Off-Shell Extension of the Chiral Reduction Formula
Hiroyuki~\textsc{Kamano}
hep-ph0412281.tex
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%
% New Commands
%
\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle {#1}|}
\newcommand{\ubexav}[2]{{\underline a}\sp{#1}\sb{#2}}
\newcommand{\ubexv}[2]{{\underline v}\sp{#1}\sb{#2}}
\newcommand{\axialjc}[2]{j\sp{#1}\sb{A{#2}}}
\newcommand{\vectorjc}[2]{j\sp{#1}\sb{V{#2}}}
\newcommand{\diff}[2]{\frac{\delta {#1}}{\delta {#2}}}
\newcommand{\fpi}{f\sb{\pi}}
\newcommand{\mpi}{m\sb{\pi}}
\newcommand{\chrf}{$\chi$RF}
\newcommand{\smatrix}{\hat{{\cal S}}}
\newcommand{\smatrixon}{\hat{\boldsymbol{S}}}
%
%
%

\markboth{H.~Kamano}{Off-Shell Extension of the Chiral Reduction Formula}

\title{Off-Shell Extension of the Chiral Reduction Formula}

\author{Hiroyuki~\textsc{Kamano}}

\inst{Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan}

%\recdate{January 26, 2006}%            %Editorial Office will fill in this.

\abst{
We present a proper off-mass-shell extension of 
the chiral reduction formula ({\chrf}) proposed by Yamagishi and Zahed. 
%
This is achieved by rewriting the {\chrf} in a
manifestly consistent form with the conventional LSZ reduction formula.
}

\begin{document}

\maketitle
%
%
%
%
\section{Introduction}
\label{sec1}
%
%
%
%
There is no doubt that chiral symmetry is important for a theoretical
understanding of the low energy properties of QCD.
%
The spontaneous, and also explicit breaking of this symmetry 
governs various aspects of hadronic processes and plays a crucial role
in the construction of effective models.

%
%
%
On the basis of chiral symmetry, Yamagishi and Zahed developed
a general framework for analyzing pion-induced hadronic processes\cite{Yam96}.
%
This framework consists of two master equations 
for the extended S-matrix, which are based on the gauge covariant
divergence equations proposed by Veltman and Bell\cite{Vel66,Bell67}.
%
Each master equation represents transformation properties of 
the extended S-matrix under the local vector and axial transformations,
subject to the following asymptotic conditions
for the axial current: 
%
%
\begin{equation}
A\sb{\mu}\sp{a}(x) 
\rightarrow 
-\fpi\partial\sb{\mu}\pi\sp{a}\sb{{\rm in,out}}(x)
+ \cdots
,\ \
(x\sp{0} \rightarrow \mp\infty)
\label{eq11}
\end{equation}
%
%
and
%
%
\begin{equation}
\partial\sp{\mu}A\sb{\mu}\sp{a}(x) 
\rightarrow \fpi\mpi\sp{2}\pi\sp{a}\sb{{\rm in,out}}(x)
+ \cdots
.\ \
(x\sp{0} \rightarrow \mp\infty)
\label{eq12}
\end{equation}
%
%
One noteworthy result obtained within this framework is that
the master equation for the axial transformation provides
a new reduction formula, the chiral reduction formula ({\chrf}),
for scattering amplitudes involving any number of pions with their 
\textit{physical} masses.
%
With this formula, the relevant on-shell Ward identities 
satisfied by those amplitudes can be derived. 
%
These identities are expressed in terms of the Green functions of 
well-defined current and density operators,
and exactly embody the consequences of the broken $SU(2)$$\times$$SU(2)$ 
chiral symmetry without the need for any model or expansion scheme.
%
A number of investigations based on the {\chrf}
have been carried out for hadron reactions 
in the resonance region~\cite{Ste97,Lee99,Kam05-1}
and hadronic matter~\cite{Ste96,Ste97-2,Lee98,Dus06}.
%
These investigations have demonstrated that the {\chrf} 
is a powerful tool for clarifying the role of chiral symmetry 
in those processes in a model-independent way.

%
%
%
To this time, there have been no studies aimed of determining
how amplitudes behave in the off-shell region
within the framework of Ref.~\citen{Yam96}. 
%
In principle, the {\chrf} can be applied not only to scattering amplitudes
but also to internal vertex functions
and potentials in which the attached pion legs can be off the mass shell.
%
Although the off-shell amplitudes themselves are artificial, and
not directly related to physical quantities, 
such amplitudes are required
in some theoretical calculations.
%
For instance, in scattering problems,
the on-shell scattering amplitudes
are calculated by using off-shell potentials 
(in the Lippmann-Schwinger equation, etc.).
%
This situation, i.e. that the off-shell amplitudes are employed 
in calculations of observables, is also found in many-body problems.
%
Thus in attempting to treat these problems on the basis of the {\chrf},
it is necessary to make clear the off-shell structure.

%
%
%
However, care must be taken in applying the {\chrf} 
presented in Ref.~\citen{Yam96} 
to off-shell pions, because this formula was originally constructed 
for the purpose of treating the on-shell pions.
%
Indeed, in contrast to on-shell cases, we find that 
its naive application is invalid off the mass shell.
%
The purpose of this paper is to carefully examine 
the derivation of {\chrf}
and to formulate a proper method of extending  
the {\chrf} off the mass shell. 

