$\Delta(1232)$ in $\pi$$N$$\rightarrow$$\pi$$\pi$$N$ reaction
Hiroyuki Kamano
The contribution of $\Delta(1232)$ to the $\pi N\rightarrow \pi\pi N$ reaction is examined by making use of the chiral reduction formula developed by Yamagishi and Zahed. % The influence of the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions on this reaction, which has not been regarded as important so far, is considered for all channels with the initial $\pi\sp{\pm}p$ states. % The total cross sections are calculated to tree level for the energy up to $T\sb{\pi}=400$ MeV. % Although the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions give a small effect on the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$, $\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$, and $\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channels, the $\pi\sp{\pm}p\rightarrow\pi\sp{\pm}\pi\sp{0}p$ channels are found to be sensitive to these interactions.
paper.tex
%\documentclass[twocolumn,showpacs,preprintnumbers,amsmath,amssymb,nofootinbib]{revtex4}
\documentclass[preprint,showpacs,preprintnumbers,amsmath,amssymb,nofootinbib]{revtex4}

% Some other (several out of many) possibilities
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{longtable}
\usepackage{latexsym}
\bibliographystyle{apsrev}

\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle {#1}|}
\newcommand{\inp}[2]{\langle{#1}\ket{#2}}
\newcommand{\g}[1]{\gamma\sp{#1}}
\newcommand{\gc}[1]{\gamma\sb{#1}}
\newcommand{\vectorjc}[2]{j\sp{{#1}}\sb{V{#2}}}
\newcommand{\axialjc}[2]{j\sp{{#1}}\sb{A{#2}}}
\newcommand{\isomtx}[3]{{I}\sp{#1}({\textstyle \frac{#2}{2},\frac{#3}{2}})}

\input{epsf}
\begin{document}

%\preprint{APS/123-QED}

\title{$\Delta(1232)$ in $\pi$$N$$\rightarrow$$\pi$$\pi$$N$ reaction}

\author{Hiroyuki Kamano}
 \email{kamano@ocunp.hep.osaka-cu.ac.jp}
\author{Masaki Arima}
 \email{arima@ocunp.hep.osaka-cu.ac.jp}
\affiliation{
 Department of Physics, Osaka City University, Osaka 558-8585, Japan }

\date{\today}

\begin{abstract}
The contribution of $\Delta(1232)$ to the $\pi N\rightarrow \pi\pi N$ 
reaction is examined by making use of the
chiral reduction formula developed by Yamagishi and Zahed.
%
The influence of the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions
on this reaction, which has not been regarded as important so far,
is considered for all channels with the initial $\pi\sp{\pm}p$ states.
%
The total cross sections are calculated 
to tree level for the energy up to $T\sb{\pi}=400$ MeV.
%
Although the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions give 
a small effect on the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$, 
$\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$, and 
$\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channels,
the $\pi\sp{\pm}p\rightarrow\pi\sp{\pm}\pi\sp{0}p$ channels are 
found to be sensitive to these interactions.
\end{abstract}

\pacs{11.30.Er, 11.30.Rd, 11.40.Ha, 12.40.Vv, 13.75.Gx}

%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle

\section{\label{sec1} INTRODUCTION}%:\protect
%

The $\pi N\rightarrow\pi\pi N$ reaction is important 
in the analysis of the low energy $\pi N$ scattering
because of its role as a major inelastic process.  
%
Since the pion mass is small owing to the spontaneous 
breaking of chiral symmetry, the contribution of the 
$\pi\pi N$ channel is observed even around the $\Delta(1232)$ resonance.

%
Many theoretical approaches have been taken to study this reaction:
for example, the phenomenological model using effective 
Lagrangian~\cite{JM97,OV85,JDV92} 
and a series of systematic analyses based on 
the chiral perturbation theory~\cite{BKM95,BKM97,FBM00}. 
%
Also, the several kinds of global fits of the experimental data 
have been performed~\cite{BL91,Ver95,BPS98}. 

%
These  studies show that 
the $\pi N\rightarrow\pi\pi N$ reaction is useful 
to extract some important aspects of hadron physics.
%
The $\pi\pi$ scattering length is obtained by 
using the $\pi N\rightarrow\pi\pi N$ data.
%
Furthermore valuable information taken from 
the $\pi N\rightarrow\pi\pi N$ reaction benefits the 
study of baryon resonances
which have a considerable influence on this reaction.
%
In particular, the contribution of $\Delta(1232)$ 
is remarkable around the threshold region.

%
Because $\Delta$(1232) contributes to 
the $\pi N\rightarrow\pi\pi N$ reaction through three typical interactions, 
i.e., the $\pi N\Delta$, $\pi\Delta\Delta$, and $\rho N\Delta$ 
interactions, the information of these interactions can 
be accessible through the analysis of this reaction.
%
However the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions 
have not been taken seriously in the theoretical analyses so far.
%
These interactions are not included in Ref.~\cite{JM97},
and their contributions to the 
$\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ channel
are shown to be negligible in Refs.~\cite{OV85,JDV92}.
%
On the other hand, many types of interactions including
the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions 
are considered in the global fits~\cite{Ver95,BPS98},
but their individual roles in each channel of 
the $\pi N\rightarrow\pi\pi N$ reaction are not discussed.
%
The $\pi\Delta\Delta$ and $\rho N\Delta$ interactions still remain 
controversial in contrast to the well known $\pi N\Delta$ interaction.

%
Chiral symmetry is a key to systematic understanding of 
the pion related reactions including the $\pi N\rightarrow\pi\pi N$ reaction.
%
This symmetry has been accepted as a fundamental concept
in the hadron physics due to the successes of
the various low energy theorems~\cite{Current} and the 
chiral perturbation theory in the threshold region~\cite{Mei93,Eck95}.
%
These studies are based on spontaneous broken chiral symmetry,
where the pions are considered as the Nambu-Goldstone bosons
realized by this symmetry breaking.

%
Recently, a general framework analyzing hadron reactions 
has been developed by Yamagishi and Zahed~\cite{YZ96}
on the basis of chiral symmetry.
%
This framework introduces a new type of reduction formula, 
i.e. the chiral reduction formula,
which manifests a requirement of chiral symmetry 
satisfied by the invariant amplitudes
not only for the threshold region but for the resonance region.
%
It is worth noting that the chiral reduction formula
offers the Ward identity satisfied by the quantum amplitudes without
relying on any model or expansion scheme at the beginning.
%
This identity allows us to take a flexible and consistent view,
which is free from restrictions given by specific model, 
of the theoretical approach to resonances~\cite{SYZ97,LYZ99}.

%
In this paper, applying the chiral reduction formula to 
the invariant amplitude of the $\pi N\rightarrow \pi \pi N$ reaction, 
we try to clarify the contribution of  $\Delta(1232)$ in this reaction
to tree level. 
%
We particularly discuss  the $\pi\Delta\Delta$ and $\rho N\Delta$ 
interactions in all channels with the initial $\pi\sp{\pm}p$ state.
%
We assume the Rarita-Schwinger field for $\Delta(1232)$
without taking account of its internal structure.
%
We do not aim to fix their coupling constants but 
to clarify their importance in the $\pi N\rightarrow\pi\pi N$ reaction. 
%Our aim is not to fix their coupling constants but 
%to show those sensitivity to the $\pi N\rightarrow\pi\pi N$ reaction 
%in the exact framework of chiral symmetry.
%
Our tree level calculation is enough to make
a qualitative discussion about the $\pi\Delta\Delta$ and $\rho N\Delta$ 
interactions because of the smallness of loop collections
in this reaction as shown in ref.~\cite{FBM00} .

%
In Sec.~II, we show the total cross section of the
$\pi N\rightarrow\pi\pi N$ reaction.
%
We take into account the isospin symmetry breaking
when we compare our numerical results with the experimental values.
%
We explain the chiral reduction of 
the $\pi N\rightarrow \pi\pi N$ reaction in Sec.~III, 
and we show the form factors of the current matrix elements 
appeared due to the reduction of the invariant amplitude in Sec.~IV.
%
Our numerical results are presented in Sec.~V 
and we discuss the contributions of $\Delta(1232)$ in this reaction. 
%
The summaries are given in Sec.~VI.
%
In the Appendix, we show the phenomenological Lagrangians which are 
necessary to evaluate the form factors appeared in the current 
matrix elements, and give a brief explanation 
of the Rarita-Schwinger field.
%
%
%
%
\section{TOTAL CROSS SECTION}
We consider the total cross section of the $\pi N\rightarrow \pi\pi N$ reaction 
in the isospin symmetric limit with the averaged values $m\sb{N}=939$ MeV 
and $m\sb{\pi}=138$ MeV for the nucleon and the pion masses, respectively
(see Fig.~\ref{fig1}).
%
%
\begin{figure}[ht]
\includegraphics[width=7cm]{fig1.ps}
\caption{ 
The $\pi N\rightarrow\pi\pi N$ reaction. 
Each pion has the isospin index $(a,b,c)$ and 
the four-momentum $k\sb{i}$ $(i=1,2,3)$,
and each nucleon has the four-momentum $p\sb{j}$ $(j=1,2)$.
}
\label{fig1}
\end{figure}
%
%

%
The isospin symmetric total cross section is given by 
%
\begin{eqnarray}
\sigma\sb{\rm sym}&=&
\frac{{\cal B}}{2\sqrt{(s-m\sp{2}\sb{+})(s-m\sp{2}\sb{-})}}
\int
\frac{d\sp{3}p\sb{2}}{(2\pi)\sp{3}2E\sb{2}}
\frac{d\sp{3}k\sb{2}}{(2\pi)\sp{3}2\omega\sb{2}}
\frac{d\sp{3}k\sb{3}}{(2\pi)\sp{3}2\omega\sb{3}}\nonumber\\
&&\times(2\pi)\sp{4}\delta\sp{(4)}(p\sb{1}+k\sb{1}-p\sb{2}-k\sb{2}-k\sb{3})
\overline{|{\cal T}|\sp{2}},\label{eq:1}
\end{eqnarray}
%
%
where $s=(p\sb{1}+k\sb{1})\sp{2}$ and $m\sb{\pm}=m\sb{N}\pm m\sb{\pi}$.
%
The energy of outgoing nucleon and pions are denoted as
$E\sb{2}$ and $\omega\sb{i}\ (i=2,3)$, respectively. 
%
The Bose factor ${\cal B}$ is equal to 1/2 
if the outgoing pions are identical, otherwise this factor is 1. 
%
The invariant amplitude is denoted as ${\cal T}$,
for which the incoming (outgoing) nucleon spin average (sum) is taken, 
$\overline{|{\cal T}|\sp{2}}=(1/2)\sum\sb{spins}|{\cal T}|\sp{2}$.

