summaryrefslogtreecommitdiff
path: root/support/hypertex/tanmoy/examples/Anomalkap.tex
diff options
context:
space:
mode:
Diffstat (limited to 'support/hypertex/tanmoy/examples/Anomalkap.tex')
-rw-r--r--support/hypertex/tanmoy/examples/Anomalkap.tex1252
1 files changed, 1252 insertions, 0 deletions
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