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\markboth{\protect{\footnotesize\it Photon Propagation in a Magnetized Medium}}
{\protect{\footnotesize\it Photon Propagation in a Magnetized Medium}}

\normalsize\textlineskip

\setcounter{page}{1}

\copyrightheading{}		%{Vol.~0, No.~0 (2000) 000--000}

\vspace*{0.88truein}

%\fpage{1}
\centerline{\bf PHOTON PROPAGATION IN A MAGNETIZED MEDIUM}
\vspace*{0.37truein}
\centerline{\footnotesize SUSHAN KONAR$^{*}$}
\baselineskip=12pt
\centerline{\footnotesize\it Inter-University Centre for Astrophysics \& Astronomy}
\baselineskip=10pt
\centerline{\footnotesize\it Pune, Maharashtra 411007, India}
\baselineskip=10pt
\centerline{\footnotesize\it $^{*}$E-mail: sushan@iucaa.ernet.in}
\vspace*{0.225truein}
\publisher{(received date)}{(revised date)}

\vspace*{0.21truein}
\abstracts{Using the real time formalism of the finite temperature field theory we calculate the 1-loop 
polarization tensor in the presence of a background magnetic field in a medium. The expression is obtained
to linear order in the background field strength. We discuss the Faraday rotation as well as the photon 
absorption probabilities in this context.} {}{}

%\textlineskip			%) USE THIS MEASUREMENT WHEN THERE IS
%\vspace*{12pt}			%) NO SECTION HEADING

\vspace*{1pt}\textlineskip	%) USE THIS MEASUREMENT WHEN THERE IS
\section{Introduction}	        %) A SECTION HEADING
\vspace*{-0.5pt}
\noindent
The propagation of electro-magnetic waves, in a magnetised plasma, is of interest in systems ranging from 
laboratory plasma to astrophysical objects~\cite{1}. Yet, the expression for Faraday Rotation, for example, 
is derived assuming the medium to consist of non-relativistic and non-degenerate particles. Since, such
assumptions may not be valid in every context we re-investigate this problem in a general framework. 

Since, almost all the physical systems have magnetic fields smaller than the QED limit ($eB < m_e^2$) a 
weak-field treatment is justified. Moreover, in compact astrophysical objects (white dwarfs/neutron stars) 
the Landau level spacings are negligible compared to the electron Fermi energy~\cite{2}. Hence, we can also 
assume that the field does not introduce any spatial anisotropy in the collective plasma behaviour. 
 
Therefore, we calculate the polarization tensor ($\Pi_{\mu \nu}$), at the 1-loop level, in the weak-field limit 
retaining terms up-to ${\cal O(B)}$. As expected, we recover Faraday rotation from the dispersive part of the 
polarization tensor and the absorptive part provides the damping/instability of the photons propagating in a 
plasma. These calculations have already been reported in detail in two recent articles by us (1999, 2001)~\cite{3,4}. 

\section{The Formalism}
\noindent The presence of an external field or a medium introduces quantum corrections to the Lagrangian of an 
electro-magnetic field. In the momentum space the quadratic part of the Lagrangian, inclusive of such corrections, 
is given by,
%
\begin{equation}
{\cal L} = \frac{1}{2} [-k^2 (g_{\mu \nu} - \frac{k_\mu k_\nu}{k^2}) + \Pi_{\mu \nu}(k)] A^\mu(k) A^\nu(k) \,,
\end{equation}
%
where $\Pi_{\mu \nu}(k)$ is the polarization tensor. Assuming the direction of photon propagation to be along 
the direction of the magnetic field, the dispersion relation for the two transverse components of the photon field 
$A^{\mu}$, is:
%
\begin{equation}
k^2 = \omega_0^2 \pm \left( a_{\rm disp} + a_{\rm abs} \right) \,,
\label{disp1}
\end{equation}
%
where $a_{\rm disp}$ and $a_{\rm abs}$ are the dispersive and the absorptive parts of $\Pi_{\mu \nu}$.
Here we assume the Lorenz gauge condition ($\partial_\mu A^{\mu} = 0$) and $\omega_0$ is the plasma frequency.
For a plane polarized electro-magnetic wave propagating with a frequency $\omega$ the rate of rotation of the 
polarization angle, per unit length $l$, (i.e, the Faraday Rotation) is then given by:
%
\begin{equation}
\frac{d\Phi}{dl} = \frac{a_{\rm disp}}{\sqrt{\omega^2 - \omega_0^2}} \,.
\end{equation}
%


