%\documentstyle[prc,aps,preprint,epsf]{revtex}
\documentstyle[twocolumn,prc,aps,epsf]{revtex}

%\topmargin=0in

\preprint{NUC-MINN-02/8-T}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}

\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}

%\def\thalf{{\textstyle{\frac{1}{2}}}}
\def\oneth{{\textstyle{\frac{1}{3}}}}

\draft
\begin{document}

\title{Optically opaque color-flavor locked phase inside compact stars} 

\author{Igor A.~Shovkovy$^{a,b,*}$ and Paul J. Ellis$^{b}$}


\address{$^{a}$Institut f\"ur Theoretische Physik,
Johann Wolfgang Goethe--Universit\"at,
60054 Frankfurt/Main, Germany}

\address{$^{b}$School of Physics and Astronomy, University of Minnesota,
        Minneapolis, Minnesota 55455, USA}

\date{\today}

\maketitle

\begin{abstract} 

The contribution of thermally excited electron-positron pairs to the bulk
properties of the color-flavor locked quark phase inside compact stars is
examined. The presence of these pairs causes the photon mean free path to
be much smaller than a typical core radius ($R_{0} \simeq 1$ km)  for all
temperatures above $25$ keV so that the photon contribution to the thermal
conductivity is much smaller than that of the NG bosons. We also find that
the electron and positron contributions to the total thermal energy of the
quark core is considerably smaller than the contribution of the NG bosons
and photons.  These electron-positron pairs dominate the electrical
conductivity since their masses are considerabley smaller than those of
the lightest charged pseudo-NG bosons.

\end{abstract}

\pacs{12.38.Aw, 21.65.+f}

% 11.30.Qc Spontaneous and radiative symmetry breaking
% 12.39.-x Phenomenological quark models
% 11.15.Ex Spontaneous breaking of gauge symmetries
% 12.38.Aw General properties of QCD
% 12.38.Mh Quark-gluon plasma
% 21.65.+f Nuclear matter

The observational study of compact stars is of prime importance because it
could potentially reveal the existance of new phases of matter at high
densities and low temperatures. Given the expected central densities it
was suggested long ago that some compact stars may be, at least partially,
made of quark matter \cite{quark-stars}. Recent observations on the
cooling \cite{quark1} of one neutron star and the radius \cite{quark2} of
another led the authors to suggest that exotic components, possibly
quarks, were required. These suggestions have been disputed
\cite{no-quark1,no-quark2}. However the current situation, where there is
no evidence that free quarks play a role in compact stars, may well change
as further observations are made.

If the central densities of compact stars are indeed sufficient to support
quark matter, it is most likely that it will be found in the color-flavor
locked (CFL) phase \cite{ARW}. In this phase the three lightest flavors of
quarks participate in a condensate on an approximately equal footing. From
a theoretical viewpoint, there already exists a rather detailed
understanding of the basic properties of CFL quark matter
\cite{ARW,CasGat,SonSt,ShoWij,Risch,other,rev}.  This brief report
augments our previous discussion \cite{cool-star} of the thermal
properties of CFL quark matter by addressing the role of thermally excited
electron-positron pairs which, hitherto, has been overlooked.

Let us first recall the symmetry properties of the CFL ground state.  The
original gauge symmetry $SU(3)_{c}$ and the global chiral symmetry
$SU(3)_{L} \times SU(3)_{R}$ break down to a global diagonal ``locked"
$SU(3)_{c+L+R}$ subgroup. Because of the Higgs mechanism gluons become
massive and decouple from the infrared dynamics. The quarks also decouple
because large gaps develop in their energy spectra. The breaking of chiral
symmetry leads to the appearance of an octet of pseudo-Nambu-Goldstone
(NG) bosons ($\pi^{0}$, $\pi^{\pm}$, $K^{\pm}$, $K^{0}$, $\bar{K}^{0}$,
$\eta$). In addition an extra NG boson $\phi$ and a pseudo-NG boson
$\eta^{\prime}$ appear in the low energy spectrum as a result of the
breaking of global baryon number symmetry and approximate $U(1)_{A}$
symmetry, respectively. The general structure of the low energy action in
the CFL phase can be established by symmetry arguments alone
\cite{CasGat}. However, the values of the parameters in such an action can
be rigorously derived only at asymptotically large baryon densities
\cite{SonSt}. Thus, in the most interesting case of intermediate densities
existing in the cores of compact stars, the details of the action are not
well known. For the purposes of the present paper, however, it suffices to
know that there are 9 massive pseudo-NG bosons and one massless NG boson
$\phi$ in the low energy spectrum. It was argued in Ref.~\cite{cool-star}
that the latter, along with photons, should dominate the kinetic
properties of dense quark matter. As we reveal below, some kinetic
properties are also affected by the presence of the thermally excited
electron-positron pairs.

