%edited by A. Bodek Sept. 9.2003
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\hyphenation{author another created financial paper re-commend-ed Post-Script}

\title{NuInt02 Conference Summary: Modeling Quasi-elastic,
Resonance and Inelastic Neutrino and Electron Scattering on 
Nucleons and Nuclei}

\author{Arie Bodek\address[roc]{Department of Physics and Astronomy,
             University of Rochester,
             Rochester, New York 14618,  USA}}
\begin{document}

\begin{abstract}
The relationships between electron scattering and neutrino scattering 
on nucleons and nuclei is reviewed, aiming at better modeling of 
neutrino and electron scattering on nucleons and nuclei in the 
quasi-elastic,
resonance,  and inelastic regions.
 (Conference Summary by Arie Bodek at NuInt02,  
 Dec. 2002, Irvine, CA, USA)
\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle
%
\section{Introduction}
%
Experimental  evidence for oscillations among the three
neutrino generations has been recently reported~\cite{Fukada_98}.
Present and future high intensity neutrino beams for
neutrino oscillations experiments are in the 0.5-5 GeV 
energy region. Since 
 quasielastic scattering(QE),
resonance production,  and inelastic scattering are all important components of 
neutrino scattering
at low energies, it is important to investigate
the relation between electron and neutrino scattering using the latest information
on nucleon elastic, resonance production and inelastic vector
and axial form factors and structure functions.

This summary is aimed at both members of the electron
scattering community and members of the neutrino community.
One thing to clarify is 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 this article,
%
\section{Quasielatic charged-current scattering}
%
This topic 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.
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, many neutrino experiment have assumed
that the vector 
form factors are 
described by the dipole approximation.
$$ G_D(q^2)=\frac{1}{\left(1-\frac{\D q^2}{\D M_V^2}\right)^2 },~~M_V^2=0.71~GeV^2
$$
$$ 
G_E^p=G_D(q^2),~~~G_E^n=0,
$$ 
$$ 
G_M^p={\mu_p}G_D(q^2),~~~ G_M^n={\mu_n}G_D(q^2).
$$
%figure 1
\begin{figure}[htb]
\begin{center}
\epsfxsize=2.91in
\mbox{\epsffile[65 470 550 710]{show_gep_ratio.eps}}
\end{center}
\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}
%
%
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}. 
HerewWe use the
 updated form factors to which we refer as `BBA-2003 Form Factors' 
 (Budd, Bodek, Arrington).
 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.
Table~\ref{parameters} summarizes the
most up to date values of the coupling constants and magnetic
moments that we use in our calculations. 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.
%
%
%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       & 3.226 & 1.508 & -0.3773 & 0.6109    & -0.1853   & 0.01596      \\
%$G_M^p$ & CS       & 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  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'.}
\label{JRA_coef}
\end{table*}
%
%figure2
\begin{figure}%[htb]
\begin{center}
\epsfxsize2.91in
\mbox{\epsffile[65 470 550 710]{show_gmp_ratio.eps}}
\end{center}
\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]
\begin{center}
\epsfxsize=2.71in
%\mbox{\epsffile[79  408  549  720]{show_gmn_ratio.eps}}
\mbox{\epsffile[65 470 550 710]{show_gmn_ratio.eps}}
\end{center}
\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}
%
\section{ `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]
\begin{center}
\epsfxsize=2.91in
%\mbox{\epsffile[79  408  549  720]{show_gepgmp_ratio.eps}}
\mbox{\epsffile[65 470 550 710]{show_gepgmp_ratio.eps}}
\end{center}
\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]
\begin{center}
\epsfxsize=2.71in
%\mbox{\epsffile[79  408  549  720]{show_gen_new.eps}}
\mbox{\epsffile[65 470 550 710]{show_gen_new.eps}}
\end{center}
\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}
Figure~\ref{show_gepgmp} shows the `BBA-2003'  
fits to $\mu_p$$G_E^p$/$G_M^p$.
The fit including only cross section data 
is roughly flat versus $Q^2$ ($Q^2=-q^2$),
while ratio decreases with
$Q^2$ in the combined fit to
cross section and polarization transfer data. 
The combined fit to both
cross section and polarization transfer data
is used as the default.
Although the polarization transfer measurement is
believed to have smaller systematic error, especially 
at high $Q^2$, the origin of this disagreement is not known.
If this disagreement comes from radiative corrections
to the electron, in particular two-photon
exchange terms, then the polarization transfer extraction
will give the correct ratio, but the overall scale
of $G_E^p$ at low $Q^2$ would be shifted down by
$\approx$$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.
%figure6
\begin{figure}
\begin{center}
\epsfxsize=2.91in
\mbox{\epsffile[7 75 550 360]{ratio_JhaKJhaJ_D0DD.eps}}
\end{center}
\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.
%
%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}
%
\section{A re-extraction of the axial 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, $M_A$ 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. 
%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{ (a)  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}
%
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.
%
\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}[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{mark}.}
\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{mark}.}
\label{binding1}
\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}

\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}.
Note that this model is only relevant for $Q^2$ 
less than 1 $GeV^2$, and that
experiments from Jlab indicate that
the binding effects on the form factors
are expected to be very small at higher $Q^2$ (see next section).

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{mark}.
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.

\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}

\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~\cite{Manly} 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{mark} by $Q^2$=2 GeV$^2$.

The interpertation of M. Strikman and others~\cite{mark}
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.
As discussed in a later section,
the MiniBoone and JPARC experiments can help 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.

\section{Plane Wave Impulse Approximation (PWIA)}

A.  Free protons

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

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}

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:
	   $$ |\phi(\vec P)|^2 =   {1\over C} \;{\rm for} \; 0 < |\vec P|<K_F \;,$$ 
	    $$ |\phi(\vec P)|^2 = 0 \; {\rm for}\; |\vec P|> K_F \;$$

%
%figure 15	   
\begin{figure*}
\begin{center}
\epsfxsize=6.51in
\mbox{\epsffile[0 0 567 240]{elas_JhaKJhaJ_nu_tsush_g.eps}}
\epsfxsize=6.51in
\mbox{\epsffile[0 30 567 260]{elas_JhaKJhaJ_nu_2GeV_tsush.eps}}
\end{center}
\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=6.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*}	  


	  
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).

\section{Future extractions of the axial form factor}

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).
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 
investigate
the various nuclear and binding effects in Carbon in
neutrino scattering versus those observed in electron scattering
at Jlab.


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~\cite{MINERvA}
at Fermilab (which can span the 2-8 GeV energy neutrino range).

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.

\section{Comparison to Experimental Data}

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}). 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}).

The updated form factors improve the agreement with neutrino QE cross section data 
and give a reasonable description of the cross sections from deuterium. 

Bodek, Budd and Arrington plan to continue to study the nuclear corrections,
adopting models
which have been used in precision electron scattering measurements
from nuclei at SLAC and JLab.  For example,  investigation of the
Pauli blocking correction using an improved Fermi
Gas model with a high momentum tail~\cite{Bodek_81},
and more sophisticated
nuclear spectral functions. In addition, the 
 the extraction of $M_A$ from previous neutrino experiments,
using the updated versions of the input parameters 
and electromagnetic form factors is continuing.


\section{Inelastic Scattering: Origin of Higher Twist Terms}

  The quark distributions in the proton
and neutron are parametrized as
Parton Distribution Functions (PDFs) obtained from
global fits to various sets of data at very high energies.
These fits are done within
the theory of Quantum Chromodynamics (QCD) in either leading order
(LO) or next to leading order (NLO). The most important data
come from deep-inelastic
e/$\mu$ scattering experiments on hydrogen and deuterium,
and $\nu_\mu$ and $\overline\nu_\mu$ experiments on nuclear targets.
In previous publications~\cite{highx,nnlo,yangthesis}
Bodek and Yang have compared the predictions of the
NLO MRSR2 PDFs to deep-inelastic e/$\mu$ scattering data~\cite{slac}
on hydrogen
and deuterium from SLAC, BCDMS and
NMC. In order to get agreement with the lower
energy SLAC data  for
 $F_2$ and $R$ down to $Q^{2}$=1 GeV$^2$, and
at the highest values of $x$ ($x = 0.9$), Bodek and Yang found
that the following modifications  to the NLO MRSR2 PDFs must be
included.
\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=nufact_fig1.ps,width=5.0in}}
    \caption{Electron and muon data 
(SLAC, BCDMS and NMC) for $F_{2p}$ [a] and $R$ [b] compared
to the predictions with MRSR2 NLO
PDFs including both NNLO and target mass corrections with (solid line) 
and without (dashed line)
higher twist corrections (From Yang and Bodek Ref.~\cite{nnlo}). These 
studies indicate that in QCD LO or NLO fits, the extracted  
higher twist corrections originate from target mass effects and 
the missing QCD NNLO higher order terms (for $Q^2>~1~$GeV$^2$). }
\label{fig:NNLOfig}
\end{figure*}
\begin{enumerate}
       \item The relative normalizations between the various
             data sets and the BCDMS systematic error shift must
              be included~\cite{highx,nnlo}.
        \item Deuteron binding corrections need to be applied and
%         discussed in ref.~\cite{highx}.
%        \item 
	the ratio of $d/u$ at high $x$ must be increased as
          discussed in ref.~\cite{highx}.
\item Kinematic higher-twist originating from target mass effects~\cite{gp}
are very large and
must be included.
\item Dynamical higher-twist corrections are smaller but also need
          to be included~\cite{highx,nnlo}.
\item In addition, the  analysis including QCD Next to NLO
(NNLO) terms shows~\cite{nnlo} that most of the dynamical 
higher-twist corrections
 needed to fit the data within a NLO QCD analysis  originate from
the $missing$ $NNLO$ $higher$ $order$ $terms$.
\end{enumerate} 