%
We will show the numerical result as a function of 
the incoming pion kinetic energy in the laboratory system, $T\sb{\pi}$,
which incorporates the isospin symmetry breaking observed in  
the mass difference in each reaction channel. 
%
The invariant amplitude is sensitive to this difference through
the pion kinetic energy of each channel near the threshold.
%
We consider this symmetry breaking 
by shifting the value of the kinetic energy from 
the isospin symmetric threshold to the physical threshold~\cite{BKM97}. 
%
The total cross section $\sigma\sb{{\rm expt}}$ 
measured by the experiments is related to 
Eq.~(\ref{eq:1}) by 
%
%
\begin{equation}
\sigma\sb{{\rm expt}}(T\sb{\pi})=
\sigma\sb{\rm sym}({\bar T}\sb{\pi}),\label{eq:2}
\end{equation}
%
%
where $\bar{T}\sb{\pi}$ is the isospin symmetric  
pion kinetic energy and $\delta T\sb{\pi}=T\sb{\pi}-{\bar T}\sb{\pi}$.
%
In Table \ref{tab1}, we summarize 
the value of $\delta T\sb{\pi}$ in each reaction channel.
%
%
\begin{table}[ht]
\caption{
The values of $\delta T\sb{\pi}$ in each reaction channel.
}
\label{tab1}
\begin{ruledtabular}
\begin{tabular}{cc}
channel & $\delta T\sb{\pi}$ MeV\\ \hline
$\pi\sp{+}p \rightarrow \pi\sp{+}\pi\sp{+}n$ & $+3.97$\\
$\pi\sp{+}p \rightarrow \pi\sp{+}\pi\sp{0}p$ & $-3.66$\\
$\pi\sp{-}p \rightarrow \pi\sp{+}\pi\sp{-}n$ & $+3.97$\\
$\pi\sp{-}p \rightarrow \pi\sp{0}\pi\sp{0}n$ & $-7.92$\\
$\pi\sp{-}p \rightarrow \pi\sp{-}\pi\sp{0}p$ & $-3.66$\\
\end{tabular}
\end{ruledtabular}
\end{table}
%
%
%
%
\section{CHIRAL REDUCTION FOR $\pi N\rightarrow\pi\pi N$ REACTION}
In this section, we explain the chiral reduction of 
the invariant amplitude for the $\pi N\rightarrow\pi\pi N$ reaction,
%
%
\begin{equation}
(2\pi)\sp{4}\delta\sp{(4)}(p\sb{1}+k\sb{1}-p\sb{2}-k\sb{2}-k\sb{3})i{\cal T}
=
\bra{N(p\sb{2})}a\sp{c}(k\sb{3})a\sp{b}(k\sb{2})
\hat{\cal S}a\sp{a \dag}(k\sb{1})\ket{N(p\sb{1})}|\sb{\phi=0},
\label{eq:3}
\end{equation}
%
%
where $a\sp{a}(k)$ [$a\sp{a\dag}(k)$] is  
an annihilation (creation) operator of the pion 
with the isospin component $a$ and the four-momentum $k$. 
%
$\hat{\cal S}=\hat{\cal S}[\phi]$ is the extended $S$ matrix operator which 
is a functional of $\phi=(a,v,s,J)$; 
the axial vector, vector, scalar, and pseudoscalar 
$c$-number external fields~\cite{YZ96}.
%
At $\phi=0$, $\hat{\cal S}$ is reduced to 
the ordinary $S$ matrix operator. 
%
These external fields play an important role 
in the formulation of the chiral reduction formula.
%

%
Using the chiral reduction formula, we decompose 
the invariant amplitude $i{\cal T}$ as~\cite{SYZ98}
%
%
\begin{eqnarray}
i{\cal T}&=&(i{\cal T}_{\pi}+
i{\cal T}_{A}+i{\cal T}_{SA}+i{\cal T}_{VA})\nonumber\\ 
&&+(k\sb{1},a\leftrightarrow -k\sb{3},c)
+(k\sb{2},b\leftrightarrow k\sb{3},c)\nonumber\\
&&+i{\cal T}_{AAA},\label{eq:4}
\end{eqnarray}
%
%
where $(\ \leftrightarrow\ )$ represents a permutation of 
the momentum and isospin indices of the pion in the first four terms. 
%
Each term on the right-hand side of Eq.~(\ref{eq:4}) 
is explicitly written as
%
%
\begin{eqnarray}
i{\cal T}_\pi&=&
\frac{i}{f\sb{\pi}\sp{2}}[(k\sb{1}-k\sb{2})\sp{2}-m\sb{\pi}\sp{2}]\delta\sp{ab}
\bra{N(p_2)}\hat\pi\sp{c}(0)\ket{N(p_1)},\label{eq:5}\\
%
i{\cal T}_A&=&
\frac{1}{2f\sb{\pi}\sp{3}}(k_2-k_1)^\mu\delta\sp{ab}
\bra{N(p_2)}\axialjc{c}{\mu}(0)\ket{N(p_1)},\label{eq:6}\\
%
i{\cal T}_{SA} &=&
-i\frac{m\sb{\pi}\sp{2}}{f\sb{\pi}\sp{2}}k\sb{3}\sp{\mu}
\delta\sp{ab}\int d^4x e\sp{-i(k_1-k_2)\cdot x}
\bra{N(p_2)}T^\ast (\hat{\sigma}(x)\axialjc{c}{\mu}(0))\ket{N(p_1)},
\label{eq:7}\\
%
i{\cal T}_{VA} &=&
\frac{1}{2f\sb{\pi}\sp{3}}(k\sb{1}+k\sb{2})\sp{\mu}k\sb{3}\sp{\nu}
\varepsilon\sp{abe}\int d^4x e\sp{-i(k_1-k_2)\cdot x}
\bra{N(p_2)}T^\ast (\vectorjc{e}{\mu}(x)\axialjc{c}{\nu}(0))\ket{N(p_1)},
\label{eq:8}\\
%
i{\cal T}_{AAA} &=&
-\frac{1}{f\sb{\pi}\sp{3}}k\sb{1}\sp{\mu}k\sb{2}\sp{\nu}k\sb{3}\sp{\lambda}\nonumber\\
&&\times\int d^4x_1d^4x_2 e\sp{-ik_1\cdot x_1+ik_2\cdot x_2}
\bra{N(p_2)}T^\ast (\axialjc{a}{\mu}(x_1)\axialjc{b}{\nu}(x_2)
\axialjc{c}{\lambda}(0))\ket{N(p_1)}.\label{eq:9}
\end{eqnarray}
%
%
The pseudoscalar density $\hat{\pi}\sp{a}(x)$ is identified with
the asymptotic pion field $\pi\sb{as}(x)$ as $x\sb{0}\rightarrow\pm \infty$. 
%
The one-pion reduced axial current $\axialjc{a}{\mu}(x)$ is 
defined by 
$\axialjc{a}{\mu}(x)={\bf A}\sp{a}\sb{\mu}(x)
+f\sb{\pi}\partial\sb{\mu}\hat{\pi}\sp{a}(x)$ 
where ${\bf A}\sp{a}\sb{\mu}(x)$ 
is the ordinary axial current with the asymptotic form
${\bf A}\sp{a}\sb{\mu}(x)\rightarrow 
-f\sb{\pi}\partial\sb{\mu}\pi\sp{a}\sb{as}(x)+\cdots$
as $x\sb{0}\rightarrow \pm\infty$.
%
The vector current and the scalar density are represented by
$\vectorjc{a}{\mu}(x)$ and $\hat{\sigma}(x)$, 
respectively\footnote
{We call $\hat{\pi}\sp{a}(x)$ and $\hat{\sigma}(x)$ 
the {\textquoteleft current\textquoteright} instead of
the {\textquoteleft density\textquoteright}.}.

%
Equation~(\ref{eq:4}) is an exact relation 
between the invariant amplitude and the current correlations
constrained by chiral symmetry with the 
partially conserved axial-vector current (PCAC) condition, 
$\partial\sp{\mu}{\bf A}\sp{a}\sb{\mu}(x)\rightarrow
f\sb{\pi}m\sb{\pi}\sp{2}\pi\sp{a}\sb{as}(x)$ as 
$x\sb{0}\rightarrow \pm\infty$.
%
Thus the chiral reduction formula can naturally deal 
with the explicit breaking of chiral symmetry.
%
For example, ${\cal T}\sb{SA}$ is due to 
this breaking and vanishes in chiral limit $m\sb{\pi}\rightarrow0$.

%
In Fig.~\ref{fig2}, we show the diagrammatical 
interpretation of Eq.~(\ref{eq:4}) to tree level.
%
The solid line corresponds to the external nucleon, and 
the double line to the propagation of 
the nucleon or $\Delta(1232)$.
%
The cross denotes that the hadron couples to the current. 
%
The pion pole appears in the $\hat\pi$ current,  
while the $\axialjc{}{}$ current is free of this pole by definition.
%
We assume the vector meson dominance (VMD) for the 
$\vectorjc{}{}$ current, and
take into account the $\rho$ meson pole in ${\cal T}\sb{VA}$.
%
The $\Delta(1232)$ resonance 
does not propagate in ${\cal T}\sb{SA}$ because
the $N$-$\Delta$ transition is not brought about 
by the $\hat\sigma$ current.
%
%
\begin{figure}[ht]
  \includegraphics[width=15cm]{fig2.ps}
\caption{Diagrammatical interpretation for the chiral reduction of 
$\pi N\rightarrow\pi\pi N$ reaction to tree level.
}
\label{fig2}
\end{figure}
%
%
%
%
\section{CURRENT MATRIX ELEMENTS}
Taking account of the Lorentz, isospin, parity and time reversal
invariances and the vector current conservation 
$\partial\sp{\mu}\vectorjc{a}{\mu}(x)=0$,
we can write the general forms of the current matrix elements 
necessary to evaluate Eqs.~(\ref{eq:5})-(\ref{eq:9}).
%
We summarize those matrix elements 
in Table \ref{tab2}. 
%
%
\begin{table}[ht]
\caption{
The current matrix elements necessary to evaluate each diagram
in Fig.~\ref{fig2}.
}
\label{tab2}
\begin{ruledtabular}
\begin{tabular}{ccccc}
${\cal T}\sb{\pi}$		&
${\cal T}\sb{A}$		&
${\cal T}\sb{SA}$		&
${\cal T}\sb{VA}$		&
${\cal T}\sb{AAA}$		\\ \hline
$\bra{N}\hat{\pi}\ket{N}$ 	&
$\bra{N}\axialjc{}{}\ket{N}$	&
$\bra{N}\axialjc{}{}\ket{N}$ 	&
$\bra{N}\vectorjc{}{}\ket{N}$ 	&
$\bra{N}\axialjc{}{}\ket{N}	$ \\ 
				&
				&
$\bra{N}\hat{\sigma}\ket{N}$ 	&
$\bra{\Delta}\vectorjc{}{}\ket{N}$& 
$\bra{\Delta}\axialjc{}{}\ket{N}$ \\
				&
				&
				&
$\bra{N}\axialjc{}{}\ket{N}$ 	&
$\bra{\Delta}\axialjc{}{}\ket{\Delta}$ \\
				&
				&
				&
$\bra{\Delta}\axialjc{}{}\ket{N}$&
				\\
\end{tabular}
\end{ruledtabular}
\end{table}
%
%
%
%
\begin{center}
\bf{A. Vector-isovector part}
\end{center}

%
The vector current matrix elements are generally written as
%
%
\begin{equation}
\bra{N(p\sp{\prime})}\vectorjc{a}{\mu}(0)\ket{N(p)}=
\bar{u}(p\sp{\prime})\left[ F\sp{N}\sb{V,1}(t)\gc{\mu}+
F\sp{N}\sb{V,2}(t)\frac{i}{2m\sb{N}}\sigma\sb{\mu\nu}q\sp{\nu}\right]
 \frac{\tau\sp{a}}{2}u(p)\label{eq:10}
\end{equation}
%
for the nucleon, and
%
%
\begin{eqnarray}
\bra{\Delta(p\sp{\prime})}\vectorjc{a}{\mu}(0)\ket{N(p)}&=&
\bar{U}\sp{\nu}(p\sp{\prime})\left[F\sp{N\Delta}\sb{V,1}(t)g\sb{\nu\mu}
+F\sp{N\Delta}\sb{V,2}(t)Q\sb{\nu}\gc{\mu}\right.\nonumber\\
&&\left.+F\sp{N\Delta}\sb{V,3}(t)Q\sb{\nu}Q\sb{\mu}+
iF\sp{N\Delta}\sb{V,4}(t)Q\sb{\nu}\sigma\sb{\mu\lambda}
Q\sp{\lambda}
\right]\gc{5}
\isomtx{a}{3}{1}u(p)\label{eq:11}
\end{eqnarray}
%
%
for the $N$-$\Delta$ transition, 
where $q\sp{\mu}=(p\sp{\prime}-p)\sp{\mu}$, $Q\sp{\mu}=-q\sp{\mu}$
and $t=(p\sp{\prime}-p)\sp{2}$,
$\tau\sp{a}$ is the isospin Pauli matrix and $I\sp{a}(i,j)$ is 
the $j\rightarrow i$ isospin transition $(2i+1)\times(2j+1)$ matrix.
%
The isoquadruplet Rarita-Schwinger vector-spinor 
and the isodoublet Dirac spinor are denoted as
$U\sp{\mu}(p)$ and $u(p)$, respectively. 
%
We note that $\gc{5}$ appears in Eq.~(\ref{eq:11})
because the Rarita-Schwinger field with spin-parity $(3/2)\sp{+}$
is employed for $\Delta(1232)$. 
%
The form factors ( $F\sp{N}\sb{V,i}(t)$ with $i=1,2$ 
and $F\sp{N\Delta}\sb{V,j}(t)$ with $j=1\sim 4$) 
are the functions of $t$.

