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\begin{document}

\title{{\bf A SKYRMION FLUID}}
\vspace{1 true cm}
\author{G. K\"ALBERMANN \\
Faculty of Agriculture \\
and \\
Racah Institute of Physics \\
Hebrew University, 91904 Jerusalem, Israel \\}
%\date{MAY, 1996}

\maketitle

\begin{abstract}
\baselineskip 1.5 pc
A fluid of Skyrmions coupled to the dilaton field and the $\o $ meson
field is considered. A mean field theory is developed in which the
dilaton and the $\o$ field acquire a mean value determined by the
Skyrmions. The influence of the background fields on
the Skyrmion profile is determined and consequently the scaling
properties of the Skyrmion follow.
The model obeys chiral invariance and scale invariance broken only
by the dilaton trace anomaly potential. 
The dilaton plays the role of the scalar field as in the $\s + \o$ model.
The dilaton potential is augmented by terms that do not spoil
the trace anomaly while allowing a fit to nuclear matter properties.
The phase diagram of the model shows unique features, like a lack of
solutions for certain densities and temperatures, signalling the
appearance of a new phase that can not be described in terms of Skyrme baryons.
\end{abstract}




{\bf PACS} 12.39Dc, 21.10Dr, 21.65.+f, 24.10.Pa 
%\newpage
%\tableofcontents
\newpage
\baselineskip 1.5 pc

\section{\label{Introduction} Introduction: Application of Skyrmions
to the many body problem}

Around 35 years ago Skyrme \ci{Skyrme} proposed to describe baryons as
topological solitons in a meson field theory.
More recently, studies of the large $N_c$ approximation to QCD \ci{tHooft}
suggested that indeed the meson degrees of freedom are dominant
and baryons might arise as solitons \ci{Witten,ANW}.
In the last decade there has been a large body of research on these
solitons, or Skyrmions \ci{Bal,ZB,HS86a,ek1995}.
The research has mainly focused on single baryon and few baryon systems.

Skyrme himself was interested in applying his model to systems of many
nucleons aimed at the description of nuclei. After gaining support from
QCD it appears that a model of nuclei based on Skyrmions would be better
related to more fundamental theories of matter. This is especially
appealing due to the difficulty of solving QCD explicitly.

The Skyrme model has had moderate success in dealing with the 
nucleon-nucleon interaction, including the attractive isoscalar central
potential for which several mechanisms were proposed. It appears then
natural to investigate more formal issues involving Skyrmions,
such as their behavior in nuclear matter.

In the present work a mean field model of dilatons and $\o$ mesons coupled 
to Skyrmions is investigated.
The model is inspired in the classical $\s + \o$ Walecka approach \ci{ser1}.
The original $\s + \o$ model has been extended in order to include scale 
invariance by means of the dilaton field in refs.\ci{ser2,ru1}. In these works
two scalar fields are used, the chiral partner of the pion represented
by the $\s$ field and the dilaton. The models try to reproduce correctly
both the bulk properties of nuclei and their excitation spectra. Chiral
symmetry is introduced in its nonlinear realization in the former and
in its linear one in the latter, while the nucleons are considered as
pointlike Dirac objects. A different attempt to include chiral symmetry in the
mean field theory of Walecka was done by Lynn \ci{lynn1}.
The virtue of the Skyrme model is that it enables to include both chiral
symmetry and baryons in a consistent way, the baryons being solitons of
finite extent in space. These aspects make the model a good tool for the
investigation of baryonic nuclear matter,
but at the same time the treatment becomes more involved.
We will see, however that in a dilute fluid approximation it is
possible to treat the Skyrmions as essentially free particles
interacting with a medium that  carries the information of density and
temperature. This medium acts through
the mean fields of the dilaton and the $\o$ that are determined by the
Skyrmions themselves.
Mishustin \ci{mi1} has treated a single Skyrmion immersed in a bath of
Dirac particles. Our approach is to generate the bath from the Skyrmions
themselves in a self-consistent manner.
Section 2 deals with the dilute fluid approximation. Section 3 describes
a specific Skyrmion fluid. Section 4 treats the choice of dilaton potential.
Section 5 shows the results of the mean field approximation for finite
temperature and density.

\section{\label{fluid} Dilute Skyrmion fluid}

Consider a field theory lagrangian of Skyrmions, the dilaton $\s$ and the
$\o$ meson  \ci{ek1995,AN}
\bea \label{skydil}
 {\cal L} & = & {\cal L}_{2\ {\rm dilaton}} +
 {\cal L}_2 + {\cal L}_4 - V_{\rm interaction}
 - V(\s) + {\cal L}_\o\nono
& = & \e^{2\s}\bigg[\half\Gamma_0^2\,
\dd_\mu\s\,\dd^\mu\s
- {F_\pi^2 \over 16}\tr(L_\mu L^\mu)\bigg]
+{1 \over 32 e^2}\tr[L_\mu,\, L_\nu]^2 -  g_V~~ \o_\mu B^\mu\nono
& - & B [1 + \e^{4\s}(4\s-1)] - {1\over 4} {(\dd_\mu\o_
\nu - \dd_\nu\o_\mu)^2} + \half\e^{2\s} m_\o^2~~\o_\mu^2.
\eea
Here
\be \label{Lmu}
L_\mu \equiv U^\da\dd_\mu U,
\ee
where $U({\bf r},t)$ is the chiral field, $F_\pi$ is the pion decay constant 
and $e$ the Skyrme parameter.