We start by mentioning that it is now commonly accepted that the CFL phase
is electrically neutral \cite{neutral}. The actual meaning of this
statement is that the chemical potential related to the electric charge is
zero. This simply implies that at finite temperature the (nonvanishing)
densities of electrons and positrons are equal. Then it is natural to ask
whether such electron-positron pairs have any significant impact on the
physical properties of CFL quark matter.

The first issue to address is the photon mean free path. Since photons
scatter quite efficiently from charged leptons, even small numbers of
electrons and positrons could substantially reduce the transparency of CFL
quark matter. Here we have in mind the quark core of a compact star.  Now
the photon mean free path can be rather well approximated by the simple
expression:
\be
\ell_{\gamma} \simeq \frac{1}{2 n_{e}(T) \sigma_{\rm T}}\;,
\ee
where $n_{e}(T)$ is the equilibrium density of electrons in a plasma at
temperature $T$, the factor of 2 takes into account the equal density of
positrons and
\be
\sigma_{\rm T} = \frac{8\pi}{3} \frac{\alpha^{2}}{m_{e}^{2}}
\approx 66.54 \mbox{~fm}^{2}
\ee
is the well known expression for the Thomson cross section in terms of the
fine structure constant $\alpha$ and the electron mass, $m_e$. This
expression is the limiting case of the more complicated Compton cross
section for low photon energies. Since this limit works rather well for
$\omega_{\gamma} \ll m_{e}$ and $m_{e} \simeq 0.5$ MeV, this is sufficient
for the purposes of this paper.

The average equilibrium electron or positron density in a 
finite temperature neutral plasma is \cite{thermo-asympt}:
\be
n_e = \frac{m_{e}^2 T}{\pi^{2}} \sum_{k=1}^{\infty}
\frac{(-1)^{k+1}}{k} K_{2}\!\left(\frac{m_{e}k}{T}\right),
\label{n_e}
\ee
where $K_2$ is a modified Bessel function.
By making use of this result we find the 
temperature dependence of the photon mean free path in the 
CFL phase shown in Fig.~\ref{fig-mfp}. We see that 
\be
\ell_{\gamma} \alt 220\mbox{~m}, \quad \mbox{for}   \quad
T \agt 25\mbox{~keV}.
\label{ell_gamma}
\ee
Since the radius of the CFL core, $R_0$, is of order 1 km the photon 
mean free path is short for temperatures above $25$ keV so that 
the quark core of a compact star is opaque to light. Conversely,
transparency can be considered to set in when the mean free path 
exceeds $1$ km which occurs for temperatures below
$23.4$ keV.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize=8.0cm
\epsffile[88 14 595 285]{phot_mfp.eps}
\caption{Photon mean free path as a function of temperature 
in the color-flavor locked phase.}
\label{fig-mfp}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now, let us discuss the implications of the fact that the photon mean free
path is very short in comparison to the characteristic quark core radius.
In our previous study \cite{cool-star}, based on the assumption of a very
long mean free path for photons, we concluded that photons and massless NG
bosons give about equal contributions to the thermal conductivity of CFL
matter. In the light of the above results we should restrict this
statement to temperature of less than $25$ keV or so. At higher
temperatures the photon contribution to the thermal conductivity becomes
negligible and only the contribution \cite{cool-star} of massless NG
bosons, $\phi$, needs to be considered so that
\be
\kappa_{\rm CFL} =\kappa_{\phi} \simeq \frac{2\pi^{2}}{15} T^{3} R_{0},
\quad \mbox{for} \quad T \agt 25\mbox{~keV},
\label{kappa-CFL}
\ee
where for the velocity we used $v_{\phi}=1/\sqrt{3}$. Numerically, this 
leads to the following estimate
\be
\kappa_{\rm CFL} \simeq 7.2 \times 10^{31} T_{\rm MeV}^{3} R_{0,{\rm km}}
\mbox{~erg~cm}^{-1} \mbox{~sec}^{-1} \mbox{~K}^{-1},
\label{kappa-CFL-num}
\ee
where the notation indicates that $T$ is measured in units of MeV and
$R_0$ is measured in units of km. This differs from our previous result
\cite{cool-star} by a factor $3/5$. Of course this does not change the
qualitative conclusion that the thermal conductivity of the CFL matter is
extremely high.

