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\documentclass[twocolumn,prl,aps,floats,epsf,psfig,colordvi,showpacs]{revtex4}
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\newcommand{\gep}{G_{Ep}}
\newcommand{\gmp}{G_{Mp}}
\newcommand{\gmn}{G_{Mn}}
\newcommand{\gmpmu}{G_{Mp}/\mu_{p}}
\newcommand{\gepK}{G^{Kelly}_{Ep}}
\newcommand{\gmpK}{G^{Kelly-upd}_{Mp}}
\newcommand{\gen}{G_{En}}
\newcommand{\gmnmu}{G_{Mn}/\mu_{n}}
\newcommand{\gepnew}{G_{Ep}^{new}}
\newcommand{\gmpnewmu}{G_{Mp}^{new}/\mu_{p}}
\newcommand{\gmpKmu}{G^{Kelly-upd}_{Mp}/\mu_{p}}
\newcommand{\gennew}{G_{En}^{new}}
\newcommand{\gmnnewmu}{G_{Mn}^{new}/\mu_{n}}
\newcommand{\numu}{\nu_{\mu}}
\newcommand{\muminus}{\mu^{-}}
\newcommand{\muplus}{\mu^{+}}
\newcommand{\numubar}{\overline{\nu}_{\mu}}
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\begin{document}

\title{Duality Constrained Parameterization of Vector and Axial
Nucleon Form Factors}
\author{ A. Bodek, S. Avvakumov, R. Bradford, and  H. Budd}
\affiliation{Department of Physics and Astronomy, University of Rochester, Rochester, NY  14627-0171}

%\date{\today}

\begin{abstract}
We present new
parameterizations of vector and axial nucleon form factors.
We maintain an excellent descriptions of the
form factors at low momentum transfers, where the spatial
structure of the nucleon is important, and
use the Nachtman scaling variable $\xi$
to relate elastic and inelastic form factors
and impose quark-hadron duality constraints at high momentum
transfers where the quark structure dominates. 
We use the new vector form factors to re-extract
updated values of the axial form factor from neutrino
experiments on deuterium.
 We obtain an updated world
average value from $\numu$d
and pion electroproduction experiments of 
$M_{A}$ = $1.0144 \pm 0.0136~GeV/c^2$. Our parameterizations are useful 
in modeling neutrino interactions at low energies
(e.g. for neutrino oscillations experiments).  The predictions
for high momentum transfers can be tested in the 
next generation electron and neutrino scattering experiments.
\end{abstract}


\pacs{PACS numbers:13.40.Gp, 13.15.+g, 13.85.Dz, 14.20.Dh, 24.30.Bf, 25.30.Pt} 
\maketitle

% \section{Introduction}

The nucleon vector and axial elastic form factors have been measured
for more than 50 years in $e^- N$ and $\nu N$ scattering.
At low $Q^2$, a reasonable  description of the proton and neutron
elastic form factors is given by the dipole approximation.
The dipole approximation is a lowest-order attempt to
incorporate the non-zero size of the proton into the form
factors. The approximation assumes that the proton has
a simple exponential spatial charge distribution, $\rho(r)=\rho_0
e^{-r/r_0}$, where $r_0$ is the scale of the proton radius.
Since the form factors are related in
the non-relativistic limit to the Fourier transform of the
charge and magnetic moment distribution, the above $\rho(r)$
yields the dipole form defined by:
\begin{eqnarray}
%\begin{equation}
G_D^{V,A}(Q^2) =
\frac{C^{V,A}}{\left(1+\frac{Q^2}{M_{V,A}^2}\right)^2} \nonumber.
\end{eqnarray}
%\end{equation}
Here $C^{V,A}$= (1,$g_{A}$), $g_{A}$ = -1.267, $M_{V}^2$ = 0.71 $(GeV/c)^2$,
  and $M_{A}$ = 1.015 $GeV/c^{2}$ (as discussed below). 
  
Since $M_{A}$ is not equal to $M_{V}$, the 
distribution of electric and axial charge are different.
However, the magnetic moment distributions were assumed to have the
same spatial dependence as the charge distribution ({\it
i.e.}, form factor scaling). Recent measurements from Jefferson Lab 
show that the ratio of $\frac{\mu _p G_{Ep}}{G_{Mp}}$ 
falls at high $Q^2$
challenging the validity of form factor scaling \cite{jlab1} and  resulting in new updated parameterizations of
the form factors\cite{nuint05},\cite{kelly}). In this paper we present
parameterizations that simultaneously satisfy constraints at low $Q^2$ where the
spatial structure of the nucleon is important, and at high
$Q^2$ where the quark structure is important. A violation of
form-factor scaling is expected from quark-hadron
duality.
%\cite{qduality}. 
We use our
new vector form factors to re-extract updated values of the axial form factor
from a re-analysis of previous neutrino scattering data 
on deuterium and present a new parameterization for the axial form
factor within the framework of quark-hadron duality.