The trace of the energy-momentum tensor in QCD is given 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. 
In the lagrangian of eq.~(\ref{skydil}) $\psi \approx \Gamma_0 \e^{\s}$.
The trace of the energy momentum tensor of eq.~(\ref{skydil})
-generated by $V_\s$- is constructed in
accordance with eq.~(\ref{trace}).

The lagrangian incorporates scale invariance broken only by the anomaly.
All other terms are scale invariant, including the 
$\o$ mass term that carries appropriate factors of $e^{\s}$.

In the present work
we focus on the SU(2) version of the Skyrmion. The lagrangian
possesses chiral as well as isospin symmetry. We have omitted a pion
mass term that manifestly breaks chiral symmetry because it is
relatively unimportant. We have also omitted $\rho$ meson coupling as it
is our intention to treat symmetric nuclear matter only.
The two parameters $\Gamma_0$ and $B$ are related to the glueball---or
scalar field---mass through $m_\s^2 = 16 B/\Gamma_0^2.$  An
accepted value of the mass is around 1.5 GeV, although the mixing of
the glueball with $\pi\pi$-exchange may bring it down to the below GeV
range.(A critique to the use of the dilaton in nuclear matter applications
can be found in ref. \ci{Birse}.)

Analogously to the mean field theory of pointlike baryons, we consider
an ensemble of essentially free Skyrmions. Each Skyrmion will be
accompanied by its own dilaton and $\o$ fields. Although we deal with
free Skyrmions, the average properties of the ensemble are still
included in distribution functions that depend on the 
density and temperature.

For the $N$ Skyrmions we use the product ansatz

\be \label{pa}
U_{B=N}({\bf r},\R_1,\R_2,\cdots,\R_N) = U({\bf r}-\R_1)\,U({\bf r}-\R_2)\cdots 
U({\bf r}-\R_N),
\ee
whereas for the scalar fields we use an additive ansatz

\bea \label{additivity}
\s_{B=N} & = & \s_1 + \s_2 + \cdots+ \s_N, \nono
& = &\s_0 + \delta\s_1 + \delta\s_2 +\cdots+ \delta\s_N, \\
\o_{B=N} & = & \o_1 + \o_2 + \cdots+ \o_N, \nono
& = &\o_0 + \delta\o_1 + \delta\o_2 +\cdots+ \delta\o_N,
\eea

Where $\s_0 , \o_0$ are the mean field constant values of the
dilaton and the $\o$ and $\delta\s, \delta\o$ represent the
fluctuations. The fields $\s, \delta\s, \o, \delta\o$
depend on the same arguments as the Skyrmion they are attached to.
The ans\"atze above insure total baryon number $=N$ and allow
an easy separation of the mean field degrees of freedom from the microscopic
excitations inside each Skyrmion.
In thermal equilibrium, the mean field fields will depend on the 
temperature $T$ and the chemical potential $\mu$.
Self-consistency then requires that
the values of $\s_0$ and $\o_0$ have to be determined by the properties
of the ensemble. 
For a certain distribution function ${\bf f}(\mu,T)$ 
the temperature, we then demand the phase 
space thermal average to be

\bea
<\s> & = & <\s_0 + \delta\s_1 +\cdots+ \delta\s_N> \nono
& = & {(2 \pi)}^{-3 N}\int {d \vec R_1 d \vec P_1
\cdots d \vec R_N d \vec P_N~\bf{f}}~~\s \nono
& = & \s_0
\label{average}
\eea
with a similar equation for $\o$. The mean field average value of
the fluctuations vanishes by definition. In eq.~(\ref{average}) the integration
is over the collective coordinate coordinates $\vec R_i$ of the Skyrmions and
their conjugated momenta. In the diluted fluid approximation, Skyrmion
interactions are neglected and consequently there are no potential terms
depending on the coordinates $ \R_i$ of each baryon. There remain only kinetic
energy terms for each Skyrmion individually. Standard quantization of these
terms will then determine the corresponding wave functions to be plane waves,
as for the conventional Fermi gas model.

