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\journalname{Few Body Systems}
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\begin{document}
\title{Constraints of cluster separability and covariance on current
operators \thanks{This work supported in part by the U.S. Department
of Energy, under contract DE-FG02-86ER40286.}
\thanks{Presented by W. N. Polyzou at LIGHTCONE 2011, 23 - 27 May, 2011, Dallas.
}
}
%\title{Insert your title here%\thanks{Grants or other notes
%about the article that should go on the front page should be
%placed here. General acknowledgments should be placed at the 
%end of the article.}
%}
%\subtitle{Do you have a subtitle?\\ If so, write it here}

\titlerunning{cluster separability}        % if too long for running head

\author{Polyzou, W. N. \and
Keister, B. D.
}

%\authorrunning{Short form of author list} % if too long for running head

\institute{W. N. Polyzou \at
Department of Physics and Astronomy,\\
The University of Iowa,\\ 
Iowa City, IA 52242, \\
              Tel.: 319-335-1856\\
              Fax: 319-335-1753\\
              \email{polyzou@uiowa.edu}           %  \\
%             \emph{Present address:} of F. Author  %  if needed
           \and
           B. D. Keister \at
           Division of Physics,\\
           National Science Foundation,\\ 
           Arlington, VA 22230, \\
           Tel.: 703-292-7377\\
           Fax: 703-292-9078\\
           \email{bkeister@nsf.gov}           %  \\
}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor

\maketitle

\begin{abstract}
Realistic models of hadronic systems should be defined by a dynamical
unitary representation of the Poincar\'e group that is also consistent
with cluster properties and a spectral condition.  All three of these
requirements constrain the structure of the interactions.  These
conditions can be satisfied in light-front quantum mechanics,
maintaining the advantage of having a kinematic subgroup of boosts and
translations tangent to a light front.  The most straightforward
construction of dynamical unitary representations of the Poincar\'e
group due to Bakamjian and Thomas fails to satisfy the cluster
condition for more than two particles.  Cluster properties can be
restored, at significant computational expense, using a recursive
method due to Sokolov.  In this work we report on an investigation of
the size of the corrections needed to restore cluster properties in
Bakamjian-Thomas models with a light-front kinematic symmetry.  Our
results suggest that for models based on nucleon and meson degrees of
freedom these corrections are too small to be experimentally observed.
\keywords{Relativistic quantum mechanics \and cluster properties \and light-front quantum mechanics}
\end{abstract}

%\section{Introduction}
%\label{intro}
%Your text comes here. Separate text sections with
\section{Introduction}
\label{sec:1}

Light-front quantum mechanics \cite{dirac49} has the desirable feature
that it is easy to construct realistic quantum mechanical models that
are exactly Poincar\'e invariant\cite{wigner39}, with a kinematic
subgroup that leaves a light front invariant.  The most
straightforward construction, due to Bakamjian and Thomas
\cite{bakamjian53}\cite{coester82}, achieves this by requiring both
the generators of the kinematic subgroup and the total light-front
spin to be non-dynamical.  A dynamical mass operator is defined by
adding interactions, that commute with the kinematic generators and
the kinematic light-front spin, to the non-interacting invariant mass.
The dynamical Poincar\'e generators are well-defined functions of this
mass operator and these kinematical operators.

While the original Bakamjian-Thomas (BT) construction was intended for
two-body systems, the construction gives a dynamical representation of
the Poincar\'e group for any number of particles.  However, while this
construction satisfies exact Poincar\'e invariance, the Poincar\'e
generators do not satisfy cluster properties for systems of more than
two particles.

