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\begin{document}
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\bigskip
\title {\bf P04-001: Measurement of $F_2$ and $R = \sigma_L / \sigma_T$ \\
on Nuclear-Targets in the Nucleon Resonance Region \\ 
(F2-Rnuclear-Hall-C)- Part 1 of JUPITER (updated Jan. 10, 2004) \\
First Stage of a Program to Investigate  Quark Hadron Duality in 
Electron and
Neutrino Scattering on Nucleons and Nuclei in Collaboration
with the Fermilab MINERvA Experiment\footnote[3]{
JUPITER JLab Unified Program to Investigate Targets and
Electroproduction of Resonances.  Re-submission of P03-110 (deferred with regret by
PAC24)}
}

\author{A. Bodek (co-spokesperson), S. Manly, K. McFarland, I.C.  Park (J. Chovjka,
G.B. Yu),\footnote[1]{PhD thesis students, pending funding}
(D. Koltun, L. Orr, S. Rajeev) \footnote[2]{Collaborating theorists}
}
\address {Department of Physics and Astronomy,
University of Rochester, Rochester, NY 14627}

\author{M.E. Christy, W. Hinton, (C. Jayalath)*, C. Keppel (co-spokesperson), E. 
Segbefia}
\address { Hampton University, Hampton, VA}
\author{P. Bosted, S. E.  Rock}
\address { University of Massachusetts, Amherst, MA}

\author{ G. Niculescu, I. Niculescu}
\address{ James Madison University, Harrisonburg, VA}

\author{R. Ent, D. Gaskell, M. Jones, D. Mack, S. Wood}
\address{Thomas Jefferson National Accelerator Facility, Newport
News, VA}

\author{J. Arrington}
\address{Argonne National Laboratory,  
Argonne, IL}

\author{ H. Gallagher}
\address{Tufts University, Medford, MA}

\author{J. Dunne}
\address{Mississippi State University, Mississippi State, MS}

\author{P. Markowitz, J. Reinhold}
\address{ Florida International University, University Park, FL}

\author{E. Kinney}
\address { University of Colorado, Boulder, Colorado}


\author{H.P. Blok}
\address{Vrije Universiteit, Amsterdam, Netherlands}

\author{In collaboration with}
\address{The MINERvA Neutrino Experiment at Fermilab\\
\ \\}



\date {
%Resubmitted JLab proposal P03-110 
%- updated Dec. 2,2003 - 2:00 pm)
$(http://www.pas.rochester.edu/~bodek/jlab/F2-Rnuclear-HallC.pdf$
$(http://www.pas.rochester.edu/~bodek/jlab/JUPITER.html$}


\maketitle

\newpage

%\section*{Abstract}
\section{Summary}

We propose to measure the longitudinal-transverse (L-T) separated 
structure
functions $F_2$ and $R = \sigma_L / \sigma_T$
from nuclear targets in the resonance region.
P04-001 (a re-submission of P03-110 deferred with regret by PAC24) is the first stage
of an overall program to investigate quark-hadron duality and
electromagnetic and weak structure of nucleons and 
nuclei~\cite{P04-001,eois}.
The targets proposed include Carbon, Quartz, Aluminum, Calcium, Steel
and Copper 
to compliment existing baseline data on Hydrogen and Deuterium.  
The targets
proposed are (or closely resemble) nuclear
targets commonly used in neutrino  experiments
such as the low statistics bubble chamber experiments on
Hydrogen, Deuterium, Neon, and  Argon and high statistics
neutrino experiments on Scintillator (MINERvA, MiniBONNE, K2K and JPARC), Water 
(SuperK, JPARC), Steel (MINOS) and Argon (CNGS). In addition
to 
performing studies of quark-hadron duality in electron scattering
on 
nuclear targets for the separated structure functions, these
data 
will used as input vector form factors in a future analysis
of 
neutrino data in order to investigate quark hadron-duality
in the 
axial structure function of nucleons and nuclei. This will be
done in collaboration with the MINERvA neutrino experiment~\cite{eois}
to be run in the NUMI~\cite{numi} low energy neutrino beam at Fermilab.

An immediate impact of these new measurements
with nuclear targets will be the reduction in
uncertainties in neutrino oscillation parameters for current and
near term neutrino oscillations experiments such as K2K and MINOS.
The data are even more important for the
more precise next generation neutrino
oscillations experiments such as JPARC and NUMI Off-axis.

Measurements will be made in the nucleon resonance region ($1<W^2<4$ 
GeV$^2$)
spanning the four-momentum transfer range $0.5<Q^2<4.0$ (GeV/c)$^2$ at
identical kinematic points to those to be run during JLab Deuterium 
experiment 02-109.
(Therefore, it is most efficient for both experiments to be run at
the same time).
The 
%separation 
decomposition
of the inclusive electroproduction cross sections into 
longitudinal
and transverse strengths will accomplished by performing Rosenbluth 
separations
to extract the transverse structure function $F_1(x,Q^2)$, the 
longitudinal structure function
$F_L(x,Q^2)$, and the ratio $R={{\sigma_L} / {\sigma_T}}$.  
%
%This 
%proposal represents the first
%global survey of these fundamental separated quantities in the 
%resonance
%region for nucleons bound in nuclei.

The analysis of the separated resonance region proton data from 
experiment
E94-110~\cite{e94110}, which was performed in Hall~C, has recently 
been completed and the excellent precision
obtained from that measurement is presented here.
The extension of the proton measurements to deuterium is approved and
will be run by E02-109~\cite{e02109}.
This 
proposal P04-001 (previously P03-110) represents the first
global survey of these fundamental separated quantities in the 
resonance
region for nucleons bound in nuclei.
%UPDATE
Data from all of the previous
experiments will be incorporated into the analysis.

Given the experience of previous experiments, the data taking
is straightforward and allows the precision
of the proposed measurements to be predicted with great confidence. 
In addition, the analysis
machinery which was developed for E94-110 can be used with only very 
little modification.
The great care and time invested in developing the experimental 
requirements, systematic uncertainty
measurements, and analysis machinery will be of immediate benefit to 
the proposed experiment.
%
We will also build upon the experience
from 
previous lower statistics studies with (electrons)  of the 
nuclear-dependence of the separated
structure functions in the DIS region that  were done by SLAC 
experiment E140~\cite{dasu}, as well as
the very low $Q^2$ high $W^2$ 
studies (to test the
HERMES effect) that were done by JLab experiment 
E99-118~\cite{e99118}.

\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{Low-Energy Electron and  Neutrino Scattering Overview}

\subsection{Form-Factors and Structure Functions}

 Several formalisms are used to discuss electron-nucleon and
  neutrino-nucleon scattering, and the corresponding reactions on 
  nuclear targets. 
  
  Inclusive lepton scattering can be described in the language of structure functions
  or in terms of form-factors for the production of resonant  
  final states. The two descriptions are equivalent and 
  there are expressions relating form-factors to structure functions.
  In electron scattering, the vector form-factors can be related to
  the two structure functions $W_1$ and $W_2$ (which are
  different for neutrons and protons), or equivalently  $F_{2}$ and $R$.
   
   
   In neutrino scattering, there are three structure functions $W_1$, $W_2$ and $W_3$
   (or $F_{2}$, $R$ and $xF_{3}$), different for neutrons and protons, and containing
   both vector and axial-vector components. There are also two other structure functions
   (important only at very low energies) whose contributions depend
   on the final-state lepton mass; these can be related to the 
   dominant structure functions within
   the framework of theoretical models.

\subsection{Electron versus Neutrino Scattering}

From the conservation of the vector current (CVC), the vector structure functions (or form-factors)
measured in 
electron scattering can be related to their counterparts in neutrino scattering for
specific isospin final states.  For elastic scattering from
spin-$1 \over 2$ quarks or nucleons, these relationships
between vector form-factors are simple. For production
of higher spin resonances, the relations are more
complicated and involve Clebsch-Gordon coefficients.

In contrast, the axial structure functions in neutrino
scattering cannot be related to those from electron scattering, except
in certain limiting cases (for example, within the quark/parton
model at high energies with V=A). At low $Q^2$, the axial
and vector form-factors are different, e.g. because of the
different interactions with the pion cloud around the
nucleon.

Another difference arises from nuclear effects
in inclusive neutrino vs. electron scattering. 
Nuclear effects on 
the axial and vector components of
the cross-section can differ due to shadowing, and
can also affect valence and sea quarks differently.

\subsection {Sum Rules and Constraints}

Several theoretical constraints
and sum rules can be
tested in electron and neutrino reactions (or
applied in the analysis of data).
Some of the sum rules and constraints are valid at all
values of $Q^2$, and some are valid only in certain limits.

The Adler~\cite{Adler} sum rules apply separately to the axial and vector parts of  $W_{1}$, 
$W_{2}$,  and $W_{3}$ and are valid for all values of $Q^2$ (since they are based
on current algebra considerations). At high $Q^2$, these sum 
rules are equivalent to the statement that the number of u
valence quarks in the proton minus the number of d  valence
quarks is equal to 1.

Other sum rules, such as the
momentum sum rule (sum of the momentum
carried by quarks and gluons is 1) and
the Gross/Llewelyn-Smith sum rule (number of valence quarks
is equal to 3), have QCD corrections and break down
at very low $Q^2$. 

