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\begin{document}

\title{{\bf TWO-BARYON FORCES FROM SKYRMIONS}\footnote
 {Based on lectures delivered by J.M. Eisenberg at Hadron Physics 96,
Brazil, April, 1996.}}
\vspace{1 true cm}
\author{J.M. EISENBERG \\ 
Institute for Theoretical Physics \\
University of Frankfurt \\
Frankfurt, Germany \\
and \\
School of Physics and Astronomy \\ 
Raymond and Beverly Sackler Faculty of Exact Sciences \\ 
Tel Aviv University, 69978 Tel Aviv, Israel\footnote{Permanent
address.}  \\
\vspace {1 true pc}
and \\ 
\vspace {1 true pc}
G. K\"ALBERMANN \\  
Faculty of Agriculture \\
and \\
Racah Institute of Physics \\ 
Hebrew University, 91904 Jerusalem, Israel \\}
%\date{September, 1995}

\maketitle

\begin{abstract}
\baselineskip 1.5 pc
Background material on solitons and, especially, skyrmions is provided
and the applications of the latter to the derivation of the 
nucleon--nucleon force
is reviewed with attention to the use of the product ansatz, additional
terms in the lagrangian, baryon resonance admixtures, dilatons, and exact 
two- or three-dimensional solutions for the 
$B = 2$ system in order to find the sources of attraction in the central
and spin--orbit potentials.  We discuss extensions to two-baryon 
systems with nonzero strangeness and address applications to the 
behavior of the nucleon in nuclei achieved from skyrmions. 
\end{abstract}

%\newpage
%\tableofcontents
\newpage
\baselineskip 1.5 pc

\section{\label{Introduction} Introduction: the application of skyrmions 
in nuclear physics}

The original idea of Skyrme\ci{Skyrme} to use
topological solitons to describe the nucleon was resurrected some 20
years after its original introduction when it became apparent that 
it might provide an approximate description of hadronic physics as
derived from quantum chromodynamics (QCD).  This came about because of
the realization that in the limit of a large number of colors QCD 
can be reduced to a theory involving meson fields as the relevant 
degrees of freedom\ci{tHooft} and baryons can then be
constructed out of these degrees of freedom as topological 
solitons.\ci{Witten} This history is discussed in early 
reviews\footnote{Since there are many reviews of skyrmions and
their applications, we shall here refer to original papers primarily
when they address direct treatments of the two-baryon system.} 
on skyrmions.\ci{Bal,ZB,HS86a}  Following this renewed interest\ci{ANW} a
great deal of effort has
gone into the study of single-baryon behavior based on the skyrmion,\ci{ANW}
reviewed in.\ci{Bal,ZB,HS86a}

From the time of Skyrme's original work\ci{Skyrme} there has been
interest in applying his model to systems with more than one nucleon.
Indeed the possibility of such applications holds out hope for theories
of nuclei that have their origins relatively close to the fundamental
description of the strong interactions using QCD.  This will be very
difficult to achieve working directly from QCD because the pion is an 
intrinsic ingredient of low- and medium-energy nuclear 
physics\ci{EKbook,EW} yet is not easily treated within the 
fundamental theory
or in many models inspired by it.  The skyrmion is an exception in that
it is automatically couched in terms of pion degrees of freedom.

Clearly before one can consider more elaborate nuclear
applications of the skyr\-mion it is necessary to see if this approach can
account for the nucleon--nucleon force.  Early work on this 
problem\ci{Skyrme,JJP,VMLLCL}, reviewed in\ci{VMrev,Sai,NRrev88,EKrev,
EKrev90,NRrev90,OHrev,WWrev}, was based on a method
closely linked to the usual London--Heitler approach to the calculation
of molecular potentials: it used an ansatz for the two-nucleon system
that took the product of two single-nucleon fields in order to
describe the $B = 2$ system.  The hope was that this simple form might
suffice to extract the main qualitative features of the $NN$ potential.  
To some degree success was guaranteed in that the skyrmion gives reasonable
values for the $\pi NN$ coupling constant\ci{ANW}, and thus handles well
the one-pion-exchange potential (OPEP) which is the key ingredient in
deuteron properties, say.  However, in more complex nuclei, binding is 
provided by the attractive isoscalar central part of the $NN$ potential, 
a feature sadly lacking in the early skyrmion calculations\ci{JJP,VMLLCL}.  
This failure may not be surprising in view of the fact that this central
attraction is a very subtle feature of hadronic
interactions---perhaps one of the main lessons to be learned from its
exploration with various QCD-inspired models.  The central well depth in 
modern phenomenological fits is on the scale of 10 to 20 MeV, while any
basic model of hadronic physics that is applied to two-nucleon systems is
directed at an energy scale 100 times that much.  Thus it would be surprising
if anything at the level of quantitative agreement could be achieved.
The main study of the central nucleon--nucleon potential in the context of
skyrmions thus quickly moved to a search for mechanisms that could
offer a qualitative explanation of $NN$ central
attraction.  One then hopes that this will lead to new insights into
general features of baryon--baryon interactions, and, ultimately, to a
deeper understanding of how QCD brings about the hadronic structures we
encounter: the deuteron, other bound states with two or three or four
baryons in them, and heavier nuclei as well.  

That program has now been underway for about a decade among some ten or
a dozen groups around the world and we shall
attempt to assess in greater detail below what progress has been made 
and what can be expected for the future.  Certainly, in broad summary,
the efforts in this field have
yielded a number of possible candidates for mechanisms providing central
attraction, and this has emboldened the practitioners to branch out to
more intricate manifestations of two-baryon forces---for example, the
$NN$ spin--orbit forces, where attraction is again elusive---and to more
complicated systems, such as $B = 2$ systems containing hyperons, $B = 3$
and 4 systems, nuclear exchange currents, and so forth.  Applications
of skyrmions have also been made to the study of mutual distortions in
$NN$ interactions and to other formal issues reflecting on how nucleons
may behave in nuclear matter. 

Generally, our review here will be cursory where we cover material that
has already been extensively reviewed.  We present these items again 
in order to
have a consistent notation and a reasonably continuous coverage of the
area.  Most of our effort will go into surveying, as broadly as we can,
the current status of our understanding of two-nucleon interactions, with
some digression to related topics with direct impact on that question.

\section{\label{Solitons} Soliton generalities}
\subsection{\label{Solitary_waves} Solitary waves and solitons}

We include here some introductory material on solitary waves and
solitons in general in order to avoid giving the impression that
skyrmions came into existence out of a total void with regard to the
existence of ``self-standing'' solutions of nonlinear wave equations
representing lumps of energy isolated in space.  Excellent 
books\ci{Raj,DEGM,IR} exist in this area and should be consulted for a
more complete picture.

The story starts in 1835 or so with a horseback ride of John Scott
Russell, a person with much more than average knowledge of and
interest in hydrodynamical problems.  As he rode along a barge canal,
he observed the build-up of a wave in front of a barge being dragged
along the canal.  When the barge stopped, this wave broke away and
traveled on its own for several kilometers without change in form or
velocity.  These features are what characterize a solitary wave, as
first studied by Russell after that event: the solitary wave is a
localized lump of energy that retains its shape and velocity as it
propagates.  If, in addition, the solitary wave is unchanged upon
collision with other solitary waves, then it is referred to as a
soliton.  Our own interest is in a particular soliton called the
skyrmion.

\subsection{\label{Solutions} Solutions in $d= 1 + 1$}

Let us consider the situation in one spatial dimension and one time
dimension ($d = 1 + 1$).  For the massless case $m = 0$ the wave
equation reads
\be
\dal\phi = \partial_\mu\partial^\mu\phi 
= \Bigg[\frac{\partial^2}{\partial t^2} 
- \frac{\partial^2}{\partial x^2}\Bigg]\phi(\x,t) = 0,
\ee
and has the well-known solution 
\be
f(x \pm ct); \quad \cos(kx \pm ct),\quad \sin(kx \pm ct),
\ee
where $f$ is an arbitrary function of the particular combination 
$x \pm ct$ and one can also write the solution in terms of the Fourier
transform
\be
f(x-ct) = \int\frac{dk}{2\pi}\big[a(k)\cos(kx-\omega t) 
+ b(k) \sin(kx-\omega t)\big],
\ee
where the wave travels with frequency $\omega = kc = k.$  Here $a(k)$
and $b(k)$ are arbitrary functions chosen to match initial
conditions, say.  The simple relation between $\omega$ and $k$ implies
that the wave for the massless case travels with a constant velocity 
which is the same for all components, so that the wave is undistorted.

When the mass is different from zero the wave equation becomes
\be
\quad\quad (\dal + m^2)\phi = \Bigg[\frac{\partial^2}{\partial
t^2} - \frac{\partial^2}{\partial x^2} + m^2\Bigg]\phi = 0
\ee
and thus the relationship between frequency and wave number is
$\omega^2 = k^2 + m^2$ so that different $k$s travel with different
velocities $\omega(k)/k$ and are distorted by the propagation.

It is possible to compensate for the presence of a nonvanishing mass by
introducing nonlinearities.  The massless case for the so-called
$\phi^4$-theory has a wave equation
\be
\dal\phi + \phi^3 = 0
\ee
with energy
\be
E[\phi] = \int_{-\infty}^{\infty} dx \Bigg[\half\dot\phi^2 + 
\half\phi'^2 + \fourth\phi^4\Bigg].
\ee
This is minimized by the trivial solution $\phi \equiv 0.$

When a mass-like term is introduced,
\be
\ddot\phi - \phi'' - \phi + \phi^3 = 0,
\ee
with the corresponding energy
\be
E[\phi] = \int_{-\infty}^{\infty} dx \Bigg[\half\dot\phi^2 +    
\half\phi'^2 + \fourth(\phi^2-1)^2\Bigg].
\ee
Here there occurs a minimum in the energy at $\phi \equiv \pm 1,$
corresponding to a spontaneous symmetry breaking.  The localized
solution must approach $\pm 1$ as $x \to \pm\infty$ at any $t.$

Generally, a {\sl solitary wave} is defined as a localized, nonsingular
solution of the nonlinear field equation.  Its energy density has the
form
\be
\epsilon(\x,t) = \epsilon(\x-\vec u t), 
\quad\quad \vec u = \ {\rm velocity} = \ {\rm constant}.
\ee
We further define a {\sl soliton} to be such that under mutual
scattering of the solitary waves no change occurs in speed or shape,
but only in the central positions of the waves.  Thus the initial
energy distribution
\be
\epsilon(\x,t) _{\buildrel  \longrightarrow
\over {t \to -\infty}} 
\sum_{i=1}^{N}\epsilon_0(\x-\vec a_i - \vec u_i t)
\ee
goes over after scattering has taken place to
\be
\epsilon(\x,t) _{\buildrel  \longrightarrow
\over {t \to +\infty}} 
\sum_{i=1}^{N}\epsilon_0(\x-\vec a_i - \vec u_i t + \vec\delta_i),
\ee
where $\vec\delta_i$ is a constant so that the energy distributions are
only shifted in location.

For $d = 1 + 1,$ we can solve the general equation for solitary waves.
The lagrangian in that case is
\be
{\cal L} = \half\dot\phi^2 + \half\phi'^2 - U[\phi],
\ee
and variations with respect to $\phi$ immediately lead to the
Euler--Lagrange equations of motion
\be
\delta\int dt\int_{-\infty}^{\infty} dx \Rightarrow
\dal\phi = \ddot\phi - \phi'' = -\frac{\partial U}{\partial \phi}
\ee
and energy
\be
E[\phi] = \int_{-\infty}^{\infty} dx
\Bigg[\half\dot\phi^2 + \half\phi'^2 + U[\phi]\Bigg].
\ee
Note that for $d = 1 + 1$ the wave function $\phi$ is dimensionless,
while for $d = 3 + 1$ it has the dimensions of energy; dimensional
considerations will be of help when we come to consider dilatons below.

For the static case (i.e., no time dependence), multiplying by $\phi'$
immediately yields
\be
\int_{-\infty}^{x} dx \phi'\phi''
= \int_{-\infty}^{x} dx \frac{\partial U}{\partial \phi} \phi'
\ee
or 
\be
\half\phi'^2 = U[\phi]
\ee
since $\phi'$ and $U[\phi] \to 0$ as $x \to 0.$  Thus
$\phi' = \pm\sqrt{2U}$ and
\be
x - x_0 =
\pm\int_{\phi(x_0)}^{\phi(x)}\frac{d\tilde\phi}{\sqrt{2U[\tilde\phi]}}
\ee
is the general solution for the solitary wave, to be inverted to
produce $\phi$ as a function of $x.$

\subsection{\label{Kink} The kink}

For the $\phi^4$ theory, we have
\be
U = \frac{\lambda}{4}\Bigg(\phi^2 - \frac{m^2}{\lambda}\Bigg)^2,
\quad\quad \lambda,\,m^2 > 0.
\ee
Then
\be
\ddot\phi - \phi'' = m^2\phi - \lambda\phi^3
\ee
and
\bea
x - x_0 & = & \pm\frac{1}{\sqrt{\lambda/2}}\int_{\phi_0}^{\phi}
\frac{d\tilde\phi}{\tilde\phi^2 - m^2/\lambda} \nono
& = & \mp\frac{\sqrt{2}}{m}\arctanh\frac{\sqrt{\lambda}}{m}\phi(x)
\eea
or
\be
\phi(x) = \mp\frac{m}{\sqrt{\lambda}}
\tanh\Bigg(\frac{m}{\sqrt{2}}(x-x_0)\Bigg),
\ee
which is referred to as ``the kink.''  Its energy density is
\be
\epsilon(x) = \half\phi'^2 + U =
\frac{m^4}{2\lambda}\sech^4\frac{m(x-x_0)}{\sqrt{2}}.
\ee
The kink simply rises gently from $-m/\sqrt{\lambda}$ to
$m/\sqrt{\lambda},$ the rise being centered (anti)symmetrically about $x =
x_0.$  The energy density is a lump, also centered about this value,
with width $1/m.$  The (finite!) classical mass is then
\be
M_{\rm classical} = \int_{-\infty}^{\infty} dx \epsilon(x) = 
\frac{2\sqrt{2}}{3}\frac{m^3}{\lambda}.
\ee

The kink is a solitary wave but {\sl not} a soliton.  For nonzero
speed the solution is
\be
\phi_u(x,t) = \frac{m}{\sqrt{\lambda}}\,
\tanh\Bigg[\frac{m}{\sqrt{2}}\frac{x-x_0-ut}{\sqrt{1-u^2}}\Bigg],
\quad\quad u \ne 0,
\ee
and the energy is
\be
E = \frac{M_{\rm classical}}{\sqrt{1-u^2}}.
\ee
Note that $\phi_u$ is singular in $\lambda$ and so there can be no
perturbative solution about $\lambda = 0.$  The solution for $\phi$ has
two degenerate minima at $\phi = \pm m/\sqrt{\lambda}.$

It is possible to define here what we shall refer to as a topological
charge (with one eye on future developments of this sort):  Consider
the current
\be
k^\mu = \frac{\sqrt{\lambda}}{m}\epsilon^{\mu\nu}\partial_\nu\phi,
\quad\quad\bigg(
\epsilon^{01} = -\epsilon^{10} = 1,\ \epsilon^{00} = \epsilon^{11} = 0\bigg).
\ee
This current is defined without reference to symmetries of the
lagrangian or details of the dynamics and thus is a not a current of
the variety generated by the usual considerations of Noether.
This current is conserved by the intrinsic properties of its definition
(thus in part justifying the appellation ``topological''),
\be
\partial_\mu k^\mu = 0 \quad\quad \bigg(g^{00} = -g^{11} = 1,
g^{01} = g^{10} = 0\bigg).
\ee
The corresponding charge is
\be
Q = \int_{-\infty}^{\infty} dx k^0 =
\frac{\sqrt{\lambda}}{m}\int_{-\infty}^{\infty} dx
\frac{\partial\phi}{\partial x} 
= \frac{\sqrt{\lambda}}{m}[\phi(\infty) - \phi(-\infty)],
\ee
is also conserved of course.

