= }
\nonumber \\ &
\overline{u}(p_2)\left[
\gamma_{\lambda}F_V^1(q^2)
+\frac{\D i\sigma_{\lambda\nu}q^{\nu}{\xi}F_V^2(q^2)}{\D 2M}
%\right.
% \nonumber \\&
%\left.
%~~~~~~~~~~~
+\gamma_{\lambda}\gamma_5F_A(q^2)
+\frac{\D q_{\lambda}\gamma_5F_P(q^2)}{\D M} \right]u(p_1),
\end{eqnarray*}
where $q=k_{\nu}-k_{\mu}$, $\xi=(\mu_p-1)-\mu_n$, and
$M=(m_p+m_n)/2$. Here, $\mu_p$ and $\mu_n$ are the
proton and neutron magnetic moments.
We assume that there are no second class currents, so the scalar
form factor $F_V^3$ and the tensor form factor $F_A^3$
need not be included.
Using the above current, the cross section is
\begin{eqnarray*}
% \lefteqn
{ \frac{d\sigma^{\nu,~\overline{\nu}}}{dq^2} =
\frac{M^2G_F^2cos^2\theta_c}{8{\pi}E^2_{\nu}}\times }
% \nonumber \\&
\left
[A(q^2) \mp \frac{\D (s-u)B(q^2)}{\D M^2} + \frac{\D C(q^2)(s-u)^2}{\D M^4}\right],
\end{eqnarray*}
where
%$$ s-u = 4ME_{\nu} + q^2 - m_l^2$$
\begin{eqnarray*}
\lefteqn{A(q^2)=
\frac{m^2-q^2}{4M^2}\left[
\left(4-\frac{\D q^2}{\D M^2}\right)|F_A|^2 \right.}
\nonumber \\&
\left. -\left(4+\frac{\D q^2}{\D M^2}\right)|F_V^1|^2
-\frac{\D q^2}{\D M^2}|{\xi}F_V^2|^2\left(1+\frac{\D q^2}{\D 4M^2}\right)
% \right.
% \nonumber \\ &
%\left.
-\frac{\D 4q^2ReF_V^{1*}{\xi}F_V^2}{\D M^2}
\right],
%\right. \\ & & \left.
% -\frac{m_l^2}{M^2}\left(|F_V^1+{\xi}F_V^2|^2
% \right)
\end{eqnarray*}
$$
B(q^2) = -\frac{q^2}{M^2}ReF_A^*(F_V^1+{\xi}F_V^2),
%$$
%$$
~~C(q^2) = \frac{1}{4}\left(|F_A|^2 + |F_V^1|^2 -
\frac{q^2}{M^2}\left|\frac{{\xi}F_V^2}{2}\right|^2\right).
$$
%table 1
%
\begin{table}
\begin{center}
\begin{tabular}{|l|c|}
\noalign{\vspace{-8pt}} \hline
$g_A$ & -1.267 \\
$G_F$ & 1.1803${\times}10^{-5}$ GeV$^{-2}$ \\
$\cos{\theta_c}$ & 0.9740 \\
$\mu_p$ & 2.793 $\mu_N$ \\
$\mu_n$ & -1.913 $\mu_N$ \\
$\xi$ & 3.706 $\mu_N$ \\
$M_V^2$ & 0.71 GeV$^2$\\
$M_A$ & 1.00 GeV\\
\hline
\end{tabular}
\end{center}
\caption{ The most recent values of the
parameters used in the `BBA-2003' calculations.}
\label{parameters}
\end{table}
Here $$q^2 = q^2_0 - \vec q_3^2 = -4E_0E^\prime \sin^2{\theta\over 2} =
-Q^2 \; .$$
Although we have
have not shown terms of order $(m_l/M)^2$, and
terms including $F_P(q^2)$ (which
is multiplied by $(m_l/M)^2$), these terms
%$F_P(q^2)$ and terms of order $(m_l/M)^2$
are included in our calculations~\cite{Lle_72}.)
The form factors $ F^1_V(q^2)$ and ${\xi}F^2_V(q^2)$
are given by:
$$ F^1_V(q^2)=
\frac{G_E^V(q^2)-\frac{\D q^2}{\D 4M^2}G_M^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}},
%$$
%$$
~~~{\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}}.
$$
We use the CVC to determine $ G_E^V(q^2)$ and $ G_M^V(q^2)$
from the electron scattering form factors
$G_E^p(q^2)$, $G_E^n(q^2)$, $G_M^p(q^2)$, and $G_M^n(q^2)$:
$$
G_E^V(q^2)=G_E^p(q^2)-G_E^n(q^2),
%$$
%$$
~~~G_M^V(q^2)=G_M^p(q^2)-G_M^n(q^2).
$$
The axial form factor $F_A$ and the pseudoscalar form factor $F_P$
(related to $F_A$ by PCAC) are given by
$$ F_A(q^2)=\frac{g_A}{\left(1-\frac{\D q^2}{\D M_A^2}\right)^2 },
%$$
~~F_P(q^2)=\frac{2M^2F_A(q^2)}{M_{\pi}^2-q^2}. $$
In the expression for the cross section,
$F_P(q^2)$ is multiplied by $(m_l/M)^2$.
Therefore,
in muon neutrino interactions, this effect
is very small except at very low energy, below 0.2~GeV.
$F_A(q^2)$ needs to be extracted from
QE neutrino scattering. At low $Q^2$,
$F_A(q^2)$ can also be extracted from pion
electroproduction data.