\section{1-Loop Vacuum Polarization}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
%
\begin{picture}(150,50)(0,-25)
\Photon(0,0)(40,0){2}{4}
\Text(20,5)[b]{$k\rightarrow$}
\Photon(110,0)(150,0){2}{4}
\Text(130,5)[b]{$k\rightarrow$}
\Text(75,30)[b]{$p+k\equiv p'$}
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%
\caption[]{One-loop diagram for the vacuum polarization.}\label{f:1loop}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%   
\noindent At the 1-loop level, the polarization tensor arises from the diagram in fig.~\ref{f:1loop},
with the dominant contribution coming from the electron line in the loop. To evaluate this diagram we use the 
electron propagator within a thermal medium in presence of a background electro-magnetic field. Since, we
specialize to the case of a purely magnetic field, the field can be taken to be in the $z$-direction 
without any loss of generality. The magnitude of this field is denoted by $\cal B$. In presence of such a
background field, the vacuum electron propagator is (following Schwinger~\cite{5}):
%
\begin{equation}
i S_B^V(p) = \int_0^\infty ds\; e^{\Phi(p,s)} C(p,s) \,,
\label{SV2}
\end{equation}
%
using the shorthands,
%
\begin{eqnarray}
\Phi(p,s) &\equiv& is \left( p_\parallel^2 - {\tan (e{\cal B}s) \over e{\cal B}s}
\, p_\perp^2 - m^2 \right) - \epsilon |s| \,,
\label{Phi} \\
C(p,s) &\equiv& \Big[ ( 1 + i\sigma_z \tan  e{\cal B}s ) (\rlap/p_\parallel + m )
- (\sec^2 e{\cal B}s) \rlap/ p_\perp \Big] \,.
\label{C}
\end{eqnarray}
%
where, $\sigma_z = i\gamma_1 \gamma_2$. To write $\Phi(p,s)$ and $C(p,s)$ we have used the following decomposition 
of the metric tensor : $g_{\mu \nu} = g^{\parallel}_{\mu \nu} - g^{\perp}_{\mu \nu}$, where, 
$g^{\parallel}_{\mu \nu}$~= diag(1,0,0,-1) and $g^{\perp}_{\mu \nu}$~= diag(0,1,1,0). In the presence of a background 
medium, the above propagator is modified to~\cite{6}:
%
\begin{equation}
iS(p) = iS_{\rm B}^{\rm V}(p) + S^{\eta}_{\rm B}(p) \,,
\label{fullprop}
\end{equation}
%
where $S^{\eta}_{\rm B}(p) =  - \eta_F(p) \left[ iS_{\rm B}^{\rm V}(p) - i\overline S_{\rm B}^{\rm V}(p) \right]$ and 
$\overline S_{\rm B}^{\rm V}(p) \equiv \gamma_0 S^{{\rm V} \dagger}_{\rm B}(p) \gamma_0$ for a fermion propagator. 
And $\eta_F(p)$ contains the distribution function for the fermions and the anti-fermions, given by:
%
\begin{equation}
\eta_F(p) = \Theta(p\cdot u) f_F(p,\mu,\beta) + \Theta(-p\cdot u) f_F(-p,-\mu,\beta) \,,
\label{eta}
\end{equation}
%
where $u$ and $\beta$ are the 4-velocity and the effective temperature of the background thermal medium. Here, 
$f_F(p,\mu,\beta) = (e^{\beta(p\cdot u - \mu)} + 1)^{-1}$ is the Fermi-Dirac distribution function and $\Theta$ is 
the step function. Rewriting eq.(\ref{fullprop}) in the following form:
%
\begin{equation}
iS(p) = \frac{i}{2} \left[ S_B^V(p) + \overline S_B^V(p) \right] + i (1/2 - \eta_F(p)) \left[ S_B^V(p) - \overline S_B^V(p) \right] \,
