\magnification 1200
\def\del{\partial}
\def\delR{\delta\!R}
\def\simle{\mathrel{
   \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}
\vsize=7.5in
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\rightline{TAUP 1947-92}
\rightline{March, 1992}
\vskip 1in
\centerline{\bf INSTABILITY OF BUBBLES IN THE QUARK-GLUON PLASMA}
\vskip .5in
\centerline{Gideon Lana}
\centerline{Benjamin Svetitsky}
\vskip\baselineskip
{\it
\centerline{School of Physics and Astronomy}
\centerline{Raymond and Beverly Sackler Faculty}
\centerline{of Exact Sciences}
\centerline{Tel Aviv University}
\centerline{69978 Tel Aviv, Israel} }
\vskip .5in   %\baselineskip 24pt
\noindent
{\it ABSTRACT.} The small surface tension of vacuum bubbles in the
quark-gluon plasma, along with a negative curvature tension, implies
that the uniform plasma is unstable against the formation of such
bubbles.
We show that spherical bubbles are in turn unstable against the growth
of irregularities.
 
%\vfill\eject
\vskip 36pt
Farhi and Jaffe [1] noted some time ago that the MIT bag
model gives an anomalously small surface tension between the high-
and low-temperature phases of QCD.
Consider a spherical bubble of
the vacuum phase embedded in the bulk plasma.
One defines this
surface tension $\sigma$ via an expansion of the bubble's free energy
in inverse powers of its radius $R$,
$$
F(R)=\Delta P \,V + \sigma A +
8\pi \alpha R + \cdots\ , \eqno{(1)}
$$
where $V$ and $A$ are
respectively the volume and surface area of the bubble, and $\Delta
P$ is the difference between the pressures of the two phases (zero at
the transition temperature $T=T_0$, and positive above); we have
defined as well the ``curvature coefficient'' $\alpha$.
In bag model
terms, the surface tension is considered small when $\sigma\ll
B^{3/4}$, since the bag constant $B$ determines all the bulk physics,
including the transition temperature and latent heat.
The smallness
of $\sigma$ has been confirmed in lattice gauge theory calculations
for the pure glue theory [2].
 
If $\sigma$ is small, one is led to consider the value and importance
of the curvature term, and perhaps of higher orders in $1/R$ in (1).
Mardor and Svetitsky [3] found, again in the bag model, that $\alpha$ is
sizeable and negative.
This implies that vacuum bubbles are {\it
stable}, and it was shown in [3] that
this stability also emerges from an exact bag
model calculation, not presuming the validity of the expansion (1).
It was further suggested [3,4] that this result might transcend the bag
model:
All that is required is a thin interface, the energy of which is due
primarily to the reaction of the fields on either side.
A lattice calculation of $\alpha$ in the pure glue theory [5] apparently
supports this picture, even though the interface does {\it not} turn
out to be thin.
 
The fact that one can lower the free energy of the plasma phase by
creating a vacuum bubble seems at first glance to imply that the
equilibrium plasma is a Swiss cheese of bubbles, the density of which
is determined by the (unknown) bubble--bubble interaction.
This would only be true, however, if (1) the bubbles do not affect
each other, and (2) each bubble is stable in itself.
We present in this note some simple calculations which show that
spherical bubbles are unstable against deformation, and that growth
of long, narrow structures is favored.
We do not address the problem of determining the actual equilibrium
configuration of the high-temperature phase, a problem which will
require much more sophisticated analysis.
 
In order to study the stability of the spherical bubble against small
perturbations, we expand the free energy around the equilibrium spherical
bubble, of radius $R_0$.
Minimizing (1), we find that $R_0$ satisfies
$$
4\pi\Delta P\,R_0^2 + 8\pi\sigma R_0 + 8\pi\alpha=0\ .
\eqno{(2)}
$$
The generalization of (1) to non-spherical shapes is [6]
$$
F=\Delta P \,V + \sigma\int dS +\alpha\int dS\,\left({1\over
R_1}+{1\over R_2}\right) + \cdots\ ,
\eqno{(3)}
$$
where the integrand in the third term is the local extrinsic curvature,
with $R_1$ and $R_2$ representing the principal radii of curvature.
Let the bubble surface be specified by
$$
R(\theta,\phi)=R_0 + \delR(\theta,\phi) \quad.
\eqno{(4)}
$$
We demand that $\delR$ have continuous derivatives and satisfy
$\delR \ll R_0$.
To second order in $\delR$, the change in
the curvature integral in (3) is
$$
\eqalignno{
&\int \left[ 2\,\delR \sin\theta  - {\del\over \del\theta}\left(\sin\theta
{\del \over \del\theta}\delR\right) -  {1\over \sin\theta} {\del^2
 \over \del\phi^2}\delR\right] d\theta \,d\phi \cr
&\qquad\qquad\qquad+
{1\over R_0} \int \left[\sin\theta  \left({\del \over
\del\theta}\delR\right)^2 +
{1\over \sin\theta} \left({\del \over \del\phi}\delR\right)^2 \right] d\theta\, 
d\phi\ .
&(5)\cr}
$$
The last two terms in the $O(\delR)$ contribution 
vanish on integration.
 
