Rescattering effects and two-step process\\ in kaon photoproduction on the deuteron
A. Salam\footnote{\uppercase{P}ermanent address : \uppercase{D}epartemen \uppercase{F}isika
\uppercase{FMIPA}
\uppercase{U}niversitas \uppercase{I}ndonesia
\uppercase{D}epok 16424
\uppercase{I}ndonesia.}~ and K. Miyagawa
Kaon photoproduction on the deuteron is investigated by considering $YN$ and $KN$ rescattering and the two-step process $\gamma d \rightarrow \pi NN \rightarrow KYN$. A strong enhancement in the total cross section is found from the two-step process. $YN$ rescattering has remarkable effects in the inclusive and exclusive cross section, while the effect of $KN$ rescattering is much less important.
SNP04-AgusSalam.tex
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\begin{document}

\title{Rescattering effects and two-step process\\ 
in kaon photoproduction on the deuteron}
     
\author{A. Salam\footnote{\uppercase{P}ermanent address :  
\uppercase{D}epartemen \uppercase{F}isika, \uppercase{FMIPA},  
\uppercase{U}niversitas \uppercase{I}ndonesia, \uppercase{D}epok 16424, \uppercase{I}ndonesia.}~
 and K. Miyagawa}

\address{Simulation Science Center, 
Okayama University of Science, \\
1-1 Ridai-cho, Okayama 700-0005, Japan}


\author{H. Arenh\"ovel}
\address{Institut f\"ur Kernphysik, Universit\"at Mainz,
D-55099 Mainz, Germany}

\author{T. Mart}
\address{Departemen Fisika, FMIPA, Universitas Indonesia, 
Depok 16424, Indonesia}  

\author{C. Bennhold}
\address{Center for Nuclear Studies, Department of Physics,\\ 
The George Washington University, Washington, D.C. 20052, USA}

\author{W. Gl\"ockle}
\address{Institut f\"ur Theoretische Physik II, Ruhr-Universit\"at Bochum,\\
D-44780 Bochum, Germany}

\maketitle

\abstracts{
Kaon photoproduction on the deuteron is investigated 
by considering $YN$ and $KN$ rescattering 
and the two-step process 
$\gamma d \rightarrow \pi NN \rightarrow KYN$. 
A strong enhancement in the total cross section 
is found from the two-step process. 
$YN$ rescattering has remarkable effects 
in the inclusive and exclusive cross section, 
while the effect of $KN$ rescattering is much less important.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec-introduction}

Kaon photoproduction on the deuteron has been investigated by several people. 
%The advent of new generation accelerators 
%as ELSA in Bonn or CEBAF in Newport News 
%has triggered several new
%analyses\cite{MaB00,JaR02} of 
%$\gamma p\rightarrow K^+\Lambda$ and
%$\gamma p\rightarrow K^+\Sigma^0$ 
%since the experimental data\cite{Tra98} is available. 
Li {\it et al.}\cite{LiW92} have extracted 
the elementary cross section from the deuteron target 
in the study of neutron channels. 
In a recent paper Yamamura {\it et al.}\cite{YaM99} 
have calculated the $YN$ rescattering 
for the $K^+$ channels by using the Nijmegen
$YN$ potential\cite{MaR89}. They found sizeable effects in 
the inclusive cross sections from the $YN$ rescattering. 
This work is extended by considering 
the two-step process $\gamma d \rightarrow \pi NN \rightarrow KYN$ 
and the $KN$ rescattering\cite{Sal03}. 
Other recent calculations also investigated the $YN$ rescattering\cite{Ker01} 
and the pion mediated process in lowest order\cite{Max04}.

