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\title{Neutrinos and non-perturbative nucleon structure}

\author{Anthony~W.~Thomas
%\address[MCSD]{Special Research Centre for the Subatomic Structure of
\address{Special Research Centre for the Subatomic Structure of
Matter \\ and Department of Physics and Mathematical Physics, \\ 
        The University of Adelaide, Adelaide SA 5005, Australia}}
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\begin{document}

\begin{abstract}
There are many vital questions concerning the non-perturbative structure
of the nucleon for which neutrinos are uniquely suited to give answers.
We outline some of the essential issues in hadronic physics
which can and should 
be addressed with high intensity neutrino beams.
\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{INTRODUCTION}
We concentrate on the most significant open questions concerning nucleon
structure and the way non-perturbative QCD works. We begin with the
axial form factors of the nucleon, that is the elastic form factor and
the many possible inelastic transition form factors. A comparison
between these and the corresponding electromagnetic form factors
promises to be particularly revealing. Then we turn to some of the
outstanding questions that can be probed in the deep-inelastic regime,
including the flavor and spin-flavor asymmetries and what these might
teach us.

\section{AXIAL FORM FACTORS}
Chiral symmetry plays an extremely important role in modern hadron
physics -- from low energy meson production to hadron form factors and
the extrapolation of lattice data to the physical quark mass
\cite{Thomas:2001kw}.
Through PCAC the axial form factor of the nucleon plays a special role
-- it is the most natural guide to the way the pion couples to a
nucleon \cite{Guichon:1982zk}. 
Through a series of bubble chamber experiments in the 70's and
80's (e.g. Ref. \cite{Kitagaki:vs}) 
the charged current (CC) axial form factor of the nucleon appears
to be reasonably well known. It seems to be well represented by a dipole
with mass parameter $M_A = 1.03 \pm 0.04$ GeV \cite{Thomas:2001kw}. 
However, it is clear that
the statistical errors on the data are such that the sensitivity to
deviations from a dipole was not great. It is unacceptable that such a
fundamental property of the nucleon is not better determined.

{}From the point of view of building quark models of the nucleon, the
axial form factor also has a special role. G-parity forbids any
contribution from the pion cloud itself (unless there is some
discontinuity \cite{Guichon:1982zk,Jaffe:2001eb}, so in terms of a chiral quark
model (such as the cloudy bag, CBM \cite{Theberge:1981mq,Thomas:1982kv}) 
this form factor directly probes the
quark core of the nucleon. For example, the measured axial charge radius
(obtained from $m_A$ above) corresponds to a bag radius around 1.1 fm in
the MIT bag (or CBM) -- where $r_A \sim 0.62 R$ \cite{Guichon:1982zk}. 
This interpretation is even
more suggestive when it comes to the $N \rightarrow \Delta$ transition
form factor. Certainly the experimental situation there is much worse
than for the elastic form factor, but the best indication is that in the
$N \rightarrow \Delta$ case the form factor is {\it much} harder
\cite{Kitagaki:vs} -- 
typically $m_A \sim 1.3$ GeV, but with an error of 10--20\%.

Although the comparison between the axial elastic and $N \rightarrow \Delta$
transition form factors is important, even more important is the
comparison with the electromagnetic (M1) transition form factor. There,
of course, the pion cloud does have a significant role and indeed the
best dipole mass parameter for the M1 amplitude is much {\it smaller}
than that for the elastic magnetic form factor \cite{Stoler:yk}
(which has a dipole mass parameter around 0.84 GeV). That is, 
there is almost a factor of two difference between the radius of the M1
and axial $N \rightarrow \Delta$ transition form factors! This is a
fundamental difference which must be addressed by any serious model of
baryon structure.