The total $O(\delR)$ contribution to $F$ is 
$$ \delta F=\left(\Delta P\,R_0^2 +
2\sigma R_0 + 2\alpha\right)
\int \sin \theta \,d \theta \,d\phi\,  \delR  \eqno{(6)}
$$
which is identically zero when $R_0$ is the equilibrium radius,
since the coefficient is proportional to (2).
The correction to $F$ is thus second order in $\delR$,
$$ \delta F=\int \sin \theta \,d \theta \,d\phi
\left\{ A\left(\delR\right)^2 + B\left[\left({\del \over
\del\theta}\delR\right)^2 + {1\over \sin^2\theta} \left({\del \over
\del\phi}\delR\right)^2\right] \right\}\ , \eqno{(7)}
$$
where
$$ A=\Delta P\, R_0 + \sigma\ ,
\qquad B={\sigma\over 2}+{\alpha\over R_0} \quad. \eqno{(8)}
$$
Recall that we presume $\alpha<0$.
When $R_0$ is determined by (2), it is easy to see that
$A$ is positive and $B$ negative.
Integrating the second term in (7) by parts, we obtain
$$\eqalignno{
\delta F & =\int \sin \theta \,d \theta \,d\phi\, \delR\left\{ A\,\delR -
B\left[{1\over \sin\theta} {\del\over \del\theta}
\left({\sin\theta}{\del \over \del\theta}\delR\right) +
{1\over \sin^2\theta}
\left({\del^2 \over \del\phi^2}\delR\right)\right] \right\}\cr
& = \int \sin \theta\, d \theta \,d\phi \,\delR\left( A + B L^2 \right)
\delR \ .&(9)\cr }
$$
$-L^2$ is just the angular part of the Laplacian operator.
In fact, from symmetry considerations, (9) is the only
possible form that could have emerged, and the calculation was needed
only to determine the coefficient $B$.
 
Expanding $\delR$ in a multipole expansion,
$$
\delR(\theta,\phi)=\sum_{l,m}c_l^m Y_{lm}(\theta,\phi)\ ,\eqno{(10)}
$$
we have
$$
\delta F = \sum_{l,m}[A+Bl(l+1)]\left|c_l^m\right|^2\ .\eqno{(11)}
$$
Evidently $\delta F>0$ for a monopole perturbation, since (1) has been
minimized;
for a dipole perturbation we obtain $\delta F=0$ [see (2)], consistent
with the fact that $l=1$ corresponds to a translation of the bubble.
For quadrupole and higher deformations, however, $\delta F$ is always
negative, showing instability against arbitrary changes of shape.
 
One particular deformation is the growth of a finger.
Consider pushing a polar cap of the sphere outward, so that
$\delR(\theta)=a$ for $\theta\simle\theta_0\ll1$, with the transition region
from $\delR=0$ to $\delR=a$ having a thickness $\delta\theta\ll\theta_0$.
A rough estimate of (7) gives
$$
\delta F\sim A\pi R_0^2\theta_0^2a^2 + B 2\pi R_0\theta_0
{a^2\over (\delta\theta)^2}\ .\eqno{(12)}
$$
$\delta F$ can be made negative by making $\delta\theta$ sufficiently
small, whereupon the overall proportionality to $a^2$ makes the finger
grow ever longer.
 
To see the eventual fate of such a finger, we must depart from the
expansion of $F$ in $\delR$.
Since the finger appears to grow into a cylinder, let us calculate
the free energy of a cylinder of radius $R$ and length $L$.
Neglecting the ends, we find that the volume, area, and surface terms contribute
$$F=\Delta P \pi R^2 L + \sigma 2\pi R L + \alpha 2\pi L\ .
\eqno{(13)}
$$
 
When the coefficient of $L$ is positive, the cylinder tends to shrink
in both the radial and axial directions.
However, when $R$ shrinks past $R_{\rm crit}$, which satisfies
$$
{\Delta P\over 2}R_{\rm crit}^2+\sigma R_{\rm crit} +\alpha=0\ ,
\eqno{(14)}
$$
the coefficient of $L$ becomes negative, and the cylinder lowers its free 
energy by growing longer.
Note that for $R<R_{\rm crit}$ the overall free energy of the cylinder
is negative, and thus the uniform plasma is unstable against nucleation
of such cylinders.
 
