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\documentclass[epjCONF]{svjour}
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%\usepackage{graphics}
\usepackage{graphicx}
\usepackage[varg]{txfonts} % Times fonts
\usepackage[latin1]{inputenc}
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% !!! Do not change the next session-title lines !!! %
\session-title{%
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
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\begin{document}
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\title{%
%Insert your title here:
Calculations of the Triton Binding Energy with a Lorentz Boosted
Nucleon-Nucleon Potential
}%
\author{%
% add email of the responsible author:
H. Kamada\inst{1}\fnmsep\thanks{\email{kamada@mns.kyutech.ac.jp}} , W.
Gl\"ockle\inst{2}, H. Wita\l a\inst{3}, J. Golak\inst{3}, R.
Skibi\'nski\inst{3},
W. N. Polyzou\inst{4}, Ch. Elster\inst{5}
%\and %
%N.N. Surname2\inst{2}
%\and %
%N.N. Surname3\inst{1}
}
%
\institute{%
Department of Physics, Faculty of Engineering, Kyushu Institute of
Technology, \\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,
IA 52242, USA\\
\and %
Institute of Nuclear and Particle Physics, and Department of Physics, \\
Ohio University, Athens,
Ohio 45701, USA\\
}
%
\abstract{
We study the binding energy of the three-nucleon system in
relativistic models that use two different relativistic treatments of
the potential that are phase equivalent to realistic NN interactions.
One is based on a unitary scale transformation that relates the
non-relativistic center-of-mass Hamiltonian to the relativistic mass
(rest energy) operator and the other uses a non-linear equation that
relates the interaction in the relativistic mass operator  to
the non-relativistic interaction.  In both cases Lorentz-boosted
interactions are used in the relativistic Faddeev equation to solve
for the three-nucleon binding energy.  Using the same realistic NN
potentials as input, the solution of the relativistic three-nucleon
Faddeev equation for $^3$H shows slightly less binding energy than the
corresponding nonrelativistic result.  The effect of the Wigner spin
rotation on the binding is very small.  }
%end of abstract
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\maketitle
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%----- Beginning of MAIN TEXT ---------------------------------------
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\section{Introduction}
\label{Kamada_sec1}
For up to 300 MeV proton energy, proton-deuteron (pd) scattering
measurements have been analyzed with rigorous three-nucleon (3N)
Faddeev calculations \cite{Kamada_ref6} based on the CD-Bonn potential
\cite{Kamada_ref7} and the Tucson-Melbourne 3N force
(3NF) \cite{Kamada_ref8}.  Comparing theoretical calculations to the
recent precise measurements of pd scattering data
\cite{Kamada_ref1,Kamada_ref2,Kamada_ref3,Kamada_ref4,Kamada_ref5}
indicates that theoretical predictions based on two-nucleon forces
alone are not sufficient to describe the data above about 100 MeV.
The minimum of the differential cross section has been discussed as
the first signal of the 3NF effects, which are already seen below
100 MeV \cite{Kamada_ref9,Kamada_ref10,Kamada_ref11}.  However,
presently available 3NF's only partially improve the description of
cross section data and spin observables.  Since most of the cited
calculations are based on the non-relativistic formulation of the
Faddeev equations \cite{Kamada_ref12}, one needs to question if in the
intermediate energy regime a Poincar\'e invariant formulation is
required.

There are different formulations of the relativistic few-body problem.
Our calculations are based on an exact realization of the symmetry of
the Poincar\'e group in three-nucleon quantum
mechanics \cite{Kamada_ref14}.  The mass operator (rest energy
operator) consists of the relativistic kinetic energy together with two-
and three-body interactions, including their boost corrections
\cite{Kamada_ref15}.  Our approach differs from a manifestly covariant
scheme linked to a field theoretical approach \cite{Kamada_ref13}.

The first attempt in solving the relativistic \- Faddeev equation for the
three-nucleon bound state based on the second approach has been carried out in
\cite{Kamada_ref16}, resulting in a decrease of the binding energy
compared to the nonrelativistic result.  On the other hand, similar
calculations based on the field theory approach \cite{Kamada_ref13}
increase it.  These contradictory results require more investigation.