%
%
%
This paper is organized as follows.
%
In \textsection\ref{sec2}, we briefly review the theoretical framework
developed in Ref.~\citen{Yam96}.
%
In \textsection\ref{sec3}, we explain in detail the kind of 
interpolating pion field chosen 
within the framework of Ref.~\citen{Yam96}. 
%
Generally, the interpolating pion field must be defined 
in order to relate scattering amplitudes to
the Green functions of current and density operators.
%
This is also the case for the {\chrf}.
%
The problem inherent in the {\chrf} will become clear through 
this explanation, because the off-shell behavior of amplitudes 
is fixed once an interpolating pion field is given.
%
In \textsection\ref{sec4}, we present an off-shell extension of 
the {\chrf} that is consistent with the framework of Ref.~\citen{Yam96}.
%
A summary is given in \textsection\ref{sec5}.
%
%
%
%
\section{Chiral reduction formula}
\label{sec2}
%
%
%
%
A fundamental quantity in the theoretical framework developed in
Ref.~\citen{Yam96} is the extended S-matrix,
$\smatrix = \smatrix[v\sb{\mu}\sp{a},a\sb{\mu}\sp{a},s,J\sp{a}]$,
which is a functional of the vector, axial vector, scalar 
and pseudoscalar external fields.
%
Then, the current and density operators 
$\hat{\cal O} = 
( \vectorjc{a}{\mu}, \axialjc{a}{\mu}, \fpi\hat{\sigma}, \hat{\pi}\sp{a} )$
conjugate to the corresponding external fields
$\phi = ( v\sp{a}\sb{\mu}, a\sp{a}\sb{\mu}, s, J\sp{a} )$
are defined as
%
%
\begin{equation}
\hat{\cal O}(x) = -i\smatrix\sp{\dag}\diff{\smatrix}{\phi(x)}.
\label{eq21}
\end{equation}
%
%
Here, $\hat{\pi}\sp{a}(x)$ is the pseudoscalar-isovector density operator,
and $\hat{\sigma}(x)$ is the scalar-isoscalar density operator from which 
the pion decay constant $\fpi$ has been subtracted.
%
In the presence of the external fields,
the operators $\vectorjc{a}{\mu}$, $\axialjc{a}{\mu}$ and $\hat{\pi}\sp{a}$
are related to the ordinary vector and axial currents as
%
%
\begin{equation}
V\sp{a}\sb{\mu}(x) =  \vectorjc{a}{\mu}(x) 
                    + \fpi\ubexav{ab}{\mu}(x) \hat{\pi}\sp{b}(x)
\label{eq22}
\end{equation}
%
%
and
%
%
\begin{equation}
A\sp{a}\sb{\mu}(x) =  \axialjc{a}{\mu}(x)
                    + \fpi\sp{2}a\sb{\mu}\sp{a}(x)
                    - \fpi\nabla\sp{ab}\sb{\mu}\hat{\pi}\sp{b}(x),
\label{eq23}
\end{equation}
%
%
respectively, where we use the notation
$\ubexav{ac}{\mu} \equiv \varepsilon\sp{abc}a\sp{b}\sb{\mu}$ and
$\nabla\sp{ac}\sb{\mu} \equiv \delta\sp{ac}\partial\sb{\mu} + \ubexv{ac}{\mu}$.
%
Owing to the Bogoliubov causality condition~\cite{Bog}, 
the $T\sp{\ast}$ product of the operators $\hat{\cal O}$ 
can be expressed as
%
%
\begin{equation}
T\sp{\ast}(\hat{\cal O}(x\sb{1})\cdots\hat{\cal O}(x\sb{n}))
= 
(-i)\sp{n}\smatrix\sp{\dag}
\diff{}{\phi(x\sb{1})} \cdots \diff{}{\phi(x\sb{n})} \smatrix.
\label{eq24}
\end{equation}

%
%
%
The essence of the framework presented in Ref.~\citen{Yam96} is embodied by
two linear master equations for the extended S-matrix,
%
%
\begin{equation}
T\sb{V}\sp{a}(x) \smatrix = 0,
\label{eq25}
\end{equation}
%
%
\begin{equation}
T\sb{A}\sp{a}(x) \smatrix = 0,
\label{eq26}
\end{equation}
%
%
where
%
%
\begin{equation}
T\sb{V}\sp{a}(x) =
\left(
  \nabla\sb{\mu}    \diff{}{v\sb{\mu}}
+ \ubexav{}{\mu}    \diff{}{a\sb{\mu}}
+ {\underline J}    \diff{}{J}
\right)\sp{a}(x),
\label{eq27}
\end{equation}
%
%
\begin{eqnarray}
T\sb{A}\sp{a}(x)
&=&
\left[
- (\Box + \mpi\sp{2} + K)\diff{}{J} 
+ iJ
+ \frac{1}{\fpi} X\sb{A}
- \frac{1}{\fpi} 
  \left( \nabla\sp{\mu}a\sb{\mu} - \frac{J}{\fpi} \right) \diff{}{s}
\right]\sp{a}(x)
\label{eq28}
\end{eqnarray}
%
%
and
%
%
\begin{equation}
K\sp{ab}(x) =
( \nabla\sp{\mu}\nabla\sb{\mu} - \ubexav{\mu}{}\ubexav{}{\mu}
+ s - \Box 
)\sp{ab}(x),
\ \
X\sb{A}\sp{a}(x) = 
\left(
  \nabla\sb{\mu}   \diff{}{a\sb{\mu}}
+ \ubexav{}{\mu}   \diff{}{v\sb{\mu}}
\right)\sp{a}(x).
\label{eq29}
\end{equation}
%
%
In Eqs.~(\ref{eq27})-(\ref{eq29}), the isospin indices with respect to which 
the contraction is taken are suppressed.
%
The operator $T\sb{V}\sp{a}(x)$ is the (local) isospin generator, 
and hence Eq.~(\ref{eq25}) represents the isospin invariance 
of the extended S-matrix.
%
Equation~(\ref{eq26}) represents the transformation properties of 
$\smatrix$ under axial transformations. 
%
By applying the functional derivatives of $\phi(x)$ 
to the master equations and using Eqs.~(\ref{eq21}) and~(\ref{eq24}), 
we straightforwardly obtain the vector and axial Ward identities
satisfied by the Green functions of $\hat{\cal O}(x)$.

%
%
%
A crucial point in the derivation described above is that
the asymptotic conditions (\ref{eq11}) and (\ref{eq12})
are exactly incorporated in the master equations
without the need for any specific model or expansion scheme.
%
Consequently, the resulting Ward identities incorporate
not only the spontaneous chiral symmetry breaking,
but also its explicit breaking to all orders in the quark masses. 
% 
This property of the master equations allows us 
to derive the Ward identities 
for scattering amplitudes involving pions with their 
\textit{physical} masses.
%
This contrasts with the situation in
other theoretical frameworks, 
such as the chiral perturbation theory, 
in which the pion mass is expressed as a
perturbative extrapolation from the chiral symmetric point.
%
The Ward identities are integrated into 
the form of a reduction formula,
called the chiral reduction formula ({\chrf}).