For later use in Adler sum rule, we express
the following functions used by Adler\cite{adler} in
the notation of C.H.~Llewellyn Smith~\cite{Lle_72}
(which we use here).
$$|F_V(q^2)|^{2} =|F^1_V(q^2)|^{2}-
\frac{q^2}{M^2}\left|\frac{{\xi}F_V^2(q^2)}{2}\right|^2
$$
$$g_V(q^2) =F^1_V(q^2)+{\xi}F_V^2(q^2)
$$
Note that $F_A(q^2)$ in our notation is the
same as $g_A(q^2)$ as defined by Adler,
and $F_P(q^2)$ in our notation is the same as $h_A(q^2)/M$
as defined by Adler. Also,
Adler defines $q^2$ as positive, while
we define $q^2$ as negative and $Q^2$ as
positive.
Previously, people have assumed
that the vector
form factors are
described by the dipole approximation.
$$ G_D(q^2)=\frac{1}{\left(1-\frac{\D q^2}{\D M_V^2}\right)^2 },~~M_V^2=0.71~GeV^2$$
$$
G_E^p=G_D(q^2),~~~G_E^n=0,
%$$
%$$
~~~G_M^p={\mu_p}G_D(q^2),~~~ G_M^n={\mu_n}G_D(q^2).
$$
\begin{figure}[htb]
\centerline{\psfig{figure=show_gep_ratio.eps,width=4.0in,height=3.in}}
\caption{ Fits to $G_E^p/G_D$, using cross section data only
(solid), compared with `BBA-2003' combined fits to
both the cross section and polarization transfer data
(dashed). The diamonds are the from Rosenbluth extractions
and the crosses are the Hall A polarization transfer data. Note that
the fit is to cross sections, rather than fitting directly to
the extracted values of $G_E^p$ shown here. Since the difference
between the two is
only at high $Q^2$, the two fits yield similar results for the
predicted neutrino-nucleon cross sections.}
\label{show_gep}
\end{figure}
%
%
Note that
$G_E^p$, $G_M^p$, and $G_E^n$ are positive, while
$G_M^n$ and the axial form factor $F_A$ are negative.
We refer to the above combination of form factors
as `Dipole Form Factors'. It is an approximation that
has been improved by Budd, Bodek and Arington~\cite{budd}.
Here we use the
updated form factors to which we refer as `BBA-2003 Form Factors'
(Budd, Bodek, Arrington) which are described below.
Previous neutrino
experiments used $g_A$=$-1.23$, while
the best current value is $-1.267$.
The previous
world average value from neutrino experiments for $M_A$ was
1.026 $\pm$ 0.020 GeV~\cite{Bernard_01}.
The value of $M_A$ extracted
from neutrino experiments depends on both
the value of $g_A$ and the values of
the electromagnetic form factors
which are assumed in the extraction process.
Here we use the updated value~\cite{budd} of $M_A$
1.00 $\pm$ 0.020 GeV, which has been
re-extracted from previous neutrino data
using the better known values for $g_A$ and the
updated `BBA-2003' vector
form factors.
This value of $M_A$ is in good agreement with the
theoretically corrected value
from pion electroproduction~\cite{Bernard_01} of 1.014 $\pm$ 0.016 GeV.
Figure~\ref{show_gepgmp} shows the `BBA-2003'
fits to $\mu_p$$G_E^p$/$G_M^p$.
Note that at present there is a discrepancy
between two different ways of measuring the ratio of electric
and magnetic form
factor of the proton.
The fit including only cross section data (i.e. using
Rosenbluth separation)
is roughly flat versus $Q^2$ ($Q^2=-q^2$)
and is consistent with form-factor scaling. This
is what is expected if the electric charge and magnetization
distributions in the proton are the same. However,
the new technique of polarization tranfer
yields a much lower ratio at high $Q^2$, and indicates
that there is a difference between the electric charge and
magnetization distributions in the proton. At present
the polarization transfer technique is believed to be
more reliable and less senistive to radiative effects
from two photon corrections.
If the electric-charge and magneization distribution
in the proton are very different, then a test of the
high $Q^2$ behavior of the axial form factor is of
interest as an additional input to understand the origin
of this difference. This is one of the measurementc that
can be done in MINERvA. At present, there
are several proposal at Jlab to further investigate this issue.
In addition, there are several groups investigating two photon
corrections to elastic electron scattering in order
to see if the two measurements can be reconciled.
In the BBA-2003 analysis a combined fit is done, using the
cross section data combined with the polarization
tansfer ratio, the ratio decreases with
$Q^2$ in the combined fit to
cross section and polarization transfer data.
The combined fit to both
cross section and polarization transfer data
is used as the default BBA-2003 form factors.
Although the polarization transfer measurement is
believed to have smaller systematic error, especially
at high $Q^2$, the origin of this disagreement is not known.
If this disagreement comes from radiative corrections
to the electron, in particular two-photon
exchange terms, then the polarization transfer extraction
will give the correct ratio, but the overall scale
of $G_E^p$ at low $Q^2$ would be shifted down by
$\approx$$3\%$. Because the fit is constrained as $Q^2 \rightarrow 0$,
there will not be an overall shift in $G_E^p$ at low $Q^2$, but
there will be some uncertainty in the low $Q^2$ behavior.
Current experiments at JLab aim to better understand the source of the
disagreement by looking at the recoil proton in elastic electron-proton
scattering, thus minimizing the sensitivity to the dominant sources
of uncertainty in previous Rosenbluth separations.
However, since this discrepancy is most prominent at high $Q^2$, and
the fit is constrained at low $Q^2$, it has only
a relatively small effect on the neutrino QE scattering cross section.
The fractional
contributions of $G_M^p$,$G_M^n$,$G_E^p$, $G_E^n$ and
$F_A$ to the distribution in $Q^2$ for
quasileastiv events in neutrino running and
antineutrino running with
the NUMI low energy beam configuration are shown in
Figure~\ref{FFcontributions}.
These contributions were
determined by looking at the difference between
the cross section calculated usintg BBA-2003 form factors and
the cross section when each of the following form
factors $G_M^p$,$G_M^n$,$G_E^p$, $G_E^n$ and
$F_A$ are set to zero. Because of intereference terms, the
sum of the fraction does not have to add up to 1.0.