A crucial point in the mean field theory of skyrmions is the topological baryon 
density

\be \label{Bmu}
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
The corresponding baryon number becomes
\be \label{B}
B_0 = {{\epsilon^{ijk}}\over
{24\pi^2}} \tr \left[ \left( U^{\da}\dd_i U \right)
\left( U^{\da} \dd_j U \right)
\left( U^{\da} \dd_k U \right) \right] ,
\ee

Using the product ansatz of eq.~(\ref{pa}) in the baryon number above
allows us to calculate the mean field average by integrating over the 
Skyrmions' coordinates and momenta. Consistently with the
approximation of a fluid of free Skyrmions we demand that there is no
overlap between the Skyrmions profile functions, and further assume that
the profile drops to zero at a distance smaller than the inter-Skyrmion
separation. This condition is needed in order to insure integer baryon
number within each Skyrmion cell, because the single baryon density 
 depends on the values of the
profile function at the particles' location and at infinity. In the
present case, infinity is replaced by a finite distance.  Typically,
this distance is of the order of 0.8 $fm$, whereas the interparticle
separation in normal nuclear matter is approximately 2 $fm$. It is then
not very unrealistic to assume that most of the topological baryon density is
contained within a volume smaller than the average volume available
to each Skyrmion. Note that, even if there is some overlap
between the Skyrmions, the choice of product ansatz insures total baryon
number = $N$.
To this approximation we obtain 
\bea \label{b1}
B_0 & = & b_1 + b_2 + \cdots + b_N,\nono
<B_0> & = & N/V
\eea
where $ b_i$ is the baryon density of the $i^{th}$ Skyrmion, V is the volume 
of the fluid, and the average is defined in eq.~(\ref{average}). 
The integration over the collective coordinates of the Skyrmions is 
mathematically equivalent to an integration over the coordinates $\bf r$ in
eq.~(\ref{pa}), this is the reason why the averaging above essentially
counts the number of Skyrmions. After the Fermi averaging, the localized 
baryon density becomes uniformly smeared out. There is 
no residue of the individual Skyrmion densities. We will see below that the
influence of the mean fields on the Skyrmions operates solely through
the dilaton field. Although the baryon density of each Skyrmion will vary
accordingly, the integrated density is scale invariant and will still
contribute the value $B = 1$ for each Skyrmion.
The dilute fluid approximation is therefore closely related to the mean
field Dirac model with two major differences: 1) there is a different
dynamics for the baryons dictated by the lagrangian of eq.~(\ref{skydil}) and,
2) there is a clear way to uncover the baryon response to the medium.

\section{\label{model} Mean field Skyrmion fluid}

The many body Skyrmion fluid in the dilute approximation is built from
the product ansatz of eq.~(\ref{pa}) for the Skyrmions and the additive
ansatz for the mesons of eq.~(\ref{additivity}). We therefore first
focus on the single baryon case. Each soliton is constructed as a static
solution of the equations of motion. The collective coordinate energy of
the Skyrmion translation is later introduced in an adiabatic approximation.
The single baryon ans\"atze are then
\bea
U(\bf r) & = & \exp[i\bftau\cdot\hat{\bf r} F(r)], \nono
\o^\mu(\bf r) & = & (\o(r), 0, 0, 0)
\label{hedgehog}
\eea

Substituting eq.~(\ref{hedgehog}) into eq.~(\ref{skydil}) we find
the static mass of the Skyrmion
\bea
M & = & 4\pi\int_0^\infty r^2\,dr M(r)\nono
M(r) &= &\e^{2 \s}\frac{F_\pi^2}{8}
\left[F'^2 + 2 \frac{\ssF}{r^2}\right] + \frac{1}{2 e^2}
\frac{\ssF}{r^2} \bigg[\frac{\ssF}{r^2}+2
F'^2\bigg]+V(\s)\nono
& & + 
\e^{2\s}\half\Gamma_0^2\,\s'^2 -\half \o'^2
+\frac{g_V~~\o~~F'~~\ssF}{2 \pi^2 r^2}-\half m_\o^2~\o^2~\e^{2\s} \label{M}
\eea
where primes denote derivatives with respect to $r$.
The Euler-Lagrange equations for the static profile and the mesons become

\bea
\bigg(\e^{2\s} + \frac{8~\ssF}{\tilde r^2}\bigg) F''
& + & 2 e^{2\s} F' (1/r +\s') + \frac{4 \sin 2F
F'^2}{\tilde r^2} -\frac{e^{2\s} \sin 2F }{r^2} \nono
& - & \frac{4 \ssF \sin 2F}{r^2 \tilde r^2} +
\frac{2 g_V~\o'~\ssF}{\tilde r^2} = 0 \nono
\Gamma_0^2 ~~ e^{2\s} \bigg(\s'' + 2
\s'^2 & + & \frac{2\s'}{r} \bigg)  -
\frac{F_\pi^2e^{2\s}}{4}\bigg(F'^2 + \frac{2
\ssF}{r^2}\bigg) \nono
& - & \frac{dV_\s}{d\s} +
m_\o^2~\o^2~ e^{2\s} = 0 \nono
\o'' + \frac {2 \o'}{r} & - & m_\o^2 ~ \o~ e^{2 \s} +
\frac {g_V F' \ssF}{2 \pi^2 r^2} = 0
\label{ode1}
\eea
where $\tilde r = e F_\pi r$ and again  primes denote derivatives with 
respect to $r$.