%
In order to determine the $t$ dependence of the form factors to tree level, 
we make use of the current-field identity~\cite{JJ}
%
%
\begin{equation}
\vectorjc{a}{\mu}(x)=\frac{m\sb{\rho}\sp{2}}{f\sb{\rho}}\rho\sp{a}\sb{\mu}(x),
\label{eq:12}
\end{equation}
%
%
where $m\sb{\rho}$ is the $\rho$ meson mass and 
$f\sb{\rho}$ corresponds to the gauge coupling constant of 
the hidden local symmetry model for the vector mesons~\cite{Ban85,HY03}. 
%
Based on this identity which expresses the phenomenology of the VMD,
we write the nucleon form factors as 
%
%
\begin{eqnarray}
F\sp{N}\sb{V,1}(t)&=&
\frac{m\sb{\rho}\sp{2}}
{m\sb{\rho}\sp{2}-t-im\sb{\rho}\Gamma\sb{\rho}(t)},\label{eq:13}\\ 
F\sp{N}\sb{V,2}(t)&=&\kappa\sb{V}F\sp{N}\sb{V,1}(t)\label{eq:14},
\end{eqnarray}
%
%
where we use the standard $\rho NN$ interaction (see the Appendix)
and the universality relation $g\sb{\rho NN}=f\sb{\rho}$.
%
The isovector magnetic moment is denoted as $\kappa\sb{V}$.
%
The phenomenological width of the $\rho$ meson
is parametrized as
%
%
\begin{equation}
\Gamma\sb{\rho}(t)=\Gamma\sb{\rho}\frac{m\sb{\rho}}{\sqrt{t}}
\left(\frac{t-4m\sb{\pi}\sp{2}}
{m\sb{\rho}\sp{2}-4m\sb{\pi}\sp{2}}\right)\sp{3/2}
\theta(t-4m\sb{\pi}\sp{2}),\label{eq:15}
\end{equation}
%
%
where $\Gamma\sb{\rho}$ is 
the total width at $t=m\sb{\rho}\sp{2}$~\cite{SYZ97}.
%
As for the $N$-$\Delta$ transition form factors, we obtain
%
%
\begin{eqnarray}
F\sp{N\Delta}\sb{V,1}(t)&=&\frac{f\sb{\rho N\Delta}}{f\sb{\rho}}
\left(\frac{m\sb{N}+m\sb{\Delta}}{m\sb{\rho}}\right)\frac{m\sb{\rho}\sp{2}}
{m\sb{\rho}\sp{2}-t-im\sb{\rho}\Gamma\sb{\rho}(t)},\label{eq:16}\\
F\sp{N\Delta}\sb{V,2}(t)&=&\frac{1}{m\sb{N}+m\sb{\Delta}}
F\sp{N\Delta}\sb{V,1}(t),\label{eq:17}
\end{eqnarray}
%
%
where we use the $\rho N\Delta$ interaction (\ref{eq:33}). 
%
The $\rho N\Delta$ coupling constant is denoted as 
$f\sb{\rho N\Delta}$.

%
We note that the other form factors
$F\sp{N\Delta}\sb{V,3}(t)$ and $F\sp{N\Delta}\sb{V,4}(t)$ in Eq.~(\ref{eq:11})
are fixed to zero as long as we consider 
the Lagrangian (\ref{eq:33}) for the $\rho N\Delta$ interaction.
%
%
%
%
\begin{center}
\bf{B. Axial-isovector part}
\end{center}

%
The matrix elements of the one-pion reduced axial current 
are generally written as~\cite{KZ96, Arn79}
%
%
\begin{eqnarray}
\bra{N(p\sp{\prime})}\axialjc{a}{\mu}(0)\ket{N(p)}
&=&
\bar{u}(p\sp{\prime})
\left[
F\sp{N}\sb{A,1}(t)\gc{\mu}+
F\sp{N}\sb{A,2}(t)q\sb{\mu}
\right]
\gc{5}\frac{\tau\sp{a}}{2}u(p),
\label{eq:18}\\
%
\bra{\Delta(p\sp{\prime})}\axialjc{a}{\mu}(0)\ket{N(p)}
&=&
\bar{U}\sp{\nu}(p\sp{\prime})
\left[
F\sp{N\Delta}\sb{A,1}(t)g\sb{\nu\mu}
+F\sp{N\Delta}\sb{A,2}(t)Q\sb{\nu}\gc{\mu}
\right.
\nonumber\\
&&
\left.
+F\sp{N\Delta}\sb{A,3}(t)Q\sb{\nu}Q\sb{\mu}
+iF\sp{N\Delta}\sb{A,4}(t)Q\sb{\nu}\sigma\sb{\mu\lambda}Q\sp{\lambda}
\right]
\isomtx{a}{3}{1}u(p),
\label{eq:19}\\
%
\bra{\Delta(p\sp{\prime})}\axialjc{a}{\mu}(0)\ket{\Delta(p)}
&=&
\bar{U}\sp{\nu}(p\sp{\prime})
\left[
F\sp{\Delta}\sb{A,1}(t)g\sb{\nu\lambda}\gc{\mu}
+F\sp{\Delta}\sb{A,2}(t)g\sb{\nu\lambda}q\sb{\mu}
\right.
\nonumber\\
&&
+F\sp{\Delta}\sb{A,3}(t)(q\sb{\nu}g\sb{\mu\lambda}
+g\sb{\nu\mu}q\sb{\lambda})
\nonumber\\
&&\left.+F\sp{\Delta}\sb{A,4}(t)q\sb{\nu}\gc{\mu}q\sb{\lambda}+
F\sp{\Delta}\sb{A,5}(t)q\sb{\nu}q\sb{\mu}q\sb{\lambda}\right]
\gc{5}\isomtx{a}{3}{3}U\sp{\lambda}(p).
\label{eq:20}
\end{eqnarray}
%
%
Owing to the nature of $\Delta(1232)$ in our treatment, 
$\gc{5}$ does not appear in Eq.~(\ref{eq:19}) 
in contrast to the familiar form of the axial currents given by 
Eqs.~(\ref{eq:18}) and (\ref{eq:20}). 
%
Note that the pion pole does not contribute to the above form factors
by definition. 

%
Some of the form factors in Eqs.~(\ref{eq:18})$-$(\ref{eq:20})
are exactly related to the renormalized coupling constants for 
the pion-baryon interaction,
%
%
\begin{eqnarray}
f\sb{\pi NN}(t)&=&
\frac{m\sb{\pi}}{f\sb{\pi}}\left[\frac{1}{2}F\sp{N}\sb{A,1}(t)
+\frac{t}{4m\sb{N}}F\sp{N}\sb{A,2}(t)\right],\label{eq:21}\\
%
f\sb{\pi N\Delta}(t)&=&
\frac{m\sb{\pi}}{f\sb{\pi}}\left[
F\sp{N\Delta}\sb{A,1}(t)
+(m\sb{N}-m\sb{\Delta})F\sp{N\Delta}\sb{A,2}(t)
+tF\sp{N\Delta}\sb{A,3}(t)\right],\label{eq:22}\\
%
f\sb{\pi\Delta\Delta}(t)&=&
\frac{m\sb{\pi}}{f\sb{\pi}}\left[F\sp{\Delta}\sb{A,1}(t)+
\frac{t}{2m\sb{\Delta}}F\sp{\Delta}\sb{A,2}(t)\right],\label{eq:23}
\end{eqnarray}
%
%
where we consider the effective 
$\pi NN$, $\pi N\Delta$, and $\pi\Delta\Delta$ 
interactions (\ref{eq:29})-(\ref{eq:31}) given in the Appendix.
%
The other form factors  
($F\sp{N\Delta}\sb{A,4}(t)$, $F\sp{\Delta}\sb{A,3}(t)$, 
$F\sp{\Delta}\sb{A,4}(t)$ and $F\sp{\Delta}\sb{A,5}(t)$)
do not appear in our calculation.

%
At tree level, all the form factors are reduced to constants.
%
We write $F\sp{N}\sb{A,1}=g\sb{A}$ and
$F\sp{N}\sb{A,2}=-2\overline{\Delta}\sb{\pi N}/m\sb{\pi}\sp{2}$,
%
where $g\sb{A}$ is the axial coupling constant and 
$\overline{\Delta}\sb{\pi N}$
characterizes the discrepancy between $f\sb{\pi NN}(t=0)$ 
and $f\sb{\pi NN}(t=m\sb{\pi}\sp{2})$~\cite{SYZ98}.
%
%Due to the lack of knowledges
%for the $\pi N\Delta$ and $\pi\Delta\Delta$ interactions,
%
It is difficult to determine the differences 
in their coupling constants between $t=0$ and $t=m\sb{\pi}\sp{2}$
in the present situation of experimental data. 
%
Then we eliminate 
$F\sp{N\Delta}\sb{A,3}$ and $F\sp{\Delta}\sb{A,2}$ 
naturally by using the PCAC hypothesis 
$f\sb{\pi N\Delta, \pi\Delta\Delta}(m\sb{\pi}\sp{2})\simeq 
f\sb{\pi N\Delta, \pi\Delta\Delta}(0)$.
%
In order to determine $F\sp{\Delta}\sb{A,1}$, we use the ratio 
$R\sb{N\Delta}= f\sb{\pi\Delta\Delta}(0)/f\sb{\pi NN}(0)$ and obtain
%
%
\begin{equation}
F\sp{\Delta}\sb{A,1}=\frac{R\sb{N\Delta}}{2}g\sb{A}.\label{eq:24}
\end{equation}
%
%
In the quark model, $R\sb{N\Delta}$ becomes 4/5~\cite{BW75}.
%
%
%
%
\begin{center}
\bf{C. Scalar-isoscalar part}
\end{center}

%
Since $\Delta(1232)$ dose not contribute to ${\cal T}\sb{SA}$, 
we only need the nucleon matrix element for the $\hat{\sigma}$ current,
%
%
\begin{equation}
\bra{N(p\sp{\prime})}\hat{\sigma}(0)\ket{N(p)}=
S(t)\overline{u}(p\sp{\prime})u(p).\label{eq:25}
\end{equation}
%
%
According to the definition in Ref.~\cite{YZ96}, 
the form factor $S(t)$ is equal to 
$-\sigma\sb{\pi N}(t)/ f\sb{\pi}m\sb{\pi}\sp{2}$, 
where $\sigma\sb{\pi N}(t)$ is the pion-nucleon sigma term 
which becomes independent of $t$ at tree level. 
%
%
%
%
\begin{center}
\bf{D. Pseudoscalar-isovector part}
\end{center}