This paper presents the calculation of $K^+$ and 
$K^0$ photoproduction on the deuteron by considering 
$YN$ and $KN$ rescattering and the two-step process. 
The formulations are shown in Section~\ref{sec-formulations}, 
the results in Sect.~\ref{sec-results}, 
and the conclusions in Sect.~\ref{sec-conclusions}. 
%Throughout the paper we use natural units $\hbar = c = 1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FORMULATIONS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some Formulations}
\label{sec-formulations}

In the deuteron rest frame the exclusive cross section is given by
\begin{eqnarray}
\frac{d\sigma}{dp_{K}d\Omega_{K}d\hat q_{Y}} &=&  
\frac{m_{Y}m_{N}\vert\vec p_{K}\vert^2\vert\vec q_{Y}\vert}
{4(2\pi)^2E_{\gamma}E_{K}W} 
%\nonumber \\ &\times & 
\frac{1}{6} \sum_{\mu_{Y}\mu_{N}\mu_{d}\lambda} 
\left\vert \sqrt{2}\langle\Psi^{(-)}_{\mu_{Y}\mu_{N}}
\vert t^{\gamma K}_{\lambda}\vert
\Psi_{\mu_{d}}\rangle\right\vert^2
\label{eq-gdkyn-dsigma-inclusive}
\end{eqnarray}
where $\mu_{Y}$, $\mu_{N}$, $\mu_{d}$, and $\lambda$ denote 
the spin projections of hyperon, nucleon, deuteron, and the photon 
polarization, respectively, $W^2=(P_{d}+p_{\gamma}-p_{K})^2$, 
and $\sqrt{2}$ comes from the proper antisymmetrization. 
The amplitude is approximated by the diagram in Figure~\ref{fig-gdkyn-diagram}. 

\begin{figure}[htbp]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
\centerline{\epsfxsize=4.1in\epsfbox{fig-gdkyn-diagram.eps}}   
\caption{Kaon photoproduction on the deuteron. 
Diagram (a) is impulse approximation (IA), 
(b) and (c) are $YN$ and $KN$ rescattering, respectively, and 
(d) is the two-step process ($\pi K$-process for short). 
\label{fig-gdkyn-diagram}}
\end{figure}

The impulse term and $YN$ rescattering (diagram (a) and (b), respectively) 
are calculated simultaneusly as
\begin{eqnarray}
T_{IA+YN} &=& t^{\gamma K} + t_{YN}\, G_{YN} t^{\gamma K}\,.
\label{eq-gdkyn-yn-M-IA+YN}
\end{eqnarray}
By inserting the Lippmann-Schwinger equation for $t_{YN}$, 
we get
\begin{eqnarray}
T_{IA+YN} &=& (1-V_{YN}\,G_{YN})^{-1}\,t^{\gamma K}\,.
\label{eq-gdkyn-yn-M-IA+YN-LS}
\end{eqnarray}
After solving the last equation in the partial wave decomposition, 
one obtains the $YN$ rescattering amplitude by subtraction of the
impulse term. 
The $KN$ rescattering (diagram (c)) is evaluated directly as
\begin{eqnarray}
T_{KN} &=& t_{KN}\, G_{KN}\, t^{\gamma K}\,,
\label{eq-gdkyn-kn}
\end{eqnarray}
where we use a separable potential of rank-1 for $V_{KN}$. 
The $\pi K$-process (diagram (d)) is calculated 
in the same way as in the $KN$-rescattering. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%RESULTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec-results}

