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\begin{document}


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\begin{flushright}                              FIAN/TD/96-27\\
                                                hep-th/9612225\\
                                                December 1996


\vspace{0.5cm}
\end{flushright}
\bc
\normalsize
{\large\bf Comment on

"Dynamical Chern-Simons term generation at finite
density"

and "Chern-Simons term at finite density"}

\vspace{2ex}

{\large Vadim Zeitlin\footnote{E-mail address: zeitlin@lpi.ac.ru}}

Department of Theoretical Physics, P.~N.~Lebedev Physical
Institute,

Leninsky prospect 53, 117924 Moscow, Russia

\vspace{4ex}

\ec

\centerline{{\large\bf Abstract}}

\normalsize
\begin{quote}
We comment on the calculation of the Chern-Simons coefficient in
(2+1)-dimensional gauge theories at finite chemical potential made by
A.N.Sissakian, O.Yu.Shevchenko and S.B.Solganik (hep-th/9608159   and
hep-th/9612140).
\end{quote}


\bigskip
In the recent papers A.N.Sissakian, O.Yu.Shevchenko and
S.B. Solganik have considered induced Chern-Simons
coefficient (a coefficient in front of $\i\ve_{\mu\nu\a}p^\a$ tensor structure
in the polarization operator $\Pi_{\mu\nu}(p)$ in the limit $p\rightarrow 0$)
in the (2+1)-dimensional gauge theories at finite chemical potential $\mu$ with
and without constant magnetic field $B$ [1,2].  However, results obtained in
Refs.  [1,2] contradict previous calculations of the Chern-Simons coefficient
(CSC) [3-6] and we cannot accept the former. We shall illustrate our objections
for QED$_{2+1}$.

In Ref. [1] CSC was obtained by calculating  the induced charge at
$B,\mu \ne 0$. The Chern-Simons coefficient was found to be independent of
the magnetic field and equal to (Eq. (12)):

$$
J^{cs} = \frac{e^2}{4\pi} \sn(m) [1- \sn (\mu) \theta(\mu^2 -m^2)]\quad.
\eqno{(1)}
$$

This does not agree with results of Refs. [3-6], where CSC was calculated both
directly and via induced current. The explanation of the difference is the
following: when deriving Eq. (1)  the authors suggested that
only {\it part} of the induced charge (Ref. [1],  Eq. (10) and above)
contributes to CSC, which fact is not correct. The Chern-Simons coefficient is
equal to [3]

$$
J^{cs}(B,\mu) = \dfrac{\d j_0(B,\mu)}{\d B}\quad,
\eqno{(2)}
$$

\noi
and the part omitted in Ref. [1] is $B$-dependent. With the complete expression
for the induced charge the result for $B,\mu,m >0$ is the following [3,6]

        $$
        J^{cs}(B,\mu) = \frac{e^2}{4\pi} +
        \left\{
        \begin{array}{cl}
        {}~~\dfrac{e^2}{2\pi}
        \left[
        \dfrac{\mu^2 -m^2}{2eB} \right]
        ,\quad     & \mu>\/m;
        \\
        &\\
        0,                   \quad             & |\mu|<m;\\
        &\\
        {}-
        \dfrac{e^2}{2\pi}
        \left(
        \left[
        \dfrac{\mu^2 - m^2}{2eB}      \right]
        + 1      \right) ,\quad &\mu<- m,
        \end{array}\right.\nonumber
        \eqno{(3)}
        $$

\noi
where $[ \dots ]$ denotes the integral part.

\bigskip
In Ref. [2] CSC at $B=0, \mu \ne 0$ was obtained to be the same as in Eq. (1)
by calculating the one-loop polarization operator with the external
momentum taken to be zero from the very beginning (Eqs. (4) -- (6), [2]).
However, the momentum integration should be
carried out carefully, since the integrand has singularities, and the limits
$p_0 = 0, {\bf p}^2 \rightarrow 0$ and ${\bf p}^2=0, p_0 \rightarrow 0$ at
$|\mu| > |m|$ are not the same [7].  After making the relevant calculations in
the static limit $p_0 = 0, {\bf p}^2 \rightarrow 0$ one finally obtains the
following expression for the Chern-Simons coefficient:

$$
J^{cs}(\mu) =
\frac{e^2}{4\pi}\sn (m)
(1-\theta(\mu^2-m^2))
\eqno{(4)}
$$

\noi
i.e. the induced CSC at $B,T=0$ vanishes for $|\mu|>|m|$. This is in
agreement with Ref. [8] (but we disagree with  calculations of CSC at $B\ne0$
therein).


\bigskip
This work was supported in part by
RBRF grants $N^o$ 96-02-16210-a and 96-02-16287-a.



%\baselineskip=3.5mm

\begin{thebibliography}{9}

\bibitem{SSS1} A.~N.~Sissakian, O.~Yu.~Shevchenko and S.~B.~Solganik,
{\it Dynamical Chern-Simons term generation at finite density},
preprint hep-th/9608159.


\bibitem{SSS2} A.~N.~Sissakian, O.~Yu.~Shevchenko and S.~B.~Solganik,
{\it Chern-Simons term at finite density}, preprint  hep-th/9612140.


\bibitem{LSW91} J.~Lykken, J.~Sonnenschein, and N.~Weiss, \ijmp
{A6}{1991}{5155}.

\bibitem{H93} Y.~Hosotani, \pl {B 319} {1993} {332}.

\bibitem{Sakhi}
                S.~Sakhi, {\it Effective action of a 2+1 dimensional system of
        nonrelativistic fermions in the presence of a uniform magnetic field:
        dissipation effects}, preprint UdeM-LPN-TH-179, cond-mat/9401033.

\bibitem{Z95} Vad.~Zeitlin, \pl {B 352}{1995}{422}.

\bibitem{GPY}
                D.~Gross, R.~Pisarski, and L.~Yaffe, \rmp{53}{1981}{43}.

\bibitem{Poppitz}
                E.~Poppitz, \pl{B252}{1990}{417}.

\end{thebibliography}
\end{document}