Of course, the $N \rightarrow \Delta$ transition is just the beginning.
Very precise new experiments at Mainz and JLab are yielding extremely
accurate information on the transition form factors for a large number of
baryon resonances \cite{Burkert:2001cc}. 
If this data is really to be exploited for the
maximum physics content, it would be extremely helpful to
have information of similar quality
from neutrino experiments on the corresponding axial transition form
factors. This must be a high priority for any future accelerator
producing high intensity neutrino beams.

\subsection{Axial form factor in medium}
A great deal of effort is currently being devoted at CERN, RHIC and
elsewhere to exploring the transitions between hadronic and quark
and gluon phases of matter. Equally important from the point of view of
any coherent approach to this problem is to map out the precursors to these
phase transitions. In particular, from the point of view of QCD, it is
not natural to view an atomic nucleus as built of free 
nucleons \cite{Thomas:2001iv}. Rather,
if one begins with some form of Born-Oppenheimer approximation, it is far
more natural to see the shell model orbitals filled by quasi-particles
with nucleon quantum numbers, but whose internal structure has
self-consistently adjusted to the local mean scalar and vector 
mean-fields -- the quark-meson coupling model was constructed to
explore precisely this point \cite{Guichon:1987jp,Guichon:1995ue}. 
In order to explore the precursors to deconfinement it then
becomes crucial to map out the changes in the properties of bound
``nucleons''.

Although such precursor signals have proven elusive, recent experimental
advances at Mainz and Jlab have led to a rather exciting hint of a
deviation of the ratio $G_E/G_M$ for a nucleon bound in $^4$He from the
free ratio \cite{Dieterich:2000mu,Dieterich:sj}. 
Indeed the experimental findings, over a range of momenta
up to more than 2 GeV$^2$, seem to be in quite good agreement with
predictions from the quark meson coupling model made several years
before the experiments \cite{Lu:1998tn}. 
Furthermore, the theoretical analysis of the {\it
ratio} of polarization observables is quite insensitive to effects of
distortion, exchange current corrections, and so on
\cite{Dieterich:2000mu,Udias:1999tm}. 

Obviously this phenomenon
will be explored in much greater detail for additional
nuclei and other momentum transfers over the next few years. However, it
is clear that this discovery immediately suggests that one might look for
similar in-medium changes in other hadron form factors, such as the axial form
factor. With this in mind, Lu {\it et al.} recently explored the density
dependence of $G_A(Q^2)$ within the same model that seems to describe
the density dependence of $G_E/G_M$ \cite{Lu:2001mf}. Figure \ref{gAfig3.eps} 
shows
%
\begin{figure}[htb]
\begin{center}
\includegraphics[height=14pc]{gAfig3.eps}
%\includegraphics[angle=90,height=14pc]{Fig1.ps}
\caption{The nucleon axial form factor at different densities
with $R_0=0.95$fm -- from Ref. \protect \cite{Lu:2001mf}.
The effect of medium modification, which is more important 
in the small momentum
transfer region, grows as the baryon density increases.}
\label{gAfig3.eps}
\end{center}
\end{figure}
%
the resulting change of shape of the axial form factor as the baryon density
grows. Once again the verification of such a change, along with the
corresponding change in the electromagnetic form factors, would be
extremely valuable for our understanding of nuclear matter and its
changes of phase.

\section{SPIN AND FLAVOR DEPENDENCE OF PARTON DISTRIBUTION FUNCTIONS}
The best known ambiguity in the unpolarised valence distribution is the
asymptotic $d/u$ ratio. For several decades this was believed to be
proportional to $(1 - x)$ \cite{Close:xw}. 
Indeed, in spite of the fact that this
behaviour is in disagreement with the expectation 
based perturbative QCD (pQCD) \cite{Farrar:yb}
that the ratio $d/u$ should tend to 1/5 as $x \rightarrow 1$, it
is built into most standard
parton distribution functions (PDF's). 
However, a careful relativistic treatment of binding, Fermi motion and
off-shell corrections in the deuteron by Melnitchouk and Thomas 
in 1996 \cite{Melnitchouk:1995fc}
suggested that the SLAC data \cite{Whitlow:1991uw}
was in fact consistent with the pQCD
expectation. A similar conclusion has been reached by others (e.g.
Ref. \cite{Yang:1998zb}),
but one would really like a demonstrably model independent proof for
such a fundamental property of the nucleon.

