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% 10/27/03
\hyphenation{author another created financial paper re-commend-ed Post-Script}
\begin{document}
%
\section{Quasi-Elastic Scattering}

\subsection{Quasi-Elastic Cross Section Measurement}

    Quasi-elastic scattering, either with neutrinos or with 
anti-neutrinos, provides
the largest single-channel contribution  to the total $\nu$N event rate in the
threshold regime $E_{\nu} \le$~2 GeV.  For this reason, precise 
knowledge of the
cross section for this reaction, including its energy dependence and 
its variation
with target nuclei, is important to continued progress in current and 
future neutrino
oscillation experiments.   The current state of knowledge of the 
reaction cross section
for neutrino and antineutrino beams is indicated by the compilations 
of previous
experimental measurements shown in
Figures~\ref{elas_JhaKJhaJ_nu} and~\ref{elas_JhaKJhaJ_nub}.
Among the results shown there are normalization uncertainties 
originating with flux
uncertainties which are typically 10-20\%.  It is readily seen in 
these plots that
existing measurements have large errors throughout the $E_{\nu}$ 
range accessible
by MINER$\nu$A (Fig.~\ref{elas_JhaKJhaJ_nu}, upper plot), and 
especially in the threshold
regime which is crucial to future neutrino oscillation experiments
(Fig.~\ref{elas_JhaKJhaJ_nu}, lower plot). 
Figure~\ref{elas_JhaKJhaJ_nub} shows this
problematic lack of precision to pervade the anti-neutrino 
quasi-elastic cross section
measurements as well.

%figure 15
\begin{figure*}[p]
\centerline{\psfig{figure=elas_JhaKJhaJ_nu_tsush_g.eps,width=5.in,height=3.in}}
\centerline{\psfig{figure=elas_JhaKJhaJ_nu_2GeV_tsush.eps,width=5.in,height=3.in}}
%\centerline{\missingFigure}
%\centerline{\missingFigure}
\caption[Quasi-elastic neutrino cross-section and data]
{Compilation of experimental measurements of the neutrino 
quasi-elastic cross section.
  Existing measurements are characterized by large errors throughout 
the $E_{\nu}$ range
  accessible to MINER$\nu$A (upper plot), especially so in the crucial 
threshold regime
  (lower plot).  Representative calculations are shown using BBA-2003 
form factors with
  $M_A$=1.00 GeV.  The solid curve is without nuclear corrections, the 
dashed curve
  includes a Fermi gas model~\cite{Zeller_03},
  and the dotted curve includes Pauli blocking and nuclear
  binding.  The data shown are from
  FNAL 1983~\cite{Kitagaki_83}, ANL 1977~\cite{Barish_77},
  BNL 1981~\cite{Baker_81}, ANL 1973~\cite{Mann_73},
  SKAT 1990~\cite{Brunner_90}, GGM 1979~\cite{Pohl_79},
  LSND 2002~\cite{Auerbach_02}, Serpukov 1985~\cite{Belikov_85},
  and GGM 1977~\cite{Bonetti_77}.  }
\label{elas_JhaKJhaJ_nu}
\end{figure*}

%figure16
%\begin{figure*}[p]
    \begin{figure*}[htb]
\centerline{\psfig{figure=elas_JhaKJhaJ_nub_tsush_g.eps,width=5.in,height=3.in}}
%\centerline{\missingFigure}
\caption[Quasi-elastic anti-neutrino cross section and data]
{Experimental measurements of the anti-neutrino quasi-elastic cross 
section over the
  energy range accessible to {\minerva}.  As with the neutrino 
quasi-elastic cross section
  results of Fig.~\ref{elas_JhaKJhaJ_nu}, there is large dispersion among the
  among the anti-neutrino measurements including systematic 
differences between individual
  epxeriments.   Theoretical expectations without (solid curve) and including
  nuclear corrections (dashed, dotted curves) are shown for comparison.
The data shown are from SKAT 1990~\cite{Brunner_90},
GGM 1979~\cite{Armenise_79}, Serpukov 1985~\cite{Belikov_85},
and GGM 1977~\cite{Bonetti_77}.}
\label{elas_JhaKJhaJ_nub}
\end{figure*}


The MINER$\nu$A experiment will be able to measure these quasi-elastic
cross sections with statistical precisions afforded by event sample sizes
which will exceed by two orders of magnitude the samples isolated by
the previous generation of (mostly) bubble chamber experiments.
In addition, for the first
time in neutrino scattering experimentation, precision measurements
of the form factors of the
nucleon for $Q^2$ greater than $1~\hbox{GeV}^2$ will be possible.


