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\begin{document}

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\wideabs{
\title{The Adler Sum Rules,  Quark-Hadron Duality, and
 Axial and Vector Nucleon Form Factors}


\author{A.~Bodek and H. S. Budd
%$^{1}$
}
\address{
%$^{1}$
Department of Physics and Astronomy,
University of Rochester, Rochester, NY 14627 \\ }

\date{\today}
\maketitle
\begin{abstract}
We investigate the Adler sum rules within the context
of Quark-Hadron duality. We extract 
relationships between  the axial and vector nucleon form 
factors in neutrino-nucleon and electron-nucleon scattering. 
Using these relationships, we make predictions for the electric 
and axial form factors of the neutron. These predictions
are useful in explaining the $Q^2$ distributions observed
in recent neutrino data at low energies.
%
\end{abstract}
%\pacs{PACS numbers:12.38.Qk, 13.15.+g, 24.85.+p, 25.30.Pt
%\underline {\it ; submitted to Phys ReV D.}
\twocolumn
}
%%%%%%%%%%%%%%%%%%%% ARTICLE TEXT %%%%%%%%%%%%%%%%%%%%

\section*{Introduction}

The Adler sum rules~\cite{adler} for neutrino-nucleon scattering were
first published in in 1966.
These sum rules (which include both
the elastic and inelastic contributions) are based on current-algebra.
Therefore, they are expected to be 
exact and hold for all values of momentum transfer $Q^2$ (from zero 
to infinity). The Adler sum rules for the structure function
$F_2$ are equivalent  (at high $Q^2$)  to the statement that the number of valence $u$ 
quarks minus the number of valence $d$ quarks is 2-1=1. 

In contrast,  Quantum Chromodynamics 
(QCD) sum rules break down at very low $Q^2$, and
are expected to apply above $Q^2 \approx 1-2~GeV/c^2$.  Even
at these higher values of $Q^2$, QCD sum rules have $Q^2$  dependent 
logarithmic corrections.  Example of QCD sum rules with 
 $Q^2$  dependent  logarithmic corrections include
 (a) the energy-momentum sum rule (integral of quark fractional momentum 
$ \approx $ 0.5),
(b) The Bjorken sum rule for polarized structure functions,
(c) The Gottfried sum rule(for the difference in electron/muon scattering from neutrons and 
protons) and 
(d) the Gross-Llewyllyn-Smith sum rule in neutrino 
scattering (number of
valence quarks equal to 3). 

A successful model of structure functions and nucleon form
factors in the $Q^2$ region of  0 to 5~ $GeV/c^2$  should satisfy
QCD sum rules at higher $Q^2$, and the Adler sum rules
over the entire region of $Q^2$.  This region, which spans the 
transition between the hadron and quark pictures of the nucleon, is of 
interest to neutrino oscillations experiments 
for which the  neutrino energies are in the few GeV region.  
This energy region is 
also being investigated in electron-nucleon scattering experiments at 
Jefferson Laboratory.

Recently, in has been shown by Bodek and Yang ~\cite{nnlo} that
with the inclusion of target
mass corrections~\cite{gp} and higher order NNLO QCD corrections,
the standard Parton Distribution Functions (PDFs) provide
a very good description of
deep inelastic scattering above $Q^2 \approx 1~GeV/c^{2}$ over
all regions of $x$. It was also shown that in NLO analyses, the missing 
NNLO higher order terms can be parametrized as small power corrections
(higher twist) a formulated by renormalon models~\cite{nnlo}.  Within this
framework it was shown that  
 QCD predictions in NLO (with the addition of target mass
 corrections and the renormalon higher twist terms)
 provide a good description of the average value
the structure functions in the resonance region~\cite{nnlo}.
Since the Adler and QCD sum rules include both elastic/quasielastic
and inelastic contributions, they can be used
to relate the inelastic and elastic  vector and axial 
form factors of the nucleon.

\section*{Electron-Nucleon Scattering}

	The differential cross section for scattering of an
unpolarized charged lepton with an incident energy $E_0$, final energy
$E^{\prime}$ and scattering angle $\theta$ can be written in terms of
the structure functions $F_1 = MW_{1}$ and $F_2 = \nu W_{2}$ as:

\begin{tabbing}
$\frac{d^2\sigma}{d\Omega dE^\prime}(E_0,E^{\prime},\theta)  =$ \\
   $\frac{4\alpha^2E^{\prime 2}}{Q^4} \cos^2(\theta/2)
   \left[F_2(x,Q^2)/\nu +  2 \tan^2(\theta/2) F_1(x,Q^2)/M\right]$
\end{tabbing}

where $\alpha$ is the fine structure constant, $M$ is the nucleon
mass, $\nu=E_0-E^{\prime}$ is energy of the virtual photon which
mediates the interaction, $Q^2=4E_0E^{\prime} \sin ^2 (\theta/2)$ is
the invariant four-momentum transfer squared, and $x=Q^2/2M\nu$ is a
measure of the longitudinal momentum carried by the struck partons.