%We produced two previous form
%factor parameterizations \cite{nuint05, bba, bba03}. 
The new parameterizations presented in this paper are referred to as the duality
based ``BBBA07''
parameterization. Our updated parameterizations feature the following:
\begin{figure*}
 \begin{center}
\includegraphics[width=6.0in]{{ffKellyVsOurKsi.eps}}
   \caption[Ratios to $G_D$]{Ratios of $\gep$ (a), $\gmpmu$ (b),
   $\gen$ (c) and
$\gmnmu$ (d) to $G_D$.   
The short-dashed line in each plot is
the old Kelly parameterizations (old Galster for $\gen$). 
%and the solid and dashed
%lines are our new BBBA07
%parameterization. 
The solid line is our new $BBBA07_{25}$
parameterization for $\frac{d}{u}=0.0$, and the long-dashed line is
 $BBBA07_{43}$ for $\frac{d}{u}=0.2$.
The values
of $\xi$ and the corresponding
values of $Q^2$ are shown on the bottom and top axis. 
}
 \end{center}
   \label{subfig1}
\end{figure*}
%(1) Incorporation of the BLAST results\cite{crawford};
(1) Improved functional form that adds an additional 
$Q^2$ dependence using the Nachtman scaling variable $\xi$
to relate elastic and inelastic
%vector and axial nucleon 
form factors.
For elastic scattering ($x=1$) 
$\xi^{p,n,N}=\frac{2}{(1+\sqrt{1+1/\tau_{p,n,N}})}$, where $\tau_{p,n,N} =
Q^2/4M_{p,n,N}^2$. Here
$M_{p,n,N}$ are the proton (0.9383 $GeV/c^2$), neutron (0.9396 $GeV/c^2$), 
and average nucleon mass (for proton,
neutron, and axial form factors, respectively).
 %Excellent low $Q^2$ description
%of the spatial structure of the nucleon by
(2) Yield the same values as
Arrington and Sick \cite{arringtonsick} for $Q^{2}< 0.64
(GeV/c)^{2}$,
while satisfying 
quark-hadron duality constraints at high-$Q^2$.
%for both vector and
%axial form factors.
%(3) Relating vector and axial form factor.

