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diff --git a/support/hypertex/tanmoy/examples/Anomalkap.tex b/support/hypertex/tanmoy/examples/Anomalkap.tex new file mode 100644 index 0000000000..234613653f --- /dev/null +++ b/support/hypertex/tanmoy/examples/Anomalkap.tex @@ -0,0 +1,1252 @@ +\documentstyle[preprint,aps]{revtex} +%\documentstyle[prd,aps]{revtex} +\begin{document} +%\draft + +\title{Perturbative analysis for Kaplan's lattice chiral fermions} + + +\author{S. Aoki and H. Hirose} +\address{Institute of Physics, University of Tsukuba, Tsukuba +Ibaraki-305, Japan} + +\date{\today} + +\maketitle + +\begin{abstract} +Perturbation theory for lattice fermions with domain wall mass terms +is developed and is applied to +investigate the chiral Schwinger model formulated on the lattice +by Kaplan's method. +We calculate the effective action for gauge fields to one loop, +and find that it contains a longitudinal component even for anomaly-free cases. +From the effective action we obtain +gauge anomalies and Chern-Simons current without ambiguity. +We also show that the current corresponding to the fermion number +has a non-zero divergence and it flows off the wall into the extra +dimension. +Similar results are obtained for a proposal +by Shamir, who used a constant mass term with free boundaries +instead of domain walls. +\end{abstract} +\pacs{11.15Ha, 11.30Rd, 11.90.+t} + +\narrowtext + +\newpage +\tableofcontents +\newpage + +\section{Introduction} +\label{sec:int} + +Construction of chiral gauge theories is one of the long-standing +problems of lattice field theories. Because of the fermion doubling +phenomenon, +a naively discretized lattice fermion field yields $2^d$ fermion modes, half of +one chirality and half of the other, so that the theory is non-chiral. +Several lattice approaches have been proposed to define chiral gauge theories, +but so far none of them have been proven to work successfully. + +Kaplan has proposed a new approach\cite{KP} to this problem. +He suggested that it may be possible to simulate the behavior of massless +chiral fermions in 2k dimensions by +a lattice theory of massive fermions in 2k+1 dimensions if +the fermion mass has a shape of a domain wall in the +2k+1-th dimension. +He showed for the weak gauge coupling limit +that massless chiral states arise as zero-modes bound to +the 2k-dimensional domain wall while all doublers can be given large gauge +invariant masses. If the chiral fermion content that appears +on the domain wall +is anomalous the 2k-dimensional gauge current flows off the wall +into the extra dimension so that the theory can not be 2k-dimensional. +Therefore he argued that +this approach possibly simulates the 2k-dimensional chiral fermions +only for anomaly-free cases. + +His idea was tested for smooth external gauge fields. +Jansen\cite{Jan} showed numerically +that in the case of the chiral Schwinger model in 2 dimensions +with three fermions of charge 3, 4, and 5 the anomalies +in the gauge currents cancel on the wall. +The Chern-Simons current far away from the domain wall +was calculated in Ref.\ \cite{CS}. It is shown that +the 2k+1-th component of the current is non-zero +in the positive mass region +and zero in the negative mass region +such that the derivative of the current cancels +the 2k-dimensional gauge anomaly on the wall, as was argued in Ref.\ \cite{KP}. + +In the continuum perturbation theory +Frolov and Slavnov\cite{FS} proposed a gauge invariant regularization +of the standard model through an infinite tower of regulator fields. +Some similarity between their proposal and the Kaplan's approach +was pointed out by Narayanan and Neuberger\cite{Neu}. +It has been also shown that +the chiral fermion determinant can be nonperturbatively defined as an +overlap of two vacua\cite{Neu2}, which can be extended to lattice theories. +Using Narayanan and Neuberger's point of view, +we observed in Ref.\ \cite{AK} that +non-gauge (chiral) anomalies are correctly reproduced within +the Frolov and Slavnov's regularization method. + +The results above provide positive indications that Kaplan's method for chiral +fermion may work. +There exists, however, several potential problems in his approach. +Since the original 2k+1-dimensional model is vector-like, +there always exists an anti-chiral mode, localized on an anti-domain wall +formed by periodicity of the extra dimension. +If the chiral mode and the anti-chiral mode are paired into +a Dirac mode, this approach fails to simulate chiral gauge theories. +Without dynamical gauge fields, the overlap between the chiral mode and +the anti-chiral mode is suppressed as $O (e^{-L})$ where $L$ is the size of the +extra dimension. If gauge fields become dynamical, the overlap +depends on the gauge coupling. In the original paper\cite{KP} +the strong coupling limit of the gauge coupling in the extra dimension +was proposed to suppress the overlap. However, +a mean-field calculation\cite{Alt} +in this limit indicated that the chiral mode disappears and the model becomes +vector-like. + +More recently Distler and Rey\cite{Dis} pointed out that +the Kaplan's method may have a problem in reproducing fermion number +non-conservation expected in the standard model. +Using the 2-dimensional chiral Schwinger model +they argued that either the 2-dimensional fermion number current is exactly +conserved or the light degree of freedom flows off the wall into the +extra dimension so that the model can not be 2-dimensional. + +In this paper we carry out a detailed perturbative analysis of the Kaplan's +proposal for smooth background gauge fields on a finite lattice +taking the chiral Schwinger model in 2-dimensions as a concrete example. +In sect.\ \ref{sec:model}, +we formulate the lattice perturbation theory for +the Kaplan's method with the periodic boundary condition. +Since translational invariance is violated by domain wall mass terms, +usual Feynman rules in the momentum space can not be used except +in the regions far away from the domain wall\cite{CS}. +To perform perturbative calculations near or on the wall, +we use the Feynman rules in real space of the extra dimension, +as proposed in ref.