Using eq.~(\ref{additivity}) and the fact that the fluctuation of $\s$ and
$\o$ vanishes in the mean field state we readily find that except for
small contributions coming from the $\s$ potential the
equations above are modified in the presence of a nonvanishing $\o_0$ and
$\s_0$ by the simple scaling law
\be
r \rightarrow e^{-\s_0} r ~, \quad \o \rightarrow e^{\s_0}\o
\label{scaling}
\ee
It is then unnecessary to solve the equations of motion for the single
Skyrmion with the meson fluctuations included. It suffices to solve them
for a free Skyrmion and then rescale the $\o$ field and the radial
distance. Moreover, for dilaton masses of the order of 1 GeV and more,
the fluctuation $\delta\s$ is altogether negligible. Therefore the scaling law
becomes exact.
Analogously it is easy to show that the static mass of eq.~(\ref{M}) scales as
\be
M = M_0 e^{\s_0}
\label{scaleM}
\ee
where $M_0$ is the mass for $\s = 0$.

Before proceeding to the many body case, we consider the Lorentz
boosting of the static Skyrmions. We perform a boost
on the collective coordinate $R(t)$ of each Skyrmion. For the sake
of simplicity consider a Lorentz boost along the $x$ axis with
velocity parameter $v$:

\bea\label{boost}
x\rightarrow \tilde x & = & \frac {x- R(t)}{\sqrt{1-v^2}} \nono
\tilde y & = & y \nono
\tilde z & = & z \nono
F(\vec{\bf r}) & \rightarrow & F(\vec{\tilde {\bf r}})\nono
\s(\vec{\bf r}) & \rightarrow & \s(\vec{\tilde{\bf r}}) \nono
\o^\mu & \rightarrow & \bigg( \frac{\o(\vec{\tilde{\bf
r}})}{\sqrt{1 - v^2}},\frac {v \o(\vec{\tilde{\bf r}})}{\sqrt{1 -
v^2}}, 0 , 0\bigg) \nono
B^\mu & \rightarrow & \bigg( \frac{B_0(\vec{\tilde{\bf
r}})}{\sqrt{1 - v^2}},\frac {v B_0(\vec{\tilde {\bf r}})}{\sqrt{1
- v^2}}, 0, 0\bigg)
\eea
with $\o^\mu B_\mu= \o_0(\vec{\tilde{\bf r}}) B_0(\vec{\tilde{\bf r}})$. 
Introducing the above transformation in eq.~(\ref{skydil}) and
calculating the Hamiltonian, we find the energy of a Skyrmion in motion
to be
\be\label{energy}
E_p =\bigg(E_2 + E_\s - E_\o\bigg)\frac{2 p^2+ 3 M^2}{3 \epsilon~M} +
E_4~\frac{4~p^2 + 3 M^2}{3 \epsilon~M} +\frac {M}{\epsilon}\bigg(U_\s - U_\o\
+ U_{int}\bigg)
\ee
where
$$\epsilon = \sqrt{p^2 + M^2}, \quad p=\frac{M v}{\sqrt{1 - v^2}}~,$$
$M$ is the static mass of eq.~(\ref{M}) for nonvanishing $\o_0, \s_0$ and
\bea\label{en1}
E_2 & = & \frac{4 \pi F_\pi^2}{8} {\int r^2 dr}~e^{2\s}
\bigg( F'^2 + \frac{2 \ssF}{r^2}\bigg) \nono
E_4 & = & \frac{4 \pi}{2 e^2} {\int dr}~\bigg( 2 F'^2 +
\frac{\ssF}{r^2} \bigg) \nono
E_\s & = & \frac{4 \pi \Gamma_0^2}{2} {\int r^2 dr}~e^{2\s}
\s'^2 \nono
E_\o & = & 4 \pi {\int r^2 dr}~\o'^2\nono
U_\s & = & 4\pi {\int r^2 dr}~V_\s\nono
U_\o & = & 4\pi {\int r^2 dr}~e^{2\s}\frac{m_\o^2
\o^2}{2}\nono
U_{int} & = & \frac{2 g_V}{\pi} {\int dr}~\o~F'~\ssF
\eea
The energy $E_p$ of eq.~(\ref{energy}) enters the single
particle distribution functions 
\bea\label{distribution}
n_p & = & \bigg(\exp\bigg[\frac{\epsilon_p-\mu}{kT}\bigg] + 1\bigg)^{-1} \nono
\bar n_p & = & \bigg(\exp\bigg[\frac{\bar \epsilon_p+\mu}{kT}\bigg] +
1\bigg)^{-1}
\eea
where $\epsilon_p = E_p + g_V ~\o_0, \quad \bar\epsilon_p = E_p - g_V\o_0$.

