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\begin{document}

\title{THE NUCLEON IN THE NUCLEUS AS GIVEN BY SKYRMIONS}

\author{J.M. EISENBERG}

\address{School of Physics and Astronomy \\ Raymond and Beverly Sackler
Faculty of Exact Sciences \\ Tel Aviv University, 69978 Tel Aviv,
Israel}

\author{ G. K\"ALBERMANN}

\address{Faculty of Agriculture \\ and Racah Institute of Physics \\
Hebrew University, 91904 Jerusalem, Israel}


\maketitle\abstracts{ After some ten years or more of effort towards
understanding the nucleon-nucleon force in terms of skyrmions, the way
may now be open to consider its application to various problems
involving the manifestations of quantum chromodynamics in nuclear
physics.  We here consider two such studies:  (i) aspects of color
transparency as given by skyrmions and (ii) the treatment of nucleon
swelling and shrinking in nuclei using skyrmions.}

\section{Introductory Remarks} \subsection{Historical
Perspective}\label{subsec:perspective}

During the last ten years or so, roughly ten groups around the world
have pursued the issue of correctly describing the nucleon-nucleon
force and systems with small numbers of nucleons using
skyrmions.\footnote{This work has been extensively surveyed,
practically every group involved having written a summary of the topic
from its own perspective.  We here note one review~\ci{EK_rev} which is
close to the usages and approaches of this talk.}  In offering this
modest contribution in honor of Professor Walter Greiner it was
tempting to review the impressive progress that has recently
been made in that direction.  However, it is one of Professor Greiner's
engaging qualities to be more oriented towards the future than the
past, and at the opening of this conference he asked that the speakers
not stint in addressing the future in their areas---a request which it
is a pleasure to comply with here.

Thus we shall take as our point of departure the supposition that the
skyrmion has become, or very soon will become, an adequate tool
for qualitative and semiquantitative descriptions of few-baryon
systems, so that the way is open to begin considering its use for
insight into hadronic behavior in nuclei, a topic of increasing
importance in a variety of studies of quantum chromodynamics (QCD)
effects in nuclei.  The first such topic to be considered here is color
transparency, reviewed, for example, in ref. ~\ci{FMS}, which involves
very-high-momentum transfer processes within nuclei in a test of QCD
dynamical issues.  The second question relates to nucleon swelling and
shrinking in nuclei, a matter that has received increasing attention
over the last ten years or so~\ci{No,CRS,Mul,FS85,OA87}.  More
detailed discussions of both these topics are given in the review noted
previously~\ci{EK_rev} and in the original papers~\ci{EK92,KFE}.


\section{Generalities on $B = 1$ Skyrmions}

\subsection{The Skyrme Lagrangian}

It was pointed out a number of years ago by 't Hooft that in the limit
of a large number of colors, $N_c \to \infty,$ QCD reduces to a theory
involving meson fields as the relevant degrees of freedom~\ci{tHooft},
while Witten pointed out that baryons can then be constructed out of 
these degrees of
freedom as topological solitons~\ci{Witten}.  This provided
underpinnings within QCD for a very old and prescient idea of
Skyrme~\ci{Skyrme} that baryons might be built up out of topological
solutions of a lagrangian embodying chiral symmetry, namely, 
\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, here 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.  Substituting
eq.~(\ref{hedgehog}) into eq.~(\ref{lagrangian}) yields eventually an
expression for the particle energy or mass in terms of the profile
function, 
\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
From this one sees that in order for the mass to remain finite the
profile function must approach $n\pi$ as $r \to 0$ and $m\pi$ as $r \to
\infty,$ where $n$ and $m$ are integers whose difference $n - m$ is a
topologically conserved quantity that Skyrme identified as baryon
number.  Moreover the minimum reached by varying $F$ in eq.~(\ref{M})
yields the Euler-Lagrange equation for the profile function, 
\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 
in terms of dimensionless variables
for which lengths are scaled by $eF_\pi$ and energies by $F_\pi/e.$
This dimensionless form will be particularly handy for discussing color
transparency below.