	Alternatively, one could view this scattering process 
as virtual photon
absorption.  Unlike the real photon, the virtual photon can have two
modes of polarization.  In terms of the cross section for the
absorption of transverse $(\sigma_T)$ and longitudinal $(\sigma_L)$
virtual photons, the differential cross section can be written as,
\begin{equation}
\frac{d^2\sigma}{d\Omega dE^\prime} =
   \Gamma \left[\sigma_T(x,Q^2) + \epsilon \sigma_L(x,Q^2) \right]
\end{equation}
where,
\begin{eqnarray}
 \Gamma &=& \frac{\alpha K E^\prime}{ 4 \pi^2 Q^2 E_0}  \left( \frac{2}{1-\epsilon } \right) \\
\epsilon &=& \left[ 1+2(1+\frac{Q^2}{4 M^2 x^2} ) tan^2 \frac{\theta}{2} \right] ^{-1} \\
K &=& \frac{2M \nu - Q^2 }{2M}.
\end{eqnarray}

The quantities $\Gamma$ and $\epsilon$ represent the flux and the
degree of longitudinal polarization of the virtual photons
respectively. The quantity $R$, is defined as the ratio
$\sigma_L/\sigma_T$, and is related to the structure functions by,
\begin{equation}
 R(x,Q^2)
   = \frac {\sigma_L }{ \sigma_T}
   = \frac{F_2 }{ 2xF_1}(1+\frac{4M^2x^2 }{Q^2})-1
   = \frac{F_L }{ 2xF_1}
\end{equation}
where $F_L$ is called the longitudinal structure function. The
structure functions are expressed in terms of $\sigma_L$ and
$\sigma_T$ as follows:
\begin{eqnarray}
 F_1 &=& \frac{M K }{ 4 \pi^2 \alpha} \sigma_T, \\
 F_2 &=& \frac{\nu K (\sigma_L + \sigma_T)}{4 \pi^2 \alpha (1 + 
 \frac{Q^2 }{4 M^2 x^2} )} \\
 F_L(x,Q^2) &=& F_2 \left(1 + \frac{4 M^2 x^2 }{ Q^2}\right) - 2xF_1.
\end{eqnarray}
or,
\begin{equation}
2xF_1 = F_2 \left(1 + \frac{4 M^2 x^2 }{ Q^2}\right) -  F_L(x,Q^2).
\label{eq:fl-rel}
\end{equation}

In the elastic limit, i.e. $x = 1$, these
structure functions are related to the Sach's elastic form factors
$(G_M,G_E)$ by:

\begin{eqnarray}
2xF_1^{em}(x,Q^2)&=&x^2G_M^2(Q^2)\delta(x-1) \label{eq:2xf1-rel} \\
F_2^{em}(x,Q^2)&=&x\frac{G_E^2(Q^2)+\tau G_M^2(Q^2) }{ 1+\tau} \delta(x-1)
\label{eq:f2-rel}
\end{eqnarray}

where  $\tau=Q^2/4M_p^2$.
The elastic form factors are normalized to $G_M^p(0)=\mu_p=\kappa_p+1
= 2.7928$ (the magnetic moment of the proton),
$G_E^p$(0)=1 (the charge of the proton), $G_M^n(0)=\mu_n=\kappa_n+1
= -1.913$ (the magnetic moment of the neutron),
$G_E^n$(0)=0 (the charge of the neutron) 

The above relationships between the elastic form factors
and the inelastic structure functions (equations~\ref{eq:2xf1-rel} and
~\ref{eq:f2-rel})
yield (using equation~\ref{eq:fl-rel}) a very simple
expression for $R$ at $x = 1$.  
\begin{equation}
R_{em} (x=1,Q^2) = \frac{4M^2}{ Q^2}\left(\frac{G_E^2}{ G_M^2}\right)
\end{equation}



Integrating the differential cross section over $E'$ gives:

\begin{eqnarray}
\frac{d\sigma }{ d\Omega}
=\left(\frac{d\sigma }{ d\Omega}\right)_{Mott}
    \frac{1 }{ 1+2E_0/M_p\sin^2(\theta/2)}\\
     \left ( \frac{G_E^2 + \tau G_M^2 }{ 1+\tau}
     +2\tau G_M^2\tan^2(\theta/2)\right)  \\
=\left(\frac{d\sigma }{ d\Omega}\right)_{Mott}
    \frac{1 }{ 1+2E_0/M_p\sin^2(\theta/2)}\\    
    \left(\frac{1 }{ 1+\tau}\right) \left( G_E^2+\frac{\tau }{ \epsilon}
     G_M^2\right)
\end{eqnarray}

 
 and
$\epsilon=\left(1+2(1+\tau)\tan^2(\theta/2)\right)^{-1}$ is the
longitudinal polarization of the virtual photon.

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. If we also assume that the proton has
a simple exponential spatial charge distribution:

\begin{equation}
\rho(r)=\rho_0 e^{-r/r_0}
\end{equation}

where $r_0$ is the scale of the proton radius.
The form factors are related, in
the non-relativistic limit, to the Fourier transform of the
charge and magnetic moment distribution. If it is
also assumed that the magnetic moment distribution has the
same spatial dependence as the charge distribution ({\it
i.e.}, form factor scaling), we get the dipole
approximation to the form factors:

\begin{eqnarray}
G_D(Q^2)&\equiv &  \frac{1 }{ (1+Q^2r_0^2)^2} \nonumber \\
        &\equiv &  \frac{1 }{ (1+Q^2/M_V^2)^2} \nonumber \\
        &=& G_E^p(Q^2) \nonumber \\
        &=& G_M^p(Q^2)/\mu_p \nonumber \\
        &=& G_M^n(Q^2)/\mu_n 
\end{eqnarray}

and $G_E^n = 0$. Previous measurements of {\it e-p} and
{\it e-n} elastic scattering have
indicated a best fit value of $r_0^2 =(0.24\hbox{
fm})^2=1/0.71$ (GeV)${}^{-2}$, indicating an rms radius
of $\sqrt{\langle r^2 \rangle}\approx 0.81$ fm.
Measurements of $G_E^p$, $G_M^p$, and $G_M^n$ agree with the dipole
approximation only for  $Q^2<1$ GeV$^2$. In this communication
we use updated fits to the form factor by Bodek, Budd and Arrington~\cite{BBA} 
(BBA2003). 