%
%
%
We now explain the {\chrf} in detail.
% 
First, we note that
the pseudoscalar-isovector density 
$\hat{\pi}\sp{a}(x)$ possesses the asymptotic form 
$\hat{\pi}\sp{a}(x) \rightarrow \pi\sp{a}\sb{{\rm in,out}}(x)\ 
(x\sp{0} \rightarrow \mp\infty)$, 
which follows from the construction of the master equations
and the asymptotic conditions (\ref{eq11}) and (\ref{eq12}).
%
This fact enables us to identify $\hat{\pi}\sp{a}(x)$ as the (normalized) 
interpolating pion field.
%
Then, through this identification, 
the axial Ward identities satisfied by the Green functions of 
$\hat{\cal O}(x)$ can be related to
scattering amplitudes including pions.
%
The identification is emphasized by formally solving 
the axial master equation (\ref{eq26}) 
for $\delta \smatrix / \delta J\sp{a}$
under the condition
$\hat{\pi}\sp{a}(x) \rightarrow \pi\sp{a}\sb{{\rm in,out}}(x)\ 
(x\sp{0} \rightarrow \mp\infty)$. 
%
The result is\footnote{In Ref.~\citen{Yam96}, 
Eq.~(\ref{eq210}) is called the (axial) master equation.}
% 
%
\begin{eqnarray}
\diff{\smatrix}{J\sp{a}(x)} &=& 
 i \smatrix \pi\sb{\rm in}\sp{a}(x)
+i \smatrix \int d\sp{4}y
   G\sb{R}\sp{ab}(x,y)\left(K\pi\sb{\rm in}\right)\sp{b}(y)
\nonumber\\
%
&&
+ \int d\sp{4}y G\sb{R}\sp{ab}(x,y)
  \left[
       -i J
       + \frac{1}{\fpi}
         \left( 
         \nabla\sp{\mu}a\sb{\mu} - \frac{J}{\fpi}
         \right) \diff{}{s}
       - \frac{1}{\fpi} X\sb{A}
  \right]\sp{b}(y)  \smatrix
\nonumber\\
%
&=& 
 i \pi\sb{\rm in}\sp{a}(x) \smatrix
+i \int d\sp{4}y 
   G\sb{A}\sp{ab}(x,y)\left(K\pi\sb{\rm in}\right)\sp{b}(y) \smatrix
\nonumber\\
%
&&
+ \int d\sp{4}y G\sb{A}\sp{ab}(x,y)
  \left[
       -i J
       + \frac{1}{\fpi}
         \left( 
         \nabla\sp{\mu}a\sb{\mu} - \frac{J}{\fpi}
         \right) \diff{}{s}
       - \frac{1}{\fpi} X\sb{A}
  \right]\sp{b}(y)  \smatrix .
\nonumber\\
%
&&
\label{eq210}  
\end{eqnarray}
%
%
Here, $G\sb{R}$ and $G\sb{A}$ are the retarded and advanced Green functions, 
respectively, which satisfy
%
% 
\begin{equation}
(-\Box - \mpi\sp{2} - K)\sp{ab}(x)
G\sp{bc}\sb{R,A}(x,y)
=\delta\sp{ac}\delta\sp{(4)}(x-y).
\label{eq211}
\end{equation}
%
%
Using Eq.~(\ref{eq210}), we can directly derive 
the commutation relations for the creation and annihilation operators
of the pion with the extended S-matrix $\smatrix$.
%
We find
%
%
\begin{equation}
[ a\sp{a}\sb{{\rm in}}(k), \smatrix ] = R\sp{a}(k)\smatrix, 
\ \ \ \  
[ \smatrix , a\sp{a\dag}\sb{{\rm in}}(k) ] = R\sp{a}(-k)\smatrix,
\label{eq212}
\end{equation}
%
%
where 
%
%
\begin{eqnarray}
R\sp{a}(k) &=&
\int d\sp{4}x e\sp{+ikx}
\left[
 iJ
+ \frac{1}{\fpi} X\sb{A}
- K\diff{}{J} 
- \frac{1}{\fpi} 
  \left( \nabla\sp{\mu}a\sb{\mu} - \frac{J}{\fpi} \right) \diff{}{s}
\right]\sp{a}(x)
\label{eq213}.
\end{eqnarray}
%
%
Note that we have rewritten the r.h.s. of the commutation relations 
(\ref{eq212}) in forms that do not involve the asymptotic pion field.
%
[Compare these with Eqs.~(6.2) and (6.3) in Ref.~\citen{Yam96}.]