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=FF_contribution.ps,width=4.0in,height=3.in}}
%\centerline{\psfig{figure=FF_contribution.ps,width=4.0in,height=3.in}}
\caption{The fractional
contributions of $G_M^p$,$G_M^n$,$G_E^p$, $G_E^n$ and
$F_A$ to the distribution in $Q^2$ for quasielastic events in
neutrino (top) and antineutrino (bottom) running
with the NUMI low energy beam configuration. }
\label{FFcontributions}
\end{figure}
\subsection{Input from Electron Scattering Quasielastic Scattering}
Table~\ref{parameters} summarizes the
most up to date values of the coupling constants and magnetic
moments that we use in our calculations of quasielastic cross sections.
%
%table 2
\begin{table*}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\noalign{\vspace{-8pt}} \hline
& data & $a_2$ & $a_4$ & $a_6$ & $a_8$ & $a_{10}$ & $a_{12}$ \\ \hline
$G_E^p$ & CS + Pol & 3.253 & 1.422 & 0.08582 & 0.3318 & -0.09371 & 0.01076 \\
$G_M^p$ & CS + Pol & 3.104 & 1.428 & 0.1112 & -0.006981 & 0.0003705 & -0.7063E-05 \\
$G_M^n$ & & 3.043 & 0.8548 & 0.6806 & -0.1287 & 0.008912 & \\ \hline
$G_E^p$ & CS (test) & 3.226 & 1.508 & -0.3773 & 0.6109 & -0.1853 & 0.01596 \\
$G_M^p$ & CS (test) & 3.188 & 1.354 & 0.1511 & -0.01135 & 0.0005330 & -0.9005E-05 \\ \hline
\end{tabular}
\end{center}
\caption{ The coefficients of the inverse polynomial fits for the
$G_E^p$, $G_M^p$, and $G_M^n$. The combined fits for the proton
include both the cross section data and the Hall A polarization transfer
data. Note that these different polynomials replace $G_D$
in the expression for $G_E^p$, $G_M^p$, and $G_M^n$.
The values in this the table (CS+POL) along with the fit of
$G_E^n$ Krutov {\em et. al. }~\cite{Krutov_02}
(see text) will be referred to as `BBA-2003 Vector Form Factors'. The
CS (test) is a fit which ignores the polarization transfer data.}
\label{JRA_coef}
\end{table*}
%
%figure2
\begin{figure}%[htb]
\centerline{\psfig{figure=show_gmp_ratio.eps,width=4.0in,height=3.in}}
\caption{`BBA-2003' fits to $G_M^p/{\mu}_{p}G_D$.
The lines and symbols have the same meaning as Figure~\ref{show_gep}. }
\label{show_gmp}
\end{figure}
%
%figure3
\begin{figure}%[htb]
\centerline{\psfig{figure=show_gmn_ratio.eps,width=4.0in,height=3.in}}
\caption{`BBA-2003' fit to $G_M^n/{\mu}_{n}G_D$.
The lines and symbols have the same meaning as Figure~\ref{show_gep}. }
\label{show_gmn}
\end{figure}
%
\subsection{ `BBA-2003' updated form factors}
%
The `BBA-2003' updated fit to the proton electromagnetic
form factors is similar to the one described in
Ref.~\cite{JRA_03}, but using a slightly different fitting
function (described below), and including additional data
to constrain the fit at low $Q^2$ values.
Note that in contrast to the functional form used
in Ref.~\cite{oldff}, we only include even powers
of $Q$ in our fit. This is because odd powers of $Q$ are not
theoretically allowed. For example,
one can use analyticity~\cite{paschos,mark1}
to show that there are no odd terms in $Q$ in the limit $Q\to 0$.
The vector form factors can be determined from electron
scattering cross sections using
the standard Rosenbluth separation technique~\cite{JRA_03},
which is sensitive to radiative corrections, or from polarization
measurements using the newer polarization transfer technique~\cite{halla}.
The polarization
measurements do not directly measure the form factors,
but measure the ratio $G_E$/$G_M$.
Figures~\ref{show_gep},~\ref{show_gmp},
and~\ref{show_gmn} show the ratio of
the fits divided by the dipole form, $G_D$.
%figure4
\begin{figure}%[htb]
\centerline{\psfig{figure=show_gepgmp_ratio.eps,width=4.0in,height=3.in}}
\caption{ Ratio of $G_E^p$ to $G_M^p$ as extracted by Rosenbluth
measurements and from polarization measurements.
The lines and symbols have the same meaning as Figure~\ref{show_gep}. }
\label{show_gepgmp}
\end{figure}
%figure 5
\begin{figure}%[htb]
\centerline{\psfig{figure=show_gen_new.eps,width=4.0in,height=3.in}}
\caption{
Data and fits to $G_E^n$. The dashed line is the Galster {\em et al.}
fit~\cite{Glaster_71}, and the solid line is the Krutov{\em et al.}
fit~\cite{Krutov_02}.}
\label{show_gen_new}
\end{figure}
%figure6
\begin{figure}
%\begin{center}
%\epsfxsize=2.91in
%\mbox{\epsffile[7 75 550 360]{ratio_JhaKJhaJ_D0DD.eps}}
%\end{center}
\centerline{\psfig{figure=ratio_JhaKJhaJ_D0DD.eps,width=4.0in,height=3.in}}
\caption{Ratio versus energy of predicted neutrino (antineutrino)
QE cross section using BBA-2003 Form
Factors to the prediction using
the dipole approximation with $G_E^n$=0 (with $M_A$ kept fixed).}
\label{ratio_JhaKJhaJ_D0DD}
\end{figure}
%
To account for the fact that deviations from the dipole
form are different for each of the different form
factors, the electron scattering data are fit for each of the
form factors to an inverse polynomial
$$ G_{E,M}^{N}(Q^2)=\frac{G_{E,M}^{N}(Q^2=0)}{1+a_2Q^2+a_4Q^4+a_6Q^6+...}. $$
Table~\ref{JRA_coef} shows the parameters of the `BBA-2003' fit to the
proton data using both cross section data together with
the polarization transfer data from JLab Hall A. For $G_E^p$,
the parameters in Table~\ref{JRA_coef} are used for $Q^2 < 6$~GeV$^2$.
