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\begin{document}
\title{Instanton Model of QCD Phase Transition Bubble Walls}

\author{Leonard S. Kisslinger\\
        Department of Physics,\\
       Carnegie Mellon University, Pittsburgh, PA 15213}

\maketitle
%pacs
\indent
\begin{abstract}
Using the liquid instanton model continued into Minkowski space and
modified for finite temperature, the energy momentum tensor for buble walls 
during the QCD phase transition is derived. The resulting surface tension 
of bubble walls is in agreement with lattice calculations. Application to 
bubble collisions in Minkowski space is discussed.
\end{abstract}

\vspace{0.5 in}

\noindent
PACS Indices:12.38.Lg,12.38.Mh,98.80.Cq,98.80Hw


\section{Introduction}
\hspace{.5cm}

   At the time 10$^{-5}$ s the temperature (T) of the universe was 
approximately 300 MeV.  During the time interval between 
10$^{-5}$-10$^{-4}$s the universe passed through the temperature 
T$_c \simeq$ 150 MeV, the critical temperature for the
chiral phase transition from the quark-gluon plasma to the hadronic phase,
the quark-hadron phase transition (QHT). There have been many numerical 
studies of the QHT. Some lattice gauge calculations indicate the the
transition is weakly first order\cite{lat1}, which would imply bubble 
formation during the QHT. At the present time lattice calculations cannot
determine the order of the transition\cite{lat2}. Recent numerical 
calculations in the MIT Bag model\cite{nm} find a first order transition
with hadronic bubbles stable for nucleation at a scale of 1 fm.

  Assuming that the QHT is first-order there have been many studies of the
bubble nucleation\cite{ign} and of possible observational effects of such
a phase transition. The latter include the possibility of large scale density
perturbations\cite{is} and the seeding of primary magnetic 
fields\cite{ol} that could lead to observational effects in the 
Cosmic Microwave Background Radiation(CMBR), or large scale galactic 
or extra-galactic structure. 

  The essential property of the bubbles needed to study the nucleation
is the surface tension\cite{ign,ssw}, $\sigma$. The most recent lattice 
calculations give a value of $\sigma \simeq 0.015 \rm{T_c}^3$\cite{bkp}, 
while earlier studies \cite{iw} find a larger value. 
In our present work we wish to estimate
$\sigma$ from the QCD Lagrangian, with the result that we obtain the
energy momentum tensor that can be used in future work on bubble collisions.
For the electroweak phase transition bubble collisions have been studied and
interesting estimates of magnetic fields formed during the collisions have
been made\cite{ae,cst} using an abelian Higgs model plus the QED 
Lagrangian\cite{kib}. However, because of the complexity of nonperturbative
QCD, a reliable model of the bubble walls formed in the QHT has not been
formulated. It is the goal of the present work to formulate such a model.

  Since the main structure of the bubble walls must be gluonic in nature
we use pure gluodynamics. Noting that the walls separate the 
hadronic from the quark/gluon plasma regions of the 
universe, it is natural to consider an instanton picture of the walls.
The QCD instantons are classical solutions for the color field, which
were derived in Euclidean space\cite{bev,hooft}. They connext points
in two vacuua with different winding numbers. In analogy with Coleman's
model\cite{col} of regions of true and false vacuua, we consider the 
instantons connecting vacuua with different winding numbers in the 
quark-gluon region and the hadronic region on the opposite sides of the 
bubble. The Euclidean instanton model cannot be used for the treatment 
of bubble nucleation and collisions (in fact the energy density and
surface tension vanish in this picture), but the model must be continued
to Minkowski space\cite{col}. In brief, our picture is a Minkowski space
analytic continuation of the instanton model for the bubble walls
separating the quark/gluon from the hadronic regions during the QHT.


 It is well known that the instantons of the instanton liquid model\cite{ss} 
can provide the essential nonperturbative gluonic effects of QCD at the medium
length scale of 0.3-0.5 fm, although instantons do not provide confinement.
As we show below, this seems to be the crucial length scale needed for QCD
bubble walls. In recent work an effective Lagrangian was used to estimate 
domain wall formation possibly associated with the
QCD phase transition\cite{zhit}. Although the resulting domain wall is within
the hadronic bubble it resembles the bubble wall that we obtain with the 
instanton model in the present work. That model has also
been used to examine the possibility of magnetic walls formed during the 
QHT seeding gallactic magnetic structure\cite{zhit2}. Such a magnetic
wall has also been considered for cosmic microwave background radiation
polarization correlations\cite{lsk1}. Also, there are
recent numerical investigations of the T-dependence of the properties of the 
QCD instantons\cite{chu,ss2}, which we need in the present investigation.

