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\begin{document}
\title{Unified approach for modelling neutrino and electron
nucleon scattering cross sections
from high energy to very low energy}
\author{Arie Bodek}
\address{Department of Physics and Astronomy, University of Rochester,
Rochester, New York 14618, USA}
\author{Un-ki Yang}
\address{Enrico Fermi Institute, University of Chicago,Chicago,
Illinois 60637, USA}
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\maketitle
\abstracts{
We use a new scaling variable $\xi_w$, and add low $Q^2$
modifications to GRV98 leading order
parton distribution functions such that
they can be used to model electron, muon and
neutrino inelastic scattering cross sections (and
also photoproduction) at both
very low and high energies.
}
In a previous communication~\cite{first}
we used a modified scaling variable $x_w$ and fit for modifications
to the GRV94
%~\cite{grv94}
leading order PDFs such that the PDFs
describe both high energy low energy e/$\mu$ data. In order to describe
low energy data down to the photoproduction
limit ($Q^{2} = 0$), and account for both target mass and higher twist effects,
the following modifications of the GRV94 LO PDFs are need:
%
%
%
%
\begin{enumerate}
\item We increased the $d/u$ ratio at high $x$ as described in our previous
analysis~\cite{highx}.
\item Instead of
the scaling variable $x$ we used the
scaling variable $x_w = (Q^2+B)/(2M\nu+A)$ (or =$x(Q^2 +B)/(Q^2+Ax)$).
This modification was used in early fits to SLAC data~\cite{bodek}.
The parameter A provides for an approximate way to include $both$ target
mass and higher twist effects at high $x$,
and the parameter B allows the fit to be
used all the way down to the photoproduction limit ($Q^{2}$=0).
\item In addition as was done in earlier non-QCD based
fits~\cite{DL} to low energy data, we multiplied all PDFs
by a factor $K$=$Q^{2}$ / ($Q^{2}$ +C). This was done in order for
the fits to describe low $Q^2$
data in the photoproduction limit, where
$F_{2}$ is related to the
photoproduction cross section according to
%
%
\begin{eqnarray}
\sigma(\gamma p) = {4\pi^{2}\alpha_{\rm EM}\over {Q^{2}}}
F_{2}
= {0.112 mb~GeV^{2}\over {Q^{2}}} F_{2} \nonumber
\label{eq:photo}
\nonumber
\end{eqnarray}
%
%
\item Finally, we froze
the evolution of the GRV94 PDFs at a
value of $Q^{2}=0.24$ (for $Q^{2}<0.24$),
because GRV94 PDFs are only valid down to $Q^{2}=0.23$ GeV$^2$.
\end{enumerate}
In our analyses, the measured structure functions
were corrected for the BCDMS systematic error shift and for
the relative normalizations between the SLAC, BCDMS
and NMC data~\cite{highx,nnlo}.
The deuterium data were corrected
for nuclear binding effects~\cite{highx,nnlo}.
%A simultaneous fit to both proton and deuteron SLAC, NMC and BCDMS data
%(with $x>0.07$ only)
%yields A=1.735, B=0.624 and C=0.188 (GeV$^2$) with GRV94 LO PDFs
%($\chi^{2}=$ 1351/958 DOF).
%Note that for $x_w$ the parameter
%A accounts for $both$ target mass and higher twist effects.
%
%
\begin{figure}[t]
%\centerline{\psfig{figure=Fig1Mex.ps,width=5.5in,height=5.4in}}
%\centerline{\psfig{figure=Fig1Mex.ps,width=5.0in}}
\centerline{\psfig{figure=Fig1Mex.ps,width=4.in,height=4.in}}
\caption{Electron and muon $F_2$ data (SLAC, BCDMS, NMC, H1 94)
used in our GRV98 $\xi_w$ fit compared to the predictions of the unmodified GRV98
PDFs (LO, dashed line) and the modified GRV98 PDFs
fits (LO+HT, solid line); [a] for $F_2$ proton, [b] for $F_2$ deuteron,
and [c] for the H1 and NMC proton data at low $x$.}
\label{fig:f2fit}
\end{figure}
%
\begin{figure}[t]
%\centerline{\psfig{figure=Fig2Mex.eps,width=6.0in,height=3.6in}}
%\centerline{\psfig{figure=Fig2Mex.eps,width=5.0in}}
\centerline{\psfig{figure=Fig2Mex.eps,width=5.0in,height=3.5in}}
\caption{ Comparisons to data not included in the fit.