One exciting possibility, which can be carried out after the Jefferson
Lab 12 GeV upgrade, is to use the charge symmetry of the $^3$H-$^3$He
system to extract $F_{2n}/F_{2p}$ and hence $d/u$ up to 
$x \sim 0.85$ \cite{Afnan:2000uh}.
However, a direct measurement of the ratio of $\nu$ to $\bar \nu$ deep
inelastic scattering at large-$x$ on hydrogen would yield a totally
unambiguous ratio. In combination with the JLab measurement in the very
large-$x$ region, this could yield quantitative information on possible
new physics, such as six-quark configurations in the A=3 system.

\subsection{Light sea}
In an extremely important series of experiments at CERN, HERA and
Fermilab \cite{Amaudruz:1991at,Baldit:1994jk,Hawker:1998ty}, 
it has been clearly established that there is an excess of
$\bar d$ over $\bar u$ quarks in the proton. This was actually
anticipated on the basis of chiral symmetry almost a decade before the
NMC experiment \cite{Thomas:1983fh}, 
but only recently has the unambiguous role of chiral
symmetry, in terms of a leading non-analytic (LNA) contribution to $\bar
d - \bar u \sim m^2_{\pi} \ln m_{\pi}$, been 
established \cite{Thomas:2000ny}. Such behaviour
cannot be cancelled by any other mechanism, unless by chance at one
particular quark mass. This realization has led to important advances in
connecting moments of PDF's calculated in 
lattice QCD to data \cite{Detmold:2001jb,Dolgov:2002zm}.

On the other hand, there has been an alternative suggestion concerning the
non-perturbative origins of $\bar d - \bar u$ for 
more than a decade \cite{Schreiber:tc}.
Actually this explanation has its roots in the Pauli correction of Field
and Feynman \cite{Field:ve}, but the modern view is far deeper. 
To illustrate the
concept we briefly review the recent work on related problem in 1+1
dimensions by Tsushima {\it et al.} \cite{Tsushima:2001kv}.

The problem of fermions moving in scalar and vector potentials in 1+1
dimensions has been widely studied since the work of 
Jackiw and Rebbi \cite{Jackiw:1975fn}.
In particular, one has the fascinating possibility of fermions with
fractional charge. Our concern is with a fermion moving in a purely
scalar potential. In 1+1 dimensions one can explicitly examine the
vacuum structure, which differs from the vacuum in the absence of the
scalar potential. In particular, in terms of free fermion states, 
some of the higher states in the Dirac
sea of the scalar potential will be unoccupied, 
while an equal number of positive energy states
will be occupied. In the context of deep inelastic scattering, 
the process of normal ordering 
amounts to the statement that a deep inelastic probe
will see the difference between the structure of the target and the
free vacuum. Thus the change in the sea, even in the absence of valence
quarks (of course, that would not be stable by itself 
but this is a thought
experiment), amounts to the statement that the empty scalar potential
looks like a sea of $q-\bar q$ pairs to an incident probe. Of course, this
intrinsic sea carries no net spin or flavor. 

On the other hand, if one considers a valence quark bound in this scalar
potential, one needs a complete set of free positive {\it and} negative
energy fermion levels to fully describe its wave function (unless the
system is totally non-relativistic). A consequence of adding a valence
quark to the interacting vacuum described above is that it will not
only fill some 
positive energy free states but some of the vacant negative energy
levels. Of course, the valence normalisation from the positive energy
states alone is not complete. How do we interpret this? It is simple: the
addition of the valence quark removes some of the intrinsic anti-quark
sea (of the same flavor), the corresponding sea quarks then make up the
missing valence normalisation. The result is a system with a valence
quark and some of the intrinsic sea, of the same flavor, suppressed.