At very low $Q^2$, 
vector structure functions are constrained by the
photoproduction cross-section. At high $Q^2$
it is expected that the structure functions
are described by QCD and satisfy QCD sum rules.

\subsection{Final States}

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

To describe specific final states, one can use the 
language of structure functions, combined with fragmentation functions,
at high values of $Q^2$. At low values of $Q^2$, many experiments describe
the cross-sections for specific exclusive final states.   Both of
these pictures need to be modified when the scattering takes
place on a complex nucleus.

\subsection{Current status of  Neutrino Cross Sections}

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. The current data has large
    uncertainties (from poor statistics and $20\% $ errors in the
    neutrino flux).
    
    
Since all current models for the various
components of the total neutrino cross section
are not totaly consistent with each other,
the predicted sum is constructed to be continuous in $W$ as follows.
For $W>2~\hbox{GeV}$  the Bodek-Yang model is  used 
(with the axial structure functions calculated~\cite{arie} with \mbox{$Z=0.5$}).
The Rein-Sehgal model is used used for $W<2~\hbox{GeV}$. In addition,
a fraction of the Bodek-Yang cross-section is added to the
Rein-Sehgal cross-section
between $W=1.7~\hbox{GeV}$ and $W=2~\hbox{GeV}$. The fraction increases linearly
with $W$ from 0 to 0.38 between $W=1.7$ and $W =2~\hbox{GeV}$.
These two figures also show the various contributions
to the neutrino and antineutrino total
cross-sections that will be investigated
in the next generation high statistics low energy
neutrino experiment such as MINERvA. 


 Quasi-elastic scattering and resonance
 production, either with neutrinos or with 
anti-neutrinos, provides
the largest  contributions  to the total $\nu$N event rate in the
threshold regime $E_{\nu} \le$~3 GeV. This is the region of interest
for current and next generation neutrino oscillations experiments.

For this reason, precise 
knowledge of the quasielastic and resonance production
cross sections, including the energy dependence and 
its variation
with target nuclei, is important to continued progress in current and 
future neutrino
oscillation experiments.   
The poor knowledge of the
neutrino total cross (divided by energy) shown in Figure~\ref{fig:total}
is similar to
the current state of knowledge of the quasielastic
reaction cross section.  
This is indicated by the compilations 
of previous
experimental measurements of
quasielastic cross sections with neutrino and antineutrino beams
on various nuclear targets as
shown in
Figures~\ref{elas_JhaKJhaJ_nu} and~\ref{elas_JhaKJhaJ_nub}.
Among the results shown there are normalization uncertainties 
originating with flux
uncertainties which are typically 10-20\%.  It is readily seen in 
these plots that
existing measurements have large errors throughout the $E_{\nu}$ 
range accessible with the next generation neutrino experiments such as
MINER$\nu$A (Fig.~\ref{elas_JhaKJhaJ_nu}, upper plot), and 
especially in the threshold
regime which is crucial to future neutrino oscillation experiments
(Fig.~\ref{elas_JhaKJhaJ_nu}, lower plot). 
Figure~\ref{elas_JhaKJhaJ_nub} shows this
problematic lack of precision to pervade the anti-neutrino 
quasi-elastic cross section
measurements as well.


The state of knowledge of resonance production in neutrino beams
is even worse.  Therefore, current neutrino oscillations experiment
use theoretical models which rely heavily on input from electron
scattering data on nucleons
and nuclei for the vector
contribution to all three contributions (quasielastic, resonance
and deep inelastic) to the total neutrino cross sections and
final states.  The axial contribution is now estimated using
models which are fit to low statistics neutrino data.
This is expected to continue to be the case
over the next five years. 

\subsection{Use in Neutrino Oscillations Experiments}

Current neutrino oscillations experiments compare the measured
neutrino cross section in a far detector to the expected
cross section as a function of neutrino energy. A $10\%$ dip
in the cross section as a function of energy is interpreted
as a disappearance of neutrinos originating from
oscillations of neutrinos from one flavor to another
(e.g. $\tau$ neutrinos which are below threshold).  The
amplitude of the dip gives the mixing angle, and the location
of the dip in neutrino energy gives the difference in mass
between the neutrino eigenstates.  The uncertainty in the
magnitude of the disappearance originates from the
uncertainty in the various contributions to the
neutrino cross section
as a function of neutrino energy in the
energy region between 0.5 and 5 GeV.

The uncertainty in the location of the dip originates from
uncertainties in the composition of the final states hadrons.
The response of neutrino detectors to quasielastic muons, recoil
protons, charged pions and neutral pions are all different.
Therefore, the total neutrino energy observed in a calorimetric detector
depends on assumed composition of the final state as
a function of neutrino energy. 

%figure 15
\begin{figure*}[p]
\centerline{\psfig{figure=elas_JhaKJhaJ_nu_tsush_g.eps,width=5.in,height=3.in}}
\centerline{\psfig{figure=elas_JhaKJhaJ_nu_2GeV_tsush.eps,width=5.in,height=3.in}}
%\centerline{\missingFigure}
%\centerline{\missingFigure}
\caption[Quasi-elastic neutrino cross-section and data]
{Compilation of experimental measurements of the neutrino 
quasi-elastic cross section.
  Existing measurements are characterized by large errors throughout 
the $E_{\nu}$ range
  accessible to MINER$\nu$A (upper plot), especially so in the crucial 
threshold regime
  (lower plot).  Representative calculations are shown using BBA-2003 
form factors with
  $M_A$=1.00 GeV.  The solid curve is without nuclear corrections, the 
dashed curve
  includes a Fermi gas model
  %~\cite{Zeller_03},
  and the dotted curve includes Pauli blocking and nuclear
  binding.  The data shown are from
  FNAL 1983
  %~\cite{Kitagaki_83}
  , ANL 1977
  %~\cite{Barish_77}
  ,BNL 1981
  %~\cite{Baker_81}
  , ANL 1973
%  ~\cite{Mann_73}
  , SKAT 1990
%  ~\cite{Brunner_90}
  , GGM 1979
%  ~\cite{Pohl_79}
  ,LSND 2002
%  ~\cite{Auerbach_02}
  , Serpukov 1985
%  ~\cite{Belikov_85}
  , and GGM 1977
 % ~\cite{Bonetti_77}. 
  }
\label{elas_JhaKJhaJ_nu}
\end{figure*}

%figure16
%\begin{figure*}[p]
    \begin{figure*}[htb]
\centerline{\psfig{figure=elas_JhaKJhaJ_nub_tsush_g.eps,width=5.in,height=3.in}}
%\centerline{\missingFigure}
\caption[Quasi-elastic anti-neutrino cross section and data]
{Experimental measurements of the anti-neutrino quasi-elastic cross 
section over the
  energy range accessible to MINERvA.  As with the neutrino 
quasi-elastic cross section
  results of Fig.~\ref{elas_JhaKJhaJ_nu}, there is large dispersion among the
  among the anti-neutrino measurements including systematic 
differences between individual
  experiments.   Theoretical expectations without (solid curve) and including
  nuclear corrections (dashed, dotted curves) are shown for comparison.
The data shown are from SKAT 1990
%~\cite{Brunner_90}
, GGM 1979
%~\cite{Armenise_79}
, Serpukov 1985
%~\cite{Belikov_85}
, and GGM 1977
%~\cite{Bonetti_77}
.}
\label{elas_JhaKJhaJ_nub}
\end{figure*}

Therefore, in addition to these proposed measurements
of inclusive structure functions on nuclear targets, the
composition of various hadronic final states in electron
scattering on Carbon will be also investigated
in parallel effort by members of this collaboration. 
This effort is lead by Professor Steve Manly from Rochester
and Dr. Will Brooks from Jlab, who
plan to investigate final states using existing Hall B  data on Carbon
in a similar energy range.

\subsection{Neutrino Near Detectors}

Given the large investment in the next generation neutrino oscillations
experiment (e.g. \$2B for JPARC and $250M$ for NUMI), neutrino 
oscillations experiments are taking a multiprong approach towards
predicting the expected event rates in far detectors.
 \begin{enumerate}
\item Reduce the error in the predicted neutrino flux by initiating
new hadron production experiments to precisely measure the yield of 
pions and kaons (which decay to neutrinos). These are expected to
provide a knowledge of the neutrino flux to a level of $1\%$ to $2\%$.
\item The electron and neutrino scattering communities are collaborating
in an experimental and theoretical program to measure and model lepton
vector and axial scattering on nucleons and nuclei. The
International Workshops on 
Neutrino-Nucleus Interactions in the Few GeV Region 
(NUINT~\cite{NUINT})
were started in 2001 to communicate progress
in this field.
\item New electron scattering experiments are being
initiated to investigate inclusive and exclusive processes on
nucleon and nuclear targets.
\item  Fully active near detector neutrino experiments (such as MINERvA) are 
being initiated to investigate neutrino reactions in detail.
\item  Small replicas of  the 
neutrino far detectors (which are not fully active and are constructed
of nuclear targets) are used as additional near detectors.
In principle, neutrino
flux, cross sections and nuclear effects
cancel in a comparison of rates between this kinds of
near detector and  the far detectors.
However, because of solid angle effects, the neutrino flux in these replica
near detectors is different from the neutrino flux in the far
detector, so the flux effects do not completely cancel. In
addition, because the two detectors are of such vastly
different size, detector effects do not completely cancel either.
\end{enumerate}

It is generally agreed that all of the above approaches are needed
for the next generation round of experiments.