The Bodek-Yang analysis shows that the NLO MRSR2 PDFs with target
mass and NNLO higher order terms describe electron and muon
scattering $F_2$ and $R$ data with a very small contribution from higher twists.
Figures \ref{fig:NNLOfig} shows that the NLO MRSR2 PDFs with target
mass and NNLO higher order terms describe electron and muon
scattering $F_2$ and $R$ data with a very small contribution from higher twists.
Studies by other authors~\cite{Blum} also show that in NNLO
analyses the dynamic higher twist
corrections are very small.
If (for $Q^2>$ 1 GeV$^2$) most of the higher-twist terms needed to
obtain agreement with the low energy data actually originate from target mass
effects and missing NNLO terms (i.e. not from interactions with
spectator quarks) then these
terms should be the same in  $\nu_\mu$  and e/$\mu$  scattering.
Therefore, low energy  $\nu_\mu$  data
should be described by the PDFs which are fit to high energy
data and are modified to include target mass and
higher-twist corrections that fit 
low energy e/$\mu$ scattering data.  However,  for $Q^2<$ 1 GeV$^2$
additional non-perturbative effects from spectator quarks must also be 
included~\cite{first}.


\begin{figure*}[htb]
%\centerline{\psfig{figure=Fig1Mex.ps,width=5.5in,height=5.4in}}
\centerline{\psfig{figure=Fig1Mex.ps,width=5.3in,height=4.in}}
\caption{Electron and muon $F_2$ data (SLAC, BCDMS, NMC, H1 94)
used in our  GRV98 $\xi_w$ fit compared to the predictions of the unmodified GRV98
PDFs (LO, dashed line) and the modified GRV98 PDFs 
fits (LO+HT, solid line); [a] for $F_2$ proton, [b] for $F_2$ deuteron,
and [c] for the H1 and NMC proton data at low $x$.}
\label{fig:f2fit}
\end{figure*}

\section{Previous Results with GRV94 PDFs and $x_w$}

Initially~\cite{first}, 
Bodek and Yang used a modified scaling variable $x_w$ and fit for modifications
 to the GRV94
 %~\cite{grv94}
 leading order PDFs such that the PDFs
describe both high energy and low energy e/$\mu$ data. In order to describe
 low energy data down to the photoproduction
limit ($Q^{2} = 0$), and account for both target mass and higher twist effects,
the following modifications of the GRV94 LO PDFs are need:
%
%
%
%
\begin{enumerate}
\item  Increased the $d/u$ ratio at high $x$ as described in ref.~\cite{highx}.
\item  Instead of
the scaling variable $x$ use the
scaling variable $x_w = (Q^2+B)/(2M\nu+A)$ (or =$x(Q^2 +B)/(Q^2+Ax)$).
This modification was used in early fits to SLAC data~\cite{bodek}.
The parameter A provides for an approximate way to include $both$ target
mass and higher twist effects at high $x$,
and the parameter B allows the fit to be
used all the way down to the photoproduction limit ($Q^{2}$=0).
\item  In addition as was done in earlier non-QCD based
fits~\cite{DL} to low energy data,  multipliy all PDFs
by a factor $K$=$Q^{2}$ / ($Q^{2}$ +C). This was done in order for
the fits to describe low $Q^2$
 data in the photoproduction limit, where 
$F_{2}$ is related to the
photoproduction cross section according to
%
%
\begin{eqnarray}
     \sigma(\gamma p) = {4\pi^{2}\alpha_{\rm EM}\over {Q^{2}}}
          F_{2}
           = {0.112 mb~GeV^{2}\over {Q^{2}}}   F_{2} \nonumber
\label{eq:photo}
\nonumber
\end{eqnarray}
%
%
\item Finally,  
 the evolution of the GRV94 PDFs is frozen at a
value of $Q^{2}=0.24$ (for $Q^{2}<0.24$),
because GRV94 PDFs are only valid down to $Q^{2}=0.23$ GeV$^2$.
\end{enumerate}

In the analysis, the measured structure functions
were corrected for the BCDMS systematic error shift and for
the relative normalizations between the SLAC, BCDMS
and NMC data~\cite{highx,nnlo}.
The deuterium data were corrected
for nuclear binding effects~\cite{highx,nnlo}. 
A simultaneous fit  to both proton and deuteron SLAC, NMC and BCDMS data
(with $x>0.07$ only)
yields A=1.735, B=0.624 and C=0.188 (GeV$^2$) with GRV94 LO PDFs
($\chi^{2}=$ 1351/958 DOF). 
Note that for $x_w$ the parameter
A  accounts for $both$ target mass and higher twist effects.
%
%

%
\begin{figure*}[htb]
%\centerline{\psfig{figure=Fig2Mex.eps,width=6.0in,height=3.6in}}
    %input new figure with better resolution 8/5/03
\centerline{\psfig{figure=Fig2Mex.ps,width=5.9in,height=3.5in}}
\caption{ Comparisons to proton and
iron data not included in the Bodek-Yang GRV98 $\xi_w$  fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2p}$ data
in the resonance region (or fits to these data)
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) our modifications.
(b)  Comparison of photoproduction
       data on protons  to predictions using
our modified GRV98 PDFs. 
(c)  Comparison of representative CCFR $\nu_\mu$ and $\overline\nu_\mu$
charged-current differential cross sections~\cite{yangthesis,rccfr}
on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT,
solid) and without (LO, dashed)
      our modifications. 
}
\label{fig:predict}
\end{figure*}
%
%
\section{New Analysis of Bodek-Yang with $\xi_w$, $G_D$ and GRV98 PDFs }
The initial study has been updated~\cite{second}
using a new improved scaling variable  $\xi_w$, and the fit 
done for
modifications to the more modern GRV98  LO PDFs such that the PDFs
describe both high energy and low energy electron/muon data.
In addition  NMC and H1 94 data at lower $x$ are added.
The
the evolution of the GRV98 PDFs is frozen at a
value of $Q^{2}=0.8$ (for $Q^{2}<0.8$),
because GRV98 PDFs are only valid down to $Q^{2}=0.8$ GeV$^2$.
In addition, different photoproduction limit
multiplicative factors are used for valence and sea. The proposed
new scaling variable is based on the following derivation.
Using energy momentum conservation, it can be shown that
the factional momentum $\xi$ = $(p_z + p_0)/(P_z + P_0)$
carried by a quark of 4-mometum $p$ 
in a proton target of mass M and 4-momentum P is given
by  $\xi$ = $xQ^{'2}/[0.5Q^{2}(1+[1+(2Mx)^{2}/Q^2]^{1/2})]$, where

$2Q^{'2} =[Q^2+M_f{^2}-M_i{^2}]+[(Q^2+M_f{^2}-M_i{^2})^2 
+4Q^2(M_i{^2}+P_{T}^{2})]^{1/2}.$

Here $M_i$ is the initial quark mass with average initial
transverse momentum $P_T$ and $M_f$ is the mass of the
quark in the final state.  The
above expression for $\xi$ was previously derived~\cite{gp}
for the case of $P_T=0$. Assuming $M_i=0$ we use instead:  

$\xi_w = x(Q^2+B + M_f{^2})/(0.5Q^{2}(1+[1+(2Mx)^{2}/Q^2]^{1/2})+Ax)$

Here $M_f$=0, except for charm-production processes in neutrino scattering
for which $M_f$=1.3 GeV (in leading order).(Note that
this is an effective charm quark mass which is extracted
from leading order fits to neutrino production of
dimuons. In next to leading
order analyses the charm mass is closer to 1.5 GeV).
%The $\xi_w$ scaling variable  includes the exact form
%for the target
%mass effects~\cite{gp} in the dominator. 
 For $\xi_w$ the parameter A
is expected to be much smaller than for  $x_w$ since now it only  
accounts for the  higher order (dynamic higher twist) QCD terms 
in the form of an $enhanced$ target mass term (the effects of the proton
target mass are already taken into account using the exact form
in the denominator of $\xi_w$ ).
The parameter
B accounts for the initial state quark transverse
momentum and final state quark  $effective$ $\Delta M_f{^2}$
 (originating from multi-gluon emission by quarks). 

