not too slowly. In terms of energy and angular momentum, the kinematic condition for the realization of the BSW effect is the same as for geodesic particles: $E=\\omega_H L$. In this sense, this condition by itself survives the action of the force (see also Sec. V of \\cite{insp} where, however, another physical situation was considered -- near-circular orbits around near-extremal black holes). Our approach is model-independent and is based on general properties of the horizons. \n\nFor the finite radial component of the force and the azimuthal one which tends to zero near the horizon as $r-r_H$, the BSW effect still exists. Otherwise, the effect is formally absent. The reason for the restriction on the azimuthal force seems to be clear: if azimuthal force was too large or did not tend to zero, in the infinite proper time that it takes for a critical particle to reach the horizon this force would accelerate the particle to infinite values of angular momentum. Obviously, one would not expect a force dissipative by nature, such as radiation reaction, to have such an effect. So, this only seeming restriction should be always obeyed. Even if it is not (so critical trajectories are absent), but the amplitude of the azimuthal force is small, the restrictions on $E_{c.m.}$ are shown to be inessential, and one can still attain very high energies. \n\nTo summarize, the BSW effect turned out to be more viable than one could expect.\n\nThe present work confirmed that the BSW effect relies on two main\nproperties: (i) the presence of the horizon and (ii) the existence of\nspecial types of trajectories. Thus it has geometric nature and reflects\ngeneral features of black holes irrespective of the details of the system.\nConcrete realization of the BSW effect certainly depends on particular\nproperties of a system but near the horizon these properties manifest\nthemselves in a universal way. We see that although dissipative forces in flat spacetime generically bound the values of energy peaks from above, in the strong gravitational field regime near the horizon the geometry dominates over the influence of dissipative forces on the system.\n\nThe present results refer to extremal horizons only. The nonextremal case\nand, especially, motion on circular orbits around near-extremal black holes,\nso important in astrophysical context, require separate treatment. Generalization to non-equatorial motion is also necessary. This will\nbe done elsewhere.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec1}\n\n\nIn \\cite{LR2}, the authors have formulated boundary conformal field\ntheory in real time (Lorentzian signature) in the algebraic framework\nof quantum field theory. BCFT is a local M\\\"obius covariant QFT $B_+$\non the two-dimensional Minkowski halfspace $M_+$ (given by $x>0$), which\ncontains a (given) local chiral subtheory $A$, e.g., the stress-energy\ntensor. The reward of this approach was the surprisingly simple\nformula ((\\ref{relcomm}) below), expressing the von Neumann algebras\nof local observables $B_+(O)$ in a double cone $O\\subset M_+$ in terms\nof an (in general nonlocal) chiral conformal net $B$ of localized\nalgebras associated with intervals along the boundary (the time axis\n$x=0$). The net $B$ is M\\\"obius covariant and contains the local\nchiral observables $A$: \n\\begin{eqnarray}\nA(I)\\subset B(I)\n\\end{eqnarray}\nfor each interval $I\\subset\\RR$. \n\nThe reduction to a single chiral net is responsible for a kinematical\nsimplification, explaining, e.g., Cardy's observation \\cite{C}\nthat in BCFT, bulk $n$-point correlation functions are linear\ncombinations of chiral $2n$-point conformal blocks. \n\nThe algebra $B_+(O)$ is a relative commutant of $B(K)$ within $B(L)$, \n\\begin{eqnarray}\\label{relcomm}\nB_+(O) = B(K)'\\cap B(L),\n\\end{eqnarray}\nwhere $K\\subset L$ are a pair of open intervals on the boundary $\\RR$\nsuch that the disconnected complement $L\\setminus \\overline K = I\\cup\nJ$ is the set of advanced and retarded times $t\\pm x$ associated\nwith points in $(t,x)\\in O$ (see Fig.