\label{S_reim}
\end{equation}
%
we recognise:
%
\begin{equation}
S_{\rm re} =  \frac{1}{2} \left[ S_B^V(p) + \overline S_B^V(p) \right] \; , \;\; 
S_{\rm im} = (1/2 - \eta_F(p)) \left[ S_B^V(p) - \overline S_B^V(p) \right] \,.
\end{equation}
%                                         
The subscripts {\em re} and {\em im} refer to the real and imaginary parts of the propagator. Now, the amplitude of the 
1-loop diagram of fig.~\ref{f:1loop} can be written as:
%
\begin{eqnarray}
i \Pi_{\mu\nu}(k) = - \int \frac{d^4p}{(2\pi)^4} (ie)^2 \; \mbox{tr}\,
\left[\gamma_\mu \, iS(p) \gamma_\nu \, iS(p')\right] \,,
\end{eqnarray}
%
where, for the sake of notational simplicity, we have used $p' = p+k$. Then the dispersive part of the polarisation 
tensor is given by:
%
\begin{equation}
\Pi^{\rm D}_{\mu\nu}(k) = -ie^2 \int \frac{d^4p}{(2\pi)^4} \; \mbox{tr}\,
\left[\gamma_\mu \, S^{\eta}_{\rm B}(p) \gamma_\nu \, iS^{\rm V}_{\rm B}(p^{\prime}) + 
\gamma_\mu \, iS^{\rm V}_{\rm B}(p) \gamma_\nu \, iS^{\eta}_{\rm B}(p^{\prime}) \right] \,,       
\end{equation}
%
and the (11)-component of the absorptive part, by:
%
\begin{equation}
\Pi_{\mu\nu}^{A}(k) = -ie^2 \int \frac{d^4p}{(2\pi)^4} \; \mbox{tr}\,
\left[\gamma_\mu \, iS_{\rm im}(p) \gamma_\nu \, iS_{\rm im}(p^{\prime}) \right]\,.        
\end{equation}
%
Now, for the terms odd in powers of ${\cal B}$, the explicitly gauge invariant expressions are (see~\cite{3,4} for details):
%
\begin{eqnarray}
\Pi^{\rm D}_{\mu\nu}(k) 
&=& 4ie^2 \varepsilon_{\mu \nu \alpha_{\parallel} \beta} k^\beta 
    \int \frac{d^4p}{(2\pi)^4} \; \eta_-(p)
    \int_{-\infty}^\infty ds e^{\Phi(p,s)} \int_0^\infty ds' e^{\Phi(p',s')} \nonumber \\
&\times&  \left[ p^{\widetilde{\alpha_\parallel}} \tan e{\cal B}s + p^{\prime \widetilde{\alpha_\parallel}} \tan e{\cal B}s' 
    - \frac{\tan e{\cal B}s \tan e{\cal B}s'}{\tan e{\cal B}(s+s')} (p + p')^{\widetilde{\alpha_\parallel}} \right] \,,       
\end{eqnarray}
%
and,
%
\begin{eqnarray}
\Pi^{\rm A}_{\mu\nu}(k) 
&=& - \, ie^2 \varepsilon_{\mu\nu\alpha_\parallel\beta} k^\beta \, \int \frac{d^4p}{(2\pi)^4} X(\beta, k, p) 
      \int_{-\infty}^\infty ds \; e^{\Phi(p,s)} \int_{-\infty}^\infty ds' \; e^{\Phi(p',s')} \nonumber\\* 
&\times& \Bigg[ p^{\widetilde\alpha_\parallel} \tan e{\cal B}s + p'^{\widetilde\alpha_\parallel} \tan e{\cal B}s' 
       - {\tan e{\cal B}s \; \tan e{\cal B}s' \over \tan e{\cal B}(s+s')} \; (p+p')^{\widetilde\alpha_\parallel} \Bigg] \,. 
\end{eqnarray}
%
In writing the above expressions we have used the notation of 
$p^{\widetilde{\alpha_\parallel}}$, for example. This signifies a component of $p$ which can take only the "parallel" 
indices, i.e., 0 or 3 and is moreover different from the index $\alpha$ appearing elsewhere in the expression. We have 
also used the shorthands, $\eta_-(p) = \eta_F(p) - \eta_F(-p)$ and 
$X(\beta, k, p) = (1 - 2 \eta_F(p)) \, (1 - 2 \eta_F(p^{\prime}))$.

