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\begin{document}
\title{The present status of our knowledge of the polarized parton
densities,\,or,\,why we need a neutrino factory.}
\author{Elliot Leader}

\address{Imperial College, London}
\ead{e.leader@ic.ac.uk}

\begin{abstract}
The importance of a neutrino factory for the determination of the polarized
parton densities is emphasized.

\end{abstract}

\section{NC polarized DIS:\, $l^{\pm}p\longrightarrow l^{\pm}X$}
There are two measurable asymmetries:\\
\begin{equation}\label{A}
 \fl
  \qquad \qquad \qquad \qquad A_{//}= \frac{d\sigma^{\rightrightarrows}-d\sigma^{\rightleftarrows}}
{2d\sigma_{unpold}}\qquad \qquad A_{\perp} =
\frac{d\sigma^{\rightarrow\uparrow}-d\sigma^{\rightarrow\downarrow}}
{2d\sigma_{unpold}}\qquad \qquad
\end{equation}
\medskip
which are linear combinations of the scaling functions $g_{1,2}$. Information
on the polarized parton densities $\Delta q(x,Q^{2})$ comes from
$g_{1}(x,Q^{2})$:

\begin{equation}\label{B}
  \frac{g_{1}}{F_{1}}=\frac{1}{d'}\quad [A_{//}+tan(\theta /2) A_{\perp}]
\end{equation}
\medskip
where $d'$ is a kinematical factor. There is not much data on $A_{\perp}$,
but this is unimportant for the determination of the $\Delta q(x)$.



\section{Present state of the data}
After decades of work, the quality of the data is now superb.

\begin{center}
BUT
\end{center}
a) There is no CC (neutrino) data.\quad
b) The range of $Q^{2}$ is small compared to the unpolarized case.

Consequently:\\
a) The flavour separation is poor. We have no knowledge at all about the
light antiquarks.\quad  b) The polarized gluon $\Delta G(x)$ is poorly
determined.

\section{Problems in the analysis of the data}
Instead of $\Delta u,\Delta d\cdots$ etc. it is convenient to introduce the
flavour combinations:
\begin{equation}\label{peter}
\eqalign{\Delta q_{3}& = (\Delta u + \Delta \bar u) - (\Delta d + \Delta \bar d)\cr
\Delta q_{8}&= (\Delta u + \Delta \bar u) + (\Delta d + \Delta \bar d) - 2(\Delta s + \Delta \bar s)\cr
\Delta \Sigma = flavour singlet &= (\Delta u + \Delta \bar u) + (\Delta d + \Delta \bar d) + (\Delta s + \Delta \bar s)}
\end{equation}

\medskip
Since $\Delta q_{3}$ changes sign when going from neutron to proton,it is
determined by the difference of n and p data, but the separation of
$\Delta q_{8}$ from $\Delta \Sigma$ and $\Delta G$ depends entirely on
evolution. Moreover, manifestly, there is no separate information about
the light quark sea.

\medskip
To help one usually imposes:\\
\qquad a)The Bjorken sum rule

\medskip

\begin{equation}\label{alice}
 \int_{0}^{1}\Delta q_{3}(x,Q^{2})dx = \frac{G_{A}}{G_{V}}[1 + \cdots]
\end{equation}


 b) The connection to the Cabbibo theory of weak interactions

\medskip

\begin{equation}\label{amos}
  a_{8}\equiv\int_{0}^{1}\Delta q_{8}(x,Q^{2})dx = 3F - D.
\end{equation}
\medskip
where the RHS is known from an $SU(3)_{flavour}$ analysis of hyperon
$\beta -decay$.