\subsection{\label{SG} The sine-Gordon equation}

We shall also consider briefly the sine--Gordon equation in order to
show a case which is not only a solitary wave, but also a soliton.  The
defining equation here is
\bea
{\cal L} & = & \half\partial_\mu\phi\partial^\mu\phi 
+ \frac{m^4}{\lambda}\Bigg(\cos\frac{\sqrt{\lambda}}{m}\phi - 1\Bigg)
\nono
& \sim & \half\partial_\mu\phi\partial^\mu\phi - \half m^2\phi^2
+ \frac{\lambda}{4!}\phi^4 - \frac{\lambda^2}{6!}\frac{\phi^6}{m^2},
\eea
where the first two terms are the usual free Klein--Gordon case.
Then the Euler--Lagrange equation is
\be
\dal\phi + \frac{m^3}{\sqrt{\lambda}}\sin\frac{\sqrt{\lambda}}{m}\phi =
0,
\ee
with $\bar x = mx,\ \bar t = mt,$ and $\bar\phi =
(\sqrt{\lambda}/m)\phi.$  So
\be
\bar{\cal L}(\bar x,\bar t) =
\frac{m^4}{\lambda}
\Bigg[\half\bar\partial_\mu\bar\phi\bar\partial^\mu\bar\phi
+ \cos\bar\phi - 1\Bigg].
\ee
Dropping the bars, the equation of motion is
\be
\ddot\phi - \phi'' + \sin\phi = 0,
\ee
with energy
\be
E = \frac{m^3}{\lambda}\int dx \Bigg[\half\dot\phi^2 + \half\phi'^2 + 1
- \cos\phi\Bigg].
\ee

Here there are discrete symmetries
\be
\phi\to-\phi;\quad\quad \phi\to\phi+2N\pi,\ N =\ {\rm positive\
integer,}
\ee
and the energy $E$ vanishes at $\phi = 2N\pi.$
The charge is given by
\be
Q = \frac{1}{2\pi}\int_{-\infty}^{\infty} dx
\frac{\partial\phi}{\partial x} = N_+ - N_-,
\ee
and the solution for $\phi$ is
\be
x - x_0 = \pm\int_{\phi_0}^\phi\frac{d\tilde\phi}{\sqrt{2U}} 
= \pm\int_{\phi_0}^\phi\frac{d\tilde\phi}{2\sin(\tilde\phi/2)}
= \pm\log\tan(\tilde\phi/4)\Bigg|_{\phi_0}^\phi.
\ee
Thus finally the static soliton solution is
\be
\phi_s = 4\arctan\Bigg[\exp[x-x_0]\Bigg], \quad\quad Q = 1
\ee
for the final static soliton solution.

The nonstatic version is
\be
\phi_s = 4\arctan\Bigg[\exp[\frac{x-x_0}{\sqrt{1-u^2}}]\Bigg]
\ee
and the antisoliton is given by $\phi_a = -\phi_s,\ \ Q = -1.$
The soliton energy is $E = M_s = 8m^2/\lambda.$

When one considers, for example, a soliton--antisoliton pair, the
relevant wave function is given by
\be
\phi_{sa} = 4\arctan\frac{\sinh\frac{ut}{\sqrt{1-u^2}}}
{u\cosh\frac{x}{\sqrt{1-u^2}}},
\ee
satisfying
\be
\ddot\phi - \phi'' + \sin\phi = 0.
\ee
After some small tedium one finds the asymptotic behavior
\bea
\phi_{sa} & _{\buildrel  \longrightarrow \over {t \to -\infty}} &
4\arctan\exp\Bigg[\frac{1}{r}[x-u(t+\Delta)\Bigg] \nono
& - & 4\arctan\exp\Bigg[\frac{1}{r}[-x-u(t+\Delta)\Bigg] \nono
& =  & \phi_s\Bigg(\frac{x-u(t+\Delta)}{\sqrt{1-u^2}}\Bigg)
- \phi_a\Bigg(\frac{-x-u(t+\Delta)}{\sqrt{1-u^2}}\Bigg),
\eea
where
\be
r\equiv\sqrt{1-u^2}\quad 
{\rm and}\quad\Delta\equiv\frac{\sqrt{1-u^2}}{u}\log u < 0\quad
{\rm for\ } u < 1.
\ee
Similarly,
\be
{\phi_{sa}}  _{\buildrel  \longrightarrow \over {t \to +\infty}} 
\phi_s\Bigg(\frac{x-u(t-\Delta)}{\sqrt{1-u^2}}\Bigg)
- \phi_a\Bigg(\frac{-x-u(t-\Delta)}{\sqrt{1-u^2}}\Bigg),
\ee
so that the effect of a scattering is a shift by $2\Delta,$ but no
change in speed or shape---the characteristics of a soliton.

\section{\label{N_c} The $N_c \to \infty$ limit}

In the modern  context, the skyrmion is important as a limiting solution
for QCD, namely, one that applies when $N_c \to \infty.$  As we noted,
't Hooft\ci{tHooft} showed that QCD could be treated purely in
terms of meson degrees of freedom in the sector with no baryons, and
Witten\ci{Witten} extended this to $B \ne 0$ by considering a
topological solution for the meson fields.  A natural candidate for the
lagrangian of these fields was then that of Skyrme\ci{Skyrme}.

A concise pedagogical discussion of the $N_c \to \infty$ limit is given
in the book of Bhaduri\ci{Bhaduri}, and this is summarized here only in the
briefest of terms:  The general idea is to take the coupling constant
$g$ of QCD to be proportional to $1/\sqrt{N_c}$ so that $g^2N_c$
remains constant when $N_c$ grows.  This insures that the
one-gluon-loop insertion into a gluon line remains finite (in fact of
order $N_c^0$) when $N_c \to \infty.$  This is because the coupling
constants yield a factor $1/N_c$ while the number of possibilities
for internal gluons is $\propto N_c.$  The single-quark-loop insertion
into the gluon line, on the other hand, is of order $1/N_c,$ since
there is only one possibility for an internal quark pair once the gluon is
fixed.  In general, examination of planar graphs---those in which all
crossed gluons interact, so that one need not imagine a second plane for
a crossed gluon to move on---shows them to be of order $N_c^0,$ while
nonplanar graphs are of order $1/N_c^2.$  This is again brought about
by the number of internal possibilities within the loops in question.

One can analyze the baryon mass from the viewpoint of its behavior as
$N_c \to \infty.$  In terms of its intrinsic quark content for the
$N_c$ quarks required for colorlessness, one would expect that the
baryon mass $M \propto N_c.$  In addition one can look at a single-gluon
exchange graph between two quarks within the baryon.  The combined
factors of $1/N_c$ from the coupling constants and $\half N_c(N_c-1)$
from the number of possible quark pairs that can be selected to carry
the one-gluon interaction again leads to $M \propto N_c.$

The same analysis applies to the exchange of quarks between two
interacting baryons with a gluon transferred between them.  Thus the
baryon--baryon interaction in this analysis is also of order $N_c.$
This suggests we shall have trouble ahead in considering the two-baryon
force from a skyrmion point of view: we know from the start that the
more subtle features of the $NN$ force, such as its crucial
medium-range attraction, are not of the order of the baryon mass ($\sim
1,000$ MeV) but rather some two orders of magnitude less ($\sim 20$
MeV).  The feature of $NN$ interactions which {\sl is} of order $M$ is
short-range repulsion, which indeed emerges naturally and easily with
skyrmions.

To complete the discussion we must consider meson interactions, where a
new feature enters, namely, the meson wave function in terms of
contributions from all the available quark and antiquark pairs, $N_c$
in number.  This is \be
|{\rm meson}\ra = \frac{1}{\sqrt{N_c}}|\bar ll + \bar mm + \cdots +
\bar
nn\ra
\ee
with $N_c$ terms in the ket.  Factors in the meson--nucleon coupling
are then $1/N_c$ from the gluon--quark couplings, $N_c$ from the choice
of quark on which to paste the meson (quark--antiquark pair), $N_c$
from the number of terms in the meson wave function, and $1/\sqrt{N_c}$
from the normalization of the wave function.  Thus, meson--nucleon
coupling goes as $\sqrt{N_c}.$

From considerations of the constituting quarks, one would expect the
meson mass to be of order $N_c^0$ and this is borne out also by looking
at one-gluon exchanges, where the two vertices yield $1/N_c$ and the
two meson wave functions give $\left(N_c/\sqrt{N_c} \right)^2,$ so that
overall one has $\propto N_c^0.$  Meson--meson interactions are of
order $1/N_c,$ as is seen from the fact that once the initial meson
quark--antiquark pairs are selected for each meson, the necessary
constituents for the final mesons in the exchange graph are fixed.  
Thus there is a factor of $\sqrt{N_c}$ for each
initial meson and two factors of $1/\sqrt{N_c}$ for the final mesons,
as well as the usual $\left(1/\sqrt{N_c}\right)^2$ for the coupling of
the transferred gluon: all in all a factor of $1/N_c$ for meson--meson
scattering.  

In summary, in the limit $N_c\to\infty,$ QCD acts like a theory of weakly
coupled mesons, which, if teated topologically, produces stable solitons 
that can be interpreted as baryons.  Note that the meson-meson
interaction 
\be
\propto 1/N_c\propto 1/f_\pi^2,
\ee
where $f_\pi$ is the pion decay constant, so $f_\pi\propto\sqrt{N_c}.$ 
In the space of mesons (or baryons), graphs involving gluons and quarks
as explicit degrees of freedom are of higher order in $N_c$ so meson
degrees of freedom suffice suffice for the description and no
colorless hadron connects to glue state.

\section{\label{Skyrme_genl} Skyrmion generalities}
\subsection{\label{hh} The Skyrme lagrangian and the hedgehog ansatz}

The lagrangian density originally proposed by Skyrme\ci{Skyrme} 
(see also\ci{ANW}) for the
depiction of the nucleon through a solution in terms of meson fields with
nontrivial topological properties is
\be \label{Lagrangian}
 {\cal L} = {\cal L}_2 + {\cal L}_{4}
= -{F_\pi^2 \over 16} \tr(L_\mu  L^\mu )
+ {1 \over 32 e^2} \tr[L_\mu,L_\nu]^2,
\ee
where $F_\pi$ is the pion decay constant, with experimental value 186
MeV, and $e$ is the Skyrme parameter which accompanies his stabilizing term 
with four derivatives.  Here
\be \label{Lmu}
L_\mu \equiv U^\da\dd_\mu U,
\ee
where $U(\r,t)$ is the chiral field, at this stage taken in SU(2).  For
the $B = 1$ static problem we take a time-independent hedgehog 
solution,
\be \label{hedgehog}
U(\r,t) = \exp[i\bftau\cdot\hat\r F(r)],
\ee
where $F(r)$ is referred to as the profile function.  

Let us explore briefly the features of the $B = 1$ hedgehog solution
in order to set the stage for general features of skyrmion solutions.
Substituting eq.~(\ref{hedgehog}) into eq.~(\ref{Lagrangian}) yields an
expression for the lagrangian density in terms of the profile function,
\be \label{Lagrangian_F}
 {\cal L} = -{F_\pi^2 \over 8}\left[\left(\frac{dF}{dr}\right)^2
+ 2\frac{\ssF}{r^2}\right]
-{1 \over 2 e^2}\frac{\ssF}{r^2}\left[\frac{\ssF}{r^2}
+2\left(\frac{dF}{dr}\right)^2\right];
\ee
this leads to a particle energy or mass
\bea \label{M}
M & = & 4\pi\int_0^\infty r^2\,dr \Bigg\{\frac{F_\pi^2}{8}
\bigg[\bigg(\frac{dF}{dr}\bigg)^2 + 2\frac{\ssF}{r^2}\bigg] \nono
& + & \frac{1}{2 e^2}\frac{\ssF}{r^2}
\bigg[\frac{\ssF}{r^2}+2\bigg(\frac{dF}{dr}\bigg)^2\bigg]\Bigg\}.
\eea
One can now seek the extremum for the mass under variations of the 
profile function $F(r)$; this leads to the nonlinear differential equation
\bea \label{ode1}
\bigg(\fourth F_\pi^2 r^2 & + & \frac{2}{e^2} \ssF \bigg)F''
+ \half F_\pi^2\, r F' \nono
& + & \frac{2}{e^2}\sF\,\cF F'^2 - \half F_\pi^2 \sF\,\cF \nono
& - & \frac{2}{e^2} \frac{\sin^3 F \, \cF}{r^2} = 0.
\eea
It is often convenient to cast this into a dimensionless form 
through the introduction of
\be \label{rwiggle}
\tilde r = e F_\pi r,
\ee
whence
\bea \label{ode2}
\bigg(\fourth \tilde r^2 & + & 2\, \ssF \bigg)F''
+ \half \, \tilde r F' + \sin 2F F'^2 - \fourth \sin 2F \nono
& - &  \frac{\ssF \, \sin 2F}{\tilde r^2} = 0.
\eea
Equation (\ref{rwiggle}) underscores the fact that, so long as one
works with the pure Skyrme lagrangian of eq.~(\ref{Lagrangian}) 
without symmetry breaking or terms involving higher derivatives of
$U,$ all results involve dimensionless quantities which are 
scaled by $eF_\pi$ for distances and by $F_\pi/e$ for energies.  Thus
in a sense the success of the simplest version of the skyrmion as
applied to the baryon lies in its ability to characterize the
mass scale ($\sim$ 1000 MeV) and the size scale ($\sim$ 1 fm) with
the one parameter $e,$ assuming $F_\pi$ is taken roughly at its
experimental value.  This also suggests a limitation of 
this version of the
skyrmion, namely that at this level it does not allow for subtleties
in baryon shapes beyond the mere determination of a root-mean-square
radius.