\begin{figure*}[htb]
\centerline{\psfig{figure=
Fig3Mex.ps,width=5.5in,height=3.2in}}
\caption{ Comparisons to data on deuterium
which were not included in the Bodek-Yang GRV98 $\xi_w$ fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2d}$ data
in the resonance region 
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) our modifications.
(b)  Comparison of photoproduction
       data on deuterium  to predictions using
the Bodek-Yang modified GRV98 PDFs (including shadowing corrections). 
(c) The  shadowing corrections that were applied to the PDFs for
predicting the photoproduction cross section on deuterium.}
\label{fig:predictD}
\end{figure*}
%
%For the valence quarks, we improve on the
%low $Q^2$ multiplicative factor $K$ as follows.
Using closure considerations~\cite{close} ($e.g.$the Gottfried
sum rule) and the Adler sum rule,  it can be shown
that, at low $Q^2$, 
the scaling prediction for the $valence$
quark part of $F_2$ should be multiplied by
the factor $K$=[1-$G_D^{2}$($Q^{2}$)][1+M($Q^{2}$)] where
$G_D$ = 1/(1+$Q^{2}$/0.71)$^{2}$ is the  proton elastic form factor,
and M($Q^{2})$ is related to the magnetic elastic form factors
of the proton and neutron.
At low $Q^2$, [1-$G_D^{2}$($Q^{2}$)]
is approximately $Q^{2}$/($Q^{2}$ +C) with  $C=M_{V}^{2}/4=0.71/4=0.178$ (versus
 the fit value of C=0.18 with GRV94).  In order to
 satisfy the Adler Sum rule~\cite{adler} the function M($Q^{2}$)
is added to account for  terms from the magnetic 
 elastic form factors of the nucleon).
Therefore, a more general form  
$K_{valence-vector}$=[1-$G_D^{2}$($Q^{2}$)][$Q^{2}$+$C_{2v}$]/[$Q^{2}$ +$C_{1v}$],
and   $K_{sea-vector}$=$Q^{2}$/($Q^{2}$+$C_{sea}$) is used.  Using this 
 form with the GRV98 PDFs (and now also including
 the very low $x$ NMC and H1 94 data in the fit)  yields
 $A$=0.419, $B$=0.223, and  $C_{1v}$=0.544, $C_{2v}$=0.431, and $C_{sea}$=0.380
(all in GeV$^2$, $\chi^{2}=$ 1235/1200 DOF).


As expected, A and B are now smaller with
respect to the previous fits with GRV94 and $x_w$.
 With these modifications, the GRV98 PDFs must also be multiplied
by $N$=1.011 to $normalize$ to the SLAC $F_{2p}$ data.
The fit (Figure  \ref{fig:f2fit}) yields the
following  normalizations  relative to
the SLAC $F_{2p}$ data ($SLAC_D$=0.986, $BCDMS_P$=0.964, $BCDMS_D$=0.984,
$NMC_P$=1.00, $NMC_D$=0.993, $H1_P$=0.977, and BCDMS systematic error shift of 
1.7).(Note, since the GRV98 PDFs do not include the charm sea,
for $Q^{2}>0.8$ GeV$^2$
 charm production is included using the photon-gluon fusion
model in order to fit the very high $\nu$ HERA data. This is not
needed for any of the low energy comparisons but is only needed
to describe the highest $\nu$ HERA electro and photoproduction data). 



Comparisons of $predictions$
using these modified GRV98 PDFs to other data which were $not$  
$ included$ 
in the fit are shown in Figures  \ref{fig:predict} and \ref{fig:predictD}.
From duality~\cite{bloom} considerations,
with the $\xi_w$ scaling variable, the modified
GRV98  PDFs should
also provide a reasonable
description of the average value of $F_2$
 in the resonance region. Figures \ref{fig:predict}(a) and \ref{fig:predictD}(a) show
a comparison between resonance data (from SLAC and Jefferson
Lab, or parametrizations of these data~\cite{jlab}) on protons and
deuterons versus
the predictions with the standard  GRV98 PDFs (LO)
and with the modified GRV98 PDFs (LO+HT). The 
modified GRVB98 PDFs are in good agreement with SLAC and JLab
resonance data down to $Q^{2}=0.07$ (although resonance
data were not included in the fits). 
There is also very
good agreement
of the  $predictions$ of the modified GRV98 
in the $Q^2=0$ limit with
 photoproduction  data on protons and deuterons  as shown in
Figure  \ref{fig:predict}(b)  and \ref{fig:predictD}(b). 
In predicting the photoproduction cross sections on deuterium,
 shadowing corrections~\cite{shadow} were applied
as shown in Figure  \ref{fig:predictD}(c). 
The
$predictions$ with  the modified GRV98 PDFs (LO+HT) 
are also compared to a few
representative high energy CCFR
$\nu_\mu$ and $\overline\nu_\mu$ charged-current neutrino
differential cross sections~\cite{yangthesis,rccfr} on iron
(neutrino data were not included in the fit).
In this comparison the
PDFs are used
to obtain $F_{2}$ and $xF_{3}$ and corrected for nuclear effects
in iron~\cite{first}.
The structure function $2xF_{1}$
is  obtained by using the $R_{world}$ fit from reference~\cite{slac}.
In addition, for this comparison we have assumed that
$K_{valence-axial}$=$K_{valence-vector}$,
and $K_{sea-axial}$=$K_{sea-vector}$. This assumption
is expected to be valid at high $Q^2$, but
is expected to break down as $Q^2$ approaches zero~\cite{DL} (as
discussed in a following section).
There is very good agreement of the Bodek-Yang $predictions$
 with these high energy neutrino data on iron. 
 
 \section{Using the Adler Sum Rules and Axial Structure Functions}

One of the long term goals of modeling electron and neutrino
scattering at all energies is to be able to test the
Adler~\cite{adler} sum rules.  The Adler sum rules are Current-Algebra
sum rules that are expected to be valid for all
values of $Q^2$ from $Q^2$= 0 to the highest values $Q^2$

Other sum rules such as (a) the Energy-Momentum Sum-Rule
(sum of quark and gluon fractional mometum = 1);
(b) The Bjorken Sum-Rule, (c) The Gottfried Sum-Rule
~\cite{close}
and (d) the Gross-Llewyllyn-Smith Sum-Rule (number of
valence quarks equal to 3), all have QCD corrections
and break down at low values of $Q^2$. Therefore,
a successful model of structure functions must satisfy
the QCD sum rules at high $Q^2$, and the Adler sum-rules
at $all$ $values$ of $Q^2$. In all of these sum rules, the elastic
contribution needs to be included.
As discussed earlier, Bodek and Yang
use the Adler sum rule for the vector part of 
$W_{2}$ to obtain an expression for the 
low $Q^2$ K factor multiplicative factor
to the GRV98 PDFs in the case of vector
scattering.  Here we show that a different
K factor is expected for the case of axial
scattering.

\begin{figure*}
    [htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=ariew.eps,width=5.5in,height=4.0in}}
    \caption{Comparison the prediction of
the distribution in W (in the resonance region) for the R-S model
to the predictions from the Bodek-Yang GRV98 modified PDFs
 (for neutrino energies of 2, 3 and 5 GeV). 
for different assumptions about the $K$ factor for axial
scattering. The solid line is the Rein-Sehgal prediction.
The dotted line is the Bodek-Yang predictions
assuming (incorrectly)  $K_{valence-axial}$=$K_{valence-vector}$,
and $K_{sea-axial}$=$K_{sea-vector}$. The dashed line
 is the Bodek-Yang prediction with the $K$ factors for
axial scattering  with Z=0.5.
   ( D. Casper-private communication)}
\label{wcompare}
\end{figure*}
We use the first two $\beta$ Adler sum rules for the
structure function $W_{2}$. 
Here $\beta_{p}^{-}$=$W_{2p}^{-}$ and
$\beta_{p}^{+}$=$W_{2p}^{+}$ refers to antineutrino-proton and neutrino-proton
(strangeness conserving $\Delta S=0$)
structure functions respectively.
At high $Q^2$ the basic quark model interpretation
of these two Adler sum rules is that the number of valence
$u$ quarks minus the number of valence $d$ quarks is
2-1=1 (for either vector or axial scattering). For
all  $Q^2$, these two sum rules are integrals
from $\nu=q_0=0$ to $\nu=\infty$, and $q=[1/(cos^2\theta_c)]$:
 $$\int_0^{\infty}q\left[W_{2p\Delta S=0}^{vec-}(\nu,q^2) -
 W_{2p\Delta S=0}^{vec+}(\nu,q^2)
 \right] d\nu = 1 \; $$
  $$\int_0^{\infty}q\left[W_{2p\Delta S=0}^{ax-}(\nu,q^2) - 
  W_{2p\Delta S=0}^{ax+}(\nu,q^2)
 \right] d\nu = 1 \; $$
 If we separate out the quasielastic delta function
 contributions the above two equations become. 
  $$|F_V(q^2)|^{2}+
 %\;$$
% $$
 \int_{\nu_{0}}^{\infty}q\left[W_{2p\Delta S=0}^{vec-}(\nu,q^2)-
 W_{2p\Delta S=0}^{vec+}(\nu,Q^2)
 \right] d\nu = 1 \; $$
 $$|F_A(q^2)|^2 +
% \;$$
 %$$
 \int_{\nu_{0}}^{\infty}q\left[W_{2p\Delta S=0}^{ax-}(\nu,q^2)-
 W_{2p\Delta S=0}^{ax+}(\nu,Q^2)
 \right] d\nu = 1\; $$
Where the limits of the integrals are
from pion threshold $\nu_{0}$ where $W= M_{\pi}+M_P$ to infinity.
Here