Because the result may depend on  the transformation of the nonrelativistic
potential to a relativistic potential, a momentum scale transformation
\cite{Kamada_refscale} (MST) was  introduced without any additional
parameters.  Of course, this scale transformation method is not
equivalent to the construction of a relativistic potential from a
field theory.  However, the scale transformation is a very useful and
simple parameterization of a relativistic NN potential, which
preserves the NN phase shifts exactly.  Using a s-wave approximation we
solved the relativistic Faddeev equation with Lorentz boost of the
scale transformed potential \cite{Kamada_refscale,Kamada_ref22} and it
agreed with the previous result \cite{Kamada_ref16}.

Recently, going back to the idea of the Coester-Pieper-Serduke scheme
(CPS), \cite{Kamada_ref17} we succeeded to obtain a relativistic
potential numerically, and used the same method to directly construct the 
Lorentz boosted potential.

In the following we want  to demonstrate some recent results: in
Section \ref{Kamada_sec2} we introduce the relativistic
nucleon-nucleon potentials constructed by the MST and CPS methods, in
Section \ref{Kamada_sec3} the construction of the boosted potentials
is discussed, in Section \ref{Kamada_sec4} we give numerical results
for the triton binding energy based on the  Poincar\'e invariant
Faddeev equation and in Section \ref{Kamada_sec5} we summarize.


\section{The Relativistic Potential}
\label{Kamada_sec2}

As mentioned in the introduction, our relativistic treatment 
is based on the Bakamjian-Thomas framework, where the rest Hamiltonian for the
three-body system consists of relativistic kinetic energies and two-
and three-body interactions, including their boost corrections, which
are dictated by the Poincar\'e algebra.  The boost techniques will be
discussed in the next section \ref{Kamada_sec3}.  For two-body systems
realistic interactions are designed to fit scattering data. Relativistic 
two-body calculations must fit the same data, which means that at the
two-body level the relativistic and non-relativistic models must be 
phase equivalent as a function of either the center-of-momentum momentum or the 
energy.   
Differences in relativistic and non-relativistic calculations then  appear first
in the three-body calculations.


The usual nonrelativistic Lippmann-Schwinger
equation (LS) with a potential $v $ is given in momentum space as
\begin{eqnarray}
\label{Kamada_eq1}
t(\vec p,\vec p';E) = v(\vec p, \vec p') + \int {v(\vec p, \vec p'')
t(\vec p'' , \vec p ';E) \over E- {p''}^2/m +i\epsilon} d\vec p''~~,
\end{eqnarray}
where $t$ and $m$ are the t-matrix and the nucleon mass, respectively.

On the other hand, there are relations between the c.m. kinetic energy $E$
and the relative momentum $k$ for the relativistic and nonrelativistic
formalism, namely
\begin{eqnarray}
\label{Kamada_eq2}
&&E_{r}\equiv 2\sqrt{m^2+{k_{r}}^2}-2m, \cr
&&E_{nr} \equiv {{k_{nr}}^2 \over m}.
\end{eqnarray}
Here the subscripts $r$ and $nr$ denote relativistic and
nonrelativistic relations.

Our relativistic potential $v_r$  appears in the relativistic
Lippmann Schwinger equation as
\begin{eqnarray}
\label{Kamada_eq1.2}
&&t_r (\vec p, \vec p';E) = v_r(\vec p, \vec p') \cr
&&+ \int {v_r(\vec p, \vec p'') t_r(\vec p'' , \vec p' ;E) \over
2\sqrt{m^2+k^2}-2\sqrt{m^2+{p''}^2} +i\epsilon } d\vec p''~~~~,
\end{eqnarray}
where $t_r$ is a relativistic t-matrix.
Note that in Eqs.~(\ref{Kamada_eq1}) and (\ref{Kamada_eq1.2}) there are
no subscripts ($r$ or $nr$) for $E $ and $k$ (or $\vec p$) before
entering the next
subsection.  The energies $E$ in Eq.(\ref{Kamada_eq1}) and in Eq.
(\ref{Kamada_eq1.2}) are not necessarily equal. The
momenta $k$ and $\vec p$ in Eq.(\ref{Kamada_eq1}) and in Eq.
(\ref{Kamada_eq1.2}) are not necessarily equal either.