For $Q^2>6$~GeV$^2$, the ratio of $G_E^p/G_M^p$ is assumed to be constant:
$$ G_E^p(Q^2) = G_M^p(Q^2) \frac{G_E^p(6~\mbox{GeV}^2)}{G_M^p(6~\mbox{GeV}^2)} $$
Since the neutron has no charge, $G_E^n$ must be
zero at $q^2$=0, and previous neutrino experiments
assumed $G_E^n(q^2)$=0 for all $q^2$ values.
However, it is non-zero away from $q^2$=0,
and its slope at $q^2$=0
is known precisely from neutron-electron scattering.
At intermediate $Q^2$, recent polarization transfer data give
precise values of $G_E^n(q^2)$.
Our analysis uses the parameterization of
Krutov {\em et. al.}~\cite{Krutov_02}:
$$G_E^n(Q^2) = -\mu_n\frac{a\tau}{1+b\tau}G_D(Q^2),~~~\tau=\frac{Q^2}{4M^2},$$
with $a=0.942$ and $b=4.61$.
This parameterization is very similar to that of
Galster {\em et al.}~\cite{Glaster_71}, as shown
in Figure~\ref{show_gen_new}.
%
The parameters in Table~\ref{JRA_coef},
along with the fit of
$G_E^n$ of Krutov {\em et. al. }~\cite{Krutov_02},
are referred to as `BBA-2003 Form Factors'.
For BBA-2003 Form Factors, both the cross
section and polarization data are used in the extraction
of $G_E^p$ and $G_M^p$.
Figure~\ref{ratio_JhaKJhaJ_D0DD}
shows the ratio versus neutrino energy of the
predicted neutrino (antineutrino) QE cross section on nucleons
using the `BBA-2003' Vector Form Factors to the prediction using the
Dipole Vector Form Factors (with $G_E^n$=0 and $M_A$ kept fixed).
This plot indicates that it is important to use
the updated form factors.
%
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=monizPRL.eps,width=2.91in,height=4.3in}}
\caption{Extraction of the Fermi Gas model parameters i.e.
the effective $K_f$ and nuclear
potential binding energy $\epsilon$
from 500 MeV
electron scattering data (from Moniz.~\cite{Smith_72}.
(a)Carbon, (b) Nickel and (c) Lead}
\label{monizPRL}
\end{figure}
%
%%figure8
\begin{figure}
\centerline{\psfig{figure=NukeFermiGasC12EBind25.eps,width=2.91in,height=2.91in}}
%\begin{center}
%\epsfxsize=2.91in
%\mbox{\epsffile[0 0 567 240]{NukeFermiGasC12EBind25.eps}}
%\end{center}
\caption{ The Pauli blocking suppression for
a Fermi gas model for carbon with a 25 MeV
binding energy and 220 MeV Fermi momentum. }
\label{pauli}
\end{figure}
%
%
%figure8
\begin{figure}[htb]
\centerline{\psfig{figure=Tsushima_ff.eps,width=2.91in,height=2.91in}}
%\begin{center}
%\epsfxsize=2.91in
%\mbox{\epsffile[0 0 567 260]{Tsushima_ff.eps}}
%\end{center}
\caption{
The ratio of bound nucleon (in Carbon) to free nucleon
form factors for $F_1$, $F_2$, and $F_A$ from ref ~\cite{Tsushima_03}.
Note that this model is only relevant for $Q^2$
less than 1 $GeV^2$, and that the binding effects on the form factors
are expected to be very small at higher $Q^2$. At low $Q^2$ the
effect of the nuclear binding effects from this model
are similar to what is observed in experiments at Jlab ~\cite{mark1}.}
\label{binding}
\end{figure}
%
\begin{figure}[htb]
\centerline{\psfig{figure=Tsushima_ff_file_g.eps,width=2.91in,height=2.91in}}
%\begin{center}
%\epsfxsize=2.91in
%\mbox{\epsffile[0 0 567 260]{TsushimaG.eps}}
%\end{center}
\caption{
The ratio of bound nucleon (in Carbon) to free nucleon
form factors for $G_E^p$, $G_E^n$, $G_M^p$ and $G_M^n$
from ref ~\cite{Tsushima_03}.
Note that this model is only relevant for $Q^2$
less than 1 $GeV^2$, and that the binding effects on the form factors
are expected to be very small at higher $Q^2$. At low $Q^2$ the
effect of the nuclear binding effects from this model
are similar to what is observed in experiments at Jlab ~\cite{mark1}.}
\label{binding1}
\end{figure}
%
\section{Nuclear Effects in Quasielastic Scattering
from Bound Nucelons}
There are three important effects on the inclusive
quasielastic cross section on nuclear targets.
These are (a) Fermi Motion, (b) Pauli Blocking and
(c) Binding corrections to the nucleon form factors due
to distortion of the both the nucleon size or distortions of
the pion cloud around the nucleon in the nucleus.
Figure~\ref{pauli} shows the nuclear suppression
versus $Q^2$ from
a NUANCE~\cite{Casper_02} calculation~\cite{Zeller_03} of a
Smith and Moniz~\cite{Smith_72} based Fermi gas model
for carbon. This nuclear model includes Pauli blocking
and Fermi
motion (but not final state interactions).
