\noindent I introduce two new boundary layers to the optimal domain-wall fermion action, and formulate vector gauge theories (e.g., QCD) with the new action. The generating functional for $n$-point Green's function of quark fields is derived. The bilinear and higher-order quark observables are shown to be $a_5$-invariant, to possess optimal chiral symmetry for any finite $ N_s $, as well as to preserve continuum discrete symmetries at finite lattice spacing. With 2D abelian background gauge fields, I compute the index, the axial anomaly, and the fermion determinant (ratio) of the effective lattice Dirac operator, for the optimal DWF and the conventional DWF respectively. All quantities of the optimal DWF are in good agreement with the exact solution, but the fermion determinant (ratio) of the conventional DWF disagrees with the exact solution, for topologically nontrivial sectors. \vskip 2.0cm \noindent PACS numbers: 11.15.Ha, 11.30.Rd, 12.38.Gc \noindent Keywords : Lattice QCD, Domain-wall fermions, Overlap Dirac operator, Zolotarev optimal rational approximation.
hep-lat0303008.tex
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\begin{flushright}
NTUTH-03-505A \\
March 2003
\end{flushright}
\bigskip\bigskip\bigskip
\vskip 2.5truecm
\begin{center}
\LARGE {Aspects of domain-wall fermions on the lattice}
%\vskip 7pt
\end{center}
\vskip 0.5truecm
\centerline{Ting-Wai Chiu}
\vskip5mm
%\centerline{Department of Physics, University of Washington}
%\centerline{Seattle, WA 98195-1560, USA}
%\smallskip
%\centerline{and}
%\smallskip
\centerline{Department of Physics, National Taiwan University}
\centerline{Taipei, Taiwan 106, Taiwan.}
\centerline{\it E-mail : twchiu@phys.ntu.edu.tw}
\vskip 0.5cm
\bigskip \nopagebreak \begin{abstract}
\noindent

I introduce two new boundary layers to the optimal domain-wall fermion
action, and formulate vector gauge theories (e.g., QCD) with the
new action. The generating functional for $n$-point Green's function of
quark fields is derived.
The bilinear and higher-order quark observables are shown to
be $a_5$-invariant, to possess optimal chiral symmetry
for any finite $ N_s $, as well as to preserve continuum discrete
symmetries at finite lattice spacing.
With 2D abelian background gauge fields, I compute the index,
the axial anomaly, and the fermion determinant (ratio) of the effective
lattice Dirac operator, for the optimal DWF and the conventional DWF
respectively. All quantities of the optimal DWF are in good agreement
with the exact solution, but the fermion determinant (ratio)
of the conventional DWF disagrees with the exact solution,
for topologically nontrivial sectors.


\vskip 2.0cm
\noindent PACS numbers: 11.15.Ha, 11.30.Rd, 12.38.Gc

\noindent Keywords : Lattice QCD, Domain-wall fermions,
Overlap Dirac operator, Zolotarev optimal rational approximation.


\end{abstract}
\vskip 1.5cm

\newpage\setcounter{page}1

%\baselineskip=24pt

\section{Introduction}


The basic idea of domain-wall fermions (DWF) \cite{Rubakov:bb,Callan:sa}
is to use an infinite set of coupled Dirac fermion fields
[ $ \psi_s (x), s \in (-\infty, \infty) $ ] with masses
behaving like a step function $ m(s) = m \theta(s) $ such that Weyl 
fermion states can arise as zeromodes bound to the mass defect 
at $ s = 0 $. However, if one uses a compact set of masses, then
the boundary conditions of the mass (step) function must
lead to the occurrence of both left-handed and right-handed
chiral fermion fields, i.e., a vector-like theory.
For lattice QCD with DWF \cite{Kaplan:1992bt}, in practice,
one can only use a finite number ($ N_s $) of
lattice Dirac fermion fields to set up the domain wall, thus the
chiral symmetry of the quark fields (in the massless limit) is broken.
Therefore, a relevant question is how to construct
a domain-wall fermion action such that the effective 4D lattice Dirac
operator has the optimal chiral symmetry for any finite $ N_s $.
Nevertheless, the chiral symmetry is not the only concern of lattice QCD
with domain-wall fermions.
Another crucial question is whether the effective 4D lattice Dirac operator
depends on the lattice spacing $ a_5 $ in the fifth dimension.
If the dependence on $ a_5 $ is strong, it is unclear what is the proper
value of $a_5$ for measuring physical observables, assuming $ N_s $ and
the lattice spacing $a$ are fixed.

Recently, a lattice domain-wall fermion action has been constructed
\cite{Chiu:2002ir} such that the effective 4D lattice Dirac operator
preserves the chiral symmetry optimally for any finite $ N_s $,
and is invariant for any $ a_5 $.
Further, it has been asserted that the effective 4D lattice Dirac
operator for the internal fermion loops is exponentially-local
for sufficiently smooth gauge backgrounds \cite{Chiu:2002kj}.

Explicitly, the action of optimal lattice domain-wall fermions
reads\footnote{In this paper, we suppress the lattice spacing $a$,
as well as the Dirac and color indices, which can be restored easily.}
\bea
\label{eq:ODWF}
{\cal A}_f &=& \sum_{s,s'=1}^{N_s} \sum_{x,x'}
\bar\psi(x,s)
\{ (\omega_s a_5 D_w(x,x') + \delta_{x,x'} )\delta_{s,s'} \nn
&& \hspace{10mm}   -(\delta_{x,x'}-\omega_s a_5 D_w(x,x'))
    (P_{-} \delta_{s',s+1} + P_{+} \delta_{s',s-1}) \} \psi(x',s') \nn
&\equiv& \sum_{s,s'=1}^{N_s} \sum_{x,x'}
    \bar\psi(x,s) {\cal D}(x,s;x's') \psi(x',s'),
    \hspace{6mm} P_{\pm} = \frac{1 \pm \gamma_5}{2}
\eea
with boundary conditions
\bea
\label{eq:bc1}
P_{+} \psi(x,0) &=& - r m_q P_{+} \psi(x,N_s) \ ,
\hspace{4mm} r = \frac{1}{2m_0} \\
\label{eq:bc2}
P_{-} \psi(x,N_s+1) &=& - r m_q P_{-} \psi(x,1) \ ,
\eea
where $ m_q $ is the bare quark mass, $ D_w $ is
the standard 4D Wilson-Dirac operator plus a negative parameter,
\bea
\label{eq:Dw}
D_w &=& \sum_{\mu=1}^4 \gamma_\mu t_\mu + W - m_0,
                       \hspace{6mm} m_0 \in (0,2) \\
t_\mu(x,x') &=& \frac{1}{2} \left[  U_\mu(x) \delta_{x',x+\mu}
                   - U_\mu^{\dagger}(x') \delta_{x',x-\mu} \right] \ , \\
W(x,x') &=& \sum_{\mu=1}^4 \frac{1}{2} \left[ 2 \delta_{x,x'}
                          - U_\mu(x) \delta_{x',x+\mu}
                          - U_\mu^{\dagger}(x') \delta_{x',x-\mu} \right] \ ,
\eea
and the weights
\bea
\label{eq:omega}
\omega_s a_5 = \frac{1}{\lambda_{min}} \sqrt{ 1 - \kappa'^2 \mbox{sn}^2
               \left( v_s ; \kappa' \right) }, \hspace{6mm} s=1,\cdots,N_s \ .
\eea
Here $ \mbox{sn}( v_s; \kappa' ) $ is the Jacobian elliptic function
with argument $ v_s $ (\ref{eq:vs}) and modulus
$ \kappa' = \sqrt{ 1 - \lambda_{min}^2 / \lambda_{max}^2 } $,
where $ \lambda_{max} $ and $ \lambda_{min} $ are the maximum
and the minimum of the eigenvalues of $ | H_w | $,
\bea
\label{eq:Hw}
H_w = \gamma_5 D_w \ .
\eea

The weights in (\ref{eq:omega}) are obtained from
the roots $ (u_s = (\omega_s a_5)^{-2}, s=1,\cdots,N_s) $ of the equation
\bea
\label{eq:delta_Z}
\delta_Z(u) =
     \left\{ \begin{array}{ll}
 1-\sqrt{u} R_Z^{(n,n)}(u)=0 \ ,   & \ N_s=2n+1 \nn
 1-\sqrt{u} R_Z^{(n-1,n)}(u)=0 \ , & \ N_s=2n   \nn
             \end{array} \right.
\eea
such that the operator $ S_{opt} $ (\ref{eq:S_opt})
in the effective 4D lattice Dirac operator is equal to the optimal rational
approximation of sgn($H_w$)=$ H_w (H_w^2)^{-1/2} $,
\bea
S_{opt} = S_{opt}(H_w)
= \left\{ \begin{array}{ll}
           H_w R_Z^{(n,n)}(H_w^2),   &  N_s = 2n+1,   \\
           H_w R_Z^{(n-1,n)}(H_w^2), &  N_s = 2n.     \\
          \end{array} \right.
\label{eq:S_opt_RZ}
\eea
where $ R_Z(H_w^2) $ is the Zolotarev optimal rational polynomial
\cite{Zol:1877,Akhiezer:1992} for the inverse square root of $ H_w^2 $,
\bea
\label{eq:rz_nn}
R^{(n,n)}_Z(H_w^2) = \frac{d_0}{\lambda_{min}}
\prod_{l=1}^{n} \frac{ 1+ h_w^2/c_{2l} }{ 1+ h_w^2/c_{2l-1} } \ ,
\hspace{8mm}  h_w^2 = H_w^2/\lambda_{min}^2
\eea
and
\bea
\label{eq:rz_n1n}
R^{(n-1,n)}_Z(H_w^2) = \frac{d'_0}{\lambda_{min}}
\frac{ \prod_{l=1}^{n-1} ( 1+ h_w^2/c'_{2l} ) }
     { \prod_{l=1}^{n} ( 1+ h_w^2/c'_{2l-1} ) } \ ,
\eea
and the coefficients $ d_0 $, $ d'_0 $, $ c_l $ and $ c'_l $
are expressed in terms of elliptic functions \cite{Akhiezer:1992}
with arguments depending only on $ N_s $
and $ \lambda_{max}^2 / \lambda_{min}^2 $.

The argument $ v_s $ in (\ref{eq:omega}) is
\bea
\label{eq:vs}
v_s &=& (-1)^{s-1} M \ \mbox{sn}^{-1}
    \left( \sqrt{\frac{1+3\lambda}{(1+\lambda)^3}}; \sqrt{1-\lambda^2} \right)
  + \left[ \frac{s}{2} \right] \frac{2K'}{N_s}
\eea
where
\bea
\label{eq:lambda}
\lambda &=&
\prod_{l=1}^{N_s}
\frac{\Theta^2 \left(\frac{2lK'}{N_s};\kappa' \right)}
     {\Theta^2 \left(\frac{(2l-1)K'}{N_s};\kappa' \right)} \ , \\
\label{eq:M}
M &=&
\prod_{l=1}^{[\frac{N_s}{2}]}
\frac{\mbox{sn}^2 \left(\frac{(2l-1)K'}{N_s};\kappa' \right) }
{ \mbox{sn}^2 \left(\frac{2lK'}{N_s};\kappa' \right) } \ ,
\eea
$ K' $ is the complete elliptic integral of the first kind with
modulus $ \kappa' $, and $ \Theta $ is the elliptic theta function.
From (\ref{eq:omega}), it is clear that
$ \lambda_{max}^{-1} \le a_5 \omega_s \le \lambda_{min}^{-1} $,
since $ \mbox{sn}^2(;) \le 1 $.
%The formulas (\ref{eq:omega}) and (\ref{eq:vs})
%will be derived in this paper.

Obviously, the optimal domain-wall fermion action (\ref{eq:ODWF})
is invariant for any $ a_5 $,
since its dependence on $ a_5 $ only through the product $ \omega_s a_5 $
(\ref{eq:omega}) which only depends on $ N_s $, $ \lambda_{min} $ and
$ \lambda_{max}^2 / \lambda_{min}^2 $. Therefore, {\it $ a_5 $ is a
redundant parameter in the optimal domain-wall fermion action}
(\ref{eq:ODWF}).

Now the effective 4D lattice Dirac operator of the optimal domain-wall
fermion action (\ref{eq:ODWF}) can be extracted from its fermion
determinant
\bea
\label{eq:det_5}
\det {\cal D}(m_q) = \int [d \bar\psi] [d \psi]
e^{-{\cal A}_f[\psi,\bar\psi]} \ ,
\eea
where $ {\cal D}(m_q) $ is the 5-dimensional optimal DWF operator
defined in (\ref{eq:ODWF}). The Grassman integral on the
r.h.s. of (\ref{eq:det_5}) can be evaluated by transforming
$ \{ \psi_s, \bar\psi_s \} $ to $ \{ \eta_s, \bar\eta_s \} $
according to
\bea
\label{eq:f1i}
&& \hspace{-8mm}
\psi_1 = P_{-} \eta_1 + P_{+} \eta_2 \ , \hspace{4mm}
\bar\psi_1 = \bar\eta_1 (\omega_1 a_5 H_w - 1)^{-1} \gamma_5 \ , \\
\label{eq:f2i}
&& \hspace{-8mm}
\psi_s = P_{-} \eta_s + P_{+} \eta_{s+1} \ , \hspace{4mm}
\bar\psi_s = \bar\eta_s (\omega_s a_5 H_w - 1)^{-1} \gamma_5 \ ,
           s=2,\cdots, N_s-1 \\
\label{eq:f3i}
&& \hspace{-8mm}
\psi_{N_s} = P_{-} \eta_{N_s} + P_{+} \eta_1 \ , \hspace{4mm}
\bar\psi_{N_s} = \bar\eta_{N_s} (\omega_{N_s} a_5 H_w - 1)^{-1} \gamma_5 \ ,
\eea
and then integrating $ \{ \eta_s, \bar\eta_s \} $
successively from $ s=N_s $ to $ s = 1 $.
The result (will be derived in Section 2) is
\bea
\label{eq:det_54}
\det {\cal D}(m_q) = K \ \det [r(D_c + m_q)]
\eea
where
\bea
\label{eq:K}
K &=& \prod_{s=1}^{N_s} \det \left[ \gamma_5 (\omega_s a_5 H_w - 1) \right]
      \det \left(-P_{+} + \prod_{s=1}^{N_s} T_s^{-1} P_{-} \right) \ , \\
\label{eq:rDc}
r D_c &=& \frac{1+\gamma_5 S_{opt}}{1-\gamma_5 S_{opt}}  \ , \\
\label{eq:S_opt}
    S_{opt} &=& \frac{ 1 - \prod_{s=1}^{N_s} T_s}
                            { 1 + \prod_{s=1}^{N_s} T_s} \ , \\
\label{eq:Ts}
T_s &=&  \frac{1-\omega_s a_5 H_w}{1+ \omega_s a_5 H_w}
         \ , \hspace{4mm} s = 1, \cdots, N_s \ .
\eea

Even though $ D_c $ is chirally symmetric in the limit $ N_s \to \infty $,
one still cannot identify $ ( D_c + m_q ) $ as the effective 4D
lattice Dirac operator for the internal quark fields,
since $ D_c $ is nonlocal and $ \det(D_c + m_q) $
does not reproduce $ \det [ \gamma_\mu ( \partial_\mu + i A_\mu ) + m_q ] $
in the continuum limit $ (a \to 0) $.
(It is understood that the fermion determinant is normalized by
its free fermion limit.) Also, one cannot neglect the factor $ K $
which can be written as the determinant of a 4D lattice fermion operator.

