The Nucleon Elastic Form Factors
The nucleon elastic form factors are of fundamental interest, provide a unique testing ground for QCD motivated models of nucleon structure and are of critical importance to our understanding of the electromagnetic properties of nuclei. Even after an experimental effort spanning nearly 50 years the nucleon form factors are still the subject of active investigation. Advances in polarized beams, polarized targets and recoil polarimetry have been exploited over the last decade to produce an important and precise set of data. I review the status of the experimental efforts to measure the nucleon elastic form factors.
Day-Proc-Baryons.tex
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\title{The Nucleon Elastic Form Factors}

\author{Donal Day\address{Department of Physics
   University of Virginia, Charlottesville, Virginia 22904 }}
       
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\newcommand{\gepgmp}{$G_E^p/G_M^p\:$}
\newcommand{\gen}{$G_E^n\,$}
\newcommand{\gmn}{$G_M^n\,$}
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\begin{document}

% typeset front matter
\maketitle

\begin{abstract}
   The nucleon elastic form factors are of fundamental interest, provide a unique testing ground for QCD motivated models of nucleon structure and are of critical importance to our understanding of the electromagnetic properties of nuclei.
   Even after an experimental effort spanning nearly 50 years the nucleon form factors are still the subject of active investigation.
   Advances in polarized beams, polarized targets and recoil polarimetry have been exploited over the last decade to produce an important and precise set of data.
I  review the status of the experimental efforts to measure the nucleon elastic form factors.
\end{abstract}

\section{INTRODUCTION}
%  The magnetic moments measurements by Otto Stern in 1934 were the first evidence
%that the neutron and the proton  were composite particles, ones with internal
%structure. Without compositeness, one would expect the magnetic moment of the
%proton to be one nuclear magneton and that of the neutron to be zero.

The nucleon elastic form factors encapsulate important information about  their internal structure and have been the subject of sustained experimental and theoretical investigations for almost 50 years. Recent experimental efforts, especially those exploiting spin degrees of freedom, have produced an impressive data set out to large momentum transfers, generating considerable theoretical interest.

%The source of the nucleon  anomalous  magnetic moments is the strong interaction 
%which gives rise to complex electromagnetic currents of quarks and antiquarks in
%the nucleon. 
%The non-zero value  of the neutron's magnetic moment implies that the neutron
%must have a charge distribution. 
%  Precise knowledge of this charge distribution will give important information
%about the strong force that binds quarks together in neutrons and protons and
%other composite particles. The distribution of the charge  is contained in an
%experimentally determined quantity, the electric form factor, \gen $\!$\,,  a
%function of momentum transfer.

%Nucleons are composed of quarks and gluons and information about  their
%internal structure  is critical for testing quark models. For example in a
%symmetric quark model, with all the valence quarks with the same wave function,
%the charge would everywhere be zero and \gen = 0. Any deviation from zero   
%exposes the details of the wavefunctions.  \gen is critical for any study of
%nuclear structure -- without an accurate description of the nucleon form factors
%it is almost impossible  to obtain information from the few body structure
%functions, our best testing ground for FSI, MEC, and NN potentials (see, for
%example, \cite{acg1980}).

The earliest investigations exploited elastic electron scattering to determine the Dirac and Pauli form factors, $F_1$ and $F_2$, which are functions of momentum transfer, $Q^2=4E_0E'\sin^2(\theta/2)$, alone. In single photon exchange, the elastic cross section is written as:
\begin{equation}
\frac{d \sigma}{d \Omega}   =  \sigma _{\rm{Mott}}\frac{E'}{E_{0}}\left\{ \left( F_{1}\right) ^{2}+\tau \left[ 2\left( F_{1}+F_{2}\right) ^{2}\tan ^{2}\left( \theta\right) +(F_{2}^{})^{2}\right] \right\}, \label{eq:f1f2}
 \end{equation}
 where  $\tau=\frac{Q^2}{4M^2}$, $\theta$ is the electron scattering angle, $E_0,E'$ are the incident and final electron energies respectively, and $\sigma _{\rm{Mott}}$ is the Mott cross section. Because of their  direct relation (in the Breit frame) to the Fourier transforms of the charge and magnetization distributions in the nucleon, the Sachs form factors are commonly used. They are  linear combinations of $F_1$ and $F_2$: 
$G_{E}   =  F_{1}-\tau F_{2}$ and 
$G_{M}   =  F_{1} +F_{2}$. Early measurements of the form factors established a scaling law relating three of the four nucleon elastic form factors and the dipole law describing their common $Q^2$-dependence, $G_{E}^{p}(Q^2)  \approx \frac{G_{M}^{p}(Q^2)}{\mu_{p}}\approx \frac{G_{M}^{n}(Q^2)}{\mu_{n}} \approx  G_{D}\equiv \left( 1+Q^{2}/0.71\right) ^{-2}.$ This behavior is known as form factor scaling.
%    The neutron electric form factor has been usefully parametrized by  $G_E^n = -\mu_NG_D\frac{\tau}{1+5.6\tau}$.