\subsection{The Momentum Scale Transformation}
\label{Kamada_sec2sub1}

In order to build the potential $v $ one may identify the energy $E$ of
Eq.~(\ref{Kamada_eq1}) just as a c.m. energy $E_{ex}$ which is measured in
an experiment.  One may interpret~\cite{Kamada_refscale}:
\begin{eqnarray}
\label{Kamada_eq3}
E {\buildrel ! \over = } E_{ex}{\buildrel ! \over = }E_{r}{\buildrel !
\over = }E_{nr}.
\end{eqnarray}
For this choice, $k_{r} \ne k_{nr} $.

Eq.~(\ref{Kamada_eq1}) is rewritten as
\begin{eqnarray}
\label{Kamada_eq4}
&&t(\vec p_{nr},\vec p_{nr}';E) \cr
&&= v(\vec p_{nr}, \vec p_{nr}') + \int {v(\vec p_{nr}, \vec p_{nr}'')
t(\vec p_{nr}'' , \vec p_{nr}'' ;E) \over {k_{nr}''}^2/m- {p_{nr}'}^2/m
+i\epsilon} d\vec p_{nr}''~~~~ \cr
&&=v(\vec p_{nr}, \vec p_{nr}') \cr
&&+ \int {v(\vec p_{nr}, \vec p_{nr}'') t(\vec p_{nr}'' , \vec p_{nr}'
;E) \over 2\sqrt{m^2+k_r^2}-2\sqrt{m^2+{p_r''}^2} +i\epsilon } d\vec
p_{nr}'' \cr
&&= v(\vec p_{nr}, \vec p_{nr}')
+ \int {v(\vec p_{nr}, \vec p_{nr}'') t(\vec p_{nr}'' , \vec p_{nr}' ;E)
\over 2\sqrt{m^2+k_r^2}-2\sqrt{m^2+{p_r''}^2} +i\epsilon } \cr
&& \times J (p''_r ) d\vec p_{r}''~~~~~~,
\end{eqnarray}
where $J$ is the Jacobian.
Under the interpretation of Eq. (\ref{Kamada_eq3}) the nonrelativistic
momentum $k_{nr}$ is a function
of $k_r$,
\begin{eqnarray}
\label{Kamada_eq5}
k_{nr}= k_{nr} ( k_r ) = \sqrt{2m}\sqrt{\sqrt{m^2+k_r^2}-m}.
\end{eqnarray}
One defines
\begin{eqnarray}
\label{Kamada_eq6}
&&v_r (\vec p_r, \vec p_r') \equiv {1 \over h(p_{nr})} v(\vec
p_{nr},\vec p_{nr}'){1 \over h(p'_{nr})}, \cr
&&t_r (\vec p_r, \vec p_r';E) \equiv {1 \over h(p_{nr})} t (\vec
p_{nr},\vec p_{nr}';E){1 \over h(p'_{nr})} ,
\end{eqnarray}
with
\begin{eqnarray}
\label{Kamada_eq7}
h(p_{nr})\equiv \sqrt{(1+{p_{nr}^2 \over 2m^2})\sqrt{1+{k_{nr}^2 \over
4m^2}}}\equiv {1 \over \sqrt{ J (p_r)} }.~~~~
\end{eqnarray}
The amplitudes $t_r$ and $v_r$ are related by solving the relativistic LS
equation:
\begin{eqnarray}
\label{Kamada_eq8}
&&t_r (\vec p_{r}, \vec p_{r}';E) = v_r(\vec p_{r}, \vec p_{r}') \cr
&&+ \int {v_r(\vec p_{r}, \vec p_{r}'') t_r(\vec p_{r}'' , \vec p_{r}'
;E) \over 2\sqrt{m^2+k_r^2}-2\sqrt{m^2+{p_r''}^2} +i\epsilon } d\vec
p_{r}''~~~~.
\end{eqnarray}
Therefore, one could identify them as relativistic amplitudes.
We call Eqs.~(\ref{Kamada_eq5}-\ref{Kamada_eq7}) momentum scale
transformation (MST) \cite{Kamada_refscale}.