The resolution is to introduce a set of pseudofermions (Pauli-Villars) fields
$ \{ \phi, \bar\phi \} $ similar to the domain-wall fermions but obeying
Bose statistics. Explicitly, the action of the pseudofermion fields reads
\bea
\label{eq:ODWF_pf}
& & {\cal A}_{pf} = \sum_{s,s'=1}^{N_s} \sum_{x,x'}
\bar\phi(x,s) \{ (\omega_s a_5 D_w(x,x') + \delta_{x,x'} )\delta_{s,s'} \nn
&& \hspace{8mm}  -(\delta_{x,x'}-\omega_s a_5 D_w(x,x'))
    (P_{-} \delta_{s',s+1} + P_{+} \delta_{s',s-1}) \} \phi(x',s')
\eea
with boundary conditions
\BAN
P_{+} \phi(x,0) = -P_{+} \phi(x,N_s), \hspace{6mm}
P_{-} \phi(x,N_s+1) = -P_{-} \phi(x,1) \ ,
\label{eq:bc}
\EAN
where $ m_q = 2 m_0 $ has been imposed. Then, we have
\BAN
  \det [ {\cal D}(m_q) {\cal D}^{-1}(2m_0) ]
&=& \int [d \bar\psi] [d \psi] [d \bar\phi] [d \phi]
    e^{-{\cal A}_f[\psi,\bar\psi]-{\cal A}_{pf}[\phi,\bar\phi]}  \nn
&=& \det [ r ( D_c + m_q ) ( 1 + r D_c )^{-1} ] \ .
\EAN
Thus the effective 4D lattice Dirac operator for the internal quark
fields (dressed with pseudofermions) can be identified as
\bea
\label{eq:Dm}
D(m_q) = (D_c + m_q)(1 + r D_c)^{-1}
       = m_q + \left(m_0-m_q/2\right)
               \left( 1 + \gamma_5 H_w R_Z(H_w) \right)
\eea
where $ R_Z(H_w^2) $ is the Zolotarev rational polynomial defined
in (\ref{eq:S_opt_RZ})-(\ref{eq:rz_n1n}).
The exponential locality of $ D(m_q) $
(for any $ m_q $ and $ N_s $) has been asserted for
sufficiently smooth gauge backgrounds \cite{Chiu:2002kj}.

In the limit $ N_s \to \infty $, $ S_{opt} $ becomes
the sign function of $ H_w $
\BAN
\lim_{N_s \to \infty} S_{opt} = \frac{H_w}{\sqrt{H_w^2}} \ ,
\EAN
and $ D_c $ (\ref{eq:rDc}) is exactly chirally symmetric
\bea
\label{eq:exact}
\lim_{N_s \to \infty } (D_c \gamma_5 + \gamma_5 D_c) = 0 \ ,
\eea
which is equivalent to
\bea
\label{eq:exact_v}
\lim_{ \stackrel{N_s \to \infty}{m_q \to 0} }
[ (D_c + m_q)^{-1} \gamma_5 + \gamma_5 (D_c + m_q)^{-1} ] = 0 \ ,
\eea
the chiral symmetry of the valence quark propagator $ (D_c + m_q)^{-1} $
in the massless limit.
Therefore one can use this chirally symmetric (but nonlocal) $ D_c $ to
construct an exponentially-local lattice Dirac operator \cite{Chiu:1998gp}
\bea
\label{eq:D}
D = D_c ( 1 + r D_c )^{-1}
\eea
which satisfies the Ginsparg-Wilson relation \cite{Ginsparg:1981bj}
\bea
\label{eq:gwr}
D \gamma_5 + \gamma_5 D = 2 r D \gamma_5 D \ ,
\eea
and its massive generalization \cite{Chiu:1998eu}
\bea
\label{eq:Dm_GW}
D(m) = (D_c + m )(1 + r D_c )^{-1} \ ,
\eea
which is exactly in the same form of the effective 4D lattice Dirac
operator (\ref{eq:Dm}) for any $ N_s $.

Obviously, the optimal DWF action (\ref{eq:ODWF})
in the limit $ N_s \to \infty $ realizes
the exact chiral symmetry (\ref{eq:exact_v}) as well as the
Ginsparg-Wilson relation (\ref{eq:gwr}).
Further, for any finite $ N_s $, the action (\ref{eq:ODWF}) preserves
the chiral symmetry optimally since the deviation of
$ S_{opt} $ from the sign function
%$ S_{opt}(x) $ (\ref{eq:S_opt_RZ}) from the sign function
%$ \mbox{sgn}(x)=1, x > 0 $ is $ \delta(x) = 1 - S_{opt}(x) $ which has
%$ N_s + 1 $ alternate change of sign in the interval $ [1, b] $,
%and attains its maxima and minima
%(all with equal magnitude $ (1-\lambda)/(1+\lambda) $)
%at consecutive points $ 1 = x_1 < x_2 < \cdots < x_{N_s+1} = b $,
is the minimum, and has a theoretical upper bound \cite{Chiu:2002eh},
\bea
\label{eq:Delta_Z}
\Delta_Z = \max_{ \forall \ Y \ne 0 }
\left|\frac{Y^{\dagger} S_{opt}^2 Y}{Y^{\dagger}Y}-1 \right| \le
 \frac{2( 1 - \lambda )}{1+\lambda} \equiv B_Z(N_s, b)
\eea
where $ \lambda $ is defined in (\ref{eq:lambda}), a function
of $ N_s $ and $ b = \lambda_{max}^2 / \lambda_{min}^2 $.
Thus, for any given gauge configuration, one can use
the theoretical error bound to determine what values of $ N_s $ and $ b $
(i.e., how many low-lying eigenmodes of $ H_w^2 $ should be projected out)
are required to attain one's desired accuracy in preserving the exact
chiral symmetry.
In practice, with $ N_s = 32 $, one should have no difficulties to
achieve $ \Delta_Z < 10^{-12} $ for any gauge configurations
on a finite lattice (say, $ 16^3 \times 32 $ at $ \beta = 6.0 $).
The theoretical error bound can be estimated as
\BAN
\label{eq:BZ}
B_Z(N_s,b) = A(b) e^{-c(b) N_s}
\EAN
where
\BAN
\label{eq:cb}
c(b) &\simeq& 4.27(45) \ \ln(b)^{-0.746(5)} \ , \\
\label{eq:Ab}
A(b) &\simeq& 8.11(1) \ b^{-0.0091(1)} \ln(b)^{0.0042(3)} \ .
\EAN

Note that $ D $ (\ref{eq:D}) ($N_s \to \infty$) times a factor
$ r = 1/2m_0 $ is exactly equal to the overlap
Dirac operator \cite{Neuberger:1998fp,Narayanan:1995gw},
\bea
\label{eq:D_o}
D_o = \frac{1}{2} \left( 1 + \gamma_5 \frac{H_w}{\sqrt{H_w^2}} \right) \ .
\eea
This implies that $ D $ is topologically-proper
(i.e., with the correct index and axial anomaly),
similar to the case of overlap Dirac operator.
For any finite $ N_s $, $ r $ times $ D(m_q=0) $ (\ref{eq:Dm})
is exactly equal to the overlap Dirac operator $ D_o $ with
$ (H_w^2)^{-1/2} $ approximated by Zolotarev optimal rational polynomial,
(\ref{eq:rz_nn}) ($N_s$ = odd), or (\ref{eq:rz_n1n}) ($ N_s $ = even).

Athough the effective 4D lattice Dirac operator of the optimal DWF action
is $a_5$-invariant and preserves optimal chiral symmetry for any finite
$N_s$, the correlation function of quark fields $ q(x) $
and $ \bar q (y) $ is not equal to the quark propagator
$ (D_c + m_q)^{-1}_{x,y} $, nor the inverse
of the effective 4D lattice Dirac operator, $ D(m_q)_{x,y}^{-1} $.
This is unsatisfactory from the theoretical viewpoint.
In this paper, I introduce two new boundary layers to the optimal
DWF action, and derive the generating functional for
$n$-point Green's function of quark fields $ q $ and $ \bar q $.
Then the bilinear (and higher-order) quark operators
manifest their continuum discrete symmetries,
besides the $ a_5 $-invariance
and optimal chiral symmetry for any finite $ N_s $.
With 2D abelian background gauge fields, I compute the index,
the axial anomaly, and the fermion determinant (ratio) of the effective
lattice Dirac operator, for the optimal DWF and the conventional DWF
respectively. All quantities of the optimal DWF are in good agreement
with the exact solution, but the fermion determinant (ratio)
of the conventional DWF disagrees with the exact solution,
for topologically nontrivial sectors.

The outline of this paper is as follows. In Section 2, we derive
the generating functional for $n$-point Green's function of quarks
fields in the optimal DWF as well as the conventional DWF,
and discuss the issues pertainent to finite $ N_s $ and $ a_5 $.
The discrete symmetry (V-A structure) of the left-handed quark
current in effective weak Hamiltonian is shown explicitly.
In Section 3, we compute the index, the axial anomaly, and the fermion
determinant of the effective lattice Dirac operators in 2D abelian
background gauge fields.
In Section 4, we summarize and conclude with a discussion on
the difference between the realization of exact chiral
symmetry via the domain-wall fermions and the Ginsparg-Wilson lattice
Dirac operator. In the Appendix, we derive a formula for computing
the propagator of the effective 4D lattice Dirac operator (\ref{eq:Dm})
of the optimal DWF (\ref{eq:ODWF}).


\section{Generating functional for $n$-point Green's function}

\bigskip
\bigskip

In quantum field theory, one of the fundamental quantities is the
generating functional $ Z[J] $ ($ W[J] = \ln Z[J] $)
for (connected) $n$-point Green's function.
Once it is derived, one can obtain any (connected) $n$-point Green's
function by successive differentiations with respect to the source $ J $.
Further, the generating functional for one-particle irreducible
vertices, $ \Gamma[\phi] $, can also be obtained through the Legendre
transform $ \Gamma[\phi] = W[J] - \int dx J(x) \phi(x) $.
In this Section, we derive the generating functional $ Z[J] = e^{W[J]} $
for $n$-point Green's function of quark fields $ q $ and $ \bar q $
constructed from the boundary modes of domain-wall fermions.

The generating functional for $n$-point Green's function of
quark $ q $ and $ \bar q $ is defined as
\bea
\label{eq:ZW}
e^{W[J,\bar J]}
= Z[J,\bar J]
= \frac{ \int [dU][d\bar\psi][d\psi][d\bar\phi][d\phi]
         e^{ -{\cal A}_g -{\cal A}_f -{\cal A}_{pf}
   + \sum_x \left\{ \bar J(x)q(x) + \bar q(x)J(x) \right\}} }
 {\int [dU][d\bar\psi][d\psi][d\bar\phi][d\phi]
         e^{-{\cal A}_g-{\cal A}_f-{\cal A}_{pf}} }
\eea
where $ {\cal A}_g $ is the gauge field action, $ {\cal A}_f $
is the domain-wall fermion action, $ {\cal A}_{pf} $ is the
corresponding pseudofermion action, and $ \bar J $ and $ J $ are
the Grassman sources of $ q $ and $ \bar q $ respectively.

We first derive $ Z[J,\bar J] = e^{W[J, \bar J]} $ for the optimal
DWF action, then show that it is invariant for any $ a_5 $, and
preserves the chiral symmetry optimally for any finite $ N_s $.
Then we derive $ Z[J, \bar J] $ for the usual DWF action,
and discuss its problems due to finite $ N_s $ and $ a_5 $.


\subsection{Optimal domain-wall fermions}

First, I generalize the optimal domain fermion action (\ref{eq:ODWF})
by appending two boundary layers at $ s=0 $ and $ s=N_s + 1 $,
\bea
\label{eq:ODWF1}
{\cal A}'_f &=& \sum_{s,s'=0}^{N_s+1} \sum_{x,x'}
\bar\psi(x,s)
\{ (\omega_s a_5 D_w(x,x') + \delta_{x,x'} )\delta_{s,s'} \nn
&& \hspace{10mm}   -(\delta_{x,x'}-\omega_s a_5 D_w(x,x'))
    (P_{-} \delta_{s',s+1} + P_{+} \delta_{s',s-1}) \} \psi(x',s') \nn
& \equiv & \sum_{s,s'=0}^{N_s+1} \sum_{x,x'}
           \bar\psi(x,s) {\cal D}'(x,s;x's') \psi(x',s')
\eea
with boundary conditions
\bea
\label{eq:bc1_new}
P_{+} \psi(x,-1) &=& - r m_q P_{+} \psi(x,N_s+1) \ ,
\hspace{4mm} r = \frac{1}{2m_0} \\
\label{eq:bc2_new}
P_{-} \psi(x,N_s+2) &=& -r m_q P_{-} \psi(x,0) \ ,
\eea
where the weights at the boundaries are zero,
i.e. $ \omega_0 = \omega_{N_s+1} = 0 $, and
other weights are given by (\ref{eq:omega}) as before.

The quark fields coupling to physical hadrons are constructed
from the left and right boundary modes
\bea
\label{eq:q_odwf}
q(x) = P_{-} \psi(x,0) + P_{+} \psi(x,N_s+1) \ , \\
\label{eq:bar_q_odwf}
\bar q(x) = \bar\psi(x,0) P_{+} + \bar\psi(x,N_s+1) P_{-} \ .
\eea

It will be shown that the fermion determinant of the new action
(\ref{eq:ODWF1}) is exactly equal to that (\ref{eq:det_54}) of the
original action (\ref{eq:ODWF}).
Thus the effective 4D lattice Dirac operator is exactly equal to
(\ref{eq:Dm}), after the pseudofermion action (with $ m_q = 2 m_0 $)
is incorporated. The purpose of adding two boundary layers
at $ s=0 $ and $ s=N_s + 1 $ is for the construction of
the quark fields (\ref{eq:q_odwf})-(\ref{eq:bar_q_odwf})
such that the correlation function of $ q(x) $ and $ \bar q(y) $
is equal to the valence quark propagator, i.e.,
$ r \langle q(x) \bar q(y) \rangle = ( D_c + m_q )^{-1}_{x,y} $
in any background gauge field, and the
generating functional $ Z[J, \bar J] $ can be derived.

Now we derive the generating functional $ Z[J, \bar J] $
for the optimal DWF (\ref{eq:ODWF1}),
according to the formula (\ref{eq:ZW}).

Using $ \gamma_5 P_{\pm} = \pm P_{\pm} $,
$ P_{+} + P_{-} = 1 $, and $ H_w = \gamma_5 D_w $,
we can rewrite (\ref{eq:ODWF1}) as
\bea
\label{eq:A_3a}
{\cal A}'_f &=&
   (r m_q + 1) \bar\psi_0 \gamma_5 P_{+} \psi_{N_s+1} \nn
&&
-\bar\psi_0\gamma_5(P_{-}\psi_0 + P_{+} \psi_{N_s+1})
+\bar\psi_0\gamma_5(P_{-}\psi_1 + P_{+} \psi_0) \nn
&&
\hspace{-16mm}
+ \hspace{-1mm}
\sum_{s=1}^{N_s} \left\{
  \bar\psi_s\gamma_5(\omega_s a_5 H_w - 1)(P_{-}\psi_s + P_{+} \psi_{s-1})
+ \bar\psi_s\gamma_5(\omega_s a_5 H_w + 1)(P_{-}\psi_{s+1} + P_{+} \psi_s)
      \right\} \nn
&&
    - \bar\psi_{N_s+1} \gamma_5 (P_{-}\psi_{N_s+1}+P_{+}\psi_{N_s})
    + \bar\psi_{N_s+1} \gamma_5 (P_{-}\psi_0 + P_{+} \psi_{N_s+1} ) \nn
&&  -(r m_q + 1)\bar\psi_{N_s+1}\gamma_5 P_{-} \psi_0
\eea
where all indices are suppressed except the flavor index in the
fifth dimension.

