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\begin{document}

\title{Exchange current contributions in null-plane quantum models of 
elastic electron deuteron scattering}


\author{Y. Huang}
\affiliation{
Department of Physics and Astronomy, The University of Iowa, Iowa City, IA
52242}

\author{W. N. Polyzou}
\affiliation{
Department of Physics and Astronomy, The University of Iowa, Iowa City, IA
52242}


\vspace{10mm}
\date{\today}

\begin{abstract}
We investigate exchange current contributions to elastic
electron-deuteron scattering using exactly Poincar\'e invariant
quantum mechanics with a null-plane kinematic symmetry.  Our model
exchange current is motivated by one-pion-exchange physics.  Exact
current conservation and current covariance are satisfied by using
this current to compute a linearly independent set of current matrix
elements.  The remaining current matrix elements are generated from
the independent matrix elements using the covariance and current
conservation constraints.  The presence of the exchange current
increases the sensitivity to the choice of independent current matrix
elements.  Two choices of independent matrix elements that have
distinct motivations lead to a good description of all of the elastic
scattering observables for momentum transfers below 8 (GeV)$^2$.
\end{abstract}

\vspace{10mm}

\pacs{11.80.-m,13.40.Gp,21.45.Bc,25.30.Bf}

\maketitle


%\newpage

%****************************************************************************

%\narrowtext

%****************************************************************************

\section{Introduction}

The scattering of electrons from nuclei in the impulse approximation
is based on the assumptions that the cross section can be accurately
calculated in the one-photon-exchange approximation and the nuclear
current operator is a sum of single-nucleon current operators.
Unfortunately, the dynamical constraints imposed by current
conservation and current covariance cannot be satisfied with one-body
current operators. In addition to the dynamical contributions to the
current required by covariance and current conservation there are
dynamical contributions to the nuclear current operator due to the
exchange of charged particles and eliminated degrees of freedom.  All
of these effects require that the nuclear current operator has
non-vanishing many-body contributions.  Because the constraints on the
current operator are dynamical, the structure of the current operator
is representation dependent.

In a theory satisfying cluster separability the current operator has a
well-defined cluster decomposition.  Single-nucleon current operators
and cluster properties fix the one-body contribution to the current
operator.  The additional many-body contributions to the current
operator are not as well understood.  The result is that the
interpretation of electron-scattering experiments involves theoretical
ambiguities that must be resolved in order to use electron scattering
as a precision tool for understanding nuclear structure and dynamics.

Elastic electron-deuteron scattering is the simplest reaction where
these questions arise \cite{ostebee75}.  In this paper we
investigate two-body current operators in a formulation of the
dynamics based on Poincar\'e-invariant quantum mechanics with a
null-plane kinematic symmetry\cite{Leutwyler78}
\cite{Coester82}\cite{Keister91}. We investigate the one-pion exchange
contribution to the current.  The long-range part of realistic
nucleon-nucleon interactions includes a one-pion-exchange
contribution.  There is an additional short-range part that is modeled
to fit measured nucleon-nucleon scattering cross sections.  Our model exchange
current is constructed using the same long-range pion-exchange
interaction that is used in model two-body interactions.  In this work
we demonstrate that the addition of this two-body current operator to
the impulse current operator provides an excellent description of all
of the elastic electron-deuteron observables over a wide range of
momentum transfers.

There are several theoretical approaches for calculating elastic
electron-deuteron scattering observables.  In all cases the problem is
to construct matrix elements of a conserved covariant current operator
\beq
\langle P+Q, \mu', d \vert I^{\nu}(0) \vert
P , \mu, d \rangle
\label{a.1}
\eeq
for a range of space-like momentum transfers, $Q$, where $P$ and $P+Q$
are the initial and final deuteron four momenta.  The input to
equation (\ref{a.1}) is the deuteron eigenstates, a dynamical
representation of the Lorentz group to transform the deuteron
eigenstates to different frames related by the four-momentum transfer,
and a current operator for the two-nucleon system.  For the deuteron
the current is a sum of one-body and two-body operators.  The one-body
current operators are fixed by covariance, current conservation, and
nucleon form factors.  Elastic electron-deuteron scattering has been
studied by many authors \cite{Gross65a}\cite{Terent77}
\cite{Chung88}\cite{Schiavilla91} \cite{VanOrden95}\cite{Phillips98}
\cite{Karmanov99} \cite{Arenh00}\cite{Lev00}\cite{Allen01}
\cite{VanOrden01}\cite{Phillips05}\cite{kruto06} using a variety of
different methods and assumptions.  These have been reviewed elsewhere
\cite{VanOrden01} \cite{Gilman02}.

While true impulse approximations, that replace the current in
(\ref{a.1}) by a sum of single-nucleon currents, necessarily violate
current conservation and current covariance, there are ``generalized
impulse approximations'' in each formalism that use the sum of
single-nucleon current matrix elements as input to generate
approximations that enforce current conservation and current
covariance.  These ``generalized impulse approximations'' typically
provide a qualitative description of all three elastic
electron-deuteron scattering observables, however a quantitative
agreement requires additional corrections.  Such an agreement has been
achieved in some formalisms \cite{VanOrden95}\cite{Schiavilla91}. The
presence of a one-pion exchange ``pair'' current \cite{Kloet75} or a
related ``contact current'' \cite{Riska89_2} is a common element in
corrections that explain a good part of the difference between the
``generalized impulse approximations'' and experiment.

In this paper we construct an exactly Poincar\'e invariant dynamical
model of the two-nucleon system and a model of a covariant
one-pion-exchange current operator.  The dynamical representation of
the Poincar\'e group is constructed using precision nucleon-nucleon
interactions \cite{Wiringa95} \cite{cdb} that are fit to scattering
data.  Current conservation, current covariance and discrete
symmetries are used to identify linearly independent current matrix
elements.  The one-body contributions to the linearly {\it
independent} current matrix elements are fixed by cluster
properties.  Our model exchange current is added to these independent
current matrix elements.  The remaining current matrix elements are
generated using the dynamical constraints of current conservation and
current covariance.

The Poincar\'e invariant dynamical models in this paper are formulated
in a representation with a null-plane kinematic symmetry
\cite{Leutwyler78} \cite{Coester82}\cite{Keister91}.  Unlike models
based on null-plane quantum field theory
\cite{Chang73}\cite{Chang73b}, where few-degree of freedom
approximations break rotational covariance, the class of quantum
mechanical models that we consider are {\it exactly} rotationally
invariant.
The null-plane treatment of electron
scattering has a number of notable advantages:

\begin{itemize}

\item [1.] For electron scattering the momentum transfer, $Q$, is
  spacelike, so the orientation of the null plane can be chosen so
  the $+$-component, $Q^+:= Q^0+Q^3$, of the momentum transfer is
  zero.  When $Q^+=0$ all current matrix elements can be constructed
  from a maximal independent set of matrix elements of the $+$
  component, $I^+ (0) :=I^0 (0)+ I^3 (0)$, of the current:

\beq
\langle \tilde{\mathbf{P}}', \mu',d \vert I^+(0) \vert 
\tilde{\mathbf{P}} ,\mu,d \rangle
\qquad \tilde{\mathbf{P}} := (P^+,P_1,P_2).   
\label{a.2}
\eeq

\item [2.]  There is a three-parameter subgroup of Lorentz boosts that
  leaves the null-plane invariant.  Since these boosts form a
  subgroup, boosting does not lead to Wigner rotations.  It follows
  that matrix elements of $I^+(0)$ with $Q^+=0$ and suitably
  normalized initial and final states are independent of frames
  related by null-plane boosts:

\beq
\langle \tilde{\mathbf{P}}''', \mu',d \vert I^+(0) \vert 
\tilde{\mathbf{P}}'' ,\mu, d\rangle  
= 
\langle \tilde{\mathbf{P}}', \mu',d \vert I^+(0) \vert \tilde{\mathbf{P}} ,\mu,d \rangle . 
\label{a.3}
\eeq
\item[] This means that matrix elements of $I^+(0)$ are equal to 
Breit-frame matrix elements with the same null-plane spin magnetic quantum 
numbers.

\item [3.]  A consequence of property two is that the {\it on-shell
  one-body} contributions to matrix elements of $I^+(0)$ {\it exactly}
  factor out\cite{Keister91} of the corresponding nuclear current
  matrix elements.
\[
\langle \Psi, \tilde{\mathbf{P}}+\tilde{\mathbf{Q}} 
\vert I_i^+(0) \vert \Psi, \tilde{\mathbf{P}} \rangle =
\]
\beq
\langle \tilde{\mathbf{p}}_{i0}+\tilde{\mathbf{Q}} \vert 
I_i^+ (0) \vert \tilde{\mathbf{p}}_{i0} \rangle 
\int  d\tilde{\mathbf{p}}_1 \cdots d\tilde{\mathbf{p}}_n
\langle \Psi, \tilde{\mathbf{P}}+\tilde{\mathbf{Q}} \vert
\tilde{\mathbf{p}}_1 \cdots \tilde{\mathbf{p}}_i
+\tilde{\mathbf{Q}}
\cdots
\tilde{\mathbf{p}}_n \rangle 
\langle 
\tilde{\mathbf{p}}_1 \cdots \tilde{\mathbf{p}}_i
\cdots
\tilde{\mathbf{p}}_n \vert
\Psi, \tilde{\mathbf{P}} 
\rangle .
\label{a.4}
\eeq
\item[] This means that the matrix elements of the one-body 
contributions to $I^+(0)$ can be expressed as sums of products of
Breit-frame nucleon matrix elements multiplied by functions of 
$Q^2$ that only depend on the initial and final wave functions.

\end{itemize} 

This formalism has the advantage that it is possible to satisfy exact
rotational covariance and the impulse contribution to the independent
current matrix elements involves only experimentally observable
on-shell single-nucleon matrix elements.
 

\section{Null-plane kinematics}

In this section we introduce our notation and discuss the null-plane
kinematic subgroup.  The null plane is the three-dimensional
hyperplane of points tangent to the light cone satisfying the
condition
\beq
\{ x \vert x^+ := t + \mathbf{x} \cdot \hat{\mathbf{e}}_3 = 0 \} .
\label{b.1}
\eeq
The null-plane components of the four vector $x^{\mu}$ are 
\beq
x^{\pm} :=  t \pm \mathbf{x} \cdot \hat{\mathbf{e}}_3
\qquad 
\mathbf{x}_{\perp} = (\mathbf{x} \cdot \hat{\mathbf{e}}_1,
\mathbf{x} \cdot \hat{\mathbf{e}}_2).
\label{b.2}
\eeq
Four vectors $x^{\mu}$ can be represented by $2\times 2$ 
Hermitian matrices in the null-plane components of $x^{\mu}$:
\beq
X = \left (
\begin{array}{cc} 
x^+ & x_{\perp}^* \\  
x_{\perp} & x^-
\end{array}
\right ) = x^{\mu} \sigma_{\mu} \qquad
x_{\perp}=x_1 + i x_2 \qquad x^{\mu} = {1 \over 2} \mbox{Tr}(\sigma_{\mu} X)
\label{b.3}
\eeq
where $\sigma_{\mu}= (I, \bsig )$ and $\bsig$ are the two $\times$ two
Pauli matrices.  Since the determinant of $X$ is minus the square of the
proper time, $-x^2$, if $\Lambda$ is a complex $2\times 2$ matrix with
unit determinant then the transformation
\beq
X\to X' = \Lambda X \Lambda^{\dagger} 
\label{b.4}
\eeq 
defines a real Lorentz transformation. 
\beq
\Lambda^{\mu}{}_{\nu} = 
{1 \over 2}\mbox{Tr}(\sigma_{\mu}\Lambda \sigma_{\nu} \Lambda^{\dagger} ).
%\qquad 
%\Lambda^{\mu}{}_{\nu} \leftrightarrow \Lambda, - \Laambda
\label{b.5}
\eeq
Points on the null-plane can be represented by 
triangular matrices of the form
\beq
X = \left (
\begin{array}{cc} 
0 & x_{\perp}^*  \\  
x_{\perp} & x^-
\end{array}
\right ) . 
\label{b.6}
\eeq
Poincar\'e transformations
\beq
X \to X' = \Lambda X \Lambda^{\dagger} +A  
\label{b.7}
\eeq
that leave the null plane invariant (i.e. preserve the form  of
(\ref{b.6})) have the form 
\beq
\Lambda  = \left (
\begin{array}{cc} 
\alpha  & 0  \\  
\beta & 1/\alpha
\end{array}
\right ) 
\qquad 
A  = \left (
\begin{array}{cc} 
0  & a_{\perp}^*  \\  
a_{\perp} & a^-
\end{array}
\right ), 
\label{b.8}
\eeq
where $\alpha \not= 0$, $\beta$, and $a_{\perp}$ are complex and $a^-$
is real.  These transformations form a seven-parameter subgroup of the
Poincar\'e group that leaves the null plane invariant.  This subgroup
is called the {\it kinematic subgroup} of the null plane.  This
subgroup includes the three-parameter subgroup of translations on the
null plane, a three-parameter subgroup of null plane preserving
boosts as well as rotations about the $3$-axis.

The null plane-preserving boost that transforms a rest 4-momentum 
$(m,\mathbf{0})$ to a value $p$ are matrix valued functions of 
$\tilde{\mathbf{v}}=\tilde{\mathbf{p}}/m$ given by
\beq
\Lambda_f (\tilde{\mathbf{p}}/m) := \Lambda_f (\tilde{\mathbf{v}}) :=
\left (
\begin{array}{cc} 
\sqrt{v^+}  & 0  \\  
v_{\perp}/\sqrt{v^+}  & 1/\sqrt{v^+}
\end{array}
\right ) .
\label{b.9}
%\label{b.10}
\eeq
Since the reality of $\alpha$ is preserved under matrix
multiplication, the null-plane boosts form a {\it subgroup} of the
Lorentz group.

The action of a null-plane boost on an arbitrary four momentum vector is
determined by the transformation properties of the $+$ and $\perp$
components of the four momentum
\beq
p^+ \to p^{+ \prime} = v^+ p^+ 
\qquad 
\mathbf{p}_{\perp} \to \mathbf{p}_{\perp}' + \mathbf{v}_{\perp} p^+ .
\label{b.10}
%\label{b.12}
\eeq
Since $p^-$ does not appear in (\ref{b.10}) the three components
$\tilde{\mathbf{p}}:= (p^+, \mathbf{p}_{\perp})$ are called a
``null-plane vector''.  The $-$ component can be calculated using the
mass-shell condition
\beq
p^- = {m^2 + \mathbf{p}_{\perp}^2 \over p^+} .
\label{b.11}
%\label{b.14}
\eeq
The null-plane spin of a particle of mass $m$ is defined
\cite{Polyzou89}\cite{Keister91} 
so that (1) it agrees with ordinary canonical spin in
the particle's rest frame and (2) is invariant with respect to
null-plane boosts, (\ref{b.9}).

The null-plane representation of the single-nucleon Hilbert space is
the space of square integrable functions of the null-plane vector
components of the particle's four momentum and the 3-component of its
null-plane spin:
\beq
\psi (\tilde{\mathbf{p}}, \mu) =  
\langle \tilde{\mathbf{p}}, \mu \vert \psi \rangle 
\qquad
\int_0^\infty dp^+ \int_{\mathbb{R}^2} 
d\mathbf{p}_{\perp} \sum_{\mu=-j}^j  
\vert \psi (\tilde{\mathbf{p}}, \mu) \vert^2 < \infty .
\label{b.12}
%\label{b.15}
\eeq
The Poincar\'e group acts irreducibly on single-nucleon states 
\[
\langle \tilde{\mathbf{p}}, \mu \vert U_1(\Lambda ,A) \vert \psi \rangle =
\]
\beq
\int_0^\infty dp^{+\prime} \int_{\mathbb{R}^2} 
d\mathbf{p}'_{\perp} \sum_{\mu'=-j}^j  
{\cal D}^{m,j}_{\tilde{\mathbf{p}},\mu; \tilde{\mathbf{p}}',\mu'}[\Lambda ,A]
\langle 
\tilde{\mathbf{p}}', \mu' \vert \psi \rangle ,
\label{b.13}
%\label{b.16}
\eeq
where the Poincar\'e group Wigner-${\cal D}$ function in the null-plane
irreducible basis \cite{Keister91} is
\[
{\cal D}^{m,j}_{\tilde{\mathbf{p}},\mu; \tilde{\mathbf{p}}',\mu'}
[\Lambda,A]:=  
\langle \tilde{\mathbf{p}}, \mu \vert U(\Lambda ,a) 
\vert \tilde{\mathbf{p}}', \mu' \rangle 
=
\]
\beq
\delta (\tilde{\mathbf{p}}  - \tilde{\bLam}(p') ) 
\sqrt{{p^+ \over p^{+\prime} }}
D^{j}_{\mu \mu'} [\Lambda_f^{-1} (\tilde{\mathbf{p}}/m) \Lambda 
\Lambda_f(\tilde{\mathbf{p}}'/m) ]
e^{i p \cdot a} 
\label{b.14}
%\label{b.17}
\eeq
where $D^{j}_{\mu \mu'}(R)$ is the ordinary $SU(2)$ Wigner $D$-function.

