%edited by A. Bodek Sept. 9.2003
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% 10/27/03
\hyphenation{author another created financial paper re-commend-ed Post-Script}
\begin{document}
%
\section{Low Energy Neutrino Scattering Overview}

\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=crossbreakdown.eps,width=6.in,height=3.0in}}
%\centerline{\psfig{figure=crossbreakdown.eps,width=6.in,height=3.0in}}
    \caption{Total neutrino and antinuetrino cross sections (devided by energy)
    versus energy compared to the sum of quasielastic, resonance,
    and inelastic contributions (from D. Casper, Private 
    Communication)}
\label{fig:total}
\end{figure*}


%
    
 There are several formalisms  used to discuss electron-nucleon and
  neutrino-nucleon (CC and NC) scattering, and the corresponding reactions on 
  nuclear targets. 
  
  Inclusive lepton scattering can be described in either structure function
  language or in terms of form factors for the production of resonant  
   final states. The two descriptions are equivalent and 
  there are expressions that relate form factors to structure functions.
  In electron scattering, the vector form factors can be related to
    the two structure functions $W_1$ and $W_2$ (which are
    different for neutrons and protons) or equivalently  $F_{2}$ and $R$.
   
    In neutrino scattering, there are three structure functions $W_1$, 
    $W_2$ and $W_3$
    (or $F_{2}$, $R$ and $xF_{3}$), which are different for the scattering
     from neutrons and protons, and contain both a vector
     and an axial vector component. In addition, because of the finite mass
     of the final state muon, there are additional structure functions
      (which are only important at very low energies) which are depend
        on the muon mass. These muon mass dependent structure functions
	   can be related to the other structure functions within
	   the framework of theoretical models.
	   
	   From CVC, the vector structure functions (or form factors)
	   measured in 
	   electron scattering can be related to the corresponding
	   vector structure functions in neutrino scattering for
	   specific isospin final states.  For elastic scattering from
	     spin 1/2 quarks or spin 1/2 nucleons, these relationships
	      for the vectro form factors are simpler. For production
	        of higher spin resonances, the relations are more
		  complicated and involve Clebsch-Gordon coefficients.
		  
		  In contrast, the axial structure functions in neutrino
		  scattrering cannot be related to electron scattering, except
		  in certain limits. For example, within a quark parton
		  model at high energies with V=A. At  low $Q^2$, the axial
		  and vector form factors are different, e.g. because of the
		  differnet interactions with the pion cloud around the
		   nucleon.
		   
		   Another difference is the issue of nuclear effects
		   in inclusive neutrino versus electron scattering. 
		   The nuclear effects
		   can be different for axial and vector parts of
		   the scattering (e.g. shadowing effects). They can
		    also be different for the various structure functions
		     (e.g. different for valence and sea quarks).
		     
		     There are several theoretical constraints
		     and sum rules that can be
		     tested in electron and neutrino reactions (or equivalently
		     included as a constraint in the overall anlaysis of data.
		     Sum of the sum rules and constraints are valid at all
		     values of Q2, and some are only valid in certain limits.
		     These include: (a) The adler sum rules, which are
		     separate sum rules
		     for the axial and vector parts of  $W_{1}$, 
		     and  $W_{2}$,  and for  $W_{3}$
		     and are valid at all values of $Q^2$ (since they are based
		     on current algebra considerations). At high $Q^2$, these sum 
		     rules are equivalent to the statement that the number of u
		     valence quarks in the proton minus the number of d  valence
		     quarks is equal to 1.  Other sum rules such as the
		      momentum sum rule (sum of the momentum
		      carried by quark and gluons is 1) and
		      the Gross-LLewelyn-Smith sum Rule (number of valence quarks
		      is equal to 3) have QCD corrections and break down
		      at very low $Q^2$. 
		      
		      At very low $Q^2$, there is the constraint that the
		       vector structure functions are related to the
		        photoproduction cross section. At high $Q^2$
			it is expected that the structure functions
			are described by QCD and satisfy QCD sum rules.

		      	Quasielastic scattering(QE),
resonance production,  and inelastic scattering are all important components of 
neutrino scattering
at low energies.  We should clarify  that the neutrino community uses
the term `quasielastic' to describe a charged-current
process in which a neutrino interacts with a nucleon
and produces a muon (or an electron)
in the final state. The nucleon can be a free nucleon
or a nucleon bound in the nucleus. The term `quasielastic' refers
to the fact that the initial state neutrino changes
into a different lepton, and there is a single recoil
nucleon in the final state (which changes its
charge state). In contrast, the electron scattering community
refers to the case of electron-nucleon scattering with
a single recoil nucleon as `elastic' scattering.
The term `quasielastic' scattering is used by the electron
scattering community to describe elastic electron-nucleon scattering
from bound nucleons in a nucleus.  Here the term `quasielastic'
refers to the fact that the bound nucleon is quasi-free. Both
of these interpertations are  used in the literature.