\section*{Neutrino-Nucleon Scattering}

We now expand the discussion to include neutrino quasielastic 
scattering. Here
we use the notation of C.H.~Llewellyn Smith~\cite{Lle_72} 

\begin{equation}
q^2 = q^2_0 - \vec q_3^2 = -4E_0E^\prime \sin^2{\theta\over 2} =
-Q^2  
\end {equation}

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*}
$$
B(q^2) = -\frac{q^2}{M^2}ReF_A^*(F_V^1+{\xi}F_V^2),
$$
$$
C(q^2) = \frac{1}{4}\left(|F_A|^2 + |F_V^1|^2 -
 \frac{q^2}{M^2}\left|\frac{{\xi}F_V^2}{2}\right|^2\right).
$$


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:
\begin{equation}
 F^1_V(Q^2)=
\frac{G_E^V(Q^2)+\tau G_M^V(Q^2)}{1+\tau },
\end{equation}

\begin{equation}
{\xi}F^2_V(Q^2) =\frac{G_M^V(Q^2)-G_E^V(Q^2)}{1+\tau }.
\end{equation}

Note that from  CVC we can  relate $ G_E^V(Q^2)$ and $ G_M^V(Q^2)$ 
 the electron scattering form factors
$G_E^p(Q^2)$, $G_E^n(Q^2)$, $G_M^p(Q^2)$, and $G_M^n(Q^2)$:

\begin{equation}
G_E^V(Q^2)=G_E^p(Q^2)-G_E^n(Q^2)
\end{equation}
\begin{equation}
G_M^V(Q^2)=G_M^p(Q^2)-G_M^n(Q^2). 
\end{equation}


At low $Q^2$ the axial form
factor $F_A$ and the pseudoscalar form factor $F_P$
(related to $F_A$ by PCAC) are given by dipole approximation:

\begin{equation}
F_A(Q^2)=\frac{g_A}{\left(1+\frac{\D Q^2}{\D M_A^2}\right)^2 },
\end{equation}
\begin{equation}
F_P(Q^2)=\frac{2M^2F_A(Q^2)}{M_{\pi}^2+Q^2}.
\end{equation}


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.
%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$^2$\\
$M_A$ & 1.014 GeV\\
\hline
\end{tabular}
%\end{center}
\caption{  The most recent values of the
parameters used in the our calculations.}
\label{parameters}
\end{table}

The axial form factor is normalized
to measurements from neutron decay ($F_A(0) = g_A = -1.267$).
At low $Q^2$,
$M_A$ can be extracted from pion
electroproduction data. 
The 
theoretically corrected value of $M_A$ 
from pion electroproduction~\cite{Bernard_01} is 1.014 $\pm$ 0.016 GeV.
This value is in agreement with
the updated  value~\cite{BBA} of $M_A$
1.00 $\pm$ 0.020 GeV, which has been
re-extracted from previous neutrino data
using current values for $g_A$ and the 
updated `BBA-2003' vector form factors.

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.

Experimental investigation of deviations from the dipole
form of  $F_A(Q^2)$ will be done  in
the next generation high precision
neutrino quasieastic scattering experiments such as
MINERvA~\cite{eoi}.  However, as we show below, current experiments such
as K2K, MiniBoone and NOMAD, are
already affected by  non-dipole corrections to both the vector
and axial form factors.

For neutrino scattering,  inelastic structure functions
 (in the elastic limit) are related to the quasielastic form factors
$(G_M^{V},G_E^{V})$ as follows:
%
\begin{eqnarray}
2xF_1^{\nu-vec}(x,Q^2)&=&x^2[G_M^{V}]^2(Q^2)\delta(x-1) 
\label{eq:2xf1-rel-nu} \\
F_2^{\nu-vec}(x,Q^2)&=&x\frac{[G_E^{V}(Q^2)]^2+\tau [G_M^{V}(Q^2)]^2 }
{ 1+\tau } \delta(x-1) \label{eq:f2-rel-nu} \\ 
2xF_1^{\nu-axi}(x,Q^2)&=&x^2 \frac {(1+\tau)[FA(Q^2)]^2}{\tau} \delta(x-1) 
\label{eq:2xf1-rel-ax} \\
F_2^{\nu-ax }(x,Q^2)&=&x F_A(Q^2)]^2\delta(x-1) \label{eq:f2-rel-ax} 
\end{eqnarray}
%
The above relationships between the elastic form factors
and the inelastic structure functions (equations~\ref{eq:2xf1-rel-nu} and
~\ref{eq:f2-rel-nu})
yield (using equation~\ref{eq:fl-rel}) a very simple
expression for $R_{\nu}$ at $x = 1$.  
%
\begin{eqnarray}
R_{\nu}^{vec} (x=1,Q^2) &=& \frac{4M^2}{ Q^2}
\left(\frac{[G_E^{V}(Q^2)]^2}{ [G_M^{V}(Q^2)]^2}\right) \\
R_{\nu}^{ax} (x=1,Q^2) &=& 0
\end{eqnarray}
%
\section*{The Adler Sum Rules}
%
In deep inelastic charged current scattering from quarks, the vector
and axial couplings are equal (V-A). Similarly, in electron scattering
from quarks (vector current), there is a well defined ratio ($R$)  between
electric and magnetic scattering from point-like Dirac quarks.
All of these relations break down at
low momentum transfers. For
example, in quasielastic and resonance production at low momentum
transfers, the measured
charge and anomalous magnetic moments of the neutron and
proton imply a ratio of electric and magnetic scattering for
the vector current on nucleons is not the same as for point
like spin 1/2 quarks.
Similarly, from neutron decay, we know that $|g_{a}(Q^{2}=0)|=1.267$
instead of 1.0, so vector and axial scattering are not equal at
$Q^{2}$=0.