\\ 1). Although the chiral net $B$\nis not necessarily local, the intersections (\\ref{relcomm}) do commute\nwith each other when two double cones are spacelike separated. \n\nThe main result in \\cite{LR2} is that every BCFT is contained in a\nmaximal (Haag dual) BCFT of the form (\\ref{relcomm}).\n\n\\bigskip\n\n\\hskip40mm \\epsfig{file=intervals.eps,width=50mm} \n\\nopagebreak\n\n{\\bf Figure 1:} Intervals on the boundary and double cones in the halfspace.\n\n\\bigskip\n\nThis leads to a somewhat paradoxical conclusion: on the one\nhand, each local bulk observable is defined as a (special) observable\nfrom a chiral CFT. Thus, superficially, the ``degrees of freedom'' of\na BCFT are not more than those of a chiral CFT, containing only a\nsingle chiral component of the stress-energy tensor (Virasoro\nalgebra). One might argue that such a ``reduction of degrees of\nfreedom'' is a characteristic feature of QFT with a boundary. But this\npoint of view cannot be maintained, because on the other hand, the\nresulting BCFT $B_+$ is locally equivalent to another CFT $B_{2D}$ on\nthe full two-dimensional (2D) Minkowski spacetime, which has all the\ndegrees of freedom of a 2D QFT, and in particular contains a full 2D\nstress-energy tensor (two commuting copies of the Virasoro algebra). \nEven in the simplest case, when the chiral net $B$ on the boundary\ncoincides with $A$ (sometimes known as ``the Cardy case''), the\nassociated bulk QFT contains apart from the full 2D stress-energy \ntensor more (``non-chiral'') local fields that factorize into chiral\nfields with braid group statistics. Locally, also the BCFT contains\nthe same fields. \n\\medskip\n\nThis paradoxical situation is not a contradiction; it rather shows\nthat ``counting degrees of freedom'' of a QFT is an elusive\ntask. Trivially, there is no obstruction against a proper inclusion of\nthe form ${\\mathcal B}({\\mathcal H})\\otimes{\\mathcal B}({\\mathcal H})\\subset {\\mathcal B}({\\mathcal H})$ if ${\\mathcal H}$ is an\ninfinite-dimensional Hilbert space. But ``counting degrees of\nfreedoms'', e.g.\\ by entropy arguments, requires the specification of\nthe Hamiltonian. The BCFT shares the Hamiltonian and ground state\n(vacuum) of the chiral CFT, while the associated 2D CFT has a\ndifferent Hamiltonian and a different ground state. Thus, with respect\nto different Hamiltonians, the spacetime dimension (measured through\nsome power law behaviour of the entropy) may assume different values\n(1 or 2, in the present case). \n\nLooking at the issue from a different perspective, we may start from a\nvacuum representation of the Virasoro algebra. The latter integrates\nto a unitary projective representation of the diffeomorphism group of\nthe circle ${\\mathrm {Diff}\\;}(S^1)$, which contains the diffeomorphism group of an\ninterval ${\\mathrm {Diff}\\;}(I)$ as a subgroup. For two open intervals with\ndisjoint closures, there is a canonical identification between\n${\\mathrm {Diff}\\;}(I\\cup J)$ and ${\\mathrm {Diff}\\;}(I)\\times {\\mathrm {Diff}\\;}(J)$. In terms of the\nstress-energy tensor $T$, this amounts to an isomorphism between $\\exp\niT(f+g)$ and $\\exp iT(f)\\otimes \\exp iT(g)$, when $f$ and $g$ have\ndisjoint support. It would be hard to see this local isomorphism\ndirectly in terms of the Virasoro algebra. \n\n\\medskip\n\nThe mathematical theorem underlying these facts is the well-known\nSplit