We can now write down the energy of N Skyrmions per unit volume
in the mean field approximation for symmetric nuclear matter.
\be\label{MFT}
E_V = 4{\int\frac{d^3p}{\left(2~\pi\right)^3}}~E_p (n_p + \bar n_p) + V_\s
(\s_0) -\half \, \e^{2\s_0} m_\o^2 \o_0^2 + g_V~\o_0~\rho_V
\ee
where
\be\label{rho}
\rho_V = 4 {\int\frac{d^3p}{\left(2 \pi\right)^3}}~(n_p - \bar n_p)
\ee
At $T = 0$ we have $n_p = \Theta(p_F - p), \bar n_p = 0$
yielding the equations of motion for the mean fields

\bea\label{T=0}
0 = \frac{\dd E_V}{\dd \s_0} & = & 4 {\int \frac{d^3p}{\left(2\pi\right)^3}}~
\frac{\dd E_p}{\dd \s_0} + \frac {d V_\s}{d\s_0} -m_\o^2 e^{2\s_0} \o_0^2 \nono
0 = \frac{\dd E_V}{\dd\o_0} & =& m_\o^2 e^{2 \s_0} \o_0
-\frac{2 g_V p_F^3}{3 \pi^2}
\eea
At finite temperature we have to maximize the pressure instead of
minimizing the energy. The contribution to the pressure of a single
Skyrmion is given by
\be\label{pres1}
P_p =\bigg(E_2 + E_\s - E_\o\bigg)\frac{2 p^2- 3 M^2}{9 \epsilon~ M} +
~E_4~\frac{4p^2 + 3 M^2}{9 \epsilon~ M} -\frac {M}{\epsilon}\bigg(U_\s - U_\o\bigg)
\ee
and the pressure of the ensemble per unit volume becomes

\be\label{pres2}
P_V = 4 {\int \frac{d^3p}{\left(2 \pi\right)^3}}~P_p (n_p + \bar n_p) - V_\s(\s_0) +
\half\e^{2\s_0} m_\o^2 \o_0^2
\ee

As mentioned above, due to the large dilaton mass one can neglect the
fluctuation of the dilaton field inside the Skyrmion. Numerical
evaluation of $\delta\s$ for a dilaton mass above 1 GeV indeed gives a
negligible dilaton field $\delta\s \approx 0$. Moreover, the simplest
approximation to the mean field demand of a vanishing expectation value
of the fluctuation of the $\o$ field is $ \delta\o = 0$. This first
approximation to the problem permits a large simplification of the
single particle's energy and pressure. Using the virial theorem for the
soliton profile appropriate for the case of vanishing dilaton and $\o$ 
fluctuation, $E_2 = E_4 = \half M$, we obtain   
\bea\label{simple}
M & = & \pi F_\pi^2 e^{2 \s_0} {\int {r^2 dr}}~\bigg(F'^2 +\frac{2 \ssF}{r^2}\
\bigg) = e^{\s_0} M_0 \nono
\epsilon & = & E_p = \sqrt{p^2 + e^{2 \s_0} M_0^2} \nono
P_p & = & \frac{p^2}{3 E_p}
\eea

To this approximation the mean field Skyrmion fluid becomes
identical to the Dirac mean field approach, with the bonus of knowing how to
calculate the reaction of the single Skyrmion to the bath using the
scaling properties determined by the dilaton.
As it is our goal to investigate the most important contribution to the
Skyrmion properties due to the bath, we will proceed along this line and
neglect the dilaton and the $\o$ meson fluctuations inside the Skyrmion. 
The results we will obtain will then be applicable to a conventional
mean field theory of Dirac pointlike particles coupled to the dilaton
and the $\o$ meson. The key new ingredient in such a treatment would 
then be the use of a single scalar field in order to fit the nuclear matter
properties.

\section{\label{potential} The dilaton potential}

In eq.~(\ref{skydil}) we introduced the conventional glue potential that
reflects the trace anomaly \ci{Schech}, where the `bag constant' is $B
\approx (240$ MeV)$^4$.
The potential can be supplemented by other terms that do not
affect the anomaly. Modifications of the potential have been
proposed in order to include chiral symmetry breaking corrections and
quark contributions to the trace anomaly \ci{ru1}, whereas Furnstahl et
al. \ci{ser2} opted for an approximation to the potential that takes
into account anomalous dimensions acquired by the scalar fields upon
renormalization. In any event there is a need to modify the potential in
order to fit the properties of nuclear matter.
The difficulty arises mostly in fitting the low value of the nuclear
compressibility modulus $\kappa \approx 270$ MeV. In the abovementioned
works two scalar fields come into play, the dilaton and 
a scalar partner of the pion, the $\s$ meson. In the present approach
there is no room for the latter, because we are working in the nonlinear
realization of chiral symmetry due to the use of the Skyrme lagrangian.
We therefore allow for modifications of the dilaton potential that do
not affect the anomaly, but influence the predictions for nuclear matter. Terms
of the form $ e^{n\s} - 1$ , generically referred to as no-log terms \ci{ru1},
can be added to the potential leaving the vacuum
expectation value of the $\s$ to be that dictated by the anomaly potential,
chosen here to be $\s = 0$ by a simple shift of the original dilaton field. The
actual choice of the functional form of the new terms in the potential turns out
to be relatively unimportant. (There exists also the possibility of modifying the
$\o$ potential as in ref. \ci{ru1}, introducing higher order terms in the $\o$
field).