Inspection of the lagrangian density of eq.~(\ref{Lagrangian}) at 
the level of dimensions suggests that the first term ${\cal L}_2$ 
will ultimately yield a contribution to the energy that varies as
the size of the baryon, while the second term ${\cal L}_4$ 
achieves its stabilizing effect because, having two additional
derivatives, it varies as the reciprocal of the system 
size\footnote{In fact it is easy to show by a scaling argument that 
these two 
contributions must exactly equal each other for a proper solution
of eq.~(\ref{ode1}) or (\ref{ode2}).  This is a handy check on
numerics for the simplest case, and also hints\ci{Mou} that a 
chiral expansion is inadequate if truncated at the level of four
derivatives since one must expect the fourth-order contribution 
to be much less than the second-order one for that expansion to 
be valid.}.  It is also clear from the expression for the baryon
mass, eq.~(\ref{M}), that the term $F_\pi^2/4 (\ssF/r^2)r^2$ 
coming from ${\cal L}_2$ blows up as $r\to\infty$ unless
\be \label{F_infty}
\lim_{r\to\infty} F(r) \to n\pi,\quad n={\rm integer},
\ee
while $((1/2e^2)(\ssF/r^2)^2 r^2$ becomes infinite at $r=0$
unless
\be \label{F_0}
F(0) = m\pi,\quad m = {\rm integer}.
\ee
Further, since adding an integer multiple of $\pi$ to $F(r)$
changes nothing in the differential equation for $F(r),$
only the difference between $F(0)$ and $F(r\to\infty),$
or $(m-n),$ can matter in fixing $F(r).$

Indeed, the whole strength of the topological solution for
baryons in terms of meson fields lies in the realization
that there exists a topological constant determined by
the structure of the solution for $U$ rather than from
the specifics of the dynamical solutions to the differential
equation for it.  The quantity in question is defined as
\be \label{B}
B^\mu = \frac{\epsilon^{\mu\alpha\beta\gamma}}{24\pi^2}
\tr \left[\left(U^\da\dd_\alpha U\right)
\left(U^\da\dd_\beta U\right)
\left(U^\da\dd_\gamma U\right)\right],
\ee
where $\epsilon^{\mu\alpha\beta\gamma}$ is the totally
antisymmetric tensor density, and it is easily seen that
\be \label{dB=0}
\dd_\mu\,B^\mu=0.
\ee
Thus the current $B^\mu$ is conserved without regard to 
the equations of motion (i.e., it is not a Noether
current.)  One easily sees that the corresponding charge
is conserved,
\be \label{B_conservation}
\frac{d}{dt}\int B^0(\r,t)d\r 
= -\int\bfnabla\cdot{\bf B}(\r,t)d\r 
= \int_S{\bf B}(\r,t)\cdot d{\bf a} = 0,
\ee
using Gauss's theorem and the vanishing of $U$ on the
surface $S$ at infinity.  This absolute conservation of the charge 
density $B^0$ led Skyrme to identify its integral as the baryon
number in this theory, so that $B^\mu$ is the baryon 
four-current.  For the hedgehog case
\bea \label{B^0}
B^0(r) & = & -\frac{1}{2\pi^2 r^2} \ssF \cdot F'(r) \nono
& = & -\frac{1}{4\pi^2 r^2}\frac{d}{dr}
\left[F(r) - \half\,\ssF\right],
\eea
so that the baryon number is given by
\be \label{baryon_number}
\int B^0 d\r = 4\pi\int_0^\infty\,B^0(r) r^2 dr 
= \frac{1}{\pi}[F(0) - F(\infty)] = m-n,
\ee
which is simply the difference of the boundary values of
the profile function divided by $\pi.$  Thus for a single
baryon the natural choice of boundary values is $F(0)=\pi$
and $F(r\to\infty)=0.$  In numerical solutions of 
eq.~(\ref{ode2}) one finds\ci{ANW} simply a featureless 
fall-off of $F$ from $\pi$ to 0 as $r$ is increased.

\subsection{\label{observables} The mass spectrum and one-baryon observables}

Thus far we have ignored time dependence and we also have
not attempted to distinguish different baryon states through
their different internal quantum numbers (spin and isospin
in the present SU(2) case).  Adkins, Nappi, and Witten\ci{ANW} dealt
with these issues by proposing a technique to rotate the
hedgehog solution and project onto states of good baryon 
spin and isospin.\footnote{Criticism of this method is given 
in\ci{BRK}.}  To do this, they introduce time
dependence through a unitary transformation 
\be \label{AUA}
U(\r,\,t) = A(t)\,U_0(\r)\,A^\da(t),
\ee
where $A(t)$ is a unitary SU(2) matrix.  Were $A$ fixed in time ,
the rotated $U$ is a solution of the same differential 
equation as the unrotated one.  With the introduction of
time dependence we can use standard techniques to 
project out the quantum numbers we need, converting $A$
into a dynamical variable and quantizing it (in the sense 
of first-quantization).

Substituting eq.~(\ref{AUA}) into the lagrangian density
of (\ref{Lagrangian}) we get for the approximate lagrangian
\be \label{L_dynamic}
L = \int\,{\cal L}\, d\r = -M 
+ I \, \tr\left[\frac{dA}{dt}\frac{dA^\da}{dt}\right],
\ee
where $M$ is the mass defined in eq.~(\ref{M}) and
\be \label{lambda}
I = \frac{2\pi}{3 e^3 F_\pi}\int_0^\infty \ssF 
\left\{1 + 4 \left[F'^2 
+ \frac{\ssF}{\tilde r^2}\right]\right\}\,\tilde r^2\,
d\tilde r,
\ee
with the numerical result $I = 50.9 \, (2\pi/3 e^3 F_\pi).$
That there appear in eq.~(\ref{L_dynamic}) only terms bilinear 
in the time derivatives of $A$ is assured by Skyrme's original 
choice of ${\cal L}_4.$ 

The time-dependent transformation $A(t)$ is conveniently
parametrized 
\be \label{a}
A = a_0 + i {\bf a}\cdot\bftau,
\ee
where $a_0(t)$ and $a_i(t),\ i=1,2,3,$ now become the dynamical
variables of the problem.  The unitarity of $A$ is assured by 
the side condition
\be \label{a^2}
a_0^2 + {\bf a}^2 = 1.
\ee
The lagrangian is
\be \label{La}
L = -M + 2I\sum_{i=0}^3\,\dot a_i^2,
\ee
whence the momenta conjugate to the $a_i$ are
\be \label{pi_i}
\pi_i \equiv \frac{\dd L}{\dd \dot a_i} 
= 4I \, \dot a_i.
\ee
The hamiltonian is given by
\be \label{H1}
H = \sum_{i=0}^3\,\pi_i\,\dot a_i - L = 4I\dot a_i^2 - L
= M + \frac{1}{8I}\sum_{i=0}^3\,\pi_i^2.
\ee
This is then first-quantized by taking
\be \label{pi_d/da}
\pi_i \to -i\frac{\dd}{\dd a_i},
\ee
whence
\be \label{H2}
H = M +\frac{1}{8I}\sum_{i=0}^3\left(-\frac{\dd^2}
 {\dd a_i^2}\right).  
\ee

By rotating the hedgehog form of eq.~(\ref{hedgehog}) in
configuration space and in the SU(2) space one arrives 
at expressions\ci{ANW} for the spin and isospin operators
for the baryon,
\be \label{J}
J_k = \frac{i}{2}\left[a_k\frac{\dd}{\dd a_0} 
- a_0\frac{\dd}{\dd a_k} 
- \epsilon_{klm}\,a_l\,\frac{\dd}{\dd a_m}\right]
\ee
and
\be \label{I}
I_k = \frac{i}{2}\left[a_0\frac{\dd}{\dd a_k} 
- a_k\frac{\dd}{\dd a_0} 
- \epsilon_{klm}\,a_l\,\frac{\dd}{\dd a_m}\right].
\ee

In turn one can use these to construct eigenfunctions of
good spin and isospin.  Since the hedgehog solution
conserves the grand spin ${\bf K} = {\bf I} + {\bf J},$
all the relevant solutions will have equal spin and
isospin, $j = t = 1/2,\,3/2,\,5/2,\,7/2\ldots.$  Of these, 
only $j = t = 1/2\ {\rm and}\ 3/2$ are
physically relevant, the higher $5/2,\,7/2,\ldots$ cases being
ruled out either on the grounds that in the underlying 
QCD theory the three valence quarks cannot sum to spin
or isospin higher than $3/2,$ or because the states
with $j = t = 5/2,\,7/2,\ldots$ have widths so large
that they should not be considered to be physical\ci{DHM}.  
For the nucleon states, with $j = t = \half,$
one has
\bea \label{nucleon}
|p\uparrow\ra & = & \frac{1}{\pi}(a_1+i a_2),\quad
|p\downarrow\ra = -\frac{i}{\pi}(a_0-i a_3),\nono
|n\uparrow\ra & = & \frac{i}{\pi}(a_0+i a_3),\quad
|p\downarrow\ra = -\frac{1}{\pi}(a_1-i a_2),
\eea
and some sample cases for the $\Delta,$ with $j = t
= \frac{3}{2},$ are
\bea \label{delta}
|\Delta^{++},\, s_z=\frac{3}{2}\ra 
& = & \frac{\sqrt{2}}{\pi}(a_1 + i a_2)^3, \nono
|\Delta^+,\, s_z=\half\ra 
& = & -\frac{\sqrt{2}}{\pi}(a_1 + i a_2)
\left[1 - 3\left(a_0^2 + a_3^2\right)\right].
\eea
The fact that these states are odd in $a$ is what
ensures here that they are fermionic in character\ci{FR}.  
The projection onto these states is made
in the form of expectation values with respect to
them, the relevant integration being carried
out over a unit sphere in four dimensions according
to
\bea \label{a_average}
\la\cdots\ra & = & 2 \int\,da_1\,da_2\,da_3\,da_0\ 
\delta\left(1-a_1^2-a_2^2-a_3^2-a_0^2\right)\cdots \nono
& = & \int\,da_1\,da_2\,da_3 \left.\frac{1}{|a_0|}\cdots
\right|_{a_0 = \pm \sqrt{1-a_1^2-a_2^2-a_3^2}}.
\eea
The energy spectrum (for the nucleon and delta states)
is given by
\be \label{spectrum1}
E = M + \frac{1}{2I}j(j+1),
\ee
that is, the nucleon and delta masses are
\be \label{spectrum2}
M_N = M + \frac{3}{8I},\quad\quad 
M_\Delta = M + \frac{15}{8I}.
\ee

One-baryon observables are extracted from the Noether 
currents generated by charge variations 
$\delta U = i Q U,$ where $Q$ is the charge operator.  
The $V-A$ combination is given by
\bea \label{V-A}
J_\mu^{V-A} & = & \frac{i F_\pi^2}{8} 
\tr \left[\left(\dd_\mu U\right)\,U^\da Q \right] \nono
& - & \frac{i}{8 e^2} \tr \left\{\left[\left(\dd_\nu U\right)
\,U^\da,\,Q\right]\left[\left(\dd_\mu\right)\,U^\da,\,
\left(\dd_\nu\right)\,U^\da\right]\right\},
\eea
and the $V+A$ combination is arrived at by interchanging
$U$ and $U^\da.$  More detailed expressions and numerical results
for the root-mean-square isoscalar and isovector charge and
magnetic radii,
%($r_{\rm rms}({\rm ch},T=0,1),\, r_{\rm rms}({\rm mag},T=0,1)$), 
proton and neutron magnetic moments,
%($\mu_p,\,\mu_n$), 
axial coupling constant $g_A,$ $\pi NN$ coupling constant,
$\pi N\Delta$ coupling constant, and
$N\Delta$ transition magnetic moment are
not shown here, having been extensively reviewed 
elsewhere\ci{Bal,ZB,HS86a,ANW}.  They generally agree with 
experiment to within the expected $1/N_c \sim 33\%$ error.