 $$|F_V(q^2)|^{2} =|F^1_V(q^2)|^{2}-
 \frac{q^2}{M^2}\left|\frac{{\xi}F_V^2(q^2)}{2}\right|^2
$$
$$
F_A(q^2)=\frac{g_A}{\left(1-\frac{\D q^2}{\D M_A^2}\right)^2 }
$$
$$ F^1_V(q^2)=
\frac{G_E^V(q^2)-\frac{\D q^2}{\D 4M^2}G_M^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}},
$$
$$
{\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}}.
$$
$$q^2 = q^2_0 - \vec q_3^2 = -4E_0E^\prime \sin^2{\theta\over 2} =
-Q^2 \; .$$
At $Q^2$=0 we have $|F_V(q^2)|^{2}$=1 and $|F_A(q^2)|^2$=$g_{a}^{2}$=1.605.
Therefore,  in this
limit the inelastic vector structure functions
must go to zero, while the inelastic axial structure function
remain finite. Note that the above sum rule only holds for the
case of strageness conserving process which consist of
$95\%$ of the cross section in neutrino scattering.
For the strangeness changing processes which consist of
$5\%$ of the cross section, the corresponding Adler sum rule
(discussed at the end of this talk) indicates that the production
of strange resonances is not suppresed at  $Q^2$=0. This difference
has not yet been implemented in the Bodek-Yang model.

For the vector structure functions, for
which there is a multitude of data, we used the form  
$K_{valence-vector}$=[1-$G_D^{2}$($Q^{2}$)][$Q^{2}$+$C_{2v}$]/[$Q^{2}$ +$C_{1v}$],
and   $K_{sea-vector}$=$Q^{2}$/($Q^{2}$+$C_{v-sea}$).
 Using this 
 form with the GRV98 PDFs (and now also including
 the very low $x$ NMC and H1 94 data in the fit)  yields
 $A$=0.419, $B$=0.223, and  $C_{1v}$=0.544, $C_{2v}$=0.431,
 and $C_{v-sea}$=0.380 (all in GeV$^2$)
 
 For the axial structure function, we first derive the
 general form that should satisfy the Adler axial sum rule.
 To satisfy the sum rule in the axial case, 
 the $K$ factors for the axial valence $u$
 and $d$ quarks must be different (and could also be
 different for the axial sea).
 $$K_{ax-Uv}=[Q^{2}+C_{2ax-Uv}]/[Q^{2} +C_{1ax-v}],$$
$$K_{ax-Dv}=[Q^{2}+C_{2ax-Dv}]/[Q^{2} +C_{1ax-v}],$$
The above $K$ factors for the $U$ and $D$ valence quarks 
become 1.0 at high $Q^{2}$. Therefore,
 this form automatically satisfies
 Adler sum rules at infinite $Q^{2}$.
 Note, however, that this form assumes that all
 the $K$ factors are independent of $W$ (which
 is only an approximation).
Plugging the above forms into the Adler Axial sum rule and
expanding for very low $Q^{2}$ yields a constant
term and a term linear in $Q^{2}$.  
Requiring the axial sum rule to be valid at $Q^{2}$=0 and also
at very small $Q^{2}$ yields (to within an overall scale
factor Z, which should be less than 1):
$$C_{1ax-v} = M_{A}^{2}/4 =0.25~GeV^2$$
$$C_{2ax-Uv}= Z C_{1ax-v} = Z(0.25)~GeV^2$$
$$C_{2ax-Dv}= [2Z+ (g_{a}^{2}-1)[C_{1ax-v}]\;$$
$$= [2Z+ 0.605)0.25~GeV^2$$ 
For $Z=0.1$ 
we get $C_{2ax-Uv}$=0.025 $~GeV^2$ and  $C_{2ax-Dv}$= 0.113 $~GeV^2$.
For $Z=0.5$ 
we get $C_{2ax-Uv}$=0.125 $~GeV^2$ and  $C_{2ax-Dv}$= 0.40 $~GeV^2$.
(within this model, the value of $Z$ needs to be extracted from
fits to CHORUS, CCFR, CDHS and other low energy neutrino data)

Note that the K factor for the axial sea is not constrained by
the Adler sum rule and the sea parameters need to be
extracted from fits to neutrino data.
$K_{ax-sea}= [Q^{2}+D_{ax-sea}]/[Q^{2} +C_{ax-sea}].$
for now, we just use the average of the valence parameters
used for the axial valence
quarks.  i.e. for Z=0.1, $C_{ac-sea}$=0.25 $~GeV^2$ and 
$D_{ax-sea}$=0.113 $~GeV^2$ . For Z=0.5,
$C_{ac-sea}$=0.25 $~GeV^2$ and 
$D_{ax-sea}$=0.263 $~GeV^2$. The axial K factors
are rather insensitive to the value of $M_{A}$,
and mostly depend on the vallue of $Z$. An expanded discussion of sum rules
is presented in a later section (section 21). 

\begin{figure*} 
    [htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=arieqsq.eps,width=5.5in,height=4.0in}}
    \caption{ The $Q^2$ distributions
(for a fixed neutrino energy of 5 GeV) are shown for
different bins in $W$. The solid line is the Rein-Sehgal prediction.
The dotted line is the Bodek-Yang prediction
assuming (incorrectly)  $K_{valence-axial}$=$K_{valence-vector}$,
and $K_{sea-axial}$=$K_{sea-vector}$. The dashed line
 is the Bodek-Yang prediction with the $K$ factors for
axial scattering  with Z=0.5. (D. Casper-private communication)}
\label{q2compare}
\end{figure*}
%
 \section{Comparison of Bodek-Yang PDFs to the Rein-Sehgal Resonance 
 Model}
% 
 Resonance production is discussed in detail in a later section.
As was shown in the the previous section, the Bodek-Yang model
provides a good description of the average vector cross sections
in the resonance region for the case of electron scattering. 
However, since low energy neutrino data have not yet been
included in the  Bodek-Yang fits,
the axial structure functions are not yet finalized.
Therefore, it is of interest to compare the Bodek-Yang predictions
in the resonance region  that of the Rein-Seghal
model. The  Rein-Seghal model is an old fit to low energy data
for the
electroproduction
and neutrino production of resonances within a
quark oscillator model. The model includes an
an axial form factor for resonance which is similar to the
axial form factor in quasielastic scattering.

 Figure ~\ref{wcompare} (from D. Casper) compares the prediction of
the distribution in $W$ (in the resonance region) for the Rein-Sehgal
model to the predictions from the Bodek-Yang GRV98 modified PDFs
 (for neutrino energies of 2, 3 and 5 GeV).  The comparison
 is done for two different assumptions about the $K$ factor for axial
scattering. The solid line is the Rein-Sehgal prediction.
The dotted line is the Bodek-Yang prediction
assuming (incorrectly)  $K_{valence-axial}$=$K_{valence-vector}$,
and $K_{sea-axial}$=$K_{sea-vector}$. The dashed line
 is the Bodek-Yang prediction with the $K$ factors for
axial scattering (described above) with Z=0.5.
As can be seen from the figure, 
there is reasonable agreement in the region of the second and third
resonances (as expected from duality) between the two models
if we include an axial contribution.
Note that above W of 1.7 GeV ,  the Rein-Seghal
model underestimates the cross section because  
the model does not include any resonances above $W=2~GeV$.

Figure ~\ref{q2compare} compares the $Q^2$ distributions
(for a fixed neutrino energy of 5 GeV) in 
different bins in $W$. The 
 solid line is the Rein-Sehgal prediction.
 The dotted line is the Bodek-Yang prediction
assuming (incorrectly)  $K_{valence-axial}$=$K_{valence-vector}$,
and $K_{sea-axial}$=$K_{sea-vector}$. 
The  dashed line is the Bodek-Yang prediction with the $K$ factors for
axial scattering (described above) with Z=0.5.
A reasonable prescription is to use 
current implementation of
Rein-Seghal model in the resonance region and
use the  Bodek-Yang modek (with an axial contribution)
in above the resonance region.
Note that
the Bodek-Yang model needs to be
coupled with the Lund fragmentation model
to provide the composition of the hadronic
final states).
Additional remarks on resonance production are presented
in section 18. 
A prescription
for providing a cross section which is continuous in $W$ is
described in a later section (section 19). 