The current level of theoretical understanding is indicated by curves 
from recent
calculations of the cross sections, shown superimposed in
Figures~\ref{elas_JhaKJhaJ_nu} and~\ref{elas_JhaKJhaJ_nub}.
Among multiple different considerations which go into realistic calculations,
a primary one concerns consistent, up-to-date treatment of the
vector and axial vector form factors which characterize the nucleon 
weak current.
Development of realistic cross section calculations has been underway 
by researchers
of MINER$\nu$A for some time.  Details can be found in papers by
H. Budd, A. Bodek, and J. Arrington~\cite{budd}; recent 
parameterizations and fits
published by these authors are hereafter designated as ``BBA-2003" results.
The calculated curves in Figs.~\ref{elas_JhaKJhaJ_nu} 
and~\ref{elas_JhaKJhaJ_nub}
are based upon BBA-2003 form factors with the
axial mass parameter of the axial form factors set to $M_A$=1.00 GeV.
In Figs.~\ref{elas_JhaKJhaJ_nu} and~\ref{elas_JhaKJhaJ_nub}, the 
solid curves are
calculated without nuclear corrections, while the dashed curves 
include a Fermi gas
model.  The dotted curves are calculations for target carbon
nuclei and include Fermi motion, Pauli blocking, and the effect of
nuclear binding on the nucleon form factors
as modelled by Tsushima {\em et al}~\cite{Tsushima_03}.
The calculated cross sections are seen to be diminished by
$\ge$ 10\% as result of inclusion of realistic nuclear corrections.
The sensitivity of calculations of this kind to the details of the
relevant nuclear physics,
shows that an understanding of final state nuclear effects is essential to
interpretation of quasi-elastic neutrino measurements.  As is related 
elsewhere in
this Proposal, MINER$\nu$A is a fine-grained tracking calorimeter
designed to enable comparison of quasi-elastic (as well as other 
exclusive channel)
production on a variety of target nuclei, consequently
knowledge of the contributions from various nuclear target effects 
can be greatly
improved by MINER$\nu$A investigations.
\begin{figure}[htb]
\centerline{\psfig{figure=FF_contribution.ps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
  \caption[Contributions to the $Q^2$ distribution of quasi-elastic 
scattering in \minerva.]
{Fractional contributions of $G_M^p$,$G_M^n$,$G_E^p$, $G_E^n$ and
$F_A$ to the $Q^2$ distributions for quasi-elastic
neutrino (top) and anti-neutrino (bottom) samples
with the NUMI beam configuration.}
\label{FFcontributions}
\end{figure}


\subsection{Form Factors of Quasi-Elastic Scattering}

The high statistics neutrino quasi-elastic event samples obtainable 
by {\minerva} will
enable the $Q^2$ response of the weak neutral current to be examined 
with unprecedented
accuracy.  The underlying V-A structures of the hadronic current 
include vector and
axial vector form factors whose $Q^{2}$ response is approximately 
described by dipole
amplitude forms.  The essential formalism is given by~\cite{Lle_72}

\begin{eqnarray*}
  \lefteqn{<p(p_2)|J_{\lambda}^+|n(p_1)>  =   }
  \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.
It is assumed that second-class currents are absent, and
so the scalar form-factor $F_V^3$ and the tensor form-factor $F_A^3$
do not appear.