\subsection{Projection onto $N$ and $\Delta$ States}

The hedgehog form of eq.~(\ref{hedgehog}) obviously mixes the SU(2)
space with the space of three-dimensional rotations, and Adkins, Nappi,
and Witten~\ci{ANW}, in an important paper for modern applications of
skyrmions, offered a projection method for generating states of good
spin and isospin.  This method is very reminiscent of projection
techniques long used in nuclear physics~\ci{EG} to reach states of good
quantum numbers from various sorts of mixed states.  The general idea
is to rotate the hedgehog solution using a time-dependent unitary
SU(2) matrix $A(t)$ according to 
\be \label{AUA} 
U(\r,\,t) = A(t)\,U_0(\r)\,A^\da(t).  
\ee 
This matrix $A(t)$ is then converted into
a dynamical variable and subjected to first-quantization, leading to a
dynamic lagrangian, upon substitution into eq.~(\ref{lagrangian}), of
the form 
\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 $I = 50.9 \, (2\pi/3 e^3 F_\pi).$ The
quantization procedure allows the construction of operators for spin
and isospin and of wave functions in the $A$-space for nucleon and
$\Delta$ states.  Ultimately the nucleon and delta masses are given by
\be \label{spectrum2} 
M_N = M + \frac{3}{8I},\quad\quad M_\Delta = M + \frac{15}{8I}.  
\ee 
The
two parameters in eq.~(\ref{lagrangian}) may then be fit to these two
values.  Higher states with $J=T$ are assumed to be nonphysical (or
very broad), as in nuclear rotational bands that are truncated at the
maximally allowed spin values for the number of nucleons
present~\ci{EG}; that role is here played by the three valence quarks.
All relevant observables for the nucleon and the $\Delta$ and
transitions between them can then be constructed and
calculated,~\ci{ANW} with the rather satisfactory results (to within
$1/N_c \sim 33$\%) shown in Table~\ref{tab:1}.

\begin{table}[t] 
\caption{Skyrmion results for $B = 1.$\label{tab:1}}
\vspace{0.4cm} 
\begin{center} 
\begin{tabular}{|l|l|l|} \hline & & \\
Physical quantity&Theory&Experiment \\ 
\hline 
& & \\
$F_\pi$&108 MeV&186 MeV\\
$r_{\rm rms}$(ch, $T = 0$)&0.68 fm&0.72 fm \\ 
$r_{\rm rms}$(ch, $T = 1$)&1.04 fm&0.88 fm \\ 
$r_{\rm rms}$(mag, $T = 0$)&0.95 fm&0.81 fm \\
$r_{\rm rms}$(mag, $T = 1$)&1.04 fm&0.81 fm \\
$\mu_p$&1.97 $\mu_0$&2.79 $\mu_0$ \\ 
$\mu_n$&-1.24 $\mu_0$&-1.91 $\mu_0$ \\
$g_A$&0.65&1.25 \\ $g_{\pi NN}$&11.9&13.5 \\ 
$g_{\pi N\Delta}$&17.8&20.3 \\ 
$\mu_{N\Delta}$&2.3&3.3 \\ 
& &  \\ 
\hline
\end{tabular} 
\end{center} 
\end{table}

\section{The Product Ansatz for $B = 2$ Systems}

In approaching the two-nucleon problem, the earliest~\ci{Skyrme} and
simplest line of attack was based on London--Heitler methods and
uses the so-called product ansatz in which the two-nucleon skyrmion is
assumed to be well approximated by the product of two one-nucleon
forms,
\be \label{PA} 
U_{B=2}(\r,\R) = U_{B=1}(\r+\half\R)\,U_{B=1}(\r-\half\R), 
\ee 
with profile functions
known from the $B = 1$ problem.  Here $\r$ is the general skyrmion
variable and $\R$ is the separation between the two baryon centers.
This ansatz automatically fulfills the correct baryon number condition
$B = 2.$

It is crucial for the study of the $NN$ system that we project the
product ansatz onto nucleon states.  This 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}).

\section{Skyrmion Applications in Nuclei: Color Transparency}

Color transparency is the phenomenon 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 noted below eq.~(\ref{ode2}) and taking values for
the minisculon such that its 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.  This last feature given by the skyrmion estimate is somewhat
surprising in that most discussions of color transparency have based
themselves on an assumption of quadratic dependence on minisculon size
through the whole scale from normal radii to very small ones.