Next we define
\bea
\label{eq:o1}
&& \hspace{-8mm}
\eta_0 = P_{-} \psi_0 + P_{+} \psi_{N_s+1} \ , \hspace{4mm}
\bar\eta_0 = - \bar\psi_0 \gamma_5  \ , \\
\label{eq:o2}
&& \hspace{-8mm}
\eta_s = P_{-} \psi_s + P_{+} \psi_{s-1} \ , \hspace{4mm}
\bar\eta_s = \bar\psi_s \gamma_5 (\omega_s a_5 H_w - 1) \ ,
              s=1, \cdots, N_s \\
\label{eq:o3}
&& \hspace{-8mm}
\eta_{N_s+1} = P_{-} \psi_{N_s+1} + P_{+} \psi_{N_s} \ , \hspace{4mm}
\bar\eta_{N_s+1} = -\bar\psi_{N_s+1} \gamma_5    \ ,
\eea
and the inverse transform
\bea
\label{eq:o1i}
&& \hspace{-8mm}
\psi_0 = P_{-} \eta_0 + P_{+} \eta_1 \ , \hspace{4mm}
\bar\psi_0 = - \bar\eta_0 \gamma_5 \ , \\
\label{eq:o2i}
&& \hspace{-8mm}
\psi_s = P_{-} \eta_s + P_{+} \eta_{s+1} \ , \hspace{4mm}
\bar\psi_s = \bar\eta_s (\omega_s a_5 H_w - 1)^{-1} \gamma_5 \ ,
           s=1,\cdots, N_s \\
\label{eq:o3i}
&& \hspace{-8mm}
\psi_{N_s+1} = P_{-} \eta_{N_s+1} + P_{+} \eta_0 \ , \hspace{4mm}
\bar\psi_{N_s+1} = -\bar\eta_{N_s+1} \gamma_5 \ .
\eea

Then the action (\ref{eq:A_3a}) can be expressed in terms of
$ \{ \eta, \bar\eta \} $ fields
\bea
\label{eq:A_3b}
{\cal A}'_f
&=& \bar\eta_0 ( P_{-} - r m_q P_{+} ) \eta_0
            - \bar\eta_0 \eta_1
            + \sum_{s=1}^{N_s} \left\{
              \bar\eta_s \eta_s - \bar\eta_s T_s^{-1} \eta_{s+1}
              \right\}              \nn
&&          + \bar\eta_{N_s+1} \eta_{N_s+1}
            - \bar\eta_{N_s+1} ( P_{+} - r m_q P_{-} ) \eta_0
\eea
where
\BAN
T_s = \frac{ 1 - \omega_s a_5 H_w }{ 1 + \omega_s a_5 H_w }
         \ , \hspace{4mm} s = 1, \cdots, N_s \ .
\EAN

Thus the fermionic integral in the numerator of (\ref{eq:ZW})
can be written as
\bea
\int [d\bar\psi] [d\psi]
     e^{-{\cal A}'_f[\psi,\bar\psi] + \bar J q + \bar q J }
= {\cal J}\int [d\bar\eta] [d\eta]
  e^{-{\cal A}'_f[\eta,\bar\eta] + \bar J \eta_0 - \bar\eta_0 P_{+} J
                 + \bar\eta_{N_s+1} P_{-} J }
\label{eq:Odet}
\eea
where
\bea
\bar J q + \bar q J &=& \bar J ( P_{-} \psi_0 + P_{+} \psi_{N_s+1} )
                        + ( \bar\psi_0 P_{+} + \bar\psi_{N_s+1} P_{-} ) J \nn
&=& \bar J \eta_0 - \bar\eta_0 P_{+} J + \bar\eta_{N_s+1} P_{-} J
\label{eq:jq}
\eea
and ${\cal J}$ is the Jacobian of the transformation,
\bea
\label{eq:j}
{\cal J}=\prod_{s=0}^{N_s+1} \det \left[\gamma_5 (\omega_s a_5 H_w-1) \right]
        =\prod_{s=1}^{N_s} \det \left[\gamma_5 (\omega_s a_5 H_w-1) \right]
\eea
where $ \omega_0 = \omega_{N_s + 1 } = 0 $ has been used.

Now using the Grassman integral formula
\bea
\label{eq:G_integral}
\int d \bar\chi d \chi \ e^{-\bar\chi M \chi + \bar v \chi + \bar \chi v}
= e^{\bar v M^{-1} v} \det M
\eea
one can easily evaluate the Grassman integrals in (\ref{eq:Odet}),
by integrating $ (\eta_{s}, \bar\eta_{s}) $ successively from
$ s = N_s + 1 $ to $ s = 0 $.

Explicitly, after integrating $ (\eta_{N_s+1}, \bar\eta_{N_s+1}) $,
(\ref{eq:Odet}) becomes
\BAN
&& {\cal J}\int \prod_{s=0}^{N_s}[d \bar\eta_s] [d \eta_s]
            \ \exp \left\{ - \bar\eta_0 ( P_{-} - r m_q P_{+} ) \eta_0
                          + \bar\eta_0 \eta_1
                          + \bar J \eta_0 - \bar\eta_0 P_{+} J \right. \nn
&& \hspace{40mm}
-\sum_{s=1}^{N_s-1} (\bar\eta_s \eta_s - \bar\eta_s T_s^{-1} \eta_{s+1})
-\bar\eta_{N_s} \eta_{N_s}  \nn
&& \hspace{40mm}
   \left.
   +\bar\eta_{N_s} T_{N_s}^{-1}
    [ (P_{+} - r m_q P_{-}) \eta_0 + P_{-} J ] \right\}
\EAN
then integrating $(\eta_{N_s},\bar\eta_{N_s})$, the remaining integral
becomes
\BAN
&& {\cal J}\int \prod_{s=0}^{N_s-1}[d \bar\eta_s] [d \eta_s]
        \ \exp \left\{ - \bar\eta_0 ( P_{-} - r m_q P_{+} ) \eta_0
                          + \bar\eta_0 \eta_1
                          + \bar J \eta_0 - \bar\eta_0 P_{+} J \right. \nn
&& \hspace{40mm}
-\sum_{s=1}^{N_s-2} (\bar\eta_s \eta_s - \bar\eta_s T_s^{-1} \eta_{s+1} )
-\bar\eta_{N_s-1} \eta_{N_s-1} \nn
&& \hspace{30mm}
   \left.
   +\bar\eta_{N_s-1} T_{N_s-1}^{-1} T_{N_s}^{-1}
    [(P_{+} - r m_q P_{-}) \eta_0 + P_{-} J ] \right\}
\EAN
Subsequent integrations over $ (\eta_{N_s-2}, \bar\eta_{N_s-2}) $
up to $ (\eta_1, \bar\eta_1) $ are similar to the above integration,
and the result is
\BAN
&& {\cal J}\int [d \bar\eta_0] [d \eta_0]
           \ \exp \left\{ - \bar\eta_0 ( P_{-} - r m_q P_{+} ) \eta_0
                          + \bar J \eta_0 - \bar\eta_0 P_{+} J \right. \nn
&& \hspace{40mm}
   \left.
   +\bar\eta_0 \prod_{s=1}^{N_s} T_s^{-1}
   [ (P_{+} - r m_q P_{-}) \eta_0 + P_{-} J ] \right\}
\EAN
Finally, integrating $ (\eta_0,\bar\eta_0) $ gives
\bea
\label{eq:int_f}
& & {\cal J}\ \det\left[ (P_{-}-r m_q P_{+})
         - \prod_{s=1}^{N_s} T_s^{-1} (P_{+}-r m_q P_{-}) \right]\times \nn
& & \exp \left\{
\bar J \left[ \left(-P_{+}+\prod_{s=1}^{N_s} T_s^{-1} P_{-} \right)^{-1}
\left(P_{-}-\prod_{s=1}^{N_s} T_s^{-1} P_{+} \right) + r m_q \right]^{-1} J
       \right\} \nn
&=& K \ \det [ r (D_c + m_q) ] \
       \exp \left\{ \bar J  r^{-1} (D_c + m_q )^{-1} J \right\}
\eea
where the simple identity
\bea
\label{eq:identity}
  \left(-P_{+}+\prod_{s=1}^{N_s} T_s^{-1} P_{-} \right)^{-1}
  \left( P_{-}-\prod_{s=1}^{N_s} T_s^{-1} P_{+} \right)
= \frac{1 + \gamma_5 S_{opt}}{1 - \gamma_5 S_{opt}} = r D_c \
\eea
has been used, and $ K $, $ S_{opt} $ and $ r D_c $ are defined in
(\ref{eq:K}), (\ref{eq:S_opt}) and (\ref{eq:rDc}) respectively.
Note that any two transfer matrices commute with each other
($ T_s T_{s'} = T_{s'} T_s $), thus
$ D_c $ is invariant for swapping any two weights in
$ \{ \omega_1, \cdots, \omega_{N_s} \} $.

Now setting $ \bar J = J = 0 $ in (\ref{eq:int_f}), we obtain the
fermion determinant of the optimal DWF action (\ref{eq:ODWF1})
\bea
\label{eq:det_54a}
\det {\cal D}'(m_q) = K \ \det [r(D_c + m_q)]
\eea
where $ {\cal D}'(m_q) $ is the 5-dimensional lattice DWF operator
defined in (\ref{eq:ODWF1}), while $ r D_c $ is the 4-dimensional
lattice Dirac operator defined in (\ref{eq:rDc}), which becomes
chirally symmetric in the limit $ N_s \to \infty $, the salient
feature of domain-wall fermions.

Obviously, if one repeats above steps to derive the fermion
determinant of the original optimal DWF action (\ref{eq:ODWF}),
then the result is exactly equal to (\ref{eq:det_54a}).
The vital observation is that the transfer matrix $ T_s $ at
$ s = 0 $ and $ s = N_s + 1 $ is equal to the identity (since
$ \omega_0 = \omega_{N_s+1} = 0 $), thus
\BAN
\prod_{s=0}^{N_s + 1} T_s^{-1} = \prod_{s=1}^{N_s} T_s^{-1} \ .
\EAN
Explicitly, the transformation (\ref{eq:o1i})-(\ref{eq:o3i}) is
replaced by (\ref{eq:f1i})-(\ref{eq:f3i})
and the action (\ref{eq:ODWF}) can be expressed as
\BAN
{\cal A}_f [\eta,\bar\eta]
&=& \bar\eta_1 ( P_{-} - r m_q P_{+} ) \eta_1
                  - \bar\eta_1 T_1^{-1} \eta_2 +
             \sum_{s=2}^{N_s-1} \left\{
             \bar\eta_s \eta_s - \bar\eta_s T_s^{-1} \eta_{s+1}
             \right\} +             \nn
&&           \bar\eta_{N_s} \eta_{N_s} -
             \bar\eta_{N_s} T_{N_s}^{-1} ( P_{+} - r m_q P_{-} ) \eta_1 \ .
\EAN
Thus the fermion determinant
\BAN
%\label{eq:det_odwf}
\det {\cal D}(m_q) = \int [d\bar\psi][d\psi] e^{-{\cal A}_f[\psi,\bar\psi]}
           = {\cal J}\int [d\bar\eta][d\eta] e^{-{\cal A}_f[\eta,\bar\eta]}
\EAN
can be evaluated by integrating $ \{ \eta_s, \bar\eta_s \} $
successively from $ s=N_s $ to $ s=1 $, and the result is given
in (\ref{eq:det_54}).

As discussed in the Introduction, one
cannot identify $ ( D_c + m_q ) $ as the effective 4D
lattice Dirac operator for the internal quark fields,
since $ D_c $ is nonlocal in the limit $ N_s \to \infty $,
and $ \det(D_c + m_q) $ does not reproduce \\
$ \det [ \gamma_\mu ( \partial_\mu + i A_\mu ) + m_q ] $
in the continuum limit $ (a \to 0) $.
The resolution is to introduce a set of pseudofermion fields
$ \{ \phi, \bar\phi \} $ similar to the domain-wall fermions but obeying
Bose statistics. Explicitly, the action of the pseudofermion fields
(with $ m_q=2 m_0 $) corresponding to the optimal DWF
action (\ref{eq:ODWF1}) is
\bea
\label{eq:ODWF1_pf}
{\cal A}'_{pf} &=& \sum_{s,s'=0}^{N_s+1} \sum_{x,x'}
\bar\phi(x,s)
\{ (\omega_s a_5 D_w(x,x') + \delta_{x,x'} )\delta_{s,s'} \nn
&& \hspace{8mm}   -(\delta_{x,x'}-\omega_s a_5 D_w(x,x'))
    (P_{-} \delta_{s',s+1} + P_{+} \delta_{s',s-1}) \} \phi(x',s')
\eea
with boundary conditions
\BAN
%\label{eq:bc1}
P_{+} \phi(x,-1) &=& - P_{+} \phi(x,N_s+1) \ , \\
%\label{eq:bc2}
P_{-} \phi(x,N_s+2) &=& -P_{-} \phi(x,0) \ ,
\EAN
where $ \omega_0 = \omega_{N_s+1} = 0 $, and
other weights are given by (\ref{eq:omega}) as before.
Thus the integrations over the pseudofermion fields give
\bea
\label{eq:int_pf}
\int [d \bar\phi][d \phi] e^{-{\cal A}'_{pf}}
= \pi^{N_s+2} \ K^{-1} \ \det ( 1 + r D_c )^{-1}
\eea

Substituting (\ref{eq:int_f}) and (\ref{eq:int_pf}) into (\ref{eq:ZW}),
we obtain
\bea
\label{eq:ZW_odwf}
e^{W[J,\bar J]} =
\frac{ \int [dU] e^{-{\cal A}_g }  \det [(D_c + m_q)(1+r D_c)^{-1}]
       \exp \left\{ \bar J  r^{-1} ( D_c + m_q )^{-1} J \right\}
       }
     { \int [dU] e^{-{\cal A}_g} \det [(D_c + m_q)(1+rD_c)^{-1}] }
\eea
This is one of the main results of this paper.

Now it is clear that the most crucial quantity in (\ref{eq:ZW_odwf})
is $ D_c $ which becomes chirally symmetric in the limit $ N_s \to \infty $.
At any finite $ N_s $, $ D_c $ (\ref{eq:rDc})
preserves the chiral symmetry optimally since $ S_{opt} $ (\ref{eq:S_opt})
is equal to the optimal rational polynomial of the sign
function (\ref{eq:S_opt_RZ}).
Moreover, $ \omega_s a_5 $ (\ref{eq:omega}) is independent
of $ a_5 $, since it only depends on $ N_s $, $\lambda_{min}$ and
$ \lambda_{max}^2 / \lambda_{min}^2 $. It follows that $ T_s $,
$ S_{opt} $, and $ D_c $ are all independent of $ a_5 $.
Therefore the generating functional $ Z[J,\bar J] $
for the optimal domain-wall fermion action (\ref{eq:ODWF1})
is $ a_5 $-invariant, preserves the chiral symmetry optimally for
any finite $ N_s $. Further, it also manifests the continuum
discrete symmetries, since the quark propagator $ (D_c + m_q)^{-1} $
(coupling to $ \bar J $ and $ J $) transforms like
$ [ \gamma_\mu ( \partial_\mu + i A_{\mu} ) + m_q ]^{-1} $.

From (\ref{eq:ZW_odwf}), it is clear that the effective 4D lattice Dirac
operator for the internal quark fields (dressed with pseudofermions) is
\bea
\label{eq:Dm_odwf1}
D(m_q) = (D_c + m_q)(1 + r D_c)^{-1}
       = m_q + (m_0- m_q/2) ( 1 + \gamma_5 H_w R_Z(H_w) )
\eea
which is exactly equal to that (\ref{eq:Dm}) of the original
action (\ref{eq:ODWF}).

Now any connected $n$-point Green's function can be obtained by
differentiating $ W[J,\bar J] $ with respect to $ J $ and $ \bar J $
successively. In particular, the quark propagator is
\bea
r \langle q(x) \bar q(y) \rangle &=&
- \left.
r \frac{\delta^2 W[J,\bar J]}{\delta \bar J(x) \delta J(y)}
\right|_{J=\bar J=0} \nn
&=& \frac{ \int [dU] e^{-{\cal A}_g} \det [D(m_q)] \
           (D_c + m_q)^{-1}(x,y) }
         { \int [dU] e^{-{\cal A}_g} \det [D(m_q)] }
\eea
which, in a background gauge field, becomes
\bea
\label{eq:quark_prop}
r \langle q(x) \bar q(y) \rangle
= ( D_c + m_q )^{-1}(x,y) \ ,
\eea
and it goes to the Dirac propagator
$ [ \gamma_\mu ( \partial_\mu + i A_{\mu} ) + m_q ]^{-1}(x,y) $ in the
continuum limit.