In our previous work \cite{cool-star} we evaluated the thermal energy of
the CFL core carried by the massless NG bosons, $\phi$, and photons and
concluded that it was much smaller than that of the outer nuclear layer so
that it could be efficiently removed by heat conduction. Now we need to
evaluate in addition the electron-positron contribution to the thermal
energy of the core. It is straightforward to obtain the expression
\cite{thermo-asympt}
\ba
E_{e}(T) &=& \frac{8T \left(R_{0} m_{e}\right)^{3}}{3\pi}
\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} \nonumber \\
&\times& \left[K_{1}\!\left(\frac{m_{e}k}{T}\right)
+\frac{3T}{m_{e}k} K_{2}\!\left(\frac{m_{e}k}{T}\right)\right].
\label{E_e}
\ea
In Fig.~\ref{fig-th-en} we indicate this contribution to the thermal
energy by a dashed line, while our previously calculated thermal energy
due to the massless states, $E_{\phi,\gamma}$, is indicated by the solid
line (the radius of the CFL quark core, $R_0$, is taken to be 1 km).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize=8.0cm
\epsffile[88 14 580 280]{therEnergy_phi_vs_elec.eps}
\caption{The contributions of massless states (solid line) and 
charged leptons (dashed line) to the thermal energy of a CFL quark 
core with radius $R_{0}=1$ km.}
\label{fig-th-en}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The comparison shows that the charged lepton contribution is very small,
$E_{e} \ll E_{\phi,\gamma}$, for all temperatures below $0.1$ MeV.  
Moreover, even at higher temperatures, $E_{e} \alt E_{\phi,\gamma}$.  
Thus, the presence of thermally excited electron-positron pairs does not
qualitatively affect the thermal energy of CFL matter. This finding,
together with the previous discussion of the thermal conductivity, means
that the cooling mechanism described in Ref.~\cite{cool-star} also remains
qualitatively unchanged.

A separate issue is the electrical conductivity of CFL matter. Since
electrons and positron are charged particles, they should give a
nonvanishing contribution to this transport coefficient. 
In principle charged pseudo-NG bosons will also contribute, as
mentioned in Ref.~\cite{cool-star}. In the CFL phase the lightest 
charged pseudo-NG boson is the positively charged kaon, $K^{+}$, whose 
mass is typically estimated to be at least an order of magnitude larger 
than that of the electron (see for example Refs.~\cite{SonSt,kaon-mass}). 
Assuming this to be true, it is clear that electrons and positrons rather 
than charged kaons will provide the  dominant contribution to the electrical
conductivity of CFL matter.