\section{Physics Overview, Motivation and Goals}

As is described in the following sections,  the physics emphasis of
the next generation
neutrino experiments at low
energy such as MINERvA is two fold. (a)  First, the investigation of 
neutrino cross sections  and final states at 
low energies as a tool to study neutrino oscillations, and (b)
a complete investigation of vector and
axial structure functions and form factors on
nucleons and nuclei, sum rules and
 quark-hadron duality. The second part is expected
 to be an ongoing program culminating in the end of 2009
(when MINERvA is scheduled to complete its data taking). By
the end of this decade,  combined information from
electron and neutrino scattering can be used to investigate
the difference between vector and axial form factors and
structure functions on nuclear targets with very high
precision.


Members of 
this collaboration are proposing to investigate quark-hadron duality 
in high statistics electron scattering with the same nuclear targets 
first.  This will be followed by 
%performing 
a comparison the electron scattering
data with all 
existing neutrino data, with the aim of continuing these studies
in the future with 
the high statistics  neutrino experiment in the 
the MINERvA experiment at the Fermilab NUMI low
energy neutrino beam. The 
investigation of quark-hadron duality in both the axial structure 
functions
of nucleons and nuclei with both electrons and neutrinos is
complementary. {\it  Separation of vector and axial-vector components
in neutrino scattering is only possible if the vector part
is provided from electron scattering experiments as described below.} 

\subsection{Kinematics}

Due to the small value of the electromagnetic coupling constant, the
scattering of electrons
from nucleons can be well approximated by the exchange of a single
virtual photon, which carries
the exchanged 4-momentum squared, $q^2$.  In terms of the incident
electron energy, $E$, the scattered electron
energy, $E^{'}$, and the scattering angle, $\theta$, the absolute
value of the exchanged 4-momentum squared is
given by
\begin{equation}
(-q)^2 = Q^2 = 4EE^{'}{\sin}^2 \frac{\theta}{2}.
\end{equation}
In the one photon exchange approximation, the spin-independent cross
section for inclusive
electron-nucleon scattering can be expressed in terms of the photon
helicity coupling as
\begin{equation}
\frac{d\sigma}{d\Omega dE^{'}} = \Gamma\left[\sigma_T(x,Q^2) +
\epsilon \sigma_L(x,Q^2)\right],
\label{eq:cs1}
\end{equation}
where $\sigma_T$ ($\sigma_L$) is the cross section for
photo-absorption of purely
transverse (longitudinal) polarized photons,
\begin{equation}
\Gamma = \frac{\alpha E^{'}(W^2 - M_p^2)}{2 \pi Q^2 M_p E (1 - 
\epsilon)}
\end{equation}
is the flux of transverse virtual photons, and
\begin{equation}
\epsilon = \left[1 + 2(1+\frac{\nu^2}{Q^2}) {\tan}^2
\frac{\theta}{2}\right]^{-1}
\end{equation}
is the relative flux of longitudinally polarized virtual photons.

In terms of the structure functions $F_1(x,Q^2)$ and $F_L(x,Q^2)$, 
the double
differential cross section can be written as
\begin{equation}
\frac{d\sigma}{d\Omega dE^{'}} = \Gamma \frac{4{\pi}^2 
\alpha}{x(W^2-M_p^2)}
\left[ 2xF_1(x,Q^2) + \epsilon F_L(x,Q^2) \right].
\label{eq:cs2}
\end{equation}
Comparison of equations \ref{eq:cs1} and \ref{eq:cs2} shows that 
$F_1(x,Q^2)$
is purely transverse, while the combination
\begin{equation}
F_L(x,Q^2)= \frac{1+4M_p^2 x^2}{Q^2} F_2(x,Q^2)-2xF_1(x,Q^2)
\label{eq:fl}
\end{equation}
is purely longitudinal.  The separation of the unpolarized structure
functions into longitudinal
and transverse parts from cross section measurements can be
accomplished via the Rosenbluth
technique~\cite{rosen}, by making measurements at two or more values
of $\epsilon$ for
fixed  $x$ and $Q^2$.  Fitting the reduced cross section,
$d\sigma/\Gamma$, linearly in $\epsilon$,
yields $\sigma_T$ (and therefore $F_1(x,Q^2)$) as the intercept, and
the structure function ratio
$R(x,Q^2) = \sigma_L/\sigma_T = F_L(x,Q^2)/2xF_1(x,Q^2)$ as the 
slope.

Further information on electron and neutrino structure functions
is given in Appendix D. 

As mentioned earlier, in neutrino scattering separating vector and
axial structure functions is done  by relating the neutrino
 vector structure functions to the separated vector structure functions in 
electron scattering on  protons and neutrons via CVC and 
Clebsch-Gordan coefficients which depend on the contributions of 
different isospin amplitudes to the scattering. The relations 
between the vector structure functions are well determined for 
elastic (quasi-elastic) scattering on free nucleons for which the 
isospin of the initial and final states are well determined.  
Similarly, the relationships between the vector form factors are also 
well determined for the scattering on free quarks (i.e.the  high 
$Q^2$ DIS scattering) for which the isospin states of the initial and
final state quarks are also well determined. 
%UPDATE
In general, even in these well 
understood cases, in order to extract the axial structure function
in neutrino scattering experiments, the separated vector structure functions 
from electron scattering,  $F_{2p}$, $F_{2n}$, $R_{p}$, $R_{n}$
for bound nucleons are needed.

In the 
resonance region which includes contributions from final states with 
different isospins, the relationships between the vector form factors 
in neutrino and electron scattering is more complicated. Therefore, a 
unified investigation of quark-hadron duality for the same nuclear 
targets with electrons and neutrinos is interesting. 



High  
%statistical 
statistics
samples in neutrino experiments  
%(for nuclear targets 
%such such as iron and scintillator/carbon)  
on nuclear targets such as iron and hydrocarbons
will 
%become available
%over the next two years from
be measured beginning in 2005 in
experiments at the near detector hall at 
the 
%Fermilab-NUMI/MINOS beams.
Fermilab-NUMI beam 
%

A large unknown is the value of $R$  in the resonance region
for both vector and axial scattering on nuclear
targets in the resonance region.
 In addition, the axial part of $F_2$ and 
$xF_3$  are not well known. The nuclear
dependence of $R$ in the resonance region at low $Q^2$ and $W$
has not been measured in either vector or axial scattering.
Figure ~\ref{rupra} (left) shows the world's data on
$R$ for all nuclear targets in the
DIS region. Figure ~\ref{rupra}
(right) shows all available data on the nuclear
dependence of $R$ in the DIS region. The errors are
very large and none of the data are in the resonance 
region. Note that the data in Figure ~\ref{rupra} (right)  
were taken at high values of $W^2$ in the DIS region. At these 
 high $W$ and  $Q^2$ values, the overall value of $R$ is generally
quite small. Therefore,  nuclear effects cannot be discerned by the
these DIS data. 
The new measurements proposed here are at low $Q^2$, where $R$ is larger 
and a
nuclear dependence easier to discern. As discussed in
following secxtions, theoretical models
indicate that the largest nuclear dependence of $R$ is expected
to be in the resonance region at low  $Q^2$ (see figure~\ref{pionmiller}).
The lines in Figure ~\ref{rupra}
(right)  represent the small projected uncertainties of the new
measurements proposed by here P04-001 (previously P03-110).



Figures~\ref{rfl94110} and  ~\ref{2xf1f294110}  show
preliminary results of analysis
of 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 this proposal P04-001 (previously P03-110).
    
 


%\begin{figure}[htb]
%\centerline{\psfig{figure=r_update5.ps,width=2.91in,height=4.0in}}
%    \caption{A compilation of the world's published data on $R$ at high
%    $Q^2$ on both nucleons and nuclei.}
%\label{rupdate}
%\end{figure}


	      
%	      \begin{figure}[htb]
%\centerline{\psfig{figure=Keith_r_vs_q2.eps,width=2.91in,height=4.0in}}
%    \caption{Recent data from Jlab experiment E94-110 (on hydrogen) for
%R 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 P03-110.}
%\label{r94110}
%\end{figure}

   \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{figure}
\begin{center}
%remove old figure and put new one in,
    %\epsfig{figure=emc.eps,width=14cm,height=8cm,angle=0}
\epsfig{figure=NUC_emcfe_bodek.ps,width=14cm,height=6cm,angle=0}
\caption{A comparison of the ratio of cross sections of iron to 
deuterium in the resonance region versus the ratio measured in
the DIS region  versus
    the target mass variable $\xi$.
These preliminary data from JLab
indicate that the nuclear
effects in the DIS and resonance region are the same if plotted
versus the Nachtman variable as expected from duality.
Note that since these are ratios of cross sections
and not of separated structure functions. Therefore, it is
necessary to assume that there is no nuclear
dependence of $R$ in order to interpret these data. 
Shown are the ratio of nuclear to deuterium cross
section per nucleon, corrected for neutron excess. The solid
circles (red)  are Jefferson lab data (nucl-ex/0307012) 
taken in the $resonance$ $region$
(1.2 < W$^2$ < 3.0 GeV$^2$, Q$^2$ $\approx$ 4 GeV$^2$). The hollow diamonds (blue)
are SLAC E139 data, the crosses (purple) are the SLAC E87 data, and
the hollow squares (green)  are BCDMS data, all in the $DIS$ $region$.
}
\label{emc.eps}
\end{center}
\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}
%
%	      \begin{figure}[htb]
%\centerline{\psfig{figure=Keith_fl_vs_x.eps,width=2.91in,height=4.0in}}
%    \caption{Recent data from Jlab experiment E94-110 (on hydrogen) for $F_L$
%    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 P03-110.}
%\label{fl94110}
%\end{figure}