%\subsection{Theory}
% The nucleon form factors have been described in terms of  Vector Meson
%Dominance (VMD)\cite{ijl73,hohler76} which  constructs the  interaction of the
%photon with the nucleon in terms of the coupling strengths of the virtual photon
%and the vector mesons and between  the vector mesons and the nucleon.
%Practitioners have adopted various approaches as to which vector mesons to
%include (all have included the  lowest mass mesons, 
%$\rho$ and $\omega$), which of the parameters to be set by data or fit  from the
%nucleon  form factors themselves, and to which degree the widths of the mesons
%are incorporated. The success of these models to describe the form factors at
%low and moderate $Q^2$  has been  offset by their failure to accommodate the
%pQCD fall off of the form factors at high $Q^2$. In quark-dimensional scaling
%\cite{brodsky75-81} the photon couples to a single quark and each   gluon
%exchange necessary  to share the momentum among the quarks contributes a factor
%proportional  to $Q^{-2}$  thereby requiring $F_1 \propto 1/Q^4$ and $F_2
%\propto 1/Q^6$.  Gari and Krumpelmann incorporated the asymptotic pQCD behavior
%in a VMD hybrid model \cite{gk73-86,gk92}. In the latter publication the model
%was modified to include the $\phi$ meson which has  significant consequences for
%the neutron electric form factor. 

%There are many models that are based on QCD and for any such model to survive it
%must predict the nucleon  form factors. Attempts to  extend  pQCD to
%experimentally accessible values of $Q^2$ include the work of Kroll {\it et
%al.}\cite{Kroll92}. The nucleon is manifested in this model as a quark and a
%diquark with the diquark approximating the role of correlations in the nucleon
%wave function.  Relativistic constituent quark models have been employed to
%calculate the four nucleon form factors with a varying degree of success. These
%include  Chung and Coester \cite{Chung1991} and Frank, Jennings, and Miller
%\cite{ Frank1996}.  Other QCD models include
%the chiral soliton model \cite{Holzwarth1996} which describes the basic features
%of
%nucleon form factors. It includes relativistic corrections, an extended object
%(skrymion) and coupling to the vector mesons. The high $Q^2$ data on $G_E^p$
%\cite{jones,gayou,gayou2} has generated new motivation  for the application of
%these models,  see  \cite{Cardarelli2000}. Dong {\it et al.} \cite{Dong1998}
%reported a lattice QCD calculation of \gen motivated in part by a study of the
%strangeness magnetic moment of the nucleon. Lu {\it et al.} \cite{Lu1998} has
%invoked the cloudy bag model to describe the form factors; here the nucleon is
%expressed as a bag containing three quarks, akin to the MIT bag model, with a
%pion field coupled to them in such a way that chiral symmetry is respected.


\section{EARLY EXPERIMENTAL TECHNIQUES}
\subsection{Proton Form Factors}
The proton form factors have been, until recently, only separated  through the Rosenbluth technique, which can be understood by re-writing Eq.~\ref{eq:f1f2} using the Sachs form factors,
\begin{equation}
\frac{d \sigma}{d \Omega} = \sigma_{\rm{NS}}
\left[\frac{G_E^2 + \tau G_M^2}{1+ \tau} +  2\tau G_M^2 \tan^2(\theta/2)\right],\end{equation} and rearranging, with $\epsilon^{-1} = 1 + 2(1+\tau)\tan^2(\theta/2)$ and $\sigma_{\rm{NS}}= \sigma _{\rm{Mott}}E'/E_{0}$, to give:

\begin{equation} \sigma_R \equiv \frac{d \sigma}{d \Omega} \frac{\epsilon(1+\tau)}{\sigma_{\rm{NS}}} = \tau G^2_M(Q^2)+\epsilon G_E^2(Q^2).
\end{equation} By making measurements at a fixed $Q^2$ and variable $\epsilon(\theta, E_0)$,
the reduced cross section $\sigma_{\rm{R}}$ can be fit with a straight line with slope $G_E^2$ and intercept $\tau G^2_M$.  Figure~\ref{fig:ne11} gives an example of the reduced cross section plotted in this way.
It should be noted that the Rosenbluth formula holds only for single photon exchange and it has been assumed (until recently) that any two-photon contribution is small.


\begin{figure}
  \begin{center}
    \begin{minipage}[c]{0.48\textwidth}
      %\epsfig{file=somefile.ps, width=\linewidth}
      \includegraphics[width=\textwidth]{ne11ellt.eps}
    \end{minipage}\hfill
    \begin{minipage}[c]{0.48\textwidth}
      \caption{Reduced cross section plotted against $\epsilon$ for a range of fixed momentum transfers. The data is fit with a straight line with a slope $G_E^2$ and an intercept $\tau G^2_M$. The linear dependence of the reduced cross section  on $\epsilon$ assumes that any two-photon exchange effects are small. The data is from Ref.~\cite{Andivahis:1994rq}.}\label{fig:ne11}
    \end{minipage}
  \end{center}
\end{figure}

%\begin{figure}
%\centering\includegraphics[width=0.35\textwidth]{ne11ellt.eps}
%\caption{Reduced cross section plotted against $\epsilon$ for a range of fixed momentum transfers.}\label{fig:ne11}
%\end{figure}