The Fermi gas model was run with a 25 MeV nuclear
potential binding energy $\epsilon$ and
220 MeV Fermi momentum $K_f$.
Figure~\ref{monizPRL} from Moniz et. al. ~\cite{Smith_72}
shows how the effective $K_f$ and nuclear
potential binding energy $\epsilon$
(within a Fermi-gas
model) for various nuclei was extracted from
electron scattering data. The effective $K_f$
is extracted from the width of the electron
scattered energy, and the nuclear
potential binding energy $\epsilon$ is extracted
from the shifted location of the quasielastic peak.
Figure~\ref{fermikf} shows
the effective $K_f$ for various nuclear targets.
Figures~\ref{binding} and ~\ref{binding1} show the prediction for the
nuclear binding effect on the nucleon form factors
(i.e. the ratio of bound to free nucleon
form factors for $F_1$, $F_2$, $F_A$) in neutrino scattering,
and for the vector form factors $G_E^p$, $G_E^n$, $G_M^p$ and $G_M^n$
in electron scattering
as modeled by Tsushima {\em et al}~\cite{Tsushima_03}.
At low $Q^2$ the
effect of the nuclear binding effects from this model
are similar to what is observed in experiments at Jlab ~\cite{mark1}.
Both the Pauli blocking and the nuclear
modifications to bound nucleon
form factors reduce the cross section relative to the cross section with
free nucleons. However, it is more likely
that the low $Q^2$ deviations are not actually modifications
of the actual nucleon form factors, but rather effects
of interaction with the pion cloud for $Q^2$
less than 1 $GeV^2$. Note that
experiments from Jlab indicate that
the binding effects on the form factors
are expected to be very small at higher $Q^2$ (as described
in the next section).
\section{Detection of recoil nucleons}
The calculation for the inclusive cross section assumes
that only the final state muon is detected. In neutrino
experiments, detection of the recoil nucleon is sometimes
required in order to differentiate between quasielastic
and inelastic events. Therefore, the final state interaction
of the final state proton with the remaining nucleons also needs
to be modeled (which leads to a reduction of the number of
identified quasielastic events). Similarly, quasielastic
scattering with nucleons in the high momenta region
of the spectral functions also needs to be modeled.
This requires more sophisticated models than the simple
Fermi-Gas model.
Conversely,
inelastic events (such as in resonance production) may be misidentified
as quasielastic events if the final state pion is absorbed in
the nucleus. The best way to model these effects is to do an analysis
on samples of electron scattering data on nuclear targets (including
the hadronic final states) in order to test the effects of the
experimental cuts on the final state nucleons. This kind of study
is being planned by a Rochester group in collaboration
with the Hall B CLAS collaboration.
Such an investigation
also tests the validity of the binding off-shell corrections to
the nucleon form factors for nucleons bound in a nucleus.
Current e,e'p experiments at intermediate $Q^2$ are not
well described by the
impulse approximation with
distortion effects. One is forced to introduce a
quenching factor which is large for low $Q^2 \sim 0.3$ GeV$^2$.
This effect has been modeled by Tsushima {\em et al}~\cite{Tsushima_03}
as binding corrections to nucleon form factors. However this
factor cannot be strickly interpreted as
a change of the form factor of
the nucleus because for large $Q^2$ the suppression
becomes much smaller and may
be practically gone~\cite{mark1} by $Q^2$=2 GeV$^2$.
The interpertation of M. Strikman and others~\cite{mark1}
is that one is dealing here with renormalization of the
interaction of nucleons at low energy scale (natural in
the Fermi liquid theory) which is essentially gone at large
$Q^2$. What this implies for low $Q^2$ is basically the
statement that the theory is not good enough
and hence it is difficult to calculate
cross section in $\nu A$ scattering at low $Q^2$ from first principles.
It is not clear how well the rescaling from e,e' to $\nu A$ will
work under these conditions.
There may be some differences since the pion field plays a rather
different rols in two cases (as can be seen from the different masses
entering in the axial and vector electromagnetic form factors).
Therefore, one should take the predictions of the model of
Tsushima {\em et al}~\cite{Tsushima_03} only as an indication of the
possible magnitude of these effects (and use it only at low $Q^2$). More
theoretical and experimental studies are needed.
MINERvA can address this
question by investigating
nuclear and binding effects in Carbon in
neutrino scattering, and compare the
data to nuclear effects observed in
electron scattering at Jlab.
One example of this is shown in
figure~\ref{fig:He3.eps} for Helium 3. The figure shows
the ratio of $G_M$/$G_E$ measured for nucleons bound in Helium3
(by looking at the polarization of the recoil protons) devided
by the theoretical pedictions for this ration which include all known
nuclear effects
for two models. One model (which is valid at low $Q^2$ ) uses
Optical Potentials. The other model (which is valid at higher $Q^2$)
uses the Glauber approach. The curves labled (QMC) are the
model predictions if we include the binding effects
in the nucleon form factors modeled by Tsushima {\em et al}~\cite{Tsushima_03}.
These data indicate that at high $Q^2$, there is no evidence
for for nuclear binding effects in the form factors.
%
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=He3.eps,width=5.1in,height=3.0in}}
\caption{The ratio of $G_M$/$G_E$ measured for nucleons bound in Helium 3
(from the polarization of the recoil protons) devided
by the theoretical pedictions including all known effects
for two models. One model (which is valid at low $Q^2$ ) uses
Optical Potentials. The other model (which is valid at higher $Q^2$)
uses the Glauber approach. The curves labled (QMC) are the
model predictions if we include the binding effects
in the nucleon form factors modeled by Tsushima {\em et al}~\cite{Tsushima_03}.
These data indicate that at high $Q^2$, there is no evidence
for for nuclear binding effects in the form factors.
