exp_method.tex

\section{Technique}\label{sec:exp_method}
%
This proposal is based on instrumentation, simulation, and analysis development made by the GMn/SBS collaboration for the GMn, E12-09-019, experiment~\cite{E12-09-019}.
The GMn experiment is one of several form factor experiments approved by JLab PAC. 
The SBS spectrometer was funded by DOE with large contributions provided by the collaborating institutions from USA, Italy, UK, and Canada. 
The apparatus and DAQ installation will start in 2020 and the data taking run is expected to be in summer-fall 2021.

The neutron form factors are challenging to be determine experimentally especially because there is no free neutron target. 
However, since deuterium is a loosely coupled system, it can be viewed as the sum of a proton target and a neutron target. 
In fact, quasi-elastic scattering from deuterium has been used to extract the neutron magnetic form factor, \gmn, at modestly high \qsq~for decades~\cite{Hughes:1965zza, Arnold:1988us} in the single arm (e,e') experiments. 
However, the proton cross section needs to be subtracted by applying a single-arm quasi-elastic electron-proton scattering. 
This ``proton-subtraction" technique suffers from a number of systematic uncertainties e.g. contributions from inelastic and secondary scattering processes. 

Many years ago, L.~Durand~\cite{Durand:1959zz} proposed the so-called ``ratio-method" based on the measurement of both D(e,e'n) and D(e,e'p) reactions. 
In this method, many of the systematic errors are canceled out. 
Several experiments \cite{Bruins:1995ns, Kubon:2001rj, Lachniet:2008qf} have applied the ratio-method to determine the neutron magnetic form factor.

The GMn/SBS experiment~\cite{E12-09-019} will take data for elastic $e-n$ scattering for several kinematics with \qsq~from 3.5 up to 13.5~\gevcsq.
We propose to use this method to measure the Rosenbluth slope and extract (in OPE approximation) the neutron electric form factor, \gen,~at one value of momentum transfer.
In fact, one of the required data points will be taken by the GMn experiment, so an additional measurement is needed only for one
kinematics.

Data will be collected for quasi-elastic electron scattering from deuterium in the process $D(e,e'n)p$. 
Complementary $D(e,e'p)n$ data will be taken to calibrate the experiment apparatus.
The current knowledge of the $e-p$ elastic scattering cross section (obtained in the single arm H(e,e')p and H(e,p)e' experiments) will be also used
for precision determination the experiment kinematics.

Applying the Rosenbluth technique to measure \gen~requires accurate measurement of the cross section  and suffers from large uncertainties. 
To overcome this issue, we propose to extract the value of \gen~from the ratio of quasi-elastic yields, $R_{n/p}$, in scattering from a deuteron target as follows: 

\begin{equation}
R_{n/p} \equiv R_{observed} = \frac{N_{e,e'n}}{N_{e,e'p}}
\label{eq:1}
\end{equation}
$R_{observed}$ needs to be corrected to extract the ratio of e-n/e-p scattering from nucleons:

\begin{equation}
R_{corrected} = f_{corr} \times R_{observed} \;\; ,
\label{eq:2}
\end{equation}
where the correction factor $f_{corr}$ takes into account the variation in the hadron efficiencies due to changes of the $e-N$ Jacobian, the radiative corrections, and absorption in path
from the target to the detector, and small re-scattering correction.

In one-photon approximation, $R_{corrected}$ can be presented as: 

\begin{equation}
R_{corrected} = \frac {\sigma_{_{_{Mott}}}^n \cdot (1+\tau_p)}{\sigma_{_{_{Mott}}}^p \cdot (1+\tau_n)} \times \frac{\epsilon \sigma_{_L}^n + \sigma_{_T}^n}{\epsilon \sigma_{_L}^p + \sigma_{_T}^p}
\end{equation}