Now it is straightforward to generalize the optimal DWF action
(\ref{eq:ODWF1}) to lattice QCD with $ n_f $ quark flavors, with
quark fields defined as
\bea
\label{eq:qf_odwf}
q_f(x) &=& \sqrt{r} [P_{-}\psi_f(x,0)+P_{+}\psi_f(x,N_s+1)] \ ,
\hspace{4mm} f=1,\cdots, n_f, \\
\label{eq:bar_qf_odwf}
\bar q_f(x) &=& \sqrt{r} [\bar\psi_f(x,0) P_{+}+\bar\psi_f(x,N_s+1) P_{-}] \ ,
\eea
where the normalization factor $ \sqrt{r} = 1/\sqrt{2m_0} $ has been included.
Then the generating functional (\ref{eq:ZW_odwf}) can be rewritten as
\bea
\label{eq:ZW_odwf_nf}
Z[ \{ J_f, \bar J_f \} ]
%= e^{W[ \{ J_f, \bar J_f \} ]} \nn
=
\frac{ \int [dU] e^{-{\cal A}_g }
       \prod_{f=1}^{n_f} \det [D(m_f)]
       \exp \left\{ \sum_{f=1}^{n_f}
       \bar J_f  ( D_c + m_f )^{-1} J_f \right\} }
     { \int [dU] e^{-{\cal A}_g} \prod_{f=1}^{n_f} \det [D(m_f)] }
\eea
where
\BAN
D(m_f) = (D_c + m_f)(1+r D_c)^{-1} = m_f + ( 1 - r m_f) D
\EAN

Evidently, the generating functional (\ref{eq:ZW_odwf_nf})
ensures that any bilinear (and higher-order) quark operator
manifests the continuum discrete symmetries, since the
quark fields (\ref{eq:qf_odwf})-(\ref{eq:bar_qf_odwf}),
both can be decomposed into left and right handed
components with the usual chiral projectors, i.e.
\bea
&& q_f^{L}(x) = P_{-} q_f(x), \hspace{4mm} q_f^{R}(x) = P_{+} q_f(x), \\
&& \bar q_f^{L}(x) = \bar q_f(x) P_{+}, \hspace{4mm}
   \bar q_f^{R}(x)=  \bar q_f(x) P_{-}.
\eea
Thus the V-A structure of the left-handed quark current
\bea
\label{eq:JL}
L_\mu^{f_1,f_2}(z) = \bar q_{f_1}(z) \gamma_\mu ( 1 - \gamma_5 ) q_{f_2} (z)
\eea
in the effective weak Hamiltonian is preserved exactly.
This is crucial for determining the indirect CP violation
in the $K\to\pi\pi$ decay through the $\bar K^0$-$K^0$ mixing.
The weak mixing matrix element
\bea
\label{eq:KKbar}
\langle \bar K^0 | \bar s \gamma_\mu (1-\gamma_5) d
                   \bar s \gamma_\mu (1-\gamma_5) d | K^0 \rangle
\eea
(where $ s $ and $ d $ denote the strange and down quark fields respectively)
can be extracted by evaluating the connected 3-point function
(for all $\vec{x}$ and $\vec{y}$ and a range of $x_4$ and $y_4$)
\bea
& & \langle \bar d(x) \gamma_4 \gamma_5 s(x) L_\mu^{s,d}(0) L_\mu^{s,d}(0)
            \bar d(y) \gamma_4 \gamma_5 s(y) \rangle^{\mbox{connected}}    \nn
&=&         \frac{\delta}{\delta J_d(x)} \gamma_4 \gamma_5
            \frac{\delta}{\delta \bar J_s(x)}
            \frac{\delta}{\delta J_s(0)} \gamma_\mu (1-\gamma_5)
            \frac{\delta}{\delta \bar J_d(0)} \times              \nn
& &  \left. \frac{\delta}{\delta J_s(0)} \gamma_\mu (1-\gamma_5)
            \frac{\delta}{\delta \bar J_d(0)}
            \frac{\delta}{\delta J_d(y)} \gamma_4 \gamma_5
            \frac{\delta}{\delta \bar J_s(y)}
            Z[ \{ J_f, \bar J_f \} ] \right|_{J=\bar J = 0 } \nn
& &      - \mbox{ disconnected diagrams } \nn
&=& \hspace{-2mm}
    \frac{1}{{\cal Z}}
    \int [dU] e^{-{\cal A}_g }
       \prod_{f} \det [D(m_f)] \
       \tr [ (D_c + m_d)^{-1}_{0,x} \gamma_4 \gamma_5 (D_c + m_s)^{-1}_{x,0}
             \gamma_\mu (1-\gamma_5) ] \times \nn
& & \hspace{30mm} \tr [ (D_c + m_s)^{-1}_{y,0} \gamma_\mu (1-\gamma_5)
                        (D_c + m_d)^{-1}_{0,y} \gamma_4 \gamma_5 ] \ ,\nonumber
\eea
and the 2-point functions
\BAN
& & \langle \bar d(x) \gamma_4 \gamma_5 s(x)
            \bar s(0) \gamma_4 \gamma_5 d(0) \rangle     \nn
&=& \frac{1}{{\cal Z}}
    \int [dU] e^{-{\cal A}_g }
       \prod_{f} \det [D(m_f)] \
       \tr [(D_c + m_d)^{-1}_{0,x} \gamma_4 \gamma_5
            (D_c + m_s)^{-1}_{x,0} \gamma_4 \gamma_5 ] \ ,    \nn
& & \langle \bar s(0) \gamma_4 \gamma_5 d(0)
            \bar d(y) \gamma_4 \gamma_5 s(y) \rangle     \nn
&=& \frac{1}{{\cal Z}}
    \int [dU] e^{-{\cal A}_g }
       \prod_{f} \det [D(m_f)] \
       \tr [(D_c + m_s)^{-1}_{y,0} \gamma_4 \gamma_5
            (D_c + m_d)^{-1}_{0,y} \gamma_4 \gamma_5 ] \ ,
\EAN
where
\BAN
{\cal Z}= \int [dU] e^{-{\cal A}_g} \prod_{f} \det [D(m_f)]  \ .
\EAN

It is clear that there is {\it no} right-handed currents
entering the weak mixing matrix element (\ref{eq:KKbar}), and
the $ SU(2)_L $ gauge symmetry of electroweak interactions
is preserved exactly.
Had one {\it not} introduced the boundary layers at $ s=0 $ and
$ s=N_s+1 $, then one might {\it not} have found
any $ q $ and $ \bar q $
such that Eq. (\ref{eq:quark_prop}) is satisfied and
the usual chiral projectors $ (1 \pm \gamma_5)/2 $
can work for both $ q $ and $ \bar q $.
Consequently its left-handed quark current may not manifest the V-A structure,
and the $ SU(2)_L $ gauge symmetry may be broken by the order $ O(a) $ effect.
Obviously, such left-handed quark current is not ideal
for determining the $\bar K^0$-$K^0$ mixing.

In practice, the valence quark propagator $ (D_c + m_q)^{-1} $
in a background gauge field can be computed via the 5-dimensional
optimal DWF operator $ {\cal D}' $ (\ref{eq:ODWF1}).
Using (\ref{eq:quark_prop}) and the definition of quark fields
(\ref{eq:q_odwf})-(\ref{eq:bar_q_odwf}), one immediately obtains
\bea
& & r^{-1} ( D_c + m_q )^{-1}(x,y) \nn
&=& \langle q(x) \bar q(y) \rangle \nn
&=& \sum_{s,s'} ( P_{-} \delta_{s,0} + P_{+} \delta_{s,N_s+1} )
\frac{\int [d\bar\psi] [d\psi] \psi_s(x) \bar\psi_{s'}(y) e^{-{\cal A}'_f} }
     {\int [d\bar\psi] [d\psi] e^{-{\cal A}'_f} }
    ( P_{+} \delta_{s',0} + P_{-} \delta_{s',N_s+1} ) \nn
&=& \sum_{s,s'} ( P_{-} \delta_{s,0} + P_{+} \delta_{s,N_s+1} )
                {\cal D}'^{-1}(x,s;y,s')
                ( P_{+} \delta_{s',0} + P_{-} \delta_{s',N_s+1} ) \ ,
\label{eq:quark_prop_54}
\eea
where $  {\cal D}'^{-1}(x,s;y,s') $ can be obtained by solving
the linear system
\bea
\label{eq:D1Y}
 {\cal D}'(x,s;y,s') Y = \Id \ .
\eea

Alternatively, one can also obtain the valence quark propagator
\bea
\label{eq:v_quark}
(D_c + m_q)^{-1} = ( 1 - r m_q )^{-1} [ D^{-1}(m_q) - r ] \ ,
\eea
by computing $ D^{-1}(m_q) $ via the original
optimal DWF operator (\ref{eq:ODWF}), i.e.,
solving the linear system
\bea
\label{eq:DY}
{\cal D}(m_q) Y = {\cal D}(2m_0) B^{-1} \Id \ ,
\eea
where $ B $ and $ B^{-1} $ are defined in (\ref{eq:B}) and (\ref{eq:BI})
respectively. Then the solution $ Y $ gives
\bea
\label{eq:DIBY}
D^{-1}(m_q)(x,x') = r (B Y)(x,1;x',1) \ ,
\eea
The derivation of (\ref{eq:DY}) is given in the Appendix.

Note that we can also compute the valence quark propagator
(\ref{eq:v_quark}) by solving the linear system
\bea
\label{eq:Dm_RZ}
D(m_q) Y =
\left[m_q+(m_0-m_q/2) \left(1+\gamma_5 H_w R_Z(H_w^2)\right) \right] Y
= \Id
\eea
where $ D(m_q) $ is the effective 4D lattice Dirac operator
(\ref{eq:Dm_odwf1}), and $ R_Z(H_w^2) $ is the Zolotarev rational
polynomial as defined in (\ref{eq:S_opt_RZ})-(\ref{eq:rz_n1n}).

Since solving any one of above three linear systems (\ref{eq:Dm_RZ}),
(\ref{eq:D1Y}) or (\ref{eq:DY}) yields the same quark propagator,
in principle, it does {\it not} matter which linear system one chooses.
However, in practice, one should pick the most efficient scheme
for one's computational system (hardware and software).
%At present, we find that solving the linear system (\ref{eq:Dm_RZ})
%is more efficient than solving the linear system (\ref{eq:D1Y})
%or (\ref{eq:DY}) with 5D DWF operator.


\subsection{Conventional domain-wall fermions}

Shamir's domain-wall fermion action \cite{Shamir:1993zy} can be
written as
\bea
\label{eq:shamir}
{\cal A}_{fs} &=& \sum_{s,s'=1}^{N_s} \sum_{x,x'} \bar\psi(x,s)
[ (a_5 D_w(x,x') + \delta_{x,x'}) \delta_{s,s'}   \nn
& & \hspace{16mm}
 -(P_{-}\delta_{s',s+1} + P_{+} \delta_{s',s-1})\delta_{x,x'}] \psi(x',s') \nn
&\equiv& \sum_{s,s'=1}^{N_s} \sum_{x,x'}
    \bar\psi(x,s) {\cal D}_{fs}(x,s;x's') \psi(x',s')
\eea
with boundary conditions
\bea
\label{eq:bc1_dwf}
P_{+} \psi(x,0) &=& - r m_q P_{+} \psi(x,N_s) \ ,
\hspace{4mm} r = \frac{1}{2m_0} \\
\label{eq:bc2_dwf}
P_{-} \psi(x,N_s+1) &=& - r m_q P_{-} \psi(x,1) \ ,
\eea
where $ D_w $ is defined in (\ref{eq:Dw}),
and $ m_q $ is the bare quark mass.
%(coupling to physical hadrons)
The quark fields are constructed from the boundary modes:
\bea
\label{eq:quark_s}
q(x) &=& P_{-} \psi(x,1) + P_{+} \psi(x, N_s ) \ , \\
\label{eq:antiquark_s}
\bar{q}(x) &=&  \bar\psi(x,1) P_{+} + \bar\psi(x, N_s) P_{-} \ .
\eea

The fermion determinant and the quark propagator (in gauge background)
have been derived in Refs. \cite{Neuberger:1997bg}-\cite{Edwards:2000qv}.
Now we derive the generating functional for $n$-point
Green's function of the quarks fields,
according to the formula (\ref{eq:ZW}). The
derivation is similar to that of the optimal DWF action in the
last subsection.

Using $ \gamma_5 P_{\pm} = \pm P_{\pm} $,
$ P_{+} P_{-} = 0 $, $ P_{+} + P_{-} = 1 $, and $ H_w = \gamma_5 D_w $,
we can rewrite (\ref{eq:shamir}) as
\bea
\label{eq:As_3a}
{\cal A}_{fs} &=&
   - (r m_q+1) \bar\psi_1 \gamma_5 ( a_5 H_w P_{-} - 1) P_{+} \psi_{N_s} \nn
&& \hspace{-4mm}
  + \bar\psi_1 \gamma_5 (a_5 H_w P_{-} - 1) (P_{-} \psi_1 + P_{+} \psi_{N_s})
  + \bar\psi_1 \gamma_5 (a_5 H_w P_{+} + 1) (P_{-} \psi_2 + P_{+} \psi_1) \nn
&&
\hspace{-16mm}
+
\hspace{-2mm}
\sum_{s=2}^{N_s-1} \left\{
  \bar\psi_s \gamma_5 (a_5 H_w P_{-} - 1) ( P_{-} \psi_s + P_{+} \psi_{s-1} )
+ \bar\psi_s \gamma_5 (a_5 H_w P_{+} + 1) ( P_{-} \psi_{s+1} + P_{+} \psi_s )
      \right\} \nn
&&
\hspace{-14mm}
+ \bar\psi_{N_s}\gamma_5 (a_5 H_w P_{-}-1)(P_{-}\psi_{N_s}+P_{+}\psi_{N_s-1})
+ \bar\psi_{N_s}\gamma_5 (a_5 H_w P_{+}+1)(P_{-}\psi_1 + P_{+} \psi_{N_s}) \nn
&& - (r m_q + 1) \bar\psi_{N_s} \gamma_5 (a_5 H_w P_{+}+1) P_{-} \psi_1
\eea
where all indices are suppressed except the flavor index in the
fifth dimension.