\subsection{\label{extensions} Extensions of the lagrangian}

It was early realized\ci{AN} that one could stabilize the
skyrmion with terms representing $\omega$-exchange in addition
to or in place of the four-derivative form ${\cal L}_4$ of
eq.~(\ref{Lagrangian}) originally introduced by Skyrme.  These
are most easily taken in the limit of infinite $\omega$ mass
as terms in the lagrangian density with six derivatives\ci{JJGBC} 
in either of the two forms
\be \label{L61}
 {\cal L}_{6,1} \equiv -\epsilon_{6,1}^2 
\tr\left[{\cal B}_\mu {\cal B}^\mu\right],
\ee
\be \label{L62}
 {\cal L}_{6,2} \equiv -\frac{\epsilon_{6,2}^2}{2}
\tr\left[{\cal B}_\mu\right] \tr\left[{\cal B}^\mu\right],
\ee
where
\be \label{B_no_trace}
 {\cal B}^\mu \equiv \frac{\epsilon^{\mu\alpha\beta\gamma}}{24\pi^2}
\left[\left(U^\da\dd_\alpha U\right)
\left(U^\da\dd_\beta U\right)
\left(U^\da\dd_\gamma U\right)\right]
\ee
differs from the similar definition of the baryon four-current 
of eq.~(\ref{B}) only in that the trace of eq.~(\ref{B}) is
here suppressed so that it may be shown explicitly in the
two different forms of eqs. (\ref{L61}) and (\ref{L62}).
Two new parameters, $\epsilon_{6,1}$ and $\epsilon_{6,2}$ are
here introduced.
The additional index 1 or 2 on the quantities of eqs.
(\ref{L61}) and (\ref{L62}) indicates whether one or two traces
are eventually involved.  The two forms of these equations
are equivalent in SU(2), but not in SU(3), where, for example,
the first contributes to nucleon spin content but the second does
not.\ci{KES}

There are also alternative terms at the level of four derivatives, 
such as
\be \label{L4s}
 {\cal L}_{4s} \equiv \frac{\gamma}{8 e^2} \left[\tr 
\left(\dd^\mu U\, \dd_\mu U^\da\right)\right]^2,
\ee
where $\gamma$ is again a new phenomenological parameter.  This
expression introduces terms with four time
derivatives into the lagrangian, thus confusing the usual
quantization procedures.  It also tends to
destabilize the skyrmion if its coefficient $\gamma$ is taken
large enough so as to overwhelm the other, repulsive terms 
${\cal L}_4,$ ${\cal L}_{6,1},$ or ${\cal L}_{6,2}.$   We thus
entertain the expression of eq.~(\ref{L4s})
here reluctantly as perhaps having some limited
phenomenological purpose.  For the discussion of $NN$ forces
they are of some interest as possible sources of additional 
central attraction.  In the context of single-nucleon uses of
the skyrmion, they serve to stress the peculiar property of the
skyrmion that it has a leg in each of two camps as it were: on
the one hand, the Skyrme lagrangian is originally defined in
terms of meson fields (more specifically, pion fields at the
SU(2) level we are currently discussing), and on the other it
applies to baryons through the topological aspect of the
skyrmion solution.  Thus one can extract\ci{DGH} from 
$\pi\pi$ scattering
data definite values for the parameters in the lagrangian 
${\cal L}_2 + {\cal L}_4 + {\cal L}_{4s}.$  Taking 
$F_\pi = 186$ MeV, one arrives at $\gamma = 0.16 \pm 0.04$
and $e = 5 \pm 2,$ from which one can ``predict'' a 
nucleon mass $M_N = 880 \pm 300$ MeV, where the error noted
is that arising only from uncertainties in the $\pi\pi$ data.
Eventually, when we discuss the extensions of the skyrmion to
SU(3) we shall also encounter an additional possible term\ci{PSW} 
in the lagrangian with four derivatives which is equivalent to 
${\cal L}_4$ in SU(2) but not in SU(3), in parallel to the
situation of ${\cal L}_{6,1}$ and ${\cal L}_{6,2}$ noted above. 

\subsection{\label{ambiguities} Ambiguities, parameter fitting, 
Casimir energy, etc.}

This subsection is meant to make explicit certain 
ambiguities that are bound to accompany almost any discussion
of applications of skyrmions to systems of 
baryons.  We isolate this discussion here in order to
give it appropriate emphasis, while saving the constant
repetition of these caveats in the later text.

From the early uses of the skyrmion for single-baryon properties
one finds, in broad outline, two somewhat different approaches
to the determination of parameters:  The first of these\ci{ANW}
uses the baryon mass spectrum to fix the coefficients of the
various terms in the lagrangian.  This usually leads to overly small
values for the axial coupling constant $g_A,$ for example, but this
is partly to be expected in a theory that from the very beginning 
is prepared to tolerate inaccuracies of about 30 percent because of
the assumption that $1/N_c = 1/3$ is a large number.  The second
approach, represented by ref.\ci{JR} say, attempts to fit 
various measurable baryon matrix elements---the so-called 
``observables''---letting the masses come out where they will.  
Since the agreement with the observables is bound to be partial
at best, and since refined fitting procedures make little sense
in a theory not expected to work to better than 30 percent, this
method is also intrinsically limited.

The situation is made more complicated by uncertain features in the
calculation of baryon masses.  At the level of SU(3), with its 
richer baryon spectrum, it proves necessary to make a large 
subtraction ($\sim 500\ {\rm to}\ 1000$ MeV) of a state with the 
quantum numbers of the
vacuum,\ci{YA} so that in effect one is looking at mass splittings
rather than at absolute positions.  In a somewhat different
approach,\ci{Kan} the need for an even larger subtraction is
justified in terms of ambiguities in operator ordering within the
lagrangian that can express themselves as a large constant term in
the energy.  And of course there is the question of the skyrmion
Casimir energy, which is estimated\ci{Mou,Liv} to be quite large
($\sim 1000$ MeV, again with large uncertainties).  

Thus there seems to be no possible hard and fast rule for fixing
skyrmion parameters.  In practice one is more or less limited to
using a mixture of criteria based on masses and observables.  The
extreme cases that are weighted strongly in one direction or the
other can then be seen as a measure of the inaccuracy of the overall
approach.

\section{\label{NN} Skyrmions for the $NN$ system with the product 
ansatz}

In order to set sensible goals for the skyrmion as applied to
two-nucleon systems, it is important to keep in mind the
advantages and the intrinsic limitations of the skyrmion approach 
as well as the overall expectations for our understanding of the 
nature of the nucleon--nucleon force.  As we have just stressed, the
skyrmion is not expected to be reliable at a level much better than 
30 percent, which certainly rules it out as a method for precision 
calculation of $NN$ potentials for confrontation with experimental 
data.  This restriction is made more acute by two further concerns:
The skyrmion is basically directed at the energy scale of baryons,
say 1000 MeV, whereas the $NN$ force involves features that are
typically one to two orders of magnitude smaller in energy and
thus may well arise from differences between much larger energy 
quantities, resulting in a loss of precision in the calculation.
In addition, the skyrmion is a low-energy theory; it involves an 
expansion in derivatives, or 
momenta, along the lines of that used in chiral perturbation 
theory.  Thus the short-range features of the nucleon--nucleon
interaction will surely not be well represented by it.  From
the beginning of the study of the application of the skyrmion to
$NN$ systems, we must look for a more or less qualitative 
understanding of the broad features of the $NN$ force, and we
must keep in mind that the method may ultimately fail at short 
ranges.  

On the other hand, since the skyrmion provides a reasonable 
representation of $\pi NN$ and $\pi N\Delta$ vertices while
retaining strong roots in our fundamental
understanding of strong interactions through QCD, it is in
some ways an ideal tool for application to nuclear physics,
where pion effects are often dominant\ci{EKbook,EW} and quark 
features must clearly also be understood.  In this sense it
is the natural successor to the chiral bag model\ci{Tho}, 
which attempted to incorporate these same elements in nuclear
applications.
%\footnote{The connection in the context of the
%$NN$ force is made particularly graphic by studies\ci{NTTY,SSS} 
%of the static interaction between two chiral bags.}  

Thus in turning to applications of the skyrmion to the $NN$
system we expect from it insights into the qualitative features 
of the $NN$ force, and we would do well first to sketch the
kinds of $NN$ issues that should be addressed with it.  
For many
years studies of the $NN$ force have been based on a picture
of meson exchanges,\ci{BJ,EG} which in more modern versions\ci{Hol} 
also incorporates notions of chiral dynamics.  The
most solidly based aspect of meson exchange in the 
nucleon--nucleon force is of course that of one-pion exchange
(OPE).  As the lightest of the mesons, the pion dominates the
long-range aspect of the $NN$ potential and its role is easily
verified in high-energy peripheral $NN$ collisions.  This is
the feature that the skyrmion handles easily.  Since the
description of the pion as a Goldstone boson in QCD is 
complicated, it is advantageous to be able to isolate it
before going on to other meson exchanges.  

One-pion exchange is the major effect in deuteron binding but
plays only a small role in heavier nuclei because its effects
involve both spin and isospin variables which are quenched in 
nuclear systems with $B \ge 4.$  There the crucial 
ingredients are the interplay between central attraction at
intermediate $NN$ separation distances and short-range 
repulsion.  In early times, the former was thought to be
provided by the exchange of a scalar $\sigma$ meson, but when
such a particle failed to turn up in experiment in the mass 
region (about 500 to 600 MeV) required to deal with 
intermediate ranges ($\sim 2$ fm), a largely successful 
interpretation was developed in terms of two-pion exchange
with intermediate states involving $\Delta$s.  
Short-range repulsion in the $NN$ system was attributed to
$\omega$-meson exchange.  This approach to the $NN$ force
based on two-pion exchange for attraction and 
omega-exchange for repulsion may well survive
as supplying major ingredients for our understanding of that 
force, but it also seems likely that both pieces of the
picture will be supplemented in an important degree by
quark dynamics; indeed it would be hard to understand how
quark degrees of freedom could fail to enter in some measure
at nuclear separation distances of 2 fm or less, where the
individual nucleons are beginning to have sizable spatial 
overlap.  In fact in the nonrelativistic quark model a
significant source of $NN$ central attraction at 
intermediate ranges has been found\ci{MI} in the form
of mutual excitation of the two three-quark clusters that 
represent the baryons in the model.  Such mechanisms must
thus be entertained as possible supplements to two-pion 
exchange in producing central attraction.  Similarly
nonrelativistic quark models suggest\ci{FFLSOY} auxiliary
sources of short-range repulsion stemming from the action 
of the color-magnetic force in QCD.  This favors the 
occupation by quarks of the 2s states whose higher energy
produces a repulsive effect at short ranges.

Against this background it is important to see what the 
skyrmion has to offer by way of descriptions of central
attraction and short-range repulsion.  The latter is in 
fact a very immediate consequence of the Skyrme model, which
gives\ci{JR} very large energies for superposed 
hedgehog solutions with $B \ge 2.$  The energies in question
exceed by 1000 MeV or more the combined rest-mass energies of
the baryons in these systems, and so are to be interpreted as 
large repulsion for small internucleon separations.  Thus at
the most fundamental level the main challenge for the skyrmion
as applied to two-baryon systems is to offer a source of central
attraction; this is the question that we shall address in the next
sections.  Once this has been dealt with, it is also of great
interest to see what the skyrmion can offer by way of explanation
of more subtle and more intrinsically dynamic issues, notably the
origins of an attractive spin--orbit force which we shall address 
in section \ref{SO} below.  Eventually one would hope to study the 
skyrmion
also for whatever insights it may offer into nucleon behavior in
larger nuclear systems (in the most immediate case, systems with 
$B = 3$) and in nuclear matter, topics to which we turn in section 
\ref{Bbeyond3}.
Since the present section covers older material which we ourselves 
have reviewed in the past,\ci{EKrev} our treatment of these topics
is deliberately cursory, and is meant to be read in
conjunction with that review; nonetheless we bring this material in
brief form in order to have a relatively self-contained 
treatment and to set the stage for the discussion of more recent
developments.

\subsection{\label{PA_sec} The product ansatz}

From the very first stage of the introduction of his topological
soliton, Skyrme was interested\ci{Skyrme} in possible applications 
to the two-nucleon system.  Towards this end he suggested the use
of the product ansatz, in which the two-baryon system is represented
by a topological field which is the product of two single-baryon
fields,
\be \label{PA}
U_{B=2}(\r,\R) = U_{B=1}(\r+\half\R)\,U_{B=1}(\r-\half\R),
\ee
where $\r$ is the general skyrmion variable and $\R$ is the 
separation between the two baryon centers.  This ansatz has the 
virtues that it automatically fulfills the correct baryon number
condition $B = 2$ and has well-defined baryon centers by its 
very construction, in contrast to other approaches we shall encounter 
below.  It also allows for simple physical 
interpretation when applied in the sense of a London--Heitler
calculation in $B = 2$ systems.  

It is crucial for the 
study of the $NN$ system that we project the product ansatz onto
nucleon states, if for no other reason that the dynamic admixture of
$\Delta$s is, as noted above, often adduced as an important 
ingredient in providing
central $NN$ attraction, and thus we must be able to separate the
nucleon from the $\Delta$ in addressing the $NN$ force.  This
projection is accomplished by rotating each member of the product 
ansatz separately according to
\be \label{PPA1}
U_{B=2}(\r,\R) = \left[A\,U_{B=1}(\r+\half\R)\,A^\da\right]\,
\left[B\,U_{B=1}(\r-\half\R)\,B^\da\right],
\ee
where $A$ and $B$ are the rotation matrices of eq.~(\ref{AUA});
it is sufficient to consider the relative rotation of the two 
baryons in order to project onto states with good nucleon quantum 
numbers, i.e., we may take
\be \label{PPA2}
U_{B=2}(\r,\R) = \left[U_{B=1}(\r+\half\R)\right]\,
\left[C\,U_{B=1}(\r-\half\R)\,C^\da\right],
\ee
where
\be \label{CBA}
C \equiv A^\da\,B = c_0 + i\bftau\cdot{\bf c},
\ee
with $c_0 = a_0 b_0 + {\bf a}\cdot {\bf b}$ and 
${\bf c} = a_0 {\bf b} - b_0 {\bf a} + {\bf a} \times {\bf b}.$

The product ansatz may now be used\ci{JJP,VMLLCL} in a 
London--Heitler technique
by choosing a separation distance $\R$ and calculating the resulting
two-nucleon energy from the lagrangian, taking care to subtract 
the masses of the two nucleons (i.e., the energy
at separation $\R\to\infty$).  This requires the evaluation of 
integrals over the variable $\r$ and averaging over the rotation
variables $\{c_0,\,{\bf c}\}$ in expectation values formed with the
nucleon wave functions of eq.~(\ref{nucleon}).  Details are provided
in the review of ref.\ci{VMrev}.  At this level, and using the basic 
lagrangian of eq.~(\ref{Lagrangian}), the skyrmion yields
three components of the $NN$ potential, namely,
\be \label{V}
V(\R) = V_C(R) + \bftau_1\cdot\bftau_2
\left[\bfsigma_1\cdot\bfsigma_2\,V_{SS}(R) + S_{12}\, V_T(R)\right],
\ee
where $\bfsigma$ and $\bftau$ are the Pauli spin and isospin operators
and $S_{12}$ is the usual tensor operator
\be \label{S12}
S_{12} \equiv 3\left(\bfsigma_1\cdot\r\right)
\left(\bfsigma_2\cdot\r\right) - \bfsigma_1\cdot\bfsigma_2.
\ee
The three components of the force of eq.~(\ref{V}) are the central part
$V_C(R),$ the spin--spin part $V_{SS}(R),$ and the tensor piece $V_T(R).$
These last two are dominated by OPE, and so should be well described by
the skyrmion, as indeed they prove to be\ci{VMrev} when compared with
phenomenological spin--spin and tensor components of the $NN$ force.  The
central potential is highly repulsive in its 
behavior for $R < 1$ fm, and no trace of attraction is found in that
component for 1~fm $< R < 2$ fm, where one knows it must occur from 
phenomenological potentials.  In some sense it appears that the repulsion
introduced through ${\cal L}_4$ in eq.~(\ref{Lagrangian}) to stabilize 
the skyrmion spreads beyond the 1 fm range and overcomes any possible
attraction.  This occurs because the lagrangian is governed by two
parameters which must account for baryon masses on the scale of 1000 MeV
and baryon sizes on the scale of 1 fm.  The smooth tail of the baryon 
profile then carries sizable repulsion out to the 2 fm range.