Our model Hilbert space for the two-nucleon system is the tensor product
of two single-nucleon Hilbert spaces.  The tensor product of two single-nucleon
representations of the Poincar\'e group defines the kinematic
representation of the Poincar\'e group on the two-nucleon Hilbert
space.
\beq
{\cal H}={\cal H}_1 \otimes {\cal H}_1
\qquad 
U_0 (\Lambda,A) :=   U_1 (\Lambda,A) \otimes U_1 (\Lambda,A).
\label{b.15}
\eeq
The tensor product of two one-body irreducible representations of the
Poincar\'e group is reducible.  It can be decomposed into an
orthogonal linear superposition of irreducible representations using
the Poincar\'e group Clebsch-Gordan coefficients in the null-plane
basis:
\beq
\Psi (
\tilde{\mathbf{p}}_1, \mu_1 , \tilde{\mathbf{p}}_2, \mu_2 ) 
= 
\sum \int 
\langle 
m_1, j_1, \tilde{\mathbf{p}}_1, \mu_1 ;m_2, j_2 , \tilde{\mathbf{p}}_2, 
\mu_2   
\vert k, j (l,s) \tilde{\mathbf{P}},\mu \rangle
d\tilde{\mathbf{P}} k^2 dk 
\Psi (
m,j, \tilde{\mathbf{P}}, \mu , l,s  ) 
\label{b.16}
%\label{b.19}
\eeq
where the Poincar\'e group Clebsch-Gordan coefficient \cite{Keister91}
in the null-plane basis is 
\[
\langle 
m_1, j_1, \tilde{\mathbf{p}}_1, \mu_1 ;m_2, j_2 , \tilde{\mathbf{p}}_2, \mu_2   
\vert k, j (l,s) \tilde{\mathbf{P}},\mu \rangle =
\]
\[
\delta ( \tilde{\mathbf{P}} - \tilde{\mathbf{p}}_1- \tilde{\mathbf{p}}_2)
{\delta (k - k(\tilde{\mathbf{p}}_1, \tilde{\mathbf{p}}_2))
\over k^2} 
\vert \sqrt{{P^+ \omega_1(\mathbf{k}) \omega_2(\mathbf{k})  \over
(\omega_1(\mathbf{k})+ \omega_2(\mathbf{k}))p_1^+ p_2^+ }}
\times 
\]
\[
\sum Y_{lm}(\hat{\mathbf{k}}(\tilde{\mathbf{p}}_1, \tilde{\mathbf{p}}_2)) 
D^{1/2}_{\mu_1, \mu_1'} [\Lambda_f^{-1} (\mathbf{k}/m_1)
\Lambda_c (\mathbf{k}/m_1)]
D^{1/2}_{\mu_2, \mu_2'} [\Lambda_f^{-1} (-\mathbf{k}/m_2)
\Lambda_c (-\mathbf{k}/m_2)] \times
\]
\beq
\langle {1 \over 2}, \mu_1', {1 \over 2}, \mu_2' \vert 
s, \mu_s \rangle 
\langle l, m, s, \mu_s \vert 
j, \mu \rangle  
\label{b.17}
%\label{b.20}
\eeq
where $(\omega(\mathbf{k}), \mathbf{k}) = k(p_1,p_2) = \Lambda_f^{-1} p_1$. 
In (\ref{b.17}) the two-body invariant mass $M_0$,
which has continuous spectrum, is replaced by 
$\mathbf{k}^2$ which is related to $M_0$ by
\beq
M_0=2 \sqrt{\mathbf{k}^2+ m^2}.
\label{b.18}
\eeq
The transformations $\Lambda_f ({\mathbf{k}}/m)$ and 
$\Lambda_c (\mathbf{k}/m)$ are null-plane, (\ref{b.9}), and 
canonical boosts,
\beq
\Lambda_c ({\mathbf{k}}/m) = e^{{1 \over 2} \brho \cdot \bsig},
\label{b.19}
\eeq
where $\brho$ is the rapidity 
\beq
\brho = \hat{\mathbf{k}} \vert \brho \vert \qquad  
\cosh (\vert \brho \vert ) = \sqrt{\mathbf{k}^2+ m^2}/m 
\qquad \sinh (\vert \brho \vert ) = \mathbf{k}/m .
\label{b.20}
\eeq
The combination 
\beq
\Lambda_f^{-1} (\pm\mathbf{k}/m)
\Lambda_c (\pm\mathbf{k}/m)
\label{b.21}
%\label{b.24}
\eeq 
is a rotation, called a Melosh rotation
\cite{Melosh74}\cite{Keister91} that transforms the null-plane spins
so they rotate under a single representation of $SU(2)$ which allows
them to be coupled using ordinary $SU(2)$ Clebsch-Gordan coefficients.
The absence of Wigner rotations in (\ref{b.17}) is a consequence 
of the null-plane boosts forming a subgroup.

In this basis two-nucleon wave functions have the form
\beq
\psi ( k, j (l,s) \tilde{\mathbf{P}},\mu ) =  
\langle  k, j (l,s) \tilde{\mathbf{P}},\mu \vert \psi \rangle 
\qquad
\int_0^\infty dP^+ \int_{\mathbb{R}^2} 
d\mathbf{P}_{\perp} k^2 dk \sum_{\mu=-j}^j \sum_{s=0}^1 
\sum_{l= \vert j-s\vert}^{\vert j+s \vert}      
\vert \psi (( k, j (l,s) \tilde{\mathbf{P}},\mu ) \vert^2 < \infty
\label{b.25}
%\label{b.5}
\eeq
and $U_0(\Lambda,A)$ acts irreducibly on these states: 
\[
\langle  k, j (l,s) \tilde{\mathbf{P}},\mu  \vert U_0(\Lambda ,A) \vert \psi \rangle =
\]
\beq
\int_0^\infty dP^{+\prime} \int_{\mathbb{R}^2} 
d\mathbf{P}'_{\perp} \sum_{\mu'=-j}^j  
{\cal D} ^{M_0(k),j}_{\tilde{\mathbf{P}},\mu; \tilde{\mathbf{P}}',\mu'}[\Lambda ,A]
\langle 
k, j (l,s) \tilde{\mathbf{P}}',\mu'  \vert \psi \rangle .
\label{b.26}
\eeq
This basis is used in the formulation of our 
dynamical model in the next section.

\section{dynamics}

Dynamical models of the two nucleon system in Poincar\'e invariant
quantum mechanics are defined by a dynamical unitary representation of
the Poincar\'e group acting on two-nucleon Hilbert space.  The mass
Casimir operator for this representation can be defined by adding a
realistic nucleon-nucleon interaction $v_{nn}$
\cite{Coester75b}\cite{Keister91} to the square of the non-interacting
two-nucleon mass operator as follows:
\beq
M^2 = M_0^2 + 4m v_{nn}
\label{c.1}
\eeq
where for a null-plane dynamics $v_{nn}$ has the form
\beq
\langle 
k', j', (l',s') \tilde{\mathbf{P}}',\mu' \vert v_{nn}
\vert k, j, (l,s) \tilde{\mathbf{P}},\mu \rangle =
\delta (\tilde{\mathbf{P}}'- \tilde{\mathbf{P}})
\delta_{j'j} \delta_{\mu'\mu} 
\langle k',l',s' \Vert v^j \Vert k,l,s \rangle
\label{c.2}
\eeq
in the non-interacting irreducible basis (\ref{b.17}).
The interaction $v_{nn}$ is restricted so $M^2$ is a positive operator.

Simultaneous eigenstates of $M^2$, $j^2$ ,$\tilde{\mathbf{P}}$,
$j_z$  in the non-interacting irreducible basis 
have the form
\beq
\langle 
k', j', (l',s') \tilde{\mathbf{P}}',\mu' \vert 
\lambda, j ,\tilde{\mathbf{P}},\mu  \rangle = 
\delta (\tilde{\mathbf{P}}'- \tilde{\mathbf{P}})
\delta_{j'j} \delta_{\mu'\mu} \phi_{\lambda,j}  (k,l,s)
\label{c.3}
\eeq
where the wave function $\phi_{\lambda,j} (k,l,s)$ is a solution of
the eigenvalue equation
\beq
(4k^2 + 4m^2 -\lambda^2) \phi_{\lambda,j}  (k,l,s) =
\sum_{s=0}^1 \sum_{l=\vert j-s \vert }^{l=\vert j+s \vert }  
\int_0^{\infty} 
4m \langle k,l,s \Vert v_{nn}^j \Vert k',l',s' \rangle
k^{\prime 2}dk' 
\phi_{\lambda,j}  (k',l',s') .
\label{c.4}
\eeq
The eigenfunctions, $\phi_{\lambda,j} (k',l',s')$, of this mass
operator are also solutions to the non-relativistic Schr\"odinger
equation with interaction $v_{nn}$, which can be seen by dividing both
sides of (\ref{c.4}) by $4m$.  The deuteron is an even parity bound
state with $j=s=1$ and parity limits the $l$ sum in (\ref{c.4}) to
$l\in \{ 0,2\}$.

If $\{M^2,j,\tilde{\mathbf{P}},\mu\}$ have the same interpretation as
$\{M_0^2,j,\tilde{\mathbf{P}},\mu\}$, then it follows that the
eigenstates (\ref{c.4}) transform irreducibly with respect to a {\it
  dynamical} representation of the Poincar\'e group defined by
replacing the eigenvalues of $M_0$ by the eigenvalues $\lambda$ of
$M$:

\[
\langle 
k', j', (l',s') \tilde{\mathbf{P}}',\mu' \vert U(\Lambda ,A) \vert  
\lambda, j ,\tilde{\mathbf{P}},\mu  \rangle =
\]
\[
\int \sum_{\mu''=-j}^j 
\langle 
k', j', (l',s') \tilde{\mathbf{P}}',\mu' \vert 
\lambda, j ,\tilde{\mathbf{P}}'',\mu'' \rangle d\tilde{\mathbf{P}}''
\langle \lambda, j ,\tilde{\mathbf{P}}'',\mu''  \vert
U(\Lambda ,A) \vert  
\lambda, j ,\tilde{\mathbf{P}},\mu  \rangle =
\]
\beq
\phi_{\lambda,j} (k',l',s')  
{\cal D}^{\lambda,j'}_{\tilde{\mathbf{P}}',\mu'; \tilde{\mathbf{P}},\mu}
[\Lambda,A] .
\label{c.5}
\eeq
Since the eigenstates (\ref{c.3}) are complete, $U(\Lambda ,A)$,
defined by (\ref{c.5}), 
is a dynamical unitary representation of the Poincar\'e group on the
two-nucleon Hilbert space.  {\it When} $(\Lambda ,A)$ is an element of
the kinematic subgroup of the null plane, the Poincar\'e Wigner
${\cal D}$-function in the null-plane basis, ${\cal
  D}^{\lambda,j}_{\tilde{\mathbf{P}}',\mu';
  \tilde{\mathbf{P}},\mu}[\Lambda,A]$, is independent of the mass
eigenvalue, $\lambda$, and is thus identical to the Poincar\'e Wigner
${\cal D}$-function for the non-interacting irreducible
representation.  This shows that the null-plane kinematic subgroup
defined by (\ref{b.8}) is the kinematic subgroup of the dynamical
representation (\ref{c.5}).

The rotational symmetry is {\it exact}, but because ${\cal
  D}^{\lambda,j}_{\tilde{\mathbf{P}}',\mu';
  \tilde{\mathbf{P}},\mu}[\Lambda,A]$, depends explicitly on the mass
eigenvalue $\lambda$ when $\Lambda =R$ is a rotation, the rotations
are dynamical transformations.

The eigenfunctions of the interacting mass operator, $M$, are
identical functions of $k,l,s$ to the eigenfunctions of the
non-relativistic Schr\"odinger equation with interaction $v_{nn}$.
The identity
\beq
\Omega_{\pm} (H_{nr},H_{0nr} ) = \Omega_{\pm} (H_r,H_{0r} ),
\label{c.6}
\eeq 
where $\Omega_{\pm } (H,H_0 )$ are the scattering wave operators
for the non-relativistic and relativistic system respectively,   
follows from the elementary calculation:
\[
\Omega_{\pm} (H_r,H_{0r} ) = 
\lim_{t \to \pm \infty} e^{i (P_0^+ + P^-) t/2 }
e^{-i (P_0^+ + P_0^-) t/2 } = 
\lim_{t' \to \pm \infty} e^{i  P^- t' }
e^{-i P_0^- t' } =
\]
\[
\lim_{t' \to \pm \infty} e^{i  (M^2 + \mathbf{P}_{0\perp}^2)  t'/P_0^+ }
e^{-i  (M_0^2 + \mathbf{P}_{0\perp}^2)  t'/P_0^+ } =
\lim_{t'' \to \pm \infty} e^{i M^2  t'' }
e^{-i  M_0^2  t'' } =
\]
\[
\lim_{t'' \to \pm \infty} e^{i (\mathbf{k}^2/m +v_{nn}) 4mt'' + i4m^2 t'' }
e^{-i (\mathbf{k}^2/m ) 4mt'' - i4m^2 t'' }=
\]
\[
\lim_{t''' \to \pm \infty}
e^{i (\mathbf{P}^2 /4m + \mathbf{k}^2/m +v_{nn}) t'''  }
e^{-i (\mathbf{P}^2 /4m + \mathbf{k}^2/m ) t'''  } =
\]
\beq
\Omega_{\pm} (H_{nr},H_{0nr} )
\label{c.7}
\eeq
where the limits are strong or absolute abelian \cite{baum} limits and 
the times in (\ref{c.7}) were reparameterized as follows
\beq
t''' = 4 m t'' = 4m t'/P_{0}^+ = 2m t/P_{0}^+ .
\label{c.8}
\eeq    
This proof uses kinematic symmetries that follow from (\ref{c.2}) 
and the spectral condition, $P_0^+>0$.  It follows from eq. (\ref{c.7}) 
that relativistic and non-relativistic scattering matrices are 
{\it identical} functions of $\mathbf{k}$:
\beq
S(H_{nr},H_{0nr} ) = 
\Omega_{+}^{\dagger} (H_{nr},H_{0nr} )  \Omega_{-} (H_{nr},H_{0nr} ) 
= \Omega_{+}^{\dagger} (H_r,H_{0r} ) \Omega_{-} (H_r,H_{0r} )
= S (H_r,H_{0r} ) .
\label{c.8a}
\eeq
Since realistic $nn$ interactions are constructed by correctly transforming 
experimental differential cross section data to the center of momentum 
frame and then fitting the solution of the scattering problem to 
the transformed data,  these interactions can be used in (\ref{c.1}) 
{\it without modification}.  There is a small binding
energy correction of about 1 part in 2000, which we ignore.
 
We use this method to construct the deuteron eigenstates and
representation of the Poincar\'e group that we use to evaluate the
deuteron current matrix elements (\ref{a.1}).