\ \cite{Neu}. +We calculate the fermion propagator for the periodic boundary condition, +which reproduces the fermion propagator in ref.\cite{Neu} +near the origin of the extra dimension. +A similar calculation is also made for the constant fermion mass +with {\it free} boundaries in the 2k+1-th dimension. +As shown by Shamir\cite{Sham} the constant mass term +with this boundary condition +can also produce the chiral zero mode on the 2k-dimensional +boundary. +The results are similar but simpler than those by the Kaplan's method. +In sect.\ \ref{sec:2dim}, +using the Feynman rules of sect.\ \ref{sec:model} +we calculate a fermion one-loop effective action for the U(1) gauge field +of the chiral Schwinger model simulated by the Kaplan's method. +We find that the effective action +contains the longitudinal component as well as +parity-odd terms, +and that this longitudinal component, which +breaks gauge invariance, remains non-zero even for anomaly-free cases. +This result is compared with those of the conventional Wilson fermion +formulation of this model\cite{chiral}. +In sect.\ \ref{sec:anomaly} +we derive gauge anomalies as well as the Chern-Simons current +from the effective action without ambiguity. +Then we show that the current corresponding to +the fermion number has a non-zero divergence +and the fermion number current flows off the walls into the extra dimension. +In sect.\ \ref{sec:concl}, we give our conclusions and discussions. + +\section{Action, Fermion Propagator and Chiral Zero Modes} +\label{sec:model} + +\subsection{Lattice Action} +We consider a vector gauge theory in D=2k+1 dimensions with +a domain wall mass term. For later convenience we use the +notation of ref.\cite{Neu}, where the fermionic action is written in terms of +a d=2k dimensional theory with infinitely many flavors. +Our action is denoted as +\begin{equation} +S= S_G + S_F . +\end{equation} +The action for gauge field $S_G$ is given by +\begin{eqnarray} +S_G & =& \beta \sum_{n,\mu>\nu}\sum_s {\rm Re}\{ {\rm Tr} [U_{\mu\nu}(n,s)]\} + \nonumber \\ +& +& \beta_D \sum_{n,\mu}\sum_s {\rm Re} \{ {\rm Tr} [U_{\mu D}(n,s)]\} +\end{eqnarray} +where $\mu$, $\nu$ run from 1 to $d$, +$n$ is a point on a d-dimensional lattice and $s$ a coordinate in the +extra dimension, +$\beta$ is the inverse gauge coupling for plaquettes $U_{\mu\nu}$ +and $\beta_D$ that for plaquettes $U_{\mu D}$. +In general we can take $\beta \not= \beta_D$. +The fermionic part of the action $S_F$ is given by +\widetext +\begin{eqnarray} +S_F & = & {1\over 2}\sum_{n,\mu}\sum_s \bar\psi_s(n)\gamma_\mu +[U_{s,\mu}(n)\psi_s(n+\mu ) - U^\dagger_{s,\mu}(n-\mu)\psi_s(n-\mu ) ] + \nonumber \\ + &+& \sum_n \sum_{s,t} \bar\psi_s(n) [ M_0 P_R + M_0^\dagger P_L]_{st} +\psi_t(n) +\nonumber \\ + & + & {1\over 2}\sum_{n,\mu}\sum_s \bar\psi_s(n) +[U_{s,\mu}(n)\psi_s(n+\mu ) + U^\dagger_{s,\mu}(n-\mu)\psi_s(n-\mu ) + -2\psi_s(n) ] +\label{actionf} +\end{eqnarray} +where $s$, $t$ are considered as flavor indices, +$P_{R/L} = (1 \pm \gamma_{2k+1})/2$, +\narrowtext +\begin{mathletters} +\begin{eqnarray} +(M_0)_{st} & = & U_{s,D}(n)\delta_{s+1,t}- a(s)\delta_{st} + \\ +(M_0^\dagger)_{st} & = & U^\dagger_{s-1,D}(n)\delta_{s-1,t}- a(s)\delta_{st} , +\end{eqnarray} +\end{mathletters} +and $U_{s,\mu}(n)$, $U_{s,D}(n)$ are link variables for gauge fields. +We consider the above model with periodic boundaries +in the extra dimension, +so that $s$, $t$ run from $-L$ to $L-1$, and we take +\begin{equation} +a(s) = 1 - m_0[ {\rm sign}(s+{1\over 2}) \cdot {\rm sign}(L-s- +{1\over 2})] += \left\{ +\begin{array}{ll} +1-m_0, & -{1\over 2} < s < L-{1\over 2} \\ +1+m_0, & -L-{1\over 2} < s < -{1\over 2} +\end{array} +\right. +\end{equation} +for $-L \leq s < L $. It is easy to see\cite{Neu} that $S_F$ above is +identical to the Kaplan's action in D=2k+1 dimensions\cite{KP} +with the Wilson parameter $r=1$. In fact the second term +in eq.\ (\ref{actionf}) can be rewritten as +\begin{eqnarray} +& {1\over 2}& \bar\psi_s\gamma_D [U_{s,D}\psi_{s+1}-U_{s-1,D}\psi_{s-1}] + \nonumber \\ +&+& {1\over 2}\bar\psi_s[U_{s,D}\psi_{s+1}+U_{s-1,D}\psi_{s-1} +-2\psi_s] + M(s) \bar\psi_s \psi_s +\end{eqnarray} +with $M(s)=m_0 [ {\rm sign}(s+1/2) \cdot {\rm sign}(L-s-1/2)]$. +Note that our action is slightly different from that of ref.\cite{Neu}: +we have the D-th component of the link variable $U_{s,D}(n)$ +and all link variables have $s$ dependence. With the gauge fixing condition +$U_{s,D}(n) =1$ for all $s$ and $n$\cite{Dis}, +our action becomes almost identical to that of ref.\cite{Neu}, but +still the $s$ dependence exists in our link variables in d dimensions. +The model in ref.\cite{Neu} corresponds to our model at +$\beta_D =\infty$, where $s$ dependences of gauge fields are +completely suppressed, and the model at $\beta_D = 0$ was +investigated by the mean field method\cite{Alt}. + +\subsection{Chiral Zero Modes} +We now consider chiral zero modes of the action $S_F$ in +the weak coupling limit , i.e. $\forall U_{s,\mu} = \forall U_{s,D}=1$. +According to ref.