\section{Skyrmion Applications in Nuclei: Shrinking and Swelling
Nucleons in Nuclei}

In order to make any sort of sensible estimate of nucleon swelling in
nuclei, one needs to have an $NN$ force that provides for central
attraction.  Among the many suggestions~\ci{EK_rev} for achieving this
within the product ansatz, two seem to have provided central attraction
based on physically plausible mechanisms.  These are (i) the admixture
of a nucleon vibrational state~\ci{BDT} (the Roper resonance), which
appears to be the equivalent for the product ansatz of skyrmion
distortions seen within more complete approaches~\ci{WWrev}; and (ii)
the inclusion of the dilaton,~\ci{Schech} which mocks up the trace
anomaly of QCD for theories based on meson degrees of freedom.

Attraction from the Roper is usually discussed in terms of a lagrangian
involving terms beyond the simple for of eq.~(\ref{lagrangian}).  One
often adds~\ci{EK_rev} terms involving four derivatives, for enhanced
attraction, and terms with six derivatives, simulating
$\omega$-exchange, which are repulsive.  The two-baryon states are then
taken as 
\bea \label{resonances} 
|\widetilde{NN}(R)\ra & = &
\alpha(R)|NN(R)\ra + \beta(R)|N\Delta(R)\ra +
\gamma(R)|\Delta\Delta(R)\ra \nono & + & \epsilon(R)|NN^*(R)\ra +
\zeta(R)|N^*N^*(R)\ra.  
\eea 
The Roper states $R^*$ are handled by first-quantizing a collective, 
breathing-mode, radial-scaling variable~\ci{EK_rev,BDT}.

The dilaton yields attraction by sharpening the edge of the skyrmion, a
reasonable physical expectation here.  The dilaton itself is motivated
by the need to make the lagrangian obey the same breaking of scale
invariance as found in QCD in the context of the trace
anomaly~\ci{Schech}, 
\be \label{trace} 
T_\mu^\mu = \partial_\mu\,D^\mu
= -\frac{9\alpha_s}{8\pi}G_{\mu\nu}^aG^{a\mu\nu} \equiv \sigma^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
$\sigma$ is an order-parameter field---the dilaton---which represents
the scalar glueball formed from the contraction of the two gluon
%\newpage 
fields.  The modified skyrmion lagrangian that satisfies the
trace anomaly is then 
\bea \label{L_skydil}
 {\cal L} & = & \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 & - &{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 $m_\sigma^2 = 4 C_G/\Gamma_0^2.$  
In using the dilaton for
$B = 2$ systems, one must augment the product ansatz for the skyrmion
by an assumption of additivity for the dilaton, 
$\sigma_{B=2} = \sigma_1 + \sigma_2.$

Both mechanisms are capable of yielding central attraction on the
scale of 10 MeV or more with reasonable parameters.  This in turn leads
to nucleon swelling of some 3 or 4 percent when the separation between
the nucleons places them in the attractive region.  This is illustrated
for the case of the Roper mechanism in fig.~\ref{fig:1}, the case for
dilaton-induced attraction being quite similar~\ci{KFE}.

\begin{figure} 
%\vspace{2.5 cm}
\centerline{\psfig{figure=roper.ps,height=2.5in}} 
\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. \label{fig:1}} 
\end{figure}

\section{Outlook}

The two studies we have discussed represent an early phase of the
applications of skyrmions to nucleon behavior and changes in the
nuclear medium (see also ref.~\ci{Mish} in this regard).  Much more can
clearly be done.  For example, in the context of heavy-ion collisions
there is a great deal to be learned about baryons in a hot medium and
here again the skyrmion may offer useful insights.  We look forward to
learning about these matters from experimental and theoretical work
over the next decade.

\section*{Acknowledgments} It is a pleasure to acknowledge helpful
discussions with Leonid Frankfurt on the subject matter of this
report.  This work was supported in part by the Israel Science
Foundation and in part by the Yuval Ne'eman Chair at Tel Aviv
University.

\section*{References} 
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\bi{No} J.V. Noble, \Jo{\PRL}{46}{412}{1981} and
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\bi{EK92} J.M. Eisenberg and G. K\"albermann,
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\bi{KFE} G. K\"albermann, L.L. Frankfurt, and J.M. Eisenberg,
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\bi{tHooft}  G. 't Hooft, \Jo{NPB}{72}{461}{1974} and
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\bi{EG}  J.M. Eisenberg and W. Greiner, {\em Microscopic theory of the
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\bi{BDT}  L.C. Biedenharn, Y. Dothan, and M. Tarlini,
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\end{thebibliography}

\end{document}