\section{observables}

The differential cross section for elastic-electron deuteron scattering 
in the one-photon exchange approximation,
\[
d\sigma = {(2\pi)^4 e
\over
\sqrt{(p_e \cdot p_d)^2 - m_e^2 m_d^2}} 
\vert  
\sqrt{\omega_e (\mathbf{p}'_e)}\langle p'_e, \mu'_e \vert I_{e \mu}(0)
\vert p_e, \mu_e \rangle \sqrt{\omega_e (\mathbf{p}_e)} 
{(2 \pi)^3 \over q^2} \times
\]
\beq \sqrt{\omega_d (\mathbf{p}'_d)} \langle p'_d, \mu'_d \vert
I_s^{\mu} (0) \vert p_d, \mu_d \rangle \sqrt{\omega_d (\mathbf{p}_d)}
\vert ^2 \delta^4 ( p_d' -p_d+p_e'-p_e) {d\mathbf{p}_e' \over \omega_e
  (\mathbf{p}'_e) } {d \mathbf{p}_d' \over \omega_d (\mathbf{p}'_d) } ,
\label{f.1}
\eeq 
is a quadratic function of the deuteron current matrix elements.
Because only three of the deuteron current matrix elements are
linearly independent, there are three independent elastic
scattering spin observables for a given momentum transfer.  

The observables are the structure functions
$A(Q^2)$ and $B(Q^2)$ and the tensor polarization
$T_{20}(Q^2,\theta)$ at $\theta_{lab} = 70^o$.

The quantities $A(Q^2)$, $B(Q^2)$ can be extracted from the 
unpolarized laboratory frame differential cross section:
\beq
{d\sigma \over d \Omega}({Q}^2,\theta) = {\alpha^2 \cos^2 (\theta/2)
\over 4E_i^2 \sin^4 (\theta/2)}{E_f \over E_i}
[A({Q}^2) + B({Q}^2) \tan^2 (\theta /2)]  
\label{f.2}
\eeq
while $T_{20}({Q}^2,\theta)$ is extracted from the difference in the 
cross sections for target deuterons having canonical spin polarizations 
$\mu_d=1$ and $\mu_d=0$ 
at a fixed laboratory scattering angle: 
\beq
T_{20}({Q}^2,\theta) = \sqrt{2} {{d\sigma \over d \Omega}_1({Q}^2,\theta)
-{d\sigma \over d \Omega}_0({Q}^2,\theta) \over
{d\sigma \over d \Omega}({Q}^2,\theta)  } .
\label{f.3}
\eeq

\section{Currents} 

The current operator $I^{\mu}(x)$ transforms as a four-vector 
density under the dynamical representation (\ref{c.5}) 
of the Poincar\'e group,
\beq
U(\Lambda ,A) I^{\mu}(x)U^{\dagger} (\Lambda ,A) =
(\Lambda^{-1})^{\mu}{}_{\nu} I^{\nu}(\Lambda x+ a) .
\label{d.2}
\eeq
The current operator must also satisfy current
conservation 
\beq
g_{\mu \nu} [P^{\mu},I^{\nu} (0)]_-=0,
\label{d.3}
\eeq
in addition to symmetries with respect to space reflections 
and time reversal.  Here $g_{\mu \nu}$ is the Minkowski metric
with signature $(-+++)$.
Current covariance and current conservation 
are dynamical constraints on the current operator.

Cluster properties imply that the current operator for the two-nucleon
system is the sum of two single-nucleon current operators plus a
two-body operator that vanishes when the nucleons are asymptotically
separated.  A non-vanishing two-body current is needed to satisfy 
(\ref{d.2}) and (\ref{d.3}) when the nucleons interact.  The 
two-body current is constrained, but not fixed by the requirements of 
current covariance and current conservation. 

Current matrix elements are matrix elements of $I^{\mu}(x)$ in
irreducible eigenstates of $U(\Lambda,A)$.
Combining current covariance (\ref{d.2}) with the transformation
properties (\ref{c.5}) of the irreducible eigenstates of $U(\Lambda,A)$
gives a set of linear equations that relate the different matrix elements:
\[
\langle d, j ,\tilde{\mathbf{P}}',\mu' \vert
I^{\mu} (x)  
\vert d, j ,\tilde{\mathbf{P}},\mu  \rangle =
\]
\beq
(\Lambda^{-1})^{\mu}{}_{\mu'} 
\int \sum {\cal D}^{M_d,1 *}_{
\tilde{\mathbf{P}}''',\mu''' ;
\tilde{\mathbf{P}}',\nu'}[\Lambda ,A] d\tilde{\mathbf{P}}'''
\langle \tilde{\mathbf{P}}''',\nu''' ,d\vert
I^{\mu'} (\Lambda x +a )  
\vert \tilde{\mathbf{P}}'',\nu,d  \rangle d\tilde{\mathbf{P}}'' 
{\cal D} ^{M_d,1}_{\tilde{\mathbf{P}}'',\nu''; \tilde{\mathbf{P}},\nu}[\Lambda ,A] .
\label{d.3a}
\eeq
Current conservation leads to the additional linear constraints on the 
current matrix elements:
\beq
g_{\alpha \mu} (P^{'\alpha} - P^{\alpha}) 
\langle \tilde{\mathbf{P}}',\nu',d \vert
I^{\mu} (0)  
\vert \tilde{\mathbf{P}},\nu,d  \rangle = 0 .
\label{d.4a}
\eeq
For the deuteron the constraints (\ref{d.3a}), (\ref{d.4a}), parity
and time reversal imply that all of the current matrix elements can be
constructed from three linearly independent matrix elements for each
value of $Q^2$.  If the coordinate axes are chosen so the $+$
component of the momentum transfer is zero, which can always be done
for electron scattering, then these relations imply that all of the
current matrix elements can be calculated from matrix elements of
$I^+(0)$ \cite{Chung88} .

In Poincar\'e invariant quantum mechanics `` generalized impulse
approximations'' are defined by using the one-body current operators
determined by cluster properties to compute linearly independent
current matrix elements; the remaining current matrix elements are
generated by the dynamical constraints discussed above.  Different
choices of independent current matrix elements lead to different
``generalized impulse approximations''; however {\it any} choice leads
to matrix elements of an exactly conserved covariant current operator.
The non-uniqueness of Poincar\'e invariant ``generalized impulse
approximations'' suggests that it is unreasonable to expect that the
two-body contributions to a chosen set of independent current matrix
elements will be exactly zero.

If the deuteron eigenstates $\vert \tilde{\mathbf{P}},\mu,d
\rangle $ are given a delta-function normalization, then the matrix
elements
\beq
I^+_{\mu, \nu} := \langle \tilde{\mathbf{P}}',\mu,d \vert I^+(0) \vert
\tilde{\mathbf{P}},\nu,d \rangle   
\label{d.4}
\eeq
with $Q^+= P^{+\prime}-P^+ =0$ are invariant with respect to
null-plane boosts.  There are 9 spin combination; all but four
are related by kinematic symmetries \cite{Chung88}.  The remaining four
matrix elements of $I^+(0)$, which can be taken as $I^+_{1,1}
I^+_{1,0} I^+_{0,0}$ and $ I^+_{1,-1}$, are related by rotational
covariance.  Only three linear combinations of these matrix elements
can be taken as independent.

The three independent linear combinations of the current matrix
elements used in ref. \cite{Chung88} were:
\beq
I^+_{1,1}+ I^+_{0,0},  I^+_{1,0}, I^+_{1,-1} .
\label{d.5}
\eeq
We refer to this choice of independent current matrix elements as
choice I.  These independent matrix elements can be distinguished by
the number of spin flips between the initial and final states.

The difference $ I^+_{1,1}- I^+_{0,0} $ is related to the matrix
elements (\ref{d.5}) by dynamical transformations (i.e they involve
the deuteron mass, $M_d$):
\beq
I^+_{1,1}- I^+_{0,0} = 
-{ 1 \over 1+ \eta} [ \eta (I^+_{1,1}+ I^+_{0,0}) - 2 \sqrt{2 \eta}I^+_{1,0}+
I^+_{1,-1} ]
\label{d.6}
%\label{d.5}
\eeq
where 
\beq
\eta:= {Q}^2 /4M_d^2 .
%\label{d.6}
\label{d.7}
\eeq
The difference between the direct computation of the difference 
$I^+_{1,1}- I^+_{0,0}$ and the difference determined by 
the constraint (\ref{d.6}) of current covariance
\beq 
\Delta_- := I^+_{1,1}- I^+_{0,0} +{ 1 \over 1+ \eta} [ \eta
  (I^+_{1,1}+ I^+_{0,0}) - 2 \sqrt{2 \eta}I^+_{1,0}+ I^+_{1,-1}]
\label{d.8}
%\label{d.7}
\eeq
gives a measure of the size of the additional required dynamical
contribution to the current operator that is generated by the
covariance constraint.  If this difference is small compared to the
size of independent matrix elements, then there will not be too much
sensitivity to the choice of independent matrix elements; on the other
hand, if this difference is large, there will be an increased
sensitivity to the choice of independent current matrix elements.

Both impulse or impulse plus an exchange current can be used to
compute the independent current matrix elements.  Our calculations,
shown in fig. 14, indicate that $\Delta_-$ is larger when the 
pion-exchange current contributions are added to the independent matrix
elements.  This means that the presence of the exchange current
enhances the sensitivity of the results to the choice of independent
matrix elements.  Because of this increased sensitivity, we
investigate the impact of using different choices of independent
linear combinations of current matrix elements to compute the elastic
electron-deuteron scattering observables.

Any matrix elements of a conserved covariant current in a set of
deuteron eigenstates can be expressed in terms of a rank 3 covariant
current tensor.  Since the deuteron spin operator is constructed by
boosting the Pauli-Lubanski 4-vector divided by the mass to the rest
frame \cite{Coester82}\cite{Keister91}, it is possible to label the
deuteron spins by four vectors by multiplying the current matrix
element by the inverse of the boost used to define spin of the
irreducible eigenstate.  Since the Pauli-Lubanski vector is orthogonal
to the four momentum, this gives a four vector that is orthogonal to
the four momentum.  If this boost is applied to the initial and final
deuteron states in a current matrix element then the result is a
rank-three tensor density, $T^{\mu} _{\rho \sigma}(P',P)$ satisfying
\beq
P^{\rho \prime } T^{\mu} _{\rho \sigma}(P',P) =   
T^{\mu} _{\rho \sigma} (P',P) P^{\sigma } = 
(P_{\mu}'- P_{\mu})T^{\mu} _{\rho \sigma}(P',P) =0.
\label{d.9}
\eeq
Formally this tensor density is related to the current matrix elements by
\[ 
T^{\mu}_{\rho\sigma}(P',P) :=
\Lambda_f(\tilde{\mathbf{P}}'/M_d)^i{}_\rho O^{*}_{\nu' i} \sqrt{P^{+\prime}}
\langle \tilde{\mathbf{P}}', \nu' \vert I^\mu (0) \vert
\tilde{\mathbf{P}}, \nu \rangle \sqrt{P^{+}} O_{\nu j}
\Lambda_f(\tilde{\mathbf{P}}/M_d)^j{}_{\sigma} 
\]
\beq 
\label{d.10}
\eeq 
where
$O_{\nu j}$ is the unitary
matrix
\beq
O_{\nu j} = 
\left (
\begin{array}{ccc}
-{1 \over \sqrt{2}} & -{i \over \sqrt{2}} & 0 \\
0 &0 & 1 \\ 
{1 \over \sqrt{2}} & -{i \over \sqrt{2}} & 0 \\
\end{array} 
\right )
\label{d.11} 
\eeq 
that converts Cartesian components of vectors to spherical components 
and the $i,j$ sums go from 1 to 3.   The factors  $\sqrt{P^{+}}$
give
$
\vert
\tilde{\mathbf{P}}, \nu \rangle \sqrt{P^{+}}
$
a covariant normalization.

The general form of this Lorentz covariant tensor density 
$T^{\mu}_{\rho\sigma}(P',P)$ can be used to define invariant form factors
$F_1(Q^2) ,F_2 (Q^2) $ and ,$F_3 (Q^2)$
\[
T^{\mu}_{\rho\sigma}(P',P) = 
{1 \over 2} (P'+P)^{\mu} E(P')_{\rho\alpha}
[ g^{\alpha \beta} F_1 (Q^2)  
+ {Q^{\alpha}Q^{\beta}\over M_d^2}F_2 (Q^2)
]
E(P)_{\beta \sigma}
+
\]
\beq
{1 \over 2} E(P')_{\rho\alpha}[Q^{\alpha}g^{\mu \beta} - 
g^{\mu \alpha}Q^{\beta}] F_3 (Q^2)    
E(P)_{\beta\sigma}
\label{d.12}
\eeq
where 
\beq
E(P)_{\alpha \beta} = g_{\alpha \beta} - {P_{\alpha}P_{\beta}\over P^2}  
\label{d.13} 
\eeq
is the covariant projector on the subspace orthogonal to $P$.  The
projectors are not needed when this tensor is contracted with vectors
orthogonal to the final or initial four-momentum. 

The current tensor constructed using formula (\ref{d.10}) will only
transform as a tensor if the underlying current matrix elements
satisfy current covariance, while the expression (\ref{d.12}) is
always a tensor.  In applications where the underlying current operator
is only  covariant with respect to the null-plane kinematic subgroup
we can relate the form factor to a maximal set of independent 
current matrix elements by contracting the current tensor with three 
sets of ``polarization 4-vectors''.  These are 4-momentum dependent 
vectors $v^{\nu} (P)$
that are orthogonal $P^{\mu}$.  The contraction has the form
\[
v'_{af} (P')^{\rho}  T^{\mu}_{\rho\sigma}(P',P)
v_{bi} (P)^{\sigma} = 
{1 \over 2} (P'+P)^{\mu} 
[ v_{af}'(P') \cdot v_{bi} (P) F_1 (Q^2)  
+ { (v'_{af} (P')\cdot Q )(Q\cdot v_{bi}(P)\over Q^2}F_2 (Q^2)
]
+
\]
\beq
{1 \over 2} [ 
( v'_{af} (P') \cdot Q) v_{bi}(P)^{\mu} -
 v_{af}(P')^{\mu} (Q \cdot v_{bi} (P))
 ) F_3 (Q^2) .    
\label{d.14}
\eeq
In this case the three invariant scalar products 
\beq
v_{af}'(P') \cdot v_{bi} (P)
\qquad
v'_{af} (P')\cdot Q 
\qquad 
Q\cdot v_{bi}(P)
\label{d.15}
\eeq
determine the form factors in terms of the current matrix elements.   
When the underlying current matrix elements are
only kinematically covariant then it is necessary to pick a ``frame'' to
evaluate the contractions.   The result will give consistent results 
in all frames related to the original frame by a null-plane kinematic 
transformation.  The three independent sets of contractions can be used to 
calculate the three form factors which fix the remaining matrix elements
by the assumed covariance.   This provides an equivalent means for 
constructing independent linear combinations of current matrix 
elements. 

The form factors $F_1(Q^2)$, $F_2(Q^2), F_3(Q^2)$ defined by
(\ref{d.10}) and (\ref{d.12}) are related to the 
form factors $G_1(Q^2)$, $G_2(Q^2), G_3(Q^2)$ 
\cite{ostebee75} which are defined in terms of 
canonical-spin Breit-frame matrix 
elements with momentum transfer in the 3 direction:
\beq
F_1 (Q^2) - 4 \eta^2 (1+ \eta) F_2(Q^2) =
_c\langle {\mathbf{Q} \over 2}, 0,d \vert I^0(0) \vert 
-{\mathbf{Q} \over 2}, 0 ,d\rangle_c = G_0(0) + \sqrt{2} G_2 (Q^2) 
\label{d.16a}
\eeq
\beq
F_1 (Q^2) = _c\langle {\mathbf{Q} \over 2}, 1,d \vert I^0(0) \vert 
-{\mathbf{Q} \over 2}, 1,d \rangle_c = G_0(0) - {1 \over \sqrt{2}} G_2 (Q^2) 
\label{d.17a}
\eeq
\beq
F_3 (Q^2) = \sqrt{2 \over \eta} 
_c\langle {\mathbf{Q} \over 2}, -1,d \vert I^1(0) \vert 
-{\mathbf{Q} \over 2}, 0,d \rangle_c = 
G_1 (Q^2) .
\label{d.18a}
\eeq

The deuteron elastic scattering observables
can be expressed in terms of the deuteron form factors 
$G_i(Q^2)$, (\ref{d.16a},\ref{d.17a},\ref{d.18a}):
\beq
A({Q}^2)= G_0^2({Q}^2) + {2 \over 3} \eta 
G_1^2({Q}^2) + G_2^2({Q}^2)
\label{d.22a}
\eeq
\beq
B({Q}^2)= {4 \over 3}\eta (1+\eta) G_1^2({Q}^2) 
\label{d.23a}
\eeq
\beq
T_{20}({Q}^2,\theta)=- {G_2^2({Q}^2) +
\sqrt{8} G_0({Q}^2) G_2({Q}^2) + 
{1 \over 3} \eta G_1^2({Q}^2) [1+2(1+\eta) 
\tan^2 (\theta/2)] \over \sqrt{2} [A(Q^2)+ B(Q^2) 
\tan^2 (\theta /2)]} .
\label{d.24a}
\eeq
The enhanced sensitivity to the choice of linearly independent
matrix elements when the exchange current is included, as shown in fig.
14,  suggests that in addition to the choice (\ref{d.5}) of linearly
independent matrix elements used in \cite{Chung88}, other choices
should be considered.