}
\label{fig:He3.eps}
\end{figure}
As mentioned earleir members of the Rochester MINERvA group
(Steve Manly) in collaboration with Jlab (Will Brooks)
will be working
CLAS collaboration to
study hadronic final states in electron scattering
on nuclear targets using existing Jlab Hall B CLAS data. This
analysis will provide information on hadronic final
states in quasielastic and inelastic resonance production
in electron scattering (for testing
theoretical models to use in both electron and neutrino
experiments)
In addition, the we wiil be collaborating with the Ghent
nuclear physics group in Belgium~\cite{belgium}, to model
both electron and neutrino
quasielastic scattering on nuclei over the entire range of $Q^{2}$.
This will give us the theoretical tools do a precise extraction
of the axial form factor of the nucleon using our data on
Carbon by performing the same analysis on neutrino and
electron scattrering data in the same range of $Q^{2}$
An example of this is shown in figure~\ref{fig:lowQ.eps} where
the difference between
electron scattering data in the quasielastic region (for Carbon) for
which both the final state electron and proton are detected is compared to the
prediction of theoretical models versus the momentum of
the recoil proton. Extension of these models to neutrino scattering
is currently under way.
%
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=lowQ.eps,width=5.1in,height=3.0in}}
\caption{The difference between
electron scattering data in the quasielastic region (for Carbon) for
which both the final state electron and proton are detected, compared to the
prediction of theoretical models versus the momentum of
the recoil proton. Extension of these models to neutrino scattering
is currently under way.}
\label{fig:lowQ.eps}
\end{figure}
\subsection{A re-extraction of the axial form factor from previous
neutrino data on deuterium}
Previous neutrino measurements, mostly bubble chamber experiments
on deuterium,
extracted
$M_A$ using the best known assumptions at the time. Changing
these assumptions changes
the extracted value of $M_A$. Hence, $M_A$ needs to
be updated using new form factors and up-to-date couplings.
Budd, Bodek and Arrington updated the
results from three previous deuterium bubble
chamber experiments. These are
Baker {\em et al.}~\cite{Baker_81}, Barish {\em et al.}~\cite{Barish_77},
Miller {\em et al.}~\cite{Miller_82},
and Kitagaki {\em et al.}~\cite{Kitagaki_83}.
Barish {\em et al.} and Miller {\em et al.} are the
same experiment, with the analysis of Miller {\em et al.} including
the full data set, roughly three times the statistics included in the
original analysis. On average, correcting
for the various assumptions in form factors and couplings
results in a decrease of 0.026 in the extracted value of $M_A$. This
is why we use a value of 1.00 instead of the previous world
average of 1.026.
%figure7
\begin{figure}[htb]
%\begin{center}
%\epsfxsize=2.91in
%\mbox{\epsffile[95 260 540 580]{Baker_d0dd_110_JhaKJhaJ_105.eps}}
%\end{center}
\centerline{\psfig{figure=Baker_d0dd_110_JhaKJhaJ_105.eps,width=2.91in}}
\caption{ A comparison of the $Q^2$ distribution using 2 different
sets of form factors. The data are from Baker {\em et al.}~\cite{Baker_81}.
The dotted curve uses Dipole Form Factors with $G_E^n=0$ and
$M_A=1.10~GeV$. The dashed curve uses BBA-2003 Form Factors
with $M_A=1.05~GeV$. The two curves cannot be distinguished
from one another. This illustrates that
it is important to use the most up to date
information on vector form factors
from electron scattering experiments
when extracting the axial form factor from
neutrino data.}
\label{Baker_d0dd_110_JhaKJhaJ_105}
\end{figure}
%
Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105} shows the $Q^2$ distribution
from the Baker {\em et al.}~\cite{Baker_81} neutrino experiment
compared to the prediction assuming
Dipole Form Factors with $G_E^n$=0 and $M_A$=1.10 GeV. Also shown are
the prediction using BBA-2003 Form Factors
and $M_A$=1.050 GeV. When we modify the electromagnetic form factors,
we can use a diffeernt $M_A$ to describe the same $Q^2$ distribution.
Although the overall total cross sections are different,
there is no modification of the $Q^2$ dependence when
a contribution to the distribution
is shifted between the electromagnetic and axial form factors.
Therefore, we conclude that
with the same value of $g_A$, the use of Dipole Form Factors (and Gen =0)
instead of the BBA-2003 form factors lead to an error in extracted
value of $M_A$ of 0.050
GeV, independent of the details of the experiment.
%
\subsection{Extractions of the axial form factor in MINERvA}
Current and future high statistics
neutrino experiments at low energies (such as MiniBoone, JPARC and
MINERvA) use an active nuclear target such as scintillator (e.g. Carbon).
As shown in
Figure ~\ref{fig:quasiplots}
The maximum $Q^2$ values that can be reached with neutrino energies of
0.5, 1.0, 1.5 and 2 GeV are 0.5, 1.2, 2,1 and 2.9 $~GeV^2$.
Since
MiniBoone and JPARC energies are in the 0.7 GeV range, these
experiments probes the
low $Q^2< 1 ~GeV^2$ region where nuclear effects are very large
(see Figures~\ref{pauli} and ~\ref{binding})
and where the
axial from factor is already known very well from
neutrino data
on deuterium (see Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105}).
The low $Q^2$ ($Q^2< 1 ~GeV^2$) MiniBoone and JPARC experiments can
begin to investigate
the various nuclear and binding effects in Carbon in
neutrino scattering versus those observed in electron scattering
at Jlab.
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=quasiplots.epsf,width=2.9in,height=4.in}}
\caption{ $Q^2$ distributions for different incident neutrino and
antineutrino energies illustrating the maximum $Q^2$ reach at
each energy}
\label{fig:quasiplots}
\end{figure}
At higher $Q^2~GeV^2$, as shown by the BBA-2003 fits
to the vector form factors,
the dipole approximation can be as much
as factor of 2 wrong for the vector form factors for $Q^2> 2 GeV^2$.