%
The nucleon matrix element of the $\hat{\pi}$ current appearing
in ${\cal T}\sb{\pi}$ is written as
%
%
\begin{equation}
\bra{N(p\sp{\prime})}\hat{\pi}\sp{a}(0)\ket{N(p)}=
P(t)\overline{u}(p\sp{\prime})i\gc{5}\tau\sp{a}u(p).\label{eq:26}
\end{equation}
%
%
This current satisfies 
%
%
\begin{equation}
(\Box +m\sb{\pi}\sp{2})\hat{\pi}\sp{a}(x)
=\frac{1}{f\sb{\pi}}\partial\sp{\mu}\axialjc{a}{\mu}(x),\label{eq:27}
\end{equation}
where the external fields are set to zero~\cite{YZ96}.
%
%
From Eqs.~(\ref{eq:18}),~(\ref{eq:27})~and~(\ref{eq:29}), 
we obtain the general relation between the form factor $P(t)$ and 
the form factors $F\sp{N}\sb{A,1}$, $F\sp{N}\sb{A,2}$
or the renormalized $\pi NN$ coupling constant,
%
%
\begin{eqnarray}
P(t)&=&
\frac{1}{m\sb{\pi}\sp{2}-t}\frac{1}{f\sb{\pi}}
(m\sb{N}F\sp{N}\sb{A,1}(t)+\frac{t}{2}F\sp{N}\sb{A,2}(t))\nonumber\\
&=&\frac{1}{m\sb{\pi}\sp{2}-t}
\left(\frac{2m\sb{N}}{m\sb{\pi}}\right)f\sb{\pi NN}(t),
\label{eq:28}
\end{eqnarray}
%
%
where the pion pole contribution is taken into account. 
%
%
%
%
\section{RESULTS AND DISCUSSIONS}
In this section we show the numerical results for the 
$\pi N\rightarrow\pi\pi N$ total cross section
below $T\sb{\pi}=400$ MeV.
%
We discuss the influence of 
the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions
on this reaction.
%
The free parameters are $f\sb{\rho N\Delta}$ and $R\sb{N\Delta}$.
%
In Table \ref{tab3}, we summarize the value of constants in
this paper.
%
%
\begin{table}[ht]
\caption{
The value of constants in this paper.  
The mass and width of each particle or resonance are shown in MeV.
Note that $F\sp{N\Delta}\sb{A,2}$ has 
the opposite sign with $G$ in Ref.~\cite{KZ96} 
due to the difference of definition for the 
$N$-$\Delta$ transition form factors of $\axialjc{}{}$.
}
\label{tab3}
\begin{ruledtabular}
\begin{tabular}{crcl}
Masses and widths       &(MeV)  &
Parameters              &       \\ \hline
$m\sb{N}$               &939    &
$g\sb{A}$               &\ \ \ 1.265\\
$m\sb{\pi}$             &138    &
$f\sb{\rho}$            &\ \ \ 5.80\footnotemark[1]\\
$m\sb{\Delta}$          &1232   &
$\kappa\sb{V}$          &\ \ \ 3.71\\
$m\sb{\rho}$            &770    &
$F\sp{N\Delta}\sb{A,1}$ &\ \ \ 1.382\footnotemark[2]\\
$\Gamma\sb{\Delta}$     &120    &
$\overline{\Delta}\sb{\pi N}$  &$-$54\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ MeV\footnotemark[3]\ \ \ \ \\
$\Gamma\sb{\rho}$       &149    &
$\sigma\sb{\pi N}$      &\ \ 45\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ MeV\footnotemark[4]\\
                      &       &
$f\sb{\pi}$             &\ \ 93\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ MeV\\
                        &       &
$F\sp{N\Delta}\sb{A,2}$ &\ $-4.235\times 10\sp{-4}$ MeV$\sp{-1}$\footnotemark[2]
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{See p.33 in Ref.~\cite{HY03}}
\footnotetext[2]{Ref.~\cite{KZ96}}
\footnotetext[3]{Ref.~\cite{SYZ98}}
\footnotetext[4]{Ref.~\cite{GLS91}}
\end{table}
%
%
\begin{figure}[ht]
\includegraphics[width=7cm]{fig3a.ps} 
\includegraphics[width=7cm]{fig3b.ps}\\ 
\includegraphics[width=7cm]{fig3c.ps}
\caption{
The dependence of the cross section on the $\pi\Delta\Delta$ interaction;
(a)~$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$,
(b)~$\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$, and
(c)~$\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$. 
%
It is taken $R\sb{N\Delta}=0$ for the solid line,  
$R\sb{N\Delta}=0.4$ for the dashed line, and 
$R\sb{N\Delta}=0.8$ for the dashed-dotted line. 
In addition, the dotted line is the result with the $\pi$ and $N$ only. 
The data from Refs.~\cite{Sev91,Ker90,KST62} for 
$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$, 
\cite{Ker89-1,Bjo80,JAS74,Bla70,SMC70,Bat65,Blo63,Dea61,Per60,Blo62}
for $\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$, 
and \cite{Low91,Bel80,Bel78,Bun77,Kra75}
for $\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$. 
}
\label{fig3}
\end{figure}
%
%
\begin{figure}[ht]
\includegraphics[width=7cm]{fig4a.ps} 
\includegraphics[width=7cm]{fig4b.ps}
\caption{
The dependence of the cross section on the $\pi\Delta\Delta$ interaction;
(a)~$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ and
(b)~$\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$. 
%
Each line is the same as that of Fig.~\ref{fig3}.
%
The data from Refs. \cite{Poc94,Bat75,Arm72}
for $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ and
\cite{JAS74,SMC70,Blo63,Blo62,Bar61,Ker89-2}
for $\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$.
}
\label{fig4}
\end{figure}
%
%

%
First we consider the $\pi\Delta\Delta$ interaction.
%
This interaction probes a double-$\Delta$ part of the contribution
in ${\cal T}\sb{AAA}$ when only $\Delta(1232)$ appears
as the intermediate baryon.
%
Figures~\ref{fig3} and \ref{fig4} show the dependence of each cross section
on the values of $R\sb{N\Delta}$. 
%
The $\rho N\Delta$ interaction is not included there.
%
Two values for $R\sb{N\Delta}$ are chosen for the calculation, 
i.e. 0.8 and 0.4.
%
The former is taken from the quark model result~\cite{BW75} and
the latter is selected based on the argument in Ref.~\cite{Zhu00}. 

%
In the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$ and
$\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ channels,
the cross sections are increased by about 10 - 30 \% 
around $T\sb{\pi}=300$ MeV
due to the double-$\Delta$ propagation in ${\cal T}\sb{AAA}$
compared to the results with $R\sb{N\Delta}=0$. 
%
In $\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channel,
the effect of the double-$\Delta$ propagation is less than a few percent.
%
In the same figure, we also display the experimental data taken from
Refs.~\cite{Sev91,Ker90,KST62,Ker89-1,Bjo80,JAS74,Bla70,SMC70,
Bat65,Blo63,Dea61,Per60,Blo62,Low91,Bel80,Bel78,Bun77,Kra75,
Poc94,Bat75,Arm72,Bar61,Ker89-2}.
%
As for the $\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ and 
$\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channel, 
the disagreement between the theory and data is far beyond 
the variation in the cross section
due to the $\pi\Delta\Delta$ interaction. 
%
This result is consistent with that of Ref.~\cite{OV85}
for the $\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ channel.

%
In contrast with the above results,
the $\pi\sp{\pm}p\rightarrow\pi\sp{\pm}\pi\sp{0}p$ channels are
sensitive to the $\pi\Delta\Delta$ interaction (see Fig.~\ref{fig4}).
%
The total cross section for the
$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ channel below 300 MeV
increases by about 50 \% with $R\sb{N\Delta}=0.4$, and the cross section
with $R\sb{N\Delta}=0.8$ becomes twice the result with $R\sb{N\Delta}=0$. 
%
As for the $\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$ channel,
the total cross section indicates the increase by about 25 \% 
with $R\sb{N\Delta}=0.4$ and 65 \% with $R\sb{N\Delta}=0.8$ around 
$T\sb{\pi}=300$ MeV.
%
These two channels exhibit large influence of 
the $\pi\Delta\Delta$ interaction on the total cross section. 
%
However, if the fit with data is considered, 
it is still difficult for 
the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ channel
to improve the result by including the $\pi\Delta\Delta$ interaction alone.
%
%
\begin{figure}[ht]
\includegraphics[width=7cm]{fig5a.ps} 
\includegraphics[width=7cm]{fig5b.ps}\\ 
\includegraphics[width=7cm]{fig5c.ps}
\caption{
The dependence of the total cross section on the $\rho N\Delta$ interaction;
(a)~$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$,
(b)~$\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ and
(c)~$\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$. 
%
It is taken $f\sb{\rho N\Delta}=0$ for the solid line,
$f\sb{\rho N\Delta}=3.5$ for the dashed line and
$f\sb{\rho N\Delta}=7.8$ for the dashed-dotted line.
%
The data and the dotted line are same as in Fig.~\ref{fig3}. 
}
\label{fig5}
\end{figure}
%
%
\begin{figure}[ht]
\includegraphics[width=7cm]{fig6a.ps} 
\includegraphics[width=7cm]{fig6b.ps}
\caption{
The dependence of the total cross section on the $\rho N\Delta$ interaction;
(a)~$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ and
(b)~$\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$. 
%
Each line is same as that of Fig.~\ref{fig5}.
%
The data is same as in Fig.~\ref{fig4}. 
}
\label{fig6}
\end{figure}
%
%

%
Next we show the dependence  
of each cross section on the value of $f\sb{\rho N\Delta}$
in Figs.~\ref{fig5}~and~\ref{fig6}.  
%
We do not include the $\pi\Delta\Delta$ interaction instead.
%
We take two values for $f\sb{\rho N\Delta}$, 3.5 and 7.8, 
referring to the self-consistent calculation of 
the $\rho N\Delta$ vertex function~\cite{HL94} and 
the $NN$ phase shift analysis using the meson exchange
diagram~\cite{MHE87}, respectively.

%
As for the three channels in Fig.~\ref{fig5}, 
the effect of the $\rho N\Delta$ interaction is negligible. 
%
This situation is similar to the case of Fig.~\ref{fig3}, in 
which the dependence on the $\pi\Delta\Delta$ interaction is examined.
%
From the results given in Figs.~\ref{fig3}~and~\ref{fig5}, 
we can say that the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$
and  $\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channels
are insensitive to $\Delta(1232)$ and $\rho$. 
%
Since the experimental data for the
$\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{+}n$ channel
are well reproduced by the theoretical calculation, 
this channel is saturated by the reaction 
mechanisms and contains only the pion and the nucleon. 
%
This result is in agreement with other calculations~\cite{JM97,BKM97}.

%
On the other hand, the effect of the $\rho N\Delta$ interaction 
is seen in the two channels in Fig.~\ref{fig6}.
%
In the $\pi\sp{+}p\rightarrow\pi\sp{+}\pi\sp{0}p$ channel,  
the cross section with $f\sb{\rho N\Delta}=3.5$ and $7.8$ 
around $T\sb{\pi}=200$ MeV decreases about 30 \% 
in comparison with the cross section with $f\sb{\rho N\Delta}=0$.
%
In the $\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$ channel,  
the cross section shows large increase over all values of $T\sb{\pi}$.
%
Many reports on the value of $f\sb{\rho N\Delta}$ are settled in the range
of $3.5\alt f\sb{\rho N\Delta} \alt 7.8$, and the 
$\pi\sp{-}p\rightarrow\pi\sp{-}\pi\sp{0}p$ channel is 
sensitive to the variation of $f\sb{\rho N\Delta}$ in this range.