Property \\cite{DL}, which can be derived in local QFT in any\ndimension under a suitable phase space assumption. In chiral local \nCFT, a sufficient assumption is the existence of the conformal\ncharacter $\\mathrm {Tr}\\; \\exp-\\beta L_0$. \n\nIn the algebraic framework, the {\\em chiral} observables of a BCFT\n(e.g., the stress-energy tensor) localized in a double cone $O$ are\noperators belonging to the von Neumann algebra $A_+(O) = A(I)\\vee\nA(J)$ where $I$ and $J$ are two open intervals of the time axis\n(``advanced and retarded times'') such that $ t+x\\in I$, $t-x\\in J$\nfor $(t,x)\\in O$ (this justifies the notation $O=I\\times J$), and\n$A(I)$ are the von Neumann algebras generated by the unitary\nexponentials of chiral fields smeared within $I$. In contrast, the\n{\\em chiral} observables in a 2D CFT are operators in the algebra\n$A_{2D}(O) = A_L(I)\\otimes A_R(J)$ where $I$ and $J$ are regarded as\ntwo open intervals of the lightcone axes, and $A_R(I)$ and $A_L(J)$\nare generated by left and right chiral fields. Our present\nassociation between BCFT and 2D CFT applies to the case when $A_L(I) =\nA_R(I)= A(I)$, i.e., the left chiral observables $A_L(I)\\otimes 1$ are\nisomorphic with the right chiral observables $1\\otimes A_R(I)$, and\nboth are isomorphic with the chiral observables $A(I)$ of the BCFT. \n\n\\medskip\n\nLet ${\\mathcal H}_0$ denote the vacuum Hilbert space for the chiral CFT\ndescribed by the algebras $A(I)$. The split property states that if\n$I$ and $J$ are two intervals with disjoint closures, there is a\ncanonical unitary ${\\mathcal{V}}:{\\mathcal H}_0\\to{\\mathcal H}_0\\otimes{\\mathcal H}_0$ implementing an\nisomorphism \n\\begin{eqnarray}\\label{splitiso}\n{\\mathcal{V}}\\big(A(I)\\vee A(J)\\big){\\mathcal{V}}^* = A(I)\\otimes A(J).\n\\end{eqnarray}\n\nThe split isomorphism does not preserve the vacuum vector, i.e.,\nthe canonical ``split vector'' $\\Xi={\\mathcal{V}}^*(\\Omega\\otimes\\Omega)$ is an\nexcited state in ${\\mathcal H}_0$. By construction, the split state\n$(\\Xi,\\cdot\\Xi)$ on $A(I)\\vee A(J)$ has the property that its\nexpectation values for either subalgebra $A(I)$ or $A(J)$ coincide\nwith those in the vacuum state, but the correlations between\nobservables $a_1\\in A(I)$ and $a_2\\in A(J)$ are suppressed: \n\\begin{eqnarray}\\label{splitstate}\n(\\Xi\\,,\\,a_1a_2\\,\\Xi) = (\\Xi\\,,\\,a_1\\,\\Xi)\\;(\\Xi\\,,\\,a_2\\,\\Xi) = (\\Omega\\,,\\,a_1\\,\\Omega)\\;(\\Omega\\,,\\,a_2\\,\\Omega).\n\\end{eqnarray}\nThe split isomorphism depends on the pair of intervals $I$ and \n$J$. It trivially restricts to algebras associated with subintervals,\nbut it does not, in general, extend to larger intervals. When the\nintervals touch or overlap, a split state and the split isomorphism\ncease to exist. \n\nWhile the split isomorphism is well known, we discuss in this paper\nits extension to ``non-chiral'' local observables, which do {\\em not} \nbelong to $A(I)\\vee A(J)$ in the BCFT, and to $A(I)\\otimes A(J)$ in\nthe 2D CFT. \n\n\\medskip\n\nAs a concrete demonstration for the resolution of the above ``paradox'', \nwe present two simple but nontrivial examples where the algebraic\nrelations outlined can be easily translated into the field-theoretic\nsetting, i.e., we characterize the local algebras of the various QFTs\nin terms of generating local Wightman fields. \n\nLet us translate (\\ref{relcomm}) into the field-theoretic language. \nThe intervals $I$ and $J$ shrink to the points $t\\pm x$ when\n$O=I\\times J$ shrinks to a point $(t,x)$. Thus, we have to approximate\na field $\\Phi(t,x)$ of the BCFT by observables in $A(L)$ (where the\ninterval $L$ approximates $(t-x,t+x)$ from the outside), that commute\nwith all fields localized in the interval $K$ (which approximates\n$(t-x,t+x)$ from the inside). This will be done in Sect.