The form factors $ F^1_V(q^2)$ and  ${\xi}F^2_V(q^2)$
are given by:
$$ F^1_V(q^2)=
\frac{G_E^V(q^2)-\frac{\D q^2}{\D 4M^2}G_M^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}},
%$$
%$$
~~~{\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}}.
$$

According to CVC, the form factors $ G_E^V(q^2)$ and $ G_M^V(q^2)$ are directly
related to form factors determined via electron scattering
$G_E^p(q^2)$, $G_E^n(q^2)$, $G_M^p(q^2)$, 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 quasi-elastic differential cross section,
$F_P(q^2)$ is multiplied by  $(m_l/M)^2$, consequently its contribution
in muon neutrino interactions
is very small except at very low energy, below 0.2~GeV.
In general, the axial form factor $F_A(q^2)$ can only be extracted from
quasi-elastic neutrino scattering. At low $Q^2$ however, the behavior of
$F_A(q^2)$ can also be inferred from pion electroproduction data.

Until recently, it has been common practice to assume that
form factors are described by the dipole approximation.   For example,
the vector form factors are often described:

$$ G_D(q^2)=\frac{1}{\left(1-\frac{\D q^2}{\D M_V^2}\right)^2 
},~~M_V^2=0.71~GeV^2$$
$$
G_E^p=G_D(q^2),~~~G_E^n=0,
%$$
%$$
~~~G_M^p={\mu_p}G_D(q^2),~~~ G_M^n={\mu_n}G_D(q^2).
$$

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.

\subsection{Axial Form Factor and Axial Mass}

   Electron scattering experiments continue to provide increasingly 
detailed determinations
of the vector form factors.  Neutrino scattering experiments however, 
are the only plausible
route to comparable determinations of the axial form factors, the 
principal one being
the axial form factor $F_A(Q^2)$.   The fall-off of the form factor 
strength with
increasing $Q^{2}$ is traditionally parameterized using an effective 
axial vector mass
$M_A$.  Its value is known to be $\approx$ 1.00 GeV to an accuracy of 
perhaps 5\%.
This value is in agreement with  with the theoretically corrected value
from pion electroproduction~\cite{Bernard_01}, \mbox{1.014 $\pm$ 0.016 GeV}.
Uncertainty in the value of $M_A$ contributes directly to uncertainty 
in the quasi-elastic
cross section.   Current values of other parameters which enter into
calculations (as in BBA-2003)
of $\sigma(E_{\nu})$ for quasi-elastic reactions, e.g. coupling constants and
magnetic moments, are listed in Table~\ref{parameters}.

%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[Parameters of the BBA-2003 form-factors]{Values of
parameters for the weak nucleon current used in the BBA-2003 calculation
of quasi-elastic reaction cross sections.}
\label{parameters}
\end{table}

The fractional contributions of $F_A$ and of
$G_M^p$,$G_M^n$,$G_E^p$, and $G_E^n$
to the $Q^2$ distribution for
quasi-elastic neutrino and anti-neutrino running with
the NUMI beam configuration are shown in
Figure~\ref{FFcontributions}.
The contributions were determined by comparing
the cross section calculated using BBA-2003 form-factors and
the cross section with each of the form-factors set to zero.
(Because of interference terms, the
sum of the fractions does not have to add up to 100\%.)

%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=show_gepgmp_ratio.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption[Ratio of $G_E^p$ to $G_M^p$]{Ratio of $G_E^p$ to $G_M^p$ as 
extracted by Rosenbluth
separation measurements (diamonds) and as obtained by polarization 
measurements(diamonds).
There measurements are seen to be in disagreement at high $Q^2$; only a small
uncertainty however is implied for the neutrino quasi-elastic total 
cross section. }
\label{show_gepgmp}
\end{figure}

\subsection{Vector Form Factors; Discrepancy at High $Q^{2}$}

Electron scattering experiments at SLAC and Jefferson Lab (JLab) have
measured the vector electromagnetic
form factors for the proton and neutron with high precision.
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 form factors,
but measure the ratio $G_E$/$G_M$.
These form factors can be related to the vector form factors for quasi-elastic
neutrino scattering by the conserved vector current (CVC) hypothesis.
Of course, more accurate form factors enable improved calculations for
for quasi-elastic neutrino scattering.

Recently however, discrepancies in electron scattering determinations of
some vector form factors have appeared, the origins of which may
be resolvable by study of quasi-elastic reactions in {\minerva}.
Figure~\ref{show_gepgmp} shows the BBA-2003
fits to $\mu_p$ $G_E^p$/$G_M^p$.
There appears a discrepancy
between two different methods of measuring the ratio of electric
and magnetic form factor for 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 transfer
yields a much lower ratio at high $Q^2$, and indicates
a difference between the electric charge and
magnetization distributions of the proton.
The polarization transfer technique is believed to be
more reliable and less sensitive to radiative effects
from two-photon corrections.