%
Here we comment on the Roper resonance.
%
The experimental data for 
the $\pi\sp{-}p\rightarrow\pi\sp{+}\pi\sp{-}n$ 
and $\pi\sp{-}p\rightarrow\pi\sp{0}\pi\sp{0}n$ channels 
can not be reproduced in this paper.
%
It is known that this failure 
can be cured by the Roper resonance $N\sp{\ast}(1440)$ with 
$N\sp{\ast}(1440)\rightarrow N(\pi\pi)\sb{I=0}$ decay~\cite{JM97,OV85}. 
%
However, this resonance was not included in this paper
because we concentrate our attention on the $\Delta(1232)$ resonance.
%
We note that $N\sp{\ast}(1440)\rightarrow N(\pi\pi)\sb{I=0}$ contribution 
appears, at tree level, only in ${\cal T}\sb{SA}$ 
which results from explicit breaking of chiral symmetry.
%
Although several improvements are suggested for the treatment of 
the Roper resonance~\cite{BK02}, the role of 
this famous resonance is still in question.
%
%
%
%
\section{SUMMARY}
We have calculated the total cross section 
of the $\pi N\rightarrow\pi\pi N$ reaction up to $T\sb{\pi}=400$ MeV
and discussed the contribution of $\Delta(1232)$ to this 
reaction.
%
Applying the chiral reduction formula to the invariant amplitude,
we have considered the $\pi\Delta\Delta$ and $\rho N\Delta$ interactions
which have not been taken seriously so far.
%
Using the numerical values of the coupling constants
given in the past studies, we have shown that the 
$\pi\sp{\pm}p\rightarrow\pi\sp{\pm}\pi\sp{0}p$ channels
are sensitive to these two interactions.
%
If we hope to determine concrete values of these coupling constants,
we need to extend our treatment by, for example, the systematic inclusion
of the Roper resonance.

%
The chiral reduction formula is found to be effective to use
to analyze the $\pi N\rightarrow\pi\pi N$ reaction.
%
Owing to this formula we can consider the detail of each reaction  
mechanism separately from the general framework of 
pion induced reactions constrained by chiral symmetry. 
%
This approach has an advantage to gain deeper insight for 
the nonperturbative features of the hadron physics.
%
%
%
%
\begin{acknowledgments}
The authors acknowledge the nuclear theory group of Osaka-City University
for useful discussions.
\end{acknowledgments}
%
%
%
%
\appendix*\section{}

In this Appendix, we show the phenomenological Lagrangians 
and the $\Delta$ propagator.
%
Some useful relations for the isospin matrices are listed, too. 

%
The Lagrangians are written in the following forms,
%
%
\begin{eqnarray}
{\cal L}\sb{\pi NN}&=&
\frac{f\sb{\pi NN}}{m\sb{\pi}}
\bar{N}\gc{\mu}\gc{5}
\tau\sp{a}N\partial\sp{\mu}\pi\sp{a},\label{eq:29}\\
%
{\cal L}\sb{\pi N\Delta}&=&
\frac{f\sb{\pi N\Delta}}{m\sb{\pi}}\bar{\Delta}\sp{\nu}
\Theta\sb{\nu\mu}(Z)\isomtx{a}{3}{1}
N\partial\sp{\mu}\pi\sp{a}+h.c. \ ,\label{eq:30}\\
%
{\cal L}\sb{\pi\Delta\Delta}&=&\frac{f\sb{\pi\Delta\Delta}}{m\sb{\pi}}
\bar{\Delta}\sp{\alpha}\Theta\sb{\alpha\beta}(Z\sp{\prime})\gc{\mu}\gc{5}
\isomtx{a}{3}{3}{\Theta\sp{\beta}}\sb{\delta}
(Z\sp{\prime})\Delta\sp{\delta}\partial\sp{\mu}\pi\sp{a},\label{eq:31}\\
%
{\cal L}\sb{\rho NN}&=&
g\sb{\rho NN}
\bar{N}\left[\gc{\mu}\rho\sp{\mu a}-\frac{\kappa\sb{V}}{2m\sb{N}}
\sigma\sb{\mu\nu}\partial\sp{\nu}\rho\sp{\mu a}
\right]\frac{\tau\sp{a}}{2}N,\label{eq:32}\\
%
{\cal L}\sb{\rho N\Delta}&=&
i\frac{f\sb{\rho N\Delta}}{m\sb{\rho}}\bar{\Delta}\sp{\sigma}
\Theta\sb{\sigma\mu}(Z\sp{\prime\prime})\gc{\nu}\gc{5}
\isomtx{a}{3}{1}N
(\partial\sp{\nu}\rho\sp{\mu a}-\partial\sp{\mu}\rho\sp{\nu a})+h.c. \ ,
\label{eq:33}
\end{eqnarray}
%
%
where $(\Delta\sb{\mu})\sp{\rm T}=
(\Delta\sp{++}\sb{\mu},\Delta\sp{+}\sb{\mu},
\Delta\sp{0}\sb{\mu},\Delta\sp{-}\sb{\mu})$
and $N\sp{\rm T}=(p,n)$.
%
The second-rank Lorentz tensor $\Theta\sb{\mu\nu}$ is defined by 
$\Theta\sb{\mu\nu}(Y)\equiv g\sb{\mu\nu}-
\frac{1}{2}(1+2Y)\gc{\mu}\gc{\nu}$ (where we take $A=-1$ \cite{NEK71}). 
%
The second term of this tensor vanishes if $\Delta$ is on the mass shell
(because of $\gc{\mu}U\sp{\mu}(p)=0$), 
so that $Y$ is called the off-shell parameter. 
%
In this paper, we assume 
$Z=Z\sp{\prime}=Z\sp{\prime\prime}=-\frac{1}{2}$ for simplicity, that is
$\Theta\sb{\mu\nu}\rightarrow g\sb{\mu\nu}$.

%
The $\Delta(1232)$ propagator is 
%
%
\begin{equation}
S\sb{\mu\nu}(p)=
\frac{(\not{p}+m\sb{\Delta})}{3(p\sp{2}-m\sp{2}\sb{\Delta})}
\left[-2g\sb{\mu\nu}+\frac{2p\sb{\mu}p\sb{\nu}}{m\sb{\Delta}\sp{2}}
-i\sigma\sb{\mu\nu}
+\frac{\gc{\mu}p\sb{\nu}-\gc{\nu}p\sb{\mu}}{m\sb{\Delta}}
\right].\label{eq:34}
\end{equation}
%
%
In order to include the $\Delta(1232)$ width phenomenologically, 
we modify the denominator of the $\Delta$ propagator as
$p\sp{2}-m\sb{\Delta}\sp{2}\rightarrow 
p\sp{2}-m\sb{\Delta}\sp{2}+im\sb{\Delta}\Gamma\sb{\Delta}(p\sp{2})$.
%
The width $\Gamma\sb{\Delta}(s)$ is~\cite{JM97}
%
%
\begin{equation}
\Gamma\sb{\Delta}(s)=\Gamma\sb{\Delta}
\frac{m\sb{\Delta}}{\sqrt{s}}
\frac{|{\bf q}(\sqrt{s})|\sp{3}}{|{\bf q}(m\sb{\Delta})|\sp{3}}
\theta(s-(m\sb{N}+m\sb{\pi})\sp{2}),\label{eq:35}
\end{equation}
%
%
where ${\bf q}={\bf q}(\sqrt{s})$ is the pion spatial momentum in 
the center of mass $\pi N$ system with the total energy $\sqrt{s}$.

%
Finally we list the following relations for the isospin matrices 
(see appendix A in Ref. \cite{OV85}),
%
%
\begin{eqnarray}
\isomtx{a\dag}{3}{1}\isomtx{b}{3}{1}&=&
\delta\sp{ab}-\frac{1}{3}\tau\sp{a}\tau\sp{b},
\label{eq:36}\\
\isomtx{a\dag}{3}{1}\isomtx{b}{3}{3}\isomtx{c}{3}{1}&=&
\frac{5}{6}i\varepsilon\sp{abc}
-\frac{1}{6}\delta\sp{ab}\tau\sp{c}
+\frac{2}{3}\delta\sp{ac}\tau\sp{b}
-\frac{1}{6}\delta\sp{bc}\tau\sp{a}.\label{eq:37}
\end{eqnarray}
%
%
%
%
%\bibliography{reference}% Produces the bibliography via BibTeX.
%
%
\begin{thebibliography}{51}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Jensen and Miranda}(1997)}]{JM97}
\bibinfo{author}{\bibfnamefont{T.~S.} \bibnamefont{Jensen}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~F.} \bibnamefont{Miranda}},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{55}},
  \bibinfo{pages}{1039} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Oset and Vicente-Vacas}(1985)}]{OV85}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Oset}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.~J.} \bibnamefont{Vicente-Vacas}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A446}},
  \bibinfo{pages}{584} (\bibinfo{year}{1985}).

\bibitem[{\citenamefont{J{\"a}kel et~al.}(1992)\citenamefont{J{\"a}kel, Dillig,
  and Vasconcellos}}]{JDV92}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{J{\"a}kel}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Dillig}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{C.~A.~Z.} \bibnamefont{Vasconcellos}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A541}},
  \bibinfo{pages}{675} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Bernard et~al.}(1995)\citenamefont{Bernard, Kaiser, and
  Mei{\ss}ner}}]{BKM95}
\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Kaiser}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{B457}},
  \bibinfo{pages}{147} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Bernard et~al.}(1997)\citenamefont{Bernard, Kaiser, and
  Mei{\ss}ner}}]{BKM97}
\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Kaiser}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A619}},
  \bibinfo{pages}{261} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Fettes et~al.}(2000)\citenamefont{Fettes, Bernard, and
  Mei{\ss}ner}}]{FBM00}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Fettes}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A669}},
  \bibinfo{pages}{269} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Burkhardt and Lowe}(1991)}]{BL91}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Burkhardt}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lowe}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{67}},
  \bibinfo{pages}{2622} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Vereshagin et~al.}(1995)\citenamefont{Vereshagin,
  Sherman, Manashov, Bohnert, Dillig, Eyrich, J{\"a}kel, and
  Moosburger}}]{Ver95}
\bibinfo{author}{\bibfnamefont{V.~V.} \bibnamefont{Vereshagin}},
  \bibinfo{author}{\bibfnamefont{S.~G.} \bibnamefont{Sherman}},
  \bibinfo{author}{\bibfnamefont{A.~N.} \bibnamefont{Manashov}},
  \bibinfo{author}{\bibfnamefont{U.}~\bibnamefont{Bohnert}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Dillig}},
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Eyrich}},
  \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{J{\"a}kel}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Moosburger}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A592}},
  \bibinfo{pages}{413} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Bolokhov et~al.}(1998)\citenamefont{Bolokhov, Polyakov,
  and Sherman}}]{BPS98}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bolokhov}},
  \bibinfo{author}{\bibfnamefont{M.~V.} \bibnamefont{Polyakov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.~G.}
  \bibnamefont{Sherman}}, \bibinfo{journal}{Eur.\ Phys.\ J.}
  \textbf{\bibinfo{volume}{A1}}, \bibinfo{pages}{317} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Adler and Dashen}(1968)}]{Current}
\bibinfo{author}{\bibfnamefont{S.~L.} \bibnamefont{Adler}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~F.} \bibnamefont{Dashen}},
  \emph{\bibinfo{title}{Current Algebras and Applications to Particle Physics}}
  (\bibinfo{publisher}{Benjamin}, \bibinfo{address}{New York},
  \bibinfo{year}{1968}).