 
The Rosenbluth method is problematic--it requires the  measurement of absolute cross sections and  at  large $Q^2$ the cross section is insensitive to  $G_E$ and the error propagation suffers, $\delta G_E\propto \delta(\sigma_R(\epsilon_1) -\sigma_R(\epsilon_2))(\Delta\epsilon)^{-1}  (\tau G_M^2/G_E^2).$
Fig.~\ref{fig:gepgmp} presents the Rosenbluth data set (See Ref.~\cite{bosted95} for a compilation and references) for the  proton form factors. We see that the measurement of $G_M^p$ has been successfully made out to  30 (GeV/c)$^2$, while  the measurement of  $G_E^p$ begins to endure large errors at much smaller $Q^2$.
\begin{figure}[!hb]
\centering   \includegraphics[width=3.1in]{gep-pre-jlab.eps}\hspace{0.05in}\includegraphics[width=3.1in]{gmp-pre-jlab.eps}\\
  \caption{$G_E^p/G_D$ and $G_M^p/\mu_p/G_D$ versus $Q^2$ from the Rosenbluth method. The scaling law  and the dipole law hold to a good approximation ($\approx 10\%$) for both form factors out to $Q^2= 8$ (GeV/c)$^2$. }\label{fig:gepgmp}
\end{figure}
New measurements of the proton form factors exploiting a double polarization observable have upset the form factor scaling seen here.

\subsection{Neutron Form Factors}
The lack  of a
free neutron target and the dominance of $G_M^n$ over $G_E^n$ (setting
aside recent progress) has left the data set on the
neutron form factors much less than desired.
%The deuteron serves as an approximation of a  free  neutron target but  a firm understanding of the ground and final state wavefunctions is required in order to extract reliable information about the form factors.
The traditional
techniques (restricted to the use
of unpolarized beams and targets) used to  extract information about  $G_M^n$
and $G_E^n$ have been: elastic scattering from the deuteron (D):
 D$(e,e')$D;  inclusive quasielastic scattering:  D$(e,e')X$; 
scattering from deuteron with the coincident detection of the scattered electron
and recoiling neutron:  D$(e,e'n)p$;
%  scattering from deuteron with the
%detection of the scattered electron  and the  absence of a recoiling proton
%(anticoincidence) D$(e,e' \bar{{\rm p}}$)p,
% and  ratio measurements,
 a ratio method which minimizes uncertainties in the deuteron
wave function and the role of FSI: $\frac{D(e,e' n)}{D(e,e'p)}$. The current status of the data on $G_M^n$ is shown in the LHS of Fig.~\ref{fig:gmngen}. The lack of data at large momentum transfers  will be soon remedied with data from an experiment~\cite{PR94-017,BrooksBaryons04} at Jefferson Lab which has used the ratio method to measure \gmn with small errors out to nearly 5 (GeV/c)$^2$ in the CLAS. This experiment took data on hydrogen and deuterium targets simultaneously. The neutron detector efficiency was determined in-situ via the $H(e,e'n\pi^+)$ reaction, and the CLAS's large acceptance allows the veto of any events with extra charged particles.


Until the early 1990's the extraction of \gen was done most successfully through
either small angle elastic $e$-D scattering~\cite{Galster71,platchov90} or
%\cite{Drickey62,Benaksas64,bumiller70,Galster71,simon81,platchov90} or
by quasielastic $e$-D scattering~\cite{lung93}.
%\cite{hughes65,hughes66,bartel73,lung93} %The
%deuteron is in spin-1 ground state and supports three elastic form factors,
%\mbox{$G_C$ , $G_Q$ , $G_M$}. The elastic electron deuteron cross section can be
%written, \begin{equation}
% \frac{d \sigma}{d \Omega}= \sigma_{NS}\left[ A\left( Q^{2}\right) +B\left(
%Q^{2}\right) \tan ^{2}\left( \frac{\theta _{e}}{2}\right)\right] \end{equation}
%where
%$A(Q^2) = G_c^2 + \frac{8}{9} \eta G^2_Q + \frac{2}{3} \eta^2 G_M^2$, $B(Q^2) =
%\frac{4}{3} \eta(\eta+1)G_M^2$,  and $\eta = \frac{Q^2}{4M_D^2}$.
In  the  Impulse Approximation (IA) the elastic electron-deuteron cross section is the sum of proton
and neutron responses with deuteron wave function weighting and in the small $
\theta _{e}$ approximation can be written,
\begin{equation}\frac{d \sigma}{d \Omega}\simeq \sigma_{\textrm{Mott}}(G_E^p + G_E^n)^2
\int_0^\infty\left[u(r)^2 + w(r)^2\right]j_0(\frac{qr}{2})dr. 
\end{equation}
%\begin{equation}\frac{d \sigma}{d \Omega} = \cdots (G_E^p + G_E^n)^2
%\left[u(r)^2 + w(r)^2\right]j_0(\frac{qr}{2})dr \cdots
%\end{equation}
The coherent nature of elastic scattering
gives rise to an interference term between the neutron and proton response which
allows the smaller \gen contribution to be extracted. Still,  the large proton
contribution must be removed. Experiments have been able to achieve small
statistical errors but remain very sensitive to deuteron wave function model
leaving a significant residual dependence on the NN potential.
%The extraction of \gen from  elastic $e$-D experiments require that the
%measured $A(Q^2)$ be reduced to $A_{IA}(Q^2)$ by making corrections for
%relativistic effects and MEC. The contribution of the magnetic  form factor must
%be subtracted. Calculated  S and  D state wave functions are used to unfold the
%nuclear structure for various potentials to get the isoscalar form  factor.
%Finally the proton form factor is subtracted in order to quote \gen. 
The most precise data on \gen from elastic $e$-D scattering  are shown on the right hand side of Fig.~\ref{fig:gmngen} from an experiment at Saclay, published in 1990~\cite{platchov90}. The curves (parametrizations of the data based on different NN potentials used in the extraction) form a band which is a measure of the
theoretical uncertainty ($\approx$ 50\%) that cannot be avoided.