Next we define
\BAN
%\label{eq:e1}
&& \hspace{-6mm}
\eta_1 = P_{-} \psi_1 + P_{+} \psi_{N_s} \ , \hspace{4mm}
\bar\eta_1 = \bar\psi_1 \gamma_5 ( a_5 H_w P_{-} - 1) \ , \\
%\label{eq:e2}
&& \hspace{-6mm}
\eta_s = P_{-} \psi_s + P_{+} \psi_{s-1} \ , \hspace{4mm}
\bar\eta_s = \bar\psi_s \gamma_5 ( a_5 H_w P_{-} - 1) \ , s=2, \cdots, N_s-1 \\
%\label{eq:e3}
&& \hspace{-6mm}
\eta_{N_s} = P_{-} \psi_{N_s} + P_{+} \psi_{N_s-1} \ , \hspace{4mm}
\bar\eta_{N_s} = \bar\psi_{N_s} \gamma_5 ( a_5 H_w P_{-} - 1) \ ,
\EAN
and the inverse transform
\BAN
%\label{eq:e1i}
&& \hspace{-6mm}
\psi_1 = P_{-} \eta_1 + P_{+} \eta_2 \ , \hspace{4mm}
\bar\psi_1 = \bar\eta_1 ( a_5 H_w P_{-}- 1)^{-1} \gamma_5 \ , \\
%\label{eq:e2i}
&& \hspace{-6mm}
\psi_s = P_{-} \eta_s + P_{+} \eta_{s+1} \ , \hspace{4mm}
\bar\psi_s = \bar\eta_s (a_5 H_w P_{-} - 1)^{-1} \gamma_5 \ ,
s=2,\cdots, N_s-1 \\
%\label{eq:e3i}
&& \hspace{-6mm}
\psi_{N_s} = P_{-} \eta_{N_s} + P_{+} \eta_1 \ , \hspace{4mm}
\bar\psi_{N_s} = \bar\eta_{N_s} ( a_5 H_w P_{-} - 1)^{-1} \gamma_5 \ .
\EAN

Then the action (\ref{eq:As_3a}) can be expressed in terms of
$ \{ \eta, \bar\eta \} $ fields
\bea
\label{eq:As_3b}
{\cal A}_{fs} &=& \bar\eta_1 ( P_{-} - r m_q P_{+} ) \eta_1
                  - \bar\eta_1 T^{-1} \eta_2
               + \sum_{s=2}^{N_s-1} \left\{
                 \bar\eta_s \eta_s - \bar\eta_s T^{-1} \eta_{s+1}
                 \right\}              \nn
&&             + \bar\eta_{N_s} \eta_{N_s}
               - \bar\eta_{N_s} T^{-1} ( P_{+} - r m_q P_{-} ) \eta_1
\eea
where
\bea
\label{eq:T}
T &=& (1 + a_5 H_w P_{+})^{-1} ( 1 - a_5 H_w P_{-})
   = \frac{1-a_5 H}{1+a_5 H} \ ,  \\
\label{eq:H}
H &\equiv& (2 + a_5 H_w \gamma_5 )^{-1} H_w
   = H_w ( 2 + \gamma_5 a_5 H_w )^{-1}
   = \gamma_5 \frac{D_w}{2 + a_5 D_w } \ .
\eea

At this point, it is instructive to compare the transfer matrix
$ T $ (\ref{eq:T}) with its counterpart $ T_s $ (\ref{eq:Ts}) in
the optimal DWF. As it will become clear later, one of the
difficulties in the conventional DWF is due to the fact that the
denominator of $ H $ (\ref{eq:H}) is ($ 2 + a_5 D_w $)
rather than a constant.

Now the fermionic integral in the numerator of (\ref{eq:ZW})
can be written as
\bea
\int [d\bar\psi] [d\psi] e^{-{\cal A}_{fs} + \bar J q + \bar q J }
= \tau \int [d\bar\eta] [d\eta]
  e^{-{\cal A}_{fs} + \bar J \eta_1 - \bar\eta_1 P_{+} J
                 + \bar\eta_{N_s} T^{-1} P_{-} J }
\label{eq:Sdet}
\eea
where
\BAN
\bar J q + \bar q J &=& \bar J ( P_{-} \psi_1 + P_{+} \psi_{N_s} )
                       + (\bar\psi_1 P_{+} + \bar\psi_{N_s} P_{-}) J \nn
&=& \bar J \eta_1 + \bar\eta_1 ( a_5 H_w P_{-} - 1)^{-1} P_{+} J
                  - \bar\eta_{N_s} ( a_5 H_w P_{-} - 1)^{-1} P_{-} J \nn
&=& \bar J \eta_1 - \bar\eta_1 P_{+} J + \bar\eta_{N_s} T^{-1} P_{-} J \ ,
\EAN
and $ \tau $ is the Jacobian of the transformation,
\BAN
\tau = \det \left[ \gamma_5 ( a_5 H_w P_{-} - 1) \right]^{N_s} \ .
\EAN

Again using the Grassman integral formula (\ref{eq:G_integral}),
one can easily evaluate the Grassman integrals in (\ref{eq:Sdet}),
by integrating $ (\eta_{s}, \bar\eta_{s}) $ successively from
$ s=N_s $ to $ s=1 $, similar to the steps in the last subsection.
The final result is
\bea
\label{eq:int_fs}
& & \tau \ \det\left[(P_{-}-r m_q P_{+})-T^{-N_s}(P_{+}-r m_q P_{-})\right]
        \times \nn
& & \exp \left\{
\bar J [(-P_{+}+T^{-N_s} P_{-} )^{-1} (P_{-}-T^{-N_s} P_{+} )+ r m_q]^{-1} J
\right\} \nn
&=& K_f \ \det [ r (D_c^{(s)} + m_q ) ]
       \exp \left\{ \bar J r^{-1} (D_c^{(s)} + m_q )^{-1} J \right\}
\eea
where
\bea
\label{eq:Kf}
K_f &=& \det \left[ \gamma_5 ( a_5 H_w P_{-} - 1) \right]^{N_s}
             \det (-P_{+} + T^{-N_s} P_{-}) \\
\label{eq:rDcs}
r D_c^{(s)} &=& (-P_{+}+T^{-N_s} P_{-} )^{-1} (P_{-}-T^{-N_s} P_{+} )
        =  \frac{ 1 + \gamma_5 S(H) }{ 1 - \gamma_5 S(H) } \\
\label{eq:S}
S(H) &=& \frac{1-T^{N_s}}{1+T^{N_s}}
 = \frac{(1+a_5 H)^{N_s}-(1-a_5 H)^{N_s}}{(1+a_5 H)^{N_s}+(1-a_5 H)^{N_s}} \ .
\eea

Again, since $ D_c^{(s)} $ is nonlocal in the limit $ N_s \to \infty $,
one has to introduce pseudofermion fields (similar to
the domain-wall fermions but obeying Bose statistics)
with action (\ref{eq:shamir}) ($ m_q = 2 m_0 $) such that
the effective 4D lattice Dirac
operator for the internal fermion loops is exponentially-local
for smooth gauge backgrounds, and it
goes to $ [ \gamma_\mu ( \partial_\mu + i A_\mu ) + m_q ] $ in the continuum
limit $ a, a_5 \to 0 $. Then integrating the pseudofermion fields gives
\bea
\label{eq:int_pfs}
\int [d \bar\phi][d \phi] e^{-{\cal A}_{pfs}}
= \pi^{N_s} \ K_f^{-1}  \det ( 1 + r D_c^{(s)} )^{-1} \ .
\eea

Substituting (\ref{eq:int_fs}) and (\ref{eq:int_pfs}) into
(\ref{eq:ZW}), we obtain
\bea
\label{eq:ZW_dwf}
e^{W[J,\bar J]} =
\frac{ \int [dU] e^{-{\cal A}_g} \det [(D_c^{(s)} + m_q)(1+r D_c^{(s)})^{-1}]
       \exp \left\{ \bar J r^{-1}(D_c^{(s)}+ m_q)^{-1} J \right\}  }
 {\int [dU] e^{-{\cal A}_g} \det [(D_c^{(s)} + m_q)(1+r D_c^{(s)})^{-1}] } \nn
\eea
which is exactly in the same form of (\ref{eq:ZW_odwf}) except replacing
$ D_c $ with $ D_c^{(s)} $ [i.e., replacing $ S_{opt} $ (\ref{eq:S_opt})
with $ S $ (\ref{eq:S})]. However, such a difference has significant
consequences, as we will demonstrate in the next section.

In the limit $ N_s \to \infty $, $ S(H) $ (\ref{eq:S}) becomes
the sign function of $ H $,
\BAN
\lim_{N_s \to \infty} S(H)  = \frac{H}{\sqrt{H^2}} \ ,
\EAN
thus $ D_c^{(s)} $ (\ref{eq:rDcs}) is chirally symmetric.
However, {\it it still depends on $ a_5 $ through $ H $} (\ref{eq:H}).
In other words, the lattice spacing $ a_5 $
in the conventional DWF could be a relevant parameter\footnote{
As demonstrated by the tests in Section 3, the
fermion determinant has a strong dependence on $ a_5 $ for
2D topologically nontrivial gauge configurations.}
which must be taken to zero before its effective 4D lattice Dirac operator
\bea
\label{eq:Dms}
D^{(s)}(m_q) = (D_c^{(s)} + m_q )(1+ r D_c^{(s)})^{-1}
\eea
can behave properly.

However, taking the limit $ a_5 \to 0 $ is rather tedious in practice.
Therefore, it is vital to construct a domain-wall fermion action such that
its generating functional
$ Z[J, \bar J ] $ (thus any $n$-point Green's function) is
independent of $ a_5 $, for any $ a $ and $ N_s $.

This objective can be partially fullfilled if $ H $
(\ref{eq:H}) is replaced by $ H_w = \gamma_5 D_w $.
Then, in the limit $ N_s \to \infty $, $ S(H_w) $ (\ref{eq:S})
becomes independent of $ a_5 $,
\BAN
\lim_{N_s \to \infty} S(H_w) = \frac{a_5 H_w}{\sqrt{ a_5^2 H_w^2}}
                             = \frac{H_w}{\sqrt{H_w^2}} \ .
\EAN
However, at finite $ N_s $, $ S(H_w) $ still depends on $ a_5 $.

Aiming at modifying the action (\ref{eq:shamir}) such that
$ H $ is replaced by $ H_w $ in the tranfer matrix $ T $,
one immediately observes that if all factors
$ ( a_5 H_w P_{\pm} \pm 1 ) $ in the action (\ref{eq:As_3a})
are replaced by $ (a_5 H_w \pm 1) $, then $ T $ becomes
\bea
\label{eq:T'}
T'= \frac{1 - a_5 H_w}{1 + a_5 H_w} \ .
\eea
This amounts to adding the following terms to
the DWF action (\ref{eq:As_3a}),
\BAN
\label{eq:A_3c}
\delta {\cal A}_{fs} &=&
   - (r m_q + 1) \bar\psi_1 \gamma_5 a_5 H_w P_{+} \psi_{N_s} + \nn
&& + \bar\psi_1 \gamma_5 a_5 H_w P_{+} ( P_{-} \psi_1 + P_{+} \psi_{N_s} )
   + \bar\psi_1 \gamma_5 a_5 H_w P_{-} ( P_{-} \psi_2 + P_{+} \psi_1 ) \nn
&&
\hspace{-16mm}
+ \hspace{-2mm}
\sum_{s=2}^{N_s-1} \left\{
     \bar\psi_s \gamma_5 a_5 H_w P_{+} ( P_{-} \psi_s + P_{+} \psi_{s-1} )
   + \bar\psi_s \gamma_5 a_5 H_w P_{-} ( P_{-} \psi_{s+1} + P_{+} \psi_s )
     \right\} \nn
&&
\hspace{-14mm}
+ \bar\psi_{N_s}\gamma_5 a_5 H_w P_{+}(P_{-}\psi_{N_s}+P_{+}\psi_{N_s-1})
   + \bar\psi_{N_s}\gamma_5 a_5 H_w P_{-}(P_{-}\psi_1 + P_{+} \psi_{N_s} ) \nn
&& - (r m_q + 1) \bar\psi_{N_s} \gamma_5 a_5 H_w P_{-} \psi_1 \ ,
\EAN
which can be simplified to
\bea
\label{eq:A_3d}
\delta {\cal A}_{fs} &=&
    - r m_q \bar\psi_1 \gamma_5 a_5 H_w P_{+} \psi_{N_s}
    + \bar\psi_1 \gamma_5 a_5 H_w P_{-} \psi_2  \nn
&&
+ \sum_{s=2}^{N_s-1} \left\{
      \bar\psi_s \gamma_5 a_5 H_w P_{+} \psi_{s-1}
    + \bar\psi_s \gamma_5 a_5 H_w P_{-} \psi_{s+1}
      \right\} \nn
&&
+ \bar\psi_{N_s}\gamma_5 a_5 H_w P_{+}\psi_{N_s-1}
- r m_q \bar\psi_{N_s} \gamma_5 a_5 H_w P_{-} \psi_1 \ .
\eea
Then adding (\ref{eq:A_3d}) to (\ref{eq:As_3a}), one obtains
\bea
\label{eq:borici}
&&
\hspace{-15mm}
{\cal A}'_{fs} = \sum_{s,s'=1}^{N_s} \sum_{x,x'}
\bar\psi(x,s)
\{ ( a_5 D_w(x,x') + \delta_{x,x'} ) \delta_{s,s'} \nn
&& \hspace{10mm}  -( \delta_{x,x'}- a_5 D_w(x,x') )
   ( P_{+} \delta_{s',s-1} + P_{-} \delta_{s',s+1} ) \} \psi(x',s') \ ,
\eea
which agrees with Borici's domain-wall action \cite{Borici:1999zw}.
The effective 4D lattice Dirac operator of (\ref{eq:borici})
is the same as (\ref{eq:Dms}) except (the argument of $ D_c^{(s)} $)
$ S(H) $ (\ref{eq:S}) is replaced by $ S(H_w) $.
Thus, at $ a_5 = 1 $, it (times a factor $ r=1/2m_0 $)
is exactly equal to the overlap Dirac operator
with $ (H_w^2)^{-1/2} $ approximated by the polar approximation, since
\BAN
S(H_w) =  \left\{ \begin{array}{ll}
          a_5 H_w \left( \frac{1}{N_s} +
   \frac{2}{N_s} \sum_{l=1}^{n} \frac{b_l}{a_5^2 H_w^2 + d_l} \right) \ ,
   &  \ N_s = 2n+1 \mbox{ (odd) } \nn
     a_5 H_w \
   \frac{2}{N_s} \sum_{l=1}^{n} \frac{b_l}{a_5^2 H_w^2 + d_l} \ ,
   &  \ N_s = 2n  \mbox{ (even) }  \nn
          \end{array} \right. \\
\EAN
where
\BAN
\label{eq:polar_bd}
b_l = \sec^2 \left[ \frac{\pi}{N_s} \left( l-\frac{1}{2} \right) \right],
\hspace{4mm}
d_l = \tan^2 \left[ \frac{\pi}{N_s} \left( l-\frac{1}{2} \right) \right] \ .
\EAN
Obviously, for any finite $ N_s $, $ S(H_w) $ depends on $ a_5 $.
Also, it does not preserve chiral symmetry optimally, since $ S(H_w) $
is not the optimal rational approximation of sgn($H_w$),
as shown in Ref. \cite{Chiu:2002ir}.

A theoretical inconsistency of the Borici DWF action (\ref{eq:borici})
is that the quark propagator $ (D_c' + m_q)^{-1} $ {\it cannot} be expressed
in terms of the correlation function of the quark fields constructed from the
boundary modes (\ref{eq:quark_s})-(\ref{eq:antiquark_s}), i.e.,
\BAN
r \langle q(x) \bar q(y) \rangle \ne ( D_c' + m_q )^{-1}_{x,y} \ ,
\hspace{4mm} r D_c' = \frac{1 + \gamma_5 S(H_w)}{1 - \gamma_5 S(H_w)} \ ,
\EAN
in any background gauge field. Thus the generating functional
(with localized quark sources $ J(x) $ and $ \bar J(x) $)
for $n$-point Green's function of
$ q $ and $ \bar q $ is not well-defined, and {\it cannot} be derived.

In this paper, a solution to this problem is presented in the last subsection.
Explicitly, one can modify the action (\ref{eq:borici}) by
adding two boundary layers with identity transfer matrix
(i.e., without $ a_5 D_w $ couplings) at $ s=0 $ and $ s=N_s + 1 $,
with boundary conditions (\ref{eq:bc1_new})-(\ref{eq:bc2_new}),
and defining the quark fields from the new boundary modes
(\ref{eq:q_odwf})-(\ref{eq:bar_q_odwf}),
similar to the new optimal DWF (\ref{eq:ODWF1}).
Then the generating functional for $ n $-point Green's function
of $ q $ and $ \bar q $ can be derived in the same way as that
presented in the last subsection, and the result is in the same form of
(\ref{eq:ZW_odwf}) except replacing each $ T_s (s=1, \cdots, N_s) $
(\ref{eq:Ts}) with $ T' $ (\ref{eq:T'}).



