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\title{Analysis of $\bf F_2$ and Drell-Yan data for
       nuclear parton distribution functions}

\author{S. Kumano}
\address{Department of Physics, Saga University,
                Saga, 840-8502, Japan}
\ead{kumanos@cc.saga-u.ac.jp}
\begin{abstract}
We report recent studies on a $\chi^2$ analysis of 
nuclear DIS and Drell-Yan data for obtaining optimum
nuclear parton distribution functions. 
The initial distributions at $Q^2$=1 GeV$^2$ are expressed by
a number of parameters, which are then determined by minimizing
$\chi^2$ in comparison with the data.
Obtained distributions are discussed,
and possible studies at a neutrino factory are suggested.
\end{abstract}

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\section{Introduction}

Parton distribution functions (PDFs) are important for testing both
perturbative and nonperturbative aspects of QCD. 
In particular, the $x$ dependence is associated with nonperturbative
physics, so that the PDF studies are useful for 
establishing a realistic picture of hadron structure.
Furthermore, they are important for applications, for example,
to high-energy heavy-ion reactions and neutrino oscillation experiments.
It is, however, in an unfortunate situation that there are only
a few studies on nuclear parton distribution functions (NPDFs). 

There have been many model dependent studies on the NPDFs; however,
the first serious model independent studies, as far as the author is aware, 
were reported by Eskola, Kolhinen, Ruuskanen, and Salgado (EKRS) \cite{ekrs}.
They divided the $x$ region into three parts and tried to obtain
the optimum distributions to explain high-energy experimental data.
The unpolarized and polarized PDFs in the nucleon have been
obtained by $\chi^2$ analyses. Such a $\chi^2$ analysis method was
developed for the nuclear case in Ref. \cite{saga01}.
This analysis is partly intended to understand how well the NPDFs are 
determined solely by the DIS $F_2^A/F_2^D$ data, other data such as
those of Drell-Yan (DY) are not included.
It leads to somewhat different distributions from the EKRS ones. 
Since the first publication in Ref. \cite{saga01}, we have been working
on the analysis including the DY and $Q^2$ dependent data. 
We report preliminary analysis results.
                        
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\section{Analysis and results}

For the parametrization of the NPDFs, the initial distributions are
assumed as
\begin{equation}
f_i^A (x, Q_0^2) = w_i(x,A,Z) \, f_i (x, Q_0^2),
\label{eqn:def-w}
\end{equation}
at a fixed $Q^2$ point ($Q_0^2$=1 GeV$^2$).
The large $x$ ($x>1$) nuclear distributions cannot be expressed by
this form, but there is no significant DIS data
in such an $x$ region. Therefore, this functional form
does not sacrifice any important physics at this stage.
The subscript $i$ denotes the distribution type: $i$=$u_v$, $d_v$, $\bar q$, 
or $g$. The function $f_i$ is the corresponding distribution in the nucleon.
Nuclear modification is taken into account by
the weight function $w_i$. Because the nuclear effects are typically
less than 20-30\%, it is more practical to parametrize $w_i$ instead of
$f_i^A$ itself.
Therefore, the key point of our analysis is how to take
the functional form of $w_i$. In the present analysis, the $A$
dependence is assumed to be proportional to $1/A^{1/3}$ for simplicity,
and the $x$ dependence is taken in the cubic functional form:
\begin{equation} 
  w_i(x,A,Z)  = 1 + \left( 1 - \frac{1}{A^{1/3}} \right) \, 
  \frac{a_i(A,Z) +b_i x+c_i x^2 +d_i x^3}{(1-x)^{\beta_i}}  .
\label{eqn:wi}
\end{equation}
The parameters $a_i$, $b_i$, $c_i$, $d_i$, and $\beta_i$ are 
determined by a $\chi^2$ analysis. However, three of them are fixed
by the conservation conditions of baryon-number, charge, and momentum.
We also have been investigating more complicated $A$ dependence, but
significant results are not obtained yet.

The NPDFs are evolved to experimental $Q^2$ points by the
usual DGLAP equations.
We use the data of the structure function ratios $F_2^A/F_2^{A'}$ together
with the Drell-Yan cross-section ratios $\sigma_{DY}^A/\sigma_{DY}^{A'}$.
In comparison of the theoretical ratios with the data,
the total $\chi^2$ is calculated by
\begin{equation}
\chi^2 = \sum_j \frac{(R_j^{data}-R_j^{theo})^2}
                     {(\sigma_j^{data})^2}.
\label{eqn:chi2}
\end{equation}
Here, $R$ is $F_2^A/F_2^{A'}$ or $\sigma_{DY}^A/\sigma_{DY}^{A'}$,
where $A'$ is not only the deuteron but also other nuclei:
lithium, beryllium, and carbon.
These structure functions and the DY cross sections are calculated in
the leading order. 
The experimental error is given by systematic and statistical errors as
$(\sigma_j^{data})^2 = (\sigma_j^{sys})^2 + (\sigma_j^{stat})^2$.