%	      \begin{figure}[htb]
%\centerline{\psfig{figure=Keith_2xf1_vs_x.eps,width=2.91in,height=4.0in}}
 %   \caption{Recent data from Jlab experiment E94-110 
 %   (on Hydrogen) for $2xF_1$ 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 P03-110.}
%\label{2xf194110}
%\end{figure}
%
%\begin{figure}[htb]
%\centerline{\psfig{figure=Keith_f2_vs_x.eps,width=2.91in,height=4.3in}}
 %  \caption{Recent data from Jlab experiment E94-110 
 %   (on Hydrogen) for $F_2$ 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 P03-110.}
%\label{f294110}
%\end{figure}

 
The $R_{1998}$ function provides a good
description of the world's data . However, at low $Q^2$ and low $W$ the nuclear
effects on $R$ are not known.

%\begin{figure}
%\begin{center}
%\epsfig{figure=ra.ps,width=14cm,height=8cm,angle=0}
%\caption{The SLAC E140 data set of $R_{A}-R_{D}$ on deuterium in the
%Deep Inelastic Region .}
%\label{fig1}
%\end{center}
%\end{figure}

The existing world's data measurements of the nuclear dependence of 
$R$  in the DIS region from SLAC Experiment E140~\cite{dasu} are
shown in Figure~\ref{rupra} (right).  As seen in the figure, the errors in the 
DIS region
are very large, and no data exists in the resonance 
region. Recent JLab data on tests of duality for $F_2$ in the resonance 
region~\cite{JlabEMC} at higher $Q^2$ is shown in Figure~\ref{emc.eps}.
Note that since these are ratios of cross sections
and not of separated structure functions. Therefore, it is
necessary to assume that there is no nuclear
dependence of $R$ in order to interpret these data. 
At large 
$Q^2$ the nuclear effects in the resonance region are the same as in
the DIS region for the same value of the Nachtman scaling variable. 
At lower values of $Q^2$ duality in the Nachtman variable breaks down
(as discussed in the next section),
and the nuclear effects in $F_2$, $2xF_1$ and $R$ could be very
different.

\section{The Nuclear Dependence of $R$ and its Relation to Nuclear 
Physics}

A measurement of $R$ in the region covered by
this experiment also provides information about the
dynamics of the internucleon force inside nuclei. For example,
new calculations by Miller~\cite{miller} 
predict a significant pion excess enhancement in the $\sigma_L^A / 
\sigma_L^D$ ratio at low $Q^2$ and moderate $x_Bj$, as shown in 
figure~\ref{pionmiller}. The kinematics of Miller's maximum pionic enhancement 
effects, which require a study of the A-dependence of the longitudinal cross 
section in the resonance region, fall within the proposed measurement. The 
question ``Where are the nuclear pions?'' has been an outstanding problem in 
nuclear physics for decades now. The belief that pions are the carrier of 
the nuclear force is a basic premise of nuclear physics, and yet is is not 
an experimentally verified one. While this premise {\it seemed} verified by 
the original electromagnetic scattering (EMC effect) experiments, it was 
later brought into question by deep inelastic Drell-Yan lepton-nucleus 
measurements. Further, no single theory of pionic nuclear binding has been 
able to reconcile all of the observations at all kinematics. As Miller 
concludes, the proposed new data will present ``an excellent opportunity to 
unravel a significant long-standing mystery involving the absence of nuclear 
pionic effects.''

\begin{figure}
\begin{center}
\epsfig{figure=newestpion.eps,width=10cm,height=8cm}
\caption{Plot from G. A. Miller (Phys.Rev. C 64,022201(2001)) 
%~\cite{miller} 
showing the predicted sensitivity 
of the inclusive longitudinal cross section ratio of Iron to Deuterium due 
to pion excess.}
\label{pionmiller}
\end{center}
\end{figure}

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

The expressions used in this
section are from  Appendix D.

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.  

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 (shown in Figure~\ref{rfl94110})
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. The data shown
in Figure~\ref{fig:total} span all of this range.


Therefore, new measurements of the inclusive structure functions
on nuclear targets with electrons
this energy region  can reduce the errors on
the vector cross sections by a factor of 10. This can be done
with about 5 days of running in Hall C, as described in the
rest of this proposal.


\section{Broad Impact of These Measurements on  Neutrino Physics}

A detailed study of $F_2$ and  $R=\sigma_L/\sigma_T$ on nuclear targets in the
resonance region is an important ingredient in forming an integrated
description of charged lepton and neutrino scattering cross-sections.
High rate neutrino beams now under construction or planned at Fermilab and
J-PARC will allow the first precision experimental comparisons of
electron and neutrino cross-sections and present and future neutrino
oscillation experiments will use these results to predict event rates.

The recent discoveries of neutrino oscillations in atmospheric
neutrinos~\cite{SK} and in neutrinos from the sun~\cite{SNO,KAMLAND}
motivate the detailed studies of neutrino oscillations at future high
intensity neutrino beams from accelerators.  The two disparate mass
scales observed in oscillations from these astrophysical sources,
$\dmsq_{\rms atm}\approx \evsq{2\times10^{-3}}$ and $\dmsq_{\rms solar}\sim
\evsq{10^{-4}}$, along with the stringent limits on $\nuebar$
disappearance at the atmospheric $L/E$ in the CHOOZ and Palo Verde
reactor experiments experiments\cite{CHOOZ,PaloVerde}, have raised the
possibility that there may be an observable CP-asymmetry in
$\numu\to\nue$ transitions.  This rare, sub-leading transition in the
neutrino flavor sector is analogous to to searching for first and
third generation mixing in the quark sector, which has led to a rich
phenomenology of CP-violation, meson mixing and rare decays in the
quark sector.

These neutrino oscillation experiments are very challenging, because
of the required $L/E$ of $400$~km/GeV, and require megawatt proton
sources, $1-3$~GeV neutrino beams and multi-kiloton detectors to make
the observations.  The measurements are further complicated by the low
transition probability of $\numu\to\nue$ and the need to compare to
$\numubar\to\nuebar$ at high precision.  This requires a detailed
knowledge of the neutrino interaction cross-sections both for the
dominant signal processes and for background processes, such as
$\nu N\to\nu N \pi^0$ where the $\pi^0$ is misidentified as an
electron in a many kiloton sampling detector.  

The phenomenology of neutrino cross-sections in relatively simple when
$E\nu\ll 1$~GeV or when $E\nu\gg$~few GeV since these regimes are
dominated by (quasi)-elastic and deep inelastic processes,
respectively.  However in the $1$ to few GeV region, there are
contributions to the cross-section from both of these processes as
well as resonance-dominated hadroproduction.  A successful
phenomenological approach to modeling the resonance region in electron
scattering is the use of quark-hadron duality to relate quark-model
cross-sections to the cross-section over the discrete
resonances~\cite{bodek-yang3} as shown in Fig.~\ref{fig:f1fit},
Fig.~\ref{fig:predict} and Fig.~\ref{fig:predictD}.
 Figure~\ref{fig:f1fit} shows a fit by Bodek and Yang to inelastic
 electron and muon scattering data with a modified scaling variable
 and GRV98 PDFs with additional corrections (based on consideration
of the Adler and Gilman sum rules~\cite{Adler}. 
Figure~\ref{fig:predict} and Figure~\ref{fig:predictD} compare
the predictions of the fit to data in the resonance region
(which is not included in the fit, as well as other data
%UPDATE
such a photoproduction and high energy neutrino data).  All 
predictions assume quark-model relations, and an empirical
fit to R ($R_{1998}$).
This approach requires the separation
of the $F_{2}$, which has a simple representation in the quark
model, and $R$ whose description requires a different
prescription.  We hope to build successful models of neutrino
scattering using this same prescription with the addition of a
quark-model representation of the axial vector component of the
cross-section.
As mentioned earlier, since neutrino data are
measured in nuclear targets, even in 
the quark model case,  the separated vector structure functions 
from electron scattering,  $F_{2p}$, $F_{2n}$, $R_{p}$, $R_{n}$
for bound nucleons are needed in order 
to understand the axial structure function
in neutrino scattering experiments.
%

The new precise data will also allow us to redo a combined analysis
of electron-nucleon and neutrino nucleon data in the resonance region
within the Feynman quark-oscillator model as done
years ago (with poor precision)  by Rein and Seghal~\cite{rs}, or 
use more sophisticated recent models (e.g. Sato and Lee~\cite{lee}.
In addition,
there is on-going theoretical efforts to include nuclear effects
(e.g. the Ghent group in Belgium~\cite{belgium} and the work of
Paschos and Sakuda~\cite{sakuda}.
The results
of the updated analysis within a resonance model can be compared to an analysis
which is based on duality.