\section{Tests}

In this section, we compute the index, the axial anomaly and
the fermion determinant of the effective lattice
Dirac operator of the optimal DWF (\ref{eq:ODWF1})\footnote{
Note that both (\ref{eq:ODWF}) and (\ref{eq:ODWF1}) have the same
effective lattice Dirac operator (\ref{eq:Dm}), as shown in Section 2.},
and the conventional DWF (\ref{eq:shamir}) respectively,
in two-dimensional background $ U(1) $ gauge fields.
Our notations for the two-dimensional background gauge field
are the same as those in Eqs. (7)-(11) of Ref. \cite{Chiu:1998bh}.

The effective lattice Dirac operator of the optimal DWF
can be computed from the 5-dimensional DWF operator (\ref{eq:ODWF1}).
Setting $ m_q = 2 m_0 $ ($ r m_q = 1 $) in (\ref{eq:quark_prop_54}),
one has
\BAN
\hspace{-8mm}
& & ( 1 + r D_c )^{-1}(x,y) \nn
\hspace{-8mm}
&=& \sum_{s,s'} ( P_{-} \delta_{s,0} + P_{+} \delta_{s,N_s+1} )
    \{ {\cal D}'^{-1}(2m_0) \} (x,s;y,s')
    ( P_{+} \delta_{s',0} + P_{-} \delta_{s',N_s+1} ).
\EAN
Then the massless lattice Dirac operator can be obtained by a
subtraction
\bea
D = D_c ( 1 + r D_c )^{-1} = 1 - (1 + r D_c )^{-1} \ ,
\eea
and the massive one by
\bea
\label{eq:Dmq}
D(m_q)=(D_c + m_q)(1 + rD_c)^{-1}=1+(m_q - 1)(1 + r D_c)^{-1} \ .
\eea

Similarly, we can obtain the effective lattice
Dirac operator of the conventional DWF (\ref{eq:shamir}),
with the boundary modes at $ s=1 $ and $ N_s $.

In the following tests, we fix $ m_0 = 1 $ ($ r = 1/2 $) in
$ D_w $ (\ref{eq:Dw}), and $ m_q = 0 $
(except for the fermion determinants in Table 2 with $ m_q = 0.05 $).
With $ N_s $ in the range $ 10 \le N_s \le 32 $,
we first check that the eigenvalues of the effective
lattice Dirac operators fall on a circle of radius one ($1/2r=1$)
centered at one. For nontrivial gauge backgrounds,
the real eigenmodes (zero and two) have definite ($+1$ or $-1$) chirality
and satisfy the chirality sum rule \cite{Chiu:1998bh}
\bea
\label{eq:chirality_sum_rule}
n_{+} + N_{+} = n_{-} + N_{-} \ ,
\eea
where $ n_{\pm} ( N_{\pm} ) $ stand for the numbers of zero (+2) eigenmodes
with $ \pm 1 $ chirality respectively.
These real eigenmodes are topological, i.e., robust under local fluctuations
of the gauge fields. For any (trivial or nontrivial) gauge background,
the complex eigenmodes must come in conjugate pairs, each with zero chirality.
The Atiyah-Singer index theorem ($ n_{+} - n_{-} = Q $)
holds for any gauge configuration (with topological charge $ Q $)
satisfying the topological bound \cite{Chiu:1999vk}
\bea
\label{eq:top_bd}
\forall_x \ | \bar\rho(x) | < 0.28  \hspace{2mm}
\eea
where $ \bar\rho(x) $ is the topological charge (density) inside a unit
square (of area $ a^2 $) centered at $ x $,
\bea
\label{eq:rho_bar}
\bar\rho(x) = \frac{1}{a^2} \int_{x_1 -a/2}^{x_1+a/2} dy_1
              \int_{x_2 -a/2}^{x_2+a/2} dy_2 \ \frac{1}{2\pi} F_{12} (y) \ .
\eea
Thus, we conclude that the effective lattice Dirac operators of
the optimal DWF and the conventional DWF both are
topologically-proper for sufficiently large $ N_s $.


\subsection{Axial anomaly}

Since the effective lattice Dirac operator $ D $ of domain-wall fermions
($ N_s \to \infty $) satisfies the Ginsparg-Wilson relation (\ref{eq:gwr}),
its axial anomaly (for any $ N_s $) can be defined as \cite{Luscher:1998pq}
\bea
\label{eq:ax_gw}
\anxL = \tr [\gamma_5 (1 - r D(x,x))]= - \frac{1}{2} \tr [{\cal S}(x,x)] \ ,
\eea
where the trace runs over Dirac and color indices. Here the
operator $ {\cal S} $ becomes $ S_{opt} $ (\ref{eq:S_opt}) for the
optimal DWF, and $ S $ (\ref{eq:S}) for the conventional DWF.

For sufficiently large $ N_s $ and for gauge configurations
satisfying the topological bound (\ref{eq:top_bd}), I checked that
the sum of the axial anomaly over all sites is equal to the index
of $ D $ and the topological charge $ Q $
\bea
\label{eq:anomaly_sum}
\sum_x \anxL = n_{+} - n_{-} = Q \ ,
\eea
for both the optimal DWF and the conventional DWF.
However, this does not necessarily imply that $ \anxL $
is equal to the topological charge density at each site.

In the following, I compute the axial anomaly $ \anxL $
for the overlap Dirac operator, the optimal DWF, and the
conventional DWF respectively, and compare them with
the topological charge density $ \bar\rho(x) $ (\ref{eq:rho_bar})
of the gauge background on the torus.
The deviation of the axial anomaly can be measured in terms of
\bea
\label{eq:delta}
%\delta = \frac{1}{N}
%         \sum_{x} \frac{|\anxL - a^2 \bar\rho(x)|} {a^2|\bar\rho(x)|}
\delta = \frac{1}{N} \sum_{x} |\anxL - a^2 \bar\rho(x)|
\eea
where
%$ N $ is the total number of sites of the 2D lattice, and
$ \bar\rho(x) $ is the topological charge density inside
a unit square (of area $ a^2 $) centered at $ x $.

In a nontrivial $ U(1) $ gauge background with parameters $ Q = 1 $,
$ h_1 = 0.1 $, $ h_2 = 0.2 $, $ A_1^{(0)} = 0.3 $, $ A_2^{(0)} = 0.4 $
and $ n_1 = n_2 = 1 $ (as defined in Eqs. (7) and (8) in
Ref. \cite{Chiu:1998bh}) on the $ 8 \times 8 $ lattice,
the axial anomaly $ \anxL $ and its deviation
$ \delta $ are computed for the overlap, the optimal DWF
and the conventional DWF respectively. Here the 2D overlap Dirac
operator is computed exactly via diagonalization.

The results are :
\beq
\label{eq:delta_2d}
\delta = \left\{  \begin{array}{ll}
  4.75 \times 10^{-3}, &   \mbox{overlap} \ ,   \\
  4.75 \times 10^{-3}, &   \mbox{optimal DWF } (N_s = 16)  \ ,   \\
  5.44 \times 10^{-3}, &   \mbox{conventional DWF } (N_s = 16, 32)  \ .   \\
                  \end{array}
         \right.
\eeq

The slightly larger deviation of the axial anomaly of the
conventional DWF indicates that its effective lattice Dirac operator
is less localized than that of the optimal DWF.
Note that there is no improvements by increasing
$ N_s = 16 $ to $ N_s = 32 $.

%This implies that the $ \epsilon $ in the locality bound
%\bea
%|| \Id - U(p) || < \epsilon, \hspace{4mm} \mbox{for all plaquettes}
%\eea
%for the conventional DWF is smaller than that of
%the overlap Dirac operator, and the optimal DWF.

For all background gauge configurations we have tested,
the optimal DWF always gives anomaly deviation ($ \delta $)
smaller than that of the conventional DWF.


\subsection{Fermion determinant}

In quantum field theory, the fermion determinant $ \det(D) $ is
proportional to the exponentiation of the one-loop effective action
which is the summation of any number of external sources interacting
with one internal fermion loop. Thus it is one of the most crucial
quantities to be examined in any lattice fermion formulations.
Even though a (massless) lattice Dirac operator is topologically-proper
(i.e., with the correct index and axial anomaly), it does not necessarily
imply that its non-zero complex eigenvalues behave properly.
This can be tested by computing its fermion determinant
\bea
\det(D) = ( 1 + e^{i \pi} )^{( n_{+} + n_{-} ) } \det'(D)
\eea
where $ \det'(D) $ is equal to the product of all non-zero eigenvalues.
Since the effective lattice Dirac operators of
the optimal DWF and the conventional DWF
are $\gamma_5$-hermitian, the eigenvalues of $ D $ are either real or
come in complex conjugate pairs, and $ \det'(D) $ must be real and positive.
For $ Q = 0 $, we have $ n_{+} + n_{-} = 0 $ and $ \det(D) = \det'(D) $.
For $ Q \neq 0 $, we have $ n_{+} + n_{-} \neq 0 $ and $ \det(D) = 0 $, but
$ \det'(D) $ still provides important information about the spectrum.
In continuum, exact solution of fermion determinant in the general
background $ U(1) $ gauge fields on a torus ($ L_1 \times L_2 $)
was obtained in Ref. \cite{Sachs:en}.

With nontrivial gauge backgrounds, we compute $ \det'(D) $
for the overlap Dirac operator, the optimal DWF, and the conventional DWF
respectively, and compare them with the exact solution in continuum.
For simplicity, we turn off the harmonic part
($ h_1 = h_2 = 0 $) and the local sinusoidal fluctuations
($ A_1^{(0)} = A_2^{(0)} = 0 $), and examine
the change of $ \det'(D) $ with respect to the topological charge $ Q $.
For these gauge configurations, the exact solution \cite{Sachs:en} is
\bea
\det'[ D(Q)] = {\cal N} \left( \frac{L_1 L_2 }{2 |Q| } \right)^{|Q|/2}
\label{eq:exactD}
\eea
where the normalization constant $ {\cal N} $ is fixed by
$$
{\cal N} = \sqrt{ \frac{2}{L_1 L_2 } }
$$
such that $ \det'[D(1)] = 1 $.

In Table 1, the fermion determinants $ \det'(D) $ on the
$ 8 \times 8 $ lattice are listed.
It is clear that the fermion determinant of the optimal
DWF ($N_s=10$) highly agrees with that
of the (exact) overlap Dirac operator, and both
of them are in good agreement with the exact solution on the $ 8 \times 8 $
torus. On the other hand, the fermion determinant of the conventional
DWF ($N_s=10$, $a_5=1$) does not agree with the exact solution.
Further, the discrepancies do not decrease as $ N_s $ is increased.
This suggests that the error is not due to finite $ N_s $,
but the lattice spacing $ a_5 $ which enters the effective lattice
Dirac operator (\ref{eq:Dms}) through $ S(H) $ (\ref{eq:S}).
If this is the case, then decreasing $ a_5 $ would improve the results.
This is demonstrated in the last column with $ a_5 = 0.5 $ and $ N_s = 20 $,
which gives results much better than those
of the column with $ N_s = 32, a_5=1 $.

In Table 2, the fermion determinants for massive fermion
with $ m_q = 0.05 $ are listed,
for the same nontrivial gauge backgrounds in Table 1.
The massive overlap Dirac operator is $ [m_q + ( 1 - r m_q ) D_{o}] $,
and the effective lattice Dirac operators of the optimal DWF and the
conventional DWF are computed via (\ref{eq:Dmq}).
In each case (column), the normalization constant is fixed
such that $ \det[D(m_q)]=1 $ for $ Q=0 $.
Clearly, the fermion determinant of the optimal DWF ($ N_s = 10 $)
is in excellent agreement with that of the (exact) overlap, while
the conventional DWF disagrees with the overlap and the optimal DWF.
Further, the discrepancies do not decrease as $ N_s $ is increased.
However, if $ a_5 $ is reduced from 1 to 0.5,
then the results are significantly improved, as shown in the last column.
This demonstrates that the massive fermion determinant of the
conventional DWF is also sensitive to $ a_5 $, for topologically
nontrivial gauge configurations.


{\footnotesize
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
   & exact & overlap & optimal DWF & \multicolumn{3}{c|}{conventional DWF} \\
\hline
 Q &   &   & $ N_s=10 $ & $ N_s=10 $ & $ N_s=32 $ & $N_s=20$             \\
   &   &   &            & $ a_5 = 1$ & $ a_5 =1 $ & $a_5=0.5$             \\
\hline
\hline
%     exact    overlap  optimal DWF DWF(Ns=10,32,20)
%
 1 & 1.00000 & 1.00000 & 1.00000 & 1.00000 & 1.00000 & 1.00000  \\
\hline
 2 & 2.82843 & 2.79094 & 2.79094 & 1.44349 & 1.44558 & 2.08267    \\
\hline
 3 & 6.15840 & 5.96482 & 5.96482 & 1.63969 & 1.64629 & 3.32119     \\
\hline
 4 & 11.3137 & 10.7257 & 10.7257 & 1.61127 & 1.62411 & 4.46412     \\
\hline
 5 & 18.3179 & 16.9800 & 16.9800 & 1.43207 & 1.45172 & 5.29955 \\
\hline
 6 & 26.8177 & 24.3215 & 24.3215 & 1.18079 & 1.20677 & 5.71414 \\
\hline
 7 & 36.1083 & 32.0978 & 32.0978 & 0.91741 & 0.94853 & 5.69848 \\
\hline
 8 & 45.2548 & 40.5552 & 40.5551 & 0.67521 & 0.71130 & 5.37711 \\
\hline
 9 & 53.2732 & 45.9271 & 45.9269 & 0.47955 & 0.51653 & 4.68732  \\
\hline
10 & 59.3164 & 50.6766 & 50.6761 & 0.32497 & 0.36279 & 3.91209  \\
\hline
\end{tabular}
\end{center}
\caption{The fermion determinant versus the topological charge $ Q $.
The normalization constant is chosen such that $ \det'D = 1 $ for $ Q=1 $.
The results on the $ 8 \times 8 $ lattice are listed for the overlap Dirac
operator, the effective lattice Dirac operators of the optimal DWF and
the conventional DWF respectively.
The exact solutions on the $ 8 \times 8 $ torus are computed
according to Eq. (\ref{eq:exactD}). }
\label{table:1}
\end{table}
}




{\footnotesize
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
   & overlap & optimal DWF & \multicolumn{3}{c|}{conventional DWF} \\
\hline
 Q &         & $ N_s=10 $ & $ N_s=10 $ & $ N_s=32 $ & $N_s=20$      \\
   &         &            & $ a_5=1$   & $ a_5 =1 $ & $a_5=0.5$     \\
\hline
\hline
%       overlap    optimal DWF   DWF(Ns=10,32,20)
%
  0  &  1.00000  &   1.00000 &  1.00000  &  1.00000 & 1.00000 \\
\hline
  1  &  5.27834  &   5.27834 &  10.4991  &  10.4943 & 7.00364  \\
\hline
  2  &  0.73042  &   0.73042 &  0.75612  &  0.75684 & 0.72616  \\
\hline
  3  & $ 7.768 \times 10^{-2} $ & $ 7.768 \times 10^{-2} $
     & $ 4.289 \times 10^{-2} $ & $ 4.304 \times 10^{-2} $
     & $ 5.782 \times 10^{-2} $ \\
\hline
  4  & $ 6.961 \times 10^{-3} $ & $ 6.961 \times 10^{-3} $
     & $ 2.106 \times 10^{-3} $ & $ 2.121 \times 10^{-3} $
     & $ 3.893 \times 10^{-3} $  \\
\hline
  5  & $ 5.497 \times 10^{-4} $ & $ 5.497 \times 10^{-4} $
     & $ 9.371 \times 10^{-5} $ & $ 9.474 \times 10^{-5} $
     & $ 2.332 \times 10^{-4} $ \\
\hline
  6  & $ 3.929 \times 10^{-5} $ & $ 3.929 \times 10^{-5} $
     & $ 3.883 \times 10^{-6} $ & $ 3.935 \times 10^{-6} $
     & $ 1.289 \times 10^{-5} $\\
\hline
  7  & $ 2.588 \times 10^{-6} $ & $ 2.588 \times 10^{-6} $
     & $ 1.535 \times 10^{-7} $ & $ 1.535 \times 10^{-7} $
     & $ 6.830 \times 10^{-7} $ \\
\hline
  8  & $ 1.633 \times 10^{-7} $ & $ 1.633 \times 10^{-7} $
     & $ 5.621 \times 10^{-9} $ & $ 5.794 \times 10^{-9} $
     & $ 3.407 \times 10^{-8} $\\
\hline
  9  & $ 9.235 \times 10^{-9} $ & $ 9.235 \times 10^{-9} $
     & $ 2.405 \times 10^{-10}$ & $ 2.103 \times 10^{-10}$
     & $ 2.156 \times 10^{-9} $ \\
\hline
\end{tabular}
\end{center}
\caption{The fermion determinant $ \det[D(m_q)] $ versus the topological
charge $ Q $ for effective lattice Dirac operator with bare mass
$ m_q = 0.05 $ on the $ 8 \times 8 $ lattice.
The normalization constant is chosen such that
$ \det D(m_q) = 1 $ for $ Q=0 $.
The results are listed
for the overlap, the optimal DWF ($N_s = 10 $),
and the conventional DWF with ($N_s=10$, $a_5=1$), ($N_s=32$, $a_5=1$) and
($N_s=20$, $a_5=0.5$) respectively.}
\label{table:2}
\end{table}
}

On the other hand, for 2D topologically trivial gauge configurations,
the fermion determinant of the conventional DWF only has a weak dependence
on $ a_5 $. This can be demonstrated as follows.
For simplicity, we turn off the local sinusoidal fluctuations
($ A_1^{(0)} = A_2^{(0)} = 0 $), and examine
the change of $ \det(D) $ versus the harmonic parts $ h_1=h_2 $.
In Table 3, the fermion determinants versus the harmonic parts $ (h_1=h_2) $
on a $ 8 \times 8 $ torus (lattice)
are listed for the exact solution \cite{Sachs:en},
the overlap, the optimal DWF, and the conventional DWF respectively.
Clearly, all results are in very good agreement with each other.
Further, the fermion determinant of the conventional DWF is
insensitive to $ a_5 $, as comparing the corresponding entries of the
last two columns $ (N_s=10, a_5=1) $ and $ (N_s=20,a_5=0.5) $.