Sokolov \cite{sokolov77} introduced a recursive construction that
starts with Bakamjian-Thomas two-body models and builds a many-body
unitary representation of the Poincar\'e group consistent with cluster
properties.  For the three-particle system the mass Casimir operator
operator for this unitary representation in the Sokolov construction 
is \cite{coester82}
\begin{equation}
M:=A (A_{(12)(3)}^{\dagger}M_{(12)\otimes (3)}A_{(12)(3)} +
A_{(23)(1)}^{\dagger}M_{(23)\otimes (1)}A_{(23)(1)} +
A_{(31)(2)}^{\dagger}M_{(31)\otimes (2)}A_{(31)(2)} 
- 2 M_0 ) A^{\dagger}     
\label{a.1}
\end{equation}
where 
\begin{equation}
A := \exp (\ln (A_{(12)(3)})+\ln (A_{(23)(1)})+\ln (A_{(31)(2)}))
\label{a.2}
\end{equation}
\begin{equation}
M^2_{(ij)\otimes (k)}:= 
(P^-_{ij}\otimes I_k + I_{ij}\otimes P^-_k) 
(P^+_{ij}\otimes I_k + I_{ij}\otimes P^+_k) - 
(\mathbf{P}_{\perp ij}\otimes I_k + I_{ij}\otimes \mathbf{P}_{\perp k})^2 
\label{a.3}
\end{equation}
\begin{equation}
P^-_{ij} := {M_{BT ij }^2 + \mathbf{P}_{\perp ij}^2 \over P^+_{ij} } 
\label{a.4}
\end{equation}
and $M_{BT ij }$ is the two-body Bakamjian-Thomas mass operator. 
The operators $A_{(ij)(k)}$ are $S$-matrix preserving unitary operators that
relate $M_{(ij)\otimes (k)}$ to the three-body $2+1$ Bakamjian-Thomas
mass operator $M_{BT(ij)(k)}$, so the mass operator in (\ref{a.1}) can be
expressed in terms of 2+1 Bakamjian-Thomas mass operators as
\begin{equation}
M= A M_{BT}  A^{\dagger} =    
A (M_{BT(12)(3)}+
M_{BT(23)(1)}+
M_{BT(31)(2)}
- 2 M_0 ) A^{\dagger} .    
\label{a.5}
\end{equation}
The existence of the operators $A_{(ij)(k)}$ that preserve the $2+1$
$S$ matrix and the light-front kinematic symmetry are ensured by a
theorem of Ekstein \cite{ekstein60}.  The dynamical Poincar\'e
generators are functions of the mass operator (\ref{a.1}), kinematic
Poincar\'e generators
\cite{Keister91}, \{ $\mathbf{E}_{0\perp},P_0^+,\mathbf{P}_{0\perp},
\hat{\mathbf{z}}\cdot\mathbf{J}_0,\hat{\mathbf{z}}\cdot\mathbf{K}_0\}$ 
and the kinematic light-front spin
$\mathbf{j}_0$,
\begin{equation}
P^- := {M^2 + \mathbf{P}_{0\perp}^2 \over P^+_0 } 
\label{a.6}
\end{equation}
\begin{equation}
\mathbf{J}_{\perp}:= 
{1 \over P_0^+} \left [
{(P_0^+ - P^-)\over 2} (\hat{\mathbf{z}}\times \mathbf{E}_{0\perp})
- (\hat{\mathbf{z}}\times \mathbf{P}_{0\perp})(\hat{\mathbf{z}}\cdot\mathbf{K}_0)
+ \mathbf{P}_{0\perp}(\hat{\mathbf{z}}\cdot\mathbf{j}_0)
+ M\mathbf{j}_{0\perp} 
\right ].
\label{a.7}
\end{equation}
These generators cluster into sums of tensor products in the limit
that the interactions between particle $i$ and the pair $(jk)$ are
turned off:
\begin{equation}
\mathbf{J}_{\perp} \to 
\mathbf{J}_{\perp(jk) }\otimes I_i +  I_{jk}
\otimes \mathbf{J}_{\perp i }  = 
A_{(jk)(i)} 
\mathbf{J}_{BT \perp (jk)(i)} A_{(jk)(i)}^{\dagger}  
\label{a.8}
\end{equation}
\begin{equation}
P^- \to 
P^- _{(jk) }\otimes I_i +  I_{jk}
\otimes P^-_{i }  = 
A_{(jk)(i)} 
P^-_{BT \perp (jk)(i)} A_{(jk)(i)}^{\dagger} . 
\label{a.9}
\end{equation}
These expression demonstrate that the operators 
$A_{(jk)(i)}$ are responsible for the failure of cluster properties
in the BT generators.  The relation (\ref{a.5}) shows that the mass operator 
(\ref{a.1}) gives the same $S$ matrix as the BT mass operator $M_{BT}$; however
this is no longer true for systems of four or more particles. 

Because of its complexity, there have been no dynamical few-body
calculations that utilize the Sokolov construction.  On the other
hand there have been many calculations based on the Bakamjian-Thomas
construction, some involving more than two particles.  The size of the
unitary transformations $A_{(ij)(k)}$ relative to the identity
provides an estimate of the corrections needed to restore
cluster properties in Bakamjian-Thomas models.

\section{Model and results}
\label{sec:2}

In this work we use a simple model to investigate how close the
unitary transformations, $A_{(ij)(k)}$, are to the identity.  The model
is a four-body model where an electron scatters off of a proton
(particle 3) in the presence of a bound state of a proton and neutron
(a deuteron consisting of particles 1 and 2).  We assume that the
deuteron does not interact with the struck proton or electron.  We
assume that the interaction of the proton with the electron can be
treated in the one-photon-exchange approximation.  As a final
simplification we assume the nucleons are spinless and replace the
four-vector current by a scalar one-body current, $j(x)$.