%-------------
For vector form factors our fit functions are $A_N(\xi)$ (i.e. 
$A_{Ep}(\xi^{p})$, $A_{Mp}(\xi^{p})$, $A_{En}(\xi^{n})$,
$A_{Mn}(\xi^{n})$)
multiplying an updated Kelly\cite{kelly} type parameterization
 of one of the proton form factors. 
The Kelly parameterization is:
%\begin{equation}
    \begin{eqnarray}
 G^{Kelly}(Q^2) = \frac{\sum_{k=0}^{m}a_k \tau^k_{p}}{1 +
\sum_{k=1}^{m+2}b_k \tau^k_{p}}, \nonumber 
\end{eqnarray}
%\end{equation}
where $a_{0}=1$ and $m=1$
%and $\tau = Q^2/4M_{N}^2$.
.
In our analysis, we use all the  
datasets used by Kelly\cite{kelly}, updated to include the
recent BLAST\cite{crawford} results, to
fit $\gep$, $\gen$, $\gmpmu$, and $\gmnmu$ ($\mu_p = 2.7928$, $\mu_n = -
1.9130$).
%are described in
%\cite{kelly}.
Our parameterization employs the published Kelly
functional form to $G^{Kelly}_{Ep}$, and an updated set of parameters
for $G^{Kelly-upd}_{MP}(Q^2)$.
%Our parameterization employs an updated set of parameters
%for $G^{Kelly-upd}_{MP}(Q^2)$ that includes the
%recent BLAST\cite{crawford} results.
The 
parameters used for $G^{Kelly}_{Ep}$ and $G^{Kelly-upd}_{Mp}$ are listed in Table
\ref{kellyparm}, and  $A_N(\xi)$ is given by 
%We then modulate the Kelly  by using the
%function 
%$G_N^{new}(Q^2)=A_N(\xi) \times
%G^{Kelly}(Q^2)$.
%----------------------
\begin{eqnarray}
 A_N (\xi) &=& \sum_{j=1}^{n} P_j (\xi) \nonumber  \\
 P_j (\xi) &=& p_j\prod_{k=1, k \ne j}^{n} \frac{\xi - \xi_k}{\xi_j -
\xi_k}. \nonumber
\end{eqnarray}
Each $P_j$ is a LaGrange polynomial in $\xi$.
%the Nachtman variable,.
%$\xi=\frac{2}{(1+\sqrt{1+1/\tau})}$.
%with $\tau=\frac{Q^2}{4M_N^2}$
  The $\xi_j$ are equidistant ``nodes'' on an interval $[0,1]$ and
$p_j$ are the
fit parameters that have an additional property $A_N (\xi_j) = p_j$.
The functional form $A_N(\xi)$ (for  $\gep$, $\gmp$, $\gen$, and $\gmn$)
is used with seven $p_j$ parameters
at $\xi_j$=0, 1/6, 1/3, 1/2, 2/3, 5/6, and 1.0.
In the fitting procedure described below, the parameters of $A_N(\xi)$ are
constrained
%five of the parameters. Four ($A_N(\xi_j)=p_j$ at $\xi_j$ = 0,
%0.125, 0.25, and  0.375)  are 
to give the same vector form factors as the recent low $Q^2$
fit of Arrington and Sick \cite{arringtonsick} for $Q^{2}< 0.64
(GeV/c)^{2}$ (as that analysis includes coulombs corrections which 
modify $G_{Ep}$, and two photon exchange corrections which modify
$G_{Mp}$ and $G_{Mn}$). Since the published form 
factor data do not have these corrections, this constraint is 
implemented by including additional $''fake''$ data points for $Q^{2}<
0.64 (GeV/c)^{2}$.
%and the neutron charge radius data 
%from neutron-electron scattering  which constrains $G_{En}$).
%In our fits, only the weights of the LaGrange polynomials are
%allowed to 
%vary; the parameters of the Kelly function remained fixed in our main
%fitting step.
%This section will begin by present a short overview of the Kelly
%parameterization, and then discuss the particulars of our fits to
%each form factor.
%
% use packages: array
%
%------------------------------------
\begin{table*}
    \begin{center}
\begin{tabular}{l|l|l|l|l|l}
$ $ & $a_1$ & $b_1$ & $b_2$ & $b_3$ & $\chi^2/ndf$ \\
\hline
$\gepK$ & $-0.24$ & $10.98$ & $12.82$ &
$21.97$ & $
0.78$ \\
$\gmpK$ & $0.1717$ & $11.26$ & $19.32$ &
$8.33$ & $
1.03$
\end{tabular}
\caption[Kelly fit parameters]{Parameters for $G^{Kelly}_{Ep}$ and
$G^{Kelly-upd}_{Mp}$. Our parameterization employs the as-published Kelly
parameterization to $G^{Kelly}_{Ep}$ and an updated set of parameters
for $G^{Kelly-upd}_{MP}(Q^2)$ that includes the
recent BLAST\cite{crawford} results.}
%
%-----------------------------------------
\label{kellyparm}
\end{center}
\end{table*}
%
%
\begin{table*}
\begin{center}
% use packages: array
\begin{tabular}{l|l|l|l|l|l|l|l|}
$ $ & $p_1$ & $p_2$ & $p_3$ & $p_4$ & $p_5$ &
$p_6$ & $p_7$\\
$\xi, Q^{2} $ & $0, 0$ & $0.167, 0.029$ & $0.333, 0.147$ & $0.500,
0.440$ & $0.667, 1.174,$ &
$0.833, 3.668$ & $1.0, \infty$ \\
%$Q^2 $ & $0$ & $0.029$ & $0.147$ & $0.440$ & $1.174$ &
%$3.668$ & $\infty $ \\
\hline
%
%
$A_{Ep}$ &  $1.$ &$0.9927$ & $0.9898$ & $0.9975$ & $0.9812$ & $0.9340$
& $1.$  \\
%
$A_{Mp}$  &  $1.$  &$1.0011$ & $0.9992$ & $0.9974$ & $1.0010$ &
$1.0003$ & $1.$ \\
\hline
%
$A_{Ep-dipole}$ &  $1.$ &$0.9839$ & $0.9632$ & $0.9748$ & $0.9136$ &
$ 0.5447$ & $-0.2682$  \\
%
$A_{Mp-dipole}$  &  $1.$  &$0.9916$ & $0.9771$ & $0.9801$ & $ 1.0321$ &
$1.0429$ & $0.5084$ \\
%
\hline$A^{25}_{Mn}$ &  $1.$ & $0.9958$ & $0.9877$ & $1.0193$ &$1.0350$ &
$0.9164$& $0.7300$  \\
%
$A^{43}_{Mn}$ &  $1.$ & $0.9958$ & $0.9851$ & $1.0187$ &$1.0307 $ &
$0.9080$& $ 0.9557$  \\
\hline
%
$A^{25}_{En}$ & $1.$ & $1.1011$ & $1.1392$ & $1.0203$ &$1.1093$ &
$1.5429$& $0.9706$  \\
%
$A^{43}_{En}$ & $1.$ & $1.1019$ & $1.1387$ & $1.0234$ &$1.1046$ &
$1.5395 $& $1.2708$ \\
%
\hline$A^{25-dipole}_{FA}$ &$1.0000$ & $0.9207$ & $0.9795$ & $1.0480$
&$1.0516$ & $1.2874$& $ 0.7707$ \\
\end{tabular}
\caption[LaGrange Fit Parameters]{Fit parameters for $A_N(\xi)$, the
LaGrange portion of the new parameterization.  Note
$A^{25}_{Mn}$, $A^{25}_{En}$, and $A^{25}_{FA}$ are constrained
to have  $\frac{d}{u} = 0$ at $\xi = 1$,
and $A^{43}_{Mn}$, $A^{43}_{En}$,  are constrained
to have  $\frac{d}{u} = 0.2$.}
\label{lagparm}
\end{center}
\end{table*}
%
%
%\subsection{Kelly parameterizations to $\gepmu$ and $\gmpmu$.}
%
%