Let us consider the constraints on $V_\s$ demanded by nuclear
matter phenomenology. The potential has to be such that at the
saturation density of nuclear matter $\rho_0 = .154$ baryons/fm$^3$:
a) the binding energy per nucleon is 16 MeV ; b) the binding energy is maximal
; c) the compressibility is of the order of 270 MeV; d) the
dilaton and $\o$ fields obey the mean field equations.
These conditions translate into the following equations

\bea\label{cond1}
\o_0 = \frac{g_V~\rho_0~e^{-2 \s_0}}{m_\o^2}\nono
\frac{\p V_\s}{\p\s_0} - \o_0^2 m_\o^2 e^{2 \s_0} + Q =0
\eea


where

$$Q = \frac {M^2 p_F^2}{\pi^2}\bigg[\sqrt{1 + z^2} - \half z^2 \ln
\bigg(\frac{\sqrt{1 + z^2} + 1}{\sqrt{1 + z^2} - 1}\bigg)\bigg],$$

$$p_F = \left(\frac{3}{2} {\pi}^2 \rho\right)^{1/3}, z =\frac{M}{p_F}.$$

\bea\label{cond2}
E_V & = & V_\s + \frac{m_\o^2 \o_0^2}{2} + \fourth Q
+ \frac{p_F^3 \sqrt{p_F^2 + M^2}}{2 \pi^2}\nono
& = & \rho_0 (M_0 - 16{~\rm MeV}) \nono
\frac{E_V}{\rho_0} & = & \sqrt{ p_F^2 + M^2} + \frac{g_V^2
\rho_0 e^{2 \s_0}}{m_\o^2}.
\eea
\bea\label{cond3}
\kappa & = & 9 \rho_0 \left[\frac{\dd^2 E_V}{\dd \rho^2} -
\left(\frac{\dd^2 E_V}{\dd\rho
\dd\s}\right)^2~~\left(\frac{\dd^2 E_V}{\dd\s^2}\right)^{-1}\right]
\eea

where
\bea\label{kappa}
\frac{\dd^2 E_V}{\dd \rho^2} & = & \frac{p_F^2}{3 \rho_0\sqrt{p_F^2 + M^2}}
+\frac{g_V^2}{m_\o^2 e^{2 \s_0}} \nono
\frac{\dd^2 E_V}{\dd \rho \dd \s} & = &  \frac{M^2}{\sqrt{p_F^2 + M^2}} -  
\frac{2 g_V^2 \rho_0}{m_\o^2 e^{2\s_0}} \nono
\frac{\dd^2 E_V}{\dd \s^2} & = & \frac{d^2 V_\s}{d \s^2} + 4 Q -
\frac{2 p_F^3 M^2}{\pi^2 \sqrt{p_F^2 + M^2}} -  \frac{2 g_V^2 \rho_0^2}{m_\o^2
e^{2 \s_0}}
\eea


In order to fulfill the above constraints we introduced an additional 
piece in the dilaton potential with three new free parameters that 
proved to be quite successful in reproducing the nuclear matter phenomenology

\be\label{potential1}
V_\s = B [1 + \e^{4\s}(4\s-1)] + B \bigg[a_1 (e^{-\s} - 1)+a_2 (e^{\s} - 1)
+ a_3 (e^{2 \s} -1) + a_4 (e^{3 \s} - 1)\bigg]
\ee
where B is fixed and the anomaly condition requires
\bea\label{param}
\frac {d V_\s}{ d \s} = 0\nono
\eea
at $\s=0$ implying $ a_1 = a_2 + 2 a_3 + 3 a_4$.

We have chosen the term multiplied by $a_1$ to have a negative power of
$\s$ in order to avoid the introduction of a second minimum in the
potential for $\s~< 0$. The only sensible minimum then remains the one at 
$\s$ = 0.
There is a need to introduce three new parameters in order to reproduce 
correctly the nuclear matter binding energy at the right density, and the
compressibility factor. 

The potentials for the fluctuations of the $\s$ and $\o$ fields are then
determined by the averages of eq.~(\ref{average}) to be to lowest
order a massterm interaction

\bea\label{fluctpot}
V(\delta\s) & = & B \half \delta\s^2 \bigg[ 16 e^{4\s_0}
(1+4 \s_0) + a_1 e^{-\s_0} + a_2 e^{\s_0} +2 a_3
e^{2\s_0} + a_4 \frac{9}{2} e^{3\s_0} \bigg] \nono
V(\delta\o) & = & \half e^{2\s_0} m_\o^2 \delta\o^2
\eea

The effective mass of the dilaton is not fixed, because it depends on
the parameter $\Gamma_0$ of eq.~(\ref{skydil}), that has to be prescribed
separately. The mass will enter only in finite nuclei calculations.

\section{\label{Results} Results and discussion}

The conditions on the dilaton potential of eqs.~(\ref{cond1},~\ref{cond2}
,~\ref{cond3}) were implemented by choosing a suitable effective mass at the 
nuclear matter saturation density and a certain compressibility. We took
$M^* = 694$ MeV  corresponding to $\s_0= -0.3$ and $\kappa =
300$ MeV.
The value of $\kappa$
we used is very close to the measured one \ci{sharma} - although much lower
values are suggested by the analysis of Blaizot \ci{blaizot}-
,whereas for the effective
mass we took a conservative figure, although many authors tend to select
a lower value of around 600 MeV. We  have found that 
there are no qualitative differences
in the predictions of the model when the latter mass is used.
In all the fits we kept the bag constant at  
$B =$ (240 MeV)$^4$, although we were able to fit the data wit a large
range of values of this parameter.