To study systems with nuclear physics scales we choose the 
two-body mass Casimir operator to be $M^2_{BT12}:= M_0^2 + 4mV$
\cite{cps}\cite{Keister91} where $V$ is a Malfliet-Tjon
\cite{malfliet69} nucleon-nucleon interaction, $M_0$ is the
non-interacting two-body invariant mass, and $m$ is the nucleon mass.
The three-nucleon mass operator (\ref{a.1}) for this model 
becomes
$M \to M_{(12)(3)} = A_{(12)(3)} M_{BT(12)(3)}A_{(12)(3)}^{\dagger}$.  
We calculate
\[
F_{TP}(p_3',p_3,p_{12}):=
\int \langle d, p_3' , p_{12}' \vert J(0) 
\vert d, p_3, p_{12} \rangle dp_{12}' 
\]
\begin{equation}
= \int {} _{BT}\langle d, p_3' , p_{12}' \vert A_{(12)(3)}^{\dagger} J(0) 
A_{(12)(3)} \vert d, p_3, p_{12} \rangle_{BT} dp_{12}'
\label{b.1}
\end{equation} 
and compare it the corresponding Bakamjian Thomas quantity 
\begin{equation}
F_{BT}(p_3',p_3,p_{12}):=
\int {} _{BT}\langle d, p_3' , p_{12}' \vert J(0) 
\vert d, p_3, p_{12} \rangle_{BT}  dp_{12}'.
\label{b.2}
\end{equation}
The first integral is the form factor for nucleon 3, it is independent
of $\mathbf{p}_{12}$ as expected by cluster properties.  The second
integral treats the $2+1$ three-nucleon system as a Bakamjian-Thomas
model.  This expression violates cluster properties because it has a
non-physical dependence on $\mathbf{p}_{12}$.  In addition the result
is also sensitive to mass of the deuteron and the momentum dependence
of the deuteron wave function.

In figures 1 and 2 we plot the difference $(F_{TP}-F_{BT})/F_{TP}$ as
a function of $Q= P_3'-P_3$ and $p_{12}$.  We consider frame where the $+$
component of the momentum transfer is zero, which is always possible
for spacelike momentum transfers.  We choose the light front $z+t=0$,
assume that the momentum transfer $Q$ is in the $x$ direction and
investigate the dependence on $p_{12}$ in the $x$ (parallel) or $y$
(perp) directions.

The results are shown in figures 1 and 2 and expressed as fractional
differences between Bakamjian-Thomas and tensor-product calculations.
Deviations from zero therefore exhibit the unphysical dependence upon
$Q$ and $p_{12}$ in the 
Bakamjian-Thomas case.  The figures show, for this model,  which uses
parameters that have scales expected in nuclear physics models with
meson-exchange interactions, the size of the corrections needed to
restore cluster properties is too small to be measured in laboratory
experiments.  This investigation suggests that it is reasonable 
to construct light-front quantum mechanical models of few-nucleon 
systems using only the Bakamjian-Thomas representation of the Poincar\'e 
group, without including the corrections due to the Sokolov operators.  

Had we instead constructed our deuteron out of sub-nuclear degrees 
of freedom involving stronger binding and larger internal momenta,
the size of these corrections could be large enough to be observable.

% For one-column wide figures use
\begin{figure}
\centering
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use
\includegraphics{fig1.eps}
% figure caption is below the figure
\caption{Front form - $p_{12}\perp Q$}
\label{fig:1}       % Give a unique label
\end{figure}
%
% For one-column wide figures use
\begin{figure}
\centering
% Use the relevant command to insert your figure file.
% For example, with the graphicx package use
\includegraphics{fig2.eps}
% figure caption is below the figure
\caption{Front form - $p_{12}|| Q$}
\label{fig:2}       % Give a unique label
\end{figure}


% Non-BibTeX users please use
\begin{thebibliography}{99}
%
% and use \bibitem to create references. Consult the Instructions
% for authors for reference list style.
%
% Format for Journal Reference

\bibitem{dirac49} P. A. M. Dirac, Rev. Mod. Phys. {\bf 21},392(1949).

\bibitem{wigner39} E. P.~Wigner, Ann. Math. C {\bf 40}, 149 (1939).

\bibitem{bakamjian53} B. Bakamjian and L. H. Thomas, Phys. Rev. {\bf 92},1300(1953).

\bibitem{coester82}
  F. Coester and W. N. Polyzou, Phys. Rev. D. {\bf 26},1348(1982).

\bibitem{sokolov77}
  S. N. Sokolov, Dokl. Akad. Nauk. SSSR  {\bf 233},575(1977).