% \subsection{LaGrange Fits to $\gep$ and $\gmpmu$}
%

Our fits to the  form factors are:
% $\gep$ and $\gmpmu$ are: 
\begin{eqnarray}
% \lefteqn
      {G_{Mp}(Q^2)}/{ \mu_{p}} &=& { A_{Mp}(\xi^{p})} \times 
      {G^{Kelly-upd}_{Mp}(Q^2)} \nonumber \\
 {G_{Ep}(Q^2)} &=& A_{Ep}(\xi^{p})\times {G^{Kelly}_{Ep}(Q^2)}
 \nonumber \\
    {G_{Mn}(Q^2)}/{\mu_{n}} &=& A^{25,43}_{Mn}(\xi^{n})
    \times  {G_{Mp}(Q^2)}/ {\mu_{p}}
 \nonumber \\   
   {G_{En}(Q^2)} &=& A^{25,43}_{En}(\xi^{n})\times {G_{Ep}(Q^2)} \times
   \left( {\frac{a\tau_{n}}{1+b\tau_{n}}} \right)
  \nonumber,
% {G_{En}(Q^2)} &=& A^{a,b}_{En}(\xi) \times   {\left( \frac
% {G_{Mn}(\xi)}{\mu_{n}} 
% \right) }
%     \times { \left( {\frac{a\tau}{1+b\tau}} \right) }
%   \nonumber \\,
  \end{eqnarray}
 where we use 
 our updated parameters in the Kelly parameterizations.
 For $\gen$ the parameters a=1.7 and b=3.3 are the same as
 in the Galster\cite{galster}  parametrization and ensure that
 $d\gen/dQ^2$ at for $Q^{2}=0$ is in agreement with measurements.
 %
 For convenience, we also provide
  fits for the form factors $\gep$ and $\gmpmu$ that give very
 close to the same values, but use the dipole form instead: 
 \begin{eqnarray}
 %\begin{equation}
%\lefteqn  
     {G_{Ep}(Q^2)} &=& A_{Ep-dipole}(\xi^{p})\times {G_{D}^{V}(Q^2)} \nonumber
\\
%\end{equation}
%
%\begin{equation}
  {G_{Mp}(Q^2)}/{\mu_{p}} &=&  A_{Mp-dipole}(\xi^{p})\times  {G_{D}^{V}(Q^2)} \nonumber
%  \end{equation}
  \end{eqnarray}
  %
 %The final fit parameters are
%shown in Table \ref{lagparm}
%
% \subsection{$\gmn$}
  
  The values  $A(\xi)$=$p_1$ at $\xi_1$=0  ($Q^2 =0$) for
   $\gmp$,  $\gep$,   $\gen$,   $\gmn$ are set to  
  to 1.0. The value  $A(\xi)$=$p_7$ at $\xi_j$=1 
  ($Q^2 \rightarrow \infty$) for $\gmp$ and $\gep$ is set to 1.0.
  %The value of $A(\xi)$=$p_j$ at $\xi_k$=1 for $\gep$ and $\gep$
 % is obtained from the fit to the data.
  