\begin{figure}[!t]
\centerline {
\includegraphics[width=0.45\textwidth]{gmn_data}\includegraphics[angle=90,width=0.5\textwidth]{platchov2}
%\includegraphics[width=0.5\textwidth]{gmn.ps}\includegraphics[angle=90,width=0.5\textwidth]{platchov2}
} \caption[]{{Left: $G_M^n$  from unpolarized scattering~\cite{anklin94,bruins,anklin98,kubon02,lung93,mark93} and polarized scattering~\cite{xu00,Xu:2002xc}. Right: \gen from elastic e-D~\cite{platchov90}. The band defined by the lines represents the theoretical error associated with the extraction. The (small) dashed line is the familiar Galster parametrization\cite{Galster71}.\label{fig:gmngen}}}
%,  using the wave function generated with
%Paris potential. The solid line is a fit to  the data using the Galster form
%shown in the lower right corner. The other lines are similar fits to  the  data
%set extracted with Argonne, Nijmegen and the Paris potential with  a $\Delta
%\Delta$ admixture.
\end{figure}




\section{SPIN DEPENDENT MEASUREMENTS}\label{sec:spindep}

It has been known for many years that  the nucleon electromagnetic form factors
could be measured through   spin-dependent elastic scattering from the nucleon, accomplished either through a measurement of the
scattering asymmetry of polarized electrons from a polarized nucleon target~\cite{dombey,Do89}
%{\it e.g. for the neutron form factors,} \( \vec{D}(\vec{e},e'n)p \), \( \vec{^{3}{\rm He}}(\vec{e},e'n)pp \),
 or equivalently by measuring the polarization transferred to the nucleon~\cite{akhiezer,acg}.
%$D(\vec{e},e' \vec{n})p.
% The elastic cross section in both recoil polarization and beam-target asymmetry techniques consists of an unpolarized piece and two polarized pieces that can be separated by selecting the orientation (parallel or perpendicular) of the recoil polarization direction or the initial state polarization of the target, relative to the momentum transfer. not quite right
In the scattering of polarized electrons from a polarized target, an asymmetry appears in the elastic scattering cross section when the beam helicity is  reversed. In contrast, in scattering a polarized electron from an unpolarized target, the transferred polarization to the nucleon produces an azimuthal asymmetry in the secondary scattering of the nucleon (in a polarimeter) due to its dependence on its polarization. In both cases, the perpendicular asymmetry
%\footnote{We refer to either the beam-target asymmetry  when the target polarization is perpendicular to the momentum transfer, or the azimuthal asymmetry in the secondary scattering of the recoiling nucleon when its polarization is perpendicular to its momentum.}
 is sensitive to the product $G_EG_M$.  Only in the last decade have experiments exploiting these spin degrees of freedom become possible.

% $$\frac{d\sigma}{d\Omega}  = \underbrace{\ldots (G_E^2 +\ldots G_M^2)}_{(d\sigma/d\Omega)_{\rm{unpol}}}
%+  \underbrace{\ldots P_eP_N^\perp G_E G_M}_{A_T}
%+\underbrace{\ldots P_eP_N^{\parallel}G_M^2}_{A_\parallel}
%$$

%Like spin-averaged measurements the neutron formfactors require scattering from a deuteron (in the case of beam-target asymmetry  measurements $^3$H becomes a viable target as it approximates a polarized neutron).
% In every case, the  sensitivity to the how the reaction is treated is minimized and the the sensitivity to the neutron form factors and maximized by working in quasifree kinematics. Because these involves a nucleus extraction of \gen is done indirectly - the 
%re experimental asymmetries  are compared to theoretical calculations,  generated for scaled values of the form factor adnthe form factorr is extracted by comparison of the experimental asymmetry to acceptance averaged theory.