\bibitem[{\citenamefont{Mei{\ss}ner}(1993)}]{Mei93}
\bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Rep.\ Prog.\ Phys.} \textbf{\bibinfo{volume}{56}},
  \bibinfo{pages}{903} (\bibinfo{year}{1993}).

\bibitem[{\citenamefont{Ecker}(1995)}]{Eck95}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Ecker}},
  \bibinfo{journal}{Prog.\ Part.\ Phys.} \textbf{\bibinfo{volume}{56}},
  \bibinfo{pages}{903} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Yamagishi and Zahed}(1996)}]{YZ96}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Ann.\ Phys.\ (N.Y.)} \textbf{\bibinfo{volume}{247}},
  \bibinfo{pages}{292} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Steele et~al.}(1997)\citenamefont{Steele, Yamagishi,
  and Zahed}}]{SYZ97}
\bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Steele}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A615}},
  \bibinfo{pages}{305} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Lee et~al.}(1999)\citenamefont{Lee, Yamagishi, and
  Zahed}}]{LYZ99}
\bibinfo{author}{\bibfnamefont{C.-H.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A653}},
  \bibinfo{pages}{185} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Steele et~al.}(1998)\citenamefont{Steele, Yamagishi,
  and Zahed}}]{SYZ98}
\bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Steele}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{57}},
  \bibinfo{pages}{1703} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Sakurai}(1969)}]{JJ}
\bibinfo{author}{\bibfnamefont{J.~J.} \bibnamefont{Sakurai}},
  \emph{\bibinfo{title}{Currents and Mesons}} (\bibinfo{publisher}{Univ.
  Chicago Press}, \bibinfo{address}{Chicago}, \bibinfo{year}{1969}).

\bibitem[{\citenamefont{Bando et~al.}(1985)\citenamefont{Bando, Kugo, Uehara,
  Yamawaki, and Yanagida}}]{Ban85}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Bando}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Kugo}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Uehara}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Yamawaki}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Yanagida}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{1215} (\bibinfo{year}{1985}).

\bibitem[{\citenamefont{Harada and Yamawaki}(2003)}]{HY03}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Harada}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Yamawaki}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{381}},
  \bibinfo{pages}{1} (\bibinfo{year}{2003}).

\bibitem[{\citenamefont{Kacir and Zahed}(1996)}]{KZ96}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kacir}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{5536} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Arndt et~al.}(1979)\citenamefont{Arndt, Cammarata,
  Goradia, Hackman, Teplitz, Dicus, Aaron, and Longacre}}]{Arn79}
\bibinfo{author}{\bibfnamefont{R.~A.} \bibnamefont{Arndt}},
  \bibinfo{author}{\bibfnamefont{J.~B.} \bibnamefont{Cammarata}},
  \bibinfo{author}{\bibfnamefont{Y.~N.} \bibnamefont{Goradia}},
  \bibinfo{author}{\bibfnamefont{R.~H.} \bibnamefont{Hackman}},
  \bibinfo{author}{\bibfnamefont{V.~L.} \bibnamefont{Teplitz}},
  \bibinfo{author}{\bibfnamefont{D.~A.} \bibnamefont{Dicus}},
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Aaron}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~S.} \bibnamefont{Longacre}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{20}},
  \bibinfo{pages}{651} (\bibinfo{year}{1979}).

\bibitem[{\citenamefont{Brown and Weise}(1975)}]{BW75}
\bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Weise}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{22}},
  \bibinfo{pages}{279} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Gasser et~al.}(1991)\citenamefont{Gasser, Leutwyler,
  and Sainio}}]{GLS91}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Gasser}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Leutwyler}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{M.~E.}
  \bibnamefont{Sainio}}, \bibinfo{journal}{Phys.\ Lett.}
  \textbf{\bibinfo{volume}{B253}}, \bibinfo{pages}{252} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Sevior et~al.}(1991)}]{Sev91}
\bibinfo{author}{\bibfnamefont{M.~E.} \bibnamefont{Sevior}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Rev.\ Lett.}
  \textbf{\bibinfo{volume}{66}}, \bibinfo{pages}{2569} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Kernel et~al.}(1990)}]{Ker90}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Z.\ Phys.\ C} \textbf{\bibinfo{volume}{48}},
  \bibinfo{pages}{201} (\bibinfo{year}{1990}).

\bibitem[{\citenamefont{Kirz et~al.}(1962)\citenamefont{Kirz, Schwartz, and
  Tripp}}]{KST62}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kirz}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Schwartz}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~D.} \bibnamefont{Tripp}},
  \bibinfo{journal}{Phys.\ Rev} \textbf{\bibinfo{volume}{126}},
  \bibinfo{pages}{763} (\bibinfo{year}{1962}).

\bibitem[{\citenamefont{Kernel et~al.}(1989{\natexlab{a}})}]{Ker89-1}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B216}},
  \bibinfo{pages}{244} (\bibinfo{year}{1989}{\natexlab{a}}).

\bibitem[{\citenamefont{Bjork et~al.}(1980)}]{Bjo80}
\bibinfo{author}{\bibfnamefont{C.~W.} \bibnamefont{Bjork}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Pev.\ Lett.}
  \textbf{\bibinfo{volume}{44}}, \bibinfo{pages}{62} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Jones et~al.}(1974)\citenamefont{Jones, Allison, and
  Saxon}}]{JAS74}
\bibinfo{author}{\bibfnamefont{J.~A.} \bibnamefont{Jones}},
  \bibinfo{author}{\bibfnamefont{W.~W.~M.} \bibnamefont{Allison}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.~H.} \bibnamefont{Saxon}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{B83}},
  \bibinfo{pages}{93} (\bibinfo{year}{1974}).

\bibitem[{\citenamefont{Blair et~al.}(1970)\citenamefont{Blair, M{\"u}ller,
  Torelli, Zavattini, and Mandrioli}}]{Bla70}
\bibinfo{author}{\bibfnamefont{I.~M.} \bibnamefont{Blair}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{M{\"u}ller}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Torelli}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Zavattini}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Mandrioli}},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B32}},
  \bibinfo{pages}{528} (\bibinfo{year}{1970}).

\bibitem[{\citenamefont{d.~H.~Saxon et~al.}(1970)\citenamefont{d.~H.~Saxon,
  Mulvey, and Chinowsky}}]{SMC70}
\bibinfo{author}{\bibnamefont{d.~H.~Saxon}},
  \bibinfo{author}{\bibfnamefont{J.~H.} \bibnamefont{Mulvey}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Chinowsky}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{2}},
  \bibinfo{pages}{1790} (\bibinfo{year}{1970}).

\bibitem[{\citenamefont{Batusov et~al.}(1965)\citenamefont{Batusov, Bunyatov,
  Sidorov, and Yarba}}]{Bat65}
\bibinfo{author}{\bibfnamefont{Y.~A.} \bibnamefont{Batusov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{V.~M.} \bibnamefont{Sidorov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{1}},
  \bibinfo{pages}{374} (\bibinfo{year}{1965}).

\bibitem[{\citenamefont{Blokhintseva et~al.}(1963)\citenamefont{Blokhintseva,
  Grebinnik, Zhukov, Libman, Nemenov, Selivanov, and Jung-Fang}}]{Blo63}
\bibinfo{author}{\bibfnamefont{T.~D.} \bibnamefont{Blokhintseva}},
  \bibinfo{author}{\bibfnamefont{V.~G.} \bibnamefont{Grebinnik}},
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zhukov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Libman}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{G.~I.} \bibnamefont{Selivanov}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Jung-Fang}},
  \bibinfo{journal}{Sov.\ Phys.\ JETP} \textbf{\bibinfo{volume}{17}},
  \bibinfo{pages}{340} (\bibinfo{year}{1963}).

\bibitem[{\citenamefont{Deahl et~al.}(1961)\citenamefont{Deahl, Derrick,
  Fetkovich, Fields, and Yodh}}]{Dea61}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Deahl}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Derrick}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Fetkovich}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Fields}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.~B.} \bibnamefont{Yodh}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{124}},
  \bibinfo{pages}{1987} (\bibinfo{year}{1961}).

\bibitem[{\citenamefont{Perkins et~al.}(1960)\citenamefont{Perkins, Caris,
  Kenny, and Perez-Mendez}}]{Per60}
\bibinfo{author}{\bibfnamefont{W.~A.} \bibnamefont{Perkins}},
  \bibinfo{author}{\bibfnamefont{J.~C.} \bibnamefont{Caris}},
  \bibinfo{author}{\bibfnamefont{R.~W.} \bibnamefont{Kenny}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Perez-Mendez}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{118}},
  \bibinfo{pages}{1364} (\bibinfo{year}{1960}).

\bibitem[{\citenamefont{Blokhintseva et~al.}(1962)\citenamefont{Blokhintseva,
  Grebinnik, Zhukov, Libman, Nemenov, Selivanov, and Jung-Fang}}]{Blo62}
\bibinfo{author}{\bibfnamefont{T.~D.} \bibnamefont{Blokhintseva}},
  \bibinfo{author}{\bibfnamefont{V.~G.} \bibnamefont{Grebinnik}},
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zhukov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Libman}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{G.~I.} \bibnamefont{Selivanov}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Jung-Fang}},
  \bibinfo{journal}{Sov.\ Phys.\ JETP} \textbf{\bibinfo{volume}{15}},
  \bibinfo{pages}{629} (\bibinfo{year}{1962}).

\bibitem[{\citenamefont{Lowe et~al.}(1991)}]{Low91}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lowe}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{44}},
  \bibinfo{pages}{956} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Bel'kov et~al.}(1980)\citenamefont{Bel'kov, Bunyatov,
  Zalikhanov, Kurbatov, Khalbaev, and Yarba}}]{Bel80}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bel'kov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{31}},
  \bibinfo{pages}{96} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Bel'kov et~al.}(1978)\citenamefont{Bel'kov, Bunyatov,
  Zalikhanov, Kurbatov, and Khalbaev}}]{Bel78}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bel'kov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{25}},
  \bibinfo{pages}{657} (\bibinfo{year}{1978}).

\bibitem[{\citenamefont{Bunyatov et~al.}(1977)\citenamefont{Bunyatov, Zholobov,
  Zalikhanov, Kurbatov, Musakhanov, Khalbaev, and Yarba}}]{Bun77}
\bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{G.~V.} \bibnamefont{Zholobov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibinfo{author}{\bibfnamefont{M.~M.} \bibnamefont{Musakhanov}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{25}},
  \bibinfo{pages}{177} (\bibinfo{year}{1977}).