   The value  $A(\xi)$=$p_j$ at $\xi_j$=1 
for   $\gmn$ and  $\gen$ are fixed by  constraints from quark-hadron
duality.  
%Local duality is invoked to develop an additional constraint that
%is applied to the fit of $\gmnmu$.  At $\xi = 1$ ($Q^2 \rightarrow
%\infty$), 
Quark-hadron duality implies that the ratio of neutron and
proton magnetic form factors should be the same as the ratio of the
corresponding inelastic structure functions $ \frac{F_{2n}}{F_{2p}} $
in the $\xi$=1 limit. (Here $F_2= \xi \sum_{i}e_i^2 q_i \left ( \xi \right)$)
%\begin{equation}
    \begin{eqnarray}
\frac{G_{Mn}^2}{G_{Mp}^2}= \frac{F_{2n}}{F_{2p}} =
\frac{1+4\frac{d}{u}}{4+\frac{d}{u}} = \left(
\frac{\mu_{n}^{2}}{\mu_{p}^{2}}\right)
A_{Mn}^{2}(\xi=1)  \nonumber
%\label{step1}
\end{eqnarray}
%\end{equation}
%In the $\nu \rightarrow \infty$ and $Q^2 \rightarrow \infty$ and
%fixed $x$ limits, the $F_2$ form factor becomes a simple
%quark-counting exercise
%\begin{equation}
%F_2=x \sum_{i}e_i^2 f_i \left(x \right).
%label{step2}
%\end{equation}
%Inserting (\ref{step2}) into (\ref{step1}), we arrive at our first
%constraint:
%\begin{equation}
%\left( \frac{G_{Mn}}{G_{Mp}} \right)^2 =
%\frac{1+4\frac{d}{u}}{4+\frac{d}{u}}
%\label{constraint2}
%\end{equation}
%This condition implies that the asymptotic behavior of $\gmnmu$
% is
%proportional to $\gmpmu$.  Accordingly, we fit $\gmnmu$ with the
%function:
 %\begin{equation}
 % {G_{Mn}(Q^2)} = \mu_{n}A_{Mn}(\xi){G_{Mp}(Q^2)}/\mu_{p},
 % %A_{Mp}(\xi)\times{G^{Kelly-upd}_{Mp}(Q^2)}
 % \end{equation}
 %
 %---------------------------------
\begin{figure}
\includegraphics[width=3.0 in]{{GepOverGmpAndGenOverGmn1.eps}}
\caption[$G_{En}$]{The constraint used in fitting $G_{En}$
stipulates that $\gen^2/\gmn^2=\gep^2/\gmp^2$ at high $\xi$.  
%The plot shows these
%two ratios for $\frac{d}{u}=0$ at $\xi = 1$.
The solid line is $ \frac{G_{Ep}}{|G_{Mp}|}$ and $\frac{|G_{Ep}|}{|G_{Mp}|}$, 
and the short-dashed
line is $\frac{G_{En}}{|G_{Mn}|}$  and $\frac{|G_{En}|}{|G_{Mn}|}$.
%The long-dashed line is our prediction from  a simple
%quark hadron
%duality  model[7] (valid for $\xi>0.6$).
%The values
%of $\xi$ and the corresponding
%values of $Q^2$ are shown on the bottom and top axis. 
}
\label{subfig2}
\end{figure} 
%
%-------------------------------
 We ran fits
with two different values of $\frac{d}{u}$ at the $\xi$=1 limit:
$\frac{d}{u}$ =0 and 0.2 (corresponding
to $\frac{F_{2n}}{F_{2p}}$  = 0.25 and 0.4286). 
The fit utilizing $\frac{d}{u}=0$ is $A^{25}_{Mn}$, and
 the fit utilizing $\frac{d}{u}=0.2$ is $A^{43}_{Mn}$.  
The final parameters
%, for both cases of $\frac{d}{u}$,
are given in Table \ref{lagparm} (or download computer code\cite{FF}).
%at $www.pas.rochester.edu/\sim~bodek/FF/$).

