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\begin{document}
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\bigskip
\title {\bf Can two-boson radiative Correction Explain the NuTeV 
Anomaly? - Internal Note }

\author{A. Bodek }
\address {Department of Physics and Astronomy,
University of Rochester, Rochester, NY 14627}

\date { Octber 10, 2003 - updated January 15, 2004}


\maketitle

%\newpage

\section*{Abstract}

I discuss the possibility that two-boson radiative effects are the
reason for the fact that the rate of neutral-current divided by the
rate of charged-current (NC/CC) events in the NuTeV experiment is 
one percent lower that the standard model prediction. 

Although two-boson effects in the charged-current
process are included in the
Bardin radiative corrections (used by NuTeV) at the quark-parton
level, interactions with spectator quarks
and and spectator nucleons are not included.

In the case when the second exchanged
photon in a two-boson process is soft, its wavelenth is long and
effects of interaction 
with the other spectator-quarks in the nucleon, or the spectator nucleons
in the iron target can be important.  

A one precent change in the NC/CC ratio
can caused by an additional
two-boson contribution at a level which
is about half of the level which is currently
needed to explain recent results in elastic electron-proton scattering.

This note is written in order
to motivate more theoretical effort in the calculation of these 
two-boson spectator effects, and does not claim that this effect is
the explanation the NuTeV anomaly. However, in the absense of
reliable detailed calculations, this explanation cannot be
rulled out at this time.



\section*{Two Photon Effects in Electron and Muon Scattering}

Recently, it has been suggested that the 
electric form factor of the proton as measured in electron
scattering using
the standard Rosenbluth separation technique~\cite{JRA_03},
is sensitive to  two-photon~\cite{two-photon} radiative corrections. This
has been proposed to explain a difference 
between the Rosenbluth results and the results
using the newer polarization transfer technique~\cite{halla}.
The polarization 
measurements do not directly measure the form factors,
but measure the ratio $G_E$/$G_M$,
as  as shown in Figure~\ref{show_gepgmp}. There are currently
several experiments proposed at Jefferson laboratory to study
the magnitude of the two-photon contributions, as well as
several theoretical efforts in calculating this
effect in more detail. A simple way to parametrize
these two-photon effects is in the form of an 
effective additional longitudinal contribution to
the elastic cross section, thus resulting in a larger
electric form factor as measured via a  Rosenbluth technique.


For elastic scattering the ratio $R$ is related
to the ratio of elastic form factors by the
following expression $R_{elastic}$ = $(4M^2/Q^2)(G_E/G_M)^{2}$,
or $R^{p}_{elastic}$ = $(0.481/Q^2)$$(\mu_{p}G_E^p$/$G_M^p)^2$.
For example for  $Q^2$  of 2.5 Gev$^2$ 
the data in figure~\ref{show_gepgmp} (from Bodek, Budd
and Arrington~\cite{budd}, show that $R^{p}_{elastic}$= 
(0.19)$(0.88)^2$=0.14$\pm 0.04$
from the fit to the Rosenbluth separation data and 
$R^{p}_elastic$= (0.192)$(0.72)^2$=0.10$\pm 0.02$ from
the fit to the polarization transfer data. If
this difference is to be attributed to two-photon
effects, then it implies a 4$\%$ epsilon
dependence in the radiative corrections and an
uncertainty in $R$ of 0.04 $\pm$ 0.02.
A fixed longitudinal contribution of this order
becomes much more important at larger values of $Q^2$ as
shown in the figure. As the electric form factor (or $R_{elastic}$)
becomes smaller, the  two-photon contribution becomes larger
than the electric form factor of the proton.

 {\it The uncertainty in the longitudinal contribtution from
two-photon effects in the inelastic
radiative corrections is actually lower
than the above estimate because modern radiative corrections programs
for inelastic electron
and neutrino scattering  (e.g. Bardin)
already include two photon effects at the parton level.}

At present, simple calculations of two-photon effects in the elastic
channel cannot fully explain the above experimental
difference between the two techniques. In order to explain
this difference, the calculations need to include other
intermediate states (e.g. delta resonances) in addition to
a single proton unexcited proton. 