Extraction of the neutron form factors (necessarily from a nuclear target) using polarization observables is complicated by the need to account for Fermi motion, MEC, and FSI, complications that are absent when scattering from a proton target. Fortunately it has been found for the deuteron that  in kinematics that emphasize quasi-free neutron knockout both the  transfer polarization $P_t$~\cite{arenhoevel87} and the beam-target asymmetry $A_V^{eD}$~\cite{arenhoevel88} are especially
sensitive to \gen  and  relatively  insensitive to the NN
potential describing the ground state of the deuteron and other reaction details. Calculations~\cite{golak} of the beam-target asymmetry from a polarized $^3$He target (which can be approximated as a polarized neutron) showed modest model dependence.

\subsection{Recoil polarization}
In elastic scattering of polarized electrons from a nucleon, the nucleon obtains (is transferred) a polarization whose components, $P_l$ (along the direction of the nucleon momentum)  and $P_t$ (perpendicular to the nucleon momentum) are proportional to $G_M^2$ and $G_EG_M$ respectively. The recoil polarization technique has allowed precision measurements of  $G_E^p$
to nearly 6 (GeV/c)$^2$~\cite{jones,gayou,gayou2} and of $G_E^n$ out to $Q^2 = 1.5 \rm{(GeV/c)}^2$~\cite{eden,He99,Os99,Madey:2003av}. Polarimeters are sensitive only to the perpendicular polarization components so precession of the nucleon spin before the polarimeter in the magnetic field of the spectrometer (for the proton) or a dipole (inserted in the path of neutron)  allows a measurement of the ratio $P_t/P_l$ and the form factor ratio: $\frac{G_E}{G_M} = - \frac{P_t}{P_l}\frac{(E_0+E')}{2M_N}\tan(\theta/2).$

% The polarization components are determined by measuring either a left-right (vertical polarization) or up-down asymmetry (horizontal polarization) in the secondary scattering.  By taking the difference between 2 helicity states the instrumental asymmetries cancel, $P_B$ and $A_y$ cancel providing gain access to the polarization components with a dramatic reduction in the systematic errors.

The results from Jefferson Lab for the proton are shown in Fig.~\ref{fig:jlabgep} and it can immediately be seen that the  ratio of $\mu_pG_E^p/G_M^p$ does not follow the scaling law obtained from Rosenbluth separation, rather showing a steep decline with increasing $Q^2$ and suggesting that the distribution of magnetization and charge densities in the proton are dissimilar.  Shown with  the data are a collection of calculations including several relativistic Constituent Quark  Models (rQCM), a VMD-pQCD model and a chiral soliton model. Also shown is a pQCD calculation~\cite{Brodsky:2002st}. The right hand side of Fig.~\ref{fig:jlabgep} shows the same data, now in terms of $Q^2F_2/F_1$. The data give no indication of scaling at high momentum transfer, in contradiction to the early pQCD prediction~\cite{brodsky75-81}. The recent efforts~\cite{Brodsky:2002st}, still within pQCD and including higher twist contributions, have been able to reproduce this behavior (solid line). Other pQCD calculations  which consider quark angular orbital momentum are also successful, for example Refs.~\cite{Belitsky:2002kj,Ralston:2002ep}.

\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{gepgmp_jlab_vsth_post.eps}\includegraphics[width=0.5\textwidth]{q2f2f1_pQCD.eps}
\vspace{-0.35in}
\caption{Left: Comparison of theoretical model calculations with the data
from Ref.~\cite{jones} (solid circles) and from~\cite{gayou2} (solid squares). The curves are,
black thin solid~\cite{pfsa}, green dot-dashed and dashed~\cite{Pace:1999as}, black dashed~\cite{Frank1996}, red
solid \cite{Lomon:2002jx}, brown dashed~\cite{Brodsky:2002st} and magenta dashed and solid~\cite{holzwarth}.   Right: $Q^2F_2^p/F_1^p$ versus $Q^2$. The curves shown are from Brodsky \cite{Brodsky:2002st}(solid line), and from Belitsky et al.
\cite{Belitsky:2002kj} (dashed line). The data from Ref.~\cite{jones} are shown as solid circles, from
Ref.~\cite{gayou2} as empty squares and from Ref.~\cite{Andivahis:1994rq} as empty diamonds. For a discussion of the curves see Ref.~\cite{punjabi-03-conf} (from which these figures were taken).}\label{fig:jlabgep}
\end{figure}

%  Since pioneering work on the neutron at
%Bates \cite{eden}, the development of high polarization beams and targets,
%together with high duty factor accelerators, has improved the data set (and
%outlook)  for \gen \cite{Me94,He99,Pa99,Os99,Be99,Ro99,Zhu01}. Not incidentally,

Recoil polarization has been used at both Jefferson Lab and Mainz to extract $G_E^n/G_M^n$ when scattering polarized electrons from an unpolarized deuteron target in quasi-elastic kinematics. At both labs a dipole magnet was used to precess the neutron spin thereby allowing a measurement of both polarization components. The results from \cite{Madey:2003av} are especially precise, extending our knowledge of \gen out to 1.5 GeV/c$^2$. See Fig.~\ref{fig:gendata}.