\cite{Neu}, the right-handed zero modes are given +by zero modes of the operator $M$ and the left-handed zero modes +by those of the operator $M^\dagger$, where +\begin{equation} +(M)_{st} = (M_0)_{st} + {\nabla (p)\over 2}\delta_{st}, \ +(M^\dagger )_{st} = (M_0^\dagger )_{st} + {\nabla (p)\over 2}\delta_{st} +\end{equation} +with $\nabla (p)\equiv \sum_{\mu=1}^d 2 [\cos (p_\mu a) - 1 ]$ +in momentum space of d dimensions. +It is noted that $ 0 \leq -\nabla(p) \leq 4d$ and zero modes exist +if and only if $ -\nabla (p) \le 2m_0$\cite{CS}. +Hereafter we only consider the case that $ 0 < m_0 < 2$. +In this range of $m_0$, there is only one right-handed zero mode $u_R$ +satisfying $ M \cdot u_R = 0$, which is given by +\begin{equation} +u_R(s) = \left\{ \begin{array}{ll} +\displaystyle (1-\nabla (p)/2-m_0)^s C_0^{-1} & \mbox{for $s \geq 0$} \\ + & \\ +\displaystyle (1-\nabla (p)/2+m_0)^s C_0^{-1} & \mbox{for $s < 0$} +\end{array} \right. +\end{equation} +where the d-dimensional momentum $p$ has to be restricted to +$ 0 \leq m_0+\nabla (p) /2 $ so that +$(1-\nabla/ 2 -m_0) \leq 1$. The normalization constant $C_0$ takes the value +\begin{equation} +{1-(1-\nabla (p)/2-m_0)^L\over m_0+\nabla (p)/2} + +{1-(1-\nabla (p)/2+m_0)^{-L}\over m_0 - \nabla (p)/2} . +\end{equation} +This zero mode is localized around $s=0$. +On a finite lattice (i.e. $L\not= \infty$) with the periodic boundary +condition, there exists another zero mode $u_L$ with the opposite +chirality satisfying $ M\cdot u_L = 0 $, which is given by +$ u_L(s) = u_R(L-t-1)$ and is localized around $s=L$. +The overlap between the two zero modes vanishes exponentially as +$L\rightarrow \infty$; +\begin{eqnarray} +\sum_{s=-L}^{L-1} u_R(s) u_L(s) & =& C_0^{-2} L\times + (1-{\nabla (p)\over2}-m_0)^L + \nonumber \\ +&\times & +(1-{\nabla (p)\over 2}+m_0)^{-L} +\longrightarrow 0, +\end{eqnarray} +We illustrate the shape of the two zero modes $u_R$ and $u_L$ +at $m_0= 0.1$ and 0.5 for $ p =0$ in Fig.\ \ref{zero}. + +\subsection{Fermion Propagator and Zero Modes} +The fermion propagator in d-dimensional momentum space and +in real D-th space has been obtained in ref.\cite{Neu} for +the infinite D-th space(i.e. $L=\infty$ ). +It is not difficult to obtain +the fermion propagator for a finite lattice with +periodic boundaries. We have +\begin{eqnarray} +S_F(p)_{st} & = &- \left[ [(i\sum_\mu \gamma_\mu \bar p_\mu+ M ) G_L (p)]_{st} +P_L \right. + \nonumber \\ +& + & +\left. [(i\sum_\mu \gamma_\mu \bar p_\mu+ M^\dagger ) G_R (p)]_{st}P_R \right] +\label{fprop} +\end{eqnarray} +where +\begin{equation} +G_L (p) ={1\over \bar p^2 + M^\dagger M }, +\quad +G_R (p) ={1\over \bar p^2 + M M^\dagger } +\end{equation} +with $\bar p_\mu = \sin (p_\mu a)$ . +Explicit expressions for $G_L$ and $G_R$ are complicated in general, +but they become simple for large $L$ where we neglect terms of order +$O(e^{-cL})$ with $c > 0$. +We obtain +\widetext +\begin{equation} +G_L(p)_{st} = +\left\{ \begin{array}{ll} +B e^{-\alpha_+|s-t|} + (A_L-B) e^{-\alpha_+(s+t)} + +(A_R-B) e^{-\alpha_+(2L-s-t)}, +& (s,t \ge 0) \\ + & \\ +A_Le^{-\alpha_+s+\alpha_-t} +A_Re^{-\alpha_+(L-s)-\alpha_-(L+t)}, +& (s\ge 0,\ t\le 0) \\ + & \\ +A_Le^{\alpha_-s-\alpha_+t} + A_Re^{-\alpha_-(L+s)-\alpha_+(L-t)}, +& (s\le 0,\ t\ge 0) \\ + & \\ +C e^{-\alpha_-|s-t|}+ (A_L-C) e^{\alpha_-(s+t)} + +(A_R-C) e^{-\alpha_-(2L+s+t)}, +& (s,t \le 0) +\end{array} \right. +\label{gl} +\end{equation} +\begin{equation} +G_R(p)_{st} = +\left\{ \begin{array}{ll} +B e^{-\alpha_+|s-t|} + (A_R-B) e^{-\alpha_+(s+t+2)} + +(A_L-B) e^{-\alpha_+(2L-s-t-2)}, +& (s,t \ge -1) \\ + & \\ +A_Re^{-\alpha_+(s+1)+\alpha_-(t+1)} +A_Le^{-\alpha_+(L-s-1)-\alpha_-(L+t+1)}, +& (s\ge -1,\ t\le -1) \\ + & \\ +A_Re^{\alpha_-(s+1)-\alpha_+(t+1)} +A_Le^{-\alpha_-(L+s+1)-\alpha_+(L-t-1)}, +& (s\le -1,\ t\ge -1) \\ + & \\ +C e^{-\alpha_-|s-t|} + (A_R-C) e^{\alpha_-(s+t+2)} + +(A_L-C) e^{-\alpha_-(2L+s+t+2)}, +& (s,t \le -1) +\end{array} \right. +\label{gr} +\end{equation} +\narrowtext where +\begin{mathletters} +\begin{equation} +a_{\pm} = 1 -{\nabla (p)\over 2}\mp m_0 +\end{equation} +\begin{equation} +\alpha_{\pm} = {\rm arccosh} [{1\over 2}(a_{\pm}+{1+\bar p^2\over a_{\pm}})] +\geq 0, +\end{equation} +\begin{equation} +A_L = {1\over a_+ e^{\alpha_+} - a_-e^{-\alpha_-}}, \qquad +A_R ={1\over a_- e^{\alpha_-}- a_+e^{-\alpha_+}} +\end{equation} +\begin{equation} +B = {1\over 2 a_+ {\rm sinh} \alpha_+}, \qquad +C = {1\over 2 a_- {\rm sinh} \alpha_-} . +\end{equation} +\end{mathletters} + +For $|s|, |t| \ll L$ the propagator above coincides with +that of ref.\cite{Neu}. +From the form of $A_L$, $A_R$, $B$ and $C$, +it is easy to see\cite{Neu} +that singularities occur only in $A_L$ at $p=0$; +\begin{equation} +A_L \longrightarrow {m_0(4-m_0^2) \over 4 p^2 a^2 }, +\qquad p\rightarrow 0 . +\end{equation} +Therefore the propagator $G_L$ describes a massless right-handed +fermion around $s,t = 0$ and $G_R$ a massless left-handed fermion +around $|s|,|t| = L$, which correspond to the two zero modes in the +previous subsection. +Later we use the above forms of the fermion propagator +to calculate fermion one-loop diagrams. +It is also noted that the fermion propagator away from the two domain walls +approaches the Wilson fermion propagator +with a {\it constant} mass term, i.e. , +\begin{eqnarray} +S_F(p) & \rightarrow +& \int {dp_D\over 2\pi} +{e^{ia p_D(s-t)} \over i\gamma\cdot p + i\gamma_D p_D \pm m_0 +-\nabla (p) +1-\cos (p_D a)} +\label{wil} +\end{eqnarray} +for $1 \ll |s|, |t|, |L-s|, |L-t|$ with $s\cdot t > 0$, where $+m_0$ is taken +for $s,t > 0$ and $-m_0$ for $s,t < 0$. +Therefore the calculation in ref.\cite{CS} is valid in this region +of $s$ and $t$. + +Before closing this subsection, we give the form of propagator for the Shamir's +free boundary fermions\cite{Sham}. +This is again given by eq.(\ref{fprop}) +with $M_{st}=\delta_{s+1,t}-a_+ \delta_{s,t}$ +and $M^\dagger_{st}=\delta_{s-1,t}-a_+ \delta_{s,t}$. +For large $L$, it becomes\cite{Sham} +\begin{equation} +G_L(p)_{st} = +B e^{-\alpha_+|s-t|} + A'_L e^{-\alpha_+(s+t)} ++A'_Re^{\alpha_+(s+t-2L)} +\label{shl} +\end{equation} +\begin{equation} +G_R(p)_{st} = +B e^{-\alpha_+|s-t|}+ A'_R e^{-\alpha_+(s+t)} + +A'_Le^{\alpha_+(s+t-2L)} , +\label{shr} +\end{equation} +where +\begin{equation} +A'_L = B {1-a_+e^{-\alpha_+}\over a_+e^{\alpha_+}-1}, \qquad +A'_R = - Be^{-2\alpha_+} +\end{equation} +and now $ 0 \le s,t \le L$. +Singularities occur only in $A'_L$ at $p=0$ such that +\begin{equation} +A'_L \rightarrow {m_0(2-m_0)\over p^2 a^2} . +\end{equation} +Thus, the propagator describes a right-handed massless +fermion around one boundary at $s,t = 0$ and +a left-handed massless around the other boundary at $s,t = L$. + +\subsection{Fermion Feynman Rules } +In this subsection, we write down the lattice Feynman rules for fermions +relevant for fermion one-loop calculations, +which will be performed in the next section. +We