On one hand, the situation in the case of inelastic scattering
(within the parton model) is simpler since there can be no
intermediate excited quark states.
However,  the second photon can be be rather soft and
 in the case of inelastic scattering from
a nucleon target, the second photon
can interacts not only with the initial parton, but also with
the spectator-quarks in the nucleon.  For the scattering from
a nuclear target, such as iron, the second soft photon can also
interact with other spectator-nucleons, or with nuclear-fragments.

In this note, we assume that the the two-photon
terms can be estimated by an effective change of $R$ of 0.04 as
extracted from the elastic electron-scattering data.


%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=show_gepgmp_ratio.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption[Ratio of $G_E^p$ to $G_M^p$]{Ratio $\mu_p$ $G_E^p$/$G_M^p$ as 
extracted by Rosenbluth
separation measurements (diamonds) and as obtained by polarization 
measurements(X's).
 }
\label{show_gepgmp}
\end{figure}

%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=radcor7.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption {Two-photon contribution: Diagram 1 }
\label{radcor7.eps}
\end{figure}

%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=radcor8.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption{Two-photon contribution: Diagram 2 }
\label{radcor8.eps}
\end{figure}

%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=radcor9.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption{Two-photon contribution: Diagram 3 }
\label{radcor9.eps}
\end{figure}

%figure4
\begin{figure}[htb]
\centerline{\psfig{figure=radcor10.eps,width=4.0in,height=3.in}}
%\centerline{\missingFigure}
\caption{Two-photon contribution: Diagram 10 }
\label{radcor10.eps}
\end{figure}

\begin{figure*}[htb]
\centerline{\psfig{figure=crossbreakdown.eps,width=6.in,height=3.0in}}
\centerline{\psfig{figure=acrossbreakdown.eps,width=6.in,height=3.0in}}
    \caption[Total neutrino and anti-neutrino cross-sections]
    {Total neutrino and anti-neutrino cross-sections ($divided$ $by$ 
    $energy$)
    versus energy compared to the sum of quasi-elastic, resonant,
    and inelastic contributions from the NUANCE model.}
\label{fig:total}
\end{figure*}



\section{Two-Boson Effects in  Neutrino Scattering}

Two-boson effects which include a both a $W$ and a photon are only present in
charged current scattering.  The primary diagram is the exchange
of a $W$ boson with a single quark leading to a final state muon. The 
two-boson diagram at the parton-level is for the case in which
a additional photon is exchanged between the final state muon and the final 
state quark.  That diagrem is divergent. However, the divergence is 
cancelled by diagrams in which an additional real photon is
emitted by either the final state muon, or the final state quark.

This two-boson diagram which involves
an extra photon exchange between the final state
lepton and final state quark is not present for the case of neutral current 
scattering (since here the final state lepton is a neutrino). As 
mentioned earlier,  the 
effect of the extra-photon in charged current scattering 
(which leades to a change of the NC/CC ratio) is
included in the Bardin radiative corrections at the parton
level.

Here, we assume that the effect of the extra photon can be
estimated by an effective $R$ which 0.04 larger in charged
current then in neutral current scattering.

\section{Sensitivity of the Neutrino Cross Sections to $R$}

The expressions used in this
section are from  the Appendix.

Within the quark-parton model,
the total neutrino and antineutrino cross sections on a nucleon
target are given by integrating the differential cross
sections over $x$ and $y$:

\begin{eqnarray}
\sigma^{\nu N}  =  \frac{G^2_FME_\nu }
{\pi(1+Q^2/M_W^2)^2}\left[ Q^{\nu N}
      + (1/3)\overline{Q}^{\nu N} + K^{\nu N} \right] \\
\sigma^{\overline{\nu} N}  =  \frac{G^2_FME_\nu}
{\pi(1+Q^2/M_W^2)^2}\left[ 
       \overline{Q}^{\overline{\nu} N}
   + (1/3) Q^{\overline{\nu} N} + K^{\overline{\nu}N}  \right]
\end{eqnarray}

Here $Q$ is the fractional momentum carried by all quarks in the nucleon,
$\overline{Q}$ is the fractional momentum carried by all antiquarks
in the nucleon and $R$ = 2K/($Q$+$\overline{Q}$) is the 
average ratio of longitudinal to transverse contribution. For low
energies, the antiquark contribution is small. Therefore, the 
fractional error in the predicted neutrino and antineutrino
total cross sections from an uncertainty in $R$ is $0.5\Delta R$ for
neutrinos and $1.5\Delta R$ for antineutrinos, respectively.  