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\noindent
\footnotesize
Figure 1. Nuclear modification of the parton distribution 
functions at $Q^2$=1 GeV$^2$.
%       \label{fig:npdfs}
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Obtained optimum weight functions are shown 
for the calcium nucleus in Fig. 1 at $Q^2$=1 GeV$^2$.
The solid, dashed, and dotted curves indicate valence-quark, antiquark,
and gluon weight functions, respectively.
The functional forms of the valence-quark and antiquark distributions 
are similar. The large $x$ valence distribution is determined by
the $F_2$ data. The small $x$ antiquark distribution is fixed by
the $F_2$ shadowing data. In the middle region ($x \sim 0.1$), both
$F_2$ and DY data play an important role. In particular, the DY
data indicate small modification of the antiquark distribution 
in this $x$ region, so that the $F_2$ antishadowing should be
explained by the bump of the valence-quark distribution.
Then, because of this bump together with the  baryon-number
and charge conservations, the valence-quark distribution shows
shadowing behavior at small $x$. As far as the valence-quark and antiquark
distributions are concerned, the obtained distributions are
similar to those of the EKRS parametrization \cite{ekrs}.
Because the DY data are added, the distributions in Fig. 1
are different from those of the 2001 version \cite{saga01}.
The gluon distribution is difficult to be determined
in the leading-order analysis, especially in the large $x$ region.
Therefore, the obtained gluon distribution at medium $x$ is 
different from the EKRS one.


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\section{Comments on neutrino reactions}

Neutrino deep inelastic scattering data have been obtained; however,
no serious nuclear effect is investigated so far. In future,
it could be investigated by the NuMI \cite{morfin}
and neutrino-factory projects. It is especially important
to investigate the parity-violating structure function $F_3$.
A combination of the $F_3$ structure functions is given by
$[ \, F_3^{\nu (p+n)} + F_3^{\bar \nu (p+n)} \, ] \, /4
\approx u_v+d_v$, which probes the valence-quark distributions directly.
Because the valence distributions are known by the electron and muon
scattering data of $F_2$ at large $x$, the advantage of a
future neutrino facility is to investigate the small $x$ region.
In particular, the valence shadowing is not known although
the $F_2$ shadowing, namely the antiquark shadowing, is well investigated.
The valence-quark behavior at small $x$ is restricted by the
baryon-number and charge conservations; however, these conditions are
not enough to determine the accurate functional form at small $x$.

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\noindent
\footnotesize 
Figure 2. Nuclear modification of 
           the valence-quark distributions in the calcium.
%       \label{fig:f3}
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In Fig. 2, we show the valence-quark modification at $Q^2$=1 GeV$^2$
by our parametrization as an example. The dashed curve is obtained
by using the first analysis results of 2001 \cite{saga01}.
The first version did not include the DY data, so that the $F_2$
anti-shadowing at $x\sim 0.2$ is explained partly by the antiquark
distribution and valence antishadowing is not large at $x\sim 0.2$.
Consequently, the small $x$ modification becomes small.
The solid curve indicates the recent parametrization result
by including the DY data \cite{saga01}. In this case, the antishadowing
part is explained mainly by the valence antishadowing.
Therefore, it shows shadowing behavior at small $x$ in order
to satisfy the baryon-number and charge conservations.
There are also model studies on the possible valence shadowing \cite{kulagin}.

The detailed studies of the valence-quark distributions are also connected
to electroweak physics. For example, the Paschos-Wolfenstein relation
is useful for measuring $sin^2 \theta_W$, and the current NuTeV data indicate
a significant deviation from the standard model prediction.
It was pointed out in Ref. \cite{sinth} that such a deviation could be,
at least partially, due to the nuclear modification difference between 
up- and down-quark distributions. We should be able to find the difference
by neutrino-factory measurements.

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\section*{References}
\begin{thebibliography}{99}
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\bibitem{ekrs}  K. J. Eskola, V. J. Kolhinen, and P. V. Ruuskanen,
                         Nucl. Phys. B535 (1998) 351;
                    K. J. Eskola, V. J. Kolhinen and C. A. Salgado,
                         Eur. Phys. J. C9 (1999) 61;
                K. J. Eskola {\it et. al.}, in proceedings of this workshop.
\bibitem{saga01} M. Hirai, S. Kumano, and M. Miyama,
                     Phys. Rev. D64 (2001) 034003;
                 research in progress with Drell-Yan and $Q^2$ dependent data.
\bibitem{morfin} J. G. Morfin, in proceedings of this workshop;
                     in http://neutrino.kek.jp/nuint01/,
                     to published in Nucl. Phys. B (2002).
\bibitem{kulagin} For example, S. A. Kulagin, 
                     Nucl. Phys. A640 (1998) 435; hep-ph/9812532.
\bibitem{sinth} S. Kumano, hep-ph/0209200.
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