If the electric charge and magnetization distributions
of the proton are indeed very different, a test of the
high $Q^2$ behavior of the axial form factor can provide very
useful additional input towards resolving differences among
electron scattering measurements.  An accurate mapping of axial
behavior at high $Q^2$ can be done in MINER$\nu$A.


Current experiments at JLab aim to better understand the source of the
disagreement by looking at the recoil proton in elastic electron-proton
scattering, thereby minimizing the sensitivity to the dominant sources
of uncertainty in previous Rosenbluth separations.
Fortunately, since this discrepancy is most prominent at high $Q^2$, 
it introduces
a relatively small uncertainty to the $total$ neutrino quasi-elastic cross section.


\subsection{Form Factor Deviations from Dipole Form}

Electron scattering results show that dipole amplitudes provide only a
first-order description of form factor trends at high $Q^{2}$.  For example,
Figure~\ref{show_gmp} shows the deviation of $G_M^p$ from dipole-type 
$Q^{2}$ fall-off.
In general, the deviations are different for each of the form factors.

%figure2
%\begin{figure}[p]%[htb]
    \begin{figure}[htb]
\centerline{\psfig{figure=show_gmp_ratio.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption[BBA-2003 fits to $G_M^p/{\mu}_{p}G_D$]{BBA-2003 fits to 
$G_M^p/{\mu}_{p}G_D$.
The variation of the ratio from 1.0 indicates deviation from a pure 
dipole form; the
deviation is quite pronounced for $Q^{2}$ above 1 $GeV^{2}$. }
\label{show_gmp}
\end{figure}
%figure6
\begin{figure}
\centerline{\psfig{figure=ratio_JhaKJhaJ_D0DD.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption[Comparison between BBA-2003 cross-sections and dipole with $G_E^n=0$]
{Ratio versus energy of the neutrino (anti-neutrino)
quasi-elastic cross section using BBA-2003 form factors, to the 
expectation using
the dipole approximation with $G_E^n$=0.}
\label{ratio_JhaKJhaJ_D0DD}
\end{figure}


In the fits carried out by BBA-2003,
the form factors are usually described using an inverse polynomial:

$$ G_{E,M}^{N}(Q^2)=\frac{G_{E,M}^{N}(Q^2=0)}{1+a_2Q^2+a_4Q^4+a_6Q^6+...}. $$

The one exception is for $G_{E}^{N}$, for which
a useful parameterization has been given by
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 $b=4.61$.


Figure~\ref{ratio_JhaKJhaJ_D0DD} shows the ratio of the
predicted neutrino (anti-neutrino) quasi-elastic cross section on nucleons
using the BBA-2003 vector form factors compared to the prediction using the
dipole vector form factors (with $G_E^n$=0 and $M_A$ kept fixed).
This plot shows that it is important to use
most current form factor parameterizations.  In {\minerva}, it will be possible
to test for form factor structures which are more elaborate than simple
dipole fall-off with increasing $Q^{2}$, the approximation which has been used
by all neutrino experiments to date.



\subsection{Nuclear Effects in Quasi-elastic Scattering}

There are three important nuclear effects in
quasi-elastic scattering from nuclear targets:
Fermi motion, Pauli blocking, and corrections to the nucleon form factors due
to distortion of the nucleon's size and its pion cloud in the nucleus.
Figure~\ref{pauli} shows the nuclear suppression versus $Q^2$ from
a NUANCE~\cite{Casper_02} calculation~\cite{Zeller_03} using the
Smith and Moniz~\cite{Smith_72} 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/c 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 is inferred 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 quasi-elastic peak.

%%figure8
\begin{figure}
\centerline{\psfig{figure=NukeFermiGasC12EBind25.eps,width=2.91in,height=2.91in}} 

%\centerline{\missingFigure}
\caption[Suppression of bound cross section in Fermi gas model]{Pauli 
blocking suppression for
a Fermi gas model for carbon with binding energy and Fermi momentum 
cutoff set to 25~MeV
and 220~MeV/c respectively.  A suppression of this magnitude is 
expected for quasi-elastic
reactions in MINER$\nu$A. }
\label{pauli}
\end{figure}



\begin{figure}[htb]
\centerline{\psfig{figure=monizPRL.eps,width=2.91in,height=4.3in}}
%\centerline{\missingFigure}
     \caption[Fermi gas model parameters extracted from 
electron-scattering data]
     {Extraction of 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}.
     Distributions shown represent scattering from (a) carbon, (b) 
nickel, and (c)lead.}
\label{monizPRL}
\end{figure}


The predicted distortion that nuclear binding exerts on the nucleon 
form factors in
neutrino scattering is indicated in
Figure~\ref{binding}.   The effect is displayed as ratios of
bound to free nucleon
form factors for $F_1$, $F_2$, and $F_A$.