Therefore, there is no reason for this form to also be valid
for the axial form factors.
As can be seen from Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105}
there is very little data for the axial
form factor in the high $Q^2$ region (where nuclear effects are
smaller). Both the low $Q^2$ ($Q^2< 1 ~GeV^2$) and
high $Q^2$ ($Q^2> 2 ~GeV^2$) regions
are accessible at higher energy experiments such
as MINERvA
at Fermilab (which can span the 2-8 GeV energy neutrino range).
Figure ~\ref{fig:fraction} shows the ratio of the fractional error
in the differential cross
section for neutrinos (a) and antineutrinos (b)
versus $Q^2$ to the fractional error
in the various form factors. As can be seen in the plot, the
fractional error in the axial form factor is almost the
same as the factional error in the measured differential cross section.
In contast, the differential cross section at high $Q^2$ is not
very sensitive to the values of $G_E^p$ and $G_E^n$ at high $Q^2$.
This is because the fractional contribution of the electric
form factors to the differential cross section in this region
is small as illustrated in Figure ~\ref{FFcontributions}.
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=ddsigma_dq_dff.ps,width=4.in,height=3.in}}
%\centerline{\psfig{figure=ddsigma_dq_dff.ps,width=4.in,height=3.in}}
\caption{ Ratio of the fractional error in the differential cross
section versus $Q^2$ (for neutrinos and
antineutrinos) to the fractional error
in the various form factors. As can be seen in the plot, the
fractional error in the axial form factor is almost the
same as the factional error in the measured differential cross section.}
\label{fig:fraction}
\end{figure}
\begin{figure}%[htb]
\centerline{\psfig{figure=FA_Minerva_ME_lin.ps,width=4.in,height=3.in}}
%\centerline{\psfig{figure=FA_Minerva_ME_lin.ps,width=4.in,height=3.in}}
\caption{Linear plot (a) neutrinos
and (b) antineutrinos : The extracted values and errors
of $F_{A}$ from a sample of quasielatic events in the
MINERvA active Cargon target under the
assumption that $F_{A}$ is described by the dipole form factor
with $M_{A}$ of 1 GeV. This corresponds to a total of
the currently planned 4 years
of running in the Low Energy Beam Configuration.}
\label{lin:ps}
\end{figure}
\begin{figure}%[htb]
\centerline{\psfig{figure=FA_Minerva_ME_log.ps,width=4.in,height=3.in}}
%\centerline{\psfig{figure=FA_Minerva_ME_log.ps,width=4.in,height=3.in}}
\caption{Logarithmic plot (a) neutrinos
and (b) antineutrinos: The extracted values and errors
of $F_{A}$ from a sample of quasielatic events in the
MINERvA active Cargon target under the
assumption that $F_{A}$ is described by the dipole form factor
with $M_{A}$ of 1 GeV. This corresponds to a total of
the currently planned 4 years
of running in the Low Energy Beam Configuration.}
\label{log:ps}
\end{figure}
\begin{figure}%[htb]
\centerline{\psfig{figure=GEP_rat_FA_err_polar_dip.ps,width=4.in,height=3.in}}
\caption{ (a) Neutrino, (b) Antineutrino. The extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of $G_E^p$(Cross-Section+POLARIZATION)/$G_E^p$(Dipole)
(which is the currently assumed form factor in the
BBA2003 parametrization for $G_E^p$ ).
Note a ratio of 1.0 is expected
if the axial form factor is described exactly by the
dipole form. This corresponds to a total of
the currently planned 4 years
of running in the Low Energy Beam Configuration.}
\label{polardip:ps}
\end{figure}
\begin{figure}%[htb]
\centerline{\psfig{figure=GEP_rat_FA_err_polar_CS.ps,width=4.in,height=3.in}}
\caption{ (a) Neutrino, (b) Antineutrino.The extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of
$G_E^p$(Cross-Section+POLARIZATION)/$G_E^p$(Cross-Section).
Note a ratio of 1.0 is expected
if the axial form factor is described exactly by the
dipole form. This corresponds to a total of
the currently planned 4 years
of running in the Low Energy Beam Configuration.}
\label{polarcs:ps}
\end{figure}
\begin{figure}%[htb]
\centerline{\psfig{figure=GEP_rat_FA_err_CS_dip.ps,width=4.in,height=3.in}}
\caption{ (a) Neutrino, (b) Antineutrino. The extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of
$G_E^p$(Cross-Section)/$G_E^p$(dipole), which was the
the accepted result for $G_E^p$ before the new polarization
transfer measurement. Note a ratio of 1.0 is expected
if the axial form factor is described exactly by the
dipole form. This corresponds to a total of
the currently planned 4 years
of running in the Low Energy Beam Configuration.}
\label{csdip:ps}
\end{figure}
%Here are some FA ps files. The GEP files are plotting
%ratios of GEP with the errors of FA, to see how well
%FA can be measured, as a comparision with the measurements of
%GEP
%
% GEP_rat_FA_err_polar_dip.ps - ratio of CS+polarization/dipole
% GEP_rat_FA_err_polar_CS.ps - ratio of CS+polarization/CS
% GEP_rat_FA_err_CS_dip.ps - ratio of CS/dipole
%FA_Minerva_ME_lin.ps - Extracted FA with input FA.
% FA_Minerva_ME_log.ps - same as above log plot
%
%These use the C nuclear correction, but no form factor corrections.
%I'm using what I think, but will have to check is 1 year of running
%in the medium energy beam (ME). This is 1.4m by 1.4m by 1.4m
%fiducial area. The number of events is 115000.
Note that the high $Q^2$ region does not contribute much to
the total quasielastic cross section. Therefore, it does not
contribute much to the uncertainties in the total cross section.