{\footnotesize
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
   & exact & overlap & optimal DWF & \multicolumn{2}{c|}{conventional DWF} \\
\hline
$ h_1=h_2 $  &   &   & $ N_s=10 $ & $ N_s=10 $ & $N_s=20$             \\
             &   &   &            & $ a_5 =1 $ & $a_5=0.5$             \\
\hline
\hline
%     exact    overlap  optimal DWF   DWF(Ns=10,20)
%
0.0 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000  \\
\hline
0.1 & 1.00000 & 1.00000 & 1.00000 & 1.00000 & 1.00000  \\
\hline
0.2 & 3.34414 & 3.40518 & 3.40518 & 3.39253 & 3.40883  \\
\hline
0.3 & 5.71957 & 5.97173 & 5.97137 & 5.92350 & 5.98565  \\
\hline
0.4 & 7.24506 & 7.73701 & 7.73701 & 7.64986 & 7.76372   \\
\hline
0.5 & 7.73926 & 8.34167 & 8.34166 & 8.23777 & 8.37383  \\
\hline
0.6 & 7.24506 & 7.73701 & 7.73701 & 7.64986 & 7.76372 \\
\hline
0.7 & 5.71957 & 5.97173 & 5.97137 & 5.92350 & 5.98565 \\
\hline
0.8 & 3.34414 & 3.40518 & 3.40518 & 3.39253 & 3.40883 \\
\hline
0.9 & 1.00000 & 1.00000 & 1.00000 & 1.00000 & 1.00000  \\
\hline
1.0 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000  \\
\hline
\end{tabular}
\end{center}
\caption{The fermion determinant versus the harmonic parts $ (h_1=h_2) $,
on the $ 8 \times 8 $ lattice. In each column, the normalization constant
is chosen such that $ \det(D) = 1 $ for $ h_1=h_2=0.1 $. Note that
the fermion determinants satisfy the symmetries of the exact solution:
(i) periodicities in $ h_1 $ and $ h_2 $ with period 1;
(ii) reflection symmetries with respect to $ h_1=1/2 $ and $ h_2 = 1/2 $.
}
\label{table:3}
\end{table}
}

Even though above tests are performed with 2D abelian gauge backgrounds,
one expects that the $ a_5 $ effects might also occur
in the quark propagator (as well as the fermion determinant),
in lattice QCD with the conventional DWF.
We are now computing the quark propagator for several different
values of $ a_5 $, on the $ 8^3 \times 24 $ lattice, for 100
gauge configurations. Then we will extract the
pion mass as well as other observables, and study their
dependence on $ a_5 $. These results will be presented elsewhere.




\section{Concluding Remarks}


To formulate lattice QCD with domain-wall fermions, it seems to
be inevitable to introduce parameters $ N_s $ and $ a_5 $
(in the fifth dimension) to the theory.
Presumably, only in the limit $ N_s \to \infty $ and $ a_5 \to 0 $,
the effective 4D theory agrees with the target theory (QCD), up to the
finite volume and discretization errors.
The relevant question is how to minimize the dependence on these two
extra parameters. More importantly, if the dependence on $ a_5 $ is
too strong, then at nonzero $ a_5 $, the effective 4D theory might
differ significantly from the target theory, even in the limit
$ N_s \to \infty $. Therefore, it is desirable to
eliminate $ a_5 $ completely and to preserve the chiral
symmetry optimally for any $ N_s $.

In the limit $ N_s \to \infty $ followed by $ a_5 \to 0 $,
the fermion determinant (including the contribution of pseudofermions)
of the conventional DWF gives the effective 4D lattice Dirac operator
\BAN
D(m_q) = (D_c + m_q)(1+rD_c)^{-1} = m_q + (1-rm_q) D
\EAN
where
\BAN
D_c &=& 2m_0 \frac{1+\gamma_5 H_w (H_w^2)^{-1/2}}
                  {1-\gamma_5 H_w (H_w^2)^{-1/2}} \nn
  D &=& D_c (1+rD_c)^{-1} = m_0 [ 1 + \gamma_5 H_w (H_w^2)^{-1/2} ] \ ,
\EAN
while the quark propagator in background gauge field is
\bea
\label{eq:quark_p}
\langle q(x) \bar q(y) \rangle = ( D_c + m_q )_{x,y}^{-1} \ .
\eea

Since the optimal rational approximation for
$ \mbox{sgn}(H_w) = H_w (H_w^2)^{-1/2} $ is $ H_w R_Z(H_w^2) $,
where $ R_Z(x) $ is the Zolotarev optimal rational polynomial for $ x^{-1/2}$
(i.e., $ R_Z^{(n-1,n)}(x) $ for $N_s=2n$, or
$ R_Z^{(n,n)}(x) $ for $ N_s = 2n + 1 $),
the relevant question is whether one can construct a DWF action
such that its effective lattice Dirac operator for finite $ N_s $ is
\BAN
D(m_q) = m_q + (m_0-m_q/2) [ 1 + \gamma_5 H_w R_Z(H_w^2) ] \ ,
\EAN
in view of that the $ S $ operator in the conventional DWF
is a rational polynomial approximation of $ \mbox{sgn}(H) $. The
solution to this problem is the original optimal DWF action \cite{Chiu:2002ir}.
The salient feature of the optimal DWF is that its
weights $ \{ \omega_s \} $ (\ref{eq:omega}) not only give $ S_{opt} $,
the optimal rational approximation of sgn($ H_w $), but
also eliminate $ a_5 $ completely.

However, the correlation function of $ q(x) $ and $ \bar q(y) $
is {\it not} equal to the quark propagator $ (D_c+m_q)^{-1}_{x,y} $,
nor the inverse of the effective 4D lattice Dirac operator,
$ D(m_q)^{-1}_{x,y} $. In other words, either $ (D_c + m_q)^{-1}_{x,y} $
or $ D(m_q)^{-1}_{x,y} $ {\it cannot} be expressed in terms of
the correlation function of {\it local} quark fields $ q(x) $ and
$ \bar q (y) $, but $ q(x) $ and some ``nonlocal" field variable
$ \bar Q(y) $ which depends on $ \bar q(x) $ at all $x$.
This posed a rather serious problem to the optimal DWF,
since it implies that the basic concept of {\it local} fields $ q(x) $
and $ \bar q(x) $ in quantum field theory has to be abandoned
in the optimal DWF. Even if $ \bar Q(y) $ might be exponentially-local,
it is still unsatisfactory in comparison with the exactly local
quarks fields $ q $ and $ \bar q $ in the conventional DWF.

The solution to this problem is presented in this paper:
Appending two boundary layers at $ s=0 $ and $ s=N_s + 1 $
with $ \omega_0 = \omega_{N_s+1} = 0 $. Then the correlation
function of (local) quark fields $ q(x) $ and $ \bar q(y) $
constructed from these new boundary modes
(\ref{eq:q_odwf})-(\ref{eq:bar_q_odwf}) satisfies (\ref{eq:quark_p}).
Consequently, one can derive the generating functional for $n$-point
Green's function of quark fields
(with local Grassman sources $ J $ and $ \bar J $)
\BAN
e^{W[J,\bar J]} =
\frac{ \int [dU] e^{-{\cal A}_g }  \det [(D_c + m_q)(1+r D_c)^{-1}]
       \exp \left\{ \bar J ( D_c + m_q )^{-1} J \right\} }
     { \int [dU] e^{-{\cal A}_g} \det [(D_c + m_q)(1+rD_c)^{-1}] } \ .
\EAN


Then the bilinear (and higher order) quark operators manifest their
continuum discrete symmetries at finite lattice spacing, since
the chiral projectors for both $ q $ and $ \bar q $ are the
usual $ P_{\pm} = ( 1 \pm \gamma_5 )/2 $.
In particular, the V-A structure of the left-handed quark current
in the effective weak Hamiltonian is preserved exactly at any finite
lattice spacing, which is crucial for investigating the
weak matrix elements in lattice QCD, e.g., the CP violation
in the $K\to\pi\pi$ decay through $\bar K^0$-$K^0$ mixing.

From the 2D tests in Section 3, we see that the fermion determinant
(ratio) of the effective lattice Dirac operator $ D $ of the
conventional DWF does not agree with the exact solution
for any topologically nontrivial background gauge field,
even though $ D $ can capture the topology of the gauge background
(i.e, reproducing topological zero and nonzero real eigenmodes with
definite chirality, satisfying the Atiyah-Singer index theorem on a
finite lattice, and its axial anomaly in agreement with the topological
charge density). It is interesting to see that a lattice Dirac operator
satisfies the Ginsparg-Wilson relation, $\gamma_5$-hermiticity,
doubler-free, exponential-locality, and the correct continuum (free fermion)
behavior, but could not reproduce the proper fermion determinant for
topologically nontrivial gauge backgrounds, due to its dependence
on $ a_5 (\ne 0) $. This warrants further studies on $ a_5 $ effects
in lattice QCD with the conventional DWF (e.g., to investigate to
what extent the pion mass or other observables depending on $ a_5 $).


Finally, it is instructive to compare the quark fields
in lattice QCD with Ginsparg-Wilson Dirac operator to the
exactly local quark fields in the (optimal/conventional) DWF.
In general, the quark action of lattice QCD with Ginsparg-Wilson
Dirac operator reads
\bea
\label{eq:Aq}
{\cal A}_q = \sum_{x,y} \bar q(x) D_{x,y}(m_q) q(y) \ ,
\eea
where
\BAN
D(m_q) = (D_c + m_q)(1+rD_c)^{-1} = m_q + ( 1 - r m_q ) D,
\hspace{4mm} D = D_c ( 1 + r D_c )^{-1} \  ,
\EAN
and $ D $ satisfies the Ginsparg-Wilson relation (\ref{eq:gwr}).
Here we assume that $ D $ is doubler-free,
exponentially-local for sufficiently smooth background gauge fields,
and has the correct continuum (free fermion) behavior.
Then the quark propagator in background gauge field is
\bea
\label{eq:q_bar_q}
\langle q(x) \bar q(y) \rangle =
\frac{\int [d \bar q][d q] q(x) \bar q (y) e^{-{\cal A}_q} }
     {\int [d \bar q][d q] e^{-{\cal A}_q} } = D^{-1}_{x,y}(m_q)  \ .
\eea
But this cannot be the valence quark propagator since it
breaks the chiral symmetry even in the massless ($m_q = 0$) limit.
Thus the quark propagator has to be extracted as
\BAN
(D_c + m_q)_{x,y}^{-1}
= ( 1 - r m_q )^{-1} [ D^{-1}_{x,y} (m_q) - r \delta_{x,y}] \ .
\EAN
In other words, one has to replace either $ q(x) $ or $ \bar q(y) $
in the quark bilinear of (\ref{eq:q_bar_q}) with one of the followings:
\BAN
\label{eq:q_h}
q(x) &\to& \hat{q}(x) = \sum_z ( 1 + r D_c )^{-1}_{x,z} q(z)
                      = \sum_z ( 1 - r D )_{x,z} q(z) \ ,
\EAN
or
\BAN
\label{eq:bar_q_h}
\bar q(y) &\to&  \hat{\bar q}(y) =
\sum_{z} \bar q(z) ( 1 + r D_c )^{-1}_{z,y}
 = \sum_z \bar q(z) ( 1 - r D )_{z,y} \ .
\EAN
Note that both above transformations are exponentially-local.
Suppose one replaces $ q(x) $ with $ \hat{q}(x) $, then
the quark propagator (in background gauge field) becomes
\bea
\label{eq:quark_qcd}
\langle \hat{q}(x) \bar q(y) \rangle = (D_c + m_q)_{x,y}^{-1} \ .
\eea
Even though the concept of local field is only violated exponentially
by $ \hat{q}(x) $, it is still unsatisfactory from the theoretical
viewpoint, in comparison with the exactly local $ q $ and $ \bar q $
in the (optimal/conventional) DWF. To impose the exact locality of
$ \hat{q} $, one can rewrite the quark action as
\bea
\label{eq:Aqc}
{\cal A}_q = \sum_{x,y} {\bar q}(x) (D_c + m_q)_{x,y} \hat{q}(y) \ ,
\eea
which, however, is also unsatisfactory since $ D_c $ is nonlocal.
In general, it seems to be impossible to formulate lattice QCD
with any Ginsparg-Wilson lattice Dirac operator such that the
quark fields and the action both are consistent. However,
if the Ginsparg-Wilson lattice Dirac operator
(e.g., the overlap Dirac operator with Zolotarev rational approximation)
is equal to the effective 4D lattice Dirac operator of certain
DWF action (e.g., the optimal DWF), then one can resort to the
DWF action, in which the quark fields
$ q $ and $ \bar q $ (e.g., (\ref{eq:q_odwf})-(\ref{eq:bar_q_odwf}))
are well-defined in terms of the boundary modes, with exact locality
as well as the usual chiral projectors $ P_{\pm} = ( 1 \pm \gamma_5 )/2 $.



\section*{Appendix}

In this appendix, we derive a formula for computing
the inverse of the effective 4D lattice Dirac operator,
with the original optimal DWF operator (\ref{eq:ODWF}).