first choose the axial gauge fixing $U_{s,D}=1$. +Although the full gauge symmetries in D dimensions are lost, +the theory is still invariant under gauge +transformations independent of $s$ \cite{Dis}. +We consider the limit of small d-dimensional gauge coupling, and take +\begin{equation} +U_{s,\mu}(n) = exp[i a g A_\mu (s, n+\mu/2)] +\end{equation} +where $a$ is the lattice spacing, and +$g \propto 1/\sqrt\beta$ +is the gauge coupling constant whose mass dimension is $2-D/2$ +(mass dimension of the gauge fields $A_\mu$ is $D/2-1$). +It is noted that the other gauge coupling $g_s \propto 1/\sqrt{\beta_s}$ +is not necessarily small and can be made arbitrary large. +We consider Feynman rules in momentum space for the physical +d dimensions but in real space for the extra dimension. +\begin{itemize} +\item The fermion propagator is given by +\begin{equation} +< \psi_s (-p) \bar\psi_t (p) > = S_F(p)_{st}. +\end{equation} +where $S_F$ has been given in eq.(\ref{fprop}) with +eqs.(\ref{gl},\ref{gr}) for the Kaplan's fermions or with +eqs.(\ref{shl},\ref{shr}) for the Shamir's fermions. + +\item The fermion vertex coupled to a single gauge field is given by +\begin{equation} +ag \bar\psi_s(q) \partial_\mu [ S_F^{-1}({q+p\over 2})]_{ss} +A_\mu(s,p-q)\psi_s(-p). +\end{equation} +Here +$\partial_\mu S_F^{-1}(q) = \displaystyle {\partial S_F^{-1}(q)\over +\partial (q_\mu a)} = iC_\mu(q)\gamma_\mu + S_\mu (q)$ with +$C_\mu (q)= \cos (q_\mu a)$ and $S_\mu (q)= \sin (q_\mu a)$. +From this form of the vertex it is easy to see that +the fermion tadpole diagram for an external gauge field vanishes identically. +\item The fermion vertex with two gauge fields is given by +\begin{equation} +- a^2 {g^2\over 2} +\bar\psi_s(q) [\partial^2_\mu S_F^{-1}({q+p\over 2})]_{ss} +A^2_\mu(s,p-q)\psi_s(-p) +\end{equation} +where $A_\mu^2(s,p) = A_\mu (s,p-p_1) A_\mu(s,p_1)$ with $p$ and $p_1$ fixed. +\end{itemize} + +\section{Perturbative Calculations for the Chiral Schwinger Model} +\label{sec:2dim} +In the following two sections +we analyze the chiral Schwinger model +formulated via the Kaplan's method for lattice chiral fermions. +Using the Feynman rules of the previous section +for $D=3$, +we calculate the effective action for external gauge fields, +from which we +derive gauge anomalies, Chern-Simons current, and anomaly of the +fermion number current. +We perform the calculations for the Shamir's method in parallel +with those for the Kaplan's method. + +\subsection{Effective Action at Fermion One-Loop} +Since $ [ A_\mu, A_\nu ]=0$ for U(1) gauge fields, all diagrams with +odd number of external gauge fields vanishes identically. +Furthermore diagrams with four or more external gauge fields +are all convergent. Therefore only the diagrams with two external gauge +fields are potentially divergent. The effective action +for two external gauge fields is denoted by +\begin{eqnarray} +S_{eff}^{(2)} & \equiv & -{g^2\over 2}\sum_{p,s,t} +A_\mu (s,p) A_\nu (t,-p) I^{\mu\nu}(p)_{st} + \nonumber \\ +&=& -{g^2\over 2}\sum_{p,s,t} A_\mu (s,p) A_\nu (t,-p) + [I_a^{(2)} + I_b^{(2)}]^{\mu\nu} _{st}, +\end{eqnarray} +where +\begin{eqnarray} +[I_a^{(2)}]^{\mu\nu} _{st} & = & +\int_{-\pi/a}^{\pi/a} {d^2 q\over (2\pi)^2} +{\rm tr} \left\{[\partial_\mu S_F^{-1}(q+{p\over 2})\cdot S_F(q+p)]_{st} + \right. \nonumber \\ +& \times & \left. +[\partial_\nu S_F^{-1}(q+{p\over 2})\cdot S_F(q)]_{ts}\right\}\times a^2 +\end{eqnarray} +and +\begin{equation} +[I_b^{(2)}]^{\mu\nu} _{st} = -\delta_{st} \delta_{\mu\nu} +\int_{-\pi/a}^{\pi/a} {d^2 q\over (2\pi)^2} +{\rm tr} [\partial_\mu^2 S_F^{-1}(q)\cdot S_F(q)]_{ss} \times a^2 +\end{equation} +with tr meaning trace over spinor indices. + +\subsection{Evaluation of Zero Mode Contributions} +To evaluate $I^{\mu\nu}(p)$ we decompose it into two parts as +\begin{equation} +I^{\mu\nu}(p) = I^{\mu\nu}_0(p)+ [I^{\mu\nu}(p)-I^{\mu\nu}_0(p)] +\end{equation} +where $I^{\mu\nu}_0$ is the contribution of zero modes and +$I^{\mu\nu}-I_0^{\mu\nu}$ is the remaining contribution. +For $I_0^{\mu\nu}$ we replace the integrand of $I^{\mu\nu}$ +with that in the $a\rightarrow 0$ limit, and we obtain +\begin{eqnarray} +I^{\mu\nu}_0(p)_{st} & = & +\sum_X \int_{-\nabla (q)\le 2m_0} {d^2 q\over (2\pi)^2} + \nonumber \\ +& \times & +{\rm tr} \left[ +i\gamma_\mu (-i\gamma_\alpha (q+p)_\alpha a ) G_X^0(q+p)_{st} P_X + \right. \nonumber \\ +& \times & +\left. +i\gamma_\nu (-i\gamma_\beta (q+p)_\beta a ) G_X^0(q+p)_{ts} P_X] +\right] \times a^2 , +\end{eqnarray} +with $X=L$ for $|s|,|t| \approx 0$, or $X=R$ for $|s|,|t| \approx L$. +The zero mode propagators $G_X^0$ are given by +\begin{equation} +G_X^0(q)_{st} = \lim_{a\rightarrow 0} G_X(q)_{st} += {1\over q^2 a^2 } F_X (s,t) +\end{equation} +where $ F_X(s,t) = F_X(t,s)$ and +\widetext +\begin{equation} +F_L(s,t) = {m_0(4-m_0^2)\over 4 } \times \left\{ +\begin{array} {ll} + (1- m_0)^{s+t} & \mbox{\ for $s,t \ge 0$ } \\ + & \\ + (1- m_0)^s(1+m_0)^t & \mbox{\ for $s \ge 0$ and $t < 0$ } \\ + & \\ + (1+m_0) ^{s+t} & \mbox{\ for $s,t \le 0$ } +\end{array} \right. +\end{equation} +\begin{equation} +F_R(s,t) = {m_0(4-m_0^2)\over 4 } \times \left\{ +\begin{array} {ll} + (1- m_0)^{2L-s-t-2} & \mbox{\ for $s,t \ge 0$ } \\ + & \\ + (1- m_0)^{L-s-1} (1+m_0)^{-L-t-1} & \mbox{\ for $s \ge 0$ and $t < 0$ } \\ + & \\ + (1+m_0) ^{-2L-s-t-2} & \mbox{\ for $s,t \le 0$ } +\end{array} \right. +\end{equation} +\narrowtext for the Kaplan's fermion with the domain wall mass terms, and +\begin{mathletters} +\begin{eqnarray} +F_L(s,t) &=& m_0 (2-m_0) (1-m_0)^{s+t} + \\ +F_R(s,t) &=& m_0 (2-m_0) (1-m_0)^{2L-s-t} +\end{eqnarray} +\end{mathletters} +for the Shamir's fermion with the constant mass terms and free boundaries. + +We evaluate $I^{\mu\nu}_0(p)$ in the $a\rightarrow 0$ limit. +In this limit +\begin{eqnarray} +\displaystyle +&\displaystyle \lim_{a\rightarrow 0}& \int_{-\nabla (q)\le 2m_0} +{d^2 q\over (2\pi)^2} +{\rm tr}[P_X\gamma_\mu\gamma_\alpha\gamma_\nu\gamma_\beta ] +{(q+p)_\alpha q_\beta \over (q+p)^2 q^2} + \nonumber \\ +& = & \displaystyle + \int_{-\infty}^{\infty} {d^2 