(a) Comparison of SLAC and JLab
(electron) $F_{2p}$ data
the resonance region (or fits to these data)
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) our modifications.
(b) Comparison of photoproduction
data on protons to predictions using
our modified GRV98 PDFs.
(c) Comparison of representative CCFR $\nu_\mu$ and $\overline\nu_\mu$
%charged-current differential cross sections~\cite{yangthesis,rccfr}
on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT,
solid) and without (LO, dashed)
our modifications.
}
\label{fig:predict}
\end{figure}
%
%
%\section{New Analysis with $\xi_w$, $G_D$ and GRV98 PDFs }
In this
publication we update our previous studies,~\cite{second}
which were done with a new improved scaling variable $\xi_w$, and fit for
modifications to the more modern GRV98 LO PDFs such that the PDFs
describe both high energy and low energy electron/muon data.
We now also include NMC and H1 94 data at lower $x$.
Here we freeze
the evolution of the GRV98 PDFs at a
value of $Q^{2}=0.8$ (for $Q^{2}<0.8$),
because GRV98 PDFs are only valid down to $Q^{2}=0.8$ GeV$^2$.
In addition, we use different photoproduction limit
multiplicative factors for valence and sea. Our proposed
new scaling variable is based on the following derivation.
Using energy momentum conservation, it can be shown that
the factional momentum $\xi$ = $(p_z + p_0)/(P_z + P_0)$
carried by a quark of 4-mometum $p$
in a proton target of mass M and 4-momentum P is given
by $\xi$ = $xQ^{'2}/[0.5Q^{2}(1+[1+(2Mx)^{2}/Q^2]^{1/2})]$, where
$2Q^{'2} =[Q^2+M_f{^2}-M_i{^2}]+[(Q^2+M_f{^2}-M_i{^2})^2
+4Q^2(M_i{^2}+P_{T}^{2})]^{1/2}.$
Here $M_i$ is the initial quark mass with average initial
transverse momentum $P_T$ and $M_f$ is the mass of the
quark in the final state. The
above expression for $\xi$ was previously derived~\cite{gp}
for the case of $P_T=0$. Assuming $M_i=0$ we use instead:
$\xi_w = x(Q^2+B + M_f{^2})/(0.5Q^{2}(1+[1+(2Mx)^{2}/Q^2]^{1/2})+Ax)$
Here $M_f$=0, except for charm-production processes in neutrino scattering
for which $M_f$=1.5 GeV.
%The $\xi_w$ scaling variable includes the exact form
%for the target
%mass effects~\cite{gp} in the dominator.
For $\xi_w$ the parameter A
is expected to be much smaller than for $x_w$ since now it only
accounts for the higher order (dynamic higher twist) QCD terms
in the form of an $enhanced$ target mass term (the effects of the proton
target mass are already taken into account using the exact form
in the denominator of $\xi_w$ ).
The parameter
B accounts for the initial state quark transverse
momentum and final state quark $effective$ $\Delta M_f{^2}$
(originating from multi-gluon emission by quarks).
%For the valence quarks, we improve on the
%low $Q^2$ multiplicative factor $K$ as follows.
Using closure considerations~\cite{close} ($e.g.$the Gottfried
sum rule) it can be shown
that, at low $Q^2$,
the scaling prediction for the $valence$
quark part of $F_2$ should be multiplied by
the factor $K$=[1-$G_D^{2}$($Q^{2}$)][1+M($Q^{2}$)] where
$G_D$ = 1/(1+$Q^{2}$/0.71)$^{2}$ is the proton elastic form factor,
and M($Q^{2})$ is related to the magnetic elastic form factors
of the proton and neutron.
At low $Q^2$, [1-$G_D^{2}$($Q^{2}$)]
is approximately $Q^{2}$/($Q^{2}$ +C) with $C=0.71/4=0.178$
%(versus our fit value C=0.18 with GRV94).
In order to
satisfy the Adler Sum rule~\cite{adler} we add the function M($Q^{2}$)
to account for terms from the magnetic
and axial elastic form factors of the nucleon).
