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\documentclass[allclo]{FBSart}

\usepackage{amsfonts}

\usepackage{amssymb}

\usepackage{epsfig}



\title{Relativity in the 3-nucleon system}

\author{H.Kamada\thanks{

\textit{E-mail address:}

kamada@mns.kyutech.ac.jp}\instnr{1}, W. Gl\"ockle\instnr{2}, 

H.  Wita{\l}a\instnr{3}, 

J. Golak\instnr{3},

R. Skibi\'nski\instnr{3}, \\ and W. N. Polyzou\instnr{4}}



%\institute{

\instlist{

Department of Physics, Faculty of Engineering, Kyushu Institute of Technology,\newline 

1-1 Sensuicho Tobata, Kitakyushu 804-8550, Japan \and

Institut f\"ur Theoretische Physik II, Ruhr-Universit\"at Bochum, D-44780 Bochum, Germany \and

M. Smoluchowski Institute of Physics, Jagiellonian

University, PL-30059 Krak\'ow, Poland \and

Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA

}





\runningauthor{H. Kamada}

\runningtitle{Relativity in the  3-nucleon system}

\sloppy



\begin{document}



\maketitle

\begin{abstract}



A Poincar\'e invariant formulation of the 3-body system

is used. The 2- body force embedded in the 3- particle Hilbert space

is generated out of the high-precision NN forces

by solving a nonlinear equation. The solution of the relativistic 3N

Faddeev equation for $ ^3H$ reveals less binding energy than for the

nonrelativistic one. The effect of the Wigner spin rotation on the

binding energy is very small.



\end{abstract}



\section{Introduction}

The nonrelativistic NN force $ v^{nr} $ is added to the nonrelativistic kinetic energy $

\frac{k^2}{m}$, while 

the relativistic NN force $ v$ is added to the relativistic kinetic

energy $ 2 \omega(k) = 2 \sqrt{k^2 + m^2} $.  

They give the same NN S-matrix, $S(k)$, if they are related by:

\begin{eqnarray}

v= \sqrt{4mv^{nr}+4(k^2+m^2)} -2\sqrt{k^2+m^2}= \sqrt{4mv^{nr}+4\omega^2} - 2\omega

\label{1}

\end{eqnarray}

In  former work  \cite{kam,Kamada3} 

we employed an analytic scale  transformation

of the momenta

\begin{eqnarray}

2m+(k^{nr})^2/m \equiv 2\sqrt{{q}^2+m^2}

\label{trick}

\end{eqnarray}

such that the relativistic and nonrelativistic NN phase shifts are

exactly equal at the same energy, but not at the same c.m. momenta as

they should \cite{Polyzou}.  We introduce in section 2 a new technique

\cite{Kamada4} to solve Eq.(\ref{1}) and show relativistic and

nonrelativististic results for the $^3H$ binding energy based on high

precision NN forces in section 3.



\section{The nonlinear equation and its solution} 

The NN interactions enter the 3N mass operator in the form \cite{Coester}

\begin{eqnarray}

v_p &=& \sqrt{( 2\omega( k) +v ) ^2 + p^2} - \sqrt{(2\omega( k) )^2+p^2} 

\label{4} \cr

&=&\sqrt{4mv^{nr}+4\omega^2+p^2} -\sqrt{(2\omega(k) )^2+p^2}

\cr &=& \sqrt{4mv^{nr}+4\omega_p^2} - 2\omega_p 

\end{eqnarray}

where $p$ is the invariant relative momenta 

related to the 3N invariant mass by

$M_0= 2\omega_p = 2\sqrt{k^2+m^2+p^2/4}$.

The $p$-dependence arises because 

in a 3-body system the NN subsystems are not at

rest.





\begin{table}%[H] add [H] placement to break table across pages

\caption{Convergence of the  $v_p^{(n)}$  iteration in Eq. (\ref{14}) 

at total momentum $p=0$. 

We choose the coupled partial waves ($^3S_1$-$^3D_1$) of 

the Argonne V18 potential. The momenta $k$ and $k'$

are 1.0 fm$^{-1}$ and the potential unit is [fm$^2$]. 

}

 %\begin{ruledtabular}

 \begin{tabular}{ccccc}\hline

        $n$       & $v_0^{(n)}$($^3S_1$-$^3S_1$) & 

                  $v_0^{(n)}$($^3S_1$-$^3D_1$) & 

                  $v_0^{(n)}$($^3D_1$-$^3D_1$) \\ \hline

0 &  0.084232 & 0.044709 & 0.016853 \\

1 & 0.067716  & 0.044628 & 0.016785 \\

2 & 0.059933  & 0.044597 & 0.016744 \\

3 & 0.056135  & 0.044587 & 0.016719 \\

10 & 0.052194  & 0.044595 & 0.016684 \\

20 & 0.052126  & 0.044597 & 0.016684 \\

30 & 0.052126  & 0.044597 & 0.016684 \\ \hline

 \end{tabular}

% \end{ruledtabular}

 \end{table}





The  new way \cite{Kamada4} is to directly solve the quadratic

operator equation (\ref{4}). After partial wave decomposition in momentum space it reads

\begin{eqnarray}

& & <k | v_p | k~'> =  \frac{2m < k| v^{nr} | k~'>}{\omega_p(k) + \omega_p(k~')} \cr