q\over (2\pi)^2} +{\rm tr}[P_X\gamma_\mu\gamma_\alpha\gamma_\nu\gamma_\beta ] +{(q+p)_\alpha q_\beta \over (q+p)^2 q^2} + \nonumber \\ +& = & \displaystyle +{1\over 2\pi}\left[ \delta_X +i\epsilon^{\mu\alpha}{p_\nu p_\mu\over p^2} + +(\delta^{\mu\nu}-{p_\mu p_\nu\over p^2})-{\delta^{\mu\nu}\over 2}\right] , +\end{eqnarray} +therefore we obtain +\begin{eqnarray} +\lim_{a\rightarrow 0} I^{\mu\nu}_0(p) & = & +\sum_X {1\over 2\pi}[\delta_X i\epsilon^{\mu\alpha}{p_\nu p_\mu\over p^2} + +(\delta^{\mu\nu}-{p_\mu p_\nu\over p^2})-{\delta^{\mu\nu}\over 2}] + \nonumber \\ +&\times & F_X(s,t)^2 +\end{eqnarray} +where $\delta_L=1$ and $\delta_R=-1$. +It is noted that $F_X$ satisfies +\begin{equation} +\sum_{t}F_X(s,t)^2 = F_X(s,s) , \qquad +\sum_{s,t}F_X(s,t)^2 =1 . +\label{sumf} +\end{equation} + +\subsection{Evaluation of Remaining Contributions} +We consider the remaining terms in $I^{\mu\nu}$. +Since the combination $I^{\mu\nu}(p) - I^{\mu\nu}_0(p)$ is infra-red finite, +we can change the integration variable from $q$ to $ q a$ +and take the $a\rightarrow 0$ limit in the integrand. Thus we obtain +\begin{equation} +\lim_{a\rightarrow 0} I^{\mu\nu}(p) - I^{\mu\nu}_0(p) + =I^{\mu\nu} (0) = {1\over 2\pi}\cdot +[i \epsilon^{\mu\nu}\Gamma_{CS} +\delta^{\mu\nu} K] +\end{equation} +where +\begin{eqnarray} +\Gamma_{CS}(s,t)& = & {\epsilon^{\mu\nu}\over i} +2\pi +\int {d^2q\over (2\pi)^2} {\rm tr}\left\{ [\partial_\mu S_F^{-1}(q)S_F(q)]_{st} + \right. \nonumber \\ +& \times & \left. +[\partial_\nu S_F^{-1}(q)S_F(q)]_{ts}\right\} +\label{gcs} +\end{eqnarray} +and +\begin{eqnarray} +& \displaystyle K (s,t) = 2\pi +\int {d^2q\over (2\pi)^2}\left[ +{\rm tr}\left\{ [\partial_\mu S_F^{-1}(q)S_F(q)]_{st} + \right. \right. \nonumber \\ +& \times \left. \left. +[\partial_\mu S_F^{-1}(q)S_F(q)]_{ts}\right\} +- \delta_{st}{\rm tr} [\partial_\mu^2 S_F^{-1}(q)S_F(q)]_{ss} \right] . +\end{eqnarray} +Here no summation over $\mu$, $\nu$ is implied. + +The parity-odd term $\Gamma_{CS}$ is the coefficient function of +a 3-dimensional Chern-Simons term in the axial gauge\cite{Dis}, +which satisfies $\Gamma_{CS}(s,t)= - \Gamma_{CS}(t,s)$. +It is easy to show that +\begin{eqnarray} +& \displaystyle \sum_t \Gamma_{CS}(s,t) =-{ \epsilon^{\mu\nu}\over i} +\int {d^2q\over 2\pi} {\rm tr}\left\{ \partial_\mu S_F^{-1}(q)\partial_\nu +S_F(q)\right\}_{ss} + \nonumber \\ +& = \displaystyle {\epsilon^{\mu\nu}\over i} \left[ \int {dq_\mu \over 2\pi} +{\rm tr}\left\{ \partial_\mu S_F^{-1}(q)\cdot S_F(q)\right\}_{ss} +\right]_{q_\nu =\epsilon}^\pi \nonumber +\end{eqnarray} +This would be zero if there were +no infra-red singularities in $S_F$. +However, because of the contribution from zero modes, +$S_F$ is singular at $q=0$. Therefore +\widetext +\begin{eqnarray} +\sum_t \Gamma_{CS}(s,t) & = & - \sum_t\Gamma_{CS}(t,s) + = \displaystyle +4 \lim_{\epsilon\rightarrow 0} \left[ \int_{\epsilon}^{\pi/2} +{dq_\mu\over 2\pi}\sum_X \delta_X +[S_\nu (q) C_\mu (q) G_X^0 (q)]_{ss}\right]_{q_\nu =\epsilon}^{\pi /2} + \nonumber \\ +&=& \displaystyle +-4 \sum_X \delta_X F_X(s,s) +\lim_{\epsilon\rightarrow 0} \int_{\epsilon}^{\pi/2} {dq_\mu\over 2\pi} +{\epsilon\over q_\mu^2 + \epsilon^2} + \nonumber \\ +& = & \displaystyle +-\sum_X 2\delta_X +{F_X(s,s)\over\pi} \left[\tan^{-1}{q\over\epsilon}\right]_{\epsilon}^{\pi/2} + \nonumber \\ +& = & \displaystyle +-\sum_X {\delta_X\over 2} F_X(s,s) , +\label{sum} +\end{eqnarray} +\narrowtext where $F_X(s,t)$ is given in the previous subsection. + +Since $S_F$ becomes the Wilson fermion propagator with constant mass term +for $1 \ll |s|,|t|, |L-s|, |L-t|$ with $s\cdot t > 0$ [see eq.(\ref{wil}) ], +it becomes +\begin{eqnarray} & +\sum_{s,t}\Gamma_{CS}^{\mu\nu}(s,t)A_\mu(s,p) A_\mu(t,-p) +\longrightarrow + \nonumber \\ +& -\epsilon^{\mu\nu} \int dp_3 A_\mu(p_3,p)p_3 A_\mu (-p_3,-p) +\displaystyle \int {d^3 q\over (2\pi)^2} + \nonumber \\ +& \times +{\rm tr}\left\{ [\partial_\mu S_F^{-1}\cdot S_F ] +[\partial_3 S_F^{-1}\cdot S_F ] [\partial_\nu S_F^{-1}\cdot S_F ] +\right\} , +\label{cst} +\end{eqnarray} +which coincides with the result of ref.\cite{CS}. This is a good check of our +calculation. From Ref.\ \cite{CS} we obtain +\begin{equation} += \epsilon^{\mu\nu} \int dp_3 A_\mu(p_3,p)p_3 A_\nu (-p_3,-p) +\times \left\{ +\begin{array}{ll} +\displaystyle 1 & \mbox{\quad for $+m_0$} \\ + & \\ +0 & \mbox{\quad for $-m_0$} +\end{array} . +\right. \nonumber +\end{equation} + +The parity-even term $K$ satisfies +$K(s,t)=K(t,s)$ and +\begin{eqnarray} + & \sum_{t}& K(s,t) = \sum_t K(t,s) + \nonumber \\ +& = & \displaystyle +- \int {d^2 q\over (2\pi)^2} +{\rm tr}\left\{ +[\partial_\mu S_F^{-1}\cdot\partial_\mu S_F]_{ss} + [\partial_\mu^2 S_F^{-1} +\cdot S_F ]_{ss} \right\} + \nonumber \\ + & = & \sum_X {1\over 2}\cdot F_X (s,s) +\label{sumk} +\end{eqnarray} +The derivation of the last equality is similar to that of eq.(\ref{sum}). + + +\subsection{Total Contributions} +Combining the above contributions we finally obtain +\widetext +\begin{eqnarray} +S_{eff}^{(2)} &=& +-{g^2\over 4\pi} \sum_{s,t} \int d^2 x \left\{ +\sum_X F_X(s,t)^2 +[A_\mu(s,x)(\delta^{\mu\nu}-{\partial_\mu\partial_\nu \over \Box}) A_\nu (t,x)] + \right. + \nonumber \\ +&+& +[K(s,t)- \sum_X {1\over 2}\cdot F_X(s,t)^2 ] A_\mu (s,x) A_\mu (t,x) + \nonumber \\ + &+& \left. +\sum_X i\delta_X F_X(s,t)^2 [{\partial_\mu\over \Box } +A_\mu (s,x)\epsilon^{\alpha\nu}\partial_\alpha A_\nu (t,x) ]+ +i\Gamma_{CS}(s,t) \epsilon^{\mu\nu} A_\mu(s,x) A_\nu(t,x) +\right\} . +\label{eff} +\end{eqnarray} +\narrowtext +This is the main result of this paper. +It is noted that the above formula is valid for both the Kaplan's and +the Shamir's methods. +The following consequences can be drawn from eq.