\subsection{Beam-target asymmetry}
Polarized targets have been used to extract \gen~\cite{Me94,Pa99,Be99,bermuth03,Glazier:2004ny,golak01,Zhu01,Warren:2003ma} and \gmn~\cite{gao94,xu00,Xu:2002xc}. The beam-target asymmetry can be written schematically ($a$, $b$, $c$, and $d$ are known kinematic factors) as    
\begin{equation}
A=\frac{ a  \cos\Theta^{\star} \left(G_{M}\right)^2  + 
          b  \sin\Theta^{\star} \cos\Phi^{\star} G_EG_M}
        {c \left(G_{M}\right)^2 + d \left(G_{E}\right)^2}
\end{equation} where $\Theta^{\star}$ and $\Phi^{\star}$ fix the target polarization axis.
%\begin{figure}
%\includegraphics[scale = 0.35]{scatt_plane3.eps}
%\caption{Schematic of scattering plane showing orientation of polarization axis}\label{fig:scatt-plane}
%\end{figure}
 With the target polarization axis in the scattering plane and perpendicular to $\vec{q}$, ($\Theta^{\star}, \Phi^{\star}=90^{\circ},0^{\circ}$) the asymmetry $A_{TL}$ is proportional to $G_EG_M$. With the polarization axis in the scattering plane and parallel to $\vec{q}$ ($\Theta^{\star}, \Phi^{\star}=0^{\circ},0^{\circ}$), measuring the asymmetry $A_T$ allows $G_M$ to be determined. See the LHS of Fig.~\ref{fig:gmngen}. 


 \gen has been extracted from beam-target asymmetry measurements using polarized $^3$He targets at Mainz and polarized ND$_3$ targets at Jefferson Lab,  and a polarized atomic beam target at NIKHEF. Data for \gen from both kinds of double polarization experiments are shown in Fig.~\ref{fig:gendata} along with some relevant calculations. The models, starting with the first in the  legend, include a rCQM ~\cite{Cardarelli2000}, a hybrid VMD-pQCD model~\cite{Lomon:2002jx}, a relativistic CQM calculation~\cite{pfsa}, a light-front cloudy bag model~\cite{miller}, a soliton model~\cite{holzwarth}, and a dispersion theory calculation~\cite{Hammer:2003ai}. While most of these calculations describe the $Q^2$-dependence, several badly fail to reproduce the slope at $Q^2=0, \frac{dG_E^n(0)}{dQ^2} = -\frac{1}{6} \left \langle r_{E}^2\right \rangle$. The neutron charge radius, $r_E$, has been determined through neutron electron scattering~\cite{Kopecki95}. 


\begin{figure}
%\center\includegraphics[width=0.6\textwidth]{gen-236.ps}\vspace{-0.15in}
\center\includegraphics[width=0.6\textwidth]{gen-rel-thr-baryons.ps}\vspace{-0.15in}
\caption[]{{Comparison of selected theoretical model calculations with the  data on \gen from polarized experiments.  Starting at the top of the legend the data are from~\cite{sick-sch01,Zhu01,golak01,He99,Os99,Pa99,bermuth03,Glazier:2004ny,Madey:2003av,Warren:2003ma}. Apparently, as can be seen by the  solid line starting at $Q^2$ = 1 (GeV/c)$^2$, the neutron, at large momentum transfers,  has the same  $Q^2$-dependence as the proton.}\label{fig:gendata}}
\end{figure}

%:Discrepancy
\section{THE $\mathbf {G_E^p/G_M^p\:}$ DISCREPANCY AND TWO-PHOTON EFFECTS}

The recoil polarization measurements of the form factor ratio $\mu_pG_E^p/G_M^p$  contradict the Rosenbluth measurements and it has been suggested  that the earlier experiments might have underestimated systematic errors or suffer from normalization problems. Recently the Rosenbluth measurements have been reexamined~\cite{Arrington:2003df}. This global reanalysis could find no systematic or normalization problems that could account for the discrepancy and concluded that a 5-6\% linear $\epsilon$-dependence correction (of origin yet unknown) to the cross section measurements is required to  explain the difference.  Several investigators~\cite{Blunden:2003sp,Rekalo:2003xa,Guichon:2003qm,Chen:2004tw} have explored the possibility of two-photon exchange corrections (which would be less important in the direct ratio measurement of recoil polarization) to explain the discrepancy. While only incomplete calculations exist, the results of Ref.~\cite{Blunden:2003sp,Chen:2004tw} account for part of the difference.