How well are the polarized densities known? The results of three different
analyses (see Fig.1) [1] show that $\Delta u_{V}$ and $\Delta _{d_{V}}$, in a
convention where $\Delta \bar u=\Delta \bar d=\Delta \bar s$, are well
determined.
$\Delta s + \Delta \bar s$ is fairly well known and $\Delta G$ rather poorly,
though the first moment of $\Delta G$ is significantly positive.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fig 1 is Fig8a and b  in our paper Eur Phys J C23 (2002) 479. The
%caption should be changed to: "Results of three different analyses of
%the polarized DIS data. For curves see [1]."
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \hspace{-20mm}
\begin{figure}[h]
 \centerline{\includegraphics[width=0.30\textwidth]{fig8a.ps}
  \hspace{-0.03\textwidth}
   \includegraphics[width=0.30\textwidth]{fig8b.ps}}
    \caption{\label{fig8a}
     Results of three different analyses of the polarized DIS data.
 For curves see [1].}
\end{figure}
\begin{center}
HOWEVER
\end{center}
many papers question the use of exact $SU(3)$ symmetry in hyperon
$\beta -decay$.
We have therefore studied the sensitivity of the densities to the value
for $a_{8}$ [2],
using $a_{8}=0.40,\quad 0.58 \quad (=3F-D),\quad 0.86$,
where the first and last are the extreme values found in the literature,
and are probably too pessimistic. Fig.2  show that
$\Delta s+ \Delta \bar s$ and $\Delta G$ are highly sensitive !

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fig2 refers to delta s+sbar and is Fig3  of our paper Phys Lett B488
%(2000) 283.
%  Fig3 refers to delta G and is Fig5 of that paper. The
%caption on Fig2 should be
% the same as we had on our Fig1 but clearly
%referring to delta s+sbar.
% The caption of Fig3 should be just as in %our Fig5 except that it should
%have"See caption to Fig.2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
 \centerline{\includegraphics[width=0.3\textwidth]{fig3.ps}
  \hspace{-0.03\textwidth}
   \includegraphics[width=0.3\textwidth]{fig5.ps}}
    \caption{\label{fig35}
     NLO polarized parton densities $x(\Delta s+\Delta \bar s)$ and $x\Delta G$
at $Q^2=1~GeV^2$ (JET scheme). The solid, short-dashed and long-dashed curves
correspond to the values 0.58 (SU(3) limit), 0.86 and 0.40 for the axial
charge $a_8$, respectively. The error band shown refers to the SU(3) case
and takes into account the statistical and systematic uncertainties. }
\end{figure}

\section{Conclusions}
The superb polarized DIS data yield well determined $\Delta u+\Delta \bar u$ and $ \Delta d+\Delta \bar d$. If we accept $a_{8}=3F-D$ then $\Delta s+\Delta \bar s$ is fairly well fixed and $\Delta G$ is not too bad, and the first moment of $G$ is positive. However, if we mistrust $a_{8}=3F-D$ then both strange and gluon densities are poorly determined. Moreover, we have, in any case, absolutely no information on
$\Delta \bar u$ and $\Delta \bar d$. Data from a neutrino factory would be invaluable in finally determining the polarized densities !
\ack
This work was carried out in collaboration with A.V. Sidorov (Dubna) and
D.B. Stamenov (Sofia). The author is grateful to the Leverhulme Trust for an 
Emeritus Fellowship.

\section*{References}
\begin{thebibliography}{2}
\item AAC Collaboration: Goto Y \etal 2000 {\it Phys. Rev. \rm D \textbf{62}
034017}; GRSV: Gl\"{u}ck M \etal 2001 {\it Phys. Rev. \rm D} \textbf{63}
094005; LSS: Leader E, Sidorov A V and Stamenov D B 2001
{\it Eur. Pys. J. \rm C}  \textbf{23} 479
\item Leader E, Sidorov A V and Stamenov D B {\it Phys. Lett. \rm B}
\textbf{488} 283
%\endbib
\end{thebibliography}
\end{document}

\begin{figure}
\begin{center}
\epsfbox{fig8a.ps}
\end{center}
\caption{\label{Fig1} Results of three different analyses of %the polarized DIS data.
 For curves see [1].}
\end{figure}


 \begin{figure}[tb]
\epsfxsize=100.mm % will enlarge or reduce the postscript figures based on the
\epsfbox{fig8a.ps} % postscript image file name
%\epsfxsize=52.mm % will enlarge or reduce the postscript figures based on the
%\epsfbox{SU3fig5.ps} % postscript image file name
%\epsfxsize=52.mm % will enlarge or reduce the postscript figures based on the
%\epsfbox{SU3fig3.ps} % postscript image file name
\vspace{-10.mm}
\caption{
 Results of three different analyses of %the polarized DIS data.
 For curves see [1].
\label{fig:dgdspoz}}
\end{figure}

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\centerline{\epsffile{fig3.ps}}
\caption{
Results of three different analyses of %the polarized DIS data.
 For curves see [1].
\label{fig3}}
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