\subsection{\label{higher_order} Higher-order derivatives in the lagrangian}

Hints at possible ways to improve the skyrmion description of
eq.~(\ref{Lagrangian}) in order to identify a source of $NN$ attraction
are already found in the discussion of subsection \ref{extensions}, 
where we
considered possible additional terms in the Skyrme lagrangian.  For 
example, we noted there that the symmetric fourth-order term 
${\cal L}_{4s}$ of eq.~(\ref{L4s}) is attractive---perhaps even too 
attractive in that for too large a parameter $\gamma$ it can destabilize
the skyrmion solution.  One may thus incorporate the 
attractive ${\cal L}_{4s}$ along with the repulsive ${\cal L}_{6,1}$ of 
eq.~(\ref{L61}) and attempt to play these off against each other in 
order to obtain attraction at medium $NN$ 
separations while still keeping a stable skyrmion.  This method is
reviewed extensively in ref.\ci{VMrev}.  It remains an important auxiliary
ingredient in various other devices for producing attraction, but does not
(ref.\ci{KESS}, also reviewed in\ci{EKrev}), in and of itself, 
yield attraction in 
$V_C(R),$ even after extensive search in the space of lagrangian parameters.
Moreover, hopes that mutual distortions of the skyrmions (reviewed 
in\ci{EKrev}) might alleviate the situation come to naught since the dominant
repulsion continues to prevail over the small added attraction obtained form 
this source.

One could of course extend the skyrmion lagrangian to include the exchange
of a wide variety of mesons, thus returning the problem---at least insofar
as its $NN$ applications are concerned---to the general structure of 
one-boson potentials\ci{BJ,EG}.  This has been considered by the Orsay 
group (ref.\ci{LLVMC}, also reviewed in\ci{VMrev}) and, mostly for 
single-nucleon applications, by the Syracuse group (ref.\ci{Syracuse}, also
reviewed briefly in\ci{SSSW}).  The introduction of a large
number of mesons in the skyrmion approach complicates the model somewhat, 
which leads to a partial loss of interpretive transparency and 
reduces the hope---an important motivation for studying the $B = 2$ 
system with skyrmions---of eventually applying the skyrmion in more
complex nuclear situations.

In an approach that is related to the inclusion of a variety of exchanged 
mesons, one may consider\ci{KM89,VV89,YZ} pion quantal fluctuations about
the classical skyrmion solution.  This should allow for the inclusion of
two-pion exchange effects that have long been known\ci{BJ} to be 
significant sources of central attraction,
and indeed it does lead to components with attraction.  However these 
do not readily overcome the repulsion that still persists at 2 fm within the
product ansatz at least.  Thus, while this mechanism is undoubtedly of
continuing importance in the skyrmion context as well, it does not seem that 
it solves the overall problem of central attraction within that approach.
These fluctuations also raise difficult formal and technical problems, in
that they must be carefully chosen so as to be orthogonal to the variations
of the skyrmion that lead to the classical equations of motion.

It has been noted by several authors that one should make at least a
minimal, algebraic correction for working with $N_c\to\infty$ in the 
skyrmion approach\ci{JJP,DPMF86,ABO,O,Oka88}.  Moreover, the correction in 
question acts to increase somewhat the central $NN$ attraction found with 
the skyrmion.  As we have seen in subsection \ref{observables}, rotating \
the skyrmion
leads to states with $j=t=5/2,\,7/2\,\ldots$ which are not physically
realized, an effect which it is natural to assume is an artifact of
taking $N_c\to\infty$ and thus exceeding the natural spin-isospin
structure for the three quarks involved when $N_c = 3,$ which would cut 
off at $j=t=3/2.$  Amado and his collaborators\ci{ABO,O,Oka88} (reviewed 
in\ci{OHrev}) pointed out that a natural way of handling this cutoff
could be achieved by examining the dynamical group symmetry of skyrmion
rotation:  The rotational algebra of SU(2)$_j$ $\times$ SU(2)$_t$ is
embedded in U(4) and the symmetric $(N_c)$ representation of this group
is then taken as the lowest energy band.  The nucleon belongs to this 
band, as do its rotational excitations with $j=t=1/2,\,3/2\,\ldots,
(N_c-2)/2.$  The rotational spectrum arising from this implementation
agrees with the quark model prediction of maximal spin and isospin
$j=t=N_c/2.$  This allows the determination of correction factors for
various spin-isospin operators and thus for the corresponding 
single-baryon observables as well as for the $B = 2$ potentials,
yielding increased central attraction for the $NN$ system.

Last among the refinements within the general scope of the product 
ansatz we note studies of the dynamic features of the skyrmion 
mass-tensor parameter, which may be obtained by ``dragging'' the
hedgehog\ci{Oka86,OMY,RN87,SWA,Schroers,GP} (reviewed briefly in\ci{OHrev}).  
In the context of the $B = 2$ problem, these contribute
velocity-dependent terms in the $NN$ force some of which prove to be
attractive but not very large.  The complete analysis of translational
and rotational collective motion for two skyrmions, even within the
confines of the product ansatz, proves to be a problem of some
intricacy.  This is evident even in the relatively simple 
context of the spin--orbit force (section \ref{SO}) where these 
ingredients are necessary in order to identify that component of the
force.
There have also been efforts made\ci{TH1} to study the $NN$ force in the
context of alternative stabilization mechanisms\ci{Dal} for the skyrmion 
that may greatly simplify the calculational tedium of such studies.

\subsection{\label{baryon_resonances} Baryon resonance admixtures}

It has long been accepted wisdom\ci{BJ} that the attraction in the
central $NN$ potential represented by so-called $\sigma$-exchange in fact
arises because of two-pion exchanges which can simulate the $\sigma$
effects.  It is essential that in the application of box graphs of two-pion 
exchange to the construction of the $NN$ potential at least one of the 
intermediate baryon lines refer to excited states and not to the nucleon 
itself.  This is because in the use of the Schr\"odinger equation to
solve the two-nucleon problem one includes all
iterations of graphs with two-nucleon lines, so that the construction of
the potential itself must omit these in order to avoid double counting.
The physical picture is then that at intermediate ranges the two-nucleon
system is partly an $N\Delta$ or $\Delta\Delta$ system or indeed some
other $B = 2$ system containing higher excitations of the nucleon.
This suggests that in using the skyrmion for the derivation of the $NN$
potential it is in order to consider the admixture of $N\Delta$ and
$\Delta\Delta$ states\ci{KESS} into the wave function,
\be \label{Deltas}
|\Psi(R)\ra = \alpha(R)|NN(R)\ra + \beta(R)|N\Delta(R)\ra +
\gamma(R)|\Delta\Delta(R)\ra.
\ee
Here we once again have in mind the use of the London--Heitler technique.
That is to say, the calculation\ci{KESS} proceeds by fixing the
internucleon separation distance $R$ and then minimizing the system
energy with respect to changes in the admixture coefficients $\alpha(R),$
$\beta(R),$ and $\gamma(R)$ at the value of $R$ in question.
For this mechanism to yield appreciable intermediate-range attraction, 
it is necessary\ci{DPMF87} to treat carefully the action of the tensor
force in admixing spin states.  Even then only very weak attraction
($\sim 2$ to 3 MeV)---and that only for extreme skyrmion parameters which
tend to overaccentuate the role of the $\Delta$---is achieved\ci{DPMF87,KE88a}.  

The picture in this regard is improved considerably if one entertains the
admixture of the Roper $N(1440)$ resonance by extending eq.~(\ref{Deltas})
to include these.  The initial motivation to
consider this mechanism for central attraction in the $NN$ force
arose\ci{KE88b} from the realization that a similar device appears
importantly in deriving central attraction in the nonrelativistic quark
model, where it was found\ci{MI} that the internal excitation of the
individual three-quark clusters making up each nucleon had to be allowed
in order to obtain a reasonable result for $V_C(R).$  The description of
the Roper excitation as a vibrational state was given early on by a
number of authors\ci{BN,HS86b,ZMK,LZB,WE,BDT,Carl,DDLT,Par}.  We follow
here the method and notation of ref.\ci{BDT} (see also the 
review in\ci{EKrev}).The hedgehog ansatz for the individual baryon 
is written as
\be \label{vib}
U_\lambda(\r) = U_0(\r\e^{\lambda(t)}),
\ee
where the time-dependent scaling parameter $\e^{\lambda(t)}$ allows for a
breathing-mode vibration of the hedgehog.  This new ansatz is 
then substituted into the skyrmion lagrangian and the
time-dependent terms are analyzed in a harmonic approximation for $\lambda$ 
together with the rotational prescription of eq.~(\ref{AUA}).  The
hamiltonian with rotational and vibrational degrees of freedom is then of
the form
\be \label{H_rot_vib}
H_\lambda = \Sigma(\lambda)\,\dot\lambda^2 + V(\lambda),
\ee
where 
\be \label{Sigma_lambda}
\Sigma(\lambda) = f_1(\lambda) +\frac{j(j+1)}{4I^2(\lambda)}f_3(\lambda)
\ee
and
\be \label{V_lambda}
V(\lambda) = M(\lambda) + \frac{j(j+1)}{2I(\lambda)},
\ee
with $j$ the baryon angular momentum as in eq.~(\ref{spectrum1}), and
$M(\lambda)$ and $I(\lambda)$ are the skyrmion mass and moment of
inertia.  These last were presented in eqs. (\ref{M}) and (\ref{lambda})
for the case of the basic skyrmion lagrangian of eq.~(\ref{Lagrangian}),
which here may be modified to include the effects of higher-derivative
terms (section \ref{higher_order}).  Expressions for these more complete 
forms as well as for $f_1(\lambda)$ and $f_3(\lambda)$ are given
in refs.\ci{EKrev,KE88b}.   

One then proceeds\ci{BDT} by quantizing about the classical minimum
\be \label{minimum}
\left.\frac{dV}{d\lambda}\right|_{\lambda=\lambda_0} = 0,
\ee
which defines $\lambda_0,$ and in the harmonic approximation arrives at
\be \label{H_harmonic}
H(\lambda) =  \Sigma(\lambda_0) \, \dot\lambda^2 + V(\lambda_0) 
+ \half \left.\frac{d^2V}{d\lambda^2}\right|_{\lambda=\lambda_0}
(\lambda-\lambda_0)^2 + \cdots.
\ee
The usual first quantization in $\lambda$ leads to the oscillator spectrum
\be \label{surface_vibes}
E = V(\lambda_0) + (n + \half)\Omega, \quad n = {\rm integer,}
\ee
with oscillator frequency
\be \label{Omega}
\Omega = \left[\frac{k}{2\Sigma(\lambda_0)}\right]^{1/2}, \quad
k \equiv \left.\frac{d^2V}{d\lambda^2}\right|_{\lambda=\lambda_0}.
\ee
Eigenvalues with $n = 0$ correspond to the nucleon and $\Delta$---the
latter being distinguished, of course, by its rotational motion---and
those with $n = 1$ correspond to the Roper $N(1440)$ and the
$\Delta$-like particle built upon it, $\Delta(1670).$  

With reasonable
parameters, the extension of eq.~(\ref{Deltas}) to include the admixture of 
these two states then yields about 5 MeV of attraction in $V_C(R),$ which
is a not unreasonable well depth given the fits of modern $NN$ potentials
such as that of Paris\ci{LLRVMCPDT} or of Argonne\ci{WSA} with well 
depths of 5 to 20 MeV.  There is also a certain satisfaction in the
fact that the skyrmion and the nonrelativistic quark model find that the 
same basic feature yields attraction, namely, the internal excitation of
each baryon participating in the interaction between the two.  This
qualitative feature drives home the intrinsic subtlety of the origins of
central attraction in two-baryon systems---a feature crucial to the
binding of nuclei, for instance.  It also hints at a finding of the 
exact skyrmion calculations (section \ref{full}): there too the departure 
of the skyrmion at short
internucleon separations from its structure at $R\to\infty$ is what yields
attraction, so that at least at a qualitative level many of the 
different approaches may in fact be convergent.

\section{\label{dilaton} The dilaton}
\subsection{\label{dilaton_general} Dilaton generalities}

We shall open our considerations on dilatons by recalling briefly the
ingredients of scale invariance in the case of a classical, massless
field.  We use for this purpose the scalar fields we discussed in the
general context of solitary waves, but ultimately we shall have
reference to QCD.  Thus we take a general lagrangian of the form
\be
\L = \half \p\lmu\f(x)\,\p\umu\f(x) - V[\f(x)],
\ee
where we shall consider more specifically the free lagrangian for a
particle with mass,
\be
\L_0 = \half\p\lmu\f\,\p\umu\f - \half m^2\f^2,
\ee
and a massless $\f^4$ theory,
\be
\L = \half\p\lmu\f\,\p\umu\f - {\lambda\over 4!}\f^4.
\ee
Recall that the action---dimensionless---is $ S \equiv \int\L\, d^4x,$ 
so that it is easy to read off the dimensions of the $f$ field, namely,
$[\f] = {\rm energy} = 1/{\rm length}.$

An infinitesimal dilatation is given by
\be
{x'}\umu = (1 + \alpha)x\umu, \quad\quad \f' = (1 - \alpha)\f,
\ee
and leaves the action invariant for $d = 4.$  The Noether current
corresponding to this symmetry is
\be
j\umu = -\bigg[g_\rho\umu\,\L - {\p\L\over\p(\p\lmu\f)}\p_\rho\f\bigg]
 {\delta x^\rho\over\delta\o} 
- {\p\L\over\p(\p\lmu\f)}{\delta\f\over\delta\o},
\ee
or, suppressing $\alpha,$
\bea
j\D\umu & = & \bigg[-g_\rho\umu\,\L + \p\umu\f\,\p_\rho\f\bigg]x^\rho
- \big(\p\umu\f\big)\big(-\f\big) \nono
& = & j_\rho\umu\,x^\rho + \half\p\umu\f^2,
\eea
where $j_\rho^\mu$ is the energy-momentum tensor (i.e., the Noether 
current for $x'^\mu = x\umu+\epsilon\umu$ and $\f'=\f$).  Thus
\be
\p\lmu j\D\umu = j\lmu\umu + \half\p\umu\p\lmu\f^2
\ee
since
\be
\p\lmu j_\rho\umu = 0.
\ee

For
\be
\L = \half\p\lmu\f\,\p\umu\f - {\lambda\over 4!}\f^4
\ee
we then have
\bea
j_\mu^\mu & = & -g_\mu^\mu \L + \p\umu\f\,\p\lmu\f  \nono
& = & -4\bigg(\half\p\lmu\f\,\p\umu\f - {\lambda\over 4!}\f^4\bigg) 
+ \p\umu\f\,\p\lmu\f  \nono
& = & -\p\lmu\f\,\p\umu\f + {\lambda\over 6}\f^4.
\eea
The equation of motion is
\be
\p\lmu{\p\L\over\p(\p\lmu\f)} - {\p\L\over\p\f} 
= \p\lmu\p\umu\f + {\lambda\over 6}\f^3 = 0,
\ee
so that
\bea
\p\lmu j_{\rm D}^\mu & = & j_\mu^\mu + \half\p\lmu\p\umu\f^2 \nono
& = & j_\mu^\mu + \f\p\umu\p\lmu\f + \p\lmu\f\,\p\umu\f \nono
& = & \f\,\p\umu\p\lmu\f + {\lambda\over 6}\f^4 = 0,
\eea
and the dilatation current is indeed conserved.