Frankrurt, Frederico, and Strikman \cite{Frankfrut93} defined
independent current matrix elements using polarization vectors
constructed from the last three columns
$v_{1c}(P_b),v_{2c}(P_b),v_{3c}(P_b)$, of the canonical boost in the
Breit frame
\beq
(v_{1c} (P_b) , v_{2c} (P_b)  v_{3c} (P_b))  =
\left (
\begin{array}{ccc}
-\sqrt{\eta}  & 0 & 0 \\
\sqrt{1+\eta}      & 0 & 0\\
0      & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right ) .
\label{d.16}
\eeq
These are
automatically orthogonal to the momentum.  Three independent 
current matrix elements are extracted from the contractions
$
v_{1c} (P_b')^{\rho}  T^{+}_{\rho\sigma}(P_b',P_b)
v_{1c} (P_b)^{\sigma}
$, \phantom{a} 
$v_{2c} (P_b')^{\rho}  T^{+}_{\rho\sigma}(P_b',P_b)
v_{2c} (P_b)^{\sigma}
$ and 
$
v_{3c} (P_b')^{\rho}  T^{+}_{\rho\sigma}(P_b',P_b)
v_{1c} (P_b)^{\sigma}
$.  These are consistent in all frames related to the Breit frame
by null-plane kinematic transformations.  Their preference for 
this choice is to avoid using the $3'3$ matrix element, which is
maximally suppressed in the infinite momentum frame.
We call this choice of independent current matrix elements 
choice II.

We are indebted to Fritz Coester \cite{Fc08} for suggesting a third
choice of independent current matrix elements that directly emphasizes the
kinematic symmetries of the null plane.  This choice uses columns of
the null-plane boost as polarization vectors.
\beq
(v_{1f} (P) , v_{2f} (P)  v_{3f} (P))  =
\left (
\begin{array}{ccc}
P^x/M_d  & P^y/M_d & P^0/M_d-M_d/P^+ \\
1      & 0 &  P^x/M_d\\
0      & 1 &  P^y/M_d \\
-P^x/M_d & -P^y/M_d& P^z/M_d+M_d/P^+ \\
\end{array}
\right ) 
\label{d.17}
\eeq
The independent combinations are 
$
v_{1f} (P')^{\rho}  T^{+}_{\rho\sigma}(P',P)
v_{1f} (P)^{\sigma}
$,
$v_{2f} (P')^{\rho}  T^{+}_{\rho\sigma}(P',P)
v_{2f} (P)^{\sigma}
$, and 
$
v_{3c} (P')^{\rho}  T^{+}_{\rho\sigma}(P',P)
v_{1c} (P)^{\sigma}
$.
When the first two terms are evaluated in the Breit frame the scalar
products (\ref{d.15}) that relate the contractions to the form factors
are mass independent, which is preserved under kinematic
transformations.  This means that the natural boosts in the
impulse approximation are independent of the constituent masses.  To
construct a third independent linear combination that exploits this
mass independence would require using matrix elements of $I^2 (0)$.
Rather than using $I^2(0)$, this choice also uses the $3'1$ Breit
frame canonical spin polarization vectors that are also used in
\cite{Frankfrut93}.  In this case the mass dependence is trivial in
the Breit frame with respect to rotations.  We call this choice of
independent matrix elements choice III.

These three choices lead to three different expressions for deuteron form
factors in terms of independent linear combinations of matrix elements
of $I^+(0)$:

\noindent Choice I:
\beq
(1 + \eta) G_{0}(Q^2) = ({1 \over 2} - {\eta \over 3})( I_{11}+I_{00})
+ {5\sqrt{2 \eta} \over 3} I_{10}  + ({2 \eta \over 3} - 
{1 \over 6})I_{1-1}     
\label{d.18}
\eeq
\beq
(1 + \eta) G_{1}(Q^2) = I_{11}+I_{00} - I_{1-1}
-(1 -\eta) \sqrt{{2 \over \eta}} I_{10} 
\label{d.19}
\eeq
\beq
(1 + \eta) G_{2}(Q^2) = -{\sqrt{2}\eta \over 3} (I_{11}+I_{00})
+ {4\sqrt{\eta}\over 3} I_{10}  - {\sqrt{2} \over 3} (2+ \eta) I_{1-1} 
\label{d.20}
\eeq

\noindent Choice II ($G_{0}(Q^2)$ of choice I replaced by):

\beq
(1 + \eta) G_{0}({Q}^2)=  ({2 \eta \over 3} +1)
I^+_{1,1}  - {\eta \over 3} I^+_{00} 
+ {2 \sqrt{2 \eta} \over 3}  I^+_{1,0} + 
({2 \eta +1 \over 3} ) I^+_{1,-1}] 
\label{d.21}
\eeq

\noindent Choice III ($G_{0}(Q^2)$ and $G_{2}(Q^2)$ of choice I replaced by): 

\beq
G_{0}({Q}^2) = 
(1+ {2 \eta \over 3} )  I^+_{11} +{1 \over 3} I^+_{1,-1}+ 
-{2 \eta \over 3}G_{1}({Q}^2) 
\eeq
\beq
G_{2}({Q}^2) = 
{2 \sqrt{2} \over 3}(\eta I^+_{1,1}- I^+_{1,-1}- \eta G_{1}({Q}^2) ) 
\label{d.22}
\eeq
Since $B(Q^2)$ only depends on $G_1(Q^2)$ which uses the same linear
combination of current matrix elements for all three choices, the
computation of $B(Q^2)$, is unchanged.  These choices lead to a
$B(Q^2)$ that fits the experimental data.

The different linear combinations of null-plane current matrix
elements used in choices I , II and III have a non-trivial 
effect on the scattering observables $A(Q^2)$ and $T_{20}(Q^2,\theta)$.
Selecting a choice is a model assumption.

\section{Impulse currents}

The impulse current is the sum of single nucleon current operators for the 
proton and neutron.  For spin $1/2$ systems that are eigenstates of 
parity there are two independent matrix elements.  They are related
to the Dirac nucleon form factors $F_1(Q^2)$ and $F_2(Q^2)$ by
\beq
\langle \tilde{\mathbf{p}}' , \nu' \vert I^{\mu} (0) \vert
\tilde{\mathbf{p}} , \nu \rangle
=
\sqrt{m \over p^{+\prime}} \bar{u}(p') \Gamma^{\mu} u (p) 
\sqrt{m \over p^{+}}  
\label{e.1}
\eeq
where
\beq
\Gamma^{\mu}  = \gamma^{\mu} F_1 (Q^2) + 
{1 \over 2} [\gamma^{\mu} , {1 \over 2m} \gamma\cdot Q] F_2 (Q^2) 
\label{e.2}
\eeq
where $u(p)$ is a null plane Dirac spinor if the spin in the
current matrix element is a null plane spin. 

For spacelike momentum
transfers with $Q^+=0$ the form factors can be expressed in terms of 
the independent null-plane matrix elements    
\beq
\langle {1 \over 2} \vert I^+(0) \vert 
{1 \over 2}\rangle =
\langle \tilde{\mathbf{p}}', {1 \over 2} \vert I^+(0) \vert 
\tilde{\mathbf{p}}, {1 \over 2}\rangle = F_1 (\mathbf{Q}^2) 
\label{e.3}
\eeq
\beq
\langle {1 \over 2} \vert I^+(0) \vert 
-{1 \over 2}\rangle =
\langle \tilde{\mathbf{p}}', {1 \over 2} \vert I^+(0) \vert 
\tilde{\mathbf{p}}, -{1 \over 2}\rangle =
- \sqrt{\tau}
F_2 (\mathbf{Q}^2) .
\label{e.4}
\eeq
where $\tau := {\mathbf{Q}^2 \over 4m^2}$.

The Dirac form factors $F_1(Q^2)$ and $F_2(Q^2)$  are related to the 
Sachs form factors by \cite{Sachs62}
\beq 
F_1(\mathbf{Q}^2)={G_e(\mathbf{Q}^2)+ \tau G_m(\mathbf{Q}^2) \over 1+\tau},
\label{e.5}
\eeq
\beq  
F_2(\mathbf{Q}^2)={G_m(\mathbf{Q}^2)- G_e(\mathbf{Q}^2) \over 1+\tau}.
\label{e.6}
\eeq
We
consider recent parameterizations due to Bijker and Iachello
\cite{nnformfactor-BI}; Bradford, Bodek, Budd, and Arrington
  \cite{nnff-BBBA}; Budd, Bodek, and Arrington \cite{nnff-BBA}; Kelly
  \cite{nff-kelly} and Lomon
  \cite{nnformfactor-Lomon}\cite{nnformfactor-Lomon1}
  \cite{nnff-Lomon1}.
These parameterizations all determine the proton electric form factors
using the polarization experiments \cite{Punjabi05}.  The input to our
calculations is the isoscalar form factors, which are the sum of the
proton and neutron form factors.

\section{Dynamical exchange currents} 

The realistic interactions \cite{Wiringa95}\cite{cdb} used to
construct the mass operator (\ref{c.1}) include a one-pion
exchange contribution plus a short-range contribution that is 
designed so the two-body cross sections fit experiment.

``Required'' two-body currents are needed to satisfy the constraints
of current covariance and current conservation.  These constraints can
be satisfied by directly computing a set of linearly independent
matrix elements and then using the constraints to determine the
remaining current matrix elements.  Even if the independent current
matrix elements are computed using the one-body impulse current, the
matrix elements generated by the constraints will have two-body
contributions.  The interaction dependence arises because the
covariance and current conservation constraints involve the mass
eigenvalues of the initial and final states.
 
In addition to the two-body currents directly generated by covariance
and current conservation, realistic interactions contain terms that
involve the exchange of charged mesons, leading to the exchange of the
nucleon charges, and the possibility of two-body currents associated
with these exchanges.  These currents appear as additional
contributions to the linearly independent matrix elements of $I^+(0)$
that are needed to compute electron-scattering observables in
null-plane quantum mechanics.  The form of the dynamical exchange
current contribution to $I^+(0)$ is model dependent, but we assume
that the most important contribution is motivated by one-pion exchange
physics.

The interactions used in the dynamical equation (\ref{c.2}) are
constructed from realistic rotationally invariant interactions that
are functions of relative momentum variable and spins.  These
interactions can be used to construct the invariant mass and spin
operators for scattering equivalent dynamical models with any
kinematic symmetry by multiplying by the appropriate delta
function. In (\ref{c.2}) they are made into interactions with a
null-plane kinematic symmetry by embedding them in the full
two-nucleon Hilbert space by multiplying the rotationally invariant
interaction by a null-plane momentum-conserving delta function.

Any ``derivation'' of  model interactions 
in Poincar\'e invariant quantum mechanics should result in 
a rotationally invariant kernel in terms of relative momentum
variables and spins.  ``Derivations'' of exchange current should 
have a consistent treatment of the interactions.  
Meson-nucleon field theories provide a framework that explicitly
includes the degrees of freedom whose elimination leads to exchange
currents in dynamical models involving nucleon degrees of freedom.
These theories can be used to motivate the structure of model exchange
currents.  The precise ``derivation'' of any current 
involves model assumptions.

To construct a model exchange current we begin by constructing
off-shell transition amplitudes for nucleon-nucleon scattering and
nucleon-nucleon scattering in the presence of an external field
electromagnetic field using one-pion exchange Feynman diagrams.
We define a rotationally invariant model one-pion exchange
interaction, represented as a kernel in the relative momenta and
spins (\ref{c.2}), by identifying the interaction kernel 
with this off-shell transition amplitude in the rest frame of
the two-nucleon system. 

We use a similar method to construct our model one-pion exchange current
from the off-shell transition amplitude for nucleon-nucleon scattering
in the presence of an external field.  The origin of this current is
the part of the pion-exchange interaction that couples to the photon
with the Dirac $\Lambda_-(p)$ projection operator.  The details of how
we relate this operator to the rotationally invariant kernel that
represents the one-pion exchange interaction and how we realize the
null-plane kinematic symmetry is outlined in Appendix A.  In our
model current the rotationally invariant kernel that represents the 
one-pion exchange interaction is replaced by the corresponding 
one-poin-exchange contribution to our realistic model interaction.
  
We use the relation between the consistently derived one-pion
exchange interaction and the pion exchange current to motivate the
construction of a corresponding exchange current from the one-pion
exchange part of a realistic interaction.  Details are 
given in Appendix A and Appendix B.  

The resulting matrix element of the exchange current has the following form
\[
\langle \tilde{\mathbf{P}}', \nu' , d \vert I_{ex}^\mu(0) \vert
\tilde{\mathbf{P}}, \nu , d \rangle := 
\]
\[
\int \langle \tilde{\mathbf{P}}', \nu' , d \vert 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}'_2 , \nu_2' 
\rangle 
(-{1 \over 2m}) \sqrt{m\over p_1^+} \bar{u}_f(p_1')\Gamma^{\mu} \gamma_5 
({P \cdot \gamma \over M_d}) {u}_f(p_1'') \sqrt{m\over p_1^{+\prime\prime}}
d\tilde{\mathbf{p}}_1'd \tilde{\mathbf{p}}_1'' d\tilde{\mathbf{p}}_2'
\times 
\]
\[
\langle 
\tilde{\mathbf{p}}_1'', \nu_1'', \tilde{\mathbf{p}}'_2 , \nu_2'
\vert U(\Lambda_f (P/M_d)) \tilde{v}_1 \vert 
\tilde{\mathbf{P}}_0, \nu , d \rangle +
\]
\[
\int \langle \tilde{\mathbf{P}}', \nu' , d \vert 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}'_2 , \nu_2' 
\rangle 
(-{1 \over 2m}) \sqrt{m\over p_2^+} \bar{u}_f(p_2')\Gamma^{\mu} \gamma_5 
({P \cdot \gamma \over M_d}) {u}_f(p_2'')\sqrt{m\over p_2^{+\prime\prime}}
d\tilde{\mathbf{p}}_1' d\tilde{\mathbf{p}}_2'' d\tilde{\mathbf{p}}_2'
\times 
\]
\[
\langle 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}''_2 , \nu_2''
\vert U(\Lambda_f (P/M_d)) \tilde{v}_2 \vert 
\tilde{\mathbf{P}}_0, \nu , d \rangle +
\]
\[
\int \langle \tilde{\mathbf{P}}_0', \nu' , d \vert
\tilde{v}^{\dagger}_1 U^{\dagger} (\Lambda_f (P'/M_d))  
\vert 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}'_2 , \nu_2' 
\rangle 
(-{1 \over 2m}) \sqrt{m\over p_1^{+\prime}} \bar{u}_f(p_1')
({P' \cdot \gamma \over M_d})\gamma_5 
\Gamma^{\mu}  
{u}_f(p_1'') \sqrt{m\over p_1^{+\prime\prime}}
\times 
\]
\[
d\tilde{\mathbf{p}}_1' d\tilde{\mathbf{p}}_1'' d\tilde{\mathbf{p}}_2'
\langle 
\tilde{\mathbf{p}}_1'', \nu_1'', \tilde{\mathbf{p}}'_2 , \nu_2'
\vert 
\tilde{\mathbf{P}}, \nu , d \rangle +
\]
\[
\int 
\langle \tilde{\mathbf{P}}_0' \nu' , d \vert
\tilde{v}^{\dagger}_2 U^{\dagger} (\Lambda_f (P'/M_d))  
\vert 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}'_2 , \nu_2' 
\rangle 
(-{1 \over 2m}) \sqrt{m\over p_2^{+\prime}} \bar{u}_f(p_2')
({P' \cdot \gamma \over M_d})\gamma_5 
\Gamma^{\mu}  
{u}_f(p_2'') \sqrt{m\over p_2^{+\prime\prime}}
d\tilde{\mathbf{p}}_1' d\tilde{\mathbf{p}}_2'' d\tilde{\mathbf{p}}_2'
\times 
\]
\beq
\langle 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}''_2 , \nu_2'' 
\vert 
\tilde{\mathbf{P}}, \nu , d \rangle 
\label{j.1}
\eeq
where
\beq
\langle \tilde{\mathbf{P}}', \nu' , d \vert 
\tilde{\mathbf{p}}_1', \nu_1', \tilde{\mathbf{p}}'_2 , \nu_2' 
\rangle 
\eeq
is the deuteron eigenstate in the tensor product basis, 
$\tilde{\mathbf{P}}_0 = (M_d,0,0)$ is the rest frame 
value of the deuteron null-plane momentum,  
$U(\Lambda_f (P/M_d))$ represents a kinematic null-plane boost to the 
Breit frame.  The field theory motivated argument in Appendix A leads 
to modified pion-exchange interactions $\tilde{v}_i$ of the form 
\[
\langle 
\tilde{\mathbf{P}}', {\mathbf{k}}' , \nu_1', \nu_2'
\vert v_1 \vert
\tilde{\mathbf{P}}, {\mathbf{k}} , \nu_1, \nu_2\
\rangle =
\]
\beq
\delta (\tilde{\mathbf{P}}' - \tilde{\mathbf{P}}) 
{ g^2_{\pi} \over (2 \pi)^3} 
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{u}(\mathbf{k}')\beta u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
{\btau_1 \cdot \btau_2  \over m^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \gamma_5  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
\label{j.2}
\eeq
and
\[
\langle 
\tilde{\mathbf{P}}', {\mathbf{k}}' , \nu_1', \nu_2'
\vert v_2 \vert
\tilde{\mathbf{P}}, {\mathbf{k}} , \nu_1, \nu_2
\rangle =
\]
\beq
\delta (\tilde{\mathbf{P}}' - \tilde{\mathbf{P}}) 
{ g^2_{\pi} \over (2 \pi)^3} 
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{u}(\mathbf{k}')\gamma_5 u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
{\btau_1 \cdot \btau_2  \over m^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \beta  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
\label{j.3}
\eeq
They differ from the standard one pion exchange interaction by the replacement
of one of the factor of 
$
\bar{u}(\mathbf{k}') \gamma_5  u (\mathbf{k})$ by  
$\bar{u}(\mathbf{k}') \beta  u (\mathbf{k})$. 