The results are calculated by using 
the elementary operator of Mart and Bennhold\cite{MaB00} and 
the deuteron wave function of Bonn model\cite{MaH87}. 
\begin{figure}[htbp]
\centerline{\epsfxsize=8.6cm\epsfbox{pr-K4tc.eps}}   
\caption{Total cross section vs. photon energy $E_{\gamma}$.
\label{fig-gdkyn-tc}}
\end{figure}
\begin{figure}[htbp]
\centerline{\epsfxsize=8.6cm\epsfbox{pr-K4dc.eps}}   
\caption{Differential cross section vs. kaon angle $\theta_{K}$.
\label{fig-gdkyn-dc}}
\end{figure}
Fig.~\ref{fig-gdkyn-tc} shows the total cross section 
of $\gamma d \rightarrow KYN$ as function of $E_{\gamma}$. 
The $\pi K$-process (solid line) enhances the total cross section 
in all channels (dotted line). 
The next remarkable effect comes from $YN$ rescattering (short-dash line), 
while the effect of $KN$ rescattering (dash line) is negligible. 
Fig.~\ref{fig-gdkyn-dc} shows the differential cross section 
as function of $\theta_{K}$ calculated at different photon energies. 
$YN$ rescattering has remarkable effects at forward angles, 
while the $\pi K$-process at larger angles.
\begin{figure}[htbp]
\centerline{\epsfxsize=8cm\epsfbox{pr-K4sem.eps}}   
\caption{Inclusive cross section vs. kaon momentum $p_{K}$.
\label{fig-gdkyn-inc}}
\end{figure}
The inclusive cross section as function of $p_{K}$ is shown 
in Fig.~\ref{fig-gdkyn-inc}. 
%It is calculated 
%in the forward kaon angle at photon energies 1.14 GeV (top panels) 
%and 1.3 GeV (bottom panels). 
$YN$ rescattering shows remarkable effects 
at the threshold and peak region, while the effect of $\pi K$-process appears 
at smaller kaon momenta. Some enhancement, 
whose origin is from the $S$-matrix pole of $V_{YN}$\cite{MiY99}, 
is found at the $\Sigma$-threshold in the $\Lambda$-channels 
(indicated by arrows in the figure).  
\begin{figure}[htbp]
\centerline{\epsfxsize=9cm\epsfbox{pr-K4exc.eps}}   
\caption{Exclusive cross section vs. hyperon angle $\theta_{Y}'$.
\label{fig-gdkyn-exc}}
\end{figure}
Fig.~\ref{fig-gdkyn-exc} shows the exclusive cross section in the forward kaon angle 
at photon energy 1.3 GeV as function of $\theta_{Y}'$ 
measured relative to the direction of momentum transfer 
$\vec p_{\gamma}-\vec p_{K}$ in the deuteron rest frame. 
The $\Lambda$-channels are calculated at kaon momentum 870 MeV/c 
and $\Sigma$-channels at 810 MeV/c. $YN$ rescattering dominates the effect 
for all channels at larger $\theta_{Y}'$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CONCLUSIONS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
\label{sec-conclusions}

Kaon photoproduction on the deuteron is calculated by considering 
$YN$ and $KN$ rescattering and the two step process. 
A strong enhancement in the total cross section is found from the two-step process. 
$YN$ rescattering has remarkable effects in the inclusive and exclusive cross section, 
while the effect of $KN$ rescattering is negligible.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ACKNOWLEDGMENTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
AS would like to thank the Simulation Science Center, Okayama University of Science, 
Okayama for financial support and very kind hospitality.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%REFERENCES%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}

\end{document}

nucl-th0411036.extracted_bib
\begin{thebibliography}{0}

%\bibitem{JaR02} S. Janssen {\it et al.}, {\it Phys. Rev.} {\bf C65}, 015201 (2002). 
%\bibitem{Tra98} M.Q. Tran {\it et al.}, {\it Phys. Lett.} {\bf B445}, 20 (1998). 
\bibitem{LiW92} X. Li, L.E. Wright, and C. Bennhold, {\it Phys. Rev.} {\bf C45}, 2011 (1992). 
\bibitem{YaM99} H. Yamamura {\it et al.}, {\it Phys. Rev.} {\bf C61}, 014001 (1999). 
\bibitem{MaR89} P.M.M. Maessen {\it et al.}, {\it Phys. Rev.} {\bf C40}, 2226 (1989).
\bibitem{Sal03} A. Salam, {\it Dissertation}, Johannes Gutenberg Universit\"at, Mainz, 2003.
\bibitem{Ker01} B.O. Kerbikov, {\it Phys. Atom. Nucl.} {\bf 64}, 1835 (2001).
\bibitem{Max04} O.V. Maxwell, {\it Phys. Rev.} {\bf C69}, 034605 (2004).
\bibitem{MaB00} T. Mart and C. Bennhold, {\it Phys. Rev.} {\bf C61}, 012201 (2000).
\bibitem{MaH87} R. Machleidt, K. Holinde, and Ch. Elster, {\it Phys. Rep.} {\bf 149}, 1 (1987).
\bibitem{MiY99} K. Miyagawa and H. Yamamura, {\it Phys. Rev.} {\bf C60}, 024003 (1999).

\end{thebibliography}