In order to describe specific final states, one either uses the 
langauge of structure functions, followed by fragmentation functions
at high values of Q2. At low values of Q2, many experiments describe
the cross sections for specific exclusive final states.   Both of
these pictures need to be modified when the scattering takes
place on a complex nucleus.

Figures\ref{fig:total} (a) and (b)
show the total
neutrino and antineutrino cross section (per nucleon for an isoscalar target)
    versus energy (at low energies) compared
    to the sum of quasielastic, resonance,
    and inelastic contributions.
The sum is constructed to be continuous in $W$ as follows.
For $W>2~GeV$  the Bodek-Yang model is  used 
with the axial structure functions calculated with $Z=0.5$.
The Rein-Sehgal model is used used for $W<2~GeV$. In addition,
a fraction of the Bodek-Yang cross section is added to the
Rein-Sehgal cross section
between $W=1.7~GeV$ and $W=2~GeV$. The fraction increases linearly
with $W$ from 0 to 0.38 between $W$=1.7 and 2 $W$=2 GeV.
These two figures also show the various contributions
to the neutrino and antineutrino total
cross-sections that are investigated
in this proposal.
 


%
\section{Quasielatic charged-current scattering} 

Quasielastic scatteing is discussed in detail in two recent papers
by Budd, Bodek and Arrington~\cite{budd}. Here we give a brief
summary of the results of these two papers.

%
%figure 15	   
\begin{figure*}
%\begin{center}
%\epsfxsize=6.51in
%\epsfxsize=5.5in
%\mbox{\epsffile[0 0 567 240]{elas_JhaKJhaJ_nu_tsush_g.eps}}
%\epsfxsize=5.5in
%\mbox{\epsffile[0 30 567 260]{elas_JhaKJhaJ_nu_2GeV_tsush.eps}}
%\end{center}
\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}}
\caption{The QE neutrino cross section along 
with data from various experiments. The calculation uses 
$M_A$=1.00 GeV, $g_A$=$-1.267$, $M_V^2$=0.71 GeV$^2$ and 
BBA-2003 Form Factors.
The solid curve uses no nuclear correction, 
while the dashed curve~\cite{Zeller_03} uses
a Fermi gas model for carbon with a 25 MeV binding energy 
and 220 Fermi momentum. The dotted curve is the prediction for Carbon
including both Fermi gas Pauli blocking and the 
 effect of 
nuclear binding on the nucleon form factors~\cite{Tsushima_03}.
 The lower plot is identical to the 
upper plot with the $E_{\nu}$ axis limit changed to 2~GeV.
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*}[htb]
\centerline{\psfig{figure=elas_JhaKJhaJ_nub_tsush_g.eps,width=5.in,height=3.in}}
%\begin{center}
%\epsfxsize=6.51in
%\mbox{\epsffile[0 35 567 255]{elas_JhaKJhaJ_nub_tsush.eps}}
%\end{center}
\caption{  The QE antineutrino cross section along 
with data from various experiments.
 The calculation uses 
$M_A$=1.00 GeV, $g_A$=$-1.267$, $M_V^2$=0.71 GeV$^2$ and BBA-2003 Form Factors.
The solid curve uses no nuclear correction, 
while the dashed curve~\cite{Zeller_03} uses
a Fermi gas model for carbon with a 25 MeV 
binding energy and 220 MeV Fermi momentum.
The dotted curve is the prediction for Carbon
including both Fermi gas Pauli blocking and the 
 effect of 
nuclear binding on the nucleon form factors~\cite{Tsushima_03}.
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*}	  

	  



\subsection{Total Quasielastic Cross Sections}

Figures~\ref{elas_JhaKJhaJ_nu},
and~\ref{elas_JhaKJhaJ_nub} 
show the QE cross section for 
$\nu$ and $\overline{\nu}$ with BBA-2003 Form Factors and $M_A$=1.00 GeV.
The normalization uncertainty in the data is approximately 10\%.
The solid curve uses no nuclear correction,
while the dashed curve~\cite{Zeller_03} uses
a NUANCE~\cite{Casper_02} calculation of a 
Smith and Moniz~\cite{Smith_72} based Fermi gas model
for carbon. 