In a dynamical model based on a phenomenological interaction we 
replace these interactions by the corresponding contribution to the 
model interaction:   
\[
\langle 
\tilde{\mathbf{P}}', {\mathbf{k}}' , \nu_1', \nu_2'
\vert \tilde{v}_1 \vert
\tilde{\mathbf{P}}, {\mathbf{k}} , \nu_1, \nu_2\
\rangle  \to 
\]
\beq
\delta (\tilde{\mathbf{P}}' - \tilde{\mathbf{P}}) 
{ g^2_{\pi} \over (2 \pi)^3} 
{2m \over 2m} \btau_1 \cdot \btau_2   
v_{\pi} (\mathbf {k} - \mathbf{k}' ) 
{(\mathbf{k}' - \mathbf{k}) \cdot \bsig_2 \over 2m}  
\label{j.4}
\eeq
where $v_{\pi} (\mathbf {k} - \mathbf{k}' )$ is the coefficient 
function of the one-pion-exchange contribution to the operator 
\beq
\delta (\tilde{\mathbf{P}}' - \tilde{\mathbf{P}}) 
{ g^2_{\pi} \over (2 \pi)^3} 
{(\mathbf{k} - \mathbf{k}') \cdot \bsig_1 \over 2m}   \btau_1 \cdot \btau_2   
v_{\pi} (\mathbf {k} - \mathbf{k}' ) 
{(\mathbf{k}' - \mathbf{k}) \cdot \bsig_2 \over 2m}  
\label{j.5}
\eeq
in the phenomenological interaction.  For the Argonne V18 interaction
$v_{\pi} (\mathbf {k} - \mathbf{k}' )$
is extracted from the one-pion-exchange contribution to the tensor
$\times \, (\btau_1 \cdot \btau_2)$ and spin-spin $\times \, (\btau_1 \cdot
\btau_2)$ parts of the interaction using the method 
discussed in \cite{Riska85} and \cite{Schi90}.  The extraction is discussed in
Appendix B.  The resulting interaction $v_{\pi} (\mathbf {k} -
\mathbf{k}' )$ differs from a $1/ ( m_\pi^2 + (\mathbf {k} -
\mathbf{k}' )^2)$ by the effects of the short distance cutoff in the 
AV18 interaction.

The important property of this exchange current is that it is 
constructed to transform covariantly with respect to the 
null-plane kinematic subgroup.  This current is added to the 
impulse current when computing the independent matrix elements.
These are then used to generate the remaining current matrix 
elements using the constraint of current covariance and 
current conservation.

Each term in the expression $\mathbf{j.1}$ for the 
exchange current matrix element can be represented as 
the product of $(-{1 \over 2m})$ with the modified current 
$
\sqrt{m\over p_1^+} \bar{u}_f(p_1')\Gamma^{\mu} \gamma_5 
({P \cdot \gamma \over M_d}) {u}_f(p_1'') \sqrt{m\over p_1^{+\prime\prime}}
$
evaluated between a deuteron eigenstate and a pseudo-state defined by 
applying the rotationally invariant modified interaction 
$\tilde{v}_i$ to the 
rest deuteron state and kinematically boosting the result to the 
Breit frame.  
 


\section{results} 

The input to our calculation of the elastic electron-deuteron
scattering observables is (1) the choice of nucleon form factors
\cite{nnformfactor-BI} \cite{nnformfactor-Lomon1}\cite{nnff-Lomon1}
\cite{nnff-BBA}\cite{nnff-BBBA}\cite{nff-kelly} (2) the choice of
nucleon-nucleon interaction \cite{Wiringa95}\cite{cdb} (3) the choice
of independent linear combinations of current matrix elements used to
compute the required two-body currents,
\cite{Chung88},\cite{Frankfrut93}, \cite{Terent77}\cite{Karmanov92}.

The extraction of proton electric form factors based on polarization
measurements compared with measurements based on the Rosenbluth
separation were found to be inconsistent\cite{Punjabi05} ; these
inconsistencies have been explained \cite{tgm} by including two photon
exchange corrections in the Rosenbluth separation.  This has led to a
modification of the phenomenological parameterizations of the proton
electric form factor.  All of the nucleon form factors that we use  
are consistent with the extractions based on the polarization 
measurements.

Neutron electric form factor data is only available for a limited
range of momentum transfers.  The high momentum transfer behavior
of the different parameterizations is
consequence of different theoretical assumptions.  This leads to some
variation of the parameterizations for momentum transfers above $Q^2
\sim 1 (GeV)^2$.  Different parameterizations of the neutron electric
form factor are compared to a dipole form factor in fig 1.  The form
factors that we compare are recent parameterizations given by Lomon
\cite{nnformfactor-Lomon1}, Budd, Bodeck and Arrington (BBA)
\cite{nnff-BBA}, Bradford Budd, Bodeck and Arrington (BBBA)
\cite{nnff-BBBA}, Kelly \cite{nff-kelly}, and Bijker and Iachello (BI)
\cite{nnformfactor-BI}.  All of these parameterizations agree for
$Q^2<1$.  The curves in figure 1 are given as ratios to dipole form
factors, which emphasize differences in the parameterizations.  The
input to our calculations are the isoscalar linear combinations of the
nucleon form factors, $F_{1N}(Q^2)$ and $F_{2N}(Q^2)$.  These form
factors are plotted in figures 2 and 3 for the same parameterizations
that are compared in figure 1.  These plots show no significant
variation among the various nucleon form factors.  Our calculations of
elastic scattering observables are not very sensitive to these
differences.

As shown in section VII, our model exchange current breaks up into
a product of an effective ``one-body'' current and an interaction.
The interaction is a modification of the one-pion exchange
contribution to the tensor part of the phenomenological interaction.
Most of our calculations are based on the Argonne V18 interaction.  We
extract the one-pion exchange contribution to the V18 potential by
first discarding the short-range parts of the interaction that
contributes to the spin-spin and tensor forces, then we use a method
developed by Riska \cite{Riska85} and Schiavilla, Pandharipande, and
Riska \cite{Schi90} to extract the one-pion exchange contribution to
the tensor force from the remaining parts of the interactions.  The
procedure that we use is discussed in Appendix B.  This differs from
${1 \over m_{\pi}^2 + (\mathbf{k}-\mathbf{k}')^2} $ by the effects of
the short distance configuration space cutoff that appears in the
Argonne V18 interaction.  The Fourier transform of the extracted
interaction is compared to the pion-exchange potential without the
cutoff in figure 4.  The dotted curve includes the cutoff parameters
that are used in in the Argonne V18 interaction.  The most important
differences are for momenta above 1-2 $fm^{-1}$.

Figures 5,6,7 show the three deuteron form factors, $G_0$, $G_1$ and
$G_2$ with and without the exchange current included for the
two-nucleon form factors (BI and BBBA) that have the largest high
momentum transfer difference in the neutron electric form factor.  The
independent matrix elements (\ref{d.5}) are calculated using the
one-body parts of the current and with the pion exchange current
added.  The remaining current matrix elements are determined by the
constraints of current conservation and current covariance.  The
figures show that the addition of the pion exchange current
contributions lead to an enhancement of $G_0$ above the minimum at
$Q^2=1GeV^2$, The minimum of $G_1$ shifts to the right, and $G_2$ is
enhanced.  Except for momentum transfers $Q^2$ between 6 and 7 these
form factors are not very sensitive to the different assumptions made
about the neutron electric form factor.

Data for the observables $A$ are labeled
Stanford Mark III~\cite{buchanan65}, CEA ~\cite{Elias69},
Orsay~\cite{Benak66}, SLAC E101~\cite{Arnold75}, Saclay
ALS~\cite{Platch90}, DESY ~\cite{Galster71},
Bonn~\cite{cramer85}, Mainz~\cite{simon81}, JLab Hall C
~\cite{JlabHallc99}, JLab Hall A ~\cite{JlabHalla99} and
Monterey~\cite{Berard74}.  Data for $B$ are 
labeled: SLAC NPSA NE4~\cite{Bosted90},
Martin~\cite{martin77}, Bonn~\cite{cramer85}, Saclay
ALS~\cite{auffret85}, Mainz~\cite{simon81}, Stanford Mark
III~\cite{buchanan65}.  Data for 
$T_{20}$  are labeled: Novosibirsk-85 ~\cite{Dmitr85}
~\cite{Novos86}, Novosibirsk-90~\cite{Gilman90},
Bates-84~\cite{Schul84}, Bates-91~\cite{The91} and JLab Hall
C~\cite{Jlabt20}

Calculations of $A$, $B$, and $T_{20}$ using the independent matrix
elements (\ref{d.5}) (choice I) and the five parameterizations of the
nucleon form factors used in figures 1-3 are shown in figures 8-10.
While the generalized impulse calculations give a qualitative
understanding of the data, it is clear from these calculations that
the generalized impulse approximation is inadequate.

Figures 11,12,13 show the effects of including the phenomenological
pion-exchange current defined in section VII.  We see that $A$ and $B$
provide acceptable fits to the data when the pion exchange current is
included.  The results are insensitive to the assumptions used in
parameterizing the high momentum transfer behavior of the neutron
electric form factor.  $T_{20}$ is closer to the data, but it is still
below the most recent Jlab hall C data \cite{Jlabt20} between $Q^2$ of
$.5$ and $2 GeV^2$.

The calculation displayed in figures 11,12,13 used the choice I of
independent current matrix elements given by (\ref{d.5}).  
The presence of the pion exchange current (\ref{j.1}) increases the
sensitivity to the choice of the independent current matrix elements.
This is shown in figure 14.  What is plotted is the difference
$I^+_{11}(0)-I^+_{00}(0)$ with and without the pion exchange current using a
  direct calculation of the difference or by generating the difference
  using current conservation and current covariance.  The difference
  between the dashed curve and solid curve shows that the required
  two-body contributions to the current in this difference is small
  when the independent current matrix elements are computed in the
  impulse approximation.  Comparing the dotted and dash-dot curve
  indicates that much larger required two-body contributions 
to the current are needed
  when the exchange current contributions are included in all matrix
  elements.  This suggests that there will be a non-trivial sensitivity 
to the choice in independent current matrix elements when the pair 
current contribution is added.  

To test this we examined the two other choices, II and III, of
independent matrix elements discussed in section IV.  These methods
relate form factors to independent current matrix elements by contracting 
different sets of polarization vectors into the current tensor.
In
both approaches there are preferred polarization vectors; in one case
the vectors are chosen to minimize the dependence on matrix elements
that are maximally suppressed in the infinite momentum frame ($P^+ \to
0$), while the other choice minimizes the mass dependence in the
contractions used to define the independent current matrix elements.
Both choices were discussed in section IV.

Figures 15,16,17, and 18 show the deuteron elastic scattering
observables $A(Q^2)$ and $T_{20}(Q^2)$ for both choices, II and III,
of polarization vectors.  In both case $G_1$ is computed using the
same linear combination of current matrix elements used for choice I.
Since $B$ only depends on $G_1$, $B$ is identical for all three
choices.  All three choices of independent matrix elements give
different predictions for $A$ and $T_{20}$.  For choices II and III
there is a mild enhancement of $A$ at higher momentum transfers
compared to choice of independent matrix elements given by
(\ref{d.5}).  There is also a larger effect on the tensor polarization
that brings the curve to within the experimental error bars.

The result is that both choice II and III of independent current
matrix elements give consistent results for the elastic scattering
observables and they both provide a good description of the existing
data over a wide range of momentum transfers.  It is clear that there
is a non-trivial sensitivity to the choice of independent current
matrix elements when these results are compared to the corresponding
results based on choice I.

Another potential source of sensitivity to the input is the
choice of nucleon-nucleon interaction.  Any phase equivalent change in
the nucleon-nucleon interaction is automatically accompanied by 
a corresponding change in the current operator.  For interactions 
with a long-range meson exchange tail one might expect that 
the same data could be understood by simply adjusting the 
cutoff parameter.   For typical soft interactions that are useful 
in low energy problems, one expects that a more significant 
modification of the current would be necessary.   

In figs. 19,20 and 21 we compare calculations of $A(Q^2)$, $B(Q^2)$,
and $T_{20}(Q^2)$ using the CD Bonn wave functions with and without
the exchange current.  In these calculations the exchange current is
still based on the Argonne V18 cutoffs.  The calculations show the
generalized impulse calculations and calculations where the exchange
current is added to the impulse current.  without adjusting the cutoff
parameters. The calculations clearly show that there is more
sensitivity to the choice of nucleon-nucleon interactions than to the
choice of nucleon form factor.

Good consistency with all experimental observables is obtained using
the V18 interaction with nucleon form factors \cite{nnff-BBBA} and the
choice of independent current matrix elements suggested by 
Frankfrut, Frederico and Strickman or Coester,
discussed in section IV.  These calculations are shown in 
figs. 22,23 and 24.  For
these choices the model exchange current explains the difference
between the generalized impulse approximation and the experimental
data.  While our exchange current, which used the one-pion exchange part
of the Argonne V18 interaction, required the numerical
computation of Fourier transforms, the resulting interaction
contribution to the exchange current differed very little from a simple
static momentum space pion exchange interaction, fig 4, which
indicates the simplicity of our exchange current. 

Our results indicate that this simple pion exchange current is
sufficient to provide a good quantitative understanding of elastic
electron deuteron elastic scattering for a wide range of momentum
transfers.  The sensitivities to both the choice of interaction and
the choice of independent matrix elements are the largest
uncertainties in the calculations, and these uncertainties are all
larger than the experimental uncertainties.  For momentum transfers
where data is available there is very little sensitivity to the
uncertainties in the neutron electric form factors.