A back of an envelope calculation of the sensitivity of
the ratio of antineutrino to neutrino total cross section ratio
to a change in the average value of $R$  with
neutrino energy or from nuclear effects (e.g. $R=0.3\pm0.2$)
is illustrative. At low energy, with $\overline{Q}$=0 we obtain
the following.
If $R=0$, the ratio is 0.33.  If
$R=0.5$, the ratio is $(0.33+0.25)/(1.0+0.25)$=0.46.

Figure~\ref{fig:total} shows the current data for total
neutrino and antineutrino cross-sections
 $divided$ $by$ $energy$ (per nucleon for an isoscalar target)
    versus energy (at low energies) compared
    to a model of the sum of quasi-elastic, resonant,
    and inelastic contributions.

At high energies, e.g. NuTeV, the  antineutrino to neutrino 
charged current total cross section ratio
is about 0.5. Therefore, the additional contribution
from antiquarks needs to be taken into account.

The charged current data is well reproduced by assuming  that 
on average,   $R'$ = 2K/($Q$) is 0.26 and f = 0.025/1.196=0.021/$Q$  is
about 0.2.  For this assumption we get that the ratio of
 antineutrino to neutrino cross sections is equal to 
(0.333+0.2+0.13)/(1.0+0.066+0.13) = 0.633/1.196 = 0.529.

Figure ~\ref{rupra}
(right) shows all available data on $R$ and the nuclear
dependence of $R$ in the DIS region. 
Figures~\ref{rfl94110} and  ~\ref{2xf1f294110}  show
preliminary results of analysis
of very precise data from Jlab experiment E94-110~\cite{e94110} on hydrogen
in the resonance region.
Data with  deuterium is expected to be taken in experiment
    E02-109, and data with nuclear targets is proposed to be
    taken by Jlab proposal P04-001 (previously P03-110).
    Note that a difference in $R$ of 0.04 between the predictions of 
the theory of
    QCD (including target mass corrections and NNLO higher 
    terms) and the data cannot
    be measured because  experimental errors and
      uncertainties in the theoretical predictions are larger
      than that. Even with the new precise data of Jlab E94-110,
       there still remain the theoretical uncertainties in the
         predictions which are larger than 0.04. Therefore
	   a difference of 0.04 in $R$ between charged current
	     and neutral current (if originating from two-boson terms)
	       cannot be rulled out by the data on $R$ in charged
	lepton scattering.

   \begin{minipage}{0.5\textwidth}
\epsfxsize=3.0in \epsfbox{r_update5.ps}
\end{minipage}
    \begin{minipage}{\listwidth}
\epsfxsize=3.0in \epsfbox{ra.ps}
\end{minipage}
\vspace{-0.2in}
\begin{figure}[h]
\caption{
Left: A compilation of the world's published data on $R$ at high
    $Q^2$ on  in the DIS region for both nucleons and nuclei.
    Right: The SLAC E140 data on the nuclear dependence of
      $R$ in the DIS region presented
       in the form $R_{A}-R_{D}$ (e.g. difference between
       iron and deuterium. The lines in rhw figure on the
right represent the small projected uncertainties of the new
measurements proposed by here P04-001 (previously P03-110).
}
\label{rupra}
\end{figure}

 \begin{minipage}{0.5\textwidth}
\epsfxsize=3.0in \epsfbox{Keith_f2_vs_x.eps}
\end{minipage}
    \begin{minipage}{\listwidth}
\epsfxsize=3.0in \epsfbox{Keith_2xf1_vs_x.eps}
\end{minipage}
\vspace{-0.2in}
\begin{figure}[h]
\caption{Recent data from Jlab experiment E94-110 
    (on Hydrogen) for $2xF_1$ (left) and $F_2$ (right)
    in the resonance region. Data with
    deuterium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P04-001 (previoully P03-110).}
\label{2xf1f294110}
\end{figure}