This nuclear model includes Pauli blocking
(see Figure~\ref{pauli}) and Fermi
motion, but not final state interactions. 
The Fermi gas model was run with a 25 MeV binding energy and
220 MeV Fermi momentum.  The dotted curve is the prediction for Carbon
including both Fermi gas Pauli blocking, and the 
 effect of 
nuclear binding on the nucleon form factors
as modeled by Tsushima  {\em et al}~\cite{Tsushima_03} (see 
Figure~\ref{binding}). Note that this model is only valid 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 $Q^2$.
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.

The MINERvA experiment will be able to measure these quasielastic
cross sections with much greater precision. In addition, for the first
time, precision measurements  of the axial form factors of the
nucleon for  $Q^2$ greater than 1 $GeV^2$ will be done. At lower
values of $Q^2$, where the nuclear
corrections on Carbon are larger
the qausielastic axial form factor is rather well known from
neutrino experiments on deuterium.  Here, the contribution of
MINERvA will focus on understanding the nuclear effects at low
$Q^2$ (which give the largest contribution to the integrated 
quasielastic cross section). 

%
\subsubsection{Input from Electron Scattering Experiments}

Recent experiments at SLAC and Jefferson Lab (JLab) have 
given precise measurements of the vector electromagnetic
form factors for the proton and neutron. 
These form factors can be related to the form factors for QE
neutrino scattering by conserved vector current hypothesis, CVC. 
These more recent form factors can be used to give better predictions
for QE neutrino scattering.


The hadronic current for QE neutrino scattering 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.
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 $(P<K_F)$

For momenta less than the Fermi momenta $K_F$, the nucleon interacts
with the average potential of all the nucleons in the nucleus of
atomic weight $A$. Therefore, in the impulse approximation, the
interacting nucleon has momentum $P_i$ which is balanced by a
excited
recoiling nucleus with $A-1$ nucleons and momentum $P_{A-1} =
-P_i$.  The interacting nucleon is off the mass shell, and the
recoiling excited $A-1$ nucleus is on the mass shell.  After the collision,
the recoiling nucleus remains in its excited state, $M_{A-1}^*$ and all the
particles are on the mass shell [see Figure~\ref{diagram}(c)]. 
In the laboratory
system we have
$$\vec P = \vec P_i = -\vec P_s, 
\quad E_i =(\vec P^2 +M_p^{2})^{1/2}  -\epsilon \;,$$ 
where $\epsilon$ is the effective depth of
average nucleon potential energy in GeV (e.g. 25 MeV for Carbon)
and
$$W^{\prime 2} = (E^2_i - \vec P^2_2) + 2E_i\nu-2P_3|\vec q|-q^2 \;.$$
This process leaves behind an $A-1$ nucleus in an excited state
$M_{A-1}^*$, which does not need to be calculated, but
varies with $\vec P$ such that the
following equation also holds
$$ (\vec P^2_s +
M_{A-1}^{*2})^{1/2}  = M_A - E_i\; ,$$
\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=wavefunctions.eps,width=2.91in,height=4.0in}}
    \caption{Fermi Gas wave functions, with and without a quasideuton tail.
    These momentum distributions used in the calculations of Bodek and 
    Ritchie for
	$^{12}$C, $^{28}$Si,$^{56}$Fe and $^{208}$Pb.  (a) $4\pi|\vec
	P|^2\phi(\vec P)|^2$.  (b) $|\phi (\vec P)|^2$
    }
\label{wave}
\end{figure}

\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=fermikf.eps,width=2.91in}}
\caption{Fits to the  effective $K_f$ (within a Fermi-gas
 model) for various nuclei from  Moniz.~\cite{Smith_72} used
 by Bodek and Ritchie.} 
\label{fermikf}
\end{figure}

D. Scattering from an off-shell nucleon in the nucleus $(P\gg K_F)$

In the simple Fermi-gas model the nucleons cannot have momenta greater
than the Fermi momentum $K_F$. However,  such high momenta can come from
the interaction between individual nucleons through their hard-core
potential.  In the case where the nucleon has acquired its high
momentum by interacting with another single nucleon, one may assume that
a single nucleon is recoiling against it.  This case can be treated
as having a quasideuteron in the nucleus with a
excited spectator nucleus of
 $A-2$ nucleons which is at rest in the laboratory system. 
 If we neglect the excitation energy of the spectator nucleus, the
kinematics [shown in Figure~\ref{diagram}(d)] 
are the same as the scattering from a
nucleon bound in the deuteron. However
(within this rather simplified 
model for the high mometum tails)
one can also correct for the
fact that the $A-2$ nucleons are on average in an excited state 
because two nucleons each with average binding potential energy $\epsilon$
were removed:
$$\vec P = \vec P_i = - \vec P_s \quad {\rm and} \quad E_i = M_d -
(P^2_s + M^2_p)^{1/2} - 2\epsilon \;.$$

