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\begin{document}

\title{{\bf The nucleon-nucleon force from skyrmions}\thanks
 {Talk delivered (by J.M. Eisenberg) at the Seventeenth Course of the 
Erice School on Nuclear Physics: {\sl Quarks in hadrons and nuclei}, 
September, 1995, Erice, Sicily, Italy.}}
\vspace{1 true cm}
\author{J.M. Eisenberg \\ School of Physics and Astronomy \\ 
Raymond and Beverly Sackler Faculty of Exact Sciences \\ 
\vspace {1 true pc}
Tel Aviv University, 69978 Tel Aviv, Israel  \\
\vspace {1 true pc}
and \\ 
G. K\"albermann \\  
Rothberg School for Overseas Students \\
and Racah Institute of Physics \\ 
Hebrew University, 91904 Jerusalem, Israel \\}
\date{September, 1995}

\maketitle

\begin{abstract}
\baselineskip 1.2 pc
The applications of skyrmions 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, the instanton
ansatz, 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.  
\end{abstract}

%\newpage
%\tableofcontents
%\newpage
\baselineskip 1.2 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 lay dormant for some twenty
years or more and was revitalized 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 [4--6].  Following this renewed interest a great deal of 
effort has gone into the study of single-baryon behavior based on the 
skyrmion \ci{ANW}, reviewed in [4--6].
%\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 [12--19], 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 gave reasonable
values for the $\pi NN$ coupling constant \ci{ANW}, and thus handled 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.  Its 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 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. 
Most of our effort here 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{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.  

The 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 $B^\mu$ is conserved without regard to 
the equations of motion (i.e., it is not a Noether
current) and is therefore identified as the baryon current, the
corresponding baryon charge then being unconditionally conserved.

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

Thus far time dependence has been ignored and there has been no
attempt 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.  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 which is converted 
into a dynamical variable and quantized.
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.  Eigenstates of these operators may then be constructed
in the space of the parameters of the matrices $A,$ and
projection onto these states is made
in the form of expectation values with respect to
the eigenstates, the relevant integration being carried
out over a unit sphere in four dimensions.  

Last, one-baryon observables are extracted from the Noether 
currents generated by charge variations 
$\delta U = i Q U,$ where $Q$ is the charge operator.  
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 extensively reviewed elsewhere \ci{Bal,ZB,HS86a,ANW}, and generally
agree with experiment to within about the $1/N_c \sim 33\%$ error
inherent in the skyrmion.

\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 the form
\be \label{L61}
 {\cal L}_{6} \equiv -\frac{\epsilon_{6}^2}{2} B_\mu B^\mu,
\ee
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 
${\cal L}_4$ or ${\cal L}_{6}.$   We thus
entertain the expression of eq. (\ref{L4s})
here reluctantly as perhaps having some limited
phenomenological purpose.  For the discussion of $NN$ forces
it is of some interest as possible sources of additional 
central attraction. 

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

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)
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} 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.

\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 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 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}).

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, subtracting 
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 in expectation values formed with the
nucleon wave functions.  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.
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, but 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}

In subsection \ref{extensions} we noted that the symmetric fourth-order term 
${\cal L}_{4s}$ of eq. (\ref{L4s}) is attractive.
One may thus incorporate the attractive ${\cal L}_{4s}$
along with the repulsive ${\cal L}_{6}$ 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 \ci{VMrev}.  This remains an 
important auxiliary
ingredient in various other devices for producing attraction, but does not
\ci{EKrev,KESS}, in and of itself, yield attraction in 
$V_C(R),$ even after extensive search in the space of lagrangian parameters.

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.  This has been considered by the Orsay 
group \ci{VMrev,LLVMC} and by the Syracuse group \ci{Syracuse,SSSW}.
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 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.

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

It has long been accepted wisdom 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 appropriate 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 introduced
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 of which we take as representative \ci{BDT}.
When the vibrational problem is then solved, eigenvalues with lowest 
principal quantum number $n = 0$ correspond to the nucleon and 
$\Delta$ 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.  This depth may also be further enhanced by the
correction factor for finite $N_c$ \ci{JJP}, namely $((N_c + 2)/N_c)^2
= 25/9.$  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.  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{instanton_sec} The instanton ansatz}