\\ 2. A crucial\npoint here is that generating the local algebra $A(L)$ involves\n``non-pointwise'' operations, e.g., typical observables may be\nexponentials of smeared field operators, so that an element of the\nrelative commutant is not necessarily localized in the disconnected\nset $L\\setminus \\overline K = I \\cup J$. \n\n\\medskip\n\nA second, somewhat puzzling feature of the algebraic treatment of BCFT\nis the fact that the description of the local algebras $B_+(O)$ in\nterms of the chiral boundary net (Eq.\\ (\\ref{relcomm})) is much\nsimpler than that of the local algebras $B_{2D}(O)$ of the associated\n$2D$ conformal QFT without a boundary. The latter are (rather\nclumsily) defined as Jones extensions of the tensor products\n$A(I)\\otimes A(J)$ in terms of a Q-system constructed from the\nchiral extension $A\\subset B$ with the help of $\\alpha$-induction\n\\cite{KHR}. \n\nOne purpose of this work is to present a more direct construction\nof the 2D CFT without boundary from the BCFT. The obvious idea is to\ntake a limit as the boundary is ``shifted to infinity''. But we shall\ndo more, and establish the {\\em covariant} local isomorphism between the\nsubnets $O\\mapsto B_+(O)$ and $O\\mapsto B_{2D}(O)$ as $O\\subset O_0$,\ni.e., the restriction of the AQFTs to any double cone $O_0$ within the\nhalfspace $x>0$, at finite distance from the boundary. \n\n\\medskip\n\nThe main problem here is, of course, the enhancement of the conformal\nsymmetry, i.e., the reconstruction of the unitary positive-energy\nrepresentation of the two-dimensional conformal group $\\Mb\\times \\Mb$\nfrom that of the chiral conformal group $\\Mb$. This is done by\na ``lift'' of the chiral M\\\"obius covariance of the local chiral net\n$A$, using the split property which allows to ``embed'' the 2D chiral\nalgebra $A(I)\\otimes A(J)$ into a local BCFT algebra $B_+(O)$. This\nwill be done in Sect.~3. The point is that only a single local\nalgebra of the BCFT is needed for this reconstruction of the 2D\nconformal group and the full 2D CFT.\n\n\\medskip \n\nIn Sect.~4, we show that the 2D CFT can also be obtained through a\nlimit where the boundary is ``shifted to the left'', or equivalently,\nthe BCFT observables are ``shifted to the right''. The translations in\nthe spatial direction ``away from the boundary'' do not belong to the\nchiral M\\\"obius group of the BCFT. But they are at our disposal by the\nprevious lifting of the 2D M\\\"obius group into the BCFT. Therefore, we\ncan study the behavior of correlation functions in the limit of\n``removing the boundary''. As we shift the boundary, the retarded and\nadvanced times are shifted apart from each other. The convergence of\nthe vacuum correlations of the BCFT to the vacuum correlations of the\n2D CFT is therefore a consequence of the cluster behavior of vacuum\ncorrelations of the chiral CFT $A$. \n\n\\medskip\n\nWe add three appendices containing some related observations.\n\n\\section{Example}\\label{sec2}\n\\setcounter{equation}{0}\n\nThe purpose of this section is to illustrate the construction\n(\\ref{relcomm}) in a field-theoretic setting. It is convenient to\nassume the trivial chiral extension $B=A$ since even in this case\nthe construction (\\ref{relcomm}) is nontrivial, i.e., non-chiral local\nBCFT fields that factorize into nonlocal chiral fields can be\nconstructed from local chiral fields only. We exhibit local BCFT \nfields in a region $O=I\\times J\\subset M_+$ as ``neutral'' chiral\noperators, that behave like products of ``charged'' chiral operators\nlocalized in $I$ and $J$ in the limit of large distance from the\nboundary. The limit of pointlike localization is also discussed, and\nreproduces familiar vertex operators.