%Guichon and Vanderhaeghen\cite{Guichon:2003qm} wrote down a general expression and estimated the size of the two-photon contribution required to explaining the discrepancy.
%%\small
%$$
%\frac{d\sigma }{C_{B}(\varepsilon ,Q^{2})}
%\simeq \frac{| \tilde{G}_{M} | ^{2}}{\tau } 
%\left\{  \tau +\varepsilon  
%\frac{|\tilde{G}_{E}|^2}{|\tilde{G}_{M}|^2} \right. 
% \left. +  
%  2 \varepsilon \left( \tau +
%\frac{|\tilde{G}_{E}|}{|\tilde{G}_{M}|} \right) 
%Y_{2\gamma}(\nu,Q^2)\right\} 
%$$
%$$
%\frac{P_{t}}{P_{l}}  \simeq  - \, 
%\sqrt{\frac{2\varepsilon }{\tau(1+\varepsilon )}} \,
%\left\{\, \frac{|\tilde{G}_{E}|}{|\tilde{G}_{M}|} \right. \left. +\left( 1-\frac{2\varepsilon }{1+\varepsilon }
%\frac{|\tilde{G}_{E}|}{|\tilde{G}_{M}|} \right) Y_{2\gamma}(\nu,Q^2) \right\}
%$$
%They find $Y_{2\gamma}$ to be on the order of a few \%   which would generate a 6\% correction to the $\epsilon$ slope and only slightly modify the recoil polarization results. 
 The most recent work by Chen {\it{et al.}}~\cite{Chen:2004tw} employed a different approach than that of Ref.~\cite{Blunden:2003sp} in that they describe the process in terms of hard scattering from a quark and use GPD's to describe the quark emission and absorption. They argue that when taking the recoil polarization form factors as input, the addition of the two-photon corrections reproduces the Rosenbluth data. However, Arrington~\cite{Arrington:2004ae} has shown that when the corrections of  Chen {\it{et al.}} are applied to the new Jefferson Lab Rosenbluth data,  which have small errors, (see below and Figure~\ref{fig:super}) only one-half of the discrepancy is explained. 


The Rosenbluth-polarization transfer discrepancy has been recently confirmed by a ``Super" Rosenbluth measurement \cite{E01-001} at Jefferson Lab that was designed to minimize the systematic errors that handicap Rosenbluth measurements.  This was achieved  by  detecting the proton rather than the electron in elastic kinematics. In doing so  many of the extreme rate variations and cross section sensitivities that are normally encountered  were avoided. The results \cite{Qattan:2004ht} show that the discrepancy still exists. See Figure~{\ref{fig:super}.

\begin{figure}[!ht]\includegraphics[width=0.5\textwidth]{Super-Rosen.eps}\includegraphics[width=.5\textwidth]{e01001_chen_coul.ps}
\caption{ Left: Proton form factor ratio where the (blue) triangles are from recoil polarization~\cite{jones,gayou2}, (red) crosses from the reanalysis of the world's Rosenbluth data set~\cite{Arrington:2003df} and the filled circles from the recent super-Rosenbluth experiment~\cite{Qattan:2004ht}. Right: Corrections to the three data points from the super-Rosenbluth data set move those data towards,({\it i.e} reduces the value of the form factor ratio) the recoil polarization data set shown with fit (dotted line). Starting with the largest value at each $Q^2$ is the measured ratio, the ratio with the two-photon corrections of Ref.~\cite{Chen:2004tw} applied and the ratio  with  the two-photon and with Coulomb corrections~\cite{Arrington:2004is} applied~\cite{Arrington:2004ae}.}\label{fig:super}
\end{figure}

Direct experimental tests for the existence of two-photon exchange include  measurements of the ratio $\frac{\sigma(e^+p)}{\sigma(e^-p)}$, where the real part of the two-photon exchange amplitude leads to an enhancement, and in  Rosenbluth data where it can lead to non-linearities in $\epsilon$. There is no experimental evidence of non-linearities in the Rosenbluth data and the $e^+/e^-$ ratio data~\cite{Mar:1968qd} are of only modest precision, making it difficult to absolutely confirm the presence of two-photon effects in these processes.

It is the imaginary part of the two-photon amplitude that can lead to single spin asymmetries but again the  existing data~\cite{Kirkman:1970er,Powell:1970qt} are of insufficient precision to allow one to make a statement.   There is, however, one observable that has provided unambiguous evidence for a two-photon effect in $ep$ elastic scattering. Groups in both the US~\cite{Wells:2000rx} and Europe~\cite{Maas:2004pd}
have measured the transverse polarized beam asymmetry. These measurements are significant but have limited utility in solving the $G^p_E$ discrepancy. The reader interested in more detail about the existence of two-photon effects and their role on the form factor measurements should refer to Ref.~\cite{Arrington:2004ae}.

Fortunately  experiments are planned at Jefferson Lab to look for non-linearities in the Rosenbluth data, for the presence of induced recoil polarization and for an enhancement in the $e^+/e^-$ ratio. We can expect that a concentrated effort in both experiment and theory will reveal the full extent of two-photon effects in the not too distant future.
   
\section{OUTLOOK}

The capabilities of high duty factor accelerators, polarized beams and targets, and polarimeters have produced precision data out to large momentum transfer on the proton and neutron form factors. These new data, accumulated over the last 10 years, now challenge our previously held view of the structure of the proton  and neutron and provide a rigorous test for any QCD-inspired model of nucleon structure.

%Experiments are planned at Jefferson Lab to extend the measurements of $G_E^p/G_M^p$ to 9 GeV$^2$ via recoil polarization \cite{PR01-109}. 
%High precision measurements at low and moderate  $Q^2$ are planned for Bates-MIT using the BLAST and internal polarized deuteron and $^3$He targets \cite{calarco03},  at Jefferson Lab \cite{PR02-013} out to 3.5 GeV/c$^2$ and Mainz \cite{jourdan} out to 1.5 GeV/c$^2$ using the \han reaction.