The measurement of the
axial form factor in the high $Q^2$ region (which
can be done in MINERvA)
is mostly of interest in the investigation of the vector and
axial structure of the nucleon.
For example, for
1 year of running in the medium energy beam (ME)
and a fiducial volume of 1.4m by 1.4m by 1.4m of active scintillator
the number of quasielastic events is 115,000.
Figure~\ref{lin:ps} (linear)
and Figure~\ref{log:ps} (log) show the extracted values and errors
of $F_{A}$ from a sample of 115.000 events in the
MINERvA active Cargon target under the
assumption that $F_{A}$ is described by the dipole form factor
with $M_{A}$ of 1 GeV. We use the various measurements of
of $G_E^p$ to illustrate the values and the errors on
possible outcomes of MINERvA. Figure~\ref{polardip:ps} show the
extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of $G_E^p$(Cross-Section+POLARIZATION)/$G_E^p$(Dipole)
(which is the currently assumed form factor for $G_E^p$ in the
BBA2003 parametrization).
Figure~\ref{polarcs:ps} show the extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of
$G_E^p$(Cross-Section+POLARIZATION)/$G_E^p$(Cross-Section).
Figure~\ref{csdip:ps} show the extracted ratio and error in
$F_{A}$/$F_{A}$(Dipole) under the assumptions that
this ratio is described by
the ratio of
$G_E^p$(Cross-Section)/$G_E^p$(dipole), which was the
the accepted result for $G_E^p$ before the new polarization
transfer measurement. Note a ratio of 1.0 is expected
if the axial form factor is described exactly by the
dipole form.
\section{Appendix D: An Introduction to Nuclear Effects}
\subsection{Plane Wave Impulse Approximation (PWIA)}
A. Free protons
Although we plan to use
much more sophisticated models
in our analysis (in collaboration
with the Ghent nuclear theoryh group),
it is instructive to describe the use
of the Fermi Gas model as an example of a model
in which nuclear parameters are extracted from electron
scattering data and applied to neutrino scattering
experiments.
Here we describe an treatment of Bodek and Ritchie, and also
add the effect of the nuclear binding potential $\epsilon$
as described in Moniz.~\cite{Smith_72}.
Before we discuss the kinematics of the scattering from an off-shell
nucleon in a deuterium or heavy target nucleus, we consider the case of
scattering from a free proton. We take the case of electron
scattering to represent the general lepton-nucleon scattering at high
energies. The kinematics of the scattering from a free proton of
mass $M_p$ is shown in Figure~\ref{diagram}(a). The incident electron energy is
$E_0$ and the final scattering energy in the laboratory system is
$E^\prime$. The scattering angle in the laboratory is defined as
$\theta$. The four-momentum transfer to the target proton is $q =
(\vec q_3,q_0)$. We define the following variables in terms of
laboratory energies and angles. The square of the invariant
four-momentum transfer $q$ is
$$q^2 = q^2_0 - \vec q_3^2 = -4E_0E^\prime \sin^2{\theta\over 2} =
-Q^2 \; .$$
The square of the initial target proton four-momentum $P_i$ is
$$P^2_i = M^2_p\; . $$
The square of the final-state proton momentum $P_f$ (which is equal
to the final-state invariant mass) is\begin{eqnarray*}
P^2_f & = & W^2 = (P_i + q)^2 = P^2_i+2P_i\cdot q+q^2 \\
& = & M^2_p + 2M\nu - Q^2 \; ,
\end{eqnarray*}
where $\nu = E_0 - E^\prime = q_0$ (in the laboratory system) and $x
=Q^2/2q \cdot P_i = Q^2/2M_p\nu$.
\bigskip
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=diagram.eps,width=3.0in,height=3.5in}}
\caption{Kinematics for on-shell and off-shell scattering and
scattering from off-shell nucleons in deuterium
and nuclei. (a) Free nucleons.
(b) A nucleon bound in the deuteron. (c) A nucleon with momentum
$|\vec P_i|< K_F$ in a heavy nucleus of atomic weight $A$. (d) A
nucleon bound in a heavy nucleus having momentum $|\vec P|> K_F$ due
to an interaction with another nucleon. See text for details.}
\label{diagram}
\end{figure}
B. Scattering from an off-shell nucleon in the deuteron
In the impulse approximation, the spectator nucleon in the deuteron
is free and is on the mass shell. It is totally unaffected by the
interaction. The interacting nucleon with momentum $P_i$ must be of
the mass shell in order to conserve energy and momentum in the
scattering process. The kinematics is shown in Figure~\ref{diagram}(b).
The Fermi
motion does not change $Q^2$ but it does change the final-state
invariant mass $W$ and the quantity $P_i \cdot q$. Because the
interacting nucleon is off the mass shell, its effective mass is less
than the mass of the proton and is a function of its momentum.
The on-shell spectator has momentum $\vec P_s$ and on-shell
energy $E_s = (\vec P^2_s + M^2_p)^{1/2}$. The off-shell interacting
proton has momentum $-\vec P_s$ and off-shell energy in the
laboratory $E_i = M_d - E_s$, where $M_d$ is the mass of the
deuteron, i.e.,
$$\vec P = \vec P_i = - \vec P_s \quad {\rm and} \quad E_i = M_d -
(P^2_s + M^2_p)^{1/2} \;.$$
After the scattering the invariant mass of the final state
(neglecting the free spectator) is
\begin{eqnarray*}
P^2_f & = & W^{\prime 2} = (P_i + q)^2 = P^2_i + 2P_i \cdot
q-Q^2 \; , \\
W^{\prime 2} & = & (E^2_i - \vec P^2_s) + 2E_i\nu - 2P_3|\vec
q_3|-Q^2 \; ,
\end{eqnarray*}
where $P_3$ is the momentum along the direction of the $\vec q_3$
vector.
C. Scattering from an off-shell nucleon in the nucleus $(P