%
%
%
Iterative use of Eq.~(\ref{eq212}) yields
the {\chrf} for scattering amplitudes 
involving any number of pions with their physical masses,\footnote{ 
We consider the case in which no two pions have equal momenta.}
%
%
\begin{eqnarray}
&&
\bra{\alpha;k\sb{1}a\sb{1},\cdots,k\sb{m}a\sb{m}} 
\smatrix
\ket{\beta;l\sb{1}b\sb{1},\cdots,l\sb{n}b\sb{n}}|\sb{\phi=0}
\nonumber\\
%
&=&
[ R\sp{a\sb{1}}(k\sb{1})  \cdots R\sp{a\sb{m}} (k\sb{m}) 
  R\sp{b\sb{1}}(-l\sb{1}) \cdots R\sp{b\sb{n}} (-l\sb{n}) ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0},
\label{eq214}
\end{eqnarray}
%
%
where $k\sb{i}~(l\sb{i})$ and $a\sb{i}~(b\sb{i})$ are 
the four-momentum and isospin indices of the outgoing (incoming) pions,
respectively, and $\alpha$ and $\beta$ represent the states of other particles.
%
Here, $[\cdots]\sb{{\rm S}}$ indicates 
that we take normalized symmetric permutations
of the functional derivative operators contained therein;
explicitly, we have
%
%
\begin{equation}
[{\cal D}\sb{1}\cdots{\cal D}\sb{n}]\sb{{\rm S}} =
\frac{1}{n!} \sum\sb{{\rm perms}} {\cal D}\sb{1}\cdots{\cal D}\sb{n}.
\label{eq215}
\end{equation}
%
%
This operation clearly shows the crossing symmetry in Eq.~(\ref{eq214}).
%
By using Eq.~(\ref{eq214}), together with Eq.~(\ref{eq24}),
scattering amplitudes can be expressed in terms of 
the Green functions of the operators 
$\hat{\cal O}=(\vectorjc{}{},\axialjc{}{},\fpi\hat{\sigma},\hat{\pi})$.
%
The {\chrf} takes the form of a functional derivative, 
and all constraints that stem from the broken chiral symmetry 
are contained in $R\sp{a}(k)$.
%
The {\chrf} can be understood as the following replacement of commutators:
%
%
\begin{equation}
[ a\sb{{\rm in}}\sp{a}(k), \ ] \rightarrow R\sp{a}(k),
\ \ \ \ 
[ \ , a\sb{{\rm in}}\sp{a\dag}(k) ] \rightarrow R\sp{a}(-k).
\label{eq216}
\end{equation}
%
%
For instance, the double commutator is replaced as
%
%
\begin{equation}
[ a\sb{{\rm in}}\sp{a}(k\sb{1}), 
[ \smatrix , a\sb{{\rm in}}\sp{b\dag}(k\sb{2}) ] ]
\rightarrow 
R\sp{b}(-k\sb{2}) R\sp{a}(k\sb{1}) \smatrix.
\label{eq217}
\end{equation}
%
%
%
%
\section{Interpolating pion field}
\label{sec3}
%
%
%
%
In contrast to $V\sp{a}\sb{\mu}$ and $A\sp{a}\sb{\mu}$,
which are identified with physical electroweak currents,
the operators $\hat{\cal O}$ are convention-dependent.
%
The interpolating pion field $\hat{\pi}\sp{a}$ 
is not uniquely determined: 
%
It is only required that the asymptotic condition
$\hat{\pi}\sp{a}(x) \rightarrow \pi\sp{a}\sb{{\rm in,out}}(x)\ 
(x\sp{0} \rightarrow \mp\infty)$
be satisfied.
%
Equations~(\ref{eq22}) and~(\ref{eq23}) represent 
an arbitrary way of dividing $V\sp{a}\sb{\mu}$ and $A\sp{a}\sb{\mu}$, 
respectively.
%
For the redefinition of the pion field
$\hat{\pi}\sp{a} \rightarrow \hat{\pi}\sp{\prime a}$,
with
%
%
\begin{equation}
\hat{\pi}\sp{\prime a}(x) =
\hat{\pi}\sp{a}(x) + \Lambda\sp{a}(x),
\label{eq31}
\end{equation}
%
%
$\vectorjc{a}{\mu}$ and $\axialjc{a}{\mu}$ also change in such a manner 
that Eqs.~(\ref{eq22}) and~(\ref{eq23}) remain invariant~\cite{Yam96}:
%
%
\begin{equation}
\vectorjc{\prime a}{\mu}(x) =
\vectorjc{a}{\mu}(x) - \fpi\ubexav{ac}{\mu} \Lambda\sp{c}(x),
\label{eq32}
\end{equation}
%
%
\begin{equation}
\axialjc{\prime a}{\mu}(x) =
\axialjc{a}{\mu}(x) + \fpi\nabla\sp{ac}\sb{\mu} \Lambda\sp{c}(x).
\label{eq33}
\end{equation}

%
%
%
The arbitrariness of the operators $\hat{\cal O}$ is translated into 
that of the extended S-matrix.
%
The redefined operators 
$\hat{\cal O}\sp{\prime} = 
(\vectorjc{\prime}{},\axialjc{\prime}{}, 
\fpi\hat{\sigma}\sp{\prime},\hat{\pi}\sp{\prime})$
are obtained in terms of an appropriately extended S-matrix 
$\smatrix\sp{\prime}[\phi]$, which
equals $\smatrix[\phi]$ at $\phi=0$ 
(i.e. $\smatrix|\sb{\phi=0}=\smatrix\sp{\prime}|\sb{\phi=0}\equiv\smatrixon$),
through functional derivatives as
%
%
\begin{equation}
\hat{\cal O}\sp{\prime}(x) =
-i\smatrix\sp{\prime\dag} \diff{\smatrix\sp{\prime}}{\phi(x)}.
\label{eq34}
\end{equation}
%
%
Therefore, a different way of extending the S-matrix $\smatrixon$ 
by introducing the external fields provides a different choice of 
the current and density operators.

%
%
%
The framework presented in Ref.~\citen{Yam96} only takes
account of the interpolating pion fields
given by the extended S-matrices satisfying the master equations.
%
Because the master equations fix the extended S-matrix up to 
a phase factor,\footnote{
%%%
%%%
A new extended S-matrix $\smatrix\sp{\prime}$ can be written as
$\smatrix\sp{\prime} = \smatrix \exp(if[\phi])$\cite{Yam96}.
%
Causality implies that the functional $f[\phi]$ has the form
%
%
\begin{equation*}
f[\phi] = \int d\sp{4}x P(x),
\end{equation*}
%
%
where $P(x)$ is a polynomial in $\phi(x)$ and its derivatives. 
%
[A constant term can be eliminated from $P(x)$ by imposing 
the normalization $\bra{0}\smatrixon\ket{0}=1$.]
%
Then, for \textit{any} $f[\phi]$ satisfying $T\sb{V,A}f[\phi]=0$,
the master equations (\ref{eq25}) and (\ref{eq26}) are invariant.
%%%
%%%
}
the arbitrariness in the choice of $\hat{\cal O}$ still exists.
%
However, as long as the extended S-matrices are introduced 
in such a manner that the master equations are satisfied, 
we can apply the {\chrf} to all of them.
%
The scattering amplitudes obtained from such extended S-matrices 
then satisfy the Ward identity given by the {\chrf},
and thus exhibit the same off-shell momentum dependence up to terms 
that are not fixed by the broken chiral symmetry.
%
(Of course, these amplitudes are identical on shell.)
%
A class of interpolating pion fields 
that is chosen to be consistent with the master equations
ensures the ``form invariance'' of the Ward identities.