%
%\subsection{$\gen$}
%
%
The value  $A(\xi)$=$p_j$ at $\xi_j$=1 
for   $\gen$ is set by another duality-motivated
constraint.  $R$ is defined as the ratio of deep-inelastic 
longitudinal and transverse structure functions.
For inelastic scattering,
as $Q^2 \rightarrow \infty$, $R_n=R_p$. If we assume quark-hadron
duality, the same should be true for the elastic form factors
at $\xi$=1 ( $Q^2 \rightarrow \infty$) limit:
%In the elastic limit, $R$ takes the form 
%\begin{equation}
    \begin{eqnarray}
R_n\left( x=1; Q^2 \right) &=&  \frac{4M_{n}^2}{Q^2}\left( \frac{G^2
_{En}}{G^2_{Mn}}\right) \nonumber \\ 
%\end{eqnarray}
%\end{equation}
%\begin{equation}
%    \begin{eqnarray}
  %   \frac{G^2_{En}}{G^2_{Mn}}= \frac{G^2_{Ep}}{G^2_{Mp}}
{G^2_{En}}/{G^2_{Mn}} &=& {G^2_{Ep}}/{G^2_{Mp}} \nonumber 
\label{c2equation}
\end{eqnarray}
%\end{equation}  
    %Table 3
%
\begin{table} 
\begin{center}
\begin{tabular}{|l|l|l|l|}
$Experiment $ & $M_A$& $\Delta M_A$	\\
& (published) &  new--old    \\
%
\hline 
%$Mann-D-ANL_{73}$      & 0.95 $\pm$ 0.12       & $-0.030$ \\
%$Barish-D-ANL{77}$      & 0.95 $\pm$ 0.09       & $-0.029$ \\
    $Miller-D-ANL_{82,77,73}$      & 1.00 $\pm$ 0.05      & $-0.030$ \\
     $Baker-D-BNL_{81}$        & 1.07 $\pm$ 0.06      & $-0.028$  \\
 $Kitagaki-D-FNAL_{83}$  & 1.05$_{-0.16}^{+0.12}$& $-0.025$    \\ 
 $Kitagaki-D-BNL_{90}$  & 1.070$_{-0.045}^{+0.040}$ & $-0.036$   \\ 
\end{tabular}
%
\end{center}
\caption{$M_A$ $(GeV/c^2)$ values published
by neutrino-deuterium experiments\cite{neutrinoD2} 
and corrections  $\Delta M_A$ when re-extracted with updated
$BBBA2007_{25}$ form factors. 
%The average of the corrected
%neutrino measurements is $M_{A}$  = $1.019 \pm0.016$. 
%\ref{figaxial} (a). 
 }
\label{MA_values}
\end{table}
%
%Solving for $\gen$ and inserting our form for $\gmpmu$, we fit $\gen$
%with the following functional form:
In order to constrain the fits to $\gen$ at high $Q^{2}$ we have
assumed that the values of  $\frac{G^2_{En}}{G^2_{Mn}}$
are the same as the measured $\frac{G^2_{Ep}}{G^2_{Mp}}$ for the
three highest $Q^{2}$ data points for $\gep$, and included these three
$''fake''$ data points in the $\gen$ fits. 
In addition, the $R_n=R_p$ condition 
yields the following constraint at $\xi = 1$:
%\begin{equation}
    \begin{eqnarray}
%{A^{a,b}_{En}(\xi = 1)} = P_{j} = {\left( \frac{|\mu_{n}|}{|\mu_{p}|} \frac
%{b}{a} \right)} {A_{Mn}^{a,b}(\xi=1)}  \nonumber 
{A^{25,43}_{En}(\xi = 1)} = P_{7} = \left( \frac{b}{a} \right)
\times \left ( \frac{1+4\frac{d}{u}}{4+\frac{d}{u}}
\right) ^{1/2}  \nonumber 
%
%{A^{a,b}_{En}(\xi = 1)} = P_{j} = {|\mu_{n}| \left( \frac{b}{a} \right)}
%\times {\left ( \frac {|G_{Ep}(\xi=1)|}{G_{Mp}(\xi  = 1)} \right) } \nonumber 
% {G^{a,b}_{En}(Q^2)} = A^{a,b}_{En}(\xi) \times A^{a,b}_{Mn}(\xi) \times
%{G_{Ep}(Q^2)} \times (\mu_{n} / \mu_{p})
\end{eqnarray}
%\end{equation} 
where $b/a=1.7/3.3$.  As there are two parameter sets 
$A^{25,43}_{Mn}(\xi)$, we have produced two parameter sets 
$A^{25,43}_{En}$ as shown in Table $\ref{lagparm}$.
%Note that as stated
%earlier values  $A(\xi)$ = $p_j$ at $\xi_k$ =0, 0.125, 0.25, and  0.375 
%are  fixed to give the same values for all form factor as in the very 
% recent low $Q^2$ analysis of Arrington and Sick \cite{arringtonsick}. 
%
%----------------------------------------
 \begin{figure*}
 \begin{center}
\includegraphics[width=6.0in]{{fa-1.eps}}
\caption[$F_{A}$:Axial Form factor ratios to $G_D^{A}$]{
 (a) $F_A(Q^2)$ re-extracted from neutrino-deuterium
 data divided by $G_D^{A}(Q^2)$\cite{GD}. (b)  
$F_A(Q^2)$ from pion electroproduction
  divided by $G_D^{A}(Q^2)$\cite{GD}, corrected for
  for hadronic effects\cite{pion}.
  %(note that $Bloom_{73}$ is typically 
 % excleded since it does not agree with the other experiments).
  Solid line - duality based
  fit; Short-dashed line - $F_A(Q^2)_{A2=V2}$. 
 % The long-dashed line is $F_A(Q^2)_{A1=V1}$.
  Dashed-dot line - constituent quark model\cite{quark}.
 %  The values
%of $\xi$ and the corresponding
%values of $Q^2$ are shown on the bottom and top axis. 
}
 \end{center}
 %
 %---------------------------------------------
     \label{figaxial}  
\end{figure*}
 %\section{Results}
% \label{results}
The new form factors $\gep$, $\gmpmu$, $\gmnmu$, and $\gen$ are plotted
in Figure 1 
%\ref{subfig1}
as ratios to the dipole form $G_D^{V}$.
%In all cases, the ratios deviate from 1.0, with larger 
%deviations at higher $Q^2$. 
%In all cases, the ratios deviate significantly from 1.0, with the
%deviations growing more significant at higher $Q^2$. 
%The plot of
%$\gep$ demonstrates the tell-tale linear deviation from $G_D$ as
%shown in \cite{jlab1}.  The new fits are an accurate representation
%of the available data. 
%The values
%of $\xi$ are shown on the bottom horizontal axis and the corresponding
%values of $Q^2$ are shown on top horizontal axis 

%The BBBA07 parameterizations for $\gep$ and $\gmpmu$ are similar to the
%Kelly parameterization.  In this case, the effects of the LaGrange
%polynomials are small and the our updated to the Kelly parameterization 
%are the dominant contribution to the form factor. 
As seen in Table $\ref{lagparm}$, 
%the function
$A_N(\xi)$ is not needed for  $\gmp$ as 
it is very close to 1.0.  For $\gep$ it yields a correction
of $1\%$  at low $Q^2$ (because it is required to agree with the fits
of Arrington and Sick\cite{arringtonsick} (which include two photon exchange and Coulomb
corrections. For $\gen$ and $\gmn$  it is used to impose 
 quark-hadron duality asymptotic  constraints.
%These constraints had their greatest effect at higher $Q^2$, as can
%be seen in Figure \ref{subfig1}.  Although data for $\gen$ are spares,
%the new $\gmpmu$ parameterization is an improvement over the Kelly fit,
%particularly for the higher $Q^2$ data. 
%which 
%reasonable asymptotic behavior of the form factors, 
%particularly for
%$\gen$.  The LaGrange polynomial form in the variable 
%$\xi$ easily allows for the
%incorporation of constraints simultaneously both at low $Q^2$ where the
%spatial structure of the nucleon is important and  at large 
%$Q^2$ where the quark structure is important. 
Figure \ref{subfig2} shows plots of the data
and fits to $\frac{G_{En}}{|G_{Mn}|}$ and
$\frac{G_{Ep}}{|G_{Mp}|}$ (for the  $\frac{d}{u}$ = 0 at $\xi$ = 1
case). 