  In Sec. 2 we review the instanton liquid model at T=0 and discuss the
energy density and equation of motion in Minkowski space. 
In Sec. 3 we use recent lattice 
results for instantons at finite temperature\cite{chu} as well as our own 
recent work on glueballs\cite{lsk} to estimate the surface tension.
The surface tension is derived at time t=0, and serves as
an initial condition for bubble nucleation and collisions in Minkowski space.
In Sec. 4 we discuss the energy momentum tensor for gluonic QCD in Minkowski
space and possible applications based on the instanton model for
bubble collisions and cosmological observations.

\section{Instanton Model At T=0 and Continuation to Minkowski Space}
\hspace{.5cm}

   In this section we review the instanton model of QCD at T=0. Instantons
are obtained and studied in Euclidean space. As described in early work
on the application of instantons to bubble nucleation and 
collisions\cite{col} in formulating the instanton bubble itself one proceeds
in Eucledean space, while for the study of collisions of nucleating
bubbles one must work in Minkowski space. This will be discussed further in
Sec. 4. The Lagrangian density for pure glue is
\beq
\label{glue}
  \mathcal{L}^{glue} & = & -\frac{1}{4} G \cdot G,
\eeq
where
\beq
\label{G}
    G_{\mu\nu}^n & = & \partial_\mu A_\nu -  \partial_\nu A_\mu
-i g [A_\mu,A_\nu]\\
    A_\mu & = & A_\mu^n \lambda^n/2 , \nonumber
\eeq
with $\lambda^n$ the eight SU(3) Gell-Mann matrices. The instantons are the
classical solutions of the gauge fields for a pure SU(2) gauge theory.
I.e., writing $A_\mu^n = A_\mu^{n,inst} +A_\mu^{n,qu}$, where 
$A_\mu^{n,inst}$ is a pure gauge classical field, if one keeps only the 
instanton gluonic field , and solves the equation of motion obtained by 
minimizing the classical action in Euclidean space 
\beq
      \delta [ \int G^{inst} \cdot G^{inst}] & = & 0,
\eeq
one obtains the solution\cite{bev}
\beq
\label{inst}
 A_\mu^{n,inst}(x)& = & \frac{2 \eta^{-n}_{\mu\nu}x^\nu}{(x^2 + \rho^2)}\\
        G^{n,inst}_{\mu\nu}(x) & = & -\frac{\eta^{-n}_{\mu\nu} 4 \rho^2}
{(x^2 + \rho^2)^2},
\eeq
for the instanton and a similar expression with -n for the anti-instanton, 
where $\rho$ is the instanton size and the  $\eta^{n}_{\mu\nu}$ 
are defined in Ref.\cite{hooft}. The instanton connects points in two QCD 
vacua which differ by one unit of winding number. The instanton contribution 
to the Lagtrangian density is
\beq
\label{linst}
  \frac{1}{4} G^{inst} \cdot G^{inst}  & = & 48 \frac{\rho^4}{(x^2+\rho^2)^4}.
\eeq

The energy momentum tensor, $T^{\mu\nu}$, can be obtained from the
Lagrangian, and in Minkowski space is given by (see, e.g., Ref\cite{iz})
\beq
\label{emt}
T^{\mu \nu} &=& \sum_a(G^{\mu \alpha}_a G_{\alpha a}^\nu
-\frac{1}{4}g^{\mu \nu} G^{\alpha \beta}_a G_{\alpha \beta a}).
\eeq
To get the Euclidean space expression for the energy density, $T^{44}$,
one carries out the analytic continuation giving $x^2 = x^\mu x_\mu$ =
${\vec x}^2 +x_4^2$ in Euclidean space, while it is ${\vec x}^2 -x_0^2$ in 
Minkowski space. From this one see that $g^{44}= -g^{00}$. Therefore in
Euclidean space the energy density in the instanton model is
\beq
\label{eucliden}
T_{44}^{inst} & = & G_{a 4 \alpha} G_{a \alpha 4} +\frac{1}{4}G_{a\alpha\beta}
  G_{a\alpha\beta} \nonumber \\
      & = & 0.
\eeq
The result shown in Eq.(\ref{eucliden} follows from Eq.(\ref{inst}) and
the properties of the  $\eta^{n}_{\mu\nu}$ given in Ref\cite{hooft}.