\bibitem[{\citenamefont{Kravtsov et~al.}(1975)\citenamefont{Kravtsov, Kuptsov,
  Nemenov, Starchenko, and Khazins}}]{Kra75}
\bibinfo{author}{\bibfnamefont{A.~V.} \bibnamefont{Kravtsov}},
  \bibinfo{author}{\bibfnamefont{A.~V.} \bibnamefont{Kuptsov}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{E.~A.} \bibnamefont{Starchenko}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.~M.}
  \bibnamefont{Khazins}}, \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.}
  \textbf{\bibinfo{volume}{20}}, \bibinfo{pages}{500} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Po\v{c}ani{\'c} et~al.}(1994)}]{Poc94}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Po\v{c}ani{\'c}}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Rev.\ Lett.}
  \textbf{\bibinfo{volume}{72}}, \bibinfo{pages}{1156} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Batusov et~al.}(1975)\citenamefont{Batusov, Bunyatov,
  Gulkanyan, Sidorov, Musakhanov, Ionice, Losnianu, Mihul, and
  Tuvdendorzh}}]{Bat75}
\bibinfo{author}{\bibfnamefont{Y.~A.} \bibnamefont{Batusov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{G.~R.} \bibnamefont{Gulkanyan}},
  \bibinfo{author}{\bibfnamefont{V.~M.} \bibnamefont{Sidorov}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Musakhanov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Ionice}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Losnianu}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Mihul}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Tuvdendorzh}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{21}},
  \bibinfo{pages}{162} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Arman et~al.}(1972)\citenamefont{Arman, Blasberg,
  Haddock, Leung, Nefkens, Schrock, Sober, and Sperinde}}]{Arm72}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Arman}},
  \bibinfo{author}{\bibfnamefont{D.~J.} \bibnamefont{Blasberg}},
  \bibinfo{author}{\bibfnamefont{R.~P.} \bibnamefont{Haddock}},
  \bibinfo{author}{\bibfnamefont{K.~C.} \bibnamefont{Leung}},
  \bibinfo{author}{\bibfnamefont{B.~M.~K.} \bibnamefont{Nefkens}},
  \bibinfo{author}{\bibfnamefont{B.~L.} \bibnamefont{Schrock}},
  \bibinfo{author}{\bibfnamefont{D.~I.} \bibnamefont{Sober}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~M.} \bibnamefont{Sperinde}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{29}},
  \bibinfo{pages}{962} (\bibinfo{year}{1972}).

\bibitem[{\citenamefont{Barish et~al.}(1961)\citenamefont{Barish, Kurz,
  Perez-Mendez, and Solomon}}]{Bar61}
\bibinfo{author}{\bibfnamefont{B.~C.} \bibnamefont{Barish}},
  \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Kurz}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Perez-Mendez}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Solomon}},
  \bibinfo{journal}{Bull.\ Am.\ Phys.\ Soc.\ Ser.II}
  \textbf{\bibinfo{volume}{6}}, \bibinfo{pages}{523} (\bibinfo{year}{1961}).

\bibitem[{\citenamefont{Kernel et~al.}(1989{\natexlab{b}})}]{Ker89-2}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B225}},
  \bibinfo{pages}{198} (\bibinfo{year}{1989}{\natexlab{b}}).

\bibitem[{\citenamefont{Zhu}(2000)}]{Zhu00}
\bibinfo{author}{\bibfnamefont{S.-L.} \bibnamefont{Zhu}},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{63}},
  \bibinfo{pages}{018201} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Haider and Liu}(1994)}]{HL94}
\bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Haider}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.~C.} \bibnamefont{Liu}},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B335}},
  \bibinfo{pages}{253} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Machleidt et~al.}(1987)\citenamefont{Machleidt,
  Holinde, and Elster}}]{MHE87}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Machleidt}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Holinde}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Elster}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{149}},
  \bibinfo{pages}{1} (\bibinfo{year}{1987}).

\bibitem[{\citenamefont{Beane and van Kolck}(2002)}]{BK02}
\bibinfo{author}{\bibfnamefont{S.~R.} \bibnamefont{Beane}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.}~\bibnamefont{van Kolck}}
  (\bibinfo{year}{2002}), \eprint{nucl-th/0212039}.

\bibitem[{\citenamefont{Nath et~al.}(1971)\citenamefont{Nath, Etemadi, and
  Kimel}}]{NEK71}
\bibinfo{author}{\bibfnamefont{L.~M.} \bibnamefont{Nath}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Etemadi}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~D.} \bibnamefont{Kimel}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{3}},
  \bibinfo{pages}{2153} (\bibinfo{year}{1971}).

\end{thebibliography}
%
%
%
%
\end{document}

hep-ph0310231.extracted_bib
\begin{thebibliography}{51}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Jensen and Miranda}(1997)}]{JM97}
\bibinfo{author}{\bibfnamefont{T.~S.} \bibnamefont{Jensen}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~F.} \bibnamefont{Miranda}},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{55}},
  \bibinfo{pages}{1039} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Oset and Vicente-Vacas}(1985)}]{OV85}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Oset}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.~J.} \bibnamefont{Vicente-Vacas}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A446}},
  \bibinfo{pages}{584} (\bibinfo{year}{1985}).

\bibitem[{\citenamefont{J{\"a}kel et~al.}(1992)\citenamefont{J{\"a}kel, Dillig,
  and Vasconcellos}}]{JDV92}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{J{\"a}kel}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Dillig}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{C.~A.~Z.} \bibnamefont{Vasconcellos}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A541}},
  \bibinfo{pages}{675} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Bernard et~al.}(1995)\citenamefont{Bernard, Kaiser, and
  Mei{\ss}ner}}]{BKM95}
\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Kaiser}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{B457}},
  \bibinfo{pages}{147} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Bernard et~al.}(1997)\citenamefont{Bernard, Kaiser, and
  Mei{\ss}ner}}]{BKM97}
\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Kaiser}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A619}},
  \bibinfo{pages}{261} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Fettes et~al.}(2000)\citenamefont{Fettes, Bernard, and
  Mei{\ss}ner}}]{FBM00}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Fettes}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Bernard}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A669}},
  \bibinfo{pages}{269} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Burkhardt and Lowe}(1991)}]{BL91}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Burkhardt}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lowe}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{67}},
  \bibinfo{pages}{2622} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Vereshagin et~al.}(1995)\citenamefont{Vereshagin,
  Sherman, Manashov, Bohnert, Dillig, Eyrich, J{\"a}kel, and
  Moosburger}}]{Ver95}
\bibinfo{author}{\bibfnamefont{V.~V.} \bibnamefont{Vereshagin}},
  \bibinfo{author}{\bibfnamefont{S.~G.} \bibnamefont{Sherman}},
  \bibinfo{author}{\bibfnamefont{A.~N.} \bibnamefont{Manashov}},
  \bibinfo{author}{\bibfnamefont{U.}~\bibnamefont{Bohnert}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Dillig}},
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Eyrich}},
  \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{J{\"a}kel}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Moosburger}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A592}},
  \bibinfo{pages}{413} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Bolokhov et~al.}(1998)\citenamefont{Bolokhov, Polyakov,
  and Sherman}}]{BPS98}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bolokhov}},
  \bibinfo{author}{\bibfnamefont{M.~V.} \bibnamefont{Polyakov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.~G.}
  \bibnamefont{Sherman}}, \bibinfo{journal}{Eur.\ Phys.\ J.}
  \textbf{\bibinfo{volume}{A1}}, \bibinfo{pages}{317} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Adler and Dashen}(1968)}]{Current}
\bibinfo{author}{\bibfnamefont{S.~L.} \bibnamefont{Adler}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~F.} \bibnamefont{Dashen}},
  \emph{\bibinfo{title}{Current Algebras and Applications to Particle Physics}}
  (\bibinfo{publisher}{Benjamin}, \bibinfo{address}{New York},
  \bibinfo{year}{1968}).

\bibitem[{\citenamefont{Mei{\ss}ner}(1993)}]{Mei93}
\bibinfo{author}{\bibfnamefont{U.-G.} \bibnamefont{Mei{\ss}ner}},
  \bibinfo{journal}{Rep.\ Prog.\ Phys.} \textbf{\bibinfo{volume}{56}},
  \bibinfo{pages}{903} (\bibinfo{year}{1993}).

\bibitem[{\citenamefont{Ecker}(1995)}]{Eck95}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Ecker}},
  \bibinfo{journal}{Prog.\ Part.\ Phys.} \textbf{\bibinfo{volume}{56}},
  \bibinfo{pages}{903} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Yamagishi and Zahed}(1996)}]{YZ96}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Ann.\ Phys.\ (N.Y.)} \textbf{\bibinfo{volume}{247}},
  \bibinfo{pages}{292} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Steele et~al.}(1997)\citenamefont{Steele, Yamagishi,
  and Zahed}}]{SYZ97}
\bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Steele}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A615}},
  \bibinfo{pages}{305} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Lee et~al.}(1999)\citenamefont{Lee, Yamagishi, and
  Zahed}}]{LYZ99}
\bibinfo{author}{\bibfnamefont{C.-H.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{A653}},
  \bibinfo{pages}{185} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Steele et~al.}(1998)\citenamefont{Steele, Yamagishi,
  and Zahed}}]{SYZ98}
\bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Steele}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Yamagishi}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{57}},
  \bibinfo{pages}{1703} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Sakurai}(1969)}]{JJ}
\bibinfo{author}{\bibfnamefont{J.~J.} \bibnamefont{Sakurai}},
  \emph{\bibinfo{title}{Currents and Mesons}} (\bibinfo{publisher}{Univ.
  Chicago Press}, \bibinfo{address}{Chicago}, \bibinfo{year}{1969}).

\bibitem[{\citenamefont{Bando et~al.}(1985)\citenamefont{Bando, Kugo, Uehara,
  Yamawaki, and Yanagida}}]{Ban85}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Bando}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Kugo}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Uehara}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Yamawaki}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Yanagida}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{1215} (\bibinfo{year}{1985}).

\bibitem[{\citenamefont{Harada and Yamawaki}(2003)}]{HY03}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Harada}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Yamawaki}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{381}},
  \bibinfo{pages}{1} (\bibinfo{year}{2003}).

\bibitem[{\citenamefont{Kacir and Zahed}(1996)}]{KZ96}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kacir}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Zahed}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{5536} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Arndt et~al.}(1979)\citenamefont{Arndt, Cammarata,
  Goradia, Hackman, Teplitz, Dicus, Aaron, and Longacre}}]{Arn79}
\bibinfo{author}{\bibfnamefont{R.~A.} \bibnamefont{Arndt}},
  \bibinfo{author}{\bibfnamefont{J.~B.} \bibnamefont{Cammarata}},
  \bibinfo{author}{\bibfnamefont{Y.~N.} \bibnamefont{Goradia}},
  \bibinfo{author}{\bibfnamefont{R.~H.} \bibnamefont{Hackman}},
  \bibinfo{author}{\bibfnamefont{V.~L.} \bibnamefont{Teplitz}},
  \bibinfo{author}{\bibfnamefont{D.~A.} \bibnamefont{Dicus}},
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Aaron}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~S.} \bibnamefont{Longacre}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{20}},
  \bibinfo{pages}{651} (\bibinfo{year}{1979}).

\bibitem[{\citenamefont{Brown and Weise}(1975)}]{BW75}
\bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Weise}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{22}},
  \bibinfo{pages}{279} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Gasser et~al.}(1991)\citenamefont{Gasser, Leutwyler,
  and Sainio}}]{GLS91}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Gasser}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Leutwyler}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{M.~E.}
  \bibnamefont{Sainio}}, \bibinfo{journal}{Phys.\ Lett.}
  \textbf{\bibinfo{volume}{B253}}, \bibinfo{pages}{252} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Sevior et~al.}(1991)}]{Sev91}
\bibinfo{author}{\bibfnamefont{M.~E.} \bibnamefont{Sevior}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Rev.\ Lett.}
  \textbf{\bibinfo{volume}{66}}, \bibinfo{pages}{2569} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Kernel et~al.}(1990)}]{Ker90}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Z.\ Phys.\ C} \textbf{\bibinfo{volume}{48}},
  \bibinfo{pages}{201} (\bibinfo{year}{1990}).

\bibitem[{\citenamefont{Kirz et~al.}(1962)\citenamefont{Kirz, Schwartz, and
  Tripp}}]{KST62}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kirz}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Schwartz}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~D.} \bibnamefont{Tripp}},
  \bibinfo{journal}{Phys.\ Rev} \textbf{\bibinfo{volume}{126}},
  \bibinfo{pages}{763} (\bibinfo{year}{1962}).

\bibitem[{\citenamefont{Kernel et~al.}(1989{\natexlab{a}})}]{Ker89-1}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B216}},
  \bibinfo{pages}{244} (\bibinfo{year}{1989}{\natexlab{a}}).