%These
%are the two ratios employed in the constraint applied to the fit of
%$\gen$ as shown in Equation \ref{c2equation}.  For both values of
%$\frac{d}{u}$.
%The Figure shows that with duality constraints, the 
%$\frac{ G_{En} }{|G_{Mn}|}$ and
%$\frac{ G_{Ep} }{|G_{Mp}|}$ fits assume the same value
%at high $Q^2$. This prediction can be tested in the next
%generation electron scattering experiments at Jefferson Laboratory.
%The values
%of $\xi$ are shown on the bottom horizontal axis and the corresponding
%values of $Q^2$ are shown on the top axis. 

Using our updated $BBBA2007_{25}$ form
factors and an updated
value $g_{A}$ = -1.267, we perform a complete
reanalysis of  published $\nu$ quasielastic \cite{neutrinoD2} (QE) data on
deuterium ($\numu$ n $\rightarrow$ $\muminus$p) using the procedure described in detail
in ref. \cite{deuterium}. 
%(Because of uncertain nuclear corrections, neutrino data on 
%heavier nuclear targets are not used.)
We extract new values of $M_A$ (given in
Table \ref{MA_values}), and updated values of $F_A(Q^2)$ . The
average of the corrected measurements of $M_{A}$ from Table \ref{MA_values}
is  $1.0156 \pm 0.0278$  $GeV/c^2$.  This is to be compared to the average value
of $1.0140 \pm 0.0160$  $GeV/c^2$ extracted from pion electroproduction
experiments (excluding $Bloom_{73}$)\cite{pion}
after corrections for hadronic effects. The average of the two average
values is $1.0144 \pm0.0136~GeV/c^{2}$.  This is smaller than
the recent results\cite{boone} from MiniBoone on a carbon target 
($M_{A} = 1.23 \pm 0.20$ $GeV/c^2$) and
$K2K$\cite{k2k} on oxygen ($M_{A} = 1.20 \pm 0.12$
 $GeV/c^2$). Both collaborations use
updated vector form factors. collaborations attribute a difference from 
deuterium to nuclear effects. However, there is experimental
and theoretical evidence\cite{ma-nuclear} that 
 $M_{A}$ in nuclear targets is the same (or smaller)
 than in deuterium. 
 This 
% $M_{A}$
 discrepancy is 
important for $\nu$  oscillations experiments since it
affects the normalization 
(at high energies the QE cross section
is approximately proportional to $M_{A}$) and non-linearity of the
QE cross section, which is relevant
to the extraction of  $\nu$ mass difference and mixing angle.

  For deep-inelastic scattering, the vector and axial
  parts of $F_{2}$ are equal. 
   Local quark-hadron duality 
    at large $Q^2$ implies  that
  the axial and vector components of $F_{2}^{elastic}$
  %(or $W_{1}^{elastic}$)
  are also equal,  which yields: 
  %Therefore, if local quark-hadron duality holds
 % this implies  that
 % the axial and vector components of $W_{1,2}^{elastic}$
%  are also equal at large $Q^2$. The $W_{2}^{elastic}$
%   axial=vector constraint  yields the following expressions for large $Q^2$.
   %
 % \begin{equation}
   \begin{eqnarray}
{[F_A(Q^2)_{A2=V2}]^{2}} &=& \frac {(G_E^{V})^{2}(Q^2)+\tau_{N} 
(G_M^V(Q^2))^{2}}{(1 + \tau_{N})}, \nonumber
%{ [F_A(Q^2)_{A1=V1}]^{2}} &=& [\tau /(1+\tau)] [ G_M^{V}(Q^2)]^{2},
%\end{equation}
\label{vec-ax}
 \end {eqnarray}
where $ G_E^V(Q^2)=G_{Ep}(Q^2)-G_{En}(Q^2) $ and  
  $G_M^{V}(Q^2) = G_{Mp}(Q^2)-G_{Mn}(Q^2)$.
  %Note that for
% $\tau$ in the above expressions
% we use the average of the proton and neutron mass.
%    $$[F_A(Q^2)_{V2=A}]^{2}= \frac {(G_E^{V})^{2}(Q^2)+\tau 
%(G_M^V(Q^2))^{2}}{(1 + \tau )}.$$
 %The $W_{1}^{elastic}$
 % axial=vector constraint yields the following expression for large $Q^2$.
%  \begin{equation}
%[F_A(Q^2)_{A1=V1}]^{2}= [\tau /(1+\tau)] [ G_M^{V}(Q^2)]^{2}.
%\end{equation}
%   $$[F_A(Q^2)_{V1=A}]^{2}= [\tau /(1+\tau)] [ G_M^{V}(Q^2)]^{2}$$ 
 % Applying the same constraint to $W_{1}^{elastic}$ we get:
% \begin{equation}
%[F_A(Q^2)_{A1=V1}]^{2}= [\tau /(1+\tau)] [ G_M^{V}(Q^2)]^{2}.
%\end{equation} 
%At large  $Q^2$ $F_A(Q^2)_{A1=V1} = F_A(Q^2)_{A2=V2}$ either
%if $|G_{En}|$ = $|G_{Ep}|$, or as shown in Figure 3,
%form factor scaling is violated and both are close to zero
% (as expected from duality with $|G_{En}|$ = $|G_{Ep}|\times
%|G_{Mn}|/|G_{Mp}|$). 
 