The validity of our analysis is restricted in two respects.
Firstly, we have not taken into account the finite thickness of the
interface, which is estimated [2,5] to be approximately $2\,T_0^{-1}$.
Secondly, the $1/R$ expansion (1) may be reliably truncated only
as long as $R$ is much larger than the ratio of successive
coefficients in the series; this ratio in turn must be related to
the dynamical scales in the system.
The two dynamical scales are given by the bag constant (or some other
fundamental scale of QCD) and the temperature;
near the transition, the two are comparable.
 
Near $T_0$, where $\Delta P$ may be neglected, the solution of (2) is
$R_0=-\alpha/\sigma\simeq18\,T_0^{-1}$, where we have used the values [3,5]
$\alpha\simeq -4T_0^2/9$ and [2] $\sigma\simeq 0.024\,T_0^3$.
This evidently satisfies the constraints of validity.
 
Farther from $T_0$, where the pressure term in (2) overwhelms the
surface tension term, we estimate $R_0=\sqrt{2\alpha/\Delta P}$.
The bag model gives 
$$
\Delta P = {\pi^2\over90}({\gamma_q-\gamma_h})(T^4-T_0^4)\ ,
\eqno{(15)}
$$
where $\gamma_q=37$ and $\gamma_h=3$ are the effective degeneracy factors
stemming from counting the light degrees of freedom in the quark and hadron
phases.
Near $T_0$, we write $\Delta P\simeq 16\,T_0^3(T-T_0)$; ignoring $\sigma$,
this gives $R_0\simeq \left[6\sqrt{T_0(T-T_0)}\right]^{-1}$.
Demanding $R_0>2T_0^{-1}$ gives $T-T_0< 0.01\,T_0$.
If one were to eschew the bag formula for $\Delta P$ in favor of a lattice
calculation, one would find a substantially smaller $\Delta P$ near the
transition, since $\Delta P \sim \Delta\epsilon (T-T_0)$, where
$\Delta\epsilon$ is the latent heat; lattice calculations give a much
smaller latent heat than does the bag model.
Thus the temperature range where bubbles are large may be
significantly larger than the above estimate.
 
We have here considered only static properties of the quark--hadron
interface.
Dynamical processes in the course of the phase transition will
destabilize the interface even more.
For example, the necessity of carrying heat and baryon number away
from a growing vacuum bubble will lead to a negative effective surface
tension and hence to complex phenomena of pattern formation [7].
 
\vskip24pt
\centerline{\bf Acknowledgments}
\vskip12pt
This work was supported by a Wolfson Research Award administered by the
Israel Academy of Sciences and Humanities.
Further support was given by the German-Israel Foundation.
%\vfill\eject
\vskip 24pt
\def\PRD#1{Phys. Rev. D {\bf #1}}
\def\PLB#1{Phys. Lett. B {\bf #1}}
\def\PRC#1{Phys. Rev. C {\bf #1}}
\def\ZPC#1{Z. Phys. C {\bf #1}}
\def\NPA#1{Nucl. Phys. {\bf A#1}}
\def\NPB#1{Nucl. Phys. {\bf B#1}}
\def\AP#1{Ann. Phys. (NY) {\bf #1}}
\def\ibid#1{{\it ibid.\/} {\bf #1}}
 
\centerline{\bf REFERENCES}
\vskip\baselineskip \frenchspacing
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\item{\ 2.}  K. Kajantie and L. K\"arkk\"ainen, \PLB{214} (1988) 595;
\item{\ }  K. Kajantie, L. K\"arkk\"ainen, and K.~Rummukainen, \NPB{333}
(1990) 100.
\item{\ }S.~Huang, J.~Potvin, C.~Rebbi, and S.~Sanielevici, \PRD{42} (1990)
	2864; {\bf 43} (1991) 2056(E).
\item{\ }R.~Brower, S.~Huang, J.~Potvin, and C.~Rebbi, Boston University
preprint BUHEP-92-3 (unpublished, 1992).
\item{\ 3.}I.~Mardor and B.~Svetitsky, \PRD{44} (1991) 878.
\item{\ 4.}B. Svetitsky, 
lecture given at the Workshop on QCD at Finite Temperature and Density,
Brookhaven National Laboratory, August 1991, Tel Aviv preprint TAUP 1909--91
(1991), to appear in proceedings.
\item{\ 5.}K. Kajantie, L. K\"arkk\"ainen, and K.~Rummukainen, Helsinki
	preprint HU-TFT-92-1 (unpublished, 1992).
\item{\ 6.}R.~Balian and C.~Bloch, \AP{60} (1970) 401.
\item{\ 7.}J.~S.~Langer, Rev. Mod. Phys {\bf 52} (1980) 1.
\bye