  \begin{minipage}{0.5\textwidth}
\epsfxsize=3.0in \epsfbox{Keith_r_vs_q2.eps}
\end{minipage}
    \begin{minipage}{\listwidth}
\epsfxsize=3.0in \epsfbox{Keith_fl_vs_x.eps}
\end{minipage}
\vspace{-0.2in}
\begin{figure}[h]
\caption{
Recent data from Jlab experiment E94-110 (on hydrogen) for $R$
(left) and $F_L$ (right)
    in the resonance region. Data with
    deuterium is expected to be taken in experiment
    E02-109, and data with nuclear target is proposed to be
    taken by P04-001 (previously P03-110).}
\label{rfl94110}
\end{figure}
%

If we model the two-boson effect as change  $\Delta R$ of 0.04,
for an antiquark contribution of 0.2 
it corresponds to a change $\Delta R'$ of 0.05. ( In the electron scattering
for which the charge of the final state proton is 1.)

In neutrino charged current scattering, a $W+$ boson is echanged and $d$
quarks are transformed to $u$ quarks (in the valence and sea region); and
$\overline{u}$  to $\overline{d}$ (in the sea region).  Therefore,
the charge of the final state quark is typically (2/3)

In anti-neutrino charged current scattering, a $W-$ boson is echanged and $u$
quarks are transformed to $d$ quarks (in the valence and sea region); and
$\overline{d}$  to $\overline{d}$ (in the sea region).   Therefore,
the charge of the final state quark is typically (1/3).

If the the effect of the extra-photon is due to interference between
the two-boson and single boson diagram, then  $\Delta R'$ from
this effect will be twice as large in neutrinos than in antineutrinos,
because the charged of the final state quark is larger.
However, since the total cross section for antineutrinos is about
half of that from neutrinos, the net effect is that the
fractional change in the charged current cross section from two-boson
effects is  about the same for neutrinos and antineutrinos.

A $\Delta R'$ of 0.05 would result in  the NC/CC ratio
for neutrino scattering 
of 0.5$\Delta R'$/1.196=0.021.  The NuTeV anomaly
of 0.01 could come from and extra
 contribution of two
boson effects originating from interactions with
the spectator quarks or spectator nucleons.
A change in the NC/CC ratio of 0.1 requires
that these term are about 50$\%$ in the mangitude
of the entire two-photon contribution needed to explain
the elastic electron-proton results.  This is consistent
with present calculation of two photon effects in elastic scattering
which indicate that the mangnitude needed to explain the
electric form factor measurements is about twice the value
calculated with only a single proton in the intermediate state.
In the elastic channel, the additional contribution is attributed
to diagrams which include
excited intermediate states (such as the delta).



Here a comparison of
the relative size of the two-boson contributions in the present
 Bardin radiaive corrections
codes (at the parton level) for inelastic electron, neutrino and 
an antineutrino scattering would be of great interest.


\section*{Appendix: Charged-current neutrino and antineutrino cross sections}

\subsection*{Quark-Parton Model}

Here we use the quark-parton model to illustrate some of physics concepts
behind the proposed measurements. Note that at low
energies, there are large corrections to this model which need
to be taken into account.

In neutrino ($\nu_\mu$) nucleon scattering experiments, 
the three independently measured variables in a charged-current event 
are the outgoing muon momentum ($p_\mu$), the outgoing muon 
angle ($\theta_\mu$),
and the observed energy of the final state hadrons ($E_{had}$).
From these measured variables the neutrino energy is equal to
 $E_\nu = E_{had} + E_\mu$ as required by  energy conservation.