The full program of studies requires first additional precise electron
scattering data, in particular $\sigma_L$ and $\sigma_T$ 
%UPDATE
(or equivalently $F_{2}$ and $R$)
%
on nuclear
targets (materials suited for future neutrino oscillation detectors --
water \cite{jhf}, hydrocarbons \cite{numi}, liquid argon -- and steel,
where the most precise high energy neutrino cross-sections have been
measured \cite{CCFR}) in the relevant kinematic regime.  Later, as the
new generation of high rate neutrino beams at Fermilab and J-PARC become
available, the approach can be directly validated with
comparisons to data from high rate neutrino cross-section experiments
on the same targets \cite{eois}.

With enough data to form a comprehensive modeling of
electron-nucleon/nucleus and neutrino-nucleon/nucleus interactions,
various sum-rules can then be tested. For example, the 
Adler~\cite{Adler} sum rules
which are based on current algebra, 
are expected to be valid over the entire range in $Q^2$ and
are expected to be exact. Other sum rules (mentioned earlier), which have
QCD corrections, can be compared with expectations from QCD.


\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=Fig1Mex.eps,width=6.0in,height=8.0in}
\caption{Electron and muon $F_2$ data (SLAC, BCDMS, NMC, H1 94)
used to obtain the parameters of the Bodek-Yang
modified GRV98 $\xi_w$ fit compared to the predictions of the unmodified GRV98
PDFs (LO, dashed line) and the modified GRV98 PDFs 
fits (LO+HT, solid line); [a] for $F_2$ proton, [b] for $F_2$ deuteron,
and [c] for the H1 and NMC proton data at low $x$.}
\label{fig:f1fit}
\end{center}
\end{figure}



\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=Fig2Mex.eps,width=6.0in,height=7.0in}
\caption{ Comparisons to proton and
iron data not included in the Bodek-Yang GRV98 $\xi_w$  fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2p}$ data
in the resonance region (or fits to these data)
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) the  Bodek-Yang modifications.
(b)  Comparison of photoproduction
       data on protons  to predictions using
 Bodek-Yang modified GRV98 PDFs. 
(c)  Comparison of representative CCFR $\nu_\mu$ and $\overline\nu_\mu$
charged-current differential cross sections 
on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT,
solid) and without (LO, dashed)
       Bodek-Yang modifications.  }
\label{fig:predict}
\end{center}
\end{figure}

\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=
Fig3Mex.ps,width=5.5in,height=3.2in}
\caption{ Comparisons to data on deuterium
which were not included in the  Bodek-Yang GRV98 $\xi_w$ fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2d}$ data
in the resonance region 
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) our modifications.
(b)  Comparison of photoproduction
       data on deuterium  to predictions using
Bodek-Yang modified GRV98 PDFs (including shadowing corrections). 
(c) The  shadowing corrections that were applied to the PDFs for
predicting the photoproduction cross section on deuterium.}
\label{fig:predictD}
\end{center}
\end{figure}





\begin{table}
    %[th]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
    $Target$ & $(Z, A), Z/A$     & $r.l. (g/cm^2)$   &  $r.l. (cm)$ \\
\hline
    &        &        &            \\
H &   (1,~1.00794),~ 0.99212  & 61.28& 866 \\
D &   (1,~2.01404),~  0.49652  & 122.4& 724 \\
C (approx CH2) &   (6,~12.011),~ 0.49954  &   42.7& 18.8 \\
Ne &   (10,~20.1797),~ 0.49555  & 28.94& 14.0 \\
Poly-CH2 & 0.53768 &43.72& 43.4 \\
Water-H2O & 0.55509 &36.08& 36.1 \\
Quartz-SiO2& 0.49926 & 27.05 &12.3 \\
O2(in Water, Quartz) &    (8,~15.999),~ 0.50002  &  &  \\
Si(in Quartz) &   (14,~28.0855),~  0.49848  &21.82  &9.36  \\
Al(Approx Si) &   (13,~26.9815),~ 0.48181  &24.01  &8.90  \\
Ar &   (18,~39.948),~ 0.45059  &19.55  &14.0  \\
Ca (approx Ar) &   (20,~40.078),~ 0.49903  &16.14  &10.42  \\
Fe &    (26,~55.805),~ 0.46556  &13.84  &1.76  \\
Cu (approx Fe) &    (29,~63.506),~ 0.45636  &12.86  &1.43   \\
\hline
\end{tabular}
\caption{Targets used in low energy low statistics neutrino 
experiments (H, D, Ne, Ar, Polystyrene Scintillator, Water,  Iron), 
and  additional targets
that may be used to approximate them (Carbon, 
Quartz, Silicon, Calcium, Copper). Note that future high statistics 
experiments are planned with 
%Polystyrene Scintillator
Hydrocarbon, Water and Iron targets.}
\label{table1}
\end{center}
\end{table}

%\section*{The Nuclear Targets and Kinematics of E02-109 and this Proposal}
\section*{Nuclear Targets and Kinematics}
Table~\ref{table2} shows nuclear
targets that we propose to add to the target ladder of experiment
% \section*{Kinematics of E02-109 and this proposal}
% Table~\ref{kin} lists the kinematics and cross sections proposed for
E02-109 

Table~\ref{table1} shows the kinds of nuclear targets that are
of interest to neutrino experiments. 
%
%that already 
%includes a Deuterium target and
%an Aluminum empty target replica. 
which already is approved to take data with a Deuterium target and with
an Aluminum empty target replica.
%

The differential cross sections for inclusive electron scattering
on Deuterium
will be measured in E02-109 according to the following definition:
\begin{equation}
{ {d^2 \sigma \over d\Omega dW^2 } = {{\Delta N}\over{\Delta \Omega
\Delta W^2}}{1\over{Qnd}}},
\end{equation}
where $\Delta N$ is the counts per $W^2$ bin, $n$ is the density of
deuterium, $d$ the target thickness, and Q is the integrated
number of incident electrons on target.
Table~\ref{kin} gives a breakdown of beam time requirements for 
E02-109 for the $\Delta$ running only. 
%We have added the
The
 last column  
%which 
shows 
%the 
additional time that is needed to measure
nuclear targets 
with the same number of events at each point as for Deuterium 
if this experiment  P04-001 (previously P03-110) were to run at the same time
%.
as E02-109.
%(if the number of events with each nuclear target is the
%same as that for Deuterium)

\begin{table}
   %[th]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
    $Target-Thickness~r.l.$& Running Ratio & $ Data $   &  $comment$ \\
\hline
    &        &        &            \\
$H-0.005$ &  2.0    & $All$~$Q^2$ & $Taken~in~E94-110$ \\
$D-0.005$ &  1.0  & $All$~$Q^2$  & $planned~by~E02-109$ \\
$Al-0.005$ &  $(5.1*)$   & $All$~$Q^2$ & $planned~by~E02-109$ \\
\hline
$C-0.06$ &  0.2    & $All$~$Q^2$ & $nominal$ \\
$Quartz-0.06$& 0.4   & $Q^2$~$<4$ & $nominal$ \\
$Si -  0.06$& 0.5  & $Q^2$~$<4$ & $nominal$ \\
$Ca- 0.06$& 0.6  &  $Q^2$~$<4$ & $nominal$ \\
$Fe-  0.06$& 0.7  & $Q^2$~$<4$ & $nominal$ \\
$Cu- 0.06$ &  0.8  & $All$~$Q^2$ & $nominal$ \\
\hline
$C -  0.02$&0.7   &  $All$~$Q^2$ & $Rad-Cor$ \\
$C  - 0.12$& 0.1  & $All$~$Q^2$ & $Rad-Cor$ \\
$Cu - 0.02$&  2.4 & $All$~$Q^2$ & $Rad-Cor$ \\
$Cu - 0.12$& 0.4  &  $All$~$Q^2$ & $Rad-Cor$ \\
$Fe - 0.02$&2.2  & $Q^2$~$<4$ & $Rad-Cor$ \\
$Fe - 0.12$&0.4   & $Q^2$~$<4$ & $Rad-Cor$ \\ 
$Water  -  0.02$& 0.8  &  $ Q^2$~$<4$ & $not used$ \\
\hline
\end{tabular}
\caption{ Targets to be used  in this proposal (). Running Ratio is the ratio 
of the running time  
relative to running with a $4$~cm~($0.005$~r.l.) Deuterium target
to get the same number of events (at constant  current of  80 $\mu$A,
which is planned for E02-109).  Note that data 
with Hydrogen is already available from JLab experiment E94-110, and 
Deuterium and Aluminum data 
need need NOT be taken if this experiment () 
runs at the same time as E02-109 (thus minimizing the error on the 
ratio of structure functions on nuclear targets to deuterium). All 
$Q^2$ identifies targets for which data will be taken at all values 
of $Q^2$, and $Q^2$~$<4$ identifies targets for which no high $Q^2$ 
data will be taken.  The nominal targets are identified. Rad-Cor 
indicates targets to be used for Radiative Corrections Studies,
and 
TBI indicates Option To Be Investigated. $*$ Note that the Al target is
only used for empty target, and therefore statistics
equal to D2 are not needed.}
\label{table2}
\end{center}
\end{table}