\subsection{Coester-Pieper-Serduke Scheme}
\label{Kamada_sec2sub2}

There is another way for the identification. Instead of Eq.~(\ref{Kamada_eq3}) one
employs the following relation among momenta:
\begin{eqnarray}
\label{Kamada_eq9}
k{\buildrel ! \over = } k_{ex}{\buildrel ! \over = } k_{nr} {\buildrel !
\over = }k_r,
\end{eqnarray}
where $ k_{ex}$ is the experimental momentum.  In this case it is natural 
to add an interaction to $k^2/m$, so that the square of the two-body 
invariant mass operator becomes
\begin{equation}
M^2 = 4m^2 + 4m ({p^2\over m} + \hat{v}).
\label{Kamada_eq10}
\end{equation}  
Because this is a function of the non-relativistic Hamiltonian, it has the 
same eigenfunctions as the non-relativistic Hamiltonian as a function of 
the relative momentum, $k$.  
Since the phase-shifts can be extracted from the scattering wave functions,
this mass operator has the same phases shifts at the non-relativistic 
Hamiltonian as  function of relative momentum. 

The non-relativistic Lippmann-Schwinger equation still holds, Eq.~(\ref{Kamada_eq1}), but in the relativistic case the interaction,
the energy, and the transition operator that appear in this equation have
a different interpretations than compared to the non-relativistic case.

\noindent
The relation
\begin{equation}
M^2 = (M_0 + \hat{v}_r)^2 = M_0^2 + 4m \hat{v}  
\label{Kamada_eq11}
\end{equation} 
leads to the identity 
\begin{equation}
\{ M_0,\hat{v}_{r} \} +  \hat{v}_r^2 = 4mv_{nr}  ~~,
\label{Kamada_eq12}
\end{equation}
which can be expressed in terms of the relative momentum operator
$\hat{k}$ as
\begin{equation}
4 m\hat{v} = 2 \sqrt{m^2 + \hat{k}^2} \hat{v}_{r} + 
2 \sqrt{m^2 + \hat{k}^2} \hat{v}_r + 
 (v_r)^2 = 4mv_{nr} ~~.  
\label{Kamada_eq14}
\end{equation}
Equation (\ref{Kamada_eq14}) can be expressed as the momentum space 
integral equation
\begin{eqnarray}
\label{Kamada_eq15}
&&4m v(\vec p, \vec p') = (2\sqrt{m^2+p^2}+2\sqrt{m^2+{p'}^2 } ) v_r (
\vec p, \vec p' ) \cr
&&+ \int v_r(\vec p, \vec p'') v_r (\vec p'', \vec p') d\vec p''.
\end{eqnarray}
This is a nonlinear integral equation for $v_r$ given $v$ and can
be solved by iteration~\cite{Kamada_ref18}.
The solution of $v_r$ is then used in Eq.~(\ref{Kamada_eq1.2}).
We call this methods Coester-Pieper-Serduke scheme (CPS).

\subsection{Relation to Realistic Potentials}
\label{Kamada_sec2sub3}

The relativistic potentials discussed here were not built directly
from a relativistic Lagrangian.  The nucleon-nucleon potentials were
generated by requiring that they predict the experimental phase
shifts.  Since realistic non-relativistic potentials are constructed
to fit experimental phase shifts, relativistic interactions can be
constructed by requiring that they lead to the same phase shifts as
the non-relativistic potentials as a function of center of momentum
energy (MST) or center of momentum momentum (CPS).  The MST potentials
also lead to the same deuteron binding as the non-relativistic
calculation while the (CPS) potentials produce the same deuteron wave
numbers as the non-relativistic calculation.  The quality of $v_r$
obtained by each scheme was discussed in \cite{Kamada_ref29}.

There are some realistic potentials, e.g. the Argonne V18
potential \cite{Kamada_refAV18}, which are suitable for the MST scheme  because of the
ansatz of Eq. (\ref{Kamada_eq3}).  The CD Bonn potential
\cite{Kamada_ref7} and the Nijmegen potential \cite{Kamada_refNijm} are
suitable to CPS because of the ansatz of Eq. (\ref{Kamada_eq9}) (See Fig.1).