%
%
%
We explain the situation described above more concretely 
by considering the $\pi N$ scattering as an example.
%
Using the {\chrf}~(\ref{eq214}), 
together with Eqs.~(\ref{eq21}) and~(\ref{eq24}),
we obtain
the $\pi\sp{a}(k)+N(p)\rightarrow\pi\sp{b}(k\sp{\prime})+N(p\sp{\prime})$ 
scattering amplitude $i{\cal T}\sb{\pi N}$ with two off-shell pions
(i.e.~$k\sp{2},k\sp{\prime 2}{\not =}\mpi\sp{2}$)
as
%
%
\begin{eqnarray}
i{\cal T}\sb{\pi N} 
&=&
[ R\sp{a}(-k) R\sp{b} (k\sp{\prime}) ]\sb{{\rm S}}
\bra{N(p\sp{\prime})} \smatrix \ket{N(p)}|\sb{\phi = 0}
\nonumber\\
%
&=&
i{\cal T}\sb{S} + i{\cal T}\sb{V} + i{\cal T}\sb{AA},
\label{eq35}
\end{eqnarray}
%
%
where
%
%
\begin{equation}
i{\cal T}\sb{S} =
-i\frac{k\sp{2}+k\sp{\prime 2}}{2\fpi} 
\delta\sp{ab} \bra{N(p\sp{\prime})} \hat{\sigma}(0) \ket{N(p)},
\label{eq36}
\end{equation}
%
%
\begin{equation}
i{\cal T}\sb{V} 
=
\frac{1}{2\fpi\sp{2}} (k+k\sp{\prime})\sp{\mu} \varepsilon\sp{abc}
\bra{N(p\sp{\prime})} \vectorjc{c}{\mu}(0) \ket{N(p)},
\label{eq37}
\end{equation}
%
%
and
%
%
\begin{equation}
i{\cal T}\sb{AA} =
-\frac{1}{\fpi\sp{2}}k\sp{\mu}k\sp{\prime \nu}
\int d\sp{4}x e\sp{ikx}
\bra{N(p\sp{\prime})} T\sp{\ast}(\axialjc{a}{\mu}(x)\axialjc{b}{\nu}(0)) \ket{N(p)},
\label{eq38}
\end{equation}
%
%
modulo $(2\pi)\sp{4}\delta\sp{(4)}(k+p-k\sp{\prime}-p\sp{\prime})$.
%
Now we redefine the operators as
$\hat{\cal O} \rightarrow \hat{\cal O}\sp{\prime}$,
which follows from the change of the extended S-matrix
$\hat{\cal S} \rightarrow \hat{\cal S}\sp{\prime}$.
%
Then, if $\hat{\cal S}\sp{\prime}$ satisfies the master equations, 
the {\chrf} can be applied to it.
%
The amplitude $i{\cal T}\sp{\prime}\sb{\pi N}$ 
obtained from $\hat{\cal S}\sp{\prime}$ 
takes the same form as $i{\cal T}\sb{\pi N}$, 
except that the Green functions of $\hat{\cal O}$
are replaced with those of $\hat{\cal O}\sp{\prime}$,
and thus
the overall momentum dependences of $i{\cal T}\sb{\pi N}$ and
$i{\cal T}\sp{\prime}\sb{\pi N}$ do not change.
%
The change of the amplitude resulting from 
the redefinition of the pion field only appears 
in the Green functions of $\hat{\cal O}$, 
which are not uniquely determined by the broken chiral symmetry.
%
In addition, the difference between the Green functions of 
$\hat{\cal O}$ and $\hat{\cal O}\sp{\prime}$ 
is only in some polynomials in the momenta,\footnote{
This results from the fact that
the phase $f[\phi]$ appearing in 
$\smatrix\sp{\prime}=\smatrix\exp(if[\phi])$
consists only of some polynomials in $\phi$ and their derivatives.
}
and this does not alter on-shell quantities 
that are determined by pole terms\cite{Yam96}.

%
%
%
The interpolating pion fields defined consistently with the 
master equations can be formally expressed as
%
%
\begin{equation}
\hat{\pi}\sp{a}(x) =
\hat{\pi}\sp{a}\sb{\text{PCAC}}(x) + F\sp{a}(\phi,\hat{\cal O}),
\label{eq39}
\end{equation}
%
%
where $\hat{\pi}\sp{a}\sb{\text{PCAC}}(x)$ is 
the so-called PCAC choice of the pion field
$\hat{\pi}\sb{\text{PCAC}}\sp{a} \equiv 
\partial\sp{\mu}A\sb{\mu}\sp{a}/(\fpi\mpi\sp{2})$,
and
$F\sp{a}(\phi,\hat{\cal O})$ is some function of $\phi$ and $\hat{\cal O}$
satisfying $F\sp{a}(\phi,\hat{\cal O})|\sb{\phi=0}=0$.
%
The asymptotic one-pion component is entirely included in 
$\hat{\pi}\sb{\text{PCAC}}\sp{a}$,
and therefore the second term on the r.h.s., $F\sp{a}(\phi,\hat{\cal O})$,
can generate only an off-shell effect.
%
Furthermore, $F\sp{a}(\phi,\hat{\cal O})$ is the only quantity 
that depends on the choice of the extended S-matrix,
$F\sp{a}(\phi,\hat{\cal O}) \rightarrow 
F\sp{a}(\phi,\hat{\cal O}\sp{\prime})$,
while $\hat{\pi}\sp{a}\sb{\text{PCAC}}$ 
is invariant under the redefinition of the pion field.
%
These facts imply that $F\sp{a}(\phi,\hat{\cal O})$ 
influences only the Green functions of $\hat{\cal O}$
in the Ward identity.
%
The overall kinematical dependence and symmetry structure 
of the off-shell Ward identity is determined by the first term in
Eq.~(\ref{eq39}), i.e. the PCAC pion field $\hat{\pi}\sp{a}\sb{\text{PCAC}}$.
%
Therefore, the class of interpolating pion fields
defined consistently with the master equations
and employed in the framework of Ref.~\citen{Yam96}
gives off-shell amplitudes satisfying Ward identities
of the same form as those obtained using the PCAC pion field.
%
%
%
%
\section{Extending the {\chrf} off mass shell}
\label{sec4}
%
%
%
%
The chiral Ward identity for the $\pi N$ scattering 
has also been derived by Weinberg,
using a different approach based on 
the chiral algebra satisfied by currents and 
density operators~\cite{Wei-70,Bro71}.
%
In that derivation, the PCAC choice of the pion field,
$\hat\pi\sp{a}\sb{\text{PCAC}}$, was employed to relate the $\pi N$ 
scattering amplitude
to the Ward identity among the Green functions of the currents and 
density operators.
%
If the prescription (\ref{eq214}) can be applied to 
the off-shell pions, the {\chrf} must reproduce 
Weinberg's formula, even 
off the mass shell, because the two approaches use the same definition
of the interpolating pion field.
%
However, 
for $i{\cal T}\sb{S}$ including the pion-nucleon $\sigma$-term,
Weinberg's formula gives 
%
%
\begin{equation}
i{\cal T}\sb{S} =
-i\frac{k\sp{2}+k\sp{\prime 2} - \mpi\sp{2}}{\fpi} 
\delta\sp{ab} \bra{N(p\sp{\prime})} \hat{\sigma}(0) \ket{N(p)},
\label{eq41}
\end{equation}
%
%
instead of Eq.~(\ref{eq36}).
%
This result indicates that a naive use of the {\chrf} (\ref{eq214})
proposed in Ref.~\citen{Yam96} 
does not correctly relate the \textit{off-shell} amplitudes 
to the Ward identities among
the Green functions of $\hat{\cal O}$; 
more specifically,
it does not do this in a manner
consistent with the master equations.