\ (\ref{eff}) above. + +The Parity-odd terms, which are proportional to $\epsilon^{\alpha\nu}$, +are unambiguously defined, contrary to the case of the continuum regularization +for anomaly free chiral gauge theories\cite{FS} +which only regulates the parity even terms\cite{Neu,AK}. +These parity-odd terms break gauge invariance in the 2-dimensional sense. + +For {\it anomalous} chiral Schwinger model, +the parity-odd term with $X=R$ is localized around $s=0$ +and that for $X=L$ is localized around $s=L$. +The effective action above for anomalous chiral Schwinger model via +the Kaplan's method or the Shamir's variation is different from the one +via the usual Wilson fermion in 2 dimensions\cite{chiral}: +The term proportional to $\Gamma_{CS}$, which can not be evaluated analytically +for $s$ dependent gauge fields, is special for chiral fermions +from 3-dimensional theories, and the presence of this term prevents us from +concluding whether the anomalous chiral Schwinger model can be consistently +defined via the Kaplan's (Shamir's) method. + +For anomaly free cases such that $\sum_R g_R^2 = \sum_L g_L^2$, +the parity-odd terms are exactly cancelled {\it locally} in $s$ space. +Here $g_{R(L)}$ is the coupling constant of a fermion with positive +(negative) $m_0$ which generate a right-handed (left-handed) zero mode +around $s=0$. The simplest but non-trivial example is +a Pythagorean case, $g_R=3, 4$ and $g_L = 5$\cite{Dis}. +Even for these anomaly free cases, +the longitudinal term, whose coefficient is $K-F^2/2$, +remains non-zero in the effective action, +so that gauge invariance in the {\it 2-dimensional} sense is +violated. +In this regard +the form of the effective action via the Kaplan's +(Shamir's) method is similar to +the one via the usual Wilson fermion\cite{chiral}. + +Let us consider the effective action for $s$ independent gauge fields +as in ref.\cite{Neu}. +Since $\sum_{\mu ,\nu }\epsilon^{\mu\nu} A_\mu(x) A_\nu(x) = 0$ +for $s$ independent gauge fields, the Chern-Simons term vanishes. +The other parity-odd term is cancelled between the two zero modes +since $\sum_{X,s,t} \delta_X F_X(s,t)^2=0$. +The longitudinal term also vanishes due to the identity +\begin{equation} +\sum_{t}[K(s,t)-{F_X(s,t)^2\over 2}] = +\sum_{t}[{F_X(s,t)^2\over 2}-{F_X(s,t)^2\over 2}] = 0 +\end{equation} +[see eqs.\ (\ref{sumf}) and (\ref{sumk})]. +Therefore the effective action becomes +\begin{equation} +S_{eff}^{(2)} = +-2\times {g^2\over 4\pi} \int d^2 x +[ A_\mu(x)(\delta^{\mu\nu}-{\partial_\mu\partial_\nu \over \Box}) A_\nu (x)]. +\end{equation} +This effective action is transverse and thus gauge invariant +in the 2-dimensional sense. Both zero modes around $s=0$ and $s=L$ +equally contribute so that a factor 2 appears in the above result. +This is consistent with the general formula derived in +ref.\cite{Neu2}. +The anomalous chiral Schwinger model can not be simulated +by the Kaplan's (Shamir's) method with the $s$-independent gauge fields, +since the gauge fields see {\it both} of the zero modes so that +it fails to reproduce the parity-odd term, expected to exist\cite{JR} + +\section{Anomalies in the Chiral Schwinger Model} +\label{sec:anomaly} + +\subsection{Currents and their Divergence} + +From the effective action obtained in the previous section, we +can calculate the vacuum expectation values of various currents +in the presence of back-ground gauge fields. +Let us define the fermion number current as +\begin{equation} +\langle J_\mu^g (s, x) \rangle += {\delta S_{eff}^{(2)} \over g\delta A_\mu (s, x) } +\end{equation} +where the index $g$ in the current explicitly shows the charge of the +fermion. +From eq. (\ref{eff}) we obtain +\widetext +\begin{eqnarray} + J_\mu^g(s,x) &=& i{g\over 4\pi}\sum_t + [ \sum_X \delta_X F_X(s,t)^2(\epsilon^{\alpha\nu}\partial_\mu ++\epsilon^{\alpha\mu}\partial_\nu ){\partial_\alpha\over \Box}A_\nu (t, x) +-2\Gamma_{CS}\epsilon^{\mu\nu} A_\nu (t,x)] + \nonumber \\ +&-& {g\over 4\pi}\sum_t +[2\sum_X F_X(s,t)^2 +(\delta^{\mu\nu}-{\partial_\mu\partial_\nu \over \Box}) A_\nu (t,x) ++ +(2 K(s,t)- \sum_X F_X(s,t)^2 ) A_\mu (t,x) ] + \nonumber \\ +& \equiv & J_\mu^{g,odd}+ J_\mu^{g,even} +\label{current} +\end{eqnarray} +\narrowtext +where $J_\mu^{g,odd}$ is a parity-odd current (the first two +terms) and $J_\mu^{g,even}$ is a parity-even current (the last two +terms). Hereafter +all $J_\mu$ should be understood as vacuum expectation values, +though $\langle \qquad\rangle$ is suppressed. +From the fermion number current +the gauge current for a fermion with charge $g$ is easily constructed as +$ J_\mu^G (s,x) \equiv g J_\mu^g (s,x)$. + +Divergences of the parity-odd and parity-even currents become +\begin{eqnarray} +\partial_\mu J_\mu^{g,odd}(s,x) + &= & i {g\over 4\pi} \sum_t[\sum_X \delta_XF_X(s,t)^2- 2\Gamma_{CS}(s,t)] + \nonumber \\ +&\times & \epsilon^{\mu\nu} \partial_\mu A_\nu (t,x) , +\end{eqnarray} +\begin{eqnarray} +\partial_\mu J^\mu_{g,even}(s,x) + & = & {g\over 4\pi} \sum_t [\sum_X F_X(s,t)^2- 2K(s,t)] + \nonumber \\ +&\times & \partial_\mu A_\mu (t,x) . +\end{eqnarray} + +\subsection{Gauge invariance} + +As mentioned in Sect.\ \ref{sec:model}, +the action in $A_3 = 0$ gauge is invariant +under $s$ independent gauge transformation. This invariance implies +$\sum_{s} \partial_\mu J_\mu^G (s,x) = 0$. This identity is satisfied in +our calculation of the effective action, since +\begin{eqnarray} + \sum_s & [\sum_X\delta_XF_X(s,t)^2- 2\Gamma_{CS}(s,t)] + \nonumber \\ +&= \sum_X \delta_X [F_X(t,t)-F_X(t,t)] =0 + \label{sum0a} +\end{eqnarray} +\begin{eqnarray} + \sum_s & [\sum_X F_X(s,t)^2- 2 K(s,t)] + \nonumber \\ +&= \sum_X [F_X(t,t)-F_X(t,t)] =0 +\label{sum0b} +\end{eqnarray} +from eqs.