 \section{The Longitudinal Structure Function $R$}

In leading order QCD (e.g.
GRV98 LO PDFs), $F_2$ for the scattering
of electrons and muons on proton (or neutron) targets is
given by the sum of quark
and anti-quark distributions (each weighted the
square of the quark charges):
\begin{eqnarray}
F_2(x) &=& \Sigma_i e_i^2 \left [xq_i(x)+x\overline{q}_i(x) \right]\\
2xF_1(x) &=& F_2(x) (1+4Mx^2/Q^2) / (1+R_{w}).
\end{eqnarray}

In reality, a reconstruction of $2xF_1$ from
the values of $R$ and $F_2$ in neutrino scattering
is not as simple as in the case
of charged lepton scattering (because of charm
production). For charm production, the Bjorken scaling variable $x$ no longer
represents the fractional momentum carried by the struck quark
in the infinite momentum frame due to the non-zero heavy mass of the
charm quark
($m_c \sim 1.3$ GeV).  For charm
production processes, the variable $x$ is replaced
by the slow rescaling variable $\xi=(1+m_c^2/Q^2)x$.
Therefore, the structure functions for  the charm
production (cp) and non-charm production (ncp)
components are given by the following expressions.
\begin{eqnarray}
F_2^{cp+ncp}(x) & = & F_2^{ncp}(x)+ F_2^{cp}(\xi) \\
%\label{eq:f2}
2xF_1^{cp+ncp}(x) & = & \frac{1+4M^2x^2/Q^2}{1+R^{ncp}(x)}
			F_2^{ncp}(x) \nonumber \\
                      &   &+ \frac{1+4M^2\xi ^2/Q^2}{1+R^{ncp}(\xi)}
			F_2^{ncp}(\xi)
%\label{eq:2xf1}
\end{eqnarray}
In this model, the $R_{w}$ fit is used for $R^{ncp}$
and $R^{cp}$,
%In the extraction of original
%GRV98 LO PDFs, no separate longitudinal contribution was
%included. The quark distributions
%were directly fit to $F_2$ data. A full
%modeling of electron and muon cross section requires
%also a description of $2xF_1$.
%We use a non-zero longitudinal $R$ in reconstructing $2xF_1$
%by using a fit of $R$ to measured data.
%Thus, $2xF_1$ is given by
%\begin{eqnarray}
%2xF_1(x) &=& F_2(x) (1+4Mx^2/Q^2) / (1+R_{world}).
%\end{eqnarray}
%
where $R_{w}(x,Q^2)$ is parameterized~\cite{slac} by:
\begin{eqnarray}
R_{w} & = & \frac{0.0635}{log(Q^2/0.04)} \theta(x,Q^2)
\nonumber \\
       &   &  + \frac{0.5747}{Q^2}-\frac{0.3534}{Q^4+0.09},
\end{eqnarray}
where $\theta = 1. + \frac{12 Q^2}{Q^2+1.0}
              \times \frac{0.125^2}{0.125^2 + x^2}$.
%

\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=r_update5.ps,width=2.91in,height=4.3in}}
    \caption{A compilation of the world's data on R~\cite{yangthesis,rccfr}.}
\label{rupdate}
\end{figure}


	      
	      \begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=r_w2_q2_nnn.eps,width=2.91in,height=4.3in}}
    \caption{Recent data from Jlab experiment E94-110 (on hydrogen) for
    R in the resonance region. Data with
    deutrerium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P03-110.}
\label{r94110}
\end{figure}

	      \begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=Keith_fl_vs_x.eps,width=2.91in,height=4.3in}}
    \caption{Recent data from Jlab experiment E94-110 (on hydrogen) for $F_L$
    in the resonance region. Data with
    deutrerium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P03-110.}
\label{fl94110}
\end{figure}

	      \begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=Keith_2xf1_vs_x.eps,width=2.91in,height=4.3in}}
    \caption{Recent data from Jlab experiment E94-110 
    (on Hydrogen) for $2xF_1$ in the resonance region. Data with
    deutrerium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P03-110.}
\label{2xf194110}
\end{figure}



The $R_{w}$ function provides a good
description of the world's data in the $Q^2>0.5$ and $x>0.05$ region
(see  Figure~\ref{rupdate})
Note that the $R_{w}$ function breaks down
below $Q^2=0.3$. Therefore, the function is frozen at $Q^2=0.35$ and 
another function for $R$ in the $Q^2<0.35$ region is used.
The new function provides a smooth transition
from $Q^2=0.35$ down to $Q^2=0$ by forcing $R$ to approach zero at $Q^2=0$
as expected in the photoproduction limit (while
keeping a $1/Q^2$ behavior at large $Q^2$ and matching to $R_{w}$
at  $Q^2=0.35$).
\begin{eqnarray}
R & = & 3.207 \times \frac {Q^2}{Q^4+1}
%\nonumber \\
%       &   &   
       \times R_{w}(x,Q^2=0.35).
\label{eq:rmod}
\end{eqnarray}
In neutrino scattering the value of $R$ is required to approach zero
at $Q^2=0$ only for the vector
part of the interaction.
Studies of the axial contribution at low $Q^2$ are currently under way.
%However, the overall contribution
%from $R$ is expected to be small in this region. Therefore,
%  it is reasonable to use Eq.~\ref{eq:rmod} for $R$ in
%  both the electron/muon and neutrino scattering cases.

A large unknown is the value of $R$  in the resonance region
for both vector and axial scattering, and also the nuclear
effects on $R$ in the resonance region at low $Q^2$.
Figures~\ref{r94110}, ~\ref{fl94110} ~\ref{2xf194110} show
preliminary results of analysis
of data from Jlab experiment E94-110~\cite{e94110} on hydrogen.
Data with
    deutrerium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P03-110.
The resonance region is discussed in more detail at the
end of this paper.



\section{Inelastic Scattering, Corrections for Nuclear Binding Effects}

 
In the comparison with CCFR charged-current differential cross section
on iron, a nuclear correction for iron targets is applied.
The following parameterized function, $f(x)$ (but substituting
 $\xi_w$ for $x$) is used. This
fit to the experimental electron and muon
scattering data for the ratio of iron to deuterium cross sections
in the DIS region (shown in Figure \ref{fig:nuclear})
is used
to convert deuterium structure functions to (isoscalar) iron
structure functions~\cite{first};
\begin{eqnarray}
f(x) & = & 1.096 -0.364x - 0.278e^{-21.94x}
\nonumber \\
     &   & +2.772x^{14.417}
\end{eqnarray}

Recent results from Jlab presented at 
 this conference indicate that the Fe/D ratio in the
resonance region 
is the same as the Fe/D ratio from DIS data for the same value of
Nachtman target mass variable $\xi$ (which is
the same as $\xi_w$ with A=0 and B=0). A comparison of the
iron to deuterium ratio $f(x)$ in the deep-inelastic and 
in the resonance region (at higher $Q^2$), shown in Figure \ref{fig:emc} 
indicates that in the resonance region, the ratio is the same
in the resonance and DIS region if plotted in terms of the
 $\xi$ . Note that the variable
$\xi$ is equal to $\xi_w$ for large $Q^2$. 


For the ratio of deuterium cross sections to cross
sections
on free nucleons we use the following function
obtained from a fit to SLAC data on the nuclear
dependence of electron scattering cross sections~\cite{yangthesis}
(but substituting $\xi_w$ for $x$).
\begin{eqnarray}
f & = & (0.985 \pm 0.0013)\times (1+0.422x-2.745x^2
\nonumber \\
        &   & +7.570x^3  -10.335x^4+5.422x^5).
\label{eq:nucl-d}
\end{eqnarray}
This correction  is only valid
in the $0.05<x<0.75$ region.  In neutrino scattering,
we use the same nuclear correction factor for $F_{2}$, $xF_{3}$
and $2xF_{1}$.