   There are theoretical calculations (e.g.  lattice gauge 
   calculations, consituent quark models, etc.), of
 the anomalous  magnetic moments of the
   proton and neutron and also of the $Q^{2}$ dependence of the 
   electric, magnetic, and axial form factors of the neutron and 
   proton.   The calculations of
   of the magnetic form factors at $Q^2$=0 can be compared to the 
   measured 
   charge and anomalous (vector and axial) magnetic moments.
   The calculated slopes
    at low $Q^{2}$ are related to the measured mean square charge radii of the
     proton and neutron.  The dipole form assumes that the charge
       and magnetization distributions of the various
       types of quarks and antiquarks have an exponential form.
       The fact that  $G_E^{p} /G_{D}$ is found to be close 
	  to 1.0 at low $Q^{2}$ implies that the charge and magnetization
	    distribution of the proton are the same.  For
         $Q^{2}$ above $\approx$ 0.5-1.0 $(GeV/c)^2$ this non-relativistic picture
	  breaks down. 
	    
	    Recent polarization transfer
	  experiment at Hall A at Jefferson Lab find  that at higher
	     $Q^{2}$ the ratio  $G_E^{p} /G_{D}$ becomes much smaller than 1.0.
	      This implies that the simple non-relativsitc picture
	     breaks down, and more sophisticated models need to be used.
	    Here we show that if quark-hadron duality is approximately valid above  
	     $Q^{2}$ of $\approx$  0.5-1.0 $(GeV/c)^2$, the Adler sum rules 
	     predict that the dipole forms must break down in a way which is 
	     confirmed by current experimental data.
	 
	     The Adler 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 refer 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
more than a dozen Adler sum rules for the three structure
functions on neutrons and protons for
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.
 
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 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 vector/axial pairs 
 of equations
 are equal to a difference between pairs of
  equations below ( i.e.
  equations 1 - 2, or equations 3-4)  which are equivalent to
  the statement that ($u_{v}-d_{v}=1$).
 %  equations 5 - 6 $(u_{v}$=2) , and  equations 7 - 8 ($d_{v}=1$).
(Here $+$ denotes neutrinos and $-$ denotes antineutrinos)
 
 \begin{enumerate}
    \item $\beta_{p\Delta S=0}^{-}$= $cos^2\theta_c [u +\overline 
    d]$+$(N=udd~discrete)$
    \item $\beta_{p\Delta S=0}^{+}$= $cos^2\theta_c [d +\overline 
     u]$+$(\Delta^{++}=uuu~broad)$ 
    \item $\beta_{n\Delta S=0}^{+}$= $cos^2\theta_c [u +\overline 
   d]$ +$(P=uud~discrete)$
          \item $\beta_{n\Delta S=0}^{-}$= $cos^2\theta_c [d +\overline 
    u]$+$(\Delta^{-}=ddd~broad)$

%   \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).
$$ |F_V(Q^2)|^{2} =|F^1_V(Q^2)|^{2}-
 \frac{q^2}{M^2}\left|\frac{{\xi}F_V^2(Q^2)}{2}\right|^2
$$
which reduces to:
\begin {equation}
|F_V(Q^2)|^{2} = \frac {(G_E^{V})^{2}(Q^2)+\tau 
(G_M^V(^2))^{2}}{(1 + \tau )}
\end {equation}
and
$$G_M^V(Q^2) =F^1_V(Q^2)+{\xi}F_V^2(Q^2)
$$
which reduces to:
\begin {equation}
G_M^V(Q^2)=G_M^p(Q^2)-G_M^n(Q^2). 
\end {equation}
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, and $G_M^V(Q^2)$ in our
notation is the same as $g_V(q^2)$ as defined by Adler.
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:
   $$ [\tau ][ G_{M}^{V}(Q^2)]^{2}+ [1/(cos^2\theta_c)]~~x
 \;$$
$$
 \int_{\nu_{0}}^{\infty}\left[W_{1p\Delta S=0}^{vec-}(\nu,Q^2)-
 W_{1p\Delta S=0}^{vec+}(\nu,Q^2)
 \right] d\nu = C_{I,1}  \; $$
 $$[1+ \tau ] [F_A(Q^2)]^{2}+ [1/(cos^2\theta_c)]~~x
 \;$$
 $$
 \int_{\nu_{0}}^{\infty}\left[W_{1p\Delta S=0}^{ax-}(\nu,Q^2)-
 W_{1p\Delta S=0}^{ax+}(\nu,Q^2)
 \right] d\nu = C_{I,2}\; $$
Where the limits of the integrals are
from pion threshold $\nu_{0}$ where $W= M_{\pi}+M_P$ to $\nu_=\infty$.

%The equations for the  strangness-conserving ($\Delta S=0$)
%Adler sum rule for $\gamma_{p}$ become:
% $$ [F_A(q^2)][ g_V(q^2)]/M = [1/(cos^2\theta_c)]~~x
% \;$$
%$$
% \int_{\nu_{0}}^{\infty}\left[W_{3p\Delta S=0}^{-}(\nu,q^2)-
% W_{3p\Delta S=0}^{+}(\nu,Q^2)
% \right] d\nu \; $$
%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.
	     