E.  Nuclear Momentum Distributions
  
	If the Fermi-Gas model is to used to
	predict neutrino cross sections on nuclei, it
	is best to use Fermi-gas momentum distributions that are obtained
	from fits to quasielastic electron-scattering data from heavy
	nuclei.  In the Fermi-gas model the momentum distribution is constant
	up to the maximum Fermi momentum $K_F$ and is zero above $K_F$.  One	
	can also add a high-momentum tail to the  momentum distribution
	(according to Moniz~\cite{Moniz}) which is based on calculations of
	nucleon-nucleon correlations in nuclear matter.  The normalized
	momentum distributions with tail  (e.g. if we use this model
	up to $P_{max}$= 0.75 GeV) are:
	%\begin{eqnarray*}
	   $$ |\phi(\vec P)|^2 =   {1\over C}
	    \left [ 1 - 6\left({K_Fa\over \pi} \right)^2\right] \;$$
	    $${\rm for} \; 0 < |\vec P|<K_F \;,$$ 
	   $$ |\phi(\vec P)|^2 =
	    {1\over  C}  \left[ 2R\left( {K_Fa\over\pi}  \right)^2
	    \left(  {K_F\over P} \right)^4 \right] \;$$ 
	   $${\rm for} \; K_F<|\vec P|<P_{max}\; , \\$$
	    $$ |\phi(\vec P)|^2 = 0 \; {\rm for}\; |\vec P|> P_{max} \;$$
	%\end{eqnarray*}
	    with $a = 2$ (GeV/c)$^{-1}$, $C = {4\over 3}  \pi K^3_F$, and
	    $R = 1/[1-K_F/(P_{max}]$.  These momentum distributions satisfy
	    the normalization
	    $$\int_0^{P_{max}} |\phi(\vec P)|^2 4\pi P^2 dp = 1.0 \; .$$
	    The difference in the
	    momentum distributions for protons and neutrons is taken into
	    account as follows:
	    \begin{eqnarray*}
	K^p_F & = & K_F \left({2Z\over A}\right)^{1/3}\; , \\
	K^n_F & = & K_F \left({2N\over A}\right)^{1/3}\; ,
	\end{eqnarray*}
	where $A = A + N$ is the atomic weight.  $Z$ is the number of
	protons, and $N$ is the number of neutrons.  For an isoscalar target
	$Z = N = A/2$ and $K^p_F = K^n_F=K_F$.  The momentum distribution
	for carbon ($^{12}$C), silicon ($^{28}$Si), iron ($^{56}$Fe), and
	lead ($^{208}$Pb) are shown in linear and logarithmic scales in
	Figures~\ref{wave}(b) and~\ref{diagram}(a). 
	The Fermi momenta that are shown are 0.221
	GeV/c for $^{12}$C, 0.257 GeV/c for
	$^{56}$Fe, and 0.265 GeV/c for $^{208}$Pb. These are
	from fits by Moniz, which also extracted 
	$\epsilon$ values of 25, 36  and 44 MeV, respectively.
	 	
	
	Although in
	the original paper of Bodek and Ritchie the form 
	for the additional tail was used
	 up to momenta of $P_{max}$=4 GeV/c, it is 
	  probably  better to use this tail only up to
	  $P_{max}$=0.75 GeV/c.  For a calculation which
	  does not include the additional tail, the expressions
	   are:
	   $$ |\phi(\vec P)|^2 =   {1\over C} \;{\rm for} \; 0 < |\vec P|<K_F \;,$$ 
	    $$ |\phi(\vec P)|^2 = 0 \; {\rm for}\; |\vec P|> K_F \;$$

	    F. Nuclear Spectral functions

The above Fermi Motion model, with or without tail is relatively
easy to use in Monte Carlo simulations. The Fermi gas model is
a special case of a spectral function with specific approximation
to the momentum distribution and nucleon removal energy. More 
refined models use electron scattering data to extract  spectral
functions which give the correlated information between the
momentum of the nucleon and the nucleon removal potential binding
for the different shell-model nucleons in different energy levels
in the nucleus.

However, it is important to realize that as long as the parameters
within a given nuclear model are empirically fit and extracted from
electron scattering data, these models can
be used rather reliably to predict the corresponding nuclear
effects in neutrino scattering (especially for vector scattering).










 


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%


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\end{thebibliography}


\end{document}