%
%
%
To make the {\chrf} applicable to the off-shell pions, 
first we note that the off-shell extrapolation
of scattering amplitudes is generally realized
through the LSZ reduction formula.
%
We therefore examine the derivation of {\chrf} starting from
the LSZ formalism.

%
%
%
We consider the commutation relations~(\ref{eq212}) 
on the basis of the LSZ formalism. 
%
From the asymptotic behavior of the interpolating pion field, we obtain
%
%
\begin{equation}
[ a\sb{{\rm in}}\sp{a}(k), \smatrix ] =
\int d\sp{4}x e\sp{+ikx} ( \Box + \mpi\sp{2} ) 
\diff{}{J\sp{a}(x)} \smatrix,
\label{eq42}
\end{equation}
%
%
\begin{equation}
[ \smatrix, a\sb{{\rm in}}\sp{a\dag}(k) ] =
\int d\sp{4}x e\sp{-ikx} ( \Box + \mpi\sp{2} ) 
\diff{}{J\sp{a}(x)} \smatrix,
\label{eq43}
\end{equation}
%
%
where we have used 
$a\sb{{\rm out}} = \smatrix\sp{\dag} a\sb{{\rm in}} \smatrix$
and Eq.~(\ref{eq21}).
%
Note that these exact relations are obtained independently of 
the master equations.
%
Once the commutation relations are rewritten in the form of 
Eqs.~(\ref{eq42}) and (\ref{eq43}),
the momentum $k$ can be analytically continued off the mass shell. 
%
Using these relations iteratively, 
we obtain the conventional LSZ reduction formula,
%
%
\begin{eqnarray}
&&
\bra{ \alpha; k\sb{1}a\sb{1}, \cdots, k\sb{m}a\sb{m}} 
\smatrix
\ket{ \beta;  l\sb{1}b\sb{1}, \cdots, l\sb{n}b\sb{n}}|\sb{\phi=0}
\nonumber\\
%
&=&
\left( 
\prod\sb{i=1}\sp{m} G(k\sb{i};x\sb{i}) 
\diff{}{J\sp{a\sb{i}}(x\sb{i})} 
\right)
\left( 
\prod\sb{j=1}\sp{n}
G(-l\sb{j};y\sb{j})
\diff{}{J\sp{b\sb{j}}(y\sb{j})} 
\right)
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
\nonumber\\
%
&=&
\left( \prod\sb{i=1}\sp{m} iG(k\sb{i};x\sb{i}) \right)
\left( \prod\sb{j=1}\sp{n} iG(-l\sb{j};y\sb{j}) \right)
\nonumber\\
%
&&
\times
\bra{\alpha}
\smatrixon
T\sp{\ast}
(
\hat{\pi}\sp{a\sb{1}}(x\sb{1}) \cdot\cdot \hat{\pi}\sp{a\sb{m}}(x\sb{m})
\hat{\pi}\sp{b\sb{1}}(y\sb{1}) \cdot\cdot \hat{\pi}\sp{b\sb{n}}(y\sb{n})
)
\ket{\beta},
\label{eq44}
\end{eqnarray}
%
%
where $G(k;x) = \int d\sp{4}x \exp(ikx)(\Box\sb{x}+\mpi\sp{2})$
and $\smatrixon = \smatrix|\sb{\phi=0}$, and 
we have used Eq.~(\ref{eq24}) in the last step.
%
This formula defines the off-shell extrapolation of the scattering amplitude.

%
%
%
We stress that owing to Eqs.~(\ref{eq28}) and~(\ref{eq213}), 
the \textit{identity} 
%
%
\begin{equation}
\int d\sp{4}x e\sp{+ikx} 
( \Box + \mpi\sp{2} ) \diff{}{J\sp{a}(x)}
= 
R\sp{a}(k) - T\sb{A}\sp{a}(k)
\label{eq45}
\end{equation}
%
%
exists among functional derivative operators,
where $T\sb{A}\sp{a}(k)$ is the Fourier transformation of $T\sb{A}\sp{a}(x)$.
%
Thus, we immediately find the following replacements for the commutators:
%
%
\begin{equation}
[ a\sb{in}\sp{a}(k), \ ] \rightarrow R\sp{a}(k) - T\sb{A}\sp{a}(k),
\label{eq46}
\end{equation}
%
%
\begin{equation}
[\ , a\sb{in}\sp{a\dag}(k)] \rightarrow R\sp{a}(-k) - T\sb{A}\sp{a}(-k).
\label{eq47}
\end{equation}
%
%
With these replacements, the scattering amplitude including
the $n$ incoming and $m$ outgoing (on-shell or off-shell) pions becomes
%
%
\begin{eqnarray}
&&
\bra{\alpha; k\sb{1}a\sb{1}, \cdots, k\sb{m}a\sb{m}} 
\smatrix
\ket{\beta;  l\sb{1}b\sb{1}, \cdots, l\sb{n}b\sb{n}}|\sb{\phi=0}
\nonumber\\
%
&=&
[ (R-T\sb{A})\sp{a\sb{1}}(k\sb{1})  \cdots (R-T\sb{A})\sp{a\sb{m}}(k\sb{m}) 
\nonumber\\
%
&&
\times
  (R-T\sb{A})\sp{b\sb{1}}(-l\sb{1}) \cdots (R-T\sb{A})\sp{b\sb{n}}(-l\sb{n}) 
]\sb{{\rm S}}
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
\nonumber\\
%
&=&
[ R\sp{a\sb{1}}(k\sb{1})  \cdots R\sp{a\sb{m}}(k\sb{m}) 
  R\sp{b\sb{1}}(-l\sb{1}) \cdots R\sp{b\sb{n}}(-l\sb{n}) ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
\nonumber\\
%
&&
+({\rm terms~including}\ T\sb{A}).
\label{eq48}
\end{eqnarray}
%
This is just another expression of the LSZ reduction formula.
%
Note here that we have not yet imposed any
constraints from the broken chiral symmetry.