\subsection{Details of Event Rate Calculations}

A minimum time of one half hour per kinematic setting, a maximum rate
of 1000 Hz, and a beam current of $80$ $\mu$A were used in the 
calculation of the running time for Deuterium for E02-109. 
The calculated rates listed are for bins of width $\Delta W^2 = (25
\rm MeV)^2$ averaged over a momentum acceptance of $\pm 8 \%$
and assuming an effective solid angle of the HMS of 6.5 msr at the
$\Delta$ resonance.  The data for the higher resonances comes
in at higher rates and requires $\approx 50\%$ additional beam time.
The data acquisition time listed in Table~\ref{kin} reflects
the total time required for the $\Delta$ resonance only.  The SOS
will be used to collect positron yields (from neutral pion
production) for charge symmetric background studies and will be run
in a simultaneous single arm mode with the HMS data and, so,
this adds little time to the beam time
request.  The extraction of neutron structure functions
from D$_2$ data for E02-109 requires the
subtraction of the proton data taken in E94-110.  As a cross
check on relative normalizations, E02-109 plans to do measurements of
the hydrogen resonance region cross sections for
$Q^2$ $<$ 3.0~$\rm (GeV/c^2)^2$ to compare to E94-110.  This takes
 an additional 8 hrs.
A minimum central
spectrometer momentum setting of 400 MeV/c was assumed. All proposed
measurements with deuterium in 
E02-109 use the Hall C 4 cm deuterium target.  In
addition, E02-109 plans to take
elastic proton data  at all angles and
energies for kinematic uncertainty checks.



\begin{table}
    %[tbhc]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$Q^{2}_{\Delta}$  &  $E$  &  $E^{'}_{\Delta}$ & $\theta_{\Delta}$ &
${\epsilon}_{\Delta}$ & $Rate_{\Delta}$ & D$_2$-Time &Nucl. Tgts \\
$(GeV/c)^2$ & ($GeV$) & ($GeV$) & (deg) & & (Hz) & (Hours)&(hours) \\
	        &      &      &         &     &    & E02-109 &  P04-001\\
\hline
0.5  & 1.16 & 0.55 & 52*      & 0.54   & 1 K  &   0.5&      \\
        & 1.64 & 1.0  & 33      & 0.78   & 1 K  &   0.5 &     \\
        & 4.04 & 3.4  & 11      & 0.97   & 1 K  &   0.5&      \\
1.0  & 1.64 & 0.77 & 52*      & 0.53   & 1 K  &   0.5&      \\
        & 2.28 & 1.4  & 33      & 0.77   & 1 K  &   0.5&      \\
        & 4.52 & 3.6  & 14      & 0.95   & 1 K  &   0.5&      \\
2.0  & 2.28 & 0.87 & 60*      & 0.43   & 65   &   0.5 &     \\
        & 3.24 & 1.8  & 35      & 0.73   & 285  &   0.5&      \\
        & 5.64 & 4.2  & 17      & 0.92   & 1 K  &   0.5 &     \\
3.0  & 3.24 & 1.3  & 52*      & 0.51   & 16   &   2 &       \\
        & 4.04 & 2.1  & 35*      & 0.70   & 40   &   1&        \\
        & 5.64 & 3.7  & 22      & 0.86   & 172  &   0.5&      \\
\hline
&      &      &         &        &  sub-total   &  8 & 24   \\
\hline
	4.0  & 3.24 & 0.77 & 79*      & 0.23   & 1    &   22&       \\
        & 4.04 & 1.6  & 47*      & 0.51   & 3    &   8  &      \\
        & 5.64 & 3.2  & 27      & 0.77   & 53   &   1   &     \\
5.0  & 4.04 & 1.0  & 66*      & 0.29   & 1    &   22  &     \\
        & 4.52 & 1.4  & 52*      & 0.42   & 3    &   8 &       \\
        & 5.64 & 2.5  & 35      & 0.66   & 6    &   4 &       \\
\hline
&      &      &         &        &  sub-total   &  65 & 40  \\
\hline       
&      &      &         &        &      &  Total D  &  Total-Nucl\\
        &      &      &         &        &      &   \bf{73} &  \bf{64}\\
	        &      &      &         &     &    & E02-109 &  P04-001\\
\hline
\hline
\end{tabular}
\caption{D$_2$ Running for E02-109 Beam 
time requirements for all proposed measurements, as in the
E02-109 proposal.  All kinematics
and rates shown are for
a single bin in $W^2$ of 25 $(\rm MeV)^2$ width at the $\Delta$
resonance.  Positron data will be taken
in the SOS for the angles indicated by an asterisk. The last column is
the additional time needed for heavy targets for this proposal: 6 targets at
(C, Quartz, Si, Ca, Fe and Cu) $Q^2<4$ and
2 targets (C and Cu) at all $Q^2$. The beam energies in this table
differ slightly and inconsequentially from those in this proposal
text, as the original E02-109 energies were slightly disparate
from the actual common JLab energies. The text energies are the
correct ones for both experiments.}
\label{kin}
\end{center}
\end{table}


Note that the assumption made in the calculation of the rates
for E02-109  are conservative. The 80 $\mu$A current is assumed to
minimize density difference in the deuterium target.  For some of the
solid targets in the proposed experiment P04-001 (previously P03-110) a higher current 
of 100 $\mu$A 
can be used (especially at high $Q^{2}$).
Similarly, a data taking rate of 1000 Hz is also 
conservative and data can be taken with 2000 Hz (and possibly
at 3000 Hz with some upgrades).  

The chosen beam energies in the table assume a linac energy of 1.12
GeV (1.18, 2.30, 3,42, 4.54, 5.66), with the exception of
two 
%bean 
beam energies which assume a linac energy of 0.80 GeV (1.64,
4.04).  Both base energies are standard CEBAF accelerator tunes.
The required beam time was determined such that the statistical
accuracy per $W^2$ bin was $\approx$ 3 times greater than
the systematic point-to-point accuracy expected.  The rates were
estimated based upon a fit of previous deuterium resonance region
cross section data from JLab~\cite{ioana}.
%
Using the parameters in Table~\ref{table2} we can scale the 
running time for Deuterium and obtain the corresponding
additional running time for the nuclear targets (shown in
the last columns of Table~\ref{kin} and  Table~\ref{time}.
%
The total  beam time requested for both  E02-109 (D$_2$) and for this experiment 
P04-001 (prevously P03-110) (last
column Nucl. Tgts) are listed in Table~\ref{time}). 

For 
%D2 
D$_2$
running for E02-109
the total
data acquisition time listed
%, 
reflects the total time from
Table~\ref{kin}, as well as
an additional 40 hours to complete the higher resonances, an 
%addition
additional
60 hours for dummy runs which are
needed to subtract the yield contributions from the aluminum end caps
of the target, 
an additional 24 hours needed to complete the hydrogen elastic
scattering measurements, and an additional 16 hours
to obtain hydrogen resonance region data which is needed for
cross checks with E94-110.  Also,
since the positron data comes in at a slower rate, E02-109 requested an
additional 22 hours to complete these
measurements.  The E02-109 proposal assumes
one-quarter hour for each angle change
required at a given beam energy, and
one-quarter hour for each spectrometer central momentum change not
possible to be done concurrently
with angle changes.  Combined with one day for checkout, the 
total beam time approved for E02-109 is 13 days~\cite{hallc}.

This experiment - P04-001(previously P03-110) requests an additional 5 days of running to do the
measurements with the nuclear targets.  With 2.5 days of additional 
running, data with all six targets can be measured for all of
the E02-109 data points with 
%$Q2<4$. 
$Q^2<4$. 
The additional 2.5 days are
requested for the measurements at the higher $Q^2$.  
%Therefore,
%a significant amount of physics can be done even with a
%a 2.5 day extension.

\begin{table}
    %[tbhc]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
    & D- Time Required &Nucl. Tgts. \\
    &  (Hours) &  (Hours)\\
     & E02-109& P04-001 \\
\hline
Data acquisition (Deuterium $\Delta$)/+Nucl. Tgts. & 73 & 64 \\
Data acquisition (Deuterium $W^2$ $>$ $\Delta$)/+Nucl. Tgts. & 40& 38 \\
Data acquisition (Dummy)          &  60 & \\
Data acquisition (hydrogen elastics) & 24 & \\
Data acquisition (hydrogen resonance region) & 16 & \\
Data acquisition (additional positrons) & 22 & \\
D Angle changes (12)/+Nucl. Tgt Changes   & 3 & 10 \\
Spectrometer momentum changes (60) & 15 & \\
Major beam energy changes (1) & 8 & \\
Minor beam energy changes (5) & 20 & \\
D Checkout /+ Nucl. Rad correction Tests& 24 &10 \\
\hline
\bf{Total} & \bf{305}& \bf{120} \\
 & E02-109& P04-001 \\
\hline
\end{tabular}
\caption{D$_2$ Running for E02-109: Breakdown and tabulation of the total time requested.  Based
on previous experience, we assume
one-half hour for angle changes, 15 minutes for momentum changes,
eight hours for linac energy changes (major), and four
hours for each energy change accomplished by changing the number of
cycles (minor). The last column is the additional time required for 
heavy targets in this proposal P04-001 (previously P03-110)}
\label{time}
\end{center}
\end{table}

%\clearpage



\section{The Collaboration}

%The collaboration consists of E02-109 experimenters. 
The collaboration consists primarily of members of the current
E02-109 collaboration, with the addition of collaborators from
the MINERvA, NuTeV and MINOS Neutrino Experiments at Fermilab.
These 
scientists have participated in a substantial amount of Hall C running.
The collaboration has implemented and proven successful techniques to
reduce systematical uncertainties in Hall C experiments, including
detailed studies of spectrometer optics, spectrometer survey studies,
raster phase analysis, and additional beam line
instrumentation. This collaboration has the on-site experience, knowledge and
expertise requisite to perform a precision measurement of the type proposed.
These experimenters include  spokespersons of the previous JLab
%UPDATE
experiments on which this proposal is founded (E94-110, E02-109 and 
E99-118). 