The two approaches are not equivalent, but the differences at low
energies are primarily due to off-shell effects. 

\begin{figure}[!h]
% Use the relevant command for your figure-insertion program
% to insert the figure file.
% For example, with the option graphicx use
\centering
\includegraphics[width=1.0\columnwidth,angle=0]{Kamada_fig1.eps}
\caption{Venn diagram of worlds.}
\label{Kamada_fig1} % Give a unique label
\end{figure}

\section{The Boosted Potential}
\label{Kamada_sec3}

As shown in Section \ref{Kamada_sec2}, schemes generating relativistic
potential from  nonrelativistic interactions are rather
artificial. Compared to these schemes, the boost correction within the 
Bakamjian-Thomas framework is natural and unique.

Cluster properties require that the energy is additive. Because of the
non-linear
relations between the mass and energy in special relativity, the
additivity of energies
in the rest frame implies a non-linear relation between the two-body
interactions
in the two and three-body mass operators \cite{Kamada_ref14}. We call
the two-body interaction in
the three-body mass operator the {\bf boosted potential }$v_q$:

\begin{eqnarray}
\label{Kamada_eq20}
\hat v_q&\equiv& \sqrt{ ( 2\sqrt{ m^2+\hat k^2} + \hat v_r )^2 +q^2} \cr
&&- \sqrt{ 4(m^2+\hat k^2) +q^2 }
\end{eqnarray}
where the spectator momentum $q$ in the 3-body center of mass is
simultaneously
the negative total momentum of the pair.
In the 3-body system the momentum $q$ is operator but it behaves
as c-number in the subsystem.

Using Eq. (\ref{Kamada_eq14}) this can be rewritten as potential in the
CPS scheme,
\begin{eqnarray}
\label{Kamada_eq21}
\hat v_q&=& \sqrt{ 4( m^2 + \hat k^2 +m \hat v ) +q^2} \cr
&&- \sqrt{ 4(m^2+\hat k^2) +q^2 }.
\end{eqnarray}
%where $ \hat v$ is naturally identified to the nonrelativistic
Now, Eq. (\ref{Kamada_eq21}) can be  rewritten as
\begin{eqnarray}
\label{Kamada_eq22}
4m \hat v&=& 2\sqrt{m^2 +\hat k ^2 + q^2/4 } ~~\hat v_q + 2\hat v_q
\sqrt{ m^2+\hat k^2+q^2/4} \cr
&+&(\hat v_q)^2 ~~.
\end{eqnarray}
This has a similar structure as Eq.~(\ref{Kamada_eq14}).
We have a representation in momentum space as
\begin{eqnarray}
\label{Kamada_eq23}
&&4m v(\vec p, \vec p') = \cr
&&(2\sqrt{m^2+p^2+q^2/4}+2\sqrt{m^2+{p'}^2 +q^2/4} ) v_q ( \vec p, \vec
p' ) \cr
&&+ \int v_q(\vec p, \vec p'') v_q (\vec p'', \vec p') d\vec p''.
\end{eqnarray}
This is a nonlinear integral equation for $v_q$ in terms of $v$.
Again,  Eq.(\ref{Kamada_eq23}) is solved
by the same iterative technique used in \cite{Kamada_ref18}.

We would like to emphasize again that Eq. (\ref{Kamada_eq22}) is
a natural extension from Eq. (\ref{Kamada_eq14}).
However, Eq. (\ref{Kamada_eq22}) is not only available for the CPS
scheme.
In \cite{Kamada_ref22} the MST potential was boosted by a different way
related to a M{\o}ller operator.
The boost correction to the MST potential can also be calculated by this
way, namely,
using $v_r$ in Eq. (\ref{Kamada_eq6}) of MST one gets a new $v$ through Eq.
(\ref{Kamada_eq14}).