In order to give an estimate for the electrical conductivity of the
thermally excited electron-positron plasma in the CFL phase, we need to
know the temperature dependence of the electron mean free path.
The mean free path is finite due to the scattering of electrons by
positrons, as well as annihilation, braking radiation (bremsstrahlung)
and Compton
scattering. These processes have recently been studied in the 
ultra-relativistic limit in order to determine the kinetic properties of
gauge theories \cite{AMY}. However, in the non-relativistic limit, 
the annihilation, braking radiation and Compton scattering contributions 
are expected to be negligible, 
and the corresponding expression for the electron mean free path is known 
in the so-called leading logarithm approximation. This limit would
not be appropriate in the early life of a compact star when its interior is 
hot. However restricting ourselves to compact stars where the temperature 
$T\alt m_{e}$ we use the non-relativistic approximation. We then have the
following estimate for the electron mean free path \cite{LifPit}
\be 
\ell_{e}\sim \frac{T^{2}}{4\pi \alpha^{2} n_{e} L_{e} }, 
\label{el-mfp}
\ee 
where $n_{e}$ is the number density of positrons which, according to our
assumption, is the same as that of electrons given in
Eq.~(\ref{n_e}). The so-called Coulomb logarithm $L_{e}$ appears as a
result of the long range nature of the Coulomb interaction. Its 
value is determined by the Debye screening mass $m_{D}$ inside the plasma. 
In particular, from Ref.~\cite{LifPit},
\be 
L_{e} \simeq \ln\frac{T}{m_{D}\mbox{max}(\alpha, \bar{v}_{e})}, 
\label{L-e} 
\ee
where $\bar{v}_{e}\simeq \sqrt{T/m_{e}}$ is the average thermal velocity
of the electrons. The Debye mass itself is given by \cite{LifPit}
\be 
m_{D}^{2}=\frac{8\pi \alpha n_{e}}{T}. 
\label{Debye} 
\ee 
Now, we are in a position to derive an estimate for the electrical
conductivity of the electron-positron plasma existing inside the CFL 
quark core of the star. 
We use the following classical expression:
\be 
\sigma_{e} \simeq \frac{8\pi\alpha n_{e} \ell_{e} }{ m_{e} \bar{v}_{e}},
\label{el-cond} 
\ee
relating the conductivity to the electron mean free path in 
Heaviside-Lorentz units. If the mean
free path $\ell_{e}$ is less than the typical size of the CFL quark phase,
$R_{0}\simeq 1$ km, which is the case for temperatures higher than 
about $17$ keV, this expression for the electrical conductivity
can be written
\be 
\sigma_{e} \sim \frac{2T^{3/2}}{\alpha \sqrt{m_{e}} L_{e} }. 
\ee 
It should be noted that this expression depends
on the density of electrons only through the Coulomb logarithm, $L_e$.
If this expression and the ultra-relativistic result \cite{AMY} are
extrapolated to $T\sim m_e$, it is satisfactory that they are of
similar magnitude.

For completeness we should also estimate the electron and positron 
contributions to the thermal conductivity. To this end we use 
the following classical expression \cite{cool-star}:
\be 
\kappa_{e} \simeq \oneth \bar{v}_{e} \ell_{e} c^{(e)}_{v}, 
\label{th-cond} 
\ee 
where $c^{(e)}_{v}$ denotes the electron and positron contribution to the 
specific heat. As is easily checked, compared to the NG-boson contribution 
given in Eq.~(\ref{kappa-CFL}), this contribution is always small. At
temperatures higher than about $17$ keV it is small
because the electron mean free path is small. At low temperatures
($T \alt 17$ keV)  it is small because $\ell_{e}$ is restricted by the
size of the core and does not grow indefinitely, while the density of
electrons available is decreasing exponentially with decreasing
temperature. Numerically, we estimate that the condition $\ell_{e}
\agt R_{0}$ is satisfied if $T\alt 16.7$ keV. At such low temperatures, 
the thermal conductivity takes the following form:
\be 
\kappa_{e} \simeq \frac{\sqrt{2} m_{e}^{3} R_{0}} {3 \pi^{3/2}}
\exp\left(-\frac{m_{e}}{T}\right) , \quad \mbox{for} \quad 
T\alt 17\mbox{~keV}.
\label{th-cond-R0} 
\ee 
This is negligible compared to $\kappa_{\phi}$. In the same temperature
regime the electrical conductivity in Eq. (\ref{el-cond}) becomes:
\be
\sigma_{e} \simeq \frac{2^{5/2} \alpha m_{e} T R_{0}} { \sqrt{\pi} }
\exp\left(-\frac{m_{e}}{T}\right), \quad \mbox{for} \quad 
T\alt 17\mbox{~keV}.
\label{el-cond-R0} 
\ee 
This electron contribution will dominate the electrical conductivity because 
the contribution from, say, a positively charged kaon contains
a factor $\exp \left(-m_{K^{+}}/T\right)$ which results in an enormous
suppression at low temperatures due to its larger mass.