Finally we compare the magnetic and quadrupole moments of the deuteron
with and without the exchange current and using different choices of
independent current matrix elements.  The results are shown in table
1. Our calculations do not exhibit any sensitivity to the choice of
nucleon form factor, which are sufficiently well constrained by
experiment at low momentum transfers.  The exchange current contributions
affect the results for all of the moments. In both cases they are
closer to experiment than the moments computed using the generalized
impulse approximation.  The magnetic moment is in good agreement (to
within computational accuracy) with experiment, while the quadrupole
moment differs from the experimental result by a few percent.  

The conclusion of our research is that a simple exchange current
motivated by one-pion exchange and the freedom to define a conserved
covariant current operator by choosing a preferred set of independent
current matrix elements is sufficient to provide a good fit to all
three elastic scattering observables using Poincar\'e invariant
quantum theory with a null-plane kinematic symmetry.  The model
exchange current is essentially a pair current designed with a null
plane kinematic symmetry.  The quality of the nucleon form factors has
progressed to the point that our results are insensitive to the choice
of nucleon form factor.

While is it straightforward to include low-order pion-exchange physics
in a more general class of models, the strategy for making the best
choice of independent current matrix elements in a general class of
electron-nucleus reactions requires more investigation.  The
principles used to derive choices I, II and III of independent current
matrix elements all can be generalized to treat initial and final
states with different spins.  Whether there is one consistent set
of principles that works universally for all reactions is not yet known.

A second observation is that our model, with one of the two preferred
choices of independent current matrix elements, provides a better
description of all three observables than methods based on truncations
of null-plane field theory or instant form relativistic quantum
mechanics.  Our model has features of both - unlike the instant form
model our model has the full-null plane kinematic symmetry with all of
the advantages discussed at the beginning of this paper. Unlike
truncations of a null plane field theory, which emphasizes cluster
properties at the expense of exact Poincar\'e invariance, our model is
exactly Poincar\'e invariant.  We expect that the Poincar\'e
invariance constraint to be more important for momenta near or
slightly above the deuteron mass scale, since the deuteron 
mass scale is involved in the implementation of the symmetry.
 
This research provides a useful first step in trying to devise a more
systematic treatment of model exchange currents in Poincar\'e
invariant quantum mechanics with a null plane kinematic symmetry.  It
leads to a simple current that provides a significant improvement in
all three elastic scattering observables when compared with the
corresponding impulse calculations, however additional research is
still needed in order to determine if these methods can be
successfully applied to larger class of reactions.

This work was performed under the auspices of the U.~S.  Department of
Energy, Office of Nuclear Physics, under contract
No. DE-FG02-86ER40286.  The authors would like to express our
gratitude to Fritz Coester who made many material suggestions that
significantly improved the quality of this work.

\appendix
\section{Current construction}

The steps motivating the structure of the model exchange current 
(\ref{j.1}) are
summarized in this appendix.  
We consider a pseudoscalar pion-nucleon vertex:
\beq
{\cal L}(x) = -i {g_{\pi}} :\bar{\Psi} (x)
\gamma_5 \Psi (x) \btau \cdot \bphi (x) : 
\label{g.1}
\eeq
where $g_{\pi} = 2m {f_{\pi} \over m_\pi}$ is the pseudoscalar
pion-nucleon coupling constant and $m$ is the nucleon mass.  
The off-shell transition amplitude in
the two-body rest frame is given by the rotationally invariant kernel:
\[
\langle \mathbf{k}',\mu_1', \mu_2'  \vert v_{\pi} \vert \mathbf{k},
\mu_1,\mu_2 
\rangle :=
\]
\beq
{g_{\pi}^2  \over (2 \pi)^{3}} 
\sqrt{{m \over \omega (\mathbf{k}') }}  
\bar{u}(\mathbf{k}' )\gamma_5 {u}(\mathbf{k} )
\sqrt{{m \over \omega (\mathbf{k}) }} 
{\btau_1 \cdot \btau_2
\over m_{\pi}^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}') }}  
\bar{u}(-\mathbf{k}' )\gamma_5 {u}(-\mathbf{k} ) 
\sqrt{{m \over \omega (\mathbf{k}) }} 
\label{g.3}
\eeq
where $(k'-k)^2 := (\mathbf{k}- \mathbf{k}')^2 - (\omega_m
(\mathbf{k})- \omega_m(\mathbf{k}'))^2$ and we have assumed a
plane wave normalization $\langle \mathbf{k}' \vert \mathbf{k} \rangle
= \delta (\mathbf{k}' - \mathbf{k})$.

When the external field is turned on the off-shell transition amplitude 
has the form of a current contracted into the 4-vector potential of
the field.   The expression for the current matrix element  
using the vertex (\ref{g.1}),  with 
\beq
A^{\mu}(q) := {1 \over (2 \pi)^4} \int
e^{- i q \cdot y} A_{\mu}(y) d^4y, 
\label{g.5}
\eeq
includes four terms, one of which is 
\[
\langle \mathbf{p}_1', \mu_1', \mathbf{p}_2', \mu_2' \vert I^{\mu}(0)^{\mu}
\vert \mathbf{p}_1, \mu_1, \mathbf{p}_2, \mu_2 \rangle = 
\]
\[  
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} u(\mathbf{r}) 
\sqrt{{m \over \omega (\mathbf{r})}} 
\sqrt{{m \over \omega (\mathbf{r})}} 
\bar{u}(\mathbf{r}) \gamma_5  u(\mathbf{p}_1) 
\sqrt{{m \over \omega (\mathbf{p}_1)}}
{1 \over E_{12} - \omega(\mathbf{r}) - \omega(\mathbf{p}_2') +i0^+} \times
\]
\[
{ g^2_{\pi} \over (2 \pi)^3}
{\btau_1 \cdot \btau_2  \over m^2 + (p_2-p_2')^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{p}_2')}}
\bar{u}(\mathbf{p}_2') \gamma_5  u (\mathbf{p}_2) 
\sqrt{{m \over \omega (\mathbf{p}_2)}}
+
\]
\[
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} v(-\mathbf{r}) 
\sqrt{{m \over \omega (\mathbf{r})}} 
\sqrt{{m \over \omega (\mathbf{r})}} 
\bar{v}(-\mathbf{r}) \gamma_5  u(\mathbf{p}_1) 
\sqrt{{m \over \omega (\mathbf{p}_1)}}
{1 \over E_{12} - \omega(\mathbf{p}_2') + \omega(\mathbf{r}) -i0^+}
\times
\]
\beq
{ g^2_{\pi} \over (2 \pi)^3}
{\btau_1 \cdot \btau_2  \over m^2 + (p_2-p_2')^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{p}_2')}}
\bar{u}(\mathbf{p}_2') \gamma_5  u (\mathbf{p}_2) 
\sqrt{{m \over \omega (\mathbf{p}_2)}}
+ \cdots 
\label{g.6}
\eeq
where 
\beq
E_{12} =  \omega(\mathbf{p}_1) + \omega(\mathbf{p}_2) 
\label{g.7}
\eeq
is the initial energy,
\beq
\mathbf{r}= \mathbf{p}_1 + \mathbf{p}_2 - \mathbf{p}_2' .
\label{g.8}
\eeq
and 
\beq
\Gamma^{\mu} =  \gamma^{\mu} F_1 (Q^2) + {1 \over 4 m} [\gamma^{\mu},
\gamma \cdot Q]_-
\label{g.8a}
\eeq
is the nucleon impulse current.   The $+\cdots$ represents the contribution 
from the other three terms related by either Hermitian conjugation 
and/or exchanging the proton and neutron.

To construct our model exchange current we first evaluate this
covariant expression in the rest frame of the initial two-body system
in so the pion-exchange interaction appears as a rotationally
invariant function of the relative momenta.  This allows us to relate
the interaction part of this kernel to the interaction kernel in
(\ref{g.3}).  Next we treat the initial energy of the two-body
system, $E_{12}$, as a parameter that can be expressed in terms of the
mass of the initial state (deuteron) and the kinematically conserved
momenta.  We use the assumed null-plane kinematic symmetry of the 
interaction to express the energy denominators in terms of masses 
and null-plane kinematic variables;  
$E_{12} \to {1 \over 2} (P^+ + {M^2 + \mathbf{P}_{\perp} \over
P^+})$, $E_{120} \to {1 \over 2} (P^+ + {M_0^2 + \mathbf{P}_{\perp} \over
P^+})$.  These are all model assumptions.  
In the rest frame of the initial deuteron $P^-=M_d$
which gives 
\beq E_{12} -\omega(\mathbf{r}) -\omega(\mathbf{p}_2') \to
E_{12} - E_{120}  \to {1 \over 2} {M_d^2
-M_0^2(\mathbf{k}') \over P^+_{rest}} \to {M_d^2 -M_0^2(\mathbf{k}')
\over 2 M_d } \approx {M_d^2 -M_0^2(\mathbf{k}') \over 4 m }
\label{g.9}
\eeq
and 
\beq
 E_{12} - \omega(\mathbf{p}_2') + \omega(\mathbf{r})  \to E_{12} \to 
{1 \over 2} (P^+ + {M^2 + \mathbf{P}_{\perp}
\over P^+}) \to
{1 \over 2} (M_d + {M_d^2 
\over M_d}) = M_d \approx 2m
\label{g.10}
\eeq
Using (\ref{g.9}) and (\ref{g.10}) in (\ref{g.6}) gives
\[
\langle \mathbf{p}_1', \mu_1', -\mathbf{k}', \mu_2' \vert I^{\mu}(0)^{\mu}
\vert \mathbf{k}, \mu_1, -\mathbf{k}, \mu_2 \rangle = 
\]
\[  
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} u(\mathbf{k}') 
\sqrt{{m \over \omega (\mathbf{k}')}} 
{4m \over M_d^2 - M_0^2  -i0^+}
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{u}(\mathbf{k}') \gamma_5  u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}} \times
\]
\[
{ g^2_{\pi} \over (2 \pi)^3}
{\btau_1 \cdot \btau_2  \over m_{\pi}^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \gamma_5  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
-
\]
\[
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} v(-\mathbf{k}') 
\sqrt{{m \over \omega (\mathbf{k}')}}
{1 \over 2m} 
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{v}(-\mathbf{k}') \gamma_5  u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
\times
\]
\beq
{ g^2_{\pi} \over (2 \pi)^3}
{\btau_1 \cdot \btau_2  \over m_{\pi}^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \gamma_5  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
+ \cdots 
\label{g.11}
\eeq

The terms in this expression are easy to interpret.  The first two
lines represent the product of a one-body current matrix element, the
propagator, ${4m \over M_d^2 - M_0^2 -i0^+}$, associated with
equation (\ref{c.4}) and the rotationally invariant kernel of the
pion-exchange interaction (\ref{g.3}).  This term is already included
in the one-body contribution to the current matrix element.

The last two lines have a similar form except one of the
$u(\mathbf{k})$ spinors in the current and the rotationally invariant
interaction kernel are replaced by $v(-\mathbf{k})$ spinors.  In
addition, the propagator term is replaced by the factor $1/2m$.  
We find it convenient for
computational purposes to split the $v$ spinor terms that 
give the covariant Dirac projector $\Lambda_-(-\mathbf{k})
=- v(-\mathbf{k}) \bar{v}(-\mathbf{k}) 
$ and 
to use $\gamma_5 \beta$ to convert the
$v(-\mathbf{k})$ to $u(\mathbf{k})$.  The price paid for this is the
introduction of two factors of $\beta$, neither of which are
separately covariant.  This causes no problem if we stay in the rest
frame of the initial system, but if we transform to the Breit frame,
it is necessary to replace both terms by kinematically covariant
expressions that reduce to $\beta$ in the rest frame.

Using   
\beq
v (-\mathbf{p}) \bar{v}(-\mathbf{p})  =
\gamma_5 \beta u (\mathbf{p}) \bar{u}(\mathbf{p})\beta \gamma_5  
\label{g.12}
\eeq
the last two lines of (\ref{g.11}) become 
\[
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu}   \gamma_5 \beta u (\mathbf{k}') 
\sqrt{{m \over \omega (\mathbf{k}')}}
{1 \over 2m} 
\times
\]
\beq
{ g^2_{\pi} \over (2 \pi)^3} 
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{u}(\mathbf{k}')\beta u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
{\btau_1 \cdot \btau_2  \over m^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \gamma_5  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
+ \cdots 
\label{g.13}
\eeq
This separates into the product of an effective one-body current,
\beq
\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} \gamma_5 
\beta u (\mathbf{k}') 
\sqrt{{m \over \omega (\mathbf{k}')}}
\label{g.14}
\eeq
a factor $1/2m$, and a modified one-pion-exchange interaction.
\beq
{ g^2_{\pi} \over (2 \pi)^3} 
\sqrt{{m \over \omega (\mathbf{k}')}} 
\bar{u}(\mathbf{k}')\beta u(\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
{\btau_1 \cdot \btau_2  \over m^2 + (k'-k)^2 -i0^+} 
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(-\mathbf{k}') \gamma_5  u (-\mathbf{k}) 
\sqrt{{m \over \omega (\mathbf{k})}}
\label{g.15}
\eeq
The interaction term is identical to the rotationally invariant kernel
of the one-pion-exchange interaction (\ref{g.3}) with the replacement:
$\bar{u}(\mathbf{k}')\gamma_5 u(\mathbf{k}) \to
\bar{u}(\mathbf{k}')\beta u(\mathbf{k})$.  This replacement 
preserves the rotational invariance of this kernel in the rest frame, 
but it is not covariant with respect to null-plane boosts.

This splitting has the advantage that the computation of the exchange
current matrix element has the same structure as the computation of an
impulse current matrix element with the current replaced by
(\ref{g.14}) and the deuteron wave function $\vert \psi \rangle$
replaced by ${1\over 2m} \tilde{v} 
\vert \psi \rangle$ where $\tilde{v}$ is the modified
interaction (\ref{g.15}).  The input to this calculation 
is defined in rest frame of the initial deuteron, but we need
it in the Breit frame.  This is elementary if both modified 
state and current transform covariantly with respect to the 
null-plane kinematic subgroup.

The effective one-body current (\ref{g.14}) can be extended to 
a kinematically covariant operator that agrees with 
(\ref{g.14}) in the deuteron rest frame by replacing 
the $\beta$ in (\ref{g.14}) by 
\beq
\beta \to - P\cdot \gamma /M_d    
\label{g.16}
\eeq
This leads to a kinematically covariant modified current kernel 
\beq
-\sqrt{{m \over \omega (\mathbf{p}_1')}}
\bar{u}(\mathbf{p}_1') \Gamma^{\mu} \gamma_5 {(P\cdot \gamma) 
\over M_d}   
u (\mathbf{k}') 
\sqrt{{m \over \omega (\mathbf{k}')}}.
\label{g.17}
\eeq
Kinematic covariance is all that this needed for a consistent 
computation of the current matrix elements.

The modified interaction (\ref{g.15}) is rotationally invariant 
so when it is applied to the rest deuteron eigenstate the 
resulting pesudo-state has the same spin as the deuteron.  
This can be consistently defined in any other kinematically 
related frame using null-plane boosts.  This, along with 
the replacement (\ref{g.16}) restores the kinematic covariance. 

The last step is to replace the modified interaction (\ref{g.15}) by
the corresponding modified one-pion-exchange part of a realistic
model interaction.  In a typical realistic interaction the
pseudoscalar pion exchange interaction is obtained by replacing the
spinor terms in the rotationally invariant kernel (\ref{g.3}) by
\beq
\sqrt{{m \over \omega (\mathbf{k}')}}
\bar{u}(\mathbf{k}')\gamma_5 u(\mathbf{k})
\sqrt{{m \over \omega (\mathbf{k})}}
\to {\bsig \cdot (\mathbf{k} - \mathbf{k}') \over 2m}
\label{g.18} 
\eeq
This expression is obtained by retaining the leading term in a 
$\mathbf{k}/m$ expansion of the spinor term, which is normally 
justified because the one-pion exchange interaction also includes 
a high-momentum or short-distance cutoff.  The net effect is that the 
resulting interaction, when included in the full nucleon-nucleon 
interaction, provides a good description of the two-nucleon bound state 
and scattering observables. 