%figure8
\begin{figure}[htb]
\centerline{\psfig{figure=Tsushima_ff.eps,width=2.91in,height=2.91in}}
%\centerline{\missingFigure}
\caption[Ratio of bound to free nucleon form factors $F_1$, $F_2$, and $F_A$]{
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}.
Binding effects on the form factors
are expected to be very small at higher $Q^2$ (therefore, this
model is not valid for $Q^2$ greater tha 1 $GeV^{2}$ }
\label{binding}
\end{figure}
\begin{figure}[htb]
\centerline{\psfig{figure=lowQ2.eps,width=4.in,height=2.91in}}
\caption[Electron scattering data and predictions vs. recoil proton energy]
{Comparison between $e-e'-P$ electron scattering data on carbon
taken with incident electron energy of 2.2. GeV. Here
both final-state electron and proton are both detected. Data
and two theoretical models are compared
as a function of the recoil proton momentum.}
\label{fig:lowQ2.eps}
\end{figure}


Both the Pauli blocking and the nuclear
modifications to bound nucleon
form-factors reduce the quasi-elastic cross section relative to the 
cross section with
free nucleons. However, it is possible
that the low $Q^2$ deviations are not actual modifications
of the nucleon form factors, but rather are effects
of interaction with the pion cloud for $Q^2$
less than $1~\hbox{GeV}^2$. Note that
data from JLab indicate that
the binding effects on the form factors
should be very small at higher $Q^2$.

\subsection{Detection of Recoil Nucleons}

In neutrino experiments, detection of the recoil nucleon is highly useful
in distinguishing quasi-elastic
from inelastic events. Consequently, knowledge of
probabilities for
outgoing protons to reinteract with the remaining target nucleons
is highly desirable.
Similarly, quasi-elastic
scattering with nucleons in the high momenta region
of the spectral functions needs to be understood.
More sophisticated treatments than the simple
Fermi Gas model are required.  Conversely,
inelastic events (such as in resonance production) may be misidentified
as quasi-elastic events if the final state pion is absorbed in
the nucleus.  An optimal way to model these effects is to analyze
samples of electron scattering data on nuclear targets (including
the hadronic final states) and test the effects of the
experimental cuts on the final-state nucleons.
MINER$\nu$A can address the issues arising
with proton intranuclear rescattering
by investigating nuclear binding effects on
neutrino scattering in carbon, and then comparing
the data to observations obtained in similar kinematic regions
as obtained by electron scattering at JLab.
Indeed, MINER$\nu$A researchers of the Rochester group
will be working with the 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 quasi-elastic and inelastic resonance production
in electron  scattering, and will enable the testing
of theoretical models to be used in both electron and neutrino
experiments.  In addition, collaborative work is underway with the Ghent
nuclear physics group in Belgium~\cite{belgium}, to model
both electron and neutrino
quasi-elastic scattering on nuclei over the entire range of $Q^{2}$.
This work will develop the theoretical tools needed to do a precise extraction
of the axial form-factor of the nucleon using {\minerva} quasi-elastic data on
carbon.   It is envisaged that nearly identical analyses will be carried out
on both neutrino and
electron scattering data in the same range of $Q^{2}$.

       By way of illustration, Figure~\ref{fig:lowQ2.eps} shows
electron scattering data in the quasi-elastic region (for carbon) in
which the final-state electron and proton are both detected, compared to
predictions of theoretical models, as a function of the recoil proton momentum.
Extension of the models to neutrino scattering is currently under way.


\subsection{Extraction of the Axial Vector Form Factor}

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,
a prerequisite fo new determinations of $M_A$ is that the most
recent form factor parameterizations and couplings be used.