\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=nuclear-correction.eps,width=2.91in,height=2.5in}}
    \caption{Deep-inelastic charged lepton data on
    the iron to deuterium ratio compared to the
    fit f(x) described in the text.}
\label{fig:nuclear}
\end{figure}




Although all the above expression for $R$ and the nuclear correction
factor were parametrized in terms of the variable $x$, it is probably
more correct to substitute the scaling variable $\xi_w$ for $x$ in
these expressions. Such a substitution will make the expression
more valid at low $W$ and low $Q^2$.  

\begin{figure}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=emc.eps,width=2.91in,height=2.5in}}
    \caption{ Comparison of the iron to deuterium ratio in
    the inelastic region and in the resonance region versus
    the target mass variable $\xi$}
\label{fig:emc}
\end{figure}
\section{Other Corrections to GRV98 PDFs}

The $d/u$ correction for the GRV98 LO PDFs is obtained
from the NMC data for $F_2^D/F_2^P$.
Here, Eq.~\ref{eq:nucl-d} is used to remove nuclear binding effects
in the NMC deuterium $F_2$ data. The correction term, $\delta (d/u)(x)$
is obtained
by keeping the total valence and sea quarks the same.
\begin{eqnarray}
\delta (d/u)& = & -0.00817 + 0.0506x + 0.0798x^2,
\end{eqnarray}
where the corrected $d/u$ ratio is $(d/u)'=(d/u)+\delta (d/u)$.
Thus, the modified $u$ and $d$ valence distributions are given by
\begin{eqnarray}
u_v' = \frac{u_v}{1+\delta (d/u) \frac{u_v}{u_v+d_v}} \\
d_v' = \frac{d_v+u_v \delta (d/u)}{1+\delta (d/u) \frac{u_v}{u_v+d_v}}.
\end{eqnarray}
The same formalism is applied to the modified $u$ and $d$ sea distributions.
Accidently, the modified $u$ and $d$ sea distributions (based on NMC data)
agree with the NUSEA data in the range of $x$ between 0.1 and 0.4.
Thus, we find that any further correction on sea quarks is not necessary.

\section{Resonance Production}


As mentioned earlier the value of $R$ in the resonance region
for both vector and axial scattering
is not known very well. In addition, the axial part of $F_2$ and 
$xF_3$  are not well known. Also, the nuclear
effects of $R$ in the resonance region at low $Q^2$ and $W$
has not been measured in either vector or axial scattering.

Figures~\ref{r94110}, ~\ref{fl94110} and ~\ref{2xf194110} show
preliminary results of analysis
of data on $R$ from Jlab experiment E94-110~\cite{e94110} on hydrogen.
These high precision measurements will be extended to deuterium
by Jlab experiment E02-109~\cite{e02109} (to run sometimes
in 2004). In addition, there is a proposal to add measurements
from nuclear targets at the same time (see Jlab
proposal E03-110~\cite{e03110}). Therefore,
after these experiments are run, it is expected
that data from the following Jlab experiments
experiments in the resonance region: E94-110 (H), E02-109 (D),
E02-103 (H and D at high $Q^2$) and
nuclear target experiment P03-110~\cite{e03110}
will provide
detailed information on the vector structure functions (both $F_{2}$ and $R$)
 for H, D and nuclear targets in the resonance region.
In addition, Jlab  nuclear target
experiment E99-118~\cite{e99118} measured
the low $Q^2$ values of $F2$ and $R$ above
the resonance region. Precision measurments  of the
corresponding axial
structure functions in this low energy region 
will be done by  MINERvA at Fermilab. 

In neutrino scattering, the model of Rein and Sehgal~\cite{rs}
(R-S) is the
model which is  often used to simulate neutrino
production of resonances.  The R-S model is an old
model built on the
quark harmonic oscillator model of Feynman and Randal~\cite{rs}. It has
incorporated early data on the photoproduction and electroproduction
of resonances in combination with the
limited available data on resonance production in neutrino 
and antineutrino beams. The vector form factors
were taken from electroproduction and related to neutrino
production by Celbsch-Gordon rules. 

The following are some of the
advantages of the R-S model.
\begin{enumerate}
       \item  It relates the vector form factors of various
         resonances from fits of photoproduction and 
       electroproduction data to both charged-current and 
       neutral-current neutrino processes via Clebsch-Gordon 
       coefficients for the various isopsin states of the resonances.
       \item It models the axial form factors using neutrino data  
       \item It provides a description of the decay products of the
       hadronic final states.
       \item It provides a framework for a simultaneous description of
       all leptoproduction processes (including whatever neutrino 
       data exists) by modeling a very large number of resonances.
       \item It has been implemented in a Monte Carlo simulation form.
       \item It uses information from pion-nucleon scattering to 
       constrain the various resonances (which is are not separable
       by using photoproduction and electropduction data only,
   \end{enumerate} 

  
  
The following are some of the limitations of the R-S model.

\begin{enumerate}
       \item The  vector and axial resonance form factors
are assumed to be similar to that of the quasielastic peak. Therefore
it does not include the modifications due to the pion cloud.
 \item Only the quark contribution to the longitudinal structure
 function and $R$ are calculated. Therefore $R$ is underestimated since
it does not include the modifications due to the pion cloud. For 
example, the longitudinal structure functions
$R$ and $F_{L}$ are predicted to be zero in the region of the 
first resonance. This result
is  is in disagreement with recent 
data from Jlab experiment E94-110, and requires that the
effect of the pion cloud be included.
\item  It does not include
any resonances above a mass W of 2 GeV. Therefore, it breaks down
at high energies and in the transition region between resonance
and deep-inelastic scattering.
\item The inelastic
continuum under the resonances is not very well modeled.
\item It is an old model which does not include any of the more
recent electroproduction data and neutrino production in the fits.
\item It does not include nuclear corrections such as Fermi motion
and final state interactions of the outgoing hadrons (but these
may be added separately).
\item It does not include nuclear binding corrections to the  resonance
and continuum form factors (but these may be added
separately).
\item The neutrino data that was used as input to constrain some of the
parameters of the model is of rather poor quality.
       \end{enumerate}  
       As mentioned earlier
       the production in the region of the first resonance, the R-S model
predicts $R=0$ and $F_L=0$ 
for vector/electrorpoduction. This  is in disagreement
with recent precision measurement from Jlab as shown in Figures 
~\ref{r94110} and ~\ref{fl94110}. However, this flaw 
 may be corrected, as done in the Bodek-Yang model,
by adding an empirical contributinon to $F_{L}$ as measured in 
electroproduction experiments.
       


Figure ~\ref{fig:Lee} shows the prediction from Sato and Lee~\cite{lee} for the
vector and axial form factors in neutrino production of the
first Delta resonance. The solid
    line is the full calculation including the nucleon pion cloud.
    The dashed line is without the effects of the pion cloud (which
    is similar to what one expects in the R-S quark model). 
    
Note that  duality is expected to break down in
neutrino and antineutrino reactions,
in the region
of the first and the quasielastic peak.
This is because the  $W=M$ and  $W$=1.23 GeV regions are dominated by
one and two isospin states, and the amplitudes for neutrino
versus electron scattering are related via Clebsch-Gordon rules~\cite{rs}
instead of quark charges. For example, unlike the case
for neutrino scattering from neutron target, is no
quasielatic neutrino scattering from a proton target since the
final state is a charge +2 resonance.
Although local duality may work
work better for the sum of neutrino and antineutrino
scattering.

In addition, as already
mentioned the V and A
couplings are not equal at low $W$ and $Q^2$. All
these issues can be corrected when neutrino data is included
in the Bodek-Yang fits for the axial structure functions.


In the region of higher mass resonances 
 (e.g. $W$=1.8 GeV) there is a significant
contribution from the deep-inelastic continuum which is not
well modeled by the Rein-Seghal fits ~\cite{rs} to
neutrino resonance data (which can be
corrected if resonances  with W $> 2$ GeV are added
to R-S in the future.
Therefore, using the Bodek-Yang PDFs for $W>1.8~ GeV$ (with an axial
contribution) should be better. For nuclear targets, the
nuclear corrections~\cite{first} described earlier must also be applied.
\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=Lee.eps,width=6.in,height=3.in}}
    \caption{Magnetic and axial form factors for neutrino production 
    in the Delta resonance region from the calculation
    of Sato and Lee~\cite{lee} . The solid
    line is the full calculation including the nucleon pion cloud.
    The dashed line is without the effects of the pion cloud.}
\label{fig:Lee}
\end{figure*}


\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=crossbreakdown.eps,width=6.in,height=3.in}}
    \caption{Total neutrino and antineutrino cross sections
    versus energy compared to the sum of quasielastic, resonance,
    and inelastic contributions, see text (from D. Casper, Private 
    Communication)}
\label{fig:total}
\end{figure*}

\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=nu_era.eps,width=5.in,height=3.in}}
    \caption{Total neutrino  cross sections
    \ (per nucleon for an isoscalar target
   devided by neutrino energy) versus energy
   in the high energy region (from PDG, see talk by Donna Naples, this 
   conference)}
\label{fig:nuera}
\end{figure*}

\begin{figure*}[htb]
%\centerline{\psfig{figure=f2p.ps,width=5.0in,height=4.3in}}
\centerline{\psfig{figure=nubar_era.eps,width=5.in,height=3.in}}
    \caption{Total neutrino  cross sections
    the total
neutrino and antineutrino cross sections (per nucleon for an isoscalar target
   devided by neutrino energy) versus energy
   in the high energy region (from PDG, see talk by Donna Naples, this 
   conference)}
\label{fig:nubarera}
\end{figure*}




\section{Combining Quasielastic Resonance and Inelastic Contributions}

 In order to have a full description of all charged current
$\nu_\mu$ and $\overline\nu_\mu$
processes,  the contribution from quasielastic
scattering~\cite{budd} must be added
separately at $x=1$. 