(\ref{sumf},\ref{sum},\ref{sumk}). + +\subsection{Gauge Anomalies} +The gauge anomaly for a fermion with a charge $g$, denoted $T_g$, is defined by +$ T_g = g\partial_\mu J_\mu^{g,odd}$, and it becomes +\begin{equation} +T_g (s,x) = \sum_t g^2 C(s,t)\times T^0(t,x) +\end{equation} +where +\begin{equation} +T^0(t,x) = +i{1\over 4\pi} \epsilon^{\mu\nu} \partial_\mu A_\nu (t,x) +\end{equation} +is the gauge anomaly of a {\it 2-dimensional} theory, and +\begin{equation} +C(s,t) = \sum_X \delta_XF_X(s,t)^2-2 \Gamma_{CS}(s,t) +\end{equation} +represents the spread of the gauge anomaly over the 3rd direction +due to the spread of zero modes. +This spread of the anomaly has been observed in a numerical +computation\cite{Jan}. +It is noted that the divergence of the gauge current $J_\mu^G$ also contains +parity-even contributions, given by +\begin{equation} +\sum_t D(s,t)\times {g^2\over 4\pi} \partial_\mu A_\mu(t,x) +\end{equation} +where $ D(s,t) = \sum_X F_X(s,t)^2-2 K(s,t)$. + +The one-loop integral (\ref{gcs}) defining +$\Gamma_{CS}$ is +too complicated to calculate analytically. +For $t$-independent gauge fields $A_\mu (t,x) = A_\mu (x)$ +there is a considerable simplification and +we obtain +\begin{equation} +T_g (s,x) = g^2C(s)\times T^0(x) +\end{equation} +where +\begin{equation} +T^0(x)= i{1\over 4\pi} \epsilon^{\mu\nu} \partial_\mu A_\nu (x) +\end{equation} +and +\widetext +\begin{eqnarray} +C(s) &=& \sum_t C(s,t) = 2 \sum_X \delta_X F_X(s,s) + \nonumber \\ +&=& {m_0(4-m_0^2)\over 2} +\times \left\{ +\begin{array}{ll} +\left[ (1-m_0)^{2s}-(1-m_0)^{2(L-s-1)} \right] & \mbox{\ for $s \ge 0$} \\ + & \\ +\left[ (1+m_0)^{2s}-(1+m_0)^{-2(L+s+1)}\right] & \mbox{\ for $s \le 0$} +\end{array} +\right. . +\end{eqnarray} +\narrowtext +for the Kaplan's method. We plot $C(s)$ as a function of $s$ +at $m_0= 0.1$ and 0.5 in Fig.\ref{anomalyK}. +For the Shamir's method, we obtain +\begin{equation} +C(s) = 2m_0(2-m_0)\cdot +\left[ (1-m_0)^{2s}-(1-m_0)^{2(L-s)} \right] +\end{equation} +and plot this in Fig.\ref{anomalyS}. +It is noted that there is no parity-even contribution in $\partial_\mu J_\mu^G$ +since $\sum_t D(s,t) = 0$ in this case. + +\subsection{Chern-Simons Current} +From the 3-dimensional point of view, the gauge anomaly should be cancelled +in such a way that $T_g ++ \partial_3 gJ_3^{CS}(s,x)=0$\cite{KP,CS}, where $J_3^{CS}$ is the +third component of the Chern-Simons current for the 3-dimensional +vector gauge theory. With our gauge fixing the effective action does not +depends on $A_3$, and it is difficult to calculate $J_3^{CS}$ analytically +except in the region away from domain-walls\cite{CS}. +However, for $t$-independent gauge fields, +we can obtain the Chern-Simons current everywhere via the relation +$\partial_3 gJ^3_{g,odd}(s,x) = -T_g$, which becomes +\begin{equation} +J_3^{CS}(s+{1\over 2},x)-J_3^{CS}(s-{1\over 2},x) = g^2C(s)\times T^0 (x). +\end{equation} +Taking $J_3^{CS}(s+{1\over 2},x)= g^2 I(s)\cdot T^0 (x)$, we obtain +\begin{equation} +I(s+{1\over 2})-I(s-{1\over 2}) = C(s) . +\end{equation} +We have to solve this equation with the boundary condition +$ I(s)\rightarrow -2$ as $s\rightarrow +\infty$\cite{CS}. +For a finite $s$ space $s\rightarrow +\infty$ means +$ 1\ll s \ll L$. + +For the Kaplan's method we obtain +\widetext +\begin{equation} +I(s-{1\over 2})=\left\{ +\begin{array}{ll} +\displaystyle +{2+m_0\over 2}\cdot [(1-m_0)^{2s}+(1-m_0)^{2(L-s)}]-2 & \mbox{\quad +$0\le s\le L$} \\ + & \\ +\displaystyle -{2-m_0\over 2}\cdot +[(1+m_0)^{2s}+(1+m_0)^{-2(L+s)}] & \mbox{\quad $ -L\le s\le 0$} +\end{array} +\right. . +\end{equation} +\narrowtext +This solution automatically satisfies the other boundary condition +that $ I(s)\rightarrow 0$ as $\rightarrow -\infty$\cite{CS}. +Again $\rightarrow -\infty$ means $ 1\ll -s \ll L$ for a finite $s$ space. +We plot $I(s)$ as a function of $s$ at $m_0=0.1$ and 0.5 in Fig.\ \ref{CSK}. +For the Shamir's method we obtain +\begin{eqnarray} +I(s-{1\over 2}) &= 2[(1-m_0)^{2s}+(1-m_0)^{2(L-s+1)}]-2 + \nonumber \\ + & \qquad \qquad \mbox{for $0\le s\le L-1$}, +\end{eqnarray} +which is plotted in Fig.\ \ref{CSS}. + +\subsection{Pythagorean Chiral Schwinger Model and +Anomaly in Fermion Number Current} + +Let us consider the Pythagorean chiral Schwinger model\cite{Dis}. +In this model there are +two right-handed fermions with charges $g_1$ and $g_2$, and +one left-handed fermion with charge $g_3$. +Formulation of this model via the Kaplan's method has already been discussed +in ref.\cite{KP,Jan,Dis} +( an extension to the Shamir's method is straightforward ). +We assign $+m_0$ for fermions with charge $g_1$ and $g_2$, and +$-m_0$ for a fermion with charge $g_3$. The value of $|m_0|$ should +be equal for all fermions, as will be seen below. + +The theory has a $U(1)^3$ symmetry\cite{Dis} corresponding to independent phase +rotations of three fermions. The corresponding currents are +\begin{equation} +\left\{ +\begin{array}{lll} +J_\mu^G &=& g_1 J_\mu^{g_1}+g_2 J_\mu^{g_2}+g_3 J_\mu^{g_3} \\ + & & \\ +J_\mu^R &=& g_2 J_\mu^{g_1}-g_1 J_\mu^{g_2} \\ + & & \\ +J_\mu^F &=& J_\mu^{g_1}+J_\mu^{g_2}+J_\mu^{g_3} +\end{array} +\right. +\end{equation} +The first one is the gauge current, whose divergence becomes +\begin{equation} +\partial_\mu J_\mu^G(s,x) = (g_1^2+g_2^2-g_3^2)\times \sum_t C(s,t)\cdot + T^0(t,x). +\end{equation} +Therefore, if $g_1^2+g_2^2 = g_3^2$ (Pythagorean relation) is satisfied +and if all fermions +have the same value of $|m_0|$ to give the same $C(s,t)$, +this current is conserved and there is no gauge anomaly for any +background gauge fields. +The second current is non-anomalous, since +\begin{equation} +\partial_\mu J_\mu^R (s, x)=(g_2\cdot g_1 - g_1\cdot g_2)\sum_t +C(s,t)\cdot T^0(t,x) = 0 . +\end{equation} + +The third current, which corresponds to the fermion number of the theory, +is anomalous, since +\begin{equation} +\partial_\mu J_\mu^F (s, x)=( g_1 + g_2 - g_3)\times\sum_t C(s,t) \cdot +T^0(t,x) . +\end{equation} +The Kaplan's method as well as the Shamir's one +successfully give a non-zero divergence for the fermion number current, though +the coefficient $C(s,t)$ has a finite width. +For $t$-independent gauge fields, this anomaly becomes +\begin{equation} +( g_1 + g_2 - g_3)\cdot C(s)\cdot T^0(x) +\end{equation} +where $C(s)$ is almost localized at $s=0$ and at $s=L$ as seen in +Fig.\ref{anomalyK} and Fig.