In conclusion, our present analysis shows that the presence of a
thermally excited electron-positron plasma inside the neutral CFL core
of a compact star has some interesting consequences. One of the most 
interesting, and somewhat unexpected, consequences is that the 
CFL quark matter is optically opaque at all temperatures higher than 
about $25$ keV. This suppresses the photon contribution to the thermal
conductivity, although at lower temperatures it is comparable to
that arising from the massless NG boson associated with the breaking of 
baryon number. Nevertheless the thermal conductivity of CFL matter is 
very large in all regimes so that
the thermal energy from the CFL core of a compact star is efficiently 
conducted away to the outer nuclear layer and the core remains 
nearly isothermal, in agreement with our previous conclusions
\cite{cool-star}. If bare CFL quark stars exist in the Universe
the fact that they only become transparent to photons when
they have cooled to rather low temperatures might be of some 
observational importance.

A similar interplay between the NG boson and the photon contributions
should also appear in other transport coefficients
for the CFL phase, such as the shear viscosity.

The electrical conductivity of the CFL phase will be largely
determined by the electron-positron plasma 
since contributions from other charged particles, such as kaons, 
are strongly suppressed.
This may turn out to be important in studying the dynamo mechanism 
\cite{dynamo} of magnetic field generation in compact stars with 
quark cores.


{\bf Acknowledgments.} This work was supported by the U.S. Department of
Energy Grant No.~DE-FG02-87ER40328. The work of I.A.S. was partially 
supported by Gesellschaft f\"{u}r Schwerionenforschung (GSI) and by 
Bundesministerium f\"{u}r Bildung und Forschung (BMBF).


\begin{references}

\item[*]{On leave of absence from Bogolyubov Institute for
Theoretical Physics, 252143 Kiev, Ukraine.}

\bibitem{quark-stars} E.~Witten,
%``Cosmic Separation Of Phases,''
Phys.\ Rev.\ D {\bf 30}, 272 (1984);
%%CITATION = PHRVA,D30,272;%%
C.~Alcock, E.~Farhi and A.~Olinto,
%``Strange Stars,''
Astrophys.\ J.\  {\bf 310}, 261 (1986).
%%CITATION = ASJOA,310,261;%%

\bibitem{quark1} P.~Slane, D.J.~Helfand, S.S.~Murray,
%``New Constraints on Neutron Star Cooling from Chandra ... 3C58,'' 
Astrophys.\ J.\  {\bf 571}, L45 (2002).
%%CITATION = ;%%

\bibitem{quark2} J.J.~Drake, H.L.~Marshall, S.~Dreizler, 
P.E.~Freeman, A.~Fruscione, M.~Juda, V.~Kashyap, F.~Nicastro, 
D.O.~Pease, B.J.~Wargelin, and K.~Werner,
%``Is RX J185635-375 a Quark Star?,''
Astrophys.\ J.\ {\bf 572}, 996 (2002).
%%CITATION = ;%%

\bibitem{no-quark1} 
D.G.~Yakovlev, A.D.~Kaminker, P.~Haensel and O.Y.~Gnedin,
%``The Cooling Neutron Star in 3C 58,''
Astron.\ \& Astrophys.\  {\bf 389} (2002) L24.
%%CITATION = ;%%

\bibitem{no-quark2} F.M.~Walter and J.M.~Lattimer, 
%``A Revised Parallax and its Implications for RX J185635-3754,''
Astrophys.\ J.\ {\bf 576}, L145 (2002).
%%CITATION = ;%%

\bibitem{ARW} M.~Alford, K.~Rajagopal and F.~Wilczek,
%``Color-flavor locking and chiral symmetry breaking in ...,''
Nucl.\ Phys.\ B {\bf 537}, 443 (1999).
%%CITATION = ;%%

\bibitem{CasGat} R.~Casalbuoni and R.~Gatto,
%``Effective theory for color-flavor locking in high density QCD,''
Phys.\ Lett.\ B {\bf 464}, 111 (1999).
%%CITATION = ;%%

\bibitem{SonSt} D.T.~Son and M.A.~Stephanov,
%``Inverse meson mass ordering in color-flavor-locking ...,''
Phys.\ Rev.\ D {\bf 61}, 074012 (2000);
erratum {\em ibid.} D {\bf 62}, 059902 (2000).
%%CITATION = ;%%

