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\begin{document}

\def\NPB#1#2#3{{\it Nucl.~Phys.} {\bf{B#1}} (#2) #3}
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\def\PRD#1#2#3{{\it Phys.~Rev.} {\bf{D#1}} (#2) #3}
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\def\ZPC#1#2#3{{\it Z.~Phys.} {\bf C#1} (#2) #3}
\def\PR#1#2#3{{\it Phys.~Rep.} {\bf#1} (#2) #3}

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\title{Polarized parton distributions from charged--current
deep-inelastic scattering}
       
\author{G. Ridolfi}

\address{INFN Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy}

\begin{abstract}
We investigate the capabilities of a neutrino factory in the determination of
polarized parton distributions from charged--current deep--inelastic
scattering experiments, with special attention to the accuracy 
of this kind of measurements. We show that a neutrino factory would allow
to distinguish between different theoretical
scenarios for the proton spin structure.
\end{abstract}

Our present knowledge~\cite{spinrev} on nucleon structure functions
for polarized deep-inelastic scattering (DIS) originates from the
observation of neutral-current phenomena. In this context, a precise
determination of individual flavor distributions is very difficult,
because of the weak sensitivity to the polarized strange quark
density.  Furthermore, quark-antiquark separation is impossible, since
only C-even polarized parton distributions $\Delta q^+=\Delta
q+\Delta\bar{q}$ can be measured by this kind of experiments.
Neutrino factories offer the oppurtunity of measuring both
parity-conserving and parity-violating structure functions, thus
allowing an accurate determination of C-odd parton densities $\Delta
q^-=\Delta q-\Delta\bar{q}$.

A central issue in physics of polarized nucleons is to explain the
unexpected smallness of the axial charge 
\be
a_0(Q^2)=\int_0^1dx\,\left(\Delta\Sigma^+-\frac{n_f\as}{2\pi}\,\Delta g\right),
\ee
where $\Delta g$ is the polarized gluon distribution,
and $\Delta\Sigma^+=\Delta u^++\Delta d^++\Delta s^+$.
Present data indicate
that $a_0$ is compatible with zero, but values as large as
$a_0(10$~GeV$^2)=0.3$ are not excluded.  A possible  theoretical
interpretation is to assume that a
cancellation between a large (scale--independent, or AB--scheme)
$\Delta\Sigma^+$ and a large $\Delta g$ takes
place. In this case, $\abs{\Delta u^+}, \abs{\Delta d^+} \gg
\abs{\Delta s^+}$ (in the AB--scheme), as expected in the quark model.
This is usually called the `anomaly' scenario.  Alternatively, one may
assume that $\Delta g$ is indeed small, and that $\Delta s^+$ is large
and negative.  This might be explained by invoking non--perturbative,
instanton--like vacuum configuration (`instanton' scenario). In this
case, $\Delta s =\Delta\bar s$.  In the `skyrmion' scenario, $\Delta
s^+$ large, but $\Delta s$ is significantly different from $\Delta
\bar s$.
Future experiments will aim at improving the accuracy in the
determination of the axial charge and of the polarized gluon
distribution. Furthermore, it will be important to understand whether
$\Delta s$ is large, compared to the light flavor distributions, and
how different it is from $\Delta\bar s$. In the following we will
explore the potential of a neutrino factory in answering these
questions.