This proposal brings a significant number of 
%the 
new collaborators from 
%Rochester and American University
the University of Rochester and University of Massachusetts,
%include 
including
the two spokespersons of SLAC experiment E140 and E140X
(Bodek and Rock), 
the Rochester NuTeV neutrino group, and
the Rochester Nuclear Physics group (Manly).   A significant portion of the collaboration
plans to continue these studies with the MINERvA neutrino
experiment at Fermilab (Rochester, Hampton, Jlab, and Tufts), including
two of the Spokespersons of MINERvA (Keppel and McFarland).

Steve Manly (Rochester) and Will Brooks (Jlab) will also be looking at
     existing Hall B CLAS data at Jefferson Laboratory to
study hadronic final states in electron scattering
on nuclear targets
(e.g. Carbon), and in particular final state in
the quasielastic and resonance regions. 

Two potential Rochester 
%PhD
Ph.D.\ 
students are shown as well as Rochester theorists who have expressed
interest in the results.

We are also in close contact with The Ghent (Jan 
Ryckebusch~\cite{belgium} nuclear theory
group in Belgium, and the Argonne nuclear theory group (H. 
Lee~\cite{lee} 
who are developing models 
that are applicable to both electron and neutrino reactions
on d nuclei and nucleons.
We plan to  use these theoretical tools to do a combined
analysis of the inclusive data from this experiment P04-001 
(previously P03-110) on Cargon,
the Hall B data  exclusive electron scattering data
on Carbon, and the MINERvA neutrino data
on Carbon. 
In addition,
there are other theoretical efforts (e.g. Sakuda and Paschos) on
 nuclear effects for the hadronic final states in the region
 of the first resonance.


\section{Conclusion}

Using the existing Hall C apparatus, JLab experiment
E02-109 has been approved to perform a global
survey of L-T separated unpolarized
structure functions on deuterium throughout the nucleon resonance
region with an order
of magnitude better precision than has been achieved before.  The
recent analysis of the proton data from E94-110
clearly show that these goals are both realistic and attainable.
Furthermore, the analysis machinery previously
developed can be used nearly without modification and should allow
the analysis of the data to proceed in an
accelerated fashion.

Here we propose  another experiment P04-001
%(E02-xxx) 
(previously P03-110)
that in 
only five days of
additional running
yields a substantial amount of new data with nuclear targets.
These data 
%will launch 
are a key ingredient in a new program 
linking the nuclear and high-energy physics communities
%to  investigate 
in investigating
quark-hadron duality in nuclear
targets using both electron and neutrino beams.

An immediate impact of these new measurements
with nuclear targets will be the reduction in
uncertainties in neutrino oscillation parameters for current and
near term neutrino oscillations experiments such as K2K and MINOS.
The data are even more important for the
more precise next generation neutrino
oscillations experiments such as JPARC and NUMI Off-axis.


\section*{Appendix A - Radiative Corrections Studies}

\subsection*{Thick Radiator Effects}

%SLAC Experiment
SLAC-E140~\cite{dasu} and SLAC-E139~\cite{E139} have performed tests with 0.02, 0.06 
and 0.12 radiation length targets.  Figure~\ref{radcor.eps}
illustrates that the radiative corrections can be well understood
even for these relatively thick targets using the analysis techniques
of E140.
%below shows that with
%the improved SLAC-E140 radiative corrections, 0.06 radiation length
%targets can be used.   
In this experiment, we add a variety of
target thicknesses to repeat these radiative corrections tests in
the JLab kinematic region.

In addition, in order to obtain more statistics
with Aluminum in E01-109
(using the Aluminum Empty Target runs), we plan to go back to using
 thick empty targets, 
%as originally
% invented 
a technique pioneered
at SLAC by Bodek~\cite{radcorbodek}. 
The Aluminum Empty Target
replica is made thicker to match the radiation length of the  0.005
r.l. Deuterium target.   This greatly increases the counting
rate for the empty target replica, as well as making sure that
the radiative corrections for the empty target are indeed the
same as the correction for the full Deuterium target. Therefore,
if this target is used instead, then the 
60 hours allocated for empty target replica by E02-109 can yield 
Aluminum data with about 1/6 of the statistics of the running with
deuterium.
 
\begin{figure}
\begin{center}
\epsfig{figure=radcor.eps,width=20cm,height=14cm,angle=0}
\caption{Radiative Corrections tests from SLAC E140 and E139.}
\label{radcor.eps}
\end{center}
\end{figure}

\subsection*{Two Photon Effects}

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 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}.


For elastic scattering the ratio $R$ is related
to the ratio of 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 the mean  $Q^2$ of this experiment of 2.5 Gev$^2$ 
the data in figure~\ref{show_gepgmp} 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.
 {\it The uncertainty from
two-photon effects in the inelastic
radiative corrections is actually lower
than this estimate because modern radiative corrections programs
for inelastic electron
and neutrino scattering  (e.g. Bardin)
already include two photon effects at the parton level.}

%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}



\section*{Appendix B - Practical Issues in Choices of Targets}


A consideration in the choice of targets is the maximum amount
of current that can be delivered without damaging or changing the
density of the target. This experiment as proposed can be done in 5 days
  using standard targets, but we have made some compromises and
  may wish to consider future alternatives, as outlined below.

  At present 0.02, 0.06 and 0.12 r.l. C, Fe and Cu targets already
  exist in Hall C.  The current that the Fe target can withstand depends on the thermal
  contact with the water cooled frame. 
%With a proper job one can
%  allow up to 60 $\mu$A on the Fe targets. In practice, if one does
%  not want to bank on how proper the job of thermal contact is
%  done, a safer limit may be 30-40 $\mu$A.  
  Safe current limits will likely be in a range $30$--$60\mu$A, 
  depending on the our confidence in this thermal contact.
  In contrast, C and Cu can easily withstand 80 $\mu$A,
%, probably 100 $\mu$A is also
%  reasonable
%  (if one uses the proper type of Carbon). 
%   This is why 
and therefore those targets
    were chosen for the high $Q^2$ running.
  
%Since a scintillator target can only withstand 0.5 to 2  $\mu$A, we
%do not propose to use such a target in this experiment. 
The best measurement of cross-sections on water would come from use
 of the Hall A ``waterfall'' target, which consists of
three "free-falling" water
  curtains, each of $250$~mg/cm$^2$
and can probably withstand 
$100$~$\mu$A.  Issues to be investigated with this target include
density variation with current and whether it is practical to add
this target to the Hall C scattering chamber.
% (but one needs
%  to check its density variation vs. current).
%The current Hall A waterfall target consists of  The Hall A version is
%  standalone, i.e. no possibility to combine with other targets.
%  It may be possible to add waterfall target to the Hall C scattering
%  chamber, but it would require some thinking. 
Therefore, this
  kind of target is not part of the proposed run plan, but is
  a subject of investigation.
  
   The  existing Hall C Calcium target may have a current limitation
  of 30 $\mu$A.  We plan to investigate if a new Calcium sandwich target may be
  able to handle higher current.
  





\section*{Appendix C -Other Electron Data that will be used in our analysis} 

In addition to E94-110 (Hydrogen) and
E02-103 (Deuterium), we also plan to use previous low $Q^2$
data in the inelastic region (e.g. Jlab  experiment~\cite{e99118} E99-118),
previous data from experiment  E02-109 taken in the
quasielastic region~\cite{quasi}
(for radiative corrections),
and also high $Q^2$ data from experiment E02-103 (hydrogen and
deuterium  quasielastic and resonance $F_2$ data) 
approved by Jlab PAC24 to run in 2004
(J. Arrington, spokesperson).
Other lower precision  high $Q^2$ data such as SLAC experiment E140 
and E140x on nuclear
targets will also be included in the overall analysis.

\section*{Appendix D: 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 (shown in Figure~\ref{rfl94110})
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. The data shown
in Figure~\ref{fig:total} span all of this range.