\bibitem{ShoWij} I.A.~Shovkovy and L.C.R.~Wijewardhana,
%``On gap equations and color flavor locking in cold dense ...,''
Phys.\ Lett.\ B {\bf 470}, 189 (1999);
%%CITATION = ;%%
T.~Schafer,
%``Patterns of symmetry breaking in QCD at high baryon density,''
Nucl.\ Phys.\ B {\bf 575}, 269 (2000).
%%CITATION = ;%%

\bibitem{Risch} D.H.~Rischke,
%``Debye screening and Meissner effect in a three-flavor ...,''
Phys.\ Rev.\ D {\bf 62}, 054017 (2000).
%%CITATION = ;%%

\bibitem{other} C.~Manuel and M.H.G.~Tytgat,
%``Sum rules in the CFL phase of QCD at finite density,''
Phys.\ Lett.\ B {\bf 501}, 200 (2001);
%%CITATION = ;%%
V.A.~Miransky, I.A.~Shovkovy and L.C.R.~Wijewardhana,
%``Bethe-Salpeter equation for diquarks in color-flavor locked ....,''
Phys.\ Rev.\ D {\bf 63}, 056005 (2001).
%%CITATION = ;%%

\bibitem{rev} K.~Rajagopal and F.~Wilczek,
%``The condensed matter physics of QCD,''
{\it At the Frontier of Particle Physics: Handbook of QCD}, 
edited by M. Shifman (World Scientific, Singapore, 2001) Vol. 3, p.2061;
%%CITATION = ;%%
D.K.~Hong,
%``Aspects of color superconductivity,''
Acta Phys.\ Polon.\ B {\bf 32}, 1253 (2001);
%%CITATION = ;%%
M.G.~Alford,
%``Color superconducting quark matter,''
Ann.\ Rev.\ Nucl.\ Part.\ Sci.\  {\bf 51}, 131 (2001).
%%CITATION = ;%%

\bibitem{cool-star} I.A.~Shovkovy and P.J.~Ellis,
%``Thermal conductivity of dense quark matter and cooling of stars,''
Phys. Rev. C {\bf 66}, 015802 (2002);
%%CITATION = ;%%
.
%%CITATION = ;%%

\bibitem{neutral} K.~Rajagopal and F.~Wilczek,
%``Enforced electrical neutrality of the color-flavor locked phase,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 3492 (2001);
%%CITATION = ;%%
A.W.~Steiner, S.~Reddy and M.~Prakash,
%``Color-neutral superconducting quark matter,''
;
%%CITATION = ;%%
F.~Neumann, M.~Buballa and M.~Oertel,
%``Mixed phases of color superconducting quark matter,''
.
%%CITATION = ;%%

\bibitem{thermo-asympt} S.M.~Johns, P.J.~Ellis and J.M.~Lattimer,
%``Numerical Approximation to the Thermodynamic Integrals,''
Astrophys.\ J.\  {\bf 473}, 1020 (1996).
%%CITATION = ;%%

\bibitem{kaon-mass} T.~Schafer,
%``Instanton effects in QCD at high baryon density,''
Phys.\ Rev.\ D {\bf 65}, 094033 (2002).
%%CITATION = PHRVA,D65,094033;%%

\bibitem{AMY} G.~Baym, H.~Monien, C.J.~Pethick and D.G.~Ravenhall,
%``Transverse Interactions And Transport In Relativistic ... Plasmas,'' 
Phys.\ Rev.\ Lett.\  {\bf 64}, 1867 (1990);
%%CITATION = PRLTA,64,1867;%%
G.~Baym and H.~Heiselberg,
%``The electrical conductivity in the early universe,''
Phys.\ Rev.\ D {\bf 56}, 5254 (1997);
%%CITATION = ;%%
P.~Arnold, G.D.~Moore and L.G.~Yaffe,
%``Transport coefficients in high temperature gauge theories... ,''
JHEP {\bf 0011}, 001 (2000).
%%CITATION = ;%%

\bibitem{LifPit} E.M.~Lifshitz and L.P.~Pitaevskii,
{\sl Physical kinetics} (Pergamon, New York, 1981).

\bibitem{dynamo} C.~Thompson and R.C.~Duncan, 
%``Neutron star dynamo and the origin of pulsar magnetism,''
Astrophys.\ J.\ {\bf 408}, 194 (1993). 
%%CITATION = ASJOA,408,194;%%
 

\end{references}

\end{document}