Present experimental information on polarized parton densities is
summarized in the first two columns of table~\ref{fits}, 
\begin{table}
\caption{\label{fits}
Fits to neutral--current polarized DIS data (columns 1-2) and
to the real and fake data (columns 3-5).
}
{\small
\begin{tabular}{@{}cccccc} 
\br
              & generic fit      & $\eta_g=0$ fit   &`anomaly' refit    & `instanton' refit & `skyrmion' refit\\
\mr
$\eta_\Sigma$ & $0.38\pm 0.03$   & $0.31 \pm 0.01$  & $0.39  \pm 0.01$  & $0.321\pm 0.006$  & $0.324\pm 0.008$\\
$\eta_g$      & $0.79\pm 0.19$   & $0$              & $0.86  \pm 0.10$  & $0.20 \pm 0.06$   & $0.24 \pm 0.08$ \\
$\eta_3$      & $1.110\pm 0.043$ & $1.039\pm 0.029$ & $1.097 \pm 0.006$ & $1.052\pm 0.013$  & $1.066\pm 0.014$\\
$\eta_8$      & $0.579$          & $0.579$          & $0.557 \pm 0.011$ & $0.572\pm 0.013$  & $0.580\pm 0.012$\\
\mr						                                                               
$\eta_u$      & $0.777$          & $ 0.719$         & $ 0.764\pm 0.006$ & $0.722 \pm 0.010$ & $ 0.728\pm 0.009$\\
$\eta_d$      & $-0.333$         & $-0.321$         & $-0.320\pm 0.008$ & $-0.320\pm 0.009$ & $-0.325\pm 0.009$\\
$\eta_s$      & $-0.067$         & $-0.090$         & $-0.075\pm 0.008$ & $-0.007\pm 0.007$ & $-0.106\pm 0.008$\\
\mr						                                                               
$a_0$         & $0.183\pm 0.030$ & $0.284\pm 0.012$ & $ 0.183\pm 0.013$ & $0.255\pm 0.006$  & $ 0.250\pm 0.007$\\
\br
\end{tabular}}
\end{table}
where $\eta_\Sigma,\eta_3$ and $\eta_8$ are the first moments of
$\Delta u^+ +\Delta d^+ +\Delta s^+$, $\Delta u^+ -\Delta d^+$ and
$\Delta u^+ +\Delta d^+-2\Delta s^+$ respectively (all of them are
scale-independent quantities in the AB scheme), and $\eta_g$ is the
first moment of the polarized gluon distribution at $Q^2=1$~GeV$^2$.
The value of $a_0$ at $Q^2=10$~GeV$^2$ is also given.  The second
column corresponds to a fit with $\eta_g$ forced to zero.  Previous
analyses show that this value, although disfavoured by the data, can
only be excluded at about two standard deviations.  The last three
rows in table~\ref{fits} show the values of the first moments of
$\Delta q^+$ at $Q^2=1$~GeV$^2$ for $q=u,d,s$.

In order to estimate the impact of charged--current DIS data from a
$\nu$ factory on our knowledge of the proton spin structure, we
proceed as follows.  We assume that both $\nu$ and $\bar\nu$ beams
will be available, and therefore that the four structure functions
$g_1^{W^\pm},g_5^{W^\pm}$ will be measured (at leading twist,
charged--current polarized DIS can be described in terms of two
independent structure functions).  Under this assumption, taking
suitable combinations of structure functions for different targets
below and above the charm threshold, a complete separation of all four
flavors and antiflavors is possible (see ref.~\cite{FMR} for the
details).

We have computed the expected errors on the measurements of $g_1$ and
$g_5$ (for proton and deuteron targets) in a region of the $x,Q^2$
plane compatible with the features of the device.  With the parameter
choice of ref.~\cite{FMR}, this corresponds to about $0.01 \leq x \leq
0.7$ and $1\,{\rm GeV}^2\leq Q^2 \leq 100\,{\rm GeV}^2$.  The neutrino
beam is assumed to be generated by a muon beam with energy
$E_{\mu}=50$~GeV and number of muon decays per year $N_{\mu}=10^{20}$.

We have produced sets of ``fake'' data for charged--current structure functions
according to three different assumptions:
\begin{enumerate}
\item generic fit of table~\ref{fits} and $\Delta\bar s=0$
(anomaly scenario);
\item $\eta_g=0$ fit of table~\ref{fits} and $\Delta\bar s=\Delta s$ 
(instanton scenario);
\item $\eta_g=0$ fit of table~\ref{fits} and $\Delta\bar s=0$ 
(skyrmion scenario).
\end{enumerate}
In all three cases, we have fixed $\Delta\bar u=\Delta\bar d=0$,
consistently with the indications of semi-inclusive DIS data, and with
positivity constraints.  The fake data were then gaussianly
distributed about their central values, with the errors determined as
explained above. In this way, a
total of 8 sets of fake data ($g_1$ and $g_5$ with $\nu$ and $\bar\nu$
beams and proton or deuteron targets) were generated, with $\sim 70$
data points in each set (the neutral--current real data correspond to
176 points). Real and fake data were then fitted by means of
a NLO evolution code~\cite{FMR}, based on previous work for
neutral--current structure functions~\cite{ABFR}.  The results are
shown in the last three columns of table~\ref{fits}.  The rows
labelled $\eta_u$, $\eta_d$ and $\eta_s$ now give the best--fit values
and errors on the first moments of $\Delta q^-$ for $u,d,s$.