The derivation of the formulae for inclusive charged-current
neutrino scattering is very similar to the case of $e-\mu$ scattering. 
Both do not require any knowledge of the dynamics inside the nucleon.
The unknown couplings of the lepton-current to the nucleon
are absorbed in the definition of the structure function $F_i$.
In the case of elastic (muon,electron) or quasi-elastic (neutrino)
scattering, these can be interpreted as the Fourier transforms of the spacial
charge distribution in the nucleon.

The general form of the differential cross section for neutrino-nucleon
scattering, mediated by the $W$ boson (in the case
of charged-current scattering) is given
in terms of three structure functions:
\begin{eqnarray}
{\frac{d^2\sigma}{dx dy}}^{\nu (\nub)} & = & \frac{G^2ME}{\pi} \left[
	( 1 - y - \frac{Mxy}{2E} ) F_2 + \frac{y^2}{2} 2xF_1 
	\pm ( 1 - \frac{y}{2} ) xF_3 \right]
\label{eq:master}
\end{eqnarray}
where the $+(-)$ terms correspond to neutrino (antineutrino) scattering.
Here $G_{F}$ is the Fermi weak coupling constant.
The structure function, $F_i$ are process dependent, 
and are functions of the kinematics variable, $x$ and $Q^2$.
If the cross section is re-written in terms of the absorption 
cross-sections by left-handed, right-handed, and longitudinally polarized
$W$ bosons, then the structure function $F_1$ corresponds to the contribution
from the sum of left-handed and right-handed bosons, $F_2$ corresponds
to the contribution from all boson polarizations, whereas $F_3$ corresponds
to the contribution from the difference of right-handed and left-handed 
polarized bosons. The structure
function $F_3$ is non-zero only in weak interactions (for which parity is
violated). 


The relationship
between the experimentally extracted structure functions
and the parton distributions in the nucleon (and their dependence on
kinematic variables)
is determined within the framework of the quark-parton model.
If we denote $q(x)$ as the probability to find a parton
with momentum fraction $x$ in a frame of a fast moving nucleon,
the differential cross section for the scattering from a parton is given  by
\begin{equation}
{d\sigma^2\over dx\, dy} \propto {G_F^2 ME \over \pi(1+Q^2/M_W^2)^2} \, xq(x).
\end{equation}
Therefore, the cross sections for neutrino (and antineutrino) nucleon
 scattering are the sums of all parton contributions in the nucleon, 
with the proper angular dependence factors, as follows:
\begin{eqnarray}
\frac{d^2\sigma^{\nu N}}{dx dy}  =  \frac{G^2_FME_\nu x}
{\pi(1+Q^2/M_W^2)^2}\left[ q^{\nu N}(x)
      + (1-y)^2\overline{q}^{\nu N}(x) + 2(1-y)k^{\nu N}(x) \right] \\
\frac{d^2\sigma^{\overline{\nu} N}}{dx dy}  =  \frac{G^2_FME_\nu x}
{\pi(1+Q^2/M_W^2)^2}\left[ 
       \overline{q}^{\overline{\nu} N}(x)
   + (1-y)^2 q^{\overline{\nu} N}(x) + 2(1-y)k^{\overline{\nu} N}(x) \right]
\end{eqnarray}
The contribution ($k$) of  the longitudinal contribution
e.g. from possible spin-0 constituents or quark
transverse momenta  
is also shown.  The angular dependence factor for this contribution, $(1-y)$,
is the same as for the terms which originate 
from the small intrinsic transverse ($p_t$) of spin 1/2 partons.


\subsection*{Total neutrino and antineutrino cross sections}

Integrating the above expression over x and y (from 0 to 1) yields:

\begin{eqnarray}
\sigma^{\nu N}  =  \frac{G^2_FME_\nu }
{\pi(1+Q^2/M_W^2)^2}\left[ Q^{\nu N}
      + (1/3)\overline{Q}^{\nu N} + K^{\nu N} \right] \\
\sigma^{\overline{\nu} N}  =  \frac{G^2_FME_\nu}
{\pi(1+Q^2/M_W^2)^2}\left[ 
       \overline{Q}^{\overline{\nu} N}
   + (1/3) Q^{\overline{\nu} N} + K^{\overline{\nu}N}  \right]
\end{eqnarray}

Here $Q$ is the fractional momentum carried by all quarks in the nucleon,
$\overline{Q}$ is the fractional momentum carried by all antiquarks
in the nucleon and $R$ = 2K/($Q$+$\overline{Q}$) is the 
average ratio of longitudinal to transverse contribution. For low
energies, the antiquark contribution is small. Therefore, the 
fractional error in the predicted neutrino and antineutrino
total cross sections from an uncertainty in $R$ is $0.5\Delta R$ for
neutrinos and $1.5\Delta R$ for antineutrinos, respectively.  