\section{Triton Binding Energy}
\label{Kamada_sec4}
The relativistic bound-state Faddeev equation was solved using the
boosted t-matrix
$t_r$ of Eq. (\ref{Kamada_eq1.2}).
In Tables~\ref{Kamada_tab1} and \ref{Kamada_tab2} the results for the
triton binding energy using
several potentials calculated based on the  MST and CPS methods are displayed. 
The precision of the partial wave decomposition belongs to 5ch (S-wave
Approximation).
In the case of MST (Table \ref{Kamada_tab1}) the results
\cite{Kamada_ref22} show that the
triton binding energies obtained
from the relativistic calculation are about 400~keV smaller compared
to the ones calculated nonrelativistically. As mentioned in subsection
\ref{Kamada_sec2sub3},
the Reid Soft Core potential and Argonne V18 potential can be reasonably
applied in the MST scheme,
but the other potential are forced to be substituted into MST method.

In the case of the  CPS scheme, Table~\ref{Kamada_tab2}, the differences
between the relativistic and nonrelativistic calculations are  about 100~keV. 
The CDBonn  and  the Nijmgen potentials are
naturally applied to the CPS scheme. 
%but others are forced to substituted into CPS method.
This value is significantly smaller than a MSC result
\cite{Kamada_ref22}. The reason for
this overestimation of a relativistic effect on the binding energy can
be attributed to
a different construction of the relativistic off-shell t-matrix $t_r$.
\footnote{In the former Proceedings \cite{Kamada_refAPFB}
we would have thought that MST has a sort of defect because we need a
new potential $v$ in Eq. (\ref{Kamada_eq14}).}

In Table~\ref{Kamada_tab3} we demonstrate the convergence for partial
wave decomposition using CDBonn potential and the CPS scheme.
In order to obtain  accuracy beyond 3 digits, the total spin $j$ in the
subsystem of nucleon pair, needs to be $j$=4 (34ch).

We also included the Wigner spin rotation as outlined in
\cite{Kamada_ref23}. Thereby the
Balian-Brezin method\cite{Kamada_ref24} in handling the permutations is
quite useful. In Table~\ref{Kamada_tab4} the
triton binding energy is shown allowing charge independence breaking
(CIB) \cite{Kamada_ref26}
and Wigner spin rotations. Wigner spin rotation effects reduce the
binding energy by only about 2~keV.


%For tables use
\begin{table}
\caption{The relativistic (rel.) and nonrelativistic (nonrel) triton
binding energies in MeV obtained by MST scheme from different
nonrelativistic
potentials. The quantity (diff.) indicates the difference between the rel. and
nonrel. calculations}
\label{Kamada_tab1} % Give a unique label
% For LaTeX tables use
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
potential & rel. (MST) & nonrel. & diff. \\
\noalign{\smallskip}\hline\noalign{\smallskip}
RSC \cite{Kamada_refRSC} & -6.59 & -7.02 & 0.43 \\
AV18 \cite{Kamada_refAV18}& -7.23 & -7.66 & 0.43 \\
CD-Bonn \cite{Kamada_ref7} & -7.98 & -8.33 & 0.35 \\
Nijmegen II\cite{Kamada_refNijm}& -7.22 & -7.65 & 0.43 \\
Nijmegen I \cite{Kamada_refNijm}& -7.71 & -8.00 & 0.29 \\
Nijmegen93 \cite{Kamada_refNijm}& -7.46 & -7.76 & 0.30 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
%



%For tables use
\begin{table}
\caption{The relativistic (rel.) and nonrelativistic (nonrel) triton
binding energies in MeV obtained by CPS scheme from different
nonrelativistic
potentials. The quantity (diff.) indicates the difference between the rel. and 
nonrel. calculations}
\label{Kamada_tab2} % Give a unique label
% For LaTeX tables use
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
potential & rel. (CPS) & nonrel. & diff. \\
\noalign{\smallskip}\hline\noalign{\smallskip}
RSC \cite{Kamada_refRSC} & -6.97 & -7.02 & 0.05 \\
AV18 \cite{Kamada_refAV18} & -7.59 & -7.66 & 0.07 \\
CD-Bonn \cite{Kamada_ref7} & -8.22 & -8.33 & 0.11 \\
Nijmegen II \cite{Kamada_refNijm} & -7.58 & -7.65 & 0.07 \\
Nijmegen I \cite{Kamada_refNijm} & -7.90 & -8.00 & 0.10 \\
Nijmegen93 \cite{Kamada_refNijm} & -7.68 & -7.76 & 0.08 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
%