With the above choice we find 
: $a_1$  = -5.53, $a_2$ = -54.74, $a_3$ = 48.63, $a_4$ = 16.01 and, $g_V$ = 7.29.
Although it may seem that the dilaton potential coefficients are
a bit large, one has to bear in mind that they multiply expressions very close
to zero for all the attainable values of $\s$. 
The $\o$ meson coupling constant turns out to be smaller than expected but
still within the range quoted in the literature. Other terms
in the $\o$ meson potential might be needed in order to allow
higher values of $g_V$.

Figure 1 shows the binding energy per nucleon as a function of the
density both for the $normal$ and $abnormal$ solutions.  The normal branch 
possesses all the desired requirements for nuclear matter, therefore 
at $T = 0$ it represents the physically stable phase.
The existence of an abnormal branch is well known. It arises from the
functional dependence
of the scalar field potential. This branch is characterized
by large negative values of the scalar field.
In some models the normal and abnormal branches cross each other at
some critical density, suggesting a  transition
to a dense, chiral symmetry restored phase, accompanied by
a dramatic decrease in the nucleon effective mass. 
Chiral symmetry is built-in in the Skyrmion fluid, hence, the effective nucleon 
mass is not an indicator of chiral symmetry restoration.
The abnormal branch is here totally spurious, and might be possibly reached
in a metastable situation.

At finite temperature the criterion of maximal pressure for a fixed chemical
potential determines the physical phase. Again, the $normal$ branch prevails.

Figure 2 shows the nucleon effective mass as a function of density and 
temperature. 
(The apparent cusp in figure 2 is due to the rudimentary plotting procedure).

In the Walecka model -and many others similar to it- the nucleon mass
decreases as a function of density \ci{kal}. This is not the case in the
Skyrmion fluid model.
The minimum effective mass arises here at a density of around
1.7 $\rho_0$. Contrarily, the usual parametrizations of the scalar field
potential in Walecka models, do not show this limitation. 
The reason for this difference
can be traced back to the dynamics dictated by the dilaton, especially to 
the modified trace anomaly potential.

The scalar field potential of the Walecka type models is built in terms of
powers of the scalar field, up to fourth order generally, to
satisfy renormalizability requirements. (Although this condition is usually
hard to meet due to the need for a negative coefficient for the fourth
order term in most successful fits.) 
However, the dilaton potential is made to obey the condition
demanded by the trace anomaly. Only no-log terms that do not spoil the
anomaly are allowed. This strong demand makes the potential qualitatively
different.
The solutions to the nuclear matter equations with this potential for the
normal branch, exist only in a narrow range of values of $\s$ near $\s = 0$. 

It is now possible to understand why the effective mass increases beyond
a certain density: The dilaton attractive contribution to the
mass is limited, while the $\o$ meson repulsion grows in
direct proportion to the nuclear matter density, eq.~(\ref{cond1}). 
The solution
of the nuclear matter equations then tend to push the dilaton
towards positive values in order to fulfill the scaling
properties of the model, eq.~(\ref{scaling}).

The Skyrmion properties are essentially determined by the scaling
characteristics of the model. The mass of figure 2 scales as $e^{\s_0}$ whereas
the root mean square radius scales as $e^{-\s_0}$. The Skyrmion swells
as a function of density up to densities below $1.7 \rho_0$.
This is due to the attractive effect induced by the dilaton. Above
a certain density the Skyrmion starts to shrink, due to the abovementioned 
repulsive effect in a manner independent of temperature.
This is reminiscent to the behavior of the Skyrmion interaction at
short range (high density) that is known to be repulsive and strong.
Other properties, like $g_A$, magnetic moments, etc, follow directly from 
the scaling of eq.~(\ref{scaling}).


Figures 3-5 depict the pressure, chemical potential and
entropy as a function of density and temperature. 
Although it may appear that there could be a phase transition between $T$ = 125 
MeV and $T$ = 175 MeV, calculations performed with a finer grid than the one 
shown in the graphs, does not support such a scenario. The transition between 
those temperatures is smooth.

At a temperature of 25 MeV the graphs of the pressure and chemical potential
show a rough behavior due to the proximity of this temperature to the 
liquid-gas phase transition that appears in a broad class of
Walecka type models \ci{ser1}.

The entropies of figure 5 were calculated with three different methods. The
agreement between the numbers obtained determine the accuracy of the
minimization procedure to be better than 0.5 $\%$.