In the region of the first resonance, $R$  is predicted to be zero
in the bound quark oscillator model. In fact,  the first results
from Jlab experiment E94-110 on hydrogen 
show a value of
$R$ of about 0.3 at low $W$ and low $Q^2$.  This implies
that the effects of the nucleon pion cloud are very important
in this region.
The nuclear dependence
of $R$ in the region is totally unknown, and current measurements
of the nuclear dependence of R have an error of $\Delta R$ of 0.2.
It is expected that the nuclear effects on the nucleon pion cloud
are very significant at low energies. Therefore, even if we use the 
preliminary precise measurement of $R$ on hydrogen from
experiment E94-110, we are still left with the problem
that none of the data have been taken on Carbon or iron.
An error $\Delta R$ of 0.2
implies a 10$\%$ error in the predicted
neutrino cross section in this region
and an error 30$\%$ in the predicted antineutrino cross setion.  

A back of an envelope calculation of the sensitivity of
the ratio of antineutrino to neutrino total cross section ratio
to a change in the average value of $R$  with
neutrino energy or from nuclear effects (e.g. $R=0.3\pm0.2$)
is illustrative. At low energy, with $\overline{Q}$=0 we obtain
the following.
If $R=0$, the ratio is 0.33.  If
$R=0.5$, the ratio is $(0.33+0.25)/(1.0+0.25)$=0.46. 

\subsection*{Structure Functions}

A comparison of the  above parton-level cross sections with 
Equation~\ref{eq:master} yields the following relations between the structure functions
and parton distributions:
\begin{eqnarray}
2xF_1^{\nu(\overline{\nu}) N} & = & 2\left[xq^{\nu(\overline{\nu})N}(x) +
   x\overline{q}^{\nu(\overline{\nu})N}(x) \right] \nonumber\\
F_2^{\nu(\overline{\nu}) N} & = & 2\left[xq^{\nu(\overline{\nu})N}(x) +
   x\overline{q}^{\nu(\overline{\nu})N}(x) 
+ 2xk^{\nu(\overline{\nu})N}(x) \right] 
   \label{eq:parton} \\
xF_3^{\nu(\overline{\nu}) N} & = & 2\left[xq^{\nu(\overline{\nu})N}(x) -
  x \overline{q}^{\nu(\overline{\nu})N}(x) \right] \label{eq:parton3}
\end{eqnarray}
where terms proportional to $Q^2/\nu^2$ have been neglected.
Thus, in the parton model, nucleon structure functions 
are related to the momentum distributions 
carried by the partons in the nucleon.
 
If the scattering takes place exclusively from free spin-$1\over2$
constituents, the Callan-Gross relation
\begin{equation}
2xF_1 = F_2
\end{equation}
is satisfied.
However, the partons also have non-negligible transverse 
momenta, which at present energies yields an apparent spin-0 type behavior,
in the  infinite momentum frame. This 
transverse momentum leads  to a difference between $F_2$ and $2xF_1$
that diminishes as the momentum transfer $Q^2$ increases. 
The exact relation between $2xF_1$ and $F_2$ is obtained by using $R$,
the ratio of the longitudinal structure function ($F_L$) 
and transverse structure function ($2xF_1$).
\begin{eqnarray}
R & = & \frac{F_L}{2xF_1} =  \frac{F_2}{2xF_1}(1+Q^2/\nu^2) -1
=(1+\frac{2k}{q+\overline{q}})(1+Q^2/\nu^2) -1.
\label{eq:R_eq}
\end{eqnarray}