For $W$ above the quasielastic peak,
the three available models (Bodek-Yang, Rein-
Sehgal and Sato-Lee) all have problems
in different W regions (and all need to be improved).  At
present it is best to use the Sato-Lee~\cite{lee} model for the first
resonance, the
Rein-Sehgal model in the region of the second and third
resonances and the Bodek-Yang model
(coupled with a Lund fragmentation for final states) in the region 
above the
third resonance. Note that because
of the effects of experimental resolution and Fermi motion
~\cite{Bodek_81}
(for nuclear targets), a description of the average cross
section in the region of higher resonances
is sufficient for most neutrino
experiments. Figure~\ref{fig:total} shows the total
neutrino cross section (per nucleon for an isoscalar target)
    versus energy (at low energies) compared to the sum of quasielastic, resonance,
    and inelastic contributions from D. Casper.
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.
This prescription gives pretty good agreement with the 
measured total cross-section, and provides a
continuous  $W$ distribution between the resonance and inelastic
regions. 
 

The effects of terms proportional to the muon mass and $F_4$ and 
$F_5$ structure functions in neutrino scattering are small and 
are discussed in Ref.~\cite{budd,kr}.

For completenes, Figures~\ref{fig:nuera} and ~\ref{fig:nubarera} show the total
neutrino and antineutrino cross sections (per nucleon for an isoscalar target
   devided by neutrino energy) versus energy
   in the high energy region.




\section{Outlook for the future}

A worthwhile aim is to improve the Sato-Lee, Rein-Sehgal
and Bodek-Yang models (and hopefully 
more in the future) to agree
with each other (and with all existing data) over a large region of overlap
in $W$.
If this is accomplished, the predicted cross sections
will be continuous in the  transition
region between the deep inelastic
and resonance regions.


A. Improvements to the Sato-Lee Model.

There are current plans to extend the model of Sato and Lee
(which is only available in the region of the
first resonance)
to the higher resonances. Since the model includes both quarks
states and meson clouds, it should work better than the R-S model. However,
the R-S model includes a very large number of resonances, and 
large volume of exclusive electroroduction data is needed
to constrain these resonance
parameters within the Sato and Lee model. Therefore
it may be easier to improve on the R-S model by using 
information learned from the Sato and Lee fits for
the first resonance.

B. Improvements to the Rein and Sehgal model.
\begin{enumerate}
    \item Improve the resonance parameters by repeating the fits using
    more up to
date data from Jlab in the resonance region.
    \item  Apply corrections for the missing terms from the pion cloud 
(guided by the Sato-Lee model) to improve the modeling
of the vector and axial form factors.
 \item Use new
 electroproduction data on
$R$ and $F_2$ from Jlab to improve the modeling of
the continuum.
\item Improve he modeling of $R$
\item Include  resonances and continuum with W $> 2$ GeV (so the model works
for higher values of $W$). 
\item One could use 
the Sato-Lee model for the first resonance and
the Rein-Seghal model for
the remaining resonances.

\end{enumerate}

C. Improvements to the Bodek-Yang Model

\begin{enumerate}
\item 
At very low  $Q^2$, the vector structure function
for changing conserving processes must go 
to zero, while the axial-vector part is finite. 
Plans are to including low energy data
from Chorus (on Pb), CDHS and CCFR (in Fe) and other
neutrino data in the fit, in order to constrain
the low $Q^2$ axial-vector contribution, as
previously discussed.
\item 
For strangeness-changing processes which consist of
$5\%$ of the cross section, the corresponding Adler sum rule
(discussed at the end of this talk) indicates that the production
of strange resonances is not suppresed at  $Q^2$=0. This difference
needs to be implemented in the Bodek-Yang model.
\item Improve the fit for the longitudinal
structure function $R$ for vector scattering, by including new data
taken on  Hydrogen from Jlab experiment E94-110.
\item  Improve the description of the axial
contribution to the longitudinal structure function.
\item  The PDFs can be multiplied by a modulating
function~\cite{bodek,close} A(W,$Q^{2}$)  to improve  modeling  in
the resonance region (for hydrogen) by including 
(instead of $predicting$)
the resonance  data~\cite{jlab} in the fit.
One can also include resonance 
data on deuterium~\cite{jlab} and heavier nuclear targets
in the fit, and low energy neutrino data.
\item In the  future, there are plans to investigate the effects of including
the initial
state quark $P_T$ in $\xi_w$,
and institute further improvements such as allowing for
different higher twist parameters for u, d, s, c, b  
quarks in the sea.


\end{enumerate} 

D. New electron scattering experiments that can be used to improve 
the models.

\begin{enumerate}

\item JLab hydrogen experiment E94-110 (investigate $F_2$ and $R$
    in the resonance region). C.E. Keppel spokesperson (data already
    taken).
\item JLab deuterium  experiment E02-109,
investigate $F_2$ and $R$
    in the resonance region). C.E. Keppel, M. E. Christy,
spokespersons (approved to run in 2004). 
\item JLab experiment E99-118 on the nuclear dependence
of $R$ at low $Q^2$ 
for high values of $W$. A. Brull, C.E. Keppel spokespersons (data already 
taken).
\item Jlab experiment E02-103 hydrogen and
deuterium resonance $F_2$ data at  high $Q^2$
(and EMC effect in He3, He4) - approved by Jlab PAC24 to run in 2004
(J. Arrington, spokesperon)
    \item Jlab Proposal PR03-110 to investigate $F_2$ and $R$
    in the resonance region with nuclear targets.  A. Bodek and C. E. 
    Keppel, spokespersons  (proposed to run in  Hall C together
    with E02-109 in 2004) to provide vector resonance form
    factors and $R$ on nuclear targets for use in
    neutrino experiments (collaborate with MINERvA). 
    \item Steve Manly (Rochester) and Will Brooks (Jlab) proposal to
    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 electron
    scattering  (for use in  neutrino experiments  and 
    collaborate with MINEvA)
    \end{enumerate}

E. Current and future neutrino near detector experiments

\begin{enumerate}
    \item K2K:  Near detector;  fully active scintillator (carbon) neutrino detector currently taking data in 
    2003 with 1-2 GeV neutrinos at KEK (low flux).
    \item MiniBoone:  (carbon) neutrino
    detector currently taking data in 2003 with 0.7 GeV 
    neutrinos in the Fermilab Booster Beam.
    \item MINOS:  iron-scintillator neutrino near detector. Currently being
    constructred to run in the
     high flux NUMI neutrino beam at Fermilab (between 2-8 GeV).
    \item MINERvA: A fully active scintillator (carbon) near detector
     proposed to run in both on-axis and off-axis configuration
      in the NUMI neutrino beam at Fermilab (between 2-8 GeV). K. 
      McFarland (Rochester), J. Morfin (Fermilab) and  C. Keppel 
      (Hampton/Jlab) spokespersons (collaborate with Jlab experiments
       listed above). 
        \item JPARC:  Near detector;  fully active scintillator
	(carbon) neutrino detector to take data  
    with 1-2 GeV neutrinos to run at the new  JHF facility in Japan.
  
  
    
    \end{enumerate}


\section{Testing Sum Rules}

One of the long term goals of modeling electron and neutrino
scattering at all energies is to be able to test 
various sum rules over a large range of $Q^2$.
For example, testing the
Adler and Gilman~\cite{adler} sum rules. 
The Adler sum rules are Current-Algebra
sum rules that are expected to be valid for all
values of $Q^2$ from $Q^2$= 0 to  $Q^2$=$\infty $.

We have used one of the Adler sum rules to guide us
on the choice of functional forms for our fits
for the vector and axial K factors. When
a large amount of electron and neutrino scattering data are 
available, the models can be tested to see if they
satisfy the various Adler sum rules. These sum
rules are integrals
from $\nu=q_0=0$ to $\nu =\infty $ (including the
elastic/quasielastic contribution).
The
functions $\alpha$, $\beta$ and $\gamma$ as defined
by Adler correspond to the
structure functions $W_{1}$ ,$W_{2}$, and $W_{3}$
in modern notation.
Here $W_{1}^{-}$=$\alpha^{-}$, $W_{2}^{-}$=
 $\beta^{-}$ and 
$W_{3}^{-}$=$\gamma^{-}$ refer to antineutrino 
structure functions, and
$W_{1}^{+}$=$\alpha^{+}$, $W_{2}^{+}$=$\beta^{+}$ and
$W_{3}^{+}$=$\gamma^{+}$
refer to neutrino
structure functions,  respectively. If
no subscripts are used then we refere to scattering
on protons. Otherwise the subscripts p and n refer
to scattering from protons and neutrons respectively.
The superscripts $vec$ and $ax$ refer to vector
and axial scattering respectively. There
are many Adler sum rules for the three structure
functions on neutrons and protons and for the
cases of strangeness-conserving and strangeness-changing processes.
At high $Q^{2}$ all of the
 $\beta$ sum rules are equivalent to the statement
 that the number of valence $u$ quarks is =2 and the number
 of valence d quarks is =1. A few examples
 are shown below.