\bibitem[{\citenamefont{Bjork et~al.}(1980)}]{Bjo80}
\bibinfo{author}{\bibfnamefont{C.~W.} \bibnamefont{Bjork}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Pev.\ Lett.}
  \textbf{\bibinfo{volume}{44}}, \bibinfo{pages}{62} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Jones et~al.}(1974)\citenamefont{Jones, Allison, and
  Saxon}}]{JAS74}
\bibinfo{author}{\bibfnamefont{J.~A.} \bibnamefont{Jones}},
  \bibinfo{author}{\bibfnamefont{W.~W.~M.} \bibnamefont{Allison}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.~H.} \bibnamefont{Saxon}},
  \bibinfo{journal}{Nucl.\ Phys.} \textbf{\bibinfo{volume}{B83}},
  \bibinfo{pages}{93} (\bibinfo{year}{1974}).

\bibitem[{\citenamefont{Blair et~al.}(1970)\citenamefont{Blair, M{\"u}ller,
  Torelli, Zavattini, and Mandrioli}}]{Bla70}
\bibinfo{author}{\bibfnamefont{I.~M.} \bibnamefont{Blair}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{M{\"u}ller}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Torelli}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Zavattini}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Mandrioli}},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B32}},
  \bibinfo{pages}{528} (\bibinfo{year}{1970}).

\bibitem[{\citenamefont{d.~H.~Saxon et~al.}(1970)\citenamefont{d.~H.~Saxon,
  Mulvey, and Chinowsky}}]{SMC70}
\bibinfo{author}{\bibnamefont{d.~H.~Saxon}},
  \bibinfo{author}{\bibfnamefont{J.~H.} \bibnamefont{Mulvey}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Chinowsky}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{2}},
  \bibinfo{pages}{1790} (\bibinfo{year}{1970}).

\bibitem[{\citenamefont{Batusov et~al.}(1965)\citenamefont{Batusov, Bunyatov,
  Sidorov, and Yarba}}]{Bat65}
\bibinfo{author}{\bibfnamefont{Y.~A.} \bibnamefont{Batusov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{V.~M.} \bibnamefont{Sidorov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{1}},
  \bibinfo{pages}{374} (\bibinfo{year}{1965}).

\bibitem[{\citenamefont{Blokhintseva et~al.}(1963)\citenamefont{Blokhintseva,
  Grebinnik, Zhukov, Libman, Nemenov, Selivanov, and Jung-Fang}}]{Blo63}
\bibinfo{author}{\bibfnamefont{T.~D.} \bibnamefont{Blokhintseva}},
  \bibinfo{author}{\bibfnamefont{V.~G.} \bibnamefont{Grebinnik}},
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zhukov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Libman}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{G.~I.} \bibnamefont{Selivanov}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Jung-Fang}},
  \bibinfo{journal}{Sov.\ Phys.\ JETP} \textbf{\bibinfo{volume}{17}},
  \bibinfo{pages}{340} (\bibinfo{year}{1963}).

\bibitem[{\citenamefont{Deahl et~al.}(1961)\citenamefont{Deahl, Derrick,
  Fetkovich, Fields, and Yodh}}]{Dea61}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Deahl}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Derrick}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Fetkovich}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Fields}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.~B.} \bibnamefont{Yodh}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{124}},
  \bibinfo{pages}{1987} (\bibinfo{year}{1961}).

\bibitem[{\citenamefont{Perkins et~al.}(1960)\citenamefont{Perkins, Caris,
  Kenny, and Perez-Mendez}}]{Per60}
\bibinfo{author}{\bibfnamefont{W.~A.} \bibnamefont{Perkins}},
  \bibinfo{author}{\bibfnamefont{J.~C.} \bibnamefont{Caris}},
  \bibinfo{author}{\bibfnamefont{R.~W.} \bibnamefont{Kenny}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Perez-Mendez}},
  \bibinfo{journal}{Phys.\ Rev.} \textbf{\bibinfo{volume}{118}},
  \bibinfo{pages}{1364} (\bibinfo{year}{1960}).

\bibitem[{\citenamefont{Blokhintseva et~al.}(1962)\citenamefont{Blokhintseva,
  Grebinnik, Zhukov, Libman, Nemenov, Selivanov, and Jung-Fang}}]{Blo62}
\bibinfo{author}{\bibfnamefont{T.~D.} \bibnamefont{Blokhintseva}},
  \bibinfo{author}{\bibfnamefont{V.~G.} \bibnamefont{Grebinnik}},
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zhukov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Libman}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{G.~I.} \bibnamefont{Selivanov}},
  \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Jung-Fang}},
  \bibinfo{journal}{Sov.\ Phys.\ JETP} \textbf{\bibinfo{volume}{15}},
  \bibinfo{pages}{629} (\bibinfo{year}{1962}).

\bibitem[{\citenamefont{Lowe et~al.}(1991)}]{Low91}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lowe}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{44}},
  \bibinfo{pages}{956} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Bel'kov et~al.}(1980)\citenamefont{Bel'kov, Bunyatov,
  Zalikhanov, Kurbatov, Khalbaev, and Yarba}}]{Bel80}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bel'kov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{31}},
  \bibinfo{pages}{96} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Bel'kov et~al.}(1978)\citenamefont{Bel'kov, Bunyatov,
  Zalikhanov, Kurbatov, and Khalbaev}}]{Bel78}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Bel'kov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{25}},
  \bibinfo{pages}{657} (\bibinfo{year}{1978}).

\bibitem[{\citenamefont{Bunyatov et~al.}(1977)\citenamefont{Bunyatov, Zholobov,
  Zalikhanov, Kurbatov, Musakhanov, Khalbaev, and Yarba}}]{Bun77}
\bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{G.~V.} \bibnamefont{Zholobov}},
  \bibinfo{author}{\bibfnamefont{B.~Z.} \bibnamefont{Zalikhanov}},
  \bibinfo{author}{\bibfnamefont{V.~S.} \bibnamefont{Kurbatov}},
  \bibinfo{author}{\bibfnamefont{M.~M.} \bibnamefont{Musakhanov}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khalbaev}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Yarba}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{25}},
  \bibinfo{pages}{177} (\bibinfo{year}{1977}).

\bibitem[{\citenamefont{Kravtsov et~al.}(1975)\citenamefont{Kravtsov, Kuptsov,
  Nemenov, Starchenko, and Khazins}}]{Kra75}
\bibinfo{author}{\bibfnamefont{A.~V.} \bibnamefont{Kravtsov}},
  \bibinfo{author}{\bibfnamefont{A.~V.} \bibnamefont{Kuptsov}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Nemenov}},
  \bibinfo{author}{\bibfnamefont{E.~A.} \bibnamefont{Starchenko}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.~M.}
  \bibnamefont{Khazins}}, \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.}
  \textbf{\bibinfo{volume}{20}}, \bibinfo{pages}{500} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Po\v{c}ani{\'c} et~al.}(1994)}]{Poc94}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Po\v{c}ani{\'c}}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys.\ Rev.\ Lett.}
  \textbf{\bibinfo{volume}{72}}, \bibinfo{pages}{1156} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Batusov et~al.}(1975)\citenamefont{Batusov, Bunyatov,
  Gulkanyan, Sidorov, Musakhanov, Ionice, Losnianu, Mihul, and
  Tuvdendorzh}}]{Bat75}
\bibinfo{author}{\bibfnamefont{Y.~A.} \bibnamefont{Batusov}},
  \bibinfo{author}{\bibfnamefont{S.~A.} \bibnamefont{Bunyatov}},
  \bibinfo{author}{\bibfnamefont{G.~R.} \bibnamefont{Gulkanyan}},
  \bibinfo{author}{\bibfnamefont{V.~M.} \bibnamefont{Sidorov}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Musakhanov}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Ionice}},
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Losnianu}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Mihul}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Tuvdendorzh}},
  \bibinfo{journal}{Sov.\ J.\ Nucl.\ Phys.} \textbf{\bibinfo{volume}{21}},
  \bibinfo{pages}{162} (\bibinfo{year}{1975}).

\bibitem[{\citenamefont{Arman et~al.}(1972)\citenamefont{Arman, Blasberg,
  Haddock, Leung, Nefkens, Schrock, Sober, and Sperinde}}]{Arm72}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Arman}},
  \bibinfo{author}{\bibfnamefont{D.~J.} \bibnamefont{Blasberg}},
  \bibinfo{author}{\bibfnamefont{R.~P.} \bibnamefont{Haddock}},
  \bibinfo{author}{\bibfnamefont{K.~C.} \bibnamefont{Leung}},
  \bibinfo{author}{\bibfnamefont{B.~M.~K.} \bibnamefont{Nefkens}},
  \bibinfo{author}{\bibfnamefont{B.~L.} \bibnamefont{Schrock}},
  \bibinfo{author}{\bibfnamefont{D.~I.} \bibnamefont{Sober}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~M.} \bibnamefont{Sperinde}},
  \bibinfo{journal}{Phys.\ Rev.\ Lett.} \textbf{\bibinfo{volume}{29}},
  \bibinfo{pages}{962} (\bibinfo{year}{1972}).

\bibitem[{\citenamefont{Barish et~al.}(1961)\citenamefont{Barish, Kurz,
  Perez-Mendez, and Solomon}}]{Bar61}
\bibinfo{author}{\bibfnamefont{B.~C.} \bibnamefont{Barish}},
  \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Kurz}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Perez-Mendez}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Solomon}},
  \bibinfo{journal}{Bull.\ Am.\ Phys.\ Soc.\ Ser.II}
  \textbf{\bibinfo{volume}{6}}, \bibinfo{pages}{523} (\bibinfo{year}{1961}).

\bibitem[{\citenamefont{Kernel et~al.}(1989{\natexlab{b}})}]{Ker89-2}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Kernel}} \bibnamefont{et~al.},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B225}},
  \bibinfo{pages}{198} (\bibinfo{year}{1989}{\natexlab{b}}).

\bibitem[{\citenamefont{Zhu}(2000)}]{Zhu00}
\bibinfo{author}{\bibfnamefont{S.-L.} \bibnamefont{Zhu}},
  \bibinfo{journal}{Phys.\ Rev.\ C} \textbf{\bibinfo{volume}{63}},
  \bibinfo{pages}{018201} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Haider and Liu}(1994)}]{HL94}
\bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Haider}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.~C.} \bibnamefont{Liu}},
  \bibinfo{journal}{Phys.\ Lett.} \textbf{\bibinfo{volume}{B335}},
  \bibinfo{pages}{253} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Machleidt et~al.}(1987)\citenamefont{Machleidt,
  Holinde, and Elster}}]{MHE87}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Machleidt}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Holinde}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Elster}},
  \bibinfo{journal}{Phys.\ Rep.} \textbf{\bibinfo{volume}{149}},
  \bibinfo{pages}{1} (\bibinfo{year}{1987}).

\bibitem[{\citenamefont{Beane and van Kolck}(2002)}]{BK02}
\bibinfo{author}{\bibfnamefont{S.~R.} \bibnamefont{Beane}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{U.}~\bibnamefont{van Kolck}}
  (\bibinfo{year}{2002}), \eprint{nucl-th/0212039}.

\bibitem[{\citenamefont{Nath et~al.}(1971)\citenamefont{Nath, Etemadi, and
  Kimel}}]{NEK71}
\bibinfo{author}{\bibfnamefont{L.~M.} \bibnamefont{Nath}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Etemadi}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~D.} \bibnamefont{Kimel}},
  \bibinfo{journal}{Phys.\ Rev.\ D} \textbf{\bibinfo{volume}{3}},
  \bibinfo{pages}{2153} (\bibinfo{year}{1971}).

\end{thebibliography}