The conclusion from a review article \cite{RRWilson64} published 40 years ago is still appropriate today: {\it Although the major landmarks of this field of study are now clear, we are left with the feeling that much is yet to be learned about the nucleon by refining and extending both measurement and theory.}  

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%
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%
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%%CITATION = PHLTA,B32,519;%%

%\cite{Powell:1970qt}
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T.~Powell {\it et al.},
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%%CITATION = PRLTA,24,753;%%

%\cite{Wells:2000rx}
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%[arXiv:nucl-ex/0002010].
%%CITATION = NUCL-EX 0002010;%%

%\cite{Maas:2004pd}
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% ``Measurement of the transverse beam spin asymmetry in elastic electron proton
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%%CITATION = NUCL-EX 0410013;%%

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\end{thebibliography}






\end{document}


nucl-ex0502003.extracted_bib
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%\itemsep=0cm
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%
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%
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%
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%%CITATION = NUCL-TH 0203081;%%


  
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 %\cite{Hammer:2003ai}
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%
%discrepancy
%

%\cite{Arrington:2003df}
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%%CITATION = NUCL-EX 0305009;%%



%
%two-photon stuff
%
%\cite{Blunden:2003sp}
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%[arXiv:nucl-th/0306076].
%%CITATION = NUCL-TH 0306076;%%

%\cite{Rekalo:2003xa}
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M.~P.~Rekalo and E.~Tomasi-Gustafsson,
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%[arXiv:nucl-th/0307066].
%%CITATION = NUCL-TH 0307066;%%


%\cite{Guichon:2003qm}
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P.~A.~M.~Guichon and M.~Vanderhaeghen,
 %``How to reconcile the Rosenbluth and the polarization transfer method in  the
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Phys.\ Rev.\ Lett.\  {91}, (2003) 142303.
%[arXiv:hep-ph/0306007].
%%CITATION = HEP-PH 0306007;%%

%\cite{Chen:2004tw}
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Y.~C.~Chen, A.~Afanasev, S.~J.~Brodsky, C.~E.~Carlson and M.~Vanderhaeghen,
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%electron proton scattering at large momentum transfer,''
Phys.\ Rev.\ Lett.\  {93}, (2004) 122301.
%[arXiv:hep-ph/0403058].
%%CITATION = HEP-PH 0403058;%%


%\cite{Arrington:2004ae}
\bibitem{Arrington:2004ae}
J.~Arrington,
%``Extraction of two-photon contributions to the proton form factors,''
arXiv:hep-ph/0408261 and private communication.
%%CITATION = HEP-PH 0408261;%%

\bibitem{E01-001}J. Arrington and R. Segel,
% ``New Measurement of $G_E/G_M$ for the proton",
  Jefferson Lab Experiment E01-001.

%\cite{Qattan:2004ht}
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I.~A.~Qattan {\it et al.},
%``Precision Rosenbluth measurement of the proton elastic form factors,''
arXiv:nucl-ex/0410010.
%%CITATION = NUCL-EX 0410010;%%



%\cite{Arrington:2004is}
\bibitem{Arrington:2004is}
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Phys.\ Rev.\ C70,  (2004) 028203.
%[arXiv:nucl-ex/0406014].
%%CITATION = NUCL-EX 0406014;%%



%\cite{Mar:1968qd}
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Phys.\ Rev.\ Lett.\  {21},  (1968) 482.
%%CITATION = PRLTA,21,482;%%

%\cite{Kirkman:1970er}
\bibitem{Kirkman:1970er}
H.~C.~Kirkman {\it et al.},
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%Proton Scattering At Q-Squared=1.3, 1.5 And 1.9(Gev/C)Squared,''
Phys.\ Lett.\ B {32}, (1970) 519.
%%CITATION = PHLTA,B32,519;%%

%\cite{Powell:1970qt}
\bibitem{Powell:1970qt}
T.~Powell {\it et al.},
%``Measurement Of The Polarization In Elastic Electron - Proton Scattering,''
Phys.\ Rev.\ Lett.\  {24}, (1970) 753.
%%CITATION = PRLTA,24,753;%%

%\cite{Wells:2000rx}
\bibitem{Wells:2000rx}
S.~P.~Wells {\it et al.}  [SAMPLE collaboration],
%``Measurement of the vector analyzing power in elastic electron proton
%scattering as a probe of double photon exchange amplitudes,''
Phys.\ Rev.\ C63, (2001) 064001.
%[arXiv:nucl-ex/0002010].
%%CITATION = NUCL-EX 0002010;%%

%\cite{Maas:2004pd}
\bibitem{Maas:2004pd}
F.~E.~Maas {\it et al.},
% ``Measurement of the transverse beam spin asymmetry in elastic electron proton
%scattering and the inelastic contribution to the imaginary part of the
%two-photon exchange amplitude,''
arXiv:nucl-ex/0410013.
%%CITATION = NUCL-EX 0410013;%%

\bibitem{RRWilson64}  R.R. Wilson and J.S. Levinger, Ann. Rev. of Nucl. Sci., Vol. 14,  (1964) 135.

\end{thebibliography}