 
 \section*{Predictions from Adler Sum Rule and Global Quark-Hadron Duality 
 for $F_A(Q^2)$}
 
If quark hadron duality
 holds for $W_{1}$ then the QCD PDFs describe 
 average of the resonances (above pion production threshold) for
 $Q^{2}$ above $\sim$ 0.5~$(GeV/c)^2$. Therefore
 the integral for the vector and axial parts of the inelastic
 contributions to the
  $W_{1}$ integral are equal. If we assume that this
  equality $also$ $holds$ for the contribution of the
  broad $\Delta^{++}=uuu$ 1.232 GeV resonance, then
  the Adler sum rule for vector and axial
  parts of $W_{1}$ implies also that
  the vector and axial components of $W_{1}^{elastic}$
  are equal at large $Q^2$. If we use the
  current best  prametrizations of the form factors~\cite{BBA} 
  (BBA2003), we find that the
  axial and vector contributions  become equal at $Q^{2} \approx  0.5~(GeV/c)^2$. 
  The requirement that
this ratio of vector and axial contribution is also equal to 1.0 
for higher $Q^2$ yields a 
prediction that the axial form factor is a factor of 1.4 times the
dipole dipole parametrization at higher $Q^2$. Specifically,
that for $Q^{2}$ greater than  $ \approx  0.5~ (GeV/c)^2$
$$[\tau ][ G_M^{V}Q^2)]^{2}
= [1+ \tau ] [F_A(Q2)]^{2}$$
or
\begin {equation}
[F_A(Q^2)]^{2}= [\tau /(1+\tau)] [G_M^{V}(Q^2)]^{2}
\end {equation}
Where 
$$[G_M^{V}(Q^2)]^{2}=[G_M^p(Q^2)-G_M^n(Q^2)]^{2}$$
We compare this  prediction the
Dipole Parametrization for $F_A(Q^2)$ (equation 20), with $M_{A} = 
1.014~\hbox{GeV}/c^2$, 
%
%$$
%F_A(Q^2)=\frac{g_A}{\left(1+\frac{\D Q^2}{\D M_A^2}\right)^2 },
%$$
%in Figure~\ref{f_a_log_mva:ps}~and~\ref{f_a_mva:ps}.
Note that we
only use  the dipole approximation as a scale (e.g.
to show ratio to dipole in our plots). In all of our
calculations, we use BBA2003 updated form factors. 
The dipole approximation for $G_M^p(Q^2)$ and
$G_M^n(Q^2)$ yields the following simple expression:
$$[G_M^{V}(Q^2)]^{2} \approx [4.706~G_D(Q^2)]^{2}$$.
\begin{figure}[tp]
\begin{center}
\epsfxsize=3.0in
\mbox{\epsffile{f_a_log_mva.ps}}\\
\end{center}
\caption[Extraction of $F_A$ ]{
%Estimation of $F_{A}$ from a
%sample of Monte Carlo neutrino quasi-elastic events recorded in the
% \minerva\ active carbon target.
%The simulated sample and error bars correspond to four years of NuMI running.
Axial Form Factor: shown are $|F_A(Q^2)|$ values extracted
from deuterium bubble chamber experiments
using the $d\sigma/dQ^2$ from the papers of FNAL 1983~\cite{Kitagaki_83}
BNL 1981~\cite{Baker_81}, and ANL 1982~\cite{Miller_82}. Also shown
is the expetations from a pure dipole form for $|F_{A}|$  with $M_{A} = 
1.014~\hbox{GeV}/c^2$,
(solid line) and our predictions (dotted line)
assuming $V=A$ for $Q^{2}$ greater than  $\approx$ 0.5~$(GeV/c)^2$ 
(equation 27). Also shown are the prediction from the consituent
quark model of Wagenbrunn et. al.~\cite{const}. 
}
\label{f_a_log_mva:ps}
\end{figure}


\begin{figure}[tp]
\begin{center}
\epsfxsize=3.0in
\mbox{\epsffile{fa.eps}}\\
\end{center}
\caption[sensitivity to $F_A$]
{Extracted values of $F_A(Q^2)/dipole$
(with $M_{A} = 1.014~\hbox{GeV}/c^2$)
for deuterium bubble
chamber experiments
Baker {\em et al.}~\cite{Baker_81},
Miller {\em et al.}~\cite{Miller_82},
and Kitagaki {\em et al.}~\cite{Kitagaki_83}.
Also shown are our predictions (dotted line)
assuming $V=A$ for $Q^{2}$ greater than  $\approx $ 0.5~$(GeV/c)^2$ 
(equation 27).  Also shown are the prediction from the consituent
quark model of Wagenbrunn et. al.~\cite{const}.}
\label{f_a_mva:ps}
\end{figure}

Figure~\ref{f_a_log_mva:ps}~and~\ref{f_a_mva:ps} shows the 
 re-extraction of $F_A(Q^{2})$ by Bodek, Budd and Arrington
 using published 
 neutrino data. Here neutrino data for $d{\sigma}/dQ^2$ was used 
in combination with  BBA2003 Vector Form Factors to extract values
of the axial form factor versus momentum transfer.
Figure~\ref{f_a_mva:ps} shows the same data as a ratio
to the dipole prediction $F_A$ with $M_{A} = 1.014~\hbox{GeV}/c^2$. 
The dipole form factor for
$F_A$ can only be assumed to be valid close $Q^2=0$, which the region for which
the value of $M_A$ can also be extracted from data on the
electroproduction of pions.  We conclude that the Adler sum rule and 
quark hadron duality imply that for $Q^{2}$ greater than 
$\approx $ 0.5~$(GeV/c)^2$
the axial form factor is larger than the dipole form currently
assumed in most neutrino experiments.  Our prediction 
are also in qualitative agreement the predictions of the consituent
quark model of Wagenbrunn et. al.~\cite{const}. Note however, that 
constituent quark models are not expected to be valid
at large $Q^2$, and that this specific model also yields
a value of $g_{a}=1.15$ which deviates from the experimental
value of 1.267.