From (\ref{eq:det_54}), we have
\bea
\label{eq:det_D}
\det [ {\cal D}(m_q) {\cal D}^{-1}(2m_0)]
= \det [ r( D_c + m_q )(1+rD_c)^{-1} ] = \det [ r D(m_q) ]
\eea
or equivalently,
\bea
\label{eq:det_Di}
\det [ {\cal D}^{-1}(m_q) {\cal D}(2m_0)] = \det [ r^{-1} D^{-1}(m_q) ]
\eea
where $ {\cal D} $ is the optimal DWF
operator defined in (\ref{eq:ODWF}),
while $ D(m_q) $ is the effective 4D lattice Dirac operator.
It follows that there may exist an operator $ Q $ such that
\bea
\label{eq:D54}
\{ Q {\cal D}^{-1}(m_q) {\cal D}(2m_0) Q^{-1} \} (x,1;x',1)
= r^{-1} D^{-1}(m_q) (x,x') \ .
\eea

Now we want to find an explicit expression of $ Q $.
Using the Grassman integral formula
\BAN
\frac{1}{\det{\cal D}(m_q)}
\int [d\bar\psi] [d\psi] \psi_s(y) \bar\psi_{s'}(z) e^{-{\cal A}_f}
= {\cal D}^{-1}(m_q) (y,s;z,s') \ ,
\EAN
we rewrite the l.h.s. of (\ref{eq:D54}) as
\bea
\label{eq:D54a}
\hspace{-16mm}
& & \{ Q{\cal D}^{-1}(m_q){\cal D}(2m_0) Q^{-1} \} (x,1;x',1) \nn
\hspace{-16mm}
&=& \frac{1}{\det {\cal D}(m_q)}
\int [d\bar\psi][d\psi] \{ Q \psi \}(x,1)
\{ \bar\psi {\cal D}(2m_0) Q^{-1} \}(x',1) e^{-{\cal A}_f} \ .
\eea

From the transformation (\ref{eq:f1i})-(\ref{eq:f3i}),
one immediately sees that the optimal DWF action (\ref{eq:ODWF})
can be rewritten as
\bea
\label{eq:A_f5}
{\cal A}_f &=& \sum_{s,s'=1}^{N_s} \sum_{x,x'}
\bar\psi_s(x) {\cal D}(m_q)(x,s;x's') \psi_{s'}(x')  \nn
&=& \sum_{s,s'=1}^{N_s} \sum_{x,x'}
\bar\eta_s(x) \{(\Omega a_5 H_w -1)^{-1} \gamma_5{\cal D}(m_q)
                    B^{-1}\}(x,s;x',s') \eta_{s'}(x') \nn
&=& \bar\eta_1 ( P_{-} - r m_q P_{+} ) \eta_1
                  - \bar\eta_1 T_1^{-1} \eta_2
               + \sum_{s=2}^{N_s-1} \left\{
                 \bar\eta_s \eta_s - \bar\eta_s T_s^{-1} \eta_{s+1}
                 \right\}              \nn
&&             + \bar\eta_{N_s} \eta_{N_s}
               - \bar\eta_{N_s} T_{N_s}^{-1} ( P_{+} - r m_q P_{-} ) \eta_1
\eea
where
\BAN
\Omega(x,s;x',s') = \omega_s \delta_{s,s'} \delta_{x,x'} \ ,
\EAN
and
\bea
\label{eq:B}
B(x,s;x',s') &=& \delta_{x,x'} (P_{-}\delta_{s,s'}+P_{+}\delta_{s-1,s'}) \ , \\
\label{eq:BI}
B^{-1}(x,s;x',s')
&=& \delta_{x,x'}(P_{-}\delta_{s,s'}+P_{+}\delta_{s+1,s'}) \ ,
\eea
with periodic boundary conditions in the fifth dimension.

Now it is clear that $ Q = B $ would satisfy (\ref{eq:D54}).
This can be seen as follows.
Using the transformation (\ref{eq:f1i})-(\ref{eq:f3i}),
the r.h.s. of (\ref{eq:D54a}) can be written as
\bea
\label{eq:D54b}
\frac{{\cal J}}{\det {\cal D}(m_q)}
\int [d\bar\eta][d\eta] e^{-{\cal A}_f} \eta_1(x)
\{\bar\eta (\Omega a_5 H_w -1)^{-1} \gamma_5 {\cal D}(2m_0) B^{-1} \}(x',1)
\eea
where $ {\cal J} $ is the Jacobian defined in (\ref{eq:j}).
Using (\ref{eq:A_f5}), the last expression in the integrand
of (\ref{eq:D54b}) can be simplified as
\bea
& & \{\bar\eta(\Omega a_5 H_w-1)^{-1}\gamma_5{\cal D}(2m_0)B^{-1}\}(x',1) \nn
&=& \bar\eta_1(x') (P_{-}-P_{+})
     - \{ \bar\eta_{N_s} T_{N_s}^{-1} \}(x') (P_{+}-P_{-})  \ .
\eea
Thus (\ref{eq:D54b}) becomes
\bea
\label{eq:D54c}
\frac{{\cal J}}{\det {\cal D}(m_q)} \
\int [d\bar\eta][d\eta] e^{-{\cal A}_f} \left[ - \eta_1(x) \bar\eta_1(x')
- \eta_1(x) \{ \bar\eta_{N_s} T_{N_s}^{-1} \}(x') \right]  \gamma_5 \ ,
\eea
which can be integrated successively from
$ (\eta_{N_s}, \bar\eta_{N_s}) $ to $ (\eta_1, \bar\eta_1) $.
Explicitly,
\BAN
\label{eq:D54d}
\hspace{-10mm}
& & \int [d\bar\eta][d\eta] e^{-{\cal A}_f}
 \eta_1(x) \{ \bar\eta_{N_s} T_{N_s}^{-1} \}(x') \ \gamma_5  \nn
\hspace{-10mm}
&=& \hspace{-4mm} \int [d\bar\eta_1][d\eta_1]
e^{-\bar\eta_1 \left[(P_{-}-rm_q P_{+}) - \prod_{s=1}^{N_s} T_{s}^{-1}
      (P_{+}-rm_q P_{-}) \right] \eta_1 } \
\eta_1(x) \{ \bar\eta_1 \prod_{s=1}^{N_s} T_{s}^{-1} \}(x') \gamma_5 \ ,
\EAN
which is substituted into (\ref{eq:D54c}) to give
\BAN
\label{eq:D54e}
& & \frac{{\cal J}}{\det {\cal D}(m_q)} \int [d\eta_1][d\bar\eta_1]
e^{-\bar\eta_1 \left[(P_{-}-rm_q P_{+})- \prod_{s=1}^{N_s} T_{s}^{-1}
    (P_{+}-rm_q P_{-}) \right] \eta_1 } \times \nn
& & \hspace{20mm}
    \eta_1(x)
    \left[\bar\eta_1 (-1-\prod_{s=1}^{N_s} T_{s}^{-1} ) \right] (x') \ \gamma_5 \nn
&=& \left\{ \left[(P_{-}-rm_q P_{+})-
            \prod_{s=1}^{N_s} T_{s}^{-1}(P_{+}-rm_q P_{-}) \right]^{-1}
    \left(-1-\prod_{s=1}^{N_s} T_{s}^{-1}\right) \gamma_5 \right\}(x,x') \nn
&=& r^{-1} [( D_c + m_q )^{-1} ( 1 + r D_c )](x,x')
 =  r^{-1} D^{-1} (m_q) (x,x')
\EAN
where (\ref{eq:j}), (\ref{eq:det_54}) and (\ref{eq:identity}) have been used.

Therefore, we have
\bea
\label{eq:D5D4i_odwf}
\{ B {\cal D}^{-1}(m_q) {\cal D}(2m_0) B^{-1} \} (x,1;x',1)
= r^{-1} D^{-1}(m_q) (x,x') \ ,
\eea
and $ D^{-1}(m_q) $ can be obtained by solving the linear system
\bea
\label{eq:DYB}
{\cal D}(m_q) Y = {\cal D}(2m_0) B^{-1} \Id \ .
\eea
The solution of (\ref{eq:DYB}) gives
\bea
%\label{eq:DIBY}
D^{-1}(m_q)(x,x') = r (B Y)(x,1;x',1) \ ,
\eea
and the valence quark propagator (\ref{eq:v_quark}).



\bigskip
\bigskip
\flushpar
{\bf Acknowledgement }
\bigskip

\noindent

This work was supported in part by National Science Council,
ROC, under the grant number NSC91-2112-M002-025.

\bigskip
\bigskip

\vfill\eject


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\end{thebibliography}


\end{document}



hep-lat0303008.extracted_bib
\begin{thebibliography}{15}

%\cite{Rubakov:bb}
\bibitem{Rubakov:bb}
V.~A.~Rubakov and M.~E.~Shaposhnikov,
%``Do We Live Inside A Domain Wall?,''
Phys.\ Lett.\ B {\bf 125}, 136 (1983).
%%CITATION = PHLTA,B125,136;%%

%\cite{Callan:sa}
\bibitem{Callan:sa}
C.~G.~Callan and J.~A.~Harvey,
%``Anomalies And Fermion Zero Modes On Strings And Domain Walls,''
Nucl.\ Phys.\ B {\bf 250}, 427 (1985).
%%CITATION = NUPHA,B250,427;%%


%\cite{Kaplan:1992bt}
\bibitem{Kaplan:1992bt}
D.~B.~Kaplan,
%``A Method for simulating chiral fermions on the lattice,''
Phys.\ Lett.\ B {\bf 288}, 342 (1992)
%[arXiv:hep-lat/9206013].
%%CITATION = HEP-LAT 9206013;%%


%\cite{Shamir:1993zy}
\bibitem{Shamir:1993zy}
Y.~Shamir,
%``Chiral fermions from lattice boundaries,''
Nucl.\ Phys.\ B {\bf 406}, 90 (1993)
%[arXiv:hep-lat/9303005].
%%CITATION = HEP-LAT 9303005;%%

%\cite{Chiu:2002ir}
\bibitem{Chiu:2002ir}
T.~W.~Chiu,
%``Optimal domain-wall fermions,''
Phys.\ Rev.\ Lett.\  {\bf 90}, 071601 (2003)
%[arXiv:hep-lat/0209153].
%%CITATION = HEP-LAT 0209153;%%

%\cite{Chiu:2002kj}
\bibitem{Chiu:2002kj}
T.~W.~Chiu,
%``Locality of optimal lattice domain-wall fermions,''
Phys.\ Lett.\ B {\bf 552}, 97 (2003)
%[arXiv:hep-lat/0211032].
%%CITATION = HEP-LAT 0211032;%%

%\cite{Zol:1877}
\bibitem{Zol:1877}
E.~I.~Zolotarev,
%``Application of elliptic functions to the questions of functions
%deviating least and most from zero'',
Zap. Imp. Akad. Nauk. St. Petersburg, 30 (1877), no. 5; reprinted in his
Collected works, Vol. 2, Izdat, Akad. Nauk SSSR, Moscow, 1932, p. 1-59.

%\cite{Akhiezer:1992}
\bibitem{Akhiezer:1992}
N.~I.~Akhiezer,
"Theory of approximation", Reprint of 1956 English translation, Dover,
New York, 1992;
%
%\cite{Akhiezer:1990}
%\bibitem{Akhiezer:1990}
%N.~I.~Akhiezer,
"Elements of the theory of elliptic functions",
%second revised edition, Izdat. "Nauka", Moscow, 1970,
Translations of Mathematical Monographs, 79,
American Mathematical Society, Providence, R.I. 1990.


%\cite{Chiu:1998gp}
\bibitem{Chiu:1998gp}
T.~W.~Chiu and S.~V.~Zenkin,
%``On solutions of the Ginsparg-Wilson relation,''
Phys.\ Rev.\ D {\bf 59}, 074501 (1999)
%[arXiv:hep-lat/9806019].
%%CITATION = HEP-LAT 9806019;%%

%\cite{Ginsparg:1981bj}
\bibitem{Ginsparg:1981bj}
P.~H.~Ginsparg and K.~G.~Wilson,
%``A Remnant Of Chiral Symmetry On The Lattice,''
Phys.\ Rev.\ D {\bf 25}, 2649 (1982)
%%CITATION = PHRVA,D25,2649;%%


%\cite{Chiu:1998eu}
\bibitem{Chiu:1998eu}
T.~W.~Chiu,
%``GW fermion propagators and chiral condensate,''
Phys.\ Rev.\ D {\bf 60}, 034503 (1999)
%[arXiv:hep-lat/9810052].
%%CITATION = HEP-LAT 9810052;%%


%\cite{Chiu:2002eh}
\bibitem{Chiu:2002eh}
T.~W.~Chiu, T.~H.~Hsieh, C.~H.~Huang and T.~R.~Huang,
%``A note on the Zolotarev optimal rational approximation for the overlap Dirac operator,''
Phys.\ Rev.\ D {\bf 66}, 114502 (2002)
%[arXiv:hep-lat/0206007].
%%CITATION = HEP-LAT 0206007;%%


%\cite{Neuberger:1998fp}
\bibitem{Neuberger:1998fp}
H.~Neuberger,
%``Exactly massless quarks on the lattice,''
Phys.\ Lett.\ B {\bf 417}, 141 (1998)
%[arXiv:hep-lat/9707022].
%%CITATION = HEP-LAT 9707022;%%
%
%\cite{Neuberger:1998wv}
%\bibitem{Neuberger:1998wv}
%H.~Neuberger,
%``More about exactly massless quarks on the lattice,''
%Phys.\ Lett.\ B {\bf 427}, 353 (1998)
%[arXiv:hep-lat/9801031].
%%CITATION = HEP-LAT 9801031;%%



%\cite{Narayanan:1995gw}
\bibitem{Narayanan:1995gw}
R.~Narayanan and H.~Neuberger,
%``A Construction of lattice chiral gauge theories,''
Nucl.\ Phys.\ B {\bf 443}, 305 (1995)
%[arXiv:hep-th/9411108].
%%CITATION = HEP-TH 9411108;%%


%\cite{Neuberger:1997bg}
\bibitem{Neuberger:1997bg}
H.~Neuberger,
%``Vector like gauge theories with almost massless fermions on the lattice,''
Phys.\ Rev.\ D {\bf 57}, 5417 (1998)
%[arXiv:hep-lat/9710089].
%%CITATION = HEP-LAT 9710089;%%


%\cite{Kikukawa:1999sy}
\bibitem{Kikukawa:1999sy}
Y.~Kikukawa and T.~Noguchi,
%``Low energy effective action of domain-wall fermion and the Ginsparg-Wilson relation,''
hep-lat/9902022.
%%CITATION = HEP-LAT 9902022;%%


%\cite{Edwards:2000qv}
\bibitem{Edwards:2000qv}
R.~G.~Edwards and U.~M.~Heller,
%``Domain wall fermions with exact chiral symmetry,''
Phys.\ Rev.\ D {\bf 63}, 094505 (2001)
%[arXiv:hep-lat/0005002].
%%CITATION = HEP-LAT 0005002;%%


%\cite{Borici:1999zw}
\bibitem{Borici:1999zw}
A.~Borici,
%``Truncated overlap fermions,''
Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 83}, 771 (2000)
%[arXiv:hep-lat/9909057].
%%CITATION = HEP-LAT 9909057;%%

%\cite{Luscher:1998pq}
\bibitem{Luscher:1998pq}
M.~Luscher,
%``Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation,''
Phys.\ Lett.\ B {\bf 428}, 342 (1998)
%[arXiv:hep-lat/9802011].
%%CITATION = HEP-LAT 9802011;%%


%\cite{Chiu:1998bh}
\bibitem{Chiu:1998bh}
T.~W.~Chiu,
%``The spectrum and topological charge of exactly massless fermions on the  lattice,''
Phys.\ Rev.\ D {\bf 58}, 074511 (1998)
%[arXiv:hep-lat/9804016].
%%CITATION = HEP-LAT 9804016;%%


%\cite{Sachs:en}
\bibitem{Sachs:en}
I.~Sachs and A.~Wipf,
%``Finite Temperature Schwinger Model,''
Helv.\ Phys.\ Acta {\bf 65}, 652 (1992).
%%CITATION = HPACA,65,652;%%


%\cite{Chiu:1999vk}
\bibitem{Chiu:1999vk}
T.~W.~Chiu,
%``The topological characteristics of lattice Dirac operators,''
hep-lat/9911010.
%%CITATION = HEP-LAT 9911010;%%



\end{thebibliography}