If a mass term is present,
\be
\p\lmu j_{\rm D}^\mu = m^2\f^2 \ne 0
\ee
because a dimensioned parameter appears in $\L$ and the corresponding
action behaves as
\bea
S_{\rm m} & = & -\half m^2\int\f^2 d^4x \nono
& \rightarrow & (1+\alpha)^2 \bigg(-\half m^2\int\f^2 d^4x\bigg).
\eea
For higher spin, as in the QCD case, dilatation is linked to a traceless
energy-momentum tensor.  

The crucial consideration for our purposes is that when quantal effects
are introduced dilatation invariance breaks down even if it was present
at the classical level.  This is easily seen within the same $\f^4$
theory by merely considering the lowest, one-loop corrections, which
have the effect of replacing the lowest-order $\f^4$ coupling constant
$\lambda_0$ with
\be
\lambda = \lambda_0 - {i\over 2}\lambda_0^2
\big[I(p_1+p_2) + I(p_1+p_3) + I(p_2+p_3)\big],
\ee
where
\bea
I(p) & = & \int{d^4k\over (2\pi)^4}
 {1\over (k^2+i\epsilon)[(k+p)^2+i\epsilon]}  \nono
& = & {i\over 16\pi^2}\log{\Lambda^2\over -p^2+i\epsilon}.
\eea
Thus
\be
\lambda = \lambda_0 - {3\lambda_0^2\over 32\pi^2}
\log{\Lambda^2\over \mu^2}, \quad p_1^2=p_2^2=p_3^2=p_4^2=-\mu^2,
\ee
so that the need for a cutoff $\Lambda$ introduces a scale which was 
hidden in the classical theory.  Renormalization can lead to spontaneous 
symmetry breaking and a massless classical theory may acquire a
mass in the quantum theory.  Dilatation invariance and tracelessness
are then broken.  This happens in QCD: that is why hadrons, which are made 
up of (nearly) massless quarks and thus should be expected to be
(nearly) massless themselves, have definite, nonzero masses.

\subsection{\label{dilaton_specifics} The dilaton in the context of
skyrmions}

As we have hinted, (see, e.g.,\ci{Yn,CL,Cole} for further detail) QCD 
with massless quarks is scale invariant
when treated as a classical theory and loses this property upon
quantization.  Accompanying this is the trace anomaly, whereby the
trace of the energy-momentum tensor $T^{\mu\nu}$, which vanishes
classically, is given after quantization by
\be \label{trace}
T_\mu^\mu = \partial_\mu\,D^\mu 
= -\frac{9\alpha_s}{8\pi}G_{\mu\nu}^aG^{a\mu\nu} \equiv \psi^4,
\ee
where $D^\mu (= T^{\mu\nu}x_\nu)$ is the dilatation current, 
$\alpha_s$ is the QCD coupling constant, $G_{\mu\nu}^a$ is the gluon 
field, and $\psi$ is an order-parameter 
field---the dilaton---which represents the scalar glueball formed from
the contraction of the two gluon fields; $\psi$ is taken in the fourth
power in anticipation of its use as a scalar field of dimension 1.  It
was early pointed out that an effective lagrangian based on this scalar
field can be used to incorporate the trace anomaly\ci{Schech}, and the
consequences of such an approach for scalar glueballs were studied some
time thereafter\ci{GJJS86a}, along with modifications to the Skyrme
lagrangian in order to incorporate there first the appropriate scale 
invariance and then the trace anomaly\ci{GJJS86b,JJS}.  The physical
picture for the skyrmion plus dilaton is that a kind of bag is
formed by the dilaton field such that the skyrmion is located within a
region where the gluon condensate is suppressed.  In many ways this
picture is the natural extension of previous studies based on chiral or
cloudy bags\ci{Tho}.  It also answers for one of the obvious limitations
of the skyrmion as it stands alone, namely, that there is only one length
parameter in the basic model, so that there is no well-defined concept of
a ``surface thickness'' or baryon edge.  Since the skyrmion is
characterized by the lagrangian of eq.~(\ref{Lagrangian}) in terms of the
energy unit $F_\pi/e$ and length unit $eF_\pi,$ as noted below eq.
(\ref{ode2}), the need to fit single-baryon properties automatically
introduces a scale of hundreds of MeV for the energy and about a fermi for
sizes.  Thus when we search for attraction in the $NN$ force we must
overcome the tail of the very strong repulsion at zero separation which
still persists strongly in the range of 2 fm or so of relevance
for attraction.  Obviously, the sharpening of the baryon surface will help
to reduce this problem.

Schechter\ci{Schech} and later his coworkers\ci{GJJS86a,GJJS86b,JJS}
introduced a lagrangian for the dilaton field through the
consideration of the trace anomaly.  If terms with no more than two time
derivatives appear, the most general form allowed is
\be \label{L_dilaton}
 {\cal L}_{\rm dilaton} = {\cal L}_{\rm 2\ dilaton} 
- V_{\rm dilaton}(\psi)
= \half b^2 \partial_\mu\psi\partial^\mu\psi
-\fourth \psi^4\log\left[\frac{\psi^4}{\Lambda^4}\right],
\ee
where $b$ is a dimensionless constant and $\Lambda$ is a constant with
dimensions of energy.  
For the mean-field case, eq.~(\ref{L_dilaton}) then leads to 
$\la\psi\ra = \Lambda/\e^{1/4}.$  

Given the dilaton lagrangian, it is
easy to modify\ci{GJJS86b,JJS} the basic Skyrme lagrangian
(\ref{Lagrangian}) in order to incorporate the trace anomaly, namely,
\be \label{L_skdil}
 {\cal L} = {\cal L}_{\rm dilaton} 
-{F_\pi^2 \over 16}\frac{\psi^2}{\la\psi\ra^2} \tr(L_\mu  L^\mu )
+ {1 \over 32 e^2} \tr[L_\mu,L_\nu]^2,
\ee
where the kinetic term is modified on scaling grounds by a term with
$\psi^2$ and the Skyrme stabilizing term requires no such change.
It is common practice to replace $\psi$ with an exponentiated scalar
field $\sigma,$ 
\be \label{psi_sigma}
\psi \equiv \la\psi\ra\,\e^{\sigma},
\ee
whence the dilaton potential is
\be \label{dilaton_potential}
V(\sigma) = \fourth \, C_G \, \e^{4\sigma}(1-4\sigma),
\ee
taking the pion mass to vanish.  The overall
lagrangian, including the terms with six derivatives of section
\ref{higher_order}, is then
\bea \label{L_skydil}
 {\cal L} & = & {\cal L}_{2\ {\rm dilaton}} +
 {\cal L}_2 + {\cal L}_4 + {\cal L}_6 - V(\sigma) \nono
& = & \e^{2\sigma}\bigg[\half\Gamma_0^2\,
\partial_\mu\sigma\,\partial^\mu\sigma
- {F_\pi^2 \over 16}\tr(L_\mu L^\mu)\bigg] 
+{1 \over 32 e^2}\tr[L_\mu,\, L_\nu]^2 \nono
& - & \epsilon^2 \e^{-2\sigma} B_\mu B^\mu 
- {C_G \over 4}[1 + \e^{4\sigma}(4\sigma-1)].
\eea
The two parameters $\Gamma_0$ and $C_G$ are related to the glueball---or
scalar field---mass through $m_\sigma^2 = 4 C_G/\Gamma_0^2.$  For 
frequently-used values for these parameters this may yield a rather low 
mass for the $\sigma$ glueball field, which one must assume then
mixes strongly with $\pi\pi$-exchange simulations of
$\sigma$-like effects; nonetheless, the virtue of the dilaton approach is
that is provides a natural source for this scalar rather than lying on a
completely ad hoc introduction of it into the theory.  (Other criticisms
of the use of the dilaton in nuclear-matter applications have been noted
in ref.\ci{Birse}.)

Once the dilaton is coupled to the skyrmion, it provides a very natural
source of central attraction\ci{YSH}.  The usual product ansatz must be
modified by an assumption of dilaton additivity,
\be \label{additivity}
\sigma_{B=2} = \sigma_1 + \sigma_2,
\ee
which is assumed to be of sufficient accuracy for purposes of calculating
the $NN$ force since mixed terms are cut out in the function $\psi_{B=2}
= \la\psi_{B=2}\ra\,\e^{\sigma_{B=2}}.$  Indeed, ref.\ci{YSH} 
finds some 35 MeV of
attraction in the central $NN$ potential for $r \sim 1.8$ fm.  We return
to the application of the dilaton to the $NN$ potential in section
\ref{SO} below, both with regard to its use for $V_C(R)$ and for the
spin--orbit potential.

\section{\label{full} The exact skyrmion solution for $B = 2$}

Recently solutions of the full $B = 2$ skyrmion problem in three
dimensions (or, exploiting the cylindrical symmetry of the problem, in
two dimensions) have become increasingly available.  The review of
Walhout and Wambach\ci{WWrev} devotes itself in considerable degree to
this topic, in which these authors have themselves been very much
involved.  Therefore the first part of the present section is to be read 
in conjunction with that review; more recent developments stemming from
Manton and coworkers will also be touched upon briefly below.

The earliest solutions\ci{JR} for the $B = 2$ skyrmion, 
noted briefly in section \ref{NN} above, dealt with a spherically
symmetric profile function which automatically pertains to two baryons
centered on the same point.  Apart from the fact that such a
configuration has very large energy ($\sim 3M_N$), related to the strong 
short-range
repulsion of the two-nucleon system, this configuration is of little
physical interest; it is, of course, highly unstable.  Early exact
numerical solutions\ci{VWWW,KS,SDB,BC} for the $B = 2$ system were 
obtained by relaxing the full static equations of motion on a grid within
a particular geometry.  These solutions have a
toroidal shape, the hole in the middle being, presumably, the reflection
of short-range repulsion.  They do not show any residual signs of the
presence of two separate nucleons, which raises the criticism noted by 
Manton\ci{Man}
that this solution does not allow sufficient degrees of freedom to 
describe correctly the complete dissolution into two baryons.  

Subsequent exact numerical studies were usually based on an assumption of
adiabaticity, whereby one introduces constraints to fix the collective
coordinates and solves the static equations of motion (as opposed to
trying to follow the full time evolution of two skyrmions aimed at each
other; see\ci{VWWW,AKSS,CB,CSB}).  The fixing of coordinates pertains
both to the relative orientation of the skyrmions and to the distance
between their centers.  The latter may be determined, say, through Lagrange
multipliers,\footnote{A hybrid model\ci{KM85,Kal93a}  
involving the full numerical
solution of a product ansatz form has the advantage that it automatically
places the skyrmions on well-defined centers and maintains $B = 2$
automatically.} while the relative orientations between the two skyrmions
are chosen in the same manner as for the instanton solution.  The 
solutions\footnote{Related approximative calculations involving simpler
numerical methods have also been carried 
out\ci{ALOW,Oka91,TZ,OKSS}.}\ci{WWrev,WW91} then
show two separated nucleons which coalesce at about 1.2 fm for the 
maximally attractive relative orientation of the skyrmions, and then meld
into a toroid below that ``separation'' distance.  A very small amount of
central attraction may be found in this approach; this is somewhat
increased by admixing\ci{WAH,WA,Wal93} $N\Delta,$ $\Delta N,$ and 
$\Delta\Delta$ states, as discussed in subsection 
\ref{baryon_resonances}.  Attraction is also augmented by 
incorporating\ci{WW92} the dilaton, in which case the ``bags'' 
provided by the dilaton merge at about 1.4 fm, and an 
attractive central component on the scale of 20 MeV is found.  This is
further enhanced by again considering the admixture of states
containing $\Delta$s.

The ability to carry out these difficult calculations is encouraging, and no
doubt goes far towards providing a mechanism of central attraction in
two-nucleon
interactions in the skyrmion approach, but more must be done to refine
them in order to be sure that all artifacts of the numerical procedures have
been fully eliminated.\footnote{In particular, it is even 
difficult\ci{Kal93b} fully to ascertain if the two-skyrmion 
solution remains on
the same manifold as the free skyrmions approach each other and merge
into a torus.}  This may also lead to a reliable assessment as to
which ingredients are essential to obtain central attraction.
The connections with others approaches should also be
pursued further in the hope of eventually constructing a model simple enough 
for application in more complex nuclear situations.