The maximum $Q^2$ values that can be achieved with incident neutrino energies of
0.5, 1.0, 1.5 and 2~GeV are 0.5, 1.2, 2.1 and 2.9~$\hbox{(GeV/c)}^2$, respectively.
Since K2K, MiniBooNE, and J-PARCnu energies are in the 0.7--2.0~GeV range, 
our results indicate that the there will be a relative enhancement in 
the number of events with  $Q^{2}$ greater than  $\sim$ 0.5~$(GeV/c)^2$
with respect to the number of events at lower $Q^{2}$. This is 
discussed in more detail at the end of this paper.

 

 
  \section*{Nuclear Binding Effects.  }

 %figure8
\begin{figure}[htb!]
\centerline{{\psfig{figure=q2_tsushima_0.79.eps,width=2.91in}}}
\centerline{{\psfig{figure=rat_q2_tsushima_0.7.eps,width=2.91in}}}
\caption{$d\sigma/dQ^{2}$
for a neutrino energy of 0.7 GeV, assuming various models of form factors.
We use $M_A=1.014 ~GeV/c^2$. 
The curves labeled "Free dipole FF", Free BBA FF"   
"Bounded FF, V=A", and "Bounded FF, V=A, Pauli" are
described in the text. The top set of plots shows 
$d\sigma/dQ^2$, while the bottom set of plots
shows the ratio of $d\sigma/dq^2$ to
$d\sigma/dQ^2$ with dipole free form factors.
The $Q^2$ upper limit on the plot is the kinematic limit for
an incident neutrino energy of 0.7 GeV.}
\label{dsigma_dq2A}
\end{figure}

 %figure8
\begin{figure}[htb!]
\centerline{{\psfig{figure=q2_tsushima_1.47.eps,width=2.91in}}}
\centerline{{\psfig{figure=rat_q2_tsushima_1.4.eps,width=2.91in}}}
\caption{$d\sigma/dQ^{2}$
for a neutrino energy of 1.4 GeV, assuming various models of form factors.
We use $M_A=1.014 ~GeV/c^2$. 
The curves labeled "Free dipole FF", Free BBA FF"   
"Bounded FF, V=A", and "Bounded FF, V=A, Pauli" are
described in the text. The top set of plots shows 
$d\sigma/dQ^2$, while the bottom set of plots
shows the ratio of $d\sigma/dq^2$ to
$d\sigma/dQ^2$ with dipole free form factors.
The $Q^2$ upper limit on the plot is the kinematic limit for
an incident neutrino energy of 1.4 GeV.}
\label{dsigma_dq2B}
\end{figure}

 
Current neutrino experiments such as MiniBoone, and K2K probe the
low $Q^2< 1 ~\hbox{(GeV/c)}^2$ region where nuclear effects are large.
The low $Q^2$  MiniBooNE and K2K experiments have
begun to investigate the various nuclear effects in carbon and oxygen.
Higher  $Q^2$ data from the NOMAD experiment will also come out soon.
Although the neutrino flux is not as well know for these current
experiments, a variety of effects  such non-dipole form factors,
and nuclear correction must be included in the comparison of
the $Q^2$ distributions between data and theory.

Figures~\ref{dsigma_dq2A}~and~\ref{dsigma_dq2B} show the
expected  $d\sigma/dQ^2$ assuming various models of form factors
for neutrino energies of 0.7 and 1.4 GeV, respectively.
Here we start with a dipole form factors and $M_A=1.014~GeV/c^2$. 
The curve labeled "Free dipole FF" assumes free nucleon and  dipole
fomr factors. The curve 
"Free BBA FF" assumes  BBA2003 vector form factors and
a dipole form for the axial form factor (with $M_A=1.014 ~GeV/c^2$). The cureve 
"Bounded FF" includes the binding effects on the 
form factors from ref. ~$\cite{Tsushima_03}$ (which should
be applied if the nuclear model is  a simple Fermi motion model. 
The cuve
"Bounded FF, V=A" shows what happens when the
a non-dipole form factor (equation 27) is also used for the
axial form factor, as predicted by our duality mode in
addition to nuclear binding corrections.
The last curve "Bounded FF, V=A, Pauli"
includes final state Pauli exclusion effects
at low $Q^2$ for a simple Fermi gas model on Carbon~$\cite{bodek}$.
The top set of plots shows 
$d\sigma/dQ^2$, while the bottom set of plots
shows the ratio of $d\sigma/dQ^2$ to
$d\sigma/dQ^2$ with dipole free form factors.
The $Q^2$ upper limit on the plot is the kinematic limit.

Currently, neutrino experiments such as MIniBoone, K2K and NOMAD
typically use Dipole form factors and include Fermi-Gas and Pauli blocking
 (which results in a depletion at low $Q^2$ the ther Monte Carlo 
 predictions.  Our results indicate each of the following additional
 corrections: (a) Updated non-dipole vector form factors ;
 (b) A non-dipole axial form factor, and also (c) modifications to the
 free form factors for nuclear binding corrections 
result in an  additional relative depletion at the low
 $Q^2$ with respect to high $Q^2$ events.
 
 A similar argument
can be made for the axial form factors if one assumes that $V=A$ at high
$Q^{2}$.   If we assume quark-hadron duality, we also
expect that the axial and vector contribution to $W_1$ for the $\Delta$ should
become equal with increasing $Q^{2}$. This provides a possible explanation
for the experimental observation that the axial mass for the  $\Delta$ 
is higher than the axial mass for the nucleon (equality of V=A within
the framework of the  dipole approximation requires a higher axial mass). 