\n\n\\medskip \n\nConsider the free $U(1)$ current $j$ with commutator\n$[j(x),j(y)]=2\\pi i\\delta'(x-y)$ and charge operator $Q=(2\\pi)^{-1}\\int\nj(x)dx$. The unitary Weyl operators $W(f)=e^{ij(f)}$ for real test\nfunctions $f$ satisfy the Weyl relation \n\\begin{eqnarray}\\label{weyl}\nW(f)\\,W(g) = e^{-i\\pi\\sigma(f,g)}\\cdot W(f+g) = e^{-2\\pi i\\sigma(f,g)}\n\\cdot W(g)\\,W(f)\n\\end{eqnarray}\nand have the vacuum expectation value\n\\begin{eqnarray}\\label{ground}\n\\omega(W(f)) = e^{-i\\pi\\sigma(f_-,f_+)} = e^{-\\frac\n 12\\int_{\\RR_+}k\\,dk \\vert \\hat f(k)\\vert^2}\n\\end{eqnarray}\nwhere the symplectic form is\n\\begin{eqnarray}\\label{symp}\n\\sigma(f,g)= \\frac 12\\int_\\RR dx\\;\\big(f(x)\\;g'(x) - f'(x)\\;g(x)\\big)\n= \\frac 1{2\\pi i}\\int_\\RR k\\,dk\\;\\hat f(-k)\\;\\hat g(k),\n\\end{eqnarray}\nand $f_+$ ($f_-$) correspond to the restrictions to positive\n(negative) values of $k$ of the Fourier transform $\\hat f(k) =\n\\int_\\RR dx\\,e^{ikx}\\,f(x)$. With these conventions, $W(f)\\Omega$ is a \nstate with charge density $-f'(x)$. \n\nThe vacuum correlations of Weyl operators are\n\\begin{eqnarray}\\label{corr}\n\\omega(W(f_1)\\cdots W(f_n)) =\ne^{-i\\pi\\big(\\sum_i\\sigma(f_{i-},f_{i+}) + 2\n \\sum_{i t_2+x_2 > t_2-x_2 > t_1-x_1$ or when $t_2+x_2 >\nt_1+x_1 > t_1-x_1 > t_2-x_2$, because in these cases the anyonic phase\nfactors cancel. It also commutes with $j(t_2\\pm x_2)$ if $t_2\\pm\nx_2\\neq t_1\\pm x_1$. These are precisely the requirements for locality\nof the fields $\\Phi_q(t,x)$ among each other, and relative to the\nconserved current \n\\begin{eqnarray}\nj_0(t,x) = j(t+x) + j(t-x),\\qquad j_1(t,x) = j(t+x) - j(t-x) \n\\end{eqnarray}\ndefined for $x>0$, i.e., $\\Phi_q$ and $j^\\mu$ are local fields on the\nhalfspace $M_+$. The correlation functions of $n$ fields\n$\\Phi_{q_i}(t_i,x_i)$ are correlations of $2n$ vertex operators\n($2n$-point conformal blocks).\n\n\\medskip \n\nAfter this digression to pointlike fields, let us resume the\nstudy of the correlation functions (\\ref{corr}) of the smooth\nWeyl operators $W(f_i)\\in B_+(O)$, and their behavior as $O$ is\nshifted away from the boundary. We choose $n$ test functions of the \nform \n\\begin{eqnarray}\\label{g-h}\nf_i=G_i-H_i\n\\end{eqnarray}\nwhere $G_i$, $H_i$ are smooth step functions with values $0$ at\n$-\\infty$ and $q_i$ at $+\\infty$, such that $G_i'=g_i$ is supported in\n$J$ and $H_i'=h_i$ is supported in $I$ (see Fig.\\ 2).\n\n\\bigskip\n\n\\hskip10mm \\epsfig{file=steps.eps,width=100mm} \n\\nopagebreak\n\n{\\bf Figure 2:} A test function $f$ such that $W(f)$ belongs to\n$B_+(O)$, but not to $A_+(O)$. $G$ and $H$ are smooth step functions,\n$\\mathrm{supp}\\; G' \\subset J$, $\\mathrm{supp}\\; H' \\subset I$.\n\n\\bigskip\n\nThe neutral states $W(f_i)\\Omega$ carry the charge $q_i$ in $I$ and the\ncharge $-q_i$ in $J$. \n\nThe neutrality\ncondition for each Weyl operator $W(f_i)$ can be written \n\\begin{eqnarray}\n\\label{charge}\n\\int_\\RR dx\\;g_i(x) - \\int_\\RR dx\\;h_i(x)= 0 \\qquad\\Leftrightarrow \\qquad\n\\hat g_i(0) - \\hat h_i(0) =0. \n\\end{eqnarray} \n\nThe exponent in (\\ref{corr}) is a linear combination of terms of the form \n(using $\\hat f_i = i(\\hat g_i-\\hat h_i)\/k$) \n\\begin{eqnarray}\n2\\pi i\\,\\sigma(f_{i-},f_{j_+}) = \\int_{\\RR_+} \\frac{dk}k\n\\int dx\\,\\big(g_i(x)-h_i(x)\\big)\\int dy\\, \\big(g_j(y)-h_j(y)\\big) \n \\;e^{-ik(x-y)} \\quad\n\\end{eqnarray}\nwhich are IR finite because of (\\ref{charge}). The separate\ncontributions from $g_i$ and $h_i$, however, are IR divergent. \nTherefore, we first regularize at $k=0$ by the subtraction\n$e^{-ik(x-y)}\\to e^{-ik(x-y)}-e^{-k\/\\mu}$ ($\\mu>0$ arbitrary), \nwhich does not change the result because of (\\ref{charge}), and then\ncompute the contributions from $g$ and $h$ separately. \n\nWe are interested in the behavior of the correlation function\n(\\ref{corr}) as $O$ is shifted away from the boundary. This means that\nthe functions $g_i$ are shifted by a distance $a$ to the left, and\n$h_i$ are shifted by the same distance to the right. The $g$-$g$\ncontributions and the $h$-$h$-contributions to\n$\\sigma(f_{i-},f_{j_+})$ are obviously invariant under this shift,\nwhile in the mixed $h$-$g$ contributions $x-y$ is replaced by $x-y+2a$:\n\\begin{eqnarray}\n2\\pi i\\,\\sigma_{h_i,g_j}(a) := - \\int_I dx\\, h_i(x) \\int_J dy\\, g_j(y) \n\\int_{\\RR_+} \\frac{dk}k \\Big[e^{-ik(x-y+2a)}-e^{-k\/\\mu}\\Big] \n\\end{eqnarray}\nand similar for the $g$-$h$ contributions. The last integrand\ncan be split into two parts:\n\\begin{eqnarray}\n\\big(e^{-ik(x-y+2a)}-1\\big)e^{-k\/\\mu} \\;+\\;\ne^{-ik(x-y+2a)}\\big(1-e^{-k\/\\mu}\\big) \n\\end{eqnarray}\nso that the first contribution to the momentum integral equals \n\\begin{eqnarray}\n-\\log\\big(1+i\\mu(x-y+2a)\\big)\n\\end{eqnarray}\nwhile the second (distributional) contribution is of order $O(a^{-1})$\nin the limit of large $a$. Because the remaining integrals have\ncompact support, we obtain \n\\begin{eqnarray}\n\\lim_{a\\to\\infty} \\sigma_{h_i,g_j}(a) = q_iq_j\\cdot \\log(2i\\,a\\mu)\n+ O(a^{-1}).\n\\end{eqnarray}\nTogether with the $g$-$h$ contributions $q_iq_j\\cdot \\log(-2i\\,a\\mu)$,\nthese terms in the exponent of (\\ref{corr}) cumulate up to\nthe factor \n\\begin{eqnarray}\n\\prod_{i} (2a\\mu)^{-q_i^2} \\prod_{i0$, away from the boundary), that take $I$ to $I+x$\nand $J$ to $J-x$, represented as homomorphisms from $B_+(O)$\nto $B_+(I+x\\times J+x)$, see (\\ref{beta}).\n\nIn Sect.~3 (with $O$ as the fixed reference\ndouble cone) we have given the re-inter\\-pretation of $b_i$ in the GNS\nrepresentation $\\hat\\pi$ of the state $\\xi\\circ\\mu$ as observables of\nthe associated 2D CFT, with the 2D vacuum $\\Omega_{2D}$ given by the\nGNS vector. We shall show\n\n\\medskip \n\n{\\bf Theorem 2:} {\\sl Let each $b_i\\in B_+(O)$ ($i=1,\\dots,n)$ be of the\n form $\\iota(a_1^{(i)} a_2^{(i)}) \\cdot \\psi^{(i)}$ with charged \n intertwiners $\\psi^{(i)}$ and $a^{(i)}_1\\in A(I)$ and $a^{(i)}_2\\in A(J)$. \nAs $x$ goes to $+\\infty$, the BCFT vacuum correlations (\\ref{b1bn})\nconverge to the 2D vacuum correlations }\n\\begin{eqnarray}\\label{corr2d}\n\\big(\\Omega_{2D}\\,,\\,\\hat\\pi(b_1\\cdots\nb_n)\\,\\Omega_{2D}\\big) = \\xi\\circ\\mu(b_1\\cdots b_n). \n\\end{eqnarray}\n\n\\medskip\n\n{\\it Proof:} We compute the limit and the 2D vacuum expectation value\nseparately. \n\nUsing the decomposition of products $\\psi_1\\psi_2$ into\nfinite sums of operators of the form $\\iota(T_1T_2)\\cdot \\psi$\n\\cite{LR2}, where $T_i$ are intertwiners between DHR \nendomorphisms of $A$, we see that the product $b_1\\cdots b_n$ is a\nfinite sum of operators of the same form $\\iota(a_1a_2)\\cdot \\psi$.\n\nFor the present purpose, it is more convenient to write the charged\nintertwiners as $\\psi= t\\cdot\\iota(\\bar r)$ where $r\\in\n\\mathrm{Hom}({\\mathrm {id}},\\tau\\bar\\tau)\\subset A(J)$ and $t\\in\n\\mathrm{Hom}(\\alpha^+_\\tau,\\alpha^-_\\sigma)\\subset \\mathrm{Hom}(\\iota\\tau,\\iota\\sigma)$\n(Frobenius reciprocity). Then, because $a_2=\\sigma(a_2)$, we get\n$\\iota(a_2)\\cdot \\psi = t\\cdot \\iota(\\tau(a_2)\\bar r)$. Hence, the product\n$b_1\\dots b_n$ is a finite sum of operators\nof the form\n\\begin{eqnarray} \\label{ata}\n\\iota(a_1)\\cdot t\\cdot \\iota(a_2).