It is important that the ratio $R_{Mott} = \frac {\sigma_{_{_{Mott}}}^n \cdot (1+\tau_p)}{\sigma_{_{_{Mott}}}^p \cdot (1+\tau_n)}$ could be determine with very high relative accuracy even with modest precision for the beam energy, electron scattering angle, and detector solid angle. 
Now, let us write the $R_{corrected}$ at two values of $\epsilon$ using $S^{n(p)} = \sigma_{_L}^{n(p)}/ \sigma_{_T}^{n(p)}$ as:
\begin{equation*}
  %R_{{corrected},\epsilon_1} = R_{Mott,\epsilon_1} \times \frac{\epsilon_1 \sigma_{_L}^n + \sigma_{_T}^n}{\epsilon_1 \sigma_{_L}^p + \sigma_{_T}^p}
R_{{corrected},\epsilon_1} = \frac{\epsilon_1 \sigma_{_L}^n + \sigma_{_T}^n}{\epsilon_1 \sigma_{_L}^p + \sigma_{_T}^p}
\hskip .5 in
%R_{{corrected},\epsilon_2} = R_{Mott,\epsilon_2} \times \frac{\epsilon_2 \sigma_{_L}^n + \sigma_{_T}^n}{\epsilon_2 \sigma_{_L}^p + \sigma_{_T}^p}
R_{{corrected},\epsilon_2} = \frac{\epsilon_2 \sigma_{_L}^n + \sigma_{_T}^n}{\epsilon_2 \sigma_{_L}^p + \sigma_{_T}^p}
\end{equation*}

In these two equations there are two unknown variables: $\sigma_{_L}^n$ and $\sigma_{_T}^n$.
We remind here that proton and neutron measurements are made simultaneously with the same apparatus. 
Thanks to this, the dominant contribution to the uncertainty of the Rosenbluth slope of the reduced cross section vs. $\epsilon$,  
$S^n = \sigma_{_L}^n/ \sigma_{_T}^n$, will come from the uncertainty of $S^p$.
At \qsq=4.5 \gevcsq, according to the global analysis of $e-p$ cross section~\cite{Christy2020ab}, the value of $S^p$ is close to $1/(\tau \, \mu_p^2) = 0.087$ with an uncertainty of 0.01.
The resulting equation for $S^n$ is:
\begin{equation*}
A = B \times \frac{1 + \epsilon_1 S^n}{1 + \epsilon_2 S^n} \approx B \times (1 +  \Delta \epsilon \cdot S^n),
\end{equation*}
with $\Delta \epsilon = \epsilon_1 -\epsilon_2$, and 
where the variable $A = R_{{corrected},\epsilon_1}/R_{{corrected},\epsilon_2}$ will be measured with statistical precision of 0.1\%.  
Assuming, for this estimate, equal values of \qsq~for two kinematics, the  $\tau$ and $\sigma_{_T}$ for two kinematics are canceled out, and the variable
%\mbox{$ B = {R_{M,\epsilon_1}}/{R_{M,\epsilon_2}} \times (1+ \epsilon_2 \, S^p)/( 1 + \epsilon_1 \, S^p)$}.
\begin{equation}
  B = %R_{Mott, \epsilon_1}/R_{Mott, \epsilon_2}
  (1 + \epsilon_2 S^p )/(1 + \epsilon_1 S^p )
\end{equation}

For actual small range of $\epsilon$ and small value of the slope, $B \approx (1 - \Delta \epsilon \cdot S^p)$.
The value of B will be determined from global proton $e-p$ data to a precision of $0.25 \times 0.01$.
 
At \qsq=4.5 \gevcsq~the ratio $\mu_n$\gen/\gmn~is $0.55 \pm 0.05$ based on polarization transfer data which is mostly insensitive to the two-photon exchange, see the 2015 review from Perdrisat {\it et al.}~\cite{Punjabi:2015bba}.
%
In the simplest model, the slope $S^n$ is a sum of the slope due to \gen/\gmn~and the neutron two-photon exchange nTPE contribution:
\begin{equation}
  S^n = (G_E^n/G_M^n)^2/\tau + {\rm nTPE}
\end{equation}
Without the nTPE contribution, our projected measured slope would be $S^n = (G_E^n/G_M^n)^2/\tau = 0.063$. %= (0.55/1.91)^2/(4.5*4*0.939^2) 

If we use the prediction available in~\cite{Blunden:2005ew} which is reproduced on our Fig.~\ref{pic:Fig2}, nTPE leads to increase of the neutron Rosenbluth slope $S^n$ by a factor 2;
Under this assumption, the projected measured Rosenbluth slope would now become $S^n = 0.126$, and nTPE would then be:

\begin{equation}
  {\rm nTPE} = S^n - (G_E^n/G_M^n)^2/\tau = 0.063.
\end{equation}

The projected measurement of the neutron two-photon exchange for this experiment is $nTPE = 0.063 \pm 0.012 \pm 0.01$, where the first uncertainty is due to accuracy 
of \gen/\gmn~and the second one due to projected precision of this experiment. It would be a 4-4.5 sigma observation of the two-photon exchange contribution for the neutron.