Quite recently Leese, Manton, and Schroers\ci{LMS} have produced a
skyrmion solution for the deuteron carried out on a ten-dimensional
manifold which yields properties close to those of the physical 
deuteron.\footnote{A discussion of this approach, along with criticism
of it, has been given recently in\ci{Wal96}.}
It should be emphasized that this is the first case of a realistic
representation of the deuteron within the skyrmion approach.
The $NN$ system should ideally be
calculated on a twelve-dimensional manifold in order to include full
freedom for the locations of the two skyrmions and for their SU(2)
orientations.  This calculation still proves prohibitively difficult,
however, and so ref.\ci{LMS} uses the lowest-dimensional manifold that
still includes both a toroidal configuration of minimal energy and
configurations that approximate the infinite separation of two skyrmions
oriented in the maximally attractive relative direction in SU(2).  To
perform the calculation, an instanton assumption is made
and the maximally attractive 
orientation---rotation by $\pi$ about the axis perpendicular to the line 
joining the skyrmions---is taken.  The geometry that
underlies the instanton is somewhat complicated in order to
accommodate the limiting toroidal and separated configurations.  The
technical aspect of the calculation then involves evaluating the
lagrangian after restricting to that geometry, and generating the
hamiltonian with first-quantized collective degrees of freedom.
Numerical values for deuteron properties in this approach and in that of
ref.\ci{BC} are shown in Table 1, from which it is seen that results
from the ten-dimensional manifold $M_{10}$ are far closer to those of 
experiment than are the ones arising\ci{BC} from an eight-dimensional 
one $M_8,$ because the larger dimensionality is necessary in order to
allow the skyrmions to sample a space that allows also for genuine
separation of the two nucleon centers.
\begin{table}
\centerline {
\begin{tabular}{cccc}    
\hline
&&&\\
Property & $M_{10}$ (ref.\ci{LMS}) & $M_8$ (ref.\ci{BC}) &
Experiment\\
&&&\\
\hline
&&&\\
$\epsilon_D$ [MeV] &  $-6.18$ &   $-158$ &  $-2.225$\\
&&&\\
&&&\\
$ r_c$ [fm]  & $2.18$   &   $0.92$ & $2.095$ \\
&&&\\
&&&\\
$Q$ [fm$^2$] &  $0.83$ &   $0.082$  &  $0.2859$\\
&&&\\
&&&\\
$\mu$ [n.m.]   & $0.55$   &   $0.74$  & $0.8574$ \\
&&&\\
\hline
\end{tabular}}
\caption{Deuteron properties.  Here $M_{10}$ refers to a ten-dimensional 
manifold and $M_8$ to an
eight-dimensional one; $\epsilon_D$ is the deuteron binding energy,
$r_c$ is the charge radius, $Q$ is the quadrupole moment, and $\mu$ the
magnetic moment. From ref. \protect\ci{LMS}.}
\end{table}

\section{\label{Snonzero} Systems with nonzero strangeness}

\subsection{\label{Snonzero_genl} General issues}

If nothing else, this review should by now have convinced the reader that
the origins of central attraction at medium ranges in the $NN$ force are
still unclear.  We have a loose understanding of how it might come about
in models that base themselves on QCD, but little more than that.
And the skyrmion, with perhaps the best claim amongst such models to a 
genuine QCD pedigree, mostly serves to stress that central attraction is
a subtle affair, which may derive from some or all of the proposed
mechanisms we have noted, or, indeed, from others.  Nor is it clear if 
the problems lie in the basic notion of the skyrmion (i.e., of the large 
$N_c$ limit leading to mesonic degrees of freedom
interpreted as baryons through a topological argument) or in the
particular lagrangian proposed for it (i.e., might 
medium ranges require more field derivatives?).  All of
this makes it especially interesting to consider systems with nonzero
strangeness, where very similar mechanisms may enter and one may be able
to highlight specific features of the $B = 2$ interaction by
exploiting the different quantum numbers of the $S \ne 0$ systems.  

\subsection{\label{SU3} SU(3) extensions of the skyrmion}

Early efforts to extend the skyrmion to SU(3) failed because of various
technical errors in treating flavor symmetry breaking.  With the work of 
Yabu and Ando\ci{YA} and of Callan, Hornbostel, and Klebanov\ci{CHK} 
the proper treatment of this crucial feature, within various possible 
theoretical approaches to the problem, was given,
and a good descriptions of the single-baryon system
were obtained.  We shall here follow the method of Yabu and Ando, which
is somewhat more general.  In addition to the ${\cal L}_2$ and 
${\cal L}_{4}$ terms of eq.~(\ref{Lagrangian}), these authors 
include\footnote{It was early known\ci{GL} in the context of chiral
perturbation theory, but realized\ci{PSW,PEGJ} only rather late for 
skyrmions, that a stabilizing form additional to that of 
eq.~(\ref{Lagrangian}) can occur in SU(3), namely, ${\cal L}_{4b} = 
\tr\left(L_\mu\, L_\nu\right)\tr\left(L^\mu\, L^\nu\right) -
\tr\left(L_\mu\, L^\mu\right)\tr\left(L_\nu\, L^\nu\right),$ which
is equivalent to ${\cal L}_4$ in SU(2).  Unfortunately, this new term 
offers no advantages
where the search for central $NN$ attraction is concerned\ci{KPE}.
(Recall that we encountered the existence of terms equivalent in SU(2)
but not in SU(3) also for cases with six derivatives in subsection
\ref{extensions}.)}\ci{YA} the symmetry-breaking term
\bea \label{SB}
L_{\rm SB} & = & \int d\r\ F_\pi^2 \,
\Bigg\{\frac{1}{32}\bigg(m_\pi^2 + m_\eta^2\bigg)\tr(U + U^\da - 2)
\nono
& + & \frac{\sqrt{3}}{24}\bigg(m_\pi^2 - m_K^2\bigg)
\tr\left(\lambda_8(U + U^\da)\right)\Bigg\},
\eea
where here and in the following the $\lambda_i,\,i=1,\ldots,8$ refer 
to the eight Gell-Mann
SU(3) matrices.  The SU(3) flavor symmetry is broken here by the
appearance of different masses---$m_\pi,\,m_\eta,$ and $m_K$
for the $\pi,$ $\eta,$ and $K$ mesons---of the octet.  We note that
further symmetry breaking could be included by taking also different
decay constant for the various mesons, rather than using $F_\pi$ for all
of them.
The symmetry-breaking term is taken such that when the field is expanded as
$U = \exp\left[2i/F_\pi\sum_{i=1}^8\lambda_i\,\phi_i\right]$ about the
classical vacuum $U = 1$ one can immediately read off the meson masses; the
form of eq.~(\ref{SB}) then automatically satisfies the Gell-Mann--Okubo mass
formula
\be \label{GMO}
m_\pi^2 + 3 m_\eta^2 - 4 m_K^2 = 0.
\ee

For the SU(3) case it is also necessary to include the
Wess--Zumino term (which vanishes in SU(2)),
\bea \label{WZ}
L_{\rm WZ} & = & \frac{N_c i}{240\pi^2} \int_Q
d\Sigma^{\alpha\beta\gamma\delta\epsilon} \nono
& \times &\tr\left[\big(\partial_\alpha U\big)U^\da\,
\big(\partial_\beta U\big)U^\da\,
\big(\partial_\gamma U\big)U^\da\,\big(\partial_\delta U\big)U^\da\,
\big(\partial_\epsilon U\big)U^\da\right],
\eea
which is given as an integral over a five-dimensional disk $Q,$ the
boundary
of which is the compactified space-time four-sphere.

The embedding of the SU(2) hedgehog into SU(3) is done through
\be \label{embedding}
U_0(\r) = \exp\left[i \sum_{i=1}^3 \lambda_i \hat\r_a F(r)\right].
\ee
The introduction of collective rotational coordinates goes through as for
SU(2), with, of course, the extra degrees of freedom arising in SU(3).
This leads to the hamiltonian
\bea \label{Ham}
H & = & M_{\rm cl} + \half\left[\frac{1}{\alpha^2} - \frac{1}{\beta^2}\right]
C({\rm SU}_{\rm R}(2)) - \frac{3}{8\beta^2} + \frac{1}{2\beta^2}
C({\rm SU}_{\rm L}(3)) \nono
& + & \half\gamma\left(1 - D^8(A)\right),
\eea
where $C({\rm SU}_{\rm R}(2))$ and $C({\rm SU}_{\rm L}(3))$ are the
Casimir operators for ${\rm SU}_{\rm R}(2)$ and ${\rm SU}_{\rm L}(3),$
and $D^8(A) = \half\tr\left(\lambda_8 A^\da \lambda_8 A\right),$ with $A$ 
the rotation matrix as in eq.~(\ref{AUA}) but now in SU(3).  The moment of 
inertia for SU(2) repeats that of eq.~(\ref{lambda}), $\alpha^2 = \lambda,$
while that for SU(3) is given by
\be \label{betasq}
\beta^2 = \frac{\pi}{2 e^3 F_\pi}\int_0^\infty (1 - \cos F)
\left[{\tilde r}^2 + \left(2 \sin^2 F + (\tilde r F')^2\right)\right]
d\tilde r,
\ee
and the mass term is
\be \label{gamma}
\gamma = \frac{4\pi}{3 e^3 F_\pi} \left(m_K^2 - m_\pi^2\right)\int_0^\infty
 {\tilde r}^2 (1 - \cos F) d\tilde r.
\ee

Energy splittings for the baryon octet and decuplet spectra found\ci{YA}
with this method agree remarkably well with experiment, although there is a 
persistent need to adjust the kaon mass upwards for this purpose and to
subtract the energy of a state having the quantum numbers of the vacuum, this
subtraction typically being on the scale of half the nucleon mass or so.
As with the SU(2) case, fitting to the mass spectrum yields too low a value 
for $F_\pi,$ although this situation can be improved somewhat using an
approach due to Kanazawa\ci{Kan}, which yields\ci{KE90,KES} 
reasonable results for nucleon observables.

In order to give some impression of the degree to which the skyrmion
can match SU(3) expectations we show here the results of some studies\ci{KES} 
of global parameter fits for the baryon octet and decuplet.
The lagrangian used for this purpose (motivated by
considerations of obtaining nonzero proton spin content from the
skyrmion) is
\bea
{\cal L} & = & {\cal L}_2 + {\cal L}_{4}
+ {\cal L}_{6,1} + {\cal L}_{6,2} + {\cal L}_{SB} \nono
& = & -{F_\pi^2 \over 16} \tr(L_\mu  L^\mu )
+ {1 \over 32 e^2} \tr[L_\mu,L_\nu]^2 \nono
& - &\epsilon_1 {g_\omega^2\over m_\omega^2} 
\tr ({\cal B}^\mu {\cal B}_\mu)
- \epsilon_2 {g_\omega^2\over 2 m_\omega^2} 
\tr ({\cal B}^\mu) \tr ( {\cal
B}_\mu)
\nono
& + & \bigg[{F_\pi^2 \over 32}(m_\pi^2 + m_\eta^2) \tr (U + U^\da - 2)
\nono
& + & {\sqrt{3}F_\pi^2 \over 24}(m_\pi^2 - m_K^2) \tr (\lambda_8(U +
U^\da))
\bigg],
\eea
where, as earlier, $L_\mu \equiv U^\da\partial_\mu U$ 
and ${\cal B}_\mu$ is given by eq.~(\ref{B_no_trace}).
The resulting octet and decuplet masses are shown in Table 2, along
with the deviation from experiment, which is about at the ten percent
level or less.  Octet magnetic moments are shown in Table 3 and are
also surprisingly good.  Last, we show in Table 4 a comparison with the
proton spin content, which is also not unreasonable.  Thus the application
of skyrmions to the full SU(3) baryon octet and decuplet is
generally quite successful.

\begin{table}
\centerline{
\begin{tabular}{ccccccccc}    
\hline
&&&&&&&&\\
\#&\multicolumn{3}{c}{Params}&\multicolumn{4}{c}{M}&dev\\
\hrf&\multicolumn{3}{c}{\hrf}&\multicolumn{4}{c}{\hrf}&\hrf\\
&$e$&$F_\pi$&$\epsilon_1$
&$N$/$\Delta$&$\Lambda$/$\Sigma$&$\Sigma$/$\Xi$
&$\Xi$/$\Omega$&\\
\hline
1&6.0&82&-1.00&938&1111&1240&1366&98\\
&&&&1226&1336&1485&1649&\\
2&6.0&74&-1.50&936&1114&1246&1375&120\\
&&&&1234&1338&1469&1624&\\
3&6.0&89&-0.35&928&1124&1271&1402&131\\
&&&&1233&1342&1488&1659&\\
4&6.6&82&-0.71&933&1108&1238&1364&99\\
&&&&1249&1351&1480&1633&\\
5&5.4&82&-1.26&919&1119&1272&1410&159\\
&&&&1203&1317&1469&1643&\\
6&8.0&186&-1.45&939&1120&1253&1387&120\\
&&&&1236&1344&1476&1634&\\
exp&&&&939&1116&1193&1318&\\
&&&&1232&1385&1530&1672&\\
\hline
&&&&&&&&\\
\end{tabular}}
\caption{Octet and decuplet masses in MeV.  Members of octet in first 
line of entry; decuplet in second.  Cases 1--5: $g_A \sim 1.24 \pm 0.05$;
Case 6: $g_A = 1.72.$}
\end{table}

\begin{table}
\centerline{
\begin{tabular}{ccccccccc}    
\hline
&&&&&&&&\\
\#&\multicolumn{7}{c}{$\mu$}&dev\\
\hrf&\multicolumn{7}{c}{\hrf}&\hrf\\
&p&n&$\Lambda$&$\Sigma^+$&$\Sigma^-$&$\Xi^0$&$\Xi^-$&\\
\hline
1&2.65&-1.73&-0.83&2.80&-1.10&-1.75&-0.89&0.73\\
2&2.80&-1.77&-0.90&2.98&-1.12&-1.86&-0.99&0.93\\
3&2.54&-1.72&-0.75&2.64&-1.13&-1.65&-0.79&0.58\\
4&2.49&-1.57&-0.77&2.65&-1.02&-1.62&-0.88&0.69\\
5&2.87&-1.98&-0.88&3.01&-1.24&-1.90&-0.89&0.93\\
6&2.33&-1.76&-0.78&2.42&-0.97&-1.65&-0.65&0.68\\
exp&2.79&-1.91&-0.61&2.46&-1.16&-1.25&-0.65\\
\hline
&&&&&&&&\\
\end{tabular}}
\caption{Octet magnetic moments in units of $\mu_N$}
\end{table}

\begin{table}
\centerline{
\begin{tabular}{cccc}
\hline
&&&\\
\#&$1/\alpha^2$ [MeV]&$1/\beta^2$ [MeV]&$\Delta\Sigma$\\
\hline
1&130&536&0.17\\
2&132&615&0.34\\
3&130&452&0.04\\
4&145&574&0.14\\
5&115&473&0.17\\
6&130&810&0.58\\
exp&&&$0.27\pm 0.13$\\
\hline
&&&\\
\end{tabular}}
\caption{Proton spin content}
\end{table}

\subsection{\label{YN} $YN$ and $YY$ forces}

The extension to SU(3) of the previous SU(2) methods based on the product 
ansatz and coupling to rotational and vibrational degrees of freedom shows
similar amounts of attraction as were found before\ci{KE91}.  This extension 
also allows for the study\ci{KE92,SSG} of $\Sigma N,$ $\Lambda N,$ and 
$\Lambda\Lambda$ interactions, to list only those for which there exist 
some rough data at present.  (Since the product ansatz is used, this approach
does not allow for study of the $H$ particle, which will be addressed
briefly in the next subsection.)  Once again central attraction is found if
Roper vibrations are admixed, as for the $NN$ case.  The early work\ci{KE92}
in this sector was based on Monte Carlo methods for the projection onto
baryon states with appropriate quantum numbers, which has an 
advantage of algebraic simplicity and flexibility, but inevitably
carries with it the the need for long computer runs and the imprecision of 
the statistical method.  Central attraction as strong or stronger than
for the $NN$ system was found.  