The following
are the Adler sum rules for strangeness conserving
charged-current reactions for $\beta_{p\Delta S=0}$:
 $$\int_0^{\infty}\left[\beta_{p\Delta S=0}^{vec-}(\nu,q^2) -
 \beta_{p\Delta S=0}^{vec+}(\nu,q^2)
 \right] d\nu =  cos^2\theta_c \; $$
  $$\int_0^{\infty}\left[\beta_{p\Delta S=0}^{ax-}(\nu,q^2) - 
  \beta_{p\Delta S=0}^{ax+}(\nu,q^2)
  \right] d\nu = cos^2\theta_c \; $$
  
  The following
are the Adler sum rules for strangeness conserving
charged-current reactions for $\beta_{n\Delta S=0}$:

 
  $$\int_0^{\infty}\left[\beta_{n\Delta S=0}^{vec-}(\nu,q^2) -
 \beta_{n\Delta S=0}^{vec+}(\nu,q^2)
 \right] d\nu = -cos^2\theta_c \; $$
  $$\int_0^{\infty}\left[\beta_{n\Delta S=0}^{ax-}(\nu,q^2) - 
  \beta_{n\Delta S=0}^{ax+}(\nu,q^2)
 \right] d\nu = -cos^2\theta_c \; $$

 
 The following
are the Adler sum rules for strangeness changing
charged-current reactions for $\beta_{p\Delta S\pm 1}$:
 $$\int_0^{\infty}\left[\beta_{p\Delta S\pm 1}^{vec-}(\nu,q^2) -
 \beta_{p\Delta S\pm 1}^{vec+}(\nu,q^2)
 \right] d\nu = 2 sin^2\theta_c \; $$
  $$\int_0^{\infty}\left[\beta_{p\Delta S\pm 1}^{ax-}(\nu,q^2) - 
  \beta_{p\Delta S\pm 1}^{ax+}(\nu,q^2)
 \right] d\nu = 2 sin^2\theta_c \; $$

  The following
are the Adler sum rules for strangeness changing
charged-current reactions for $\beta_{n\Delta S\pm 1}$:
 $$\int_0^{\infty}\left[\beta_{n\Delta S\pm 1}^{vec-}(\nu,q^2) -
 \beta_{n\Delta S\pm 1}^{vec+}(\nu,q^2)
 \right] d\nu =  sin^2\theta_c \; .$$
  $$\int_0^{\infty}\left[\beta_{n\Delta S\pm 1}^{ax-}(\nu,q^2) - 
  \beta_{n\Delta S\pm 1}^{ax+}(\nu,q^2)
 \right] d\nu =  sin^2\theta_c \; .$$

 
 At high $Q^{2}$ the above four vector/axial pairs 
 of equations
 are equivalent to a difference between pairs of
  equations below. i.e.
  equations 1 - 2 ($u_{v}-d_{v}=1$),  , equations 2 - 3 
  ($d_{v}-u_{v}=-1$),
   equations 5 - 6 $(u_{v}$=2) , and  equations 7 - 8 ($d_{v}=1$).
(Here $+$ denotes neutrinos and $-$ denotes antineutrinos)
 
 \begin{enumerate}
    \item $\beta_{p\Delta S=0}^{-}$= $cos^2\theta_c [u +\overline 
    d]$+$(Neutron=udd)$
    \item $\beta_{p\Delta S=0}^{+}$= $cos^2\theta_c [d +\overline 
     u]$ 
       \item $\beta_{n\Delta S=0}^{-}$= $cos^2\theta_c [d +\overline 
    u]$
    \item $\beta_{n\Delta S=0}^{+}$= $cos^2\theta_c [u +\overline 
   d]$ +$(Proton=uud)$
   \item $\beta_{p\Delta S\pm 1}^{-}$= $[sin^2\theta_c u +cos^2\theta_c\overline 
    s]$+$(\Lambda^{0} ~and~ \Sigma^{0}=usd)$
    \item $\beta_{p\Delta S\pm 1}^{+}$=$cos^2\theta_c s +sin^2\theta_c \overline 
    u$
    \item $\beta_{n\Delta S\pm 1}^{-}$= $[sin^2\theta_c d +cos^2\theta_c\overline 
    s]$+$(\Sigma^{-}=dds)$
    \item $\beta_{n\Delta S\pm 1}^{+}$=$cos^2\theta_c s + sin^2\theta_c \overline 
    d$
     \end{enumerate}

 

 At $Q^2$=0 some of the above equations have
 discrete contributions at $W=M$ (for strangeness
 conserving $\beta_{p\Delta S=0}^{-}$ and $\beta_{n\Delta S=0}^{+}$) 
 and at $W=M_{\Lambda}$ and $W=M_{\Sigma}$ 
 (for strangeness changing $\beta_{p\Delta S\pm 1}^{-}$
 and $\beta_{n\Delta S\pm 1}^{-}$).
 
To calculated the discrete contribution for
the  $W=M$ 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. 

 If we separate out the discrete quasielastic delta function
 contributions, the two equations for the
 strangness-conserving  ($\Delta S=0$)
 Adler sum rules for $\beta_{p}$ become: 
  $$|F_V(q^2)|^{2}+ [1/(cos^2\theta_c)]~~x
 \;$$
$$
 \int_{\nu_{0}}^{\infty}\left[W_{2p\Delta S=0}^{vec-}(\nu,q^2)-
 W_{2p\Delta S=0}^{vec+}(\nu,Q^2)
 \right] d\nu = 1 \; $$
 $$|F_A(q^2)|^2 + [1/(cos^2\theta_c)]~~x
 \;$$
 $$
 \int_{\nu_{0}}^{\infty}\left[W_{2p\Delta S=0}^{ax-}(\nu,q^2)-
 W_{2p\Delta S=0}^{ax+}(\nu,Q^2)
 \right] d\nu = 1\; $$
Where the limits of the integrals are
from pion threshold $\nu_{0}$ where $W= M_{\pi}+M_P$ to $\nu_=\infty$.


The two equations for the  strangness-conserving  ($\Delta S=0$)
Adler sum rules for $\alpha_{p}$ become:
   $$ [1+Q^{2}/4M^{2}][ g_V(q^2)]^{2}+ [1/(cos^2\theta_c)]~~x
 \;$$
$$
 \int_{\nu_{0}}^{\infty}\left[W_{1p\Delta S=0}^{vec-}(\nu,q^2)-
 W_{1p\Delta S=0}^{vec+}(\nu,Q^2)
 \right] d\nu = C_{I,1}  \; $$
 $$[Q^{2}/4M^{2}] [g_a(q^2)]^{2}+ [1/(cos^2\theta_c)]~~x
 \;$$
 $$
 \int_{\nu_{0}}^{\infty}\left[W_{1p\Delta S=0}^{ax-}(\nu,q^2)-
 W_{1p\Delta S=0}^{ax+}(\nu,Q^2)
 \right] d\nu = C_{I,2}\; $$
Where the limits of the integrals are
from pion threshold $\nu_{0}$ where $W= M_{\pi}+M_P$ to $\nu_=\infty$.

The equationsfor the  strangness-conserving ($\Delta S=0$)
Adler sum rule for $\gamma_{p}$ become:
 $$ [g_a(q^2)][ g_V(q^2)]/M = [1/(cos^2\theta_c)]~~x
 \;$$
$$
 \int_{\nu_{0}}^{\infty}\left[W_{3p\Delta S=0}^{-}(\nu,q^2)-
 W_{3p\Delta S=0}^{+}(\nu,Q^2)
 \right] d\nu \; $$
Where the limits of the integrals are
from pion threshold $\nu_{0}$ where $W= M_{\pi}+M_P$ to $\nu_=\infty$.
Here $C_{I,1}=C_{I,2}=1$ if
simple quark-model commutation relations are valid. However, these may
not be valid in theories in which meson fields are explicitly included
in the currents. Therefore, testing the sum rule for $W_1$ is of 
interest (which requires a knowledge of the longitunidal structure
function $R$).

The corresponding equations for $\alpha$, $\beta$
and $\gamma$ for the
strageness changing ($Delta S\pm 1$)
case for the proton and neutron
are more complicated.
The reader is refered to Adler's and Gilman's paper~\cite{adler}
 for more details about
current algebra sum rules for 
($\alpha$, $\beta$ and $\gamma$)
that could be tested
for the first time with future precision
electron scattering and neutrino scattering data.
Note that these sum rules are expected to be valid
over all values of $Q^2$. In addition to these
neutrino sum rules, there are other sum rules
based on dispersion relations that can be
tested in electron scattering as described in Ref.~\cite{baldin}


\section{ Acknowledgements}
This work is supported in part by the U. S. Department of Energy,
High Energy Physics Division under
grant DE-FG02-91ER40685 (Rochester). I thank H. Budd, G. Zeller, 
C. E. Keppel, D. Casper, J. Arrington, R. Ent and many others for 
significant contributions to this conference,

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\end{document}