For the nucleation of QCD bubles and for bubble collisions one must
work in Minkowski space. From Eqs.(\ref{emt},\ref{inst}) one finds for
the energy density in Minkowski space
\beq
\label{minen}
  T^{00,inst}  & = & \frac{1}{2} G^{inst} \cdot G^{inst}
\nonumber\\
     & = & 96 \frac{\rho^4}{(x^2+\rho^2)^4}
\eeq
Also, to evaluate the surface tension of the bubble 
walls we must consider instantons at finite temperature, which we do in the
next section.


\section{Bubble Wall surface tension}
\hspace{.5cm}

   In this section we evaluate the surface tension in the instanton model.
In order to do this we must consider the dependence of the instanton 
size and density on the temperature. We start with calculating the 
surface tension for an instanton wall. 

   The surface tension is the essential parameter for classical bubble
nucleation. For the treatment of nucleation one must work in Minkowski space. 
As discussed in detail in Ref.\cite{col}, to treat nucleation and collisions 
with instanton input one makes an analytic continuation from Euclidean to 
Minkowski space at the initial time, t=0, For our problem this is achieved
by replacing t by it, or in Minkowski space  $x_\mu x^\mu = {\vec x}^2 - t^2$.
In the present section we study the surface tension at t=0; and the
energy density in Minikowski space derived in the previous section from 
the instanton solutions can be used for the derivation of the surface tension.
The results at t=0 which we find will serve as part of
the initial conditions for the study of nucleation and collisions,
as discussed in Sec. 4. 

The surface energy is the surface tension $\times$ the area of the 
wall for a thin wall. Note that the wall thickness is of the order of 
$\rho$, the instanton size, which is a fractrion of a fermi, so that a 
thin-wall formalism is appropriate. Therefore the surface tension with one 
instanton at t=0 is
\beq
\label{iwall1}
       \sigma^{inst} & = & \int dx  T^{00,inst}(x,0,0) \nonumber \\
                     & = & \int dx \frac{1}{2}{\bf G \cdot G}.
\eeq
 From Eqs.(\ref{linst},\ref{iwall1})
\beq
\label{iwall}
  \sigma^{inst} & = & 3\cdot2^5\ \ \int dx \frac{\rho^4}
 {(x^2+\rho^2)^4} \nonumber \\
               & = & 6\cdot2^5\ \ \frac{5!!}{6!!}\frac{\pi}{\rho^3}.
\eeq

   The resulting surface tension in our instanton model is
\beq
\label{sigma}
 \sigma & = & 6\cdot5\ \ \frac{\pi \overline{N}^2}{\overline{\rho}^3},
\eeq
where $\overline{N}$ is the instanton number density at the surface and
$\overline{\rho}$ is the instanton size at T = T$_c$. Eq.(\ref{sigma}) is
obtained from Eq.(\ref{iwall}) using the model applied for the lattice 
calculations\cite{bkp}, and described in detail in Ref.\cite{iw}. In this
picture the free energy is calculated using two peaks for the two phases
on the two sides of the bubble, separated by a mixed quark/gluon-hadronic 
phase. We obtain $\overline{N}$ from the T = 0 instanton number per volume, 
n = (N/V), determined by the tunneling amplitude\cite{hooft,ss}. Defining 
n = $\overline{n}$ GeV$^4$ at finite T and using V= $\overline{\rho}^4$, 
$\overline{N}^2$ = 5.96 $\overline{n}^2$, where we use 
$\overline{\rho}$ = 0.25 fm, as discussed next.