%figure7
\begin{figure}[htb]
\centerline{\psfig{figure=Baker_d0dd_110_JhaKJhaJ_105.eps,width=2.91in,height=2.5in}}
%\centerline{\missingFigure}
\caption[Comparison of $Q^2$ distributions for neutrinos with two sets
of form-factors]{A comparison of the $Q^2$ distribution using two 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 the more accurate BBA-2003 form factors
and require $M_A=1.05~GeV$.  It is important to use the most current
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}

\begin{figure}%[htb]
\centerline{\psfig{figure=FA_Minerva_ME_log.ps,width=4.in,height=2.5in}}
%\centerline{\missingFigure}
\caption[Extraction of $F_A$ in \minerva]{Estimation of values and errors
(logarithmic scale)
of $F_{A}$ from a sample of Monte-Carlo neutrino quasi-elastic events recorded in the
{\minerva} active carbon target.  Here, $F_{A}$ is
assumed to be a pure dipole form factor
with $M_{A}$ of 1 GeV.  The reaction sample is the estimated yield from
four years of NuMI running.}
\label{log:ps}
\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 $M_A =1.100~\hbox{GeV}$. 
Also shown are
the prediction using BBA-2003 form factors
and $M_A$=1.050~GeV.  Utilization of more accurate electromagnetic form factors
requires a different $M_A$ value in order to describe the same $Q^2$ 
distribution.
Thus, with the same value of $g_A$, the use of dipole form factors 
(and $G_E^n =0$)
instead of the BBA-2003 form factors may lead to an error in extracted
value of $M_A$ of 0.050~GeV.

\subsection{Measurement of the Axial Form Factor in {\minerva}}

Current and future high-statistics
neutrino experiments at low energies (e.g. MiniBooNE, J-PARC and
{\minerva}) use an active nuclear target such as scintillator (mostly carbon).
The maximum $Q^2$ values that can be reached with incident neutrino energies of
0.5, 1.0, 1.5 and 2 GeV are 0.5, 1.2, 2.1 and 2.9 $~\hbox{GeV}^2$ respectively.
Since MiniBooNE and J-PARC energies are in the 0.7~GeV range, these
experiments probe the
low $Q^2< 1 ~\hbox{GeV}^2$ region where nuclear effects are large
(see Figures~\ref{pauli} and ~\ref{binding})
and where the
axial form factor is known rather well from
neutrino data
on deuterium (see  Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105}).
The low $Q^2$ ($Q^2< 1 ~\hbox{GeV}^2$)  MiniBooNE and J-PARC experiments can
begin to investigate the various nuclear and binding effects in carbon.

At higher $Q^2$, as shown by the BBA-2003 fits,
the dipole approximation can be in error by as much
as a factor of two for the vector form factors when $Q^2> 2~\hbox{GeV}^2$.
There is clearly no reason to assume this form will be valid
for the axial form factors.
  As shown in 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 $Q^2$ ($Q^2< 1 ~\hbox{GeV}^2$) and
high $Q^2$ ($Q^2> 2 ~\hbox{GeV}^2$) regions
are accessible in higher energy experiments such
as MINER$\nu$A which can span the 2-8~GeV energy neutrino range.
A MINER$\nu$A determination of the
axial form factor in the high $Q^2$ region
is of keen interest to on-going investigations of the vector and
axial structure of the neutron and proton.



Figure~\ref{log:ps} shows the extracted values and errors
of $F_{A}$ in bins of $Q^{2}$ from a sample of Monte-Carlo quasi-elastic
interactions recorded in the {\minerva} active carbon target,
from a four-year exposure in the NUMI beam.
It can be seen that the high $Q^{2}$ regime which is inaccessible to
MinibooNE and J-PARC, will be well-resolved in {\minerva}.

Figure~\ref{csdip:ps} shows the these results as a ratio of
 $F_{A}$/$F_{A}$(Dipole).
The lower plot assumes 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.  The upper plot shows a ratio of 1.0 which is expected
if the axial form factor is described exactly by the
dipole form


\begin{figure}%[htb]
\centerline{\psfig{figure=GEP_rat_FA_err_CS_dip.ps,width=4.in,height=3.in}}
    \caption{
    The extracted ratio and error of
 $F_{A}$/$F_{A}$(Dipole)  from a
     sample of Monte-Carlo quasi-elastic
interactions recorded in the {\minerva} active carbon target,
from a four-year exposure in the NUMI beam.
The lower plot assumes 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.  The upper plot
shows a ratio of 1.0 which is expected
if the axial form factor is described exactly by the
dipole form.}
\label{csdip:ps}
\end{figure}

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\end{document}