Comparing these values with those of our original fits (first and
second columns), we see that the precision on the singlet quark first
moment is significantly improved by the charged--current data: the
error on the first moment of $\Delta\Sigma^+$ is now of a few percent
in comparison to about 10\% with neutral--current DIS.  With this
accuracy, the `anomaly' scenario can be experimentally distinguished
from other scenarios, at the level of several standard
deviations. This is also reflected by the improvement in the precision
on the gluon first moment.

The determination of the singlet axial charge is improved by an amount
comparable to the improvement in the determination of the singlet
quark first moment. It will be possible to establish at the level of a
few percent whether the axial charge differs from zero or not.  The
determination of the isotriplet axial charge is also significantly
improved: the improvement is comparable to that on the singlet quark.
This would allow an extremely precise test of the Bjorken sum rule,
and an accurate determination of the strong coupling.  Finally, the
octet C--even combination is now also determined with an uncertainty
of a few percent. Therefore, the strange C--even component can be
determined with an accuracy which is better than 10\%. Comparing this
direct determination of the octet axial charge to the value obtained
from baryon decays would allow a test of different existing models of
SU(3) violation.

Let us now consider the results for the first moments of the C--odd
distributions. We see that in the case of $u$ and $d$ quarks they can
be measured with an accuracy of a few percent, just sufficient to
establish whether the up and down antiquark distributions, which are
constrained by positivity to be quite small, differ from zero, and
whether they are equal to each other or not. Furthermore, the strange
C--odd component can be determined at a level of about 10\%, which is
enough to distinguish between the two cases $\Delta s^-\sim 0$ and
$\Delta s^-\sim \Delta s^+$.  The instanton and skyrmion scenarios
can thus also be distinguished at the level of several standard
deviations.

In conclusion, we have explored the capabilities of a neutrino factory
in the study of polarized DIS. With such a facility
the accuracy in the determination of the first moments of C--even
distributions can be improved by about an order of magnitude with
respect to the current uncertaintiesthus allowing a definitive
confirmation or refutation of the `anomaly' scenario for the proton
spin structure, compared to the `instanton' or `skyrmion' scenarios.
The first moments of the C--odd distributions $\Delta u^-,\Delta d^-$
can be determined at the level of few percent, while $\Delta s^-$
could only be measured at the 10\% level.  This would be sufficient to
test for instrinsic strangeness, and distinguish between skyrmion and
instanton scenarios.

The determination of the shapes of the distributions is severely
limited by the uncertainty on the shape of
the gluon distribution. It turns out that $\Delta u(x)$ and $\Delta
d(x)$ can only be measured with a precision around 15-20\%, while no
significant shape information can be obtained for $\Delta s(x)$.\\
\noindent {\bf Acknowledgement}: I thank S. Forte and M. Mangano
for the fruitful collaboration on the subject of this talk.
\bigskip

\begin{thebibliography}{99}
\bibitem{spinrev}
  G.~Ridolfi, {\it Nucl. Phys.} {\bf A666} (2000) 278;
  R.D.~Ball and H.A.~M.~Tallini, {\it J. Phys.} {\bf G25} (1999) 1327;
  S.~Forte, {\tt hep-ph/9409416}, {\tt hep-ph/9610238};
  E.W.~Hughes and R.~Voss, {\it  Ann. Rev. Nucl. Part. Sci.} {\bf 49}
  (1999)  303; R.~Windmolders, {\tt hep-ph/9905505}.
\bibitem{FMR}
  S. Forte, M. Mangano and G. Ridolfi,
  \NPB{602}{2001}{585}, {\tt hep-ph/0101192}.
\bibitem{ABFR}
  G.~Altarelli, R.D.~Ball, S.~Forte and G.~Ridolfi, \NPB{496}{1997}{337};
  {\it Acta Phys. Pol.} {\bf B29} (1998) 1145.
\end{thebibliography}

\end{document}