\subsection*{Relation to Electromagnetic Structure Functions}

The analogous expressions for charged-lepton scattering via virtual
photon exchange follow from the pure vector nature 
of the electromagnetic current. Thus,  electromagnetic scattering 
probes the charge  of the  partons, whereas neutrino
scattering probes the flavor composition of the nucleon constituents.
\begin{eqnarray}
2xF_1^{\ell N} & = & \sum_i \ e_i^2 \left[ xq^{\ell N}_i(x) + 
                     x\overline{q}^{\ell N}_i(x) \right] \\
F_2^{\ell N} & = & \sum_i \ e_i^2 \left[ xq^{\ell T}_i(x) + 
                     x\overline{q}^{\ell N}_i + 2k^{\ell N}_i(x) \right]
\label{eq:f2el}
\end{eqnarray}
where $e_i$ is  electric charge of parton $i$.
Comparison of neutrino and charged-lepton scattering data provides 
the measurement of the mean-square charge of the nucleon's 
interacting constituents.

Neutrino scattering has the ability to resolve the flavor of the nucleon
constituents.
Because of charge conservation at the quark vertex, charged current
neutrino scattering happens
only with $d$, $s$, $\overline u$ and $\overline c$ quarks.
Similarly, antineutrinos can scatter
only from $\overline d$, $\overline s$, $u$ and $c$ quarks. 
For a proton target, 
the parton densities that contribute to the structure functions are:
\begin{eqnarray}
q^{\nu p}(x)   =  d^p(x) + s^p(x) & ; & 
 \qbar^{\nu p} (x)= \ubar^p(x) + \cbar^p(x) \\
q^{\nub p}(x)    =  u^p(x) + c^p(x) & ; & 
\qbar^{\nub p} (x)= \dbar^p(x) + \sbar^p(x)
\end{eqnarray}
Isospin invariance (also called  charge symmetry) requires symmetry between the 
light quark densities in the proton and neutron:
\begin{eqnarray}
d^p(x) = u^n(x), ~~u^p(x) = d^n(x), ~~\dbar^p(x) = \ubar^n(x), ~~\ubar^p(x)= \dbar^n(x).
\end{eqnarray}
Using these symmetries, 
the quark distributions in the neutron are described in terms 
the quark distributions in the proton. 
All of the parton distributions are defined with respect to the proton.
\begin{eqnarray}
q^{\nu n}(x)   =  u(x) + s(x) & ; & 
\qbar^{\nu n}(x)= \dbar(x) + \cbar(x) \\
q^{\nub n}(x)    =  d(x) + c(x) & ; & 
\qbar^{\nub n}(x)= \ubar(x) + \sbar(x)
\end{eqnarray}
Finally, the parton densities for an isoscalar nucleon, 
$\frac{1}{2}$(proton + neutron), are given by:
\begin{eqnarray}
q^{\nu N}(x)     =   \frac{1}{2}\left[ u(x)+d(x)+2s(x) \right] & ; &
\qbar^{\nu N}(x) =   \frac{1}{2}\left[ \ubar(x) + \dbar(x) +2\cbar(x) \right] \\
q^{\nub N}(x)    =   \frac{1}{2}\left[ u(x)+d(x)+2c(x)\right] & ; & 
\qbar^{\nub N}(x)=   \frac{1}{2}\left[	\ubar(x) + \dbar(x) +2\sbar(x) \right] 
\end{eqnarray}
The quark content of the isoscalar structure function 
$2xF_1$ for neutrino scattering is obtained
by substituting these densities into Equations~\ref{eq:parton}:
\begin{eqnarray}
2xF_1^{\nu N}(x) & = & xu(x) + x\overline{u}(x) +xd(x) + x\overline{d}(x) \nonumber \\  
                 &   & + xs(x) + x\overline{s}(x) + xc(x) + x\overline{c}(x)  \nonumber \\
     & = & 2xF_1^{\overline{\nu}N}(x).
\label{eq:f2nu}
\end{eqnarray}
In the following discussion (for simplicity) 
we assume $xs(x)=x\overline{s}(x)$. The charm quark
distributions, which are small when compared to the strange quark
distributions, are also neglected.