%
%
%
The {\chrf} in our case is obtained as a combination of Eq.~(\ref{eq48}) 
and the master equation~(\ref{eq26}), which includes any effects
from the broken chiral symmetry.
%
To see our {\chrf} concretely, 
we consider the case $n+m=2$, i.e. 
that in which the sum of the incoming and outgoing pions is 2.
%
In this case, Eq.~(\ref{eq48}) gives
%
%
\begin{eqnarray}
&&
[ (R - T\sb{A})\sp{a} (R - T\sb{A})\sp{b} ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
\nonumber\\
%
&=&
  [ R\sp{a} R\sp{b} ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
+\left(
  [ T\sb{A}\sp{a} T\sb{A}\sp{b} ]\sb{{\rm S}}
- [ T\sb{A}\sp{a} R\sp{b} ]\sb{{\rm S}}
- [ R\sp{a} T\sb{A}\sp{b} ]\sb{{\rm S}}
\right)
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0},
\label{eq49}
\end{eqnarray}
%
%
where we have suppressed the momentum indices.
%
Note again that this is simply the LSZ reduction formula
expressed in terms of $R$ and $T\sb{A}$.
%
Next, we move all $T\sb{A}$ to the right of all $R$,
using the relation $T\sb{A}R = [ T\sb{A} , R ] + RT\sb{A}$,
and then we use the master equation~(\ref{eq26}).
%
As a result of this operation,
Eq.~(\ref{eq49}) becomes
%
%
\begin{eqnarray}
&&
[ (R - T\sb{A})\sp{a} (R - T\sb{A})\sp{b} ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
\nonumber\\
%
&=&
  [ R\sp{a} R\sp{b} ]\sb{{\rm S}} 
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}
-\frac{1}{2}
\left(
  [ T\sb{A}\sp{a}, R\sp{b} ]
+ [ T\sb{A}\sp{b}, R\sp{a} ]
\right)
\bra{\alpha} \smatrix \ket{\beta}|\sb{\phi=0}.
\label{eq410}
\end{eqnarray}
%
%
This is the desired {\chrf} for the case $n+m=2$.
%
The above procedure is applied similarly for all 
values of $n$ and $m$.
%
The {\chrf} obtained in this manner is manifestly consistent with 
the LSZ reduction formula,
and there is no ambiguity in its derivation.

%
%
%
We note the presence of ``(terms including $T\sb{A}$)'' 
appearing in Eq.~(\ref{eq48}). 
%
In the following, we show that 
(i) such additional terms vanish if all pions are on shell, 
and therefore in that case our {\chrf} properly 
reduces to that proposed in Ref.~\citen{Yam96}, 
and that 
(ii) those terms are essential for deriving the correct
identities for the \textit{off-shell} amplitudes within the framework of
Ref.~\citen{Yam96}.

%
%
%
The assertion (i) can be readily demonstrated by considering 
the explicit expression of the commutation relation
between $T\sb{A}$ and $R$,
%
%
\begin{eqnarray}
[ T\sb{A}\sp{a}(k), R\sp{b}(k\sp{\prime}) ]
&=&
- \varepsilon\sp{abc}\frac{1}{\fpi\sp{2}}
  \int d\sp{4}x e\sp{i(k+k\sp{\prime})x}  
 T\sb{V}\sp{c}(x)
\nonumber\\
%
&&
- i\delta\sp{ab}(-k\sp{\prime 2}+\mpi\sp{2})
  (2\pi)\sp{4}\delta\sp{(4)}(k+k\sp{\prime})
\nonumber\\
%
&&
- \delta\sp{ab}(-k\sp{\prime 2}+\mpi\sp{2}) \frac{1}{\fpi\sp{2}} 
  \int d\sp{4}x e\sp{i(k+k\sp{\prime})x} \diff{}{s(x)}.
\label{eq411}
\end{eqnarray}
%
%
As found from Eqs.~(\ref{eq26}) and (\ref{eq411}), 
``(terms including $T\sb{A}$)'' in Eq.~(\ref{eq48}) 
is always proportional to $-k\sp{2}+\mpi\sp{2}$,
where $k$ is the four momentum of some external pion.
%
[By virtue of the symmetric permutation~(\ref{eq215}), 
the contribution from 
the first term in Eq.~(\ref{eq411}) involving the antisymmetric tensor 
$\varepsilon\sp{abc}$ vanishes.]
%
As a result, our {\chrf}, i.e. Eq.~(\ref{eq48}) 
together with Eq.~(\ref{eq26}), reduces to 
the {\chrf} proposed in Ref.~\citen{Yam96} if all pions are on shell.

%
%
%
To verify the assertion (ii), we return to the investigation of 
the $\pi N$ scattering.
%
With the one-nucleon states for $\bra{\alpha}$ and $\ket{\beta}$,
using Eqs.~(\ref{eq21}) and (\ref{eq411}), 
it can be shown that Eq.~(\ref{eq410}) becomes
%
%
\begin{eqnarray}
&&
[ (R - T\sb{A})\sp{a}(-k) (R - T\sb{A})\sp{b}(k\sp{\prime}) ]\sb{{\rm S}} 
\bra{N(p\sp{\prime})} \smatrix \ket{N(p)}|\sb{\phi=0}
\nonumber\\
%
&=&
[ R\sp{a}(-k) R\sp{b}(k\sp{\prime}) ]\sb{{\rm S}} 
\bra{N(p\sp{\prime})} \smatrix \ket{N(p)}|\sb{\phi=0}
\nonumber\\
%
&&
-i\frac{\delta\sp{ab}}{\fpi}
\left( \frac{k\sp{2} + k\sp{\prime 2}}{2} -\mpi\sp{2} \right)
\bra{N(p\sp{\prime})} \hat{\sigma}(0) \ket{N(p)},
\label{eq412}
\end{eqnarray}
%
%
modulo $(2\pi)\sp{4}\delta\sp{(4)}(k+p-k\sp{\prime}-p\sp{\prime})$,
where we drop the non-scattering parts arising from the second term
on the r.h.s. of Eq.~(\ref{eq411}).
%
Now we obtain the additional contribution to Eq.~(\ref{eq35}) 
arising from ``(terms including $T\sb{A}$)''.
%
Our result is identical with Weinberg's formula for the $\pi N$ scattering,
even off the mass shell, as should be the case.
%
We can thus conclude that our {\chrf}, being manifestly
consistent with the LSZ reduction formula, is
a proper off-shell extension of the on-shell {\chrf} 
presented in Ref.~\citen{Yam96}. 