\section{Results}

\noindent Our results are obtained in the rest frame of the background medium. We also take the long wavelength limit,
i.e, $K \ll \omega$, where ($k_0 = \omega$). Finally, we assume the magnetic field to be small such that we can retain 
terms only linear in ${\cal B}$. In this limit, the dispersive part of the polarization tensor is given by:
%
\begin{eqnarray}
\Pi^{\rm D}_{\mu\nu}(k) 
&=& 8ie^2 \varepsilon_{\mu \nu \alpha_{\parallel} \beta} {\cal B} \omega 
    \int \frac{d^4p}{(2\pi)^4} \; \eta_-(p) p_0
    \int_{-\infty}^\infty ds e^{is(p^2 - m^2) - \epsilon|s|} \nonumber \\
&\times& \int_0^\infty ds' e^{is(p'^2 - m'^2) - \epsilon|s'|} \left\{s+s' - \frac{ss'}{s+s'}\right\},
\end{eqnarray}
%
where we have made a further assumption that $\omega \ll m_e$ (see~\cite{3} for details). Surprisingly, in the above 
mentioned limit, the absorptive part of the polarization tensor has two terms with different signs which opens up the 
possibility that for a given magnetic field, depending on the chemical potential and the external photon momentum 
$\Pi^{\rm A}_{\mu \nu}$ can be either positive or negative giving rise to damping or instability of the propagating 
photon (see~\cite{4} for details). Also, the absorption of the photons happen between two limiting values of $p$ given by,
%
\begin{eqnarray}
P_{\rm min} =  - \frac{K}{2} + \frac{k_0}{2} \left(1- \frac{4 m^2}{k^2}\right)^{1/2}, \; \;
P_{\rm max} =  \frac{K}{2} + \frac{k_0}{2} \left(1- \frac{4 m^2}{k^2}\right)^{1/2}.
\end{eqnarray}
%
Since, $P$ is real, the condition $k^2 \ge 4 m^2$ must be satisfied. This is an important kinematic constraint which ensures
the conservation of energy-momentum in the weak-field limit.


\nonumsection{Acknowledgments}
\noindent
Participation to this conference was made possible through a travel grant (No. TG/390/01-HRD) 
from the Council of Scientific and Industrial Research of India.

\nonumsection{References}

\begin{thebibliography}{000}
\bibitem{1}
E. Parker, {\bibit Cosmic Magnetic Fields}, (Oxford University Press, Oxford, 1979).

\bibitem{2}
G. Chanmugam, {\bibit Ann. Rev. Astron. Astrophys.} {\bf 30}, 143 (1992).

\bibitem{3}
A.~K. Ganguly, S. Konar and P.~B. Pal, {\bibit Phys. Rev.} {\bf D60}, 105014 (1999).

\bibitem{4}
A.~K. Ganguly and S. Konar, {\bibit Phys. Rev. D} {\bf D63}, 65001 (2001).

\bibitem{5}
J. Schwinger, {\bibit Phys. Rev.} {\bf 82}, 664 (1951).

\bibitem{6}
P. Elmfors, D. Grasso and G. Raffelt, {\bibit Nucl. Phys.} {\bf B479}, 3 (1996).

\end{thebibliography}

\end{document}

hep-ph0111104.extracted_bib
\begin{thebibliography}{000}
\bibitem{1}
E. Parker, {\bibit Cosmic Magnetic Fields}, (Oxford University Press, Oxford, 1979).

\bibitem{2}
G. Chanmugam, {\bibit Ann. Rev. Astron. Astrophys.} {\bf 30}, 143 (1992).

\bibitem{3}
A.~K. Ganguly, S. Konar and P.~B. Pal, {\bibit Phys. Rev.} {\bf D60}, 105014 (1999).

\bibitem{4}
A.~K. Ganguly and S. Konar, {\bibit Phys. Rev. D} {\bf D63}, 65001 (2001).

\bibitem{5}
J. Schwinger, {\bibit Phys. Rev.} {\bf 82}, 664 (1951).

\bibitem{6}
P. Elmfors, D. Grasso and G. Raffelt, {\bibit Nucl. Phys.} {\bf B479}, 3 (1996).

\end{thebibliography}