More precise higher $Q^2$ data will become accessible 
in higher-energy experiments such as MINERvA
which can span the 2--8~GeV neutrino energy range.
MINERvA's measurement of the axial form-factor at high $Q^2$
will greatly contribute to a better  understanding of the vector and
axial structure of the neutron and proton.


\begin{figure}%[htb]
\begin{center}
\epsfxsize=3.01in
\mbox{\epsffile[65 420 550 710]{show_gep_ratio.eps}}
\end{center}
\caption{Data for $|G_E^p/G_M^{p}|$,  the diamonds are the from Rosenbluth extractions
and the crosses are the  polarization transfer data.  Also
shown are of values  $|G_E^n/G_M^{n}|$ using polarization trasfer data. 
The duality 
prediction that  $|G_E^p/G_M^{p}|$ $\approx$ $|G_E^n/G_M^{n}|$ with increasing $Q^{2}$
is in  agreement with the polarization transfer data.
 Also shown are the prediction from the consituent
quark model of Wagenbrunn et. al.
}
\label{show_gep}
\end{figure}


\section*{Global versus  Local Quark-Hadron Duality}
 
In the previous section  we  used global quark-hadron duality,
which assumes that the integral of
the structure function $W_{1}$ for inelastic scattering scattering
(including the resonance region) is equal to the integral of the
QCD predictions from Parton Distribution Function (including target 
mass effects). Global quark-hadron
duality implies that the lowest order moments of
the QCD predictions  for scattering on quarks are the
same as those for the measured structure functions.  This is equivalent
to the assumption that the lowest moment integrals of the structure functions
as predicted from QCD 
are not modified by the final state interactions which result
in final state resonances.



Local quark-hadron duality implies that the same holds in the region
of specific resonances.  Local quark-hadron duality taken to an extreme
implies that the integral of the QCD predictions above pion threshold
are equal to the measured elastic form factors.


However,  local quark hadron duality, even if found to be approximately valid
in electron nucleon scattering, must be modified in the case of  
charged current neutrino  and antineutrino reactions on nucleons. 
Another example is the 5/18 rule for the ratio of
the structure functions of neutrino and electron
scattering on an isoscalar (e.g. D) target, for which local duality cannot
hold unless the contribution of the $\Delta$  is also included. Near
x=1, the inelastic struture functions 
fpr electron scattering on D=n+p is proportional to
\begin {equation}
F_2^{e-[n+p]}(x,Q^2) \sim (5/9)[xu(x)+xd(x)]
%=[(2/3)^{2}xu(x)+(1/3)^{2}xd(x)+(2/3)^{2}xd(x)+(1/3)^{2}xu(x)]
\end {equation}

while neutrino scattering from D=n+l is proporational to
\begin {equation}
F_2^{\nu -[n+p]}(x,Q^2) \sim 2[xu(x)+xd(x)]
\end {equation}

In the case of elastic scattering in 
the electromagnetic case the final state in
elastic scattering is always a neutron or a proton. In contract, in
neutrino and antineutrino charged-current scattering, the hadronic
final states include (in addition to
the discrete $(neutron=udd)$ and $(proton=uud)$ M= 
0.938~$GeV/c^{2}$) the broad  $\Delta^{++}=uuu$  and $\Delta^{-}$ 
resonances a mass of  1.232 
~$GeV/c^{2}$.  Therefore, any comparisons between electron and 
neutrino scattering within the framework of local quark-hadron duality
must account for these additional transitions.  A recent 
article by Steffens and Tsuchima \cite{CSV} has concluded that
there are violations of local 
quark-hadron in the comparison between electron and neutrino 
scattering. These violations were incorrectly taken to be evidence for Charge 
Symmetry Violations of nucleon Parton Distribution Functions. 
However, since  all the vector form factors are extracted under
the assumption of charge symmetry, charge symmetry (at least for
the vector form factors) should be valid by definition.
Therefore, comparisons of neutrino and electron-nucleon interactions
must include transitions to both the 
nucleon and $\Delta$ resonances final states.

Here we investigate additional predictions that rely on local 
duality. Steffens and Tsuchima \cite{CSV}, state that
at high $Q^2$ local duality 
for electron scattering from neutrons and protons implies the 
following:

%\int_0^{\infty}

 \begin{eqnarray}
\int_{\xi_{th}}^{1}   F_1^{eP }(\xi,Q^2)
 d \xi &=& \eta \left[ G_M^{p}(Q^2) \right]^2
\label{eq:f1p-dual} \\
\int_{\xi_{th}}^{1}  F_2^{eP }(\xi,Q^2)
 d\xi &=& \eta \left[ \frac{ [G_E^{p}(Q^2)]^2+\tau [G_M^{p}(Q^2)]^2}
{ (1+\tau}) \right]
\label{eq:f2p-dual}\\
\int_{\xi_{th}}^{1}  F_1^{eN }(\xi,Q^2)
 d\xi &=& \eta \left[ G_M^{n}(Q^2) \right]^2 
\label{eq:f1n-dual} \\
\int_{\xi_{th}}^{1}   F_2^{eN }(\xi,Q^2)
 d\xi &=& \eta \frac{ [G_E^{n}(Q^2)]^2+\tau [G_M^{n}(Q^2)]^2 }
{ (1+\tau) } 
\label{eq:f2n-dual} 
\end{eqnarray}

where,  
$\xi$ = $xQ^{2}/[0.5Q^{2}(1+[1+(2Mx)^{2}/Q^2]^{1/2})]$
is the Nachtmann variable and $\xi_{0}$ is the
Nachtman variable at the nucleon pole  with x=1, $\xi_{th}$ is
Nachtman variable at the pion threshold with 
$x_{th} = Q^{2}/(Q^{2}+m_{\pi}(2M+m_{\pi}))$,
and  $\eta = \xi_{0}^{2}/(4-2\xi_{0})$.