\bibitem{ekstein60} H. Ekstein, Phys. Rev. {\bf 117},1590(1960).

\bibitem{Keister91}
  B. D. Keister and W. N. Polyzou,
  Advances in Nuclear Physics, Volume 20,
  Ed. J. W. Negele and E. W. Vogt, Plenum Press 1991.

\bibitem{cps} F. Coester, S. C. Pieper and F. J. D Serduke, 
Phys. Rev. C{\bf 11},1(1975).

\bibitem{malfliet69}
  R. A. Malfliet and J. A. Tjon,
  Nucl. Phys. {\bf A127},161(1969).

\end{thebibliography}
\end{document}


  For the comparison we integrate both of these expressions over 
the final $p'_{12}$ and study the dependence this difference on 
$ p_3, p_{12}, p_3'$ and he momentum transferred 
to particle 3.  We choose axes so the $+$-component of the momentum
transfer to the current is zero, which is the preferred frame for
electron scattering calculations. 



$\langle \Psi'_{TP} \vert J (0) \vert \Psi_{TP} \rangle = 
\langle \Psi_{BT}' A^{\dagger}_{(12)(3)} 
\vert J (0) \vert A_{(12)(3)} \Psi_{BT} \rangle$ which we compare to 
$\langle \Psi_{BT}'  
\vert J (0) \vert \Psi_{BT} \rangle$
where $\vert \Psi \rangle$ and $\vert \Psi_{BT} \rangle$ are four 
momentum eigenstates of the tensor product or BT generators.
The difference is due to the operators $A$ that restore cluster 
properties to the BT representation.

The virtue of this model is that the operator $A_{(12)(3)}$ does not
have to be explicitly computed. This is because  plane-wave eigenstates 
$\vert \Psi \rangle$ and $\vert \Psi_{BT} \rangle$ are related by
replacing a delta function the light front momentum of particle 3 by a
delta function in the light-front momentum of particle 3 in the rest
frame of the non-interacting three-body system.  


The Bakamjian-Thomas and tensor product initial and final states are related by
\begin{equation}
A \vert d, p_3, p_{12} \rangle_{bt} =\vert d, p_3, p_{12} \rangle_{tp},
\label{1}
\end{equation}
where $A=A_{(12)(3)}$, $bt$ stands for the 
Bakamjain-Thomas states and $tp$ stands for the tensor product states.
For both models we calculate the quantity 
\begin{equation}
F_x(p_3',p_3,p_{12}):=
\int{} _x\langle d, p_3' , p_{12}' \vert J(0) \vert d, p_3, p_{12} \rangle_x
dp_{12}' 
\label{2}
\end{equation}
where $x \in \{bt,tp\}$.  We use the same variables in both models,
even though the variables used in eq.~(\ref{2}) are not the most natural
choice in the Bakamjian-Thomas model. 

In the tensor product model $F_{tp} (\cdots
)$ is independent of $p_{12}$ and only depends on the momentum
transfer $Q=p_3'-p_3$.  This is the expected behavior.  In the
Bakamjian-Thomas model we find a non-trivial dependence on $p_{12}$.
Because of equation (\ref{1}) we can express the correct tensor 
product result in terms of the Bakamjian-Thomas states and the 
unitary operators $A$:
\begin{equation}
F_{tp}(p_3',p_3,p_{12}):=
\int {}_{bt}\langle d, p_3' , p_{12}' \vert A^{\dagger} J(0) 
A \vert d, p_3, p_{12} \rangle_{bt}
dp_{12}' .
\label{3}
\end{equation}
In the limit $A \to I$ this becomes the Bakamjian-Thomas result.

%\begin{figure}
%\begin{minipage}[t]{7.3cm}
%\begin{center}
%\includegraphics[width=7.0cm,clip]{fig_1.eps}
%\caption[Q $\perp$ Q]{\label{labelFig1}                          
%Front form  - $p_{12}\perp Q$ %  
%}
%\end{center}
%\label{fig.1}
%\end{minipage}
%\hfill
%\begin{minipage}[t]{7.3cm}
%\begin{center}
%\includegraphics[width=7.0cm,clip]{fig_2.eps}
%\caption[Q $\perp$ Q]{\label{labelFig2}                          
%Front form  - $p_{12} \Vert  Q$   
%}
%\end{center}
%\end{minipage}
%\label{fig.2}
%\end{figure}

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\centering
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\caption{Front form - $p_{12}\perp Q$}
\label{fig:1}       % Give a unique label
\end{figure}
%
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%
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%Citations \cite{Ref1} and \cite{Ref2} and \cite{Ref3}.
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