At T=0 $\rho$ = 0.33 fm in the liquid instanton model\cite{ss}. This has
also been found in lattice calculations\cite{neg}.
There have been a number of numerical investigations on the modification of
the instanton system at finite T.  Based on these calculation\cite{chu,ss2} 
we take $\overline{\rho}$ = 0.25 fm. There is uncertainty in the value of 
the instanton density, n, at T=0.  In our work on scalar 
mesons/glueballs\cite{lsk} we found values of $\overline{n}$ = 0.0008-0.0015, 
with 0.0008 used in the instanton liquid model. From the results of the 
finite T calculations the n seems to decrease by about a factor of 
2 at T$_c$, so we use the range $\overline{n}$ = 0.0004-0.00075. This gives 
us the result
\beq
             \sigma & = & (0.013 \rightarrow 0.046) T_c^3,
\eeq
compared to the value $\sigma =$  0.015 T$_c^3$ in the most recent lattice
calculation\cite{bkp}. We note that the instanton liquid model of the QCD 
bubble wall gives a value $\sigma =$  0.013 T$_c^3$ in very good agreement 
with the lattice calculation, and that our model for the bubble wall is 
quite promising.

\section{Discussion of Applications to Collisions}
\hspace{.5cm}

   As was discussed in Sec. 2 the instanton model itself describes a static
bubble.  For the study of bubble nucleation and collisions one works in 
Minkowski space. Starting from the instanton Lagrangian and using standard
field theory methods one obtains the color ${\bf E}$
and  ${\bf B}$ fields,
\beq
         E^n_i & = &  G^n_{i0} \nonumber \\
         B^n_i & = &  -\frac{1}{2} \epsilon^{ijk} G^n_{jk},
\eeq
with the i,j,k indices running 1,2,3. From this the energy-momentum
tensor, ${\bf T}$ for an instanton in Minkowski space is given as\cite{iz}
\beq
\label{00}
      T^{00,inst} & = & \frac{1}{2}({\bf E}^n\cdot{\bf E}^n + 
 {\bf B}^n\cdot{\bf B}^n) \\
\label{01}     
      T^{0i,inst} & = & \epsilon^{ijk} E^n_j B^n_k \\
\label{ij}      
      T^{ij,inst} & = & (E^{ni}E^{nj} +B^{ni}B^{nj}) -T^{00,inst}\,\delta_{ij}
\eeq
  From the results of Sec. 2 and with a straightforward calculation we obtain
for the stress energy tensor in the instanton model from Eq.(\ref{ij})
\beq
  T^{ij,inst} & = & - \, \left( \frac{4 \overline{\rho}^2}
{(x^2 +\overline{\rho}^2)^2} \right)^2 \, \delta_{ij} .
\eeq

From this one can get the force
\beq
      \frac{d{\bf P}}{dt} & = & \int\!\!\int {\bf T} \cdot d{\bf a}
\eeq
through the surface. One can use this to study the collisions of QCD bubbles,
illustrated in Fig. 1.
\begin{figure}
\begin{center}
\epsfig{file=qcdbub1.eps,height=5cm,width=12cm}
%\caption{}
{\label{Fig.1}}
\end{center}
\end{figure}
Solutions to the equations of motion have been found for a pure SU(2)
gauge theory in Minkowski space\cite{fks}. Using the general picture
of Coleman\cite{col} in which one sets the instanton initial conditions
at time t=0 and then uses the equations of motion in Minkowski for the 
evolution, a SU(2) model of of QCD instantons has been developed and
applied to heavy relativistic heavy ion collisions\cite{ocs}. We are 
developing a similar program for our cosmological studies of the QCD 
chiral phase transition in the early universe.

During the collision of two bubbles the colliding walls might form a wall
within the merged bubbles. Note that in Ref.\cite{zhit} an estimate of the
lifetime of the gluonic domain walls in their model was 10$^{-5}$s, which
allows magnetic walls to be formed\cite{zhit2}. Although the present theory
is quite different, a two-dimensional model of bubble collisions starting
from the gluonic QCD Lagrangian finds that an internal gluonic wall is
formed\cite{jck}, and this suggests that magnetic structures could form.
From helicity conservation a magnetic structure would remain until the last 
scattering of the CMBR. Using the standard evolution methods as used for
tangled magnetic fields for the scattering \cite{sub} or for
metric perturbations\cite{dfk} one can investigate possible observational
effects in the CMBR. A preliminary investigation of  this has been carried 
out\cite{lsk1} and this is a subject of future research.

\vspace{1mm}
{\bf Acknowledgements}\\
The author would like to acknowledge helpful discussions with Mikkel Johnson, 
Ernest Henley, Pauchy Hwang and Ho-Meoyng Choi.
This work was supported in part by NSF grant 8 and by the Taiwan
CosPA Project, Taiwan Ministry of Education 89-N-FA01-1-3

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\end{document}