Therefore, we try a more general form
$K_{valence}$=[1-$G_D^{2}$($Q^{2}$)][$Q^{2}$+$C_{2v}$]/[$Q^{2}$ +$C_{1v}$],
and $K_{sea}$=$Q^{2}$/($Q^{2}$+$C_{sea}$). Using this
form with the GRV98 PDFs (and now also including
the very low $x$ NMC and H1 94 data in the fit) we find
$A$=0.419, $B$=0.223, and $C_{1v}$=0.544, $C_{2v}$=0.431, and $C_{sea}$=0.380
(all in GeV$^2$, $\chi^{2}=$ 1235/1200 DOF).
%As expected, A and B are now smaller with
%respect to our previous fits with GRV94 and $x_w$.
With these modifications, the GRV98 PDFs must also be multiplied
by $N$=1.011 to $normalize$ to the SLAC $F_{2p}$ data.
The fit (Figure \ref{fig:f2fit}) yields the
following normalizations relative to
the SLAC $F_{2p}$ data ($SLAC_D$=0.986, $BCDMS_P$=0.964, $BCDMS_D$=0.984,
$NMC_P$=1.00, $NMC_D$=0.993, $H1_P$=0.977, and BCDMS systematic error shift of
1.7).(Note, since the GRV98 PDFs do not include the charm sea,
for $Q^{2}>0.8$ GeV$^2$
we also include charm production using the photon-gluon fusion
model in order to fit the very high $\nu$ HERA data. This is not
needed for any of the low energy comparisons but is only needed
to describe the highest $\nu$ HERA electro and photoproduction data).
\begin{figure}[t]
%\centerline{\psfig{figure=Fig3Mex.ps,width=5.0in}}
\centerline{\psfig{figure=Fig3Mex.ps,width=5.0in,height=3.2in}}
\caption{ Comparisons to data on deutrerium
which were not included in our 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
our modified GRV98 PDFs (including shadowing corrections).
(c) The shadowing corrections that were applied to the PDFs for
predicting the photoproduction cross section on deuterium.}
\label{fig:predictD}
\end{figure}
%
Comparisons of $predictions$
using these modified GRV98 PDFs to other data which were $not$
$ included$
in the fit is shown in Figures \ref{fig:predict} and \ref{fig:predictD}.
From duality~\cite{bloom} considerations,
with the $\xi_w$ scaling variable, the modified
GRV98 PDFs should
also provide a reasonable
description of the average value of $F_2$
in the resonance region. Figures \ref{fig:predict}(a) and \ref{fig:predictD}(a) show
a comparison between resonance data (from SLAC and Jefferson
Lab, or parametrizations of these data~\cite{jlab}) on protons and
deuterons versus
the predictions with the standard GRV98 PDFs (LO)
and with our modified GRV98 PDFs (LO+HT). The
modified GRVB98 PDFs are in good agreement with SLAC and JLab
resonance data down to $Q^{2}=0.07$ (although resonance
data were not included in our fits).
There is also very
good agreement
of the $predictions$ of our modified GRV98
in the $Q^2=0$ limit with
photoproduction data on protons and deuterons as shown in
Figure \ref{fig:predict}(b) and \ref{fig:predictD}(b).
In predicting the photoproduction cross sections on deuterium,
we have applied shadowing corrections~\cite{shadow}
as shown in Figure \ref{fig:predictD}(c).
We also compare the
$predictions$ with our modified GRV98 PDFs (LO+HT) to a few
representative high energy CCFR
$\nu_\mu$ and $\overline\nu_\mu$ charged-current
differential cross sections~\cite{yangthesis,rccfr} on iron
(neutrino data were not included in our fit).
In this comparison we use the
PDFs to obtain $F_{2}$ and $xF_{3}$ and correct for nuclear effects
in iron~\cite{first}.
The structure function $2xF_{1}$
is obtained by using the $R_{world}$ fit from reference~\cite{slac}.
There is very good agreement of our $predictions$
with these neutrino data on iron at 55 GeV (assuming that vector and
axial structure functions are the same). We are currently working on
further corrections to account for the fact that at low energies,
the vector and axial structure functions are different.
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\end{thebibliography}
\end{document}