\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}


\begin{figure}[htb]
\centerline{\psfig{figure=kinematics_full.eps,width=4.0in,height=3.in}}
\caption{Kinematic coverage of E02-109 and P04-001}
\label{kinematics_full}
\end{figure}

\section*{Appendix E -Reply to PAC Questions} 


\subsection*{Question 1}

Delivered-To: bodek@pas.rochester.edu

Cc: bodek@pas.rochester.edu, nicola.bianchi@lnf.infn.it

From: "Peter Kroll" <kroll@physik.uni-wuppertal.de>

To: keppel@jlab.org

Organization: University of Wuppertal - Department of Physics

date: Fri, 26 Dec 2003 11:28:48 +0100

Dear Cynthia, 

I have read your proposal PR 04-001 and found it fine in general. I would
however like to refer to the issues mentioned in the PAC24 report on your
proposal. 
Although I appreciate your estimates of the sensitivity of the (anti) neutrino 
cross section to a change in R, it would be helpful in my opinion to provide
some more details on this estimate. Thus, for instance, your statement on the 
errors of the vector cross section is not easy to understand.

Best regards, Peter Kroll 

\subsection*{Reply 1}

From: Arie Bodek <bodek@pas.rochester.edu>

Date: Fri, 26 Dec 2003 09:48:05 -0500 (EST)

To: Peter Kroll <kroll@physik.uni-wuppertal.de>

Cc: <keppel@jlab.org>, <nicola.bianchi@lnf.infn.it>,
	Arie Bodek <bodek@pas.rochester.edu>


Dear Peter

   Thanks for your Email.  The appendix on PDFs (Appendix D) gives a  derivation
of the sensitivity of the vector cross sections to R, in the low
energy limit where the number of antiquarks is small. We can
 update the derivation to include the fraction of antiquarks
say $10\%$ at high energy (which will have little  effect on the results.

 Please clarify what additional details beyond those provided in
the appendix would be helpful.

Best regards
  Arie Bodek

\subsection*{Question 2}

Delivered-To: bodek@pas.rochester.edu

Date: Mon, 29 Dec 2003 14:44:14 +0100 (MET)

From: Nicola Bianchi <Nicola.Bianchi@lnf.infn.it>
To: Arie Bodek <bodek@pas.rochester.edu>

Cc: Peter Kroll <kroll@physik.uni-wuppertal.de>, keppel@jlab.org,
	nicola.bianchi@lnf.infn.it
	

Dear Arie and Cynthia,

I have few questions on the determination of neutrino
cross section from electron scattering data.

1 - you mention that for (quasi)elastic scattering and DIS
the relation is simple for the vector part. 
It is not clear how this relation can be establish in the
nucleon resonance region where different resonances and
non-resonant background overlap and mix so that the final
state cannot be identified. 
In this respect how you can verify and use the duality
in quantitative way to extract the neutrino cross section? 

2 - nuclear effects like the ones from pion excess or
from photon shadowing at high energy may affect the 
electron scattering data.
How you will consider these effects for the determination of
the neutrino-nucleus cross section?

3 - it will be nice to see a kinematic plot ($Q^2$ vs $\nu$) of your
proposed measurement.

Best regards and best wishes for a Happy New Year,

	Nicola


\subsection*{Reply to question 2, item 3: Kinematics plot}

Date: Tue, 6 Jan 2004 15:56:34 -0500 (EST)

From: Cynthia Keppel <keppel@jlab.org>

To: Nicola Bianchi <Nicola.Bianchi@lnf.infn.it

Cc: ksmcf@pas.rochester.edu, kroll@physik.uni-wuppertal.de,
	bodek@pas.rochester.edu, manly@pas.rochester.edu
	
Subject: kinematics plot

In reply to '' 3 - it will be nice to see a kinematic plot ($Q^2$ vs $\nu$) of your
proposed measurement.''

Dear Nicola,

As Arie noted, we will get back to you while he is here at JLab this week.
In the meantime, though, I thought the attached file might be somewhat
useful to you. These are the ($W^2$, $Q^2$) kinematics of the completed
hydrogen L/T experiment. Our current proposal, and the deuterium proposal
E02-109 which it proposes to run concurrently with, will overlap the
hydrogen kinematics. 

In  Figure~\ref{kinematics_full}, the points are the central spectrometer kinematics, while
the lines are meant to indicate the "scans" in $W^2$ at fixed central 
scattering angles, accomplished by changing the spectrometer momenta. The
actual data extend at the low / high $W^2$ ends out to the elastic and deep
inelastic, respectively, as well. Overlapping ($Q^2$, $W^2$) points are at
differing epsilon values.

Happy New Year, Thia



\subsection*{Reply to question 2, items 1 and 2}

In reply to:''  1 - you mention that for (quasi)elastic scattering and DIS
the relation is simple for the vector part. 
It is not clear how this relation can be establish in the
nucleon resonance region where different resonances and
non-resonant background overlap and mix so that the final
state cannot be identified. 
In this respect how you can verify and use the duality
in quantitative way to extract the neutrino cross section? ``

This comparison needs to be done within a framework of theoretical 
models for nucleon resonances. For example, Rein-Sehgal
~\cite{rs}, Sato-Lee~\cite{lee} and others.   A good model to use in the region
of the first resonance
is the model of Sato and Lee~\cite{lee}, which fits final state 
angular distributions in electron scattering data for the vector part
of both resonance and non-resonant parts, and makes a simultaneous
prediction for the axial vector part. The axial vector part is 
predicted via a meson cloud model for the first resonance.
In that model, $R$ is not zero, because of the meson cloud,
(in agreement with recent data), and the predictions from electron
scattering to neutrino scattering are well defined.

In the region of
higher resonances, one needs to compare the data to predictions based on a 
model that includes all the resonances (refitted to electro-production 
data) such as an extension of the Sato-Lee model, an 
Rein-Sehgal~\cite{rs} model (updated to include
recent electrproduction data) or other resonance models. 

In order to have a complete understanding of the vector
and axial parts, one needs to compare to high precision neutrino 
data (which will be taken at Fermilab by the MINERvA collaboration with
a Carbon target).


In reply to `` 2 - nuclear effects like the ones from pion excess or
from photon shadowing at high energy may affect the 
electron scattering data.
How you will consider these effects for the determination of
the neutrino-nucleus cross section?''

Since CVC holds for both nucleon and nuclear
targets, the nuclear effects for the vector part are expected to be the
same for the vector part of electron scattering and neutrino
scattering (note that in both electron
and neutrino scattering  the nuclear effects
for the vector-longitudinal and the vector-trasverrse cross
sections are not be the same).

The primary reason that the 
nuclear affects can be different in neutrino scattering is  because
such effects 
can be different for the axial and vector contributions (since the axial current 
coupling to the pion cloud is different).  Therefore,  both electron 
scattering and neutrino scattering data are needed.  Note that neutrino
data by itself is not sufficient.  If one mistakenly uses incorrect 
vector form factors, one extracts incorrect axial form factors from 
the neutrino data, which has a large impact on the neutrino cross 
section.  

Let me refer to a recent illustrative
example. The K2K collaboration has taken data
with a near neutrino detector. These 
are in the 1 GeV region where the cross section is dominated by 
quasielastic events.   In order to fit their quasielastic $Q^2$ 
distribution, they had to change the axial form factor of the proton in
their Monte Carlo by a large factor.  They thought that this was due to a 
large nuclear binding correction the nucleon axial elastic form factor.

They were indeed able to modify the axial form factor to fit the
near detector neutrino data (as shown in Figure~\ref{k2k1.eps}. However,
this also resulted in a change of about $15-20\%$ in the predicted neutrino
cross section and its energy dependence in the far detector.

I was at the NuInt02 conference when they presented these results.
I realized that a large part of the
the difference in the $Q^2$ distribution between their
data and Monte Carlo, was because they were using outdated vector
elastic  form factors.  They used pure dipole form factors, and a value of
zero for the electric form factor of the neutron. (i.e. they used
an incorrect value of $R_{elastic}$. 

This was subserquently shown in detail in a later
 paper by Bodek, Budd and Arrington. By using modern
up-to-date vector elastic form factors for the proton and the
neutron, a major systematic error has
been removed in the K2K analysis.

For details, see  Figure~\ref{k2k1.eps}, Figure~\ref{k2k2.eps},
and Figure~\ref{k2k3.eps} (attached after ther 
reference sectino).  These three figurs correspond to
pages 8, 9 and 10 of the responses to the
questions of the Fermilab PAC to the MINERvA collaboration posted
at $http://www.pas.rochester.edu/~ksmcf/minerva/$. The PDF file is
$http://www.pas.rochester.edu/~ksmcf/minerva/PAC-Question-Response.pdf$

This example illustrates how important is it to have precise vector (elastic,
resonance, and inelastic) form
factors in order to
properly model neutrino cross sections, even when the neutrino events
are measured in both a near and a far neutrino detector.



%
\section*{References} 

%\vfill
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\begin{figure}[htb]
\centerline{\psfig{figure=k2k1.eps,width=7.0in,height=8.in}}
\caption{Effect of incorrect vector form factor on K2K neutrino oscillations 
analysis Slide A.}
\label{k2k1.eps}
\end{figure}

\begin{figure}[htb]
\centerline{\psfig{figure=k2k2.eps,width=7.0in,height=8.in}}
\caption{Effect of incorrect vector form factor on K2K neutrino oscillations 
analysis Slide B.}
\label{k2k2.eps}
\end{figure}

\begin{figure}[htb]
\centerline{\psfig{figure=k2k3.eps,width=7.0in,height=8.in}}
\caption{Effect of incorrect vector form factor on K2K neutrino oscillations 
analysis Slide C.}
\label{k2k3.eps}
\end{figure}

\pagebreak
\end{document}
\bye