The electromagnetic structure functions 
$2xF_1^{\ell p}$ and $2xF_1^{\ell n}$ are constructed 
from Equation~\ref{eq:f2el} 
using the same parton densities as above, and including the quark charges:
\begin{eqnarray}
2xF_1^{\ell p} & = & \left(\frac{1}{3} \right)^2 \left[ xd(x) + x\overline{d}x
   + xs(x) + x\overline{s}(x) \right] \nonumber \\ 
& & + \left(\frac{2}{3} \right)^2
   \left[ xu(x) + x\overline{u}(x) + xc(x) + x\overline{c}(x)  \right] \\
2xF_1^{\ell n} & = & \left(\frac{1}{3} \right)^2 \left[ xu(x) + x\overline{u}x
   + xs(x) + x\overline{s}(x) \right] \nonumber \\
&  & + \left(\frac{2}{3} \right)^2
   \left[ xd(x) + x\overline{d}(x) + xc(x) + x\overline{c}(x) \right]. \\
\end{eqnarray}
The $2xF_1^{\ell N}$ for an isoscalar nucleon is found by averaging: 
\begin{eqnarray}
2xF_1^{\ell N} & = & \frac{1}{2}\left( 2xF_1^{\ell p} 
+ 2xF_1^{\ell n} \right) \nonumber \\
          &  = & \frac{5}{18}\left(xu+x\overline{u} + xd + x\overline{d}
\right) \nonumber  \\
&  & \;\; + \:\frac{1}{9} \left( xs + x\overline{s} \right)
+ \frac{4}{9}\left( xc+x\overline{c} \right).
\end{eqnarray}
Under the assumption
that the value of $R$ is the same for electromagnetic neutral-current 
and weak charged-current structure functions,
the ratio of electromagnetic and neutrino
structure functions for $2xF_1$ is equal to the ratio for $F_2$:
\begin{equation}
\frac{F_2^{\ell N}}{F_2^{\nu N}} = \frac{5}{18}\left (1-\frac{3}{5}
       \frac{xs+x\overline{s}-xc-x\overline{c}}{xq+x\overline{q}} \right)
\end{equation}
where $xq+x\overline q = 2xF_1^{\nu N}$.
This relationship is known as the 5/18$ths$ rule.
The observation that  charged-lepton scattering and 
neutrino-scattering structure functions are approximately related by 
a factor of $\sim 5/18$, 
was a significant triumph for the QPM.

The structure function 
$xF_3$ (which is only present in parity violating weak interactions) represents
the momentum density of valence quarks.  Substitution of the isoscalar parton 
densities into Equation~\ref{eq:parton3} yields:
\begin{eqnarray}
xF_3^{\nu N}(x) & = & xu_V(x) + xd_V(x) + 2xs(x) - 2xc(x)\\
xF_3^{\overline{\nu} N}(x) & = & xu_V(x) + xd_V(x) - 2x\sbar(x) + 2x\cbar(x)
\end{eqnarray}
where $u_V \equiv u-\overline{u}$ and $d_V \equiv d - \overline{d}$ are the
valence densities in the proton.
The average value of $xF_3^{\nu N}$ and $xF_3^{\overline{\nu} N}$ 
yields the total valence quarks distribution. The
difference of $xF_3^{\nu N}$ and $xF_3^{\overline{\nu} N}$ is very sensitive to
both the strange sea and charm sea in the nucleon as shown below:
\begin{eqnarray}
xF_3(x) & = & \left[xF_3^{\nu N}(x)+xF_3^{\overline{\nu} N}(x) \right]/2 = xu_v(x) + xd_d(x) \\
\Delta xF_3(x)  & = & \left[xF_3^{\nu N}(x)-xF_3^{\overline{\nu} N}(x) \right] =  2(s(x)+\sbar(x)- c(x)-\cbar(x))
\end{eqnarray}





%
\section*{References} 

%\vfill
%\pagebreak

\begin{thebibliography}{22}
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%

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%
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\end{thebibliography}
%{22}

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\bye