\ref{anomalyS}. +Since the fermion number is conserved in the 3-dimensional +theory, the third component of the fermion number current should satisfy +$\partial_3 J_3^F + ( g_1 + g_2 - g_3)\cdot C(s)\cdot T^0(x) = 0$\cite{Dis}. +Therefore we obtain +\begin{equation} +J_3^F(s,x)= (g_1+g_2-g_3)\cdot I(s)\cdot T^0(x) . +\end{equation} + +\section{conclusions} +\label{sec:concl} +In this paper we have formulated a lattice perturbative expansion for +the Kaplan's chiral fermion theories, extending the suggestion +by Narayanan and Neuberger\cite{Neu}. +Applying our perturbative technique to the chiral Schwinger model +formulated via the Kaplan's or the Shamir's method, +we have calculates the fermion one-loop effective action for gauge fields. +The effective action contains parity-odd terms and +longitudinal terms, both of which break 2-dimensional gauge invariance, +and the anomaly of the gauge current is obtained from the effective action. +The gauge anomaly is calculable in the Kaplan's (Shamir's) +method if the perturbative expansion is carefully formulated. +For the anomaly-free Pythagorean chiral Schwinger model, +the fermion number current is anomalous. To obtain this anomaly +the fermion number current should not be summed over $s$, in contrast +to the case of the continuum calculation\cite{AK}, where +the anomaly comes from an infinite summation over $s$. + +The main conclusions drawn from the results are as follows. + +\begin{enumerate} +\item +Anomaly of the fermion number current is shown to be +non-zero in this method, though the current flows off walls +into the extra dimension. +Since the current is external we feel that this +does not affect the dynamics of the model +and therefore does not spoil the 2-dimensional nature of the chiral +zero mode, contrary to the suggestion of Ref.\cite{Dis}. +The 3-dimensional nature of the Kaplan's (Shamir's) +formulation manifests itself only in the +non-conservation of the fermion number, which is expected to occur in Nature. + +\item Two-dimensional gauge invariance at low energy +can not be assured by the Kaplan's (Shamir's) method, +except for $s$-independent +gauge fields, even for anomaly-free cases. +This is similar to the situation with +lattice chiral gauge theories formulated with the ordinary Wilson mass +term\cite{chiral}. +In this point the Kaplan's (Shamir's) method does not seem better than the +conventional approaches. +At this moment it is not clear whether this violation of gauge invariance +spoils the whole program of this method. In particular the effects of the +longitudinal component of gauge fields has to be analyzed further. + +\item If the theory is anomaly free and gauge fields are +$s$-independent\cite{Neu}, the gauge invariance as a 2-dimensional theory +can be maintained. However, the gauge fields feel both of the zero modes +even in the $L\rightarrow \infty$ limit, and the fermion loop contribution +to the effective action is twice as large as the one expected from a single +chiral fermion. Therefore we have to take a square-root of +the fermion determinant +to obtain the correct contribution. +For fermion quantities such as the fermion number current, however, +it seems possible +to separate the contribution of the chiral zero mode at $s=0$ from +that of the anti-chiral zero mode at $s=L$, as seen in the previous section. +\end{enumerate} + +Perturbative calculations performed in +this paper can be extended to 4+1 dimensional theories. Of course +actual calculations become much more complicated and difficult +because of severe ultra-violet divergences in 4+1 dimensions than +in 2+1 dimensions. +Work in this direction is in progress. + +\acknowledgements +We would like to think Prof. Ukawa for discussions and the careful reading of +the manuscript. + +After finishing this work, a new paper by Narayanan and Neuberger\cite{Neu3} +appeared. In the paper +the gauge anomaly for the chiral Schwinger model was calculated +semi-analytically via the overlap formula of ref.\cite{Neu2}. + + +\begin{references} +\bibitem{KP}D.\ B.\ Kaplan, Phys.\ Lett.\ {\bf B288}, 342 (1992). + +\bibitem{Jan}K.\ Jansen, Phys.\ Lett.\ {\bf B288}, 348 (1992). + +\bibitem{CS}M.\ F.\ L.\ Golterman, K.\ Jansen, and D.\ B.\ Kaplan, +Phys.\ Lett.\ {\bf B301}, 219 (1993). + +\bibitem{FS}S.\ A.\ Frolov and A.\ A.\ Slavnov, +Phys.\ Lett.\ {\bf B309}, 344 (1993). + +\bibitem{Neu}R.\ Narayanan and H.\ Neuberger, +Phys.\ Lett.\ {\bf B302}, 62 (1993). + +\bibitem{Neu2}R.\ Narayanan and H.\ Neuberger, RU-93-25, +Rutgers University preprint, July 1993. + +\bibitem{AK}S.\ Aoki and Y.\ Kikukawa, UTHEP-258/KUNS-1204, +University of Tsukuba preprint, June 1993. + +\bibitem{Alt}C.\ P.\ Korthals-Altes, S.\ Nicolis and J.\ Prades, +CPT-93/P.2920, Center de Physique Th\'{e}orique preprint, June 1993. + +\bibitem{Dis}J.\ Distler and S.-J.\ Rey, PUPT-1386/NSF-ITP-93-66/SNUTP 93-27, +Princeton University preprint, May 1993. + +\bibitem{Sham}Y.\ Shamir, WIS-93/20/FEB-PH, Weizmann Institute preprint, +February 1993. + +\bibitem{chiral}S.\ Aoki, Phys.\ Rev\ Lett.\ {\bf 60} 2109 (1988); +K.\ Funakubo and T.\ Kashiwa,{\it ibid} {\bf 60} 2113 (1988); +T.\ D.\ Kieu, D.\ Sen, S.-S.\ Xue, {\it ibid} {\bf 60} 2117 (1988); +S.\ Aoki, Phys.\ Rev.\ {\bf D38} 618 (1988). + +\bibitem{JR}R.\ Jackiw and R.\ Rajaraman, +Phys.\ Rev\ Lett.\ {\bf 54} 1219 (1985). + +\bibitem{Neu3}R.\ Narayanan and H.\ Neuberger, RU-93-34, +Rutgers University preprint, August 1993. + +\end{references} + +\newpage + +\begin{figure} +\caption{ Two zero modes $u_L$ and $u_R$ +as a function of $s$ at +$m_0=0.1$ and 0.5 for $p_1=p_2=0$. We take $L=100$.} +\label{zero} +\end{figure} + +\begin{figure} +\caption{ The coefficient of the anomaly $C(s)$ +for the Kaplan's method as a function of $s$ +at $m_0=0.1$ and 0.5 for $L=100$.} +\label{anomalyK} +\end{figure} + +\begin{figure} +\caption{ The coefficient of the anomaly $C(s)$ +for the Shamir's method +as a function of $s$ at $m_0=0.1$ and 0.5 for $L=100$.} +\label{anomalyS} +\end{figure} + +\begin{figure} +\caption{The coefficient of the Chern-Simons current $I(s)$ +for the Kaplan's method as a function of $s$ +at $m_0=0.1$ and 0.5 for $L=100$.} +\label{CSK} +\end{figure} + +\begin{figure} +\caption{ The coefficient of the Chern-Simons current $I(s)$ +for the Shamir's method as a function of $s$ +at $m_0=0.1$ and 0.5 for $L=100$.} +\label{CSS} +\end{figure} + +\end{document} +\bye |