We extract values of $F_A(Q^2)$  from
the differential
 cross sections using the procedure
 of ref. \cite{deuterium}.
 The overall normalization is set by the theoretical QE
 cross section\cite{GD}.
  We  then do a duality based fit
  %\cite{norm}
  to 
  %the updated values of the axial form factor
 $F_A(Q^2)$ (including pion
 electroproduction data, excluding $Bloom_{73}$) of the form:
% LaGrange polynomials, $A^{25}_{FA}$, multiplied by $G_D^{A}(Q^2)$\cite{GD}.
%\begin{equation}
    \begin{eqnarray}
F_A (Q^2)=A^{25}_{FA} (\xi^{N}) \times G_D^{A}(Q^2).\nonumber 
\end{eqnarray}
%\end{equation}
We impose the constraint  $A^{25}_{FA} (\xi_1=0) = p_1 = 1.0$. We also
constrain the fit
%$p_{8,9}$ ($\xi$ = 0.875,1.0 or $Q^2$= 5.5,$\infty$) 
by requiring
that $A^{25}_{FA}(\xi^{N})$ yield  
$F_A(Q^2)=F_A(Q^2)_{A2=V2}$ by including additional $''fake''$ data
points) for $\xi > 0.9$  ($Q^{2} >7.2 (GeV/c)^2$).
 %We then  fit the values for $p_{ 2,3,4,5,6,7}$
%$ 
%($\xi$ = 0.125, 0.25, 0.375, 0.5, 0.625, 0.75). In 
%addition, we let the normalization\cite{neutrinoD2} of the four 
%neutrino experiments
%float within their errors. 
%Figure \ref{figaxial}
Figure 3(a) shows 
  $F_A(Q^2)$  extracted from neutrino-deuterium experiments 
  divided by $G_D^{A}(Q^2)$\cite{GD}.
%Figure \ref{figaxial}
Figure 3(b)  shows 
 $F_A(Q^2)$  extracted from pion electroproduction
 experiments 
  divided by $G_D^{A}(Q^2)$\cite{GD}. These pion electroproduction
  values can be directly compared to the neutrino results because they
  are  multiplied  by a factor
  $F_{A}(Q^{2},M_{A}=1.014~GeV/c^{2})$/$F_{A}(Q^{2},M_{A}=1.069~GeV/c^{2})$  
  to correct  for $\Delta M_{A} = 0.055~GeV/c^{2}$ originating from 
   hadronic effects\cite{pion}.  The solid line is our duality based
 fit. The short-dashed line is  $F_A(Q^2)_{A2=V2}$. 
 % The long-dashed line is $F_A(Q^2)_{A1=V1}$.
  The dashed-dot line
  is a constituent-quark model\cite{quark} prediction.
 
%In conclusion, our new
%parameterizations of vector and axial nucleon form factors
%use quark-hadron duality constraints at high momentum
%transfers and maintain a very good descriptions of the
%form factors at low momentum transfers.
In summary, our new parameterizations
are useful in modeling $\nu$ interactions
%at low energies.
%(e.g. 
for oscillations experiments.
Our predictions\cite{comment}
for $\gen (Q^{2}) $ and $F_{A}(Q^{2})$ 
% at high $(Q^{2}$)
can be tested in future $e-N$ and $\nu$-N
%electron
% and neutrino scattering
 experiments.
%at Jefferson Laboratory and at Fermilab (MINERvA). 




  
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%
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%
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%   --------362  events -------

%BNL D2  Kitakagi 90
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%   --------2544  events -------

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\bibitem{FF} $www.pas.rochester.edu/\sim~bodek/FF/$

\bibitem{GD} Evaluated with $M_{A} = 1.015~GeV/c^{2}$.

\bibitem{comment} Local duality  may not hold for the elastic peak alone, as we
assumed, and the analysis may require using the sum of elastic and
first resonance combined.

\end{thebibliography}

\end{document}