The more accurate algebraic approach\ci{SSG}
found sizable attraction in systems with identical baryons, for example,
$NN$ and $\Lambda\Lambda$, when further symmetry breaking was introduced 
over and above that of eq.~(\ref{SB}), i.e., when using $F_\pi \ne F_K$.  
This attraction increases with increasing strangeness content, so that at
small interbaryon separation the central $\Xi\Xi$ potential is about
four times more attractive than the $NN$ one.  The bulk of the attraction
from this mechanism occurs at values of $R \le 1$ fm, however, which
reduces somewhat its relevance for conventional central baryon--baryon 
attraction.
More recently, efforts are underway\ci{TH2} to study $YN$ interactions
using stabilization mechanisms\ci{Dal} that may allow for somewhat
simpler calculational procedures.

\subsection{\label{H_sec} The $H$-particle}

An early suggestion by Jaffe\ci{Jaffe}, based on the MIT 
bag model, predicts the existence of a bound particle, called the $H,$ with 
quantum numbers $J^\pi = 0^+,\ I = 0, Y = 0$ and a mass around 2150 MeV;
this is a few tens of MeV below the $\Lambda\Lambda$ threshold for
strong decay.  In the quark model this particle, called the $H,$ involves
the six quarks $\{u,\,u,\,d,\,d,\,s,\,s\}$ in a fully-symmetric configuration.
It is the object of experimental search (reviewed, e.g., in 
refs.\ci{Imai,Quinn}) to the present day.  The possible existence of 
such a state
was studied within the context of skyrmion approaches almost from the
beginning of this model\ci{BBLRS,JK,BLR,YN} and also more recently\ci{TSW}.  
The earlier work used the ansatz
\be \label{U_H}
U_H(\r) = \exp\left\{i\hat\r\cdot\bfLambda\,f(r) 
+ i\left[(\hat\r\cdot\bfLambda)^2 - \frac{2}{3}\right]\,g(r)\right\},
\ee
where $\bfLambda = \{\lambda_7,-\lambda_5,\lambda_2\}$ and 
$f(0) = g(0) = \pi$ and $f(r\to\infty) = g(r\to\infty) = 0.$  More recently,
the kaon bound-state ansatz of ref.\ci{CHK} has been used\ci{TSW} in this
context.  Unfortunately, the uncertainties in the skyrmion approach are
sufficient so that it is difficult to reach an unambiguous statement as to
whether the $H$ is bound or not in this model.  Worse still, other features
such as the interaction with the continuum and vacuum effects
(the Casimir energy---see subsection \ref{ambiguities}) make the situation 
even less clear.

\section{\label{SO} The spin--orbit force}

The study of the $NN$ isoscalar spin--orbit force parallels in some respects
the earlier work on the isoscalar central force:  Lowest-order 
product-ansatz results again\ci{RN87,RD,OSYKS} show the wrong sign, that
is, they find repulsion instead of attraction.  Riska and Schwesinger\ci{RS}
then suggested that the term ${\cal L}_{6,2}$ of eq.~(\ref{L62}) would yield 
attraction, which indeed it does, but only at the cost of an unusually large 
coupling constant for this term\ci{Abada}.  The introduction of the 
dilaton\ci{RS} can enhance this attraction, as has been worked out more 
fully\ci{KE95b} using the lagrangian of eq.~(\ref{L_dilaton}).   In the 
course of carrying out such calculations one must deal with 
the ``dragged'' skyrmion,
that is, the spin--orbit force is identified through the appearance of a term
involving the motion of the baryon itself which must enter in the orbital
angular momentum operator ${\bf L} = \R \times {\bf P}$ coupled to the spin 
$\bfsigma.$
All this comes about through a collective coordinate referring to the baryon
position, in addition to the collective rotation.  One has
\be \label{Udot}
\dot U(\r-\r_k(t)) = A_k(t)\left\{\frac{i}{2 I}[\bftau\cdot{\bf j}_k,\,
U_0(\r-\r_k)] - \frac{\p_k}{M}\cdot\bfnabla U_0(\r-\r_k)\right\}A_k^\da(t),
\ee
where $M$ and $I$ are the skyrmion mass and moment of inertia as 
hitherto.

\begin{figure} 
\centerline{\psfig{figure=brazil_fig1.ps,height=3.0in}}
\caption{Spin-orbit potential and central potential (dashed line for 
$r \ge 2$ fm) for skyrmion parameters $F_\pi = 143$ MeV and $e = 20.0,$
and dilaton parameters $\Gamma_0 = 137$ MeV and $C_G = (121\ {\rm
MeV})^4,$ yielding a nucleon mass $M_N = 942$ MeV and a $\Delta$ mass 
$M_\Delta = 1174$ MeV.  This result requires the presence of an 
${\cal L}_{6,2}$ term and a dilaton. \label{fig:1}}
\end{figure}

\begin{figure} 
\centerline{\psfig{figure=brazil_fig2.ps,height=3.0in}}
\caption{Spin-orbit potential and central potential as in fig. 1, but
with dilaton parameters $\Gamma_0 = 61.2$ MeV and $C_G = (157.3\ {\rm
MeV})^4$ yielding a glueball mass of 809 MeV. \label{fig:2}}
\end{figure}

Choosing the
dilaton parameters $\Gamma_0 = 137$ MeV and $C_G = (121\ {\rm MeV})^4,$ with
the empirical value for the $\omega NN$ coupling constant,
$g_\omega^2/4\pi \sim 10,$ sizable attraction is found\ci{KE95b}, as seen 
in fig.~1, both for the central and the spin--orbit isoscalar components.
These dilaton parameters imply a very low value for the glueball mass
($\sim 214$ MeV) which is inconsistent with empirical expectations.  If
we take $\Gamma_0 = 61.2$ MeV and $C_G = (157.3\ {\rm MeV})^4$ for a
higher glueball mass of 809 MeV we get the results of fig.~2, with about
an order of magnitude less attraction, or about the same well depth as
implied by vibrational admixtures (section \ref{baryon_resonances}).
Since there are sizable technical 
difficulties\ci{Oka86,OMY,RN87,SWA,Schroers,GP,KE95b,ASW} in considering 
the moving 
skyrmion---both at the computational level and in matters of 
principle---it would be premature to argue that the spin--orbit problem
is solved.  Indeed in addition to this special aspect of the spin--orbit force 
it also suffers from all the uncertainties noted for the more ``static'' 
central force.  The particular choice of dilaton parameters also leads to
a very low glueball mass.  Although the dilaton mechanism itself is
probably a well-motivated ingredient of skyrmion physics by virtue of the
need to encompass the QCD trace anomaly, the appropriate values for the
glueball mass involved are not clear and may well have to be substantially
higher than the one used here (cp.\ci{TR}.).

\section{\label{Bbeyond3} The nucleon in the nucleus}

In this section we wish briefly to hint at possible applications of
skyrmions to problems of a baryon in a medium which will become
accessible once the two-baryon problem is under good control.  The cases
noted all relate to problems of active interest in nuclear physics at
large.  (We do not here address the large domain of problems involving
skyrmion at nonzero temperatures which are of relevance to the study of 
hot hadronic matter and of the quark-gluon plasma.)

\subsection{\label{Bis3} The $B = 3$ system}

One of the characteristic features of the strongly-interacting many-body
system---the nucleus---is the possibility of important
contributions from three-body forces.  These were early studied\ci{MK} 
using skyrmions, but without projection onto nucleon states through the 
methods discussed in eqs. (\ref{nucleon})--(\ref{a_average}).  As a
result, the forces found were very large ($\sim 100$ or
200 MeV).  Projection onto nucleon states immediately tames these
forces, producing\ci{KE93} potentials of 10 MeV or less, as one would
have expected for three-nucleon forces.  In addition to applications in
systems of zero strangeness, the three-body force is also of interest in
$\Lambda NN$ systems\ci{Us}, among others, for studies of the binding of
hypernuclei.

\subsection{\label{CT} Color transparency}

There has been great interest in recent years (reviewed, e.g., in
ref.\ci{FMS}) in studying the phenomenon of color transparency, in
which a nucleon is ejected from the nucleus with very high momentum
transfer and experiences little final-state interaction as it exits:
The high momentum transfer ``catches'' the nucleon in a small
configuration, and, because the color charges of a small QCD object
partially neutralize each other, the nucleon emerges while interacting
weakly with the other nucleons.  Thus color dynamics makes
the nucleus transparent to the ``small'' nucleon (referred to here as a 
minisculon $m$).  The
crucial QCD ingredient in this process is precisely the mutual color
neutralization, and thus it is of interest to examine this in
QCD-based models.  The skyrmion allows one to
study this\ci{EK92} by artificially shrinking the skyrmion and
examining the consequent $mN$ potential and total $mN$ cross section.  The
shrinking is accomplished by using the scaling of eq.~(\ref{rwiggle}) and
taking values for the minisculon such that its inverse size parameter 
$(eF_\pi)_m$ becomes large.  The ratio of minisculon size to the
size of a normal nucleon is then
\be \label{minisculon}
x \equiv \frac{r_m}{r_N} = \frac{eF_\pi}{(eF_\pi)_m}.
\ee
The result of the skyrmion calculation using the product ansatz is
that the central potential tends to zero more or less linearly with $x$
for $x \le 0.3.$  On the other hand, the range of the potential tends to
fall very gradually until $x \sim 0.05$ and then plunges rapidly to zero.
The total $NN$ cross section, in a rough high-energy approximation, falls
linearly to zero as $x$ drops from unity to zero, showing quadratic
behavior only when $x$ is below 0.3.

\subsection{\label{shsh} Nucleon shrinking and swelling}

There has been intense interest over the 
years\ci{No,CRS,Mul,COP,Ban,FS85,OA87} in a basic question of nucleon 
behavior 
in nuclei:  Does the nucleon shrink or swell when placed in the nuclear 
medium?  A related question is, Does the nucleon shrink or swell when in
interaction with a second nucleon? Of course, if the skyrmion is used
with the simple product ansatz the nucleon will surely shrink in
a two-nucleon system, since we have already seen that this simple
prescription leads to repulsion.  However, when mechanisms for attraction 
are present, such as the admixture of higher baryon resonances
(subsection \ref{baryon_resonances}) or the inclusion of a
dilaton interaction (subsection \ref{dilaton}), this
situation changes, and one would expect a slight swelling for nucleons
separated by 2 to 3 fm, followed by shrinking as they are brought closer
together.  This is indeed what is found in detailed calculation\ci{KFE},
as can be seen in figs.~3 and 4.  The swelling effect is on the scale of
3 or 4 percent, which may be sufficient to explain present EMC 
results\ci{FS88}; below about 1.5 fm the nucleons act to reduce each other's
size.  Such effects should show up as changes in the form factor of the
nucleon within the nucleus for different ranges of momentum within the
Fermi sea, and one may hope to see such effects in high-energy electron
scattering, for example at CEBAF.

\begin{figure} 
\centerline{\psfig{figure=brazil_fig3.ps,height=3.0in}}
\caption{The $NN$ potential and ratio of the radius for the 
interacting nucleon to that of the free nucleon $r/r_0$
for the usual skyrmion with product ansatz and baryon-resonance
admixtures.
(see subsections \protect\ref{extensions} and 
\protect\ref{baryon_resonances}).  
The parameters are 
taken to have the values $F_\pi=130$ MeV, $e=20,$ $\gamma=0.50,$ 
$\epsilon_{6,2}=2.58,$ and $m_\pi=139$ MeV, yielding 
the masses $M_N=998$ MeV, $M_\Delta= 1211$ MeV, 
and $M_{N^*}=1270$ MeV for the nucleon, $\Delta,$ and Roper. \label{fig:3}}
\end{figure}

\begin{figure} 
\centerline{\psfig{figure=brazil_fig4.ps,height=3.0in}}
\caption{The $NN$ potential and ratio of the radius for the 
interacting nucleon to that of the free nucleon $r/r_0$
for the skyrmion coupled to a dilaton.
(see subsection \protect\ref{dilaton}).
The parameters used 
were $\Gamma_0 = 306$ MeV, $C_G = (121\ {\rm MeV})^4,$ 
and $F_{\pi} = 186$ MeV, which produce $M_{N} = 1335$ MeV. \label{fig:4}}
\end{figure}

\section{\label{summary}  Summary}

The first decade of studies of the two-baryon system with skyrmions has
yielded an understanding of several different mechanisms that may
contribute to central attraction and to the attractive isoscalar
spin--orbit force.  Thus the way is open to sort out, within the next few
years, which of these effects are essential and dominant.  It would be
over-optimistic to claim that clear evidence for answers to that question
exist today.  In all likelihood, several of the candidates may enter
together, thus strengthening the one overarching conclusion to be reached
from QCD-related studies of the $NN$ interaction so far, namely, that central
attraction---however crucial it may be for nuclei to exist as we know
them---is the result of an extremely subtle play of QCD effects.


\vspace{3 pc}

It is a pleasure to thank Jos\'e S\'a Borges, Erasmo Ferreira, 
and Victoria Herscovitz for an invitation to deliver a series of 
lectures on skyrmions at Hadron Physics 96 in Brazil in April, 1996.
We also acknowledge useful conversations on the subject matter of 
refs.\ci{Dal,TH1,TH2} with Victoria Herscovitz, Juan Mignaco, and Gilberto
Thomas, and regarding ref.\ci{BRK} with Michel Betz.
This work was supported in part by the Deutsche Forschungsgemeinschaft,
the Israel Science Foundation, and the Yuval Ne'eman Chair at Tel Aviv 
University.

\newpage

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