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\title{Vacuum Polarization around Black Holes with Quantum Hair%
\footnote{This talk is based on ref.~\cite{KS4}.}}
\author{Kiyoshi SHIRAISHI\\
Akita Junior College\\
Shimokitade-sakura, Akita-shi, Akita 010, Japan}
\date{}
\begin{document}
\maketitle
%\begin{abstract}

%\end{abstract}


Coleman, Preskill and Wilczek have discussed (in ref.~\cite{PC}) the role
of virtual cosmic strings around black holes with screened charge in the
Higgs phase. The electric field induced by the charge of the black hole
(quantum electric hair) is present in the locus of the virtual (thin)
strings near the horizon membrane. Thus we can measure the effect of the
screened charge of the black hole in the vicinity of the horizon surface.

The authors of ref.~\cite{PC} have also shown that the charge of the
system reduces the temperature of the black hole by the discussion on the
partition function. It is worth studying the possibility that we can
observe the quantum effects of the screened charge of black holes at a
spatially-distant place.

In this talk we explicitly evaluate the expression for the Hawking
radiation of charged particles in the Hartle-Hawking vacuum~\cite{HH} at
spatial infinity, taking the virtual string effect into account. The
``effective'' temperature will be read from the amount of the Hawking
radiation.

Now suppose we live in the Higgs (screened) phase of Abelian Higgs model
in which the Higgs has $U(1)$ charge $N e$ $(N>1)$. we consider the ``thin
string'' case, for a definite discussion. The ``motion'' of the virtual
string as a topological defect generates the following gauge
configuration at spatial infinity~\cite{PC}:
\be
\int_0^{\beta\hbar}A_{\tau}d\tau\rightarrow\frac{2\pi k}{Ne}\qquad{\rm
as~}r\rightarrow\infty\, ,
\ee
where $\beta^{-1}=hc/(4\pi r_g)$ is the Hawking temperature $(k_B=1)$;
$r_g = 2GM/c^2$ is the Schwarzschild radius. The integer $k$ is the
wrapping number of the string loop around the hole. Various physical
quantities are characterized by the ``topological'' number $k$. Therefore
the (quantum-mechanical) physical quantity at a distance from the black
hole can seemingly be affected by the ``topological'' gauge configuration
induced by the quantum fluctuation of cosmic strings near the black hole.

We consider a conformally coupled scalar field with unit $U(1)$ charge
$e$. Note that although the charged particle feels no force field around a
cosmic string classically, the scalar with charge $e$ undergoes influence
of the AB effect. The scalar field with charge $Ne$ still feels no force
in any respect in the present case.

The free energy density of the Hawking radiation of the scalar particles
in the Hartle-Hawking vacuum at spatial infinity is written as follows in
the presence of the extemal field (1)~\cite{KS4,KS3}:
\be
f_k =\frac{\hbar
c\pi^2}{45(\beta\hbar c)^4}\left\{1-30\left(\frac{k}{N}\right)^2
\left(\frac{k}{N}-1\right)^2\right\} \, .
\ee
The partition function labelled by the wrapping number
$k$ is given by
\be
Z_k=Z_{rad}\times Z_{string}\, ,
\ee
where $Z_{string}$ is the partition function of the cosmic string and is
approximately given by~\cite{PC}
\be
Z_{string} \approx \exp \{- \kappa|k| \times (4\pi r_g^2)\}\equiv
e^{- \kappa|k|A}, ,
\ee
in the thin string limit, or the string tension $\kappa$
is very large. $Z_{rad}$ is given by 
\be
Z_{rad} = \exp\{- \beta V f_k\}\, ,
\ee
where $V=L^3$ is the volume of the system. In the present case, since we
must treat the quantity at a far distance from the black hole, we must
take $L\gg r_g$. If $L\approx r_g$, we must seriously consider the effect
of
string                                                                                                                                                                                
dynamics as well as the curved space effect. We should also note that the
inclusion of the partition function of the black hole is unnecessary in
(3), because that is independent of $k$ in the Schwarzschild case and can
be treated as a direct product on the final result.