Expanding
$\bar{u}(\mathbf{k}')\beta u(\mathbf{k}) $ to the same order in
$\mathbf{k}/m$
gives 1.
This suggests that the modified interaction (\ref{g.15}) 
can be modeled by replacing the one-pion exchange 
contribution to the realistic interaction  
\beq
\bsig_1 \cdot (\mathbf{k} - \mathbf{k}') v_\pi (\mathbf{k}-\mathbf{k}')  
\bsig_2 \cdot (\mathbf{k}' - \mathbf{k}) \btau_1 \cdot \btau_2 
\label{g.17a}
\eeq
by
\beq 
2m v_\pi (\mathbf{k}-\mathbf{k}')
\bsig_2 \cdot (\mathbf{k}' - \mathbf{k}) \btau_1 \cdot \btau_2 
\label{g.18a}
\eeq 
The one-pion exchange interaction (\ref{g.3}) only contributes to the
part of the tensor force in the nucleon-nucleon interaction that
multiplies the isospin exchange operator $\btau_1 \cdot \btau_2$.  The
tensor interaction also has contributions from the vector exchanges
which also contribute to the spin-spin interaction.  Riska
\cite{Riska85} and Schiavilla, Pandharipande, and Riska \cite{Schi90}
introduced a method for isolating the pion-exchange contribution to
the tensor force of a phenomenological interaction using linear
combinations of the tensor and spin-spin interactions. The resulting
interaction is
\beq
v_{ps}(\mathbf{k}-\mathbf{k}') = {1 \over 3} (2 v_t
(\mathbf{k}-\mathbf{k}') - v_{ss}(\mathbf{k}-\mathbf{k}')) 
\label{g.18b}
\eeq
where $v_t$ and $v_{ss}$ are tensor and spin-spin contributions to the
charge exchange part of the Argonne V18 interaction.
We extract this from the pion-exchange contribution to the Argonne V18
potential.  This is compared to ${1 \over m_{\pi}^2 +
  (\mathbf{k}-\mathbf{k}')^2}$ in figure 4.  The difference between
these curves is due to the short-distance cutoff used in the AV18
interaction.

When this interaction is applied to the deuteron bound state vector
the result is a spin 1 ``pseudo wave function''. The resulting
``pseudo vector'' can be defined in the Breit frame by requiring that
it transforms covariantly with respect to the null-plane kinematic
subgroup.  This kinematic covariance ensures that the current kernel
is kinematically covariant provided the pseudo-current is modified
following (\ref{g.16}).  The kinematic covariance of the current is
needed for a consistent calculation matrix elements of $I^+(0)$.

We can now write the form of the Breit frame matrix elements of this
one-pion exchange contribution to the exchange current:
\[
\langle \tilde{\mathbf{P}}',  \mu',d \vert I_{ex}^+ (0) \vert 
\tilde{\mathbf{P}}, \mu,d \rangle   =
\]
\[
\sum \int \langle \tilde{\mathbf{P}}', \mu' ,d\vert 
\tilde{\mathbf{P}}''' , k' ,j', l',s', \mu''' \rangle 
k^{\prime 2} dk' d\tilde{\mathbf{P}}'''
\times
\]
\[
\langle \tilde{\mathbf{P}}''' , k' ,j', l',s', \mu''' \vert
\tilde{\mathbf{p}}_p', \mu_p' , \tilde{\mathbf{p}}_n', \mu_n' 
\rangle 
d\tilde{\mathbf{p}}_p' d\tilde{\mathbf{p}}_n'
\times
\]
\[
\langle  \tilde{\mathbf{p}}'_p, \mu'_p , \tilde{\mathbf{p}}'_n, \mu'_n
\vert I_{ex-eff}^+(0) \vert 
\tilde{\mathbf{p}}_p, \mu_p , \tilde{\mathbf{p}}_n, \mu_n
\rangle 
d\tilde{\mathbf{p}}_p d\tilde{\mathbf{p}}_n
\times 
\]
\beq
\langle \tilde{\mathbf{p}}_p, \mu_p , \tilde{\mathbf{p}}_n, \mu_n
\vert \tilde{\mathbf{P}}'' , k ,j, l,s, \mu''
\rangle 
k^{2} dk d\tilde{\mathbf{P}}''
\langle  \tilde{\mathbf{P}}'' , k ,j, l,s, \mu'' 
\vert  \tilde{\mathbf{P}},  \mu ,\chi 
\rangle 
\label{g.19}
\eeq
where the terms in this expression are the deuteron wave 
function in the free-particle irreducible null-plane basis 
\beq
\langle \tilde{\mathbf{P}}',  \mu',d \vert 
\tilde{\mathbf{P}}''' , k' ,j', l',s', \mu''' \rangle
=
\delta (\tilde{\mathbf{P}}'- \tilde{\mathbf{P}}''')
\delta_{j'j'''} \delta_{\mu'\mu'''}
\phi^*_{j'} (k',l',s') \qquad (j'=s'=1),  
\label{g.20}
\eeq
the Poincar\'e group Clebsch-Gordan coefficients in the null-plane basis 
\[
\langle \tilde{\mathbf{P}}''' , k' ,j', l',s', \mu''' \vert
\tilde{\mathbf{p}}_p', \mu_p' , \tilde{\mathbf{p}}_n', \mu_n' 
\rangle = 
\]
\[
\delta ( \tilde{\mathbf{P}}'''- 
\tilde{\mathbf{p}}_p'- \tilde{\mathbf{p}}_n') 
{\delta (k' - k(\tilde{\mathbf{p}}_p',\tilde{\mathbf{p}}_n')) 
\over k^{\prime 2}}  
\sqrt{{\partial ( \tilde{\mathbf{P}} , \mathbf{k} ) \over
\partial (\tilde{\mathbf{p}}_p', \tilde{\mathbf{p}}_n') }}
\langle j, \mu \vert l, m_l, s, m_s \rangle
\times
\]
\[
\langle s, m_s \vert {1 \over 2}, \mu_p, {1 \over 2}, \mu_n \rangle 
Y_{l m_l}^* (\hat{\mathbf{k}}( \tilde{\mathbf{p}}_p' \tilde{\mathbf{p}}_n'))
\times
\]
\beq  
D^{1/2}_{\mu_p \mu_p'} [\Lambda_c^{-1} (\mathbf{k}/m) \Lambda_f (\mathbf{k}/m)]
D^{1/2}_{\mu_n \mu_n'} [\Lambda_c^{-1} (-\mathbf{k}/m) \Lambda_f (-\mathbf{k}/m)], 
\label{g.21}
\eeq
the proton effective current
\beq
\langle  \tilde{\mathbf{p}}'_p, \mu'_p , \tilde{\mathbf{p}}'_n, \mu'_n
\vert I_{ex-eff}^+(0) \vert 
\tilde{\mathbf{p}}_p, \mu_p , \tilde{\mathbf{p}}_n, \mu_n
\rangle =
\label{g.22}
\eeq
\beq
\delta (\tilde{\mathbf{p}}_n - \tilde{\mathbf{p}}_n)
\sqrt{{m \over p_p^{+\prime} }} 
\bar u_f(\tilde{\mathbf{p}}'_p,\mu'_p) 
\Gamma_p^\mu (q) \gamma^5 (\eta^0 \gamma^0 - \eta^1 \gamma^1) 
u_f(\tilde{\mathbf{p}}_p,\mu_p)  
\sqrt{{m \over p_p^{+} }},
\label{g.23}
\eeq
where the null plane Dirac spinors are related to the 
canonical Dirac spinors by a Melosh rotation
\beq
u_f(\tilde{\mathbf{p}}_p,\mu_p)_{\mu}   =
u_c(\tilde{\mathbf{p}}_p,\mu_p)_{\mu'}  
D^{1/2}_{\mu' \mu} [\Lambda_c^{-1} (\mathbf{p}_p/m) \Lambda_f (-\mathbf{p}_p/m)]
\label{g.24}
\eeq
another Poincar\'e Clebsch-Gordan coefficient
\beq
\langle \tilde{\mathbf{p}}_p, \mu_p , \tilde{\mathbf{p}}_n, \mu_n
\vert \tilde{\mathbf{P}}'' , k ,j, l,s, \mu''
\rangle = 
\label{g.25}
\eeq
\beq
\delta ( \tilde{\mathbf{P}}'''- 
\tilde{\mathbf{p}}_p'- \tilde{\mathbf{p}}_n') 
{\delta (k' - k(\tilde{\mathbf{p}}_p', \tilde{\mathbf{p}}_n')) 
\over k^{\prime 2}}  
\sqrt{{\partial ( \tilde{\mathbf{P}} , \mathbf{k} ) \over
\partial (\tilde{\mathbf{p}}_p', \tilde{\mathbf{p}}_n') }}
\times
\label{g.26}
\eeq
\[
D^{1/2}_{\mu_p \mu_p'} [\Lambda_f^{-1} (\mathbf{k}/m) \Lambda_c (\mathbf{k}/m)]
D^{1/2}_{\mu_n \mu_n'} [\Lambda_f^{-1} (-\mathbf{k}/m) \Lambda_c (-\mathbf{k}/m)]
\times
\]
\beq 
\langle s, m_s \vert {1 \over 2}, \mu_p', {1 \over 2}, \mu_n' \rangle 
\langle j, \mu \vert l, m_l, s, m_s \rangle 
Y_{l m_l} (\hat{\mathbf{k}}( \tilde{\mathbf{p}}_p', \tilde{\mathbf{p}}_n')),  
\label{g.27}
\eeq
and the pseudo wave function
\[
\langle  \tilde{\mathbf{P}}'' , k ,j, l,s, \mu'' 
\vert  \tilde{\mathbf{P}}, \mu ,\chi 
\rangle =  
\]
\[
\delta (\tilde{\mathbf{P}}'' - \tilde{\mathbf{P}})
\int \sum
\langle j, \mu'' \vert l, m_l, s ,m_s \rangle 
\langle s, m_s  \vert {1 \over 2},  \mu_p ,{1 \over 2}, \mu_n \rangle  
{1 \over (2 \pi)^{3/2}} 
Y^*_{l m_l}( \hat{\mathbf{k}})  d\hat{\mathbf{k}}
\times 
\]
\beq
(\mathbf{k} - \mathbf{k}')\cdot \bsig_{\mu_n \mu'_n} 
v_{ps}(\mathbf{k}' - \mathbf{k}) \btau_1 \cdot \btau_2
d \mathbf{k}' 
Y_{l' m_l'}( \hat{\mathbf{k}}')
\langle s', m'_s, \vert  {1 \over 2},  \mu_p' ,{1 \over 2}, \mu_n' \rangle 
\langle j, \mu, \vert, s', m'_s, l', m'_l \rangle .
\phi_{lsj} (k')  
\label{g.28}
\eeq
where $v_{ps}$ is given by (\ref{g.18}).
The full exchange current is the sum of the above quantity and
the three other terms related by taking Hermitian conjugates or 
exchanging the particle that couples to the photon.
The details of how these quantities are computed is discussed in 
Appendix B.

\section{Calculation of pseudo wave function}

The calculation of the pseudo wave function$~(\ref{g.28})$ requires
the one-pion exchange part of the tensor interaction.
Riska~\cite{Riska85} and Schiavilla, Pandharipande, and Riska
\cite{Schi90}outlined a method to extract the pseudoscalar
contribution to the tensor force as a linear combination of the radial
coefficient functions in the Argonne V18 interaction.  The method is
based on the observation that both pseudoscalar and vector meson
exchange contribute to both the tensor and spin-spin interaction in
the static limit\cite{Riska85}.

The Fourier transform of the interaction has the structure 
\beq
v_{nn}(\bkappa) = \Omega_t v_t (\bkappa^2) + 
\Omega_{ss} v_{ss} (\bkappa^2)
+ \cdots 
\label{i.1}
\eeq
where $\bkappa:= \mathbf{k}'-\mathbf{k}$,
\beq 
\Omega_{t}:= (\bsig_{1} \cdot \bsig_2 \bkappa^2-3 \bsig_1 \cdot \bkappa \bsig_2 \cdot  \bkappa)(   \btau_1 \cdot \btau_2)
\label{i.2}
\eeq
\beq
\Omega_{s}:=\bkappa^2 (\bsig_1 \cdot \bsig_2)(  \btau_1 \cdot \btau_2)
\label{i.3}
\eeq
and we have explicitly exhibited the tensor and spin-spin contributions
to the interaction.

The coefficient functions in (\ref{i.1}) are Fourier transforms 
of the coefficient functions the corresponding 
terms in the Argonne V18 interaction, 
\beq 
v_{ss}(\bkappa^2)=\frac{4\pi}{\bkappa^2}
\int^{\infty}_{0} dr r^2 v^{\sigma \tau}(r)(j_0(\kappa r)-1),
\label{i.4}
\eeq
\beq 
v_{t}(\bkappa^2)=\frac{4 \pi}{\bkappa^2}\int_{0}^{\infty} 
dr r^2 v^{t\tau}(r)j_2(\kappa r),
\label{i.5}
\eeq
where the factor $1/ \bkappa^2$ is due to the difference in
the conventions used to define tensor operator in  
momentum and configuration space.

These coefficient functions have contributions
from both pseudoscalar and vector meson exchange.  To separate
them, following \cite{Riska85}, define the tensor and spinor operators
and note that 
\beq
(\bsig_1 \cdot \bkappa \bsig_2 \cdot \bkappa) ( (  \btau_1 \cdot \btau_2)) = 
- \frac{1}{3}
({\Omega}_{t} -  \Omega_{ss})
\label{i.6}
\eeq
while vector meson exchange gives the combination \cite{Riska85}
\beq
((\bsig_1 \times \bkappa) \cdot (\bsig_2 \times \bkappa)
)(  \btau_1 \cdot \btau_2) =
\frac {1}{3} [2\Omega_{ss} + \Omega_{t} ] .
\label{i.7}
\eeq
This implies that 
\beq
v_t \Omega_t + v_{ss} \Omega_{ss} 
= v_{ps} \frac{1}{3}
({\Omega}_{ss} -  \Omega_{t}) + 
v_{v} \frac {1}{3} (2\Omega_{s} + \Omega_{t} ) =
\label{i.8}
\eeq
\beq
\frac{1}{3} (v_{v}-v_{ps}) \Omega_t + {1\over 3} 
(v_{ps} + 2 v_{v}) \Omega_s .
\label{i.9}
\eeq
Using these expressions we can isolate the pseudoscalar and vector
contribution to the interaction using the following linear combinations of the 
tensor and spin-spin interactions:
\beq
v_{ps} = v_{ss} - 2 v_t
\label{i.10}
\eeq
\beq
v_{v} = v_{ss} + v_t .
\label{i.11}
\eeq
For the Argonne V18 interaction, which is used in our calculations,
there remains a small difference between the pseudoscalar interaction
calculated using (\ref{i.10}) and the contribution that comes
directly from terms that can be directly identified with the one-pion
exchange contribution.  We only retain the one-pion exchange
contribution to these terms, this still
includes the short-distance cutoff used
in the Argonne V18 r-space potential.

The interactions 
$v^{\sigma \tau}(r)$ and $v^{t\tau}(r)$ have the form 
\beq
v^{\sigma \tau}(r) \equiv {f^2\over 9}\left\{\left({m_0\over m_\pm}\right)^2m_0 Y(\mu_0,r)
+2m_\pm Y(\mu_\pm,r)\right\}+v^c(r)
\label{i.12}
\eeq
\beq
v^{t\tau}(r) \equiv {f^2\over 9}\left\{\left({m_0\over m_\pm}\right)^2m_0 T(\mu_0,r)
+2m_\pm T(\mu_\pm,r)\right\}+v^t(r) .
\label{i.13}
\eeq
Here $Y(\mu ,r)$ and $T(\mu ,r)$ are the usual Yukawa and tensor
functions with the exponential cutoff of the Urbana and Argonne
$V_{14}$ models:
\beq
Y(\mu ,r) = {e^{-\mu r} \over \mu r} (1 - e^{-cr^2}),
\label{i.14}
\eeq
\beq
T(\mu ,r) = (1 + {3 \over \mu r} + {3 \over (\mu r)^2 } )Y(\mu r)
(1 - e^{-cr^2}),
\label{i.15}
\eeq
The terms $v^c(r)$ and $v^t(r)$ are short-range phenomenological 
interactions that we discard for the computation of the
exchange current.
 