&&-\frac{1}{2(\omega_p(k) + \omega_p(k~'))} \int_0^\infty dk~'' {k~''}^2

< k|  v_p | k~''> < k~''| v_p | k~'>

\end{eqnarray}

We verified numerically that for all the realistic high precision potentials 

the following

 very simple iterative scheme works

 \begin{eqnarray}

&&<k | v_p | k~'>^{(0)} = \frac{2m < k| v^{nr} | k~'>}{\omega_p(k) + \omega_p(k~')},

% \end{eqnarray}

%\begin{eqnarray}

\cr && <k | v_p | k~'>^{(n+1)} = 

\frac{1}{2(\omega_p(k) + \omega_p(k~'))}

%\times

\{ 4m < k |v^{nr}|{k~'}>  \cr

&&-

\int_0^\infty dk~'' {k~''}^2

< k|  v_p | k~''>^{(n)}< k~''| v_p | k~'>^{(n)}

\}

\label{14}

\end{eqnarray}

with $ n=0,1,2,\cdots$. We demonstrate in Table 1. 

the convergence of  of this Eq. (\ref{14}) in the case of  $p=0$.





\section{Relativistic versus nonrelativistic binding energies of $^3$H}



In our calculation of the $^3$H binding energy the relativistic Faddeev equation 

(Eq. (3.7) in \cite{Kamada3} ) is solved in a partial wave basis. 

The feasibility of our 

approach is demonstrated  in Table 2 for a five-channel calculation (NN forces only in the states

$^1$S$_0$ and $^3$S$_1$-$^3$D$_1$).

There are two kinds of  relativistic results, $E_b^{(1)}$ and $E_b^{(2)}$,

in comparison to the  nonrelativistic result $E_b^{nr}$. The present results, $E_b^{(1)} $,

 based on 

(\ref{14}) show a significantly smaller reduction of the binding energy than $E_b^{(2)}$, what 

we obtained by the

momentum scale transformation in \cite{Kamada1} and \cite{Kamada3}. The Wigner spin rotation is

handled using the Balian-Brezin method \cite{bal}. Its effect in reducing the binding energy is

 about

1 Kev and thus marginal.



  For the relativistic calculation of 3N scattering and the basic 

relativistic formulation see~\cite{1}, 

H.Witala, and W.N.Polyzou in this conference.     







 \begin{table}%[H] add [H] placement to break table across pages

 \caption{The relativistic ($E_b$)and nonrelativistic ($E_b^{nr}$)triton binding energies in MeV together with 

$\Delta \equiv E_b - E_b^{nr}$. The superscripts of $(1)$ and $(2)$ correspond to

 Eqs. (1) (direct connection)  and (2) (scale transformation), respectively. }

% \begin{ruledtabular}

 \begin{tabular}{ccccccc}

\hline

        Potential        & $E_b^{nr}$  &  $E_b^{(1)} $&  $\Delta^{ (1)}$  

&  $E_b^{(2)} $   & $\Delta^{ (2)}$   \\ \hline

RSC         &   -7.02 & -6.97 & 0.05  & -6.59 & 0.43  \\

CD-Bonn     &   -8.33 & -8.22 & 0.11  & -7.98 & 0.35  \\

Nijmegen II &   -7.65 & -7.58 & 0.07  & -7.22 & 0.43  \\

Nijmegen I  &   -8.00 & -7.90 & 0.10  & -7.71 & 0.29   \\

Nijmegen 93 &   -7.76 & -7.68 & 0.08  & -7.46 & 0.30  \\

AV18        &   -7.66 & -7.59 & 0.07  & -7.23 & 0.43  \\

\hline

Exp.  (-8.48)       &         &       &      &    & \\

\hline

 \end{tabular}

% \end{ruledtabular}

 \end{table}



\begin{acknowledge}

The numerical calculations were performed on the 

IBM Regatta p690+ of the NIC in J\"ulich,

Germany.

\end{acknowledge}



\begin{thebibliography}{3}

\bibitem{kam} Kamada, H., Gl\"ockle, W.: Phys. Rev. Lett. {\bf 80}, 2547 (1998).

\bibitem{Kamada3} Kamada, H., et al.: Phys. Rev.

C {\bf 66}, 044010 (2002).

\bibitem{Polyzou} Allen, T. W., Payne, G. L., Polyzou, W. N.: Phys. Rev. C

{\bf 62},054002 (2000).

\bibitem{Kamada4} Kamada, H., Gl\"ockle, W.: Phys. Letter. {\bf B 655}, 119 (2007).

\bibitem{Kamada1} Kamada, H., et al.: Proceedings of

17th Conference on Few-Body Problem in Physics, W. Gl\"ockle and W. Tornow (Eds.),  

Elsevier B.V.,  S263  (2004).

\bibitem{Coester} Coester, F., Helv.Phys. Acta {\bf 38} , 7 (1965).



\bibitem{bal} Balian, R., et al.: Il Nuovo Cim. {\bf B2}, 403 (1969).



\bibitem{1} Wita{\l}a, H., et al.: Phys. Rev. {\bf C71}, 054001 (2005).

%\bibitem{2} Wita{\l}a, H., et al.: Phys. Lett. {\bf B634}, 374 (2006).

%\bibitem{3} Skibi\'nski, R., et al.: Eur. Phys. J. {\bf A30}, 369 (2006).







%\bibitem{tunl1} Tornow, W., et al.: Phys. Lett. {\bf B 257}, 273 (1991).

%\bibitem{bunker} Bunker, S.N., et al.: Nucl. Phys. {\bf A113}, 461 (1968).

%\bibitem{schenk} Miller, G.A., et al.: nucl-th/0703018v2.



\received{First received October 10, 1998; accepted in final form  April 10, 1999.}



\end{thebibliography}



\end{document}