%For tables use
\begin{table}
\caption{The theoretical predictions of the trition binding energies
resulting from
the solutions of the nonrelativistic (first row) and relativistic
(second row) Faddeev
equations as function of the number of partial waves (ch) taken into
account.
The note of (S) means S-wave approximation as Tables \ref{Kamada_tab1}
and \ref{Kamada_tab2}.
The numbers inside of braket are the maximum of the total spin in the
subsystem.
The last
line indicates the absolute difference between the nonrelativistic and
relativistic
result. In the calculations only the np force of the CD-Bonn potential
was used.
Unit is in MeV.}
\label{Kamada_tab3} % Give a unique label
% For LaTeX tables use
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
& 5ch (S) & 18ch (2) & 26ch (3) & 34ch (4)\\
\noalign{\smallskip}\hline\noalign{\smallskip}
nonrel. & -8.331 & -8.220 & -8.241 & -8.247\\
rel. & -8.219 & -8.123 & -8.143 & -8.147\\
diff. & 0.112 & 0.107 & 0.098 & 0.100 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
%


%For tables use
\begin{table}
\caption{The theoretical predictions for the relativistic and
nonrelativistic triton
binding energies in MeV. All numbers are 34 channels results. The second
column is
the same as the last column in Table \ref{Kamada_tab3}. The results in
the third column take charge
dependence\cite{Kamada_ref26} into account. In addition the result of
the fourth column contains Wigner spin rotation effects.}
\label{Kamada_tab4} % Give a unique label
% For LaTeX tables use
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}
& np only & np+nn & Wigner rot. & diff. \\
\noalign{\smallskip}\hline\noalign{\smallskip}
nonrel. & -8.247 & -8.005 & - & -\\
rel. & -8.147 & -7.916 & -7.914 & -0.002\\
diff. & 0.100 & 0.089 & - & - \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
%

\section{Summary}
\label{Kamada_sec5}

A phase-shift equivalent 2N potential $v_r$ in the relativistic 2N
Schr\"odinger equation is related to the potential $v$ in the
nonrelativistic Schr\"odinger equation by the momentum scale
transformation scheme and the Coester-Pieper-Serduke scheme.  The
boosted potential $v_q$ is related to $v_r$ by Eq.(\ref{Kamada_eq20}).
With these potentials we generate the relativistic fully-off-shell
t-matrix $t_q$, which enters into the relativistic Faddeev equation.
We solve the relativistic bound state Faddeev equation and compare the
binding energy for the triton with the one obtained from a
nonrelativistic calculation with the same input interaction.

In the case of the CPS scheme we find that the difference
between the two calculations is only about 90~keV including CIB, where
the relativistic calculation gives a slightly reduced binding.  Taking
Wigner spin rotations into account in the relativistic calculation, the
the binding energy is reduced by a very small amount, 2 keV, indicating
that Wigner rotations of the spin have essentially no effect on the
predicted value of the binding energy.  Applications to the 3-body
continuum are in progress. Recently \cite{Kamada_ref23} the
formulation lined out above has been used to study the low energy Ay
puzzle in neutron-deuteron scattering.  In the intermediate energy
regime the formulation has been applied to exclusive proton-deuteron
scattering cross sections at 508 MeV \cite{Kamada_ref20,Kamada_ref21}
based on a formulation of the Faddeev equations which does not
employ a partial wave decomposition \cite{Kamada_ref25}. The approach
can also be extended and applied to electromagnetic
processes\cite{Kamada_ref27,Kamada_ref28}.

\section*{Acknowledgments}
This work was partially supported by the 2008-2011 polish science funds
as a
research project No. N N202 077435. It was also partially supported by
the Helmholtz
Association through funds provided to the virtual institute "Spin and strong
QCD"(VH-VI-231). ~~The numerical calculations  ~were performed on the IBM
Regatta
p690+ of the NIC in J\"ulich, Germany.




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\end{document}

% end of file fb19template.tex