As shown in figure 6, above $T$ = 190 MeV there cease to exist solutions around
the the normal density of nuclear matter.
Above $T$ = 220 MeV solutions cease to exist 
for densities below $4 \rho_0$, perhaps signalling the appearance of a 
different phase that we can not describe in terms of Skyrmions and mesons. 
Other degrees of freedom must come into play, perhaps quarks and gluons. 
This behavior is different as compared to the
results of conventional models \ci{kal} for which there are solutions at
all densities and temperatures with a phase transition to a baryonic
plasma phase above a certain temperature typically of the order of 200
MeV. The reason for this difference can be traced back to the special
features of the dilaton coupling and potential.


In the present work we have paved the way for a treatment of a Skyrmion
fluid based on the principles of scale invariance and chiral invariance.
In a later work we will test the approach and especially the potential
obtained in finite nuclei calculations in the Thomas-Fermi approach as
well the corrections due to the rotational degrees of freedom of the
Skyrmion. In a finite nucleus calculation new parameters will enter, like
the dilaton mass. It is then a bit risky to venture any predictions
for such a system.
A much more complicated and ambitious task would be to tackle
the introduction of Skyrmion interactions.

\vspace{3 pc}

Acknowledgements


It is a pleasure to thank Prof. J.M. Eisenberg for his constructive criticism,
Mr. Reem Sari for his help in drawing the graphs and Ms. Shula Coster for 
her Tex guidance.
This work was supported in part by the Israel Science Foundation.

\newpage

\begin{thebibliography}{99}
\bi{Skyrme}  T.H.R. Skyrme, Proc. Roy. Soc. London, {\bf A260}
(1961) 127; {\bf A262} (1961) 237 and Nucl. Phys. {\bf 31} (1962) 556.
\bi{tHooft}  G. 't Hooft, Nucl. Phys. {\bf B72} (1974) 461; {\bf B75} (1974) 
461.
\bi{Witten}  E. Witten, Nucl. Phys. {\bf B160} (1979) 57.
\bi{Bal}  A.P. Balachandran, {\it Proc. 1985 Theoretical Advanced Study
Institute}, M.J. Bowick and F. Gursey, eds. (World Scientific, Singapore,
1985).
\bi{ZB}  I. Zahed and G.E. Brown, Phys. Rev. {\bf 142} (1986) 1.
\bi{HS86a}  G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. {\bf 49} (1986)
825.
\bi{ek1995} J. M. Eisenberg and G. K\"albermann, {\it Nuclear forces from 
Skyrmions}, Int. J. Mod. Phys. E, to be published.
\bi{ANW}  G.S. Adkins, C.R. Nappi, and E. Witten, Nucl. Phys. {\bf B228}
(1983) 552.
\bi{ser1} B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. {\bf 16} (1986) 1.
\bi{ser2} R. J. Furnstahl, B. D. Serot and H. Tang, preprint OSU-95-0326
and references therein.
\bi{ru1} G. Carter, P.J. Ellis and S. Rudaz, preprint NUC-MINN-95-21-T,
and references therein.
\bi{lynn1}B.W. Lynn, Nucl. Phys. {\bf B402} (1993) 281.
\bi{mi1}I. N. Mishustin, Sov. Phys. JETP {\bf 71}(1990) 21.
\bi{AN}  G.S. Adkins and C.R. Nappi, Phys. Lett. {\bf B137} (1984) 251.
\bi{Birse}  M.C. Birse, J. Phys. {\bf G20} (1994) 1287.
\bi{Schech}  J. Schechter, Phys. Rev. {\bf D21} (1980) 3393.
\bi{kal} G. K\"albermann, J. M. Eisenberg and B. Svetitsky, Nucl. Phys.{
\bf A600} (1996) 436.
\bi{sharma} M.V. Stoitsov, P. Ring and M. M. Sharma, Phys. Rev.{\bf C50} (1994)
 1445.
\bi{blaizot} J. P. Blaizot, Phys. Rep. {\bf 64} (1980) 171.

\end{thebibliography}

\newpage
{\bf Figure Captions}

\begin{enumerate}
\item[Fig. 1:] Binding energy per nucleon in MeV as a function of ${\rho}/
{\rho_0}$. Normal branch, dashed line and abnormal branch, full line.

\item[Fig. 2:] Effective Skyrmion mass $M^*$ in MeV as a function of
${\rho}/{\rho_0}$. Full line, $T$ = 25 MeV, dashed line $T$ = 75 MeV,
dash-dot line, $T$ = 125 MeV and dotted line, $T$ = 175 MeV.

\item[Fig. 3:] Pressure in ${\rm MeV}/{{\rm fm}^3}$ as a function of
${\rho}/{\rho_0}$. Line definitions as in Fig. (2).

\item[Fig. 4:] Chemical potential in MeV as a function of ${\rho}/{\rho_0}$.
Line definitions as in Fig. (2)

\item[Fig. 5:] Entropy per baryon as a function of ${\rho}/{\rho_0}$. Line
definitions as in Fig. (2)

\item[Fig. 6:] Pressure in ${\rm MeV}/{{\rm fm}^3}$ as a function of
${\rho}/{\rho_0}$. Dotted line, $T$ = 190 MeV, full line, $T$ = 200 MeV 
and dashed line, $T$ = 220 MeV.
\end{enumerate}
\end{document}