The following are a few predictions of local duality for electron/muon
scattering.
\begin{equation}
\frac {F_1^{inel-eN }(\xi_{th},Q^2)} {F_1^{inel-eP }(\xi,Q^2)} =
\frac {d  [G_M^{n}(Q^2)]^{2}/dQ^{2}} {d  [G_M^{n}(Q^2)]^{2}/dQ^{2}}
\end {equation}
which experimentally is in reasonable agreement with the data.

QCD calculations, including target mass effects agree with 
experimental measurements that in electron scattering, the ratio
of transverse to longitudinal cross sections, 
$R$, is the same for the neutron and the proton.  Therefore, local 
duality implies that:
\begin {equation}
\frac {|G_E^{n}(Q^2)|^{2}}{ |G_M^{n}(Q^2)|^{2}} =
\frac {|G_E^{p}(Q^2)|^{2}}{ |G_M^{p}(Q^2)|^{2}}
\end {equation}

Electron scattering experiments at SLAC and Jefferson Lab (JLab) have
measured the proton and neutron electromagnetic (vector)
form-factors with high precision.
The vector form-factors can be determined from electron
scattering cross-sections using
the standard Rosenbluth separation technique\cite{JRA_03},
which is sensitive to (two-photon) radiative corrections, or from polarization
measurements using the newer polarization transfer technique\cite{halla}.
Polarization measurements do not directly measure form-factors,
but rather the ratio $G_E$/$G_M$.
Recently, discrepancies in electron scattering measurements of
some vector form-factors have appeared.
Figure~\ref{show_gep} shows the current two sets of results for
 $|G_E^p/G_M^{p}|$.
There appears to be a difference between
the two different methods of measuring this ratio.
The newer polarization transfer technique
yields a much lower value at high $Q^2$ and indicates
a difference between the electric charge and
magnetization distributions.
The polarization transfer technique is believed to be
more reliable and less sensitive to (two-photon) radiative effects
from two-photon corrections.
 Figure~\ref{show_gep} shows
that dipole amplitudes provide only a
first-order description of form-factor behavior at high $Q^{2}$.
Also
shown are of values for the neutron $|G_E^n/G_M^n|$ using polarization trasfer 
data\cite{neutron}. Our 
prediction that  the
absolute value of the two ratios become eaual with increasing $Q^{2}$
is in  better agreement with the polarization transfer
results for $|G_E^p/G_M^{p}|$ and do not agree with
the the results of the Rosebluth separation method which
yield $G_E^p/G_M^{p}$= $(1/mu_{p}$. 
These data shows that for $Q^{2} \approx 4~(GeV/c)^2$,
both the electric form factor of the proton and
electric form factor of the neutron become small, as expected
from local duality. Also shown are the prediction from the consituent
quark model of Wagenbrunn et. al.~\cite{const}, which give similar
qualitative results.
%
\section*{More on the Adler Sum Rule and Global Quark-Hadron Duality 
 Electric Form Factors of the Proton and Neutron}
%
In previous sections we assumed that the inelastic
part of the Adler sum rule
for the vector and axial part of $W_1$ were equal to each other
as expected for scattering from point like quarks.
We assumed that the axial and vector contributions
of the $\Delta$ resonance to $W_1$ were also equal, and
obtained (for $Q^{2}$ above 
$\approx 0.5~ (GeV/c)^2$).
\begin {equation}
[F_A(Q^2)]^{2}= [\tau /(1+\tau)] [G_M^{V}(Q^2)]^{2}
\end {equation}

At low $Q^2$, the structure function $W_{2}$ includes both
transverse and longitudinal contributions, and the
contribution of the $\Delta$ resonance resonance cannot be neglected.
If we assume that the vector and axial inealstic parts of $W_{2}$ are
also equal we get the following relation at high $Q^{2}$:
$$F_A(Q^2)]^{2}+F_A^{\Delta}(Q^2)]^{2} =|F_V(Q^2)|^{2} + [F_V^{\Delta}(Q^2)]^{2}$$
with 
$$|F_V(Q^2)|^{2} = \frac {[G_E^{V}(Q^2)]^{2}+\tau [G_M^V(Q^2)^{2}}{(1 + \tau )}$$
If we neglect the contribution
of the $\Delta$,  then the only way this equation can hold is if the first term
in the above equation is much smaller than the second term, or:
$$\frac {|G_E^{V}(Q^2)|^{2}}{ |G_M^{V}(Q^2)|^{2}}~\approx  0$$
The above could be satisfied if the electric form factors
of the neutron and proton are equal:
$G_E^n(Q^2) \approx G_E^p(Q^2)$
which is in disagreement with the predictions from local duality 
$|G_E^p/G_M^{p}| \approx |G_E^n/G_M^{n}|$
(except when both form factors are zero  at very high $Q^{2}$).
This is  just one of the examples that illustrates
that the contribution of the $\Delta $
needs to be included for neutrino scattering at lower $Q^2$.

More precise higher $Q^2$ data will become accessible 
in higher-energy experiments such as MINERvA
which can span the 2--8~GeV neutrino energy range.
MINERvA's measurement of the axial form-factor at high $Q^2$
will greatly contribute to a better  understanding of the vector and
axial structure of the neutron and proton.


\section*{Conclusion}

In summary, we have shown that global and local duality imply
that vector and axial form factors of the nucleon
significantly deviate from the dipole approximation with increasing 
$Q^2$, in agreement with indications from present experimental data. 
Additional studies of the implication of global and local duality on
nucleon form factors  will be presented in a future communiction,


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\end{document}