If we take this result over to 3+1 and the proton, it is obvious that
because we add two valence $u$-quarks and only one valence $d$,
the intrinsic sea will now contain
an excess of $d-\bar d$ pairs. It has so far proven difficult to make a
quantitative estimate of the effect in models like the bag, but
indications in the original work \cite{Schreiber:tc}
suggested  that the integral of 
$\bar d - \bar u$ should be between 0.05 and 0.20 -- certainly the right
order of magnitude. This leaves us with two mechanisms for the
experimental observation and it is clearly very important for our
understanding of hadron structure to determine how much should be
attributed to each one.

The key to this separation is that the clear link between the reduction
of the sea and the quantum numbers of the valence quarks provides a
signature of the origin of the effect. Using an SU(6) wave function for
a polarised proton, one finds that the valence $u$-quarks have
predominantly positive polarisation, whereas the valence $d$'s are only
slightly, but negatively polarised. Hence one expects that the intrinsic
sea produced by the Pauli effect should have $\Delta \bar u$ positive
and a little larger (by a factor 5/3 - 1/3, for exact SU(6) symmetry)
than the Pauli contribution to $\bar d - \bar u$,
whereas $\Delta \bar d$ should be very small in magnitude (1/3 of 
the Pauli contribution to 
$\bar d - \bar u$, for exact SU(6) symmetry) and
negative. In comparison, the sea produced by chiral symmetry is
unpolarised. Thus a determination of $\Delta \bar u$ and $\Delta \bar
d$, together with the existing information on $\bar d - \bar u$, should
allow us to unambiguously understand the origins of the non-perturbative,
light-quark
sea of the nucleon. High intensity neutino beams with a polarised
hydrogen target (certainly an ambitious experiment) provide a wonderful
way to resolve this problem. 

\subsection{Strange sea}
Because the strange quark is much heavier, the Pauli effect discussed
above for $\Delta \bar q$ should be relatively small. On the other hand,
considerations of chiral SU(3) show that there is an unambiguous
difference between the shape of $s(x)$ and $\bar s(x)$
\cite{Thomas:2000ny}. This was first
pointed out by Signal and Thomas \cite{Signal:1987gz}
and later by Brodsky and Ma \cite{Brodsky:1996hc} (although
the s-wave $\bar K N$ coupling used there needs revision). Prompted by
improvements in the determination of the strange quark PDF's in recent
neutrino experiments \cite{Bazarko:1994tt},
as well as the determination of the strangeness
form factors of the nucleon, there has also been quite a bit of
theoretical activity on the problem of $s(x) - \bar s(x)$ 
recently \cite{Melnitchouk:1999mv,Ji:1995rd,Cao:1999jc}.

For the present, it is probably fair to say that there is
no convincing
experimental evidence for such an asymmetry. Nevertheless, it must be
there at some level and it is again a crucial property of the nucleon
that really ought to be pinned down. It is the archetype upon which we
can begin to build an understanding of the role of heavier quarks in
light hadrons.

\section{CONCLUSION}
As we have shown with just a few key examples, intense neutrino beams
offer the promise of crucial new insight into the non-perturbative
structure of hadronic systems. It is vital that in planning future
facilities one counters the one dimensional view of physics that values
speculation about physics beyond the standard model above everything
else. The least understood part of the standard model is QCD in the
non-perturbative regime and the physics issues outlined here need to be
pursued vigorously.

\section*{Acknowledgements}
It is a pleasure to acknowledge the collaborations with many colleagues
at CSSM and elsewhere which have led to the results presented here. I
would particularly like to express my appreciation to 
Gerald Dunne, Dinghui Lu,
Wally Melnitchouk, Tony Signal, Andreas Schreiber,
Fernando Steffens and Kazuo Tsushima.
This work was supported by the Australian Research Council and 
the University of Adelaide.

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