In order to carry out the angular part of the integral in (\ref{g.28}), we 
expand the pseudoscalar potential $v_{ps}(\mathbf{k} - \mathbf{k}')$ in terms of partial waves:    
\beq
v_{ps} (| \mathbf{k}-\mathbf{k}'|) =
\sum_{lm} v_l(\mathbf{k}^2,\mathbf{k}^{\prime 2})Y^*_{lm}(\hat{\mathbf{k}})
Y_{lm}(\hat{\mathbf{k}}'),  
\label{i.16}
\eeq
where 
\beq
v_l (\mathbf{k}^2,\mathbf{k}^{ \prime 2} ) = 2 \pi \int_{-1}^1
P_l (u) v_{ps}(\sqrt{\mathbf{k}^2 + \mathbf{k}^{\prime 2} - 2 kk'' u} )du.  
\label{i.17}
\eeq
We expand 
the vector $\mathbf{k}$ and $\mathbf{k}'$  in spherical 
harmonics
\beq
\mathbf{k} = k(\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{\mathbf{k}})
- Y_{11}(\hat{\mathbf{k}})), i\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{\mathbf{k}})
+ Y_{11}(\hat{\mathbf{k}})), \sqrt{{4 \pi \over 3}} Y_{10}(\hat{\mathbf{k}})).
\label{i.18}
\eeq
to obtain the following expression for the 
pseudo wave function:
\beqa 
&\quad & \chi_{\tilde P_d,\mu_d}^{(c)}(\vec {k}'',\mu''_p,\mu'_n) \\
&\equiv & \frac{1}{(2\pi)^3}   
\int d\hat{\mathbf{k}} k^2 dk  \sum_{l=0,2} \sum_{\mu_l=-l}^l \sum_{l'=0}^\infty 
\sum_{\mu_l'=-l'}^{l'} v_{l'} (\mathbf{k}^2,\mathbf{k}^{''2}  )
Y_{l'\mu_l'}(\hat{\mathbf{k}}'') Y^*_{l' \mu_l'} (\hat{\mathbf{k}})
Y_{l\mu_l}(\hat{\mathbf{k}}) \nonumber \\
&\times &
[k''(\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{ \mathbf{k}''})
- Y_{11}(\hat{\mathbf{k}''})), i\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{\mathbf{k}''})
+ Y_{11}(\hat{\mathbf{k}})), \sqrt{{4 \pi \over 3}} Y_{10}(\hat{\mathbf{k}})) \nonumber \\
&-&
k(\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{ \mathbf{k}})
- Y_{11}(\hat{\mathbf{k}})), i\sqrt{{2 \pi \over 3}} (Y_{1-1}(\hat{\mathbf{k}})
+ Y_{11}(\hat{\mathbf{k}})), \sqrt{{4 \pi \over 3}} 
Y_{10}(\hat{\mathbf{k}}))] \nonumber \\
&\times & {\langle \mu'_n  \vert \vec \sigma \vert \mu_n \rangle}
\langle {s_p},\mu''_p,{s_n},\mu_n \vert s,\mu_s \rangle 
\langle  s, \mu_s, l,\mu_l\vert j,\mu \rangle 
Y_{\mu_l}^l(\hat{\mathbf{k}}) u_l(k), \nonumber
\label{i.19}
\eeqa
The angular integrals are evaluated using:
\beq
\int d\hat{\mathbf{k}}
Y^{*l'}_{\mu_l'} (\hat{\mathbf{k}})
Y^l_{\mu_l} (\hat{\mathbf{k}}) = \delta_{l'l} \delta_{\mu_l'\mu_l},
\label{i.20}
\eeq
\beqa
&\quad & \int d\hat{\mathbf{k}}
Y^{*l'}_{\mu_l'} (\hat{\mathbf{k}})
Y^l_{\mu_l} (\hat{\mathbf{k}})
Y^1_{\mu_l''} (\hat{\mathbf{k}}) \\
& = & 
\sqrt{{(2\cdot 1 +1) (2 \cdot l +1) \over 4 \pi (2\cdot l'+1)}}
\langle 1, \mu_l'', l, \mu_l \vert l', \mu_l' \rangle
\langle 1, 0, l, 0 \vert l', 0 \rangle. \nonumber
\label{i.21}
\eeqa

After carrying out the integral on $\hat{\mathbf{k}}$:
the expression for the pseudo wave function becomes 
\beqa
&\quad & \chi_{\tilde P_d,\mu_d}^{(c)}(\vec {k}'',\mu''_p,\mu'_n) \\
&\equiv & \frac{1}{(2\pi)^3}   
\int k^2 dk  \sum_{l=0,2} \sum_{\mu_l=-l}^l \sum_{l'}  \sum_{\mu_l'=-l'}^{l'}
v_{l'} (\mathbf{k}^2,\mathbf{k}^{\prime \prime 2}, )
Y_{l'\mu_{l'}}(\hat{\mathbf{k}}'') u_l (k)
\nonumber \\ &\times & 
\langle {s_p},\mu''_p,{s_n},\mu_n \vert s,\mu_s \rangle 
\langle  s, \mu_s, l,\mu_l\vert j,\mu \rangle 
\nonumber \\ &\times & 
[
\sqrt{{2 \pi \over 3}} \langle \mu_n' \vert \sigma_{x} \vert \mu_n \rangle 
\quad [\quad k''(Y_{1-1}(\hat{\mathbf{k}''}) - Y_{11}(\hat{\mathbf{k}''}) )\delta_{l'l}\delta_{\mu_{l'}\mu_l}
\nonumber \\ &+& 
k\sqrt{{(2\cdot 1 +1) (2 \cdot l +1) \over 4 \pi (2\cdot l'+1)}}
\langle 1, 0, l, 0 \vert l', 0 \rangle
(-\langle 1, -1, l, \mu_l \vert l', \mu_{l'} \rangle
+\langle 1, 1, l, \mu_l \vert l', \mu_{l'} \rangle)
\quad ]
\nonumber \\ &+& 
i\sqrt{{2 \pi \over 3}} 
\langle \mu_n' \vert \sigma_{y} \vert \mu_n \rangle
\quad [ \quad
k''( Y_{1-1}(\hat{\mathbf{k}''}) + Y_{11}(\hat{\mathbf{k}''}) )\delta_{l'l}\delta_{\mu_{l'}\mu_l}
\nonumber \\ &+& 
k \sqrt{{(2\cdot 1 +1) (2 \cdot l +1) \over 4 \pi (2\cdot l'+1)}}
\langle 1, 0, l, 0 \vert l', 0 \rangle
(-\langle 1, -1, l, \mu_l \vert l', \mu_{l'} \rangle
-\langle 1, 1, l, \mu_l \vert l', \mu_{l'} \rangle)
\quad ]
\nonumber \\ &+&
\sqrt{{4 \pi \over 3}} 
\langle \mu_n' \vert \sigma_{z} \vert \mu_n \rangle
\quad [ \quad 
k'' Y_{10}(\hat{\mathbf{k}''})\delta_{l'l} \delta_{\mu_{l'}\mu_l} 
\nonumber \\ &-&
k \sqrt{{(2\cdot 1 +1) (2 \cdot l +1) \over 4 \pi (2\cdot l'+1)}}
\langle 1, 0, l, 0 \vert l', 0 \rangle 
\langle 1, 0, l, \mu_l \vert l', \mu_{l'} \rangle
\quad ]\quad 
]\nonumber
\label{i.22}
\eeqa
The sum over $l'$ includes only a finite number of values.  It can
only be $0$ or $1$ for $l=0$ and $1$, $2$ or $3$ when $l=2$.  
This limits the sum on $l'$ to the first four partial waves,
$l'=0,1,2,3$.  The final expression of the pseudo wave function can be
written in the form:
\beq
\chi_{\tilde P_d,\mu_d}^{(c)}(\vec {k}'',\mu''_p,\mu'_n) 
= \sum _{ll'}I_{ll'}(k'') f_{ll'}(\hat k'',\mu''_p,\mu'_n) ,
\label{i.23}
\eeq
where $f_{ll'}(\hat k'',\mu''_p,\mu'_n)$ are angle-dependent coefficients 
and $I_{ll'}(k'')$ are the relevant scalar quantities which have the following form: 
\beq
I_{ll'}(k'') \equiv \int k^2 dk v_l (k,k'') u_{l'} (k) 
\label{i.24}
\eeq
or
\beq
I_{ll'}(k'') \int k'^3 dk v_l (k,k'') u_{l'}(k).
\label{i.25}
\eeq
The allowed combinations of $l$ and $l'$ pairs requires the 
following integrals $I_{ll'}(k'')$:
\beq
I_{00}(k'')\equiv \int v_0(k,k'') u_0 (k) k^2 dk 
\label{i.26}
\eeq
\beq
I_{10}(k'')\equiv \int v_1(k,k'') u_0 (k) k^3 dk 
\label{i.27}
\eeq
\beq
I_{12}(k'')\equiv \int v_1(k,k'') u_2 (k) k^3 dk 
\label{i.28}
\eeq
\beq
I_{22}(k'')\equiv \int v_2(k,k'') u_2 (k) k^2 dk 
\label{i.29}
\eeq
\beq
I_{32}(k'')\equiv \int v_3(k,k'') u_2 (k) k^3 dk 
\label{i.30}
\eeq

The total exchange current contribution:

\begin{eqnarray}
& \quad & \langle  \tilde{\mathbf{P}}_d', \mu_d',d 
\vert I^+_{ex}(0) \vert  \tilde{\mathbf{P}}_d , \mu_d ,d \rangle 
=  \sum \int d{\mathbf{k}'} J_i  
\\ & \times &
[  
\Psi^*_{\tilde{\mathbf{P}}'_d, \mu'_d} (\mathbf{k}', \mu'_p,  \mu_n)
[ \bar u_f(\tilde{\mathbf{p}}'_p,\mu'_p) 
\Gamma_p^\mu (q) \gamma^5 
%(\eta^0 \gamma^0 - \eta^1 \gamma^1) 
(- P \cdot \gamma/M_d)
u_f(\tilde{\mathbf{p}}_p,\mu_p) ]
\chi_{\tilde{\mathbf{P}}_d,\mu_d}^{(f)}(\mathbf{k}, \mu_p,  \mu_n) \nonumber \\
&+& 
\chi^{(f)*}_{\tilde{\mathbf{P}}'_d, \mu'_d}(\mathbf{k},  \mu'_p,  \mu_n) 
[ \bar u_f(\tilde{\mathbf{p}}'_p,\mu'_p)
%(\eta^0 \gamma^0 - \eta^1 \gamma^1)
(- P \cdot \gamma/M_d)
\gamma^5 \Gamma_p^\mu (q)u_f(\tilde p_p,\mu_p)] %  
\Psi_{\tilde{\mathbf{P}}_d,\mu_d}(\mathbf{k},  \mu_p,  \mu_n)
] \nonumber  \\
&+&
[ p \leftrightarrow n ] \nonumber
\label{i.31}
\end{eqnarray}
where $J_i$ is the jacobian 
\beq 
J_i=\vert \frac {\partial (\tilde{\mathbf{p}}''_p, \tilde{\mathbf{p}}''_n)} 
{\partial (\tilde{\mathbf{P}}'',\mathbf{k}'')}\vert^{\frac{1}{2}}  
\vert \frac {\partial (\tilde{\mathbf{P}}',\mathbf{k}')}{\partial 
(\tilde{\mathbf{p}}'_p, \tilde{\mathbf{p}}'_n)} \vert^{\frac{1}{2}}
\label{i.32}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliography{deut_paper_3x}

\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\setlength{\captionmargin}{23pt}
\begin{table}%[t]
\begin{center}
\begin {tabular}{rrrrr}
\hline
\hline
$\quad$  & IM & IM+Exchange  & IM(WSS) & IM+MEC(WSS) \\
\hline
$Q_d$  & 0.2698 & 0.2752   &  0.270 &  0.275 \\
\hline
$\mu_d $  & 0.8535 & 0.8596  & 0.847 & 0.871 \\
\hline
\hline
\end{tabular}%}
\caption
%[deuteron magnetic and quadrupole moments evaluated in the impulse approximation and including the ``pair'' current contribution constructed with(WS) and without(NS) the residual short-range potential removed. ]
{deuteron magnetic and quadrupole moments evaluated in the impulse
  approximation and including the exchange current contribution. 
The values are the same using all six different
  nucleon form factor parameterizations and three combinations of
  independent current matrix elements. Argonne $V18$ potential is used
  in the calculation.  The values labeled with WSS are
  from~\cite{Wiringa95}. The experimental values are $0.2860\pm
  0.0015 fm^2$ ~\cite{Bishop79} and $0.857406\pm 0.000001 \mu_N$
  ~\cite{Lindg65}.}
\label{momts}
\end{center}
\end{table}

\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig1.ps}
}
\end{center}
\caption{Neutron electric form factor parameterizations 
\label{fig1}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig2.ps}
}
\end{center}
\caption{Isoscalar nucleon form factor $F_{1N}(Q^2)$ 
\label{fig2}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig3.ps}
}
\end{center}
\caption{Isoscalar nucleon form factor $F_{2N}(Q^2)$ 
\label{fig3}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig4.ps}
}
\end{center}
\caption{One $\pi$-exchange interactions. 
\label{fig4}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig5.ps}
}
\end{center}
\caption{Deuteron form factor $G_0(Q^2)$ with and without exchange current 
\label{fig5}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig6.ps}
}
\end{center}
\caption{Deuteron form factor $G_1(Q^2)$ with and without exchange current
\label{fig6}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig7.ps}
}
\end{center}
\caption{Deuteron form factor $G_2(Q^2)$ with and without exchange current 
\label{fig7}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig8.ps}
}
\end{center}
\caption{$A(Q^2)$: Generalized impulse approximation 
\label{fig8}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig9.ps}
}
\end{center}
\caption{$B(Q^2)$: Generalized impulse approximation  
\label{fig9}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig10.ps}
}
\end{center}
\caption{$T_{20}(Q^2, 70^o)$: Generalized impulse approximation  
\label{fig10}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig11.ps}
}
\end{center}
\caption{$A(Q^2)$: Generalized impulse approximation plus exchange current I
\label{fig11}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig12.ps}
}
\end{center}
\caption{$B(Q^2)$: Generalized impulse approximation plus exchange current I
\label{fig12}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig13.ps}
}
\end{center}
\caption{$T_{20}(Q^2,70^o)$: Generalized impulse approximation plus exchange current I 
\label{fig13}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig14.ps}
}
\end{center}
\caption{Calculated vs. derived $I^{+}_{++} (0) - I^+_{00} (0)$ with and without
exchange currrent I 
\label{fig14}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig15.ps}
}
\end{center}
\caption{$A(Q^2)$: Generalized impulse approximation plus exchange current II 
\label{fig15}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig16.ps}
}
\end{center}
\caption{$T_{20}(Q^2,70^o)$: Generalized impulse approximation plus exchange current II  
\label{fig16}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig17.ps}
}
\end{center}
\caption{$A(Q^2)$: Generalized impulse approximation plus exchange current III  
\label{fig17}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig18.ps}
}
\end{center}
\caption{$T_{20}(Q^2,70^o)$: Generalized impulse approximation plus exchange current III   
\label{fig18}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig19_2.ps}
%\includegraphics[width=13cm]{fig19.ps}
}
\end{center}
\caption{$A(Q^2)$: CD Bonn with and without exchange current
\label{fig19}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig20_2.ps}
%\includegraphics[width=13cm]{fig20.ps}
}
\end{center}
\caption{$B(Q^2)$: CD Bonn with and without exchange current 
\label{fig20}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig21_2.ps}
%\includegraphics[width=13cm]{fig21.ps}
}
\end{center}
\caption {$T_{20}(Q^2,70^o)$: CD Bonn with and without exchange current
\label{fig21}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig22.ps}
}
\end{center}
\caption{$A(Q^2)$, V18,II, III, BBBA Form factors with and without exchange current  
\label{fig22}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig23.ps}
}
\end{center}
\caption{$B(Q^2)$ , V18,II, III, BBBA Form factors with and without exchange current
\label{fig23}
}
\end{figure}

\clearpage

\begin{figure}
\begin{center}
\rotatebox{270}{
\includegraphics[width=13cm]{fig24.ps}
}
\end{center}
\caption{$T_{20}(Q^2,70^o)$ , V18,II, III, BBBA 
Form factors with and without exchange current 
\label{fig24}
}
\end{figure}

\end{document}