It was argued by Manton \ci{Man} and then more concretely by
Atiyah and Manton \ci{AM} that special care needs to be taken in
generating skyrmion solutions for the $NN$ system in order to ensure 
that they pertain to a
manifold that includes well-separated skyrmions with 
arbitrary SU(2) orientations.  The possibility of infinite separation 
is important so that one can handle the problem of removing the two
baryons far away from each other, as is inherent in a scattering
situation or in the definition of a potential.  The freedom to
orient the skyrmions in arbitrary directions is necessary so that one
can project onto states with nucleon quantum numbers.  To describe two
well-separated skyrmions with independent positions and orientations
requires 12 degrees of freedom.  The product
ansatz satisfies this requirement, of course, and so easily succeeds
in giving a reasonable asymptotic description of the $NN$ system, but its
``product'' nature is overly restrictive, as we have seen in the
previous section, and thus fails at small distances.  Other complete 
solutions do not always fulfill
these conditions and thus may lead to descriptions of $B = 2$ systems
with unphysical properties, as we shall discuss further in section 
\ref{full}.
This led to a search for skyrmion ans\"atze for the $B = 2$ system
which possess these two essential requirements while still 
allowing for relatively simple treatment.

The full implementation of this program for the calculation 
of the $NN$ potential was carried out
(\ci{HGOA,HOA}, reviewed in \ci{OHrev}; see also \ci{Wal95} and
applications to systems with $B = 3,\,4$ in \ci{LM}).
The procedure produces excellent results \ci{HOA} for the
OPEP-dominated spin-spin and tensor parts of the $NN$ force, but
ultimately fails to find attraction for the central part of the
potential.  Indeed, the results for $V_C(R)$ are not very different 
from those obtained using the original product ansatz, which 
thus finds some substantiation in the instanton ansatz.

\section{\label{dilaton} The dilaton}

It is well known that 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.  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.  This 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 $eF_\pi,$ 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 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 this dilaton lagrangian, it is
easy to modify \ci{GJJS86b,JJS} the basic Skyrme lagrangian
(\ref{Lagrangian}) in order to incorporate the trace anomaly.
It is common practice to replace $\psi$ with an exponentiated scalar
field $\sigma,$ $\psi \equiv \la\psi\ra\,\e^{\sigma}.$ 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,
$\sigma_{B=2} = \sigma_1 + \sigma_2,$
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 exact numerical
solutions of the full $B = 2$ skyrmion problem (on restricted
manifolds) 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.  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} (discussed briefly in subsection \ref{instanton_sec}) 
that the restricted manifold of
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 fixed through Lagrange
multipliers (see also \ci{KM85,Kal93a}) while the relative orientations 
between the two skyrmions are chosen in the same manner 
(subsection \ref{instanton_sec}) 
as for the instanton solution.  The solutions \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 significantly
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 \ci{Kal93b}.
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.
It should be emphasized that this is the first case of a realistic
representation of the deuteron within the skyrmion approach.  As 
noted in section \ref{instanton_sec}, 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 again made
(section \ref{instanton_sec}) and the maximally attractive 
orientation---rotation by $\pi$ about the axis perpendicular to the line 
joining the skyrmions---is taken (see the end of section
\ref{instanton_sec}).  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} have been calculated, from which one sees 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.


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

As we have seen,
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 several different mechanisms.
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.  

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.  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 the basic SU(3) lagrangian, 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.
\newpage
\begin{figure}[h]
\vskip 9.0 cm
\special{psfile=v2new.ps
angle=0  voffset=-150 hscale=75 vscale=75}
\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.}
\end{figure}
\begin{figure}[h]
\vskip 9.0 cm
\special{psfile=v6.ps
angle=0  voffset=-150 hscale=75 vscale=75}
\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.}
\end{figure}
\newpage

\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 ${\cal L}_{6}$ of eq. (\ref{L61}) 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.  

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{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
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 
higher than the one used here (cp. \ci{TR}).

\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 appears clear  
from QCD-related studies of the $NN$ interaction so far that central
attraction is the result of an extremely subtle play of QCD effects.


\vspace{3 pc}

It is a pleasure to thank Professor Amand Faessler for the invitation
to contribute to the International School of Nuclear Physics, 17th
Course, ``Quarks in hadrons and nuclei,'' Erice, Sicily, Italy,
September, 1995, which led to the writing of this brief review.  
We are also pleased to acknowledge a useful conversation on this 
material with Robert Vinh Mau.
This work was supported in part by the Israel Science Foundation and in
part by the Yuval Ne'eman Chair at Tel Aviv University.

\newpage

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\end{document}