The partition function of the system with charge $Q$ is obtained by the
projection
\be
Z_Q=\sum_{k=-\infty}^{\infty}\exp\left(-i\frac{2\pi kQ}{N\hbar
e}\right)Z_k\, .
\ee

Using (2), (4), and (5), we can get $Z_Q$. If $V/(\beta\hbar c)^3\gg 1$,
we can consistently approximate $Z_Q$ as
\be
Z_Q\approx\exp\left(\frac{\pi^2 V}{45(\beta\hbar c)^3}\right)\left\{
1+2\cos\left(\frac{2\pi Q}{Ne\hbar}\right) e^{-\kappa A} \exp\left(-
\frac{4VX}{\pi^2(\beta\hbar c)^3}\right)\right\}\, , 
\ee
where $X$ is $(\pi^2/6)\{(N-1)^2/N^4\}$ $(\ge 0)$. Note also
that $X=0$ if $N=1$.

For the massless particles, we get the expression for the energy density:
\bea
\epsilon&=&3\beta^{-1}\frac{\partial\ln Z_Q}{\partial V}\nn
&\approx&\frac{\pi^2 \hbar c}{15(\beta\hbar c)^4}\left\{
1+\frac{720 X}{\pi^4}\cos\left(\frac{2\pi Q}{Ne\hbar}\right) e^{-\kappa A}
\exp\left(-
\frac{4VX}{\pi^2(\beta\hbar c)^3}\right)\right\}\, .
\eea

The amount of the energy density is modified by the virtual string
effect. The deviation is, however, very small. (In ref.~cite{KS3}, we have
unfortunately missed the last exponential factor in (8).) 

Since the temperature of the black hole (at the zero-loop level)
$\beta^{-1}$ is of the order of $r_g$, $V/(\beta\hbar c)^3\gg 1$ must
hold. Therefore the last factor in (8) gives rise to the very severe
suppression.

Similarly, we can estimate the vacuum value
$\langle\phi^\dagger\phi\rangle$ at a far distance from the hole. To this
end, we introduce the mass of the scalar for a device. Then the partition
function is given by
\bea
Z_Q&\approx&\exp\left(\frac{\pi^2 V}{45(\beta\hbar c)^3}-
\frac{m^2c^2V}{12h^2(\beta\hbar c)}\right)\nn
&\times&\left\{
1+2\cos\left(\frac{2\pi Q}{Ne\hbar}\right) e^{-\kappa A}
\exp\left(-
\frac{4VX}{\pi^2(\beta\hbar c)^3}+
\frac{m^2c^2V(N-1)/N^2}{2\hbar^2(\beta\hbar c)}\right)\right\}\, .
\eea

Using the result above, $\langle\phi^\dagger\phi\rangle$ is expressed as
\bea
\langle\phi^\dagger\phi\rangle&=&(-\beta V)^{-1}c^{-2}\left[
\frac{\partial\ln Z_Q}{\partial(m^2c^2/\hbar^2)}\right]_{m=0}\nn
&\approx&\frac{1}{12\beta^2\hbar c^3}\left\{
1-\frac{12(N-1)}{N^2}\cos\left(\frac{2\pi Q}{Ne\hbar}\right) e^{-\kappa A}
\exp\left(-
\frac{4VX}{\pi^2(\beta\hbar c)^3}\right)\right\}\, .
\eea

The modification is again suppressed by the exponential factor.

To summarise, we have found that the Hawking radiation of charged
particles suffers from very tiny correction from the cloud of virtual
cosmic strings and thus we can hardly ``detect'' the modification in the
distance far from the black hole. A naive expectation of the
effect of the ``topological'' configuration (1) failed us. Since the
problem of ``detection'' in the curved space is subtle, we could not give
a definitive statement actually. We can safely conclude, at least, that
the black hole temperature is not shifted by the virtual string effect,
according to the usual interpretation of the Hartle-Hawking vacuum
((unstable) equilibrium between black holes and radiation).

Of course we may observe the quantum process in the vicinity of the black
hole horizon. The evaporation of the black hole may be affected by the
cloud of the virtual string loops. To study the effect, we must
investigate the string dynamics in the curved space-time more closely. It
is a fascinating problem in the future.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{KS4} K. Shiraishi, preprint AJCHEP-4 (Feb. 1992).

\bibitem{PC} J. Preskill, Physica Scripta T36 (1991) 258.

S. Coleman, J. Preskill and F. Wilczek, Mod. Phys. Lett. A6 (1991) 1631;
Phys. Rev. Lett. 67 (1991) 1975; preprint IASSNS-HEP-91/64.

\bibitem{HH} J. B. Hartle and S. W. Hawking, Phys. Rev. D13 (1976) 2188.

\bibitem{KS3} K. Shiraishi, preprint AJCHEP-3 (Nov. 1991).

\end{thebibliography}

\end{document}
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