\n\\end{eqnarray}\nThus, the above vacuum correlation function is a finite sum\nof expectation values\n\\begin{eqnarray} F(x) = \\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,\\Omega\\big) = \\qquad\\qquad\\qquad\\qquad \\nonumber \n\\\\ = \\big(\\Omega\\,,\\,\\iota(\\alpha_x(a_1) z^\\sigma(x))\\cdot t\\cdot \n\\iota(z^\\tau(-x)^* \\alpha_{-x}(a_2)) \\,\\Omega\\big) = \\\\\n= \\big(\\Omega\\,,\\,\\alpha_x(a_1) z^\\sigma(x)\\cdot \\varepsilon(t)\\cdot \nz^\\tau(-x)^* \\alpha_{-x}(a_2)\\,\\Omega\\big). \\nonumber \\end{eqnarray}\nHere, $\\varepsilon$ is the global conditional expectation $B\\to A$, which\npreserves the vacuum state \\cite{LR1}. In particular, $\\varepsilon(t)\\in\n\\mathrm{Hom}(\\tau,\\sigma)$. Therefore, the expression vanishes identically\nunless $\\sigma$ and $\\tau$ belong to the same sector.\n\nIn the latter case, we express the cocycles as\n$z^\\rho(g)=U_0(g)U_\\rho(g)^*$, and $\\alpha_g={\\mathrm {Ad}}_{U_0(g)}$, giving \n\\begin{eqnarray}\nF(x) = \\big(\\Omega\\,,\\,a_1 U_\\sigma(x)^* \\cdot \\varepsilon(t) \\cdot U_\\tau(x)^*\na_2\\,\\Omega\\big) =\\big(\\Omega\\,,\\,a_1 \\cdot U_\\sigma(-2x) \\cdot \\varepsilon(t) a_2\\,\\Omega\\big) ,\n\\end{eqnarray}\nbecause the intertwiners between DHR endomorphisms also intertwine the \nrepresentations of the M\\\"obius group \\cite{FRS2}. \nBy the spectrum condition, $F(x)$ has a bounded analytic\ncontinuation to the lower complex halfplane. $U_\\sigma(-z)$ weakly\nconverges in every direction $z=re^{i\\varphi}$ ($-\\pi < \\varphi <0$,\n$r\\to\\infty$) to the projection onto the zero eigenspace of the\ngenerator, and the latter projection is nonzero only if $\\sigma={\\mathrm {id}}$\nis the vacuum representation; in this case $t=\\varepsilon(t)=1$. Thus, $F(z)$\nconverges in these directions to the vacuum expectation value \n\\begin{eqnarray}\\label{limit}\n\\delta_{\\sigma,0}\\delta_{\\tau,0}\n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega).\n\\end{eqnarray}\n\nNext, we consider \n\\begin{eqnarray}\n\\overline{F(x)} = \\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(a_2^*)\\cdot t^*\\cdot\n\\iota(a_1^*)\\big)\\,\\Omega\\big).\n\\end{eqnarray}\nLet $r_\\sigma\\in \\mathrm{Hom}({\\mathrm {id}},\\bar\\sigma\\sigma)\\subset A(I)$\nand $r_\\tau\\in \\mathrm{Hom}({\\mathrm {id}},\\bar\\tau\\tau)\\subset A(J)$. Then we can write \n$t^* = \\iota(r_\\sigma^*)\\cdot \\bar t\\cdot \\iota(r_\\tau)$, where \n$\\bar t \\in \\mathrm{Hom}(\\alpha^+_{\\bar\\tau},\\alpha^-_{\\bar\\sigma})\\subset \n\\mathrm{Hom}(\\iota\\bar\\tau,\\iota\\bar\\sigma)$. Using the locality\nproperties of $a_1\\in A(I)$, $a_2\\in A(J)$, we can rewrite\n\\begin{eqnarray}\n\\overline{F(x)} =\n\\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(r_\\sigma^*\\bar\\sigma(a_1^*))\\cdot \\bar t\\cdot\n\\iota(\\bar\\tau(a_2^*)r_\\tau)\\big)\\,\\Omega\\big). \n\\end{eqnarray}\nThis expression can be computed in the same way as $F(x)$ before,\ngiving\n\\begin{eqnarray}\n\\overline{F(x)} = \\big(\\Omega\\,,\\,\nr_\\sigma^*\\bar\\sigma(a_1^*))\\cdot U_{\\bar\\sigma}(-2x) \\cdot \\varepsilon(\\bar t)\n\\bar\\tau(a_2^*)r_\\tau\\,\\Omega\\big).\n\\end{eqnarray}\nThus $F(x)$ also has a bounded analytic continuation to the upper\ncomplex halfplane, and converges to the same limit (\\ref{limit}) also\nin the directions $z=re^{i\\varphi}$ ($0 < \\varphi < \\pi$, $r\\to\\infty$). \nFrom this, we may conclude the cluster limit \n\\begin{eqnarray}\\label{cluster}\n\\lim_{x\\to\\infty}\\big(\\Omega\\,,\\,\\beta_{x}\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,\\Omega\\big) = \\delta_{\\sigma,0}\\delta_{\\tau,0}\n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega).\n\\end{eqnarray}\n\nOn the other hand, we now compute (\\ref{corr2d}) and show that it\ncoincides with the factorizing cluster limit of\n(\\ref{b1bn}). For each contribution of the form\n(\\ref{ata}), we have\n\\begin{eqnarray}\n(\\Omega_{2D}\\,,\\,\\hat\\pi\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,,\\,\\Omega_{2D}) = \\xi\\circ\\mu\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big) = \\xi\\big(a_1\\cdot \\mu(t)\\cdot a_2\\big). \\qquad\n\\end{eqnarray}\nBut $\\mu(t)\\in A(I)\\vee A(J)$ is an intertwiner in $\\mathrm{Hom}(\\sigma,\\tau)$\nwhich vanishes unless $\\sigma={\\mathrm {id}}$ and $\\tau={\\mathrm {id}}$ both belong to the\nvacuum sector. In the latter case, $t=\\mu(t)=1$. Thus,\n\\begin{eqnarray}\\label{cblock}\n\\langle\\hat\\Xi\\vert \\hat\\pi\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\vert \\hat\\Xi\\rangle = \\delta_{\\sigma,0}\\delta_{\\tau,0} \\;\n\\xi(a_1a_2) = \\delta_{\\sigma,0}\\delta_{\\tau,0} \n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega). \\qquad\n\\end{eqnarray} \nThis coincides with the cluster limit (\\ref{cluster}) ``far away from\nthe boundary''. \\hspace*{\\fill}Q.E.D.\n\n\\medskip \n\nRecall that $a_1$ and $a_2$ in (\\ref{ata}) were obtained by\nmultiplying $b_1\\cdots b_n$ and successively decomposing the products\nof the charged intertwiners. Thus, the vacuum expectation values\n$(\\Omega,a_i\\Omega)$ in (\\ref{cblock}) are precisely the chiral\nconformal blocks of the corresponding 2D correlation functions. \n\nA variant of the conformal cluster theorem \\cite{FJ} should also give\na quantitative estimate for the rate of the convergence, depending on\nthe charges of the operators involved through the corresponding\nspectrum of $L_0$.\n\n\n\\section{Conclusion}\nWe have studied the passage from a local conformal quantum field\ntheory defined on the halfspace $x>0$ of two-dimensional Minkowski\nspacetime (boundary CFT, BCFT) to an associated local conformal\nquantum field defined on the full Minkowski spacetime (2D CFT). There\nare essentially two ways: the first is to consider BCFT vacuum\ncorrelations of observables localized far away from the boundary. In\nthe limit of infinite distance, these correlation factorize into\nchiral correlations (conformal blocks) of charged fields. We have\ntraced this effect back to the cluster property of the underlying\nlocal chiral subtheory. \n\nThe second method exploits the split property, i.e., the existence of\nstates of the underlying local chiral CFT in which correlations between\nobservables in two fixed intervals at a finite distance are suppressed. \nWith the help of the split property one can algebraically identify a\nfixed local algebra of the BCFT with a fixed local algebra of the 2D\nCFT, and one can generate a unitary representation of the 2D M\\\"obius\ngroup in the GNS Hilbert space of a suitable ``extended split state''\nof this algebra. Its ground state is different from the BCFT\nvacuum. Then, by acting with the 2D M\\\"obius group, one can obtain\n{\\em all} local algebras of the 2D CFT in the same Hilbert space. \n\nThe converse question: can one consistently ``add'' a boundary in any\n2D CFT (without affecting the algebraic structure away from the\nboundary), is not addressed here. However, there arises a necessary \ncondition from the discussion in App.~\\ref{apphmi}: the 2D partition\nfunction should be either modular invariant, or at least it should be\nintermediate between the vacuum partition function and some modular\ninvariant partition function. We hope to return to this problem, and\nfind also a sufficient condition. \n\n\\bigskip\n\n\\noindent\n{\\bf Acknowledgements:} KHR thanks the Dipartimento di Matematica of\nthe Universit\\`a di Roma ``Tor Vergata'' for hospitality and financial\nsupport, and M. Weiner and I. Runkel for discussions related to the subject.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}