%
%
%
Before closing this section, several comments are in order:
%
%
\begin{itemize}
\item
A naive application of the replacement of
the single commutators expressed in Eq.~(\ref{eq216}) 
to multi-commutator cases results in
the nontrivial replacement of the functional derivative operator
$\int d\sp{4}x \exp(ikx) ( \Box + \mpi\sp{2} )(\delta / \delta J\sp{a}) 
\rightarrow R\sp{a}(k)$.
%
Such replacement is consistent with the LSZ reduction formula
for on-shell amplitudes, but it is incompatible off the mass shell,
because $R$ and $T\sb{A}$ do not commute,
and thus ``(terms including $T\sb{A}$)'' does not vanish.
%
This is the reason that the {\chrf} proposed in Ref.~\citen{Yam96}
cannot be naively applied to off-shell pions.
%
%
\item
Practically, the {\chrf} formulated in Ref.~\citen{Yam96} gives 
the same off-shell result
as ours for single pion emission and absorption processes
[see Eqs.~(6.4) and (6.5) in Ref.~\citen{Yam96}]. 
%
According to our results, however, it seems to be a rather special case,
because in this case, ``(terms including $T\sb{A}$)''
survives in neither the on-shell nor off-shell case:
$\bra{\alpha}[a\sb{\text{in}}\sp{a}(k),\smatrix]\ket{\beta}
=(R-T\sb{A})\sp{a}(k)\bra{\alpha}\smatrix \ket{\beta}
=R\sp{a}(k)\bra{\alpha}\smatrix\ket{\beta}$.
%
%
\item
In principle, the {\chrf} presented in this paper is
applicable for any value of (on-shell or off-shell) pion momenta.
%
This is due to the fact that
the master equations are exact formulas essentially equivalent to 
the current conservation law or Noether's theorem.
%
The nature of specific points, such as the chiral limit and 
the soft pion limit, is relevant only 
if extra assumptions are introduced, for example, 
the possibility of a loop expansion around those points\cite{Yam96}.
\end{itemize}
%
%
%
%
\section{Summary}
\label{sec5}
%
%
%
%
We have investigated the off-shell structure of the {\chrf}
because of its importance in actual calculations and in the 
pursuit of a deeper
understanding of the theoretical framework developed 
in Ref.~\citen{Yam96}.
%
We have seen that the on-shell {\chrf} proposed in Ref.~\citen{Yam96}
cannot be naively applied to off-shell pions.
%
We then found its proper extension off the mass shell.
%
This was achieved by reconstructing the {\chrf}
in a form manifestly consistent with 
the conventional LSZ reduction formula.

%
%
%
The analysis employed in this work can be 
readily applied to the three flavor case studied in 
Refs.~\citen{Lee99} and \citen{Lee98}.
%
%
%
%
\section*{Acknowledgments}
%
%
%
%
The author would like to thank Dr. Masaki Arima 
for discussions and his helpful advice in completing the manuscript.
%
The author would also like to thank Dr. Toru Sato
for valuable comments.
%
This work was supported by Research Fellowships 
of the Japan Society for the Promotion of Science (JSPS)
for Young Scientists.
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Some macros are available for the bibliography:
%  o for general use
%    \JL : general journals                 \andvol : Vol (Year) Page
%  o for individual journal 
%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
\begin{thebibliography}{14}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\bibitem{Yam96}
H.~Yamagishi and I.~Zahed, \ANN{247,1996,292}.

\bibitem{Vel66}
M.~Veltman, \PRL{17,1966,553}.

\bibitem{Bell67}
J.~S.~Bell, \JL{Nuovo~Cim. (Ser.~X) A,50,1967,129}.

\bibitem{Ste97}
J.~V. Steele, H.~Yamagishi and I.~Zahed, \NPA{615,1997,305}.

\bibitem{Lee99}
C.-H. Lee, H.~Yamagishi and I.~Zahed, \NPA{653,1999,185}.

\bibitem{Kam05-1}
H.~Kamano, M.~Morishita and M.~Arima, \PRC{71,2005,045201}.

\bibitem{Ste96}
J.~V. Steele, H.~Yamagishi and I.~Zahed, \PLB{384,1996,255}.

\bibitem{Ste97-2}
J.~V. Steele, H.~Yamagishi and I.~Zahed, \PRD{56,1997,5605}.

\bibitem{Lee98}
C.-H. Lee, H.~Yamagishi and I.~Zahed, \PRC{58,1998,2899}.

\bibitem{Dus06}
K.~Dusling, D.~Teaney and I.~Zahed, nucl-th/0604071.

\bibitem{Bog}
N.~N. Bogoliubov and D.~V. Shirkov, 
\textit{Introduction to the Theory of Quantized Fields}, 3rd ed.
(Wiley, New York, 1980).

\bibitem{Wei-70}
S.~Weinberg, 
\textit{Lectures on Elementary Particles and Quantum Field Theory}, 
S.~Deser, M.~Grisaru and H.~Pendleton, Vol.~1
(MIT Press, 1970).

\bibitem{Bro71}
L.~S. Brown, W.~J. Pardee and R.~D. Peccei, \PRD{4,1971,2801}.


\end{thebibliography}
\end{document}

hep-ph0412281.extracted_bib
\begin{thebibliography}{14}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\bibitem{Yam96}
H.~Yamagishi and I.~Zahed, \ANN{247,1996,292}.

\bibitem{Vel66}
M.~Veltman, \PRL{17,1966,553}.

\bibitem{Bell67}
J.~S.~Bell, \JL{Nuovo~Cim. (Ser.~X) A,50,1967,129}.

\bibitem{Ste97}
J.~V. Steele, H.~Yamagishi and I.~Zahed, \NPA{615,1997,305}.

\bibitem{Lee99}
C.-H. Lee, H.~Yamagishi and I.~Zahed, \NPA{653,1999,185}.

\bibitem{Kam05-1}
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