\documentclass[preprint,aps]{revtex4}
\usepackage{graphicx,amsfonts} 
\begin{document}
\title{Radiative corrections to the Chern-Simons term at finite
temperature in the noncommutative Chern-Simons-Higgs model}
\author{L. C. T. de Brito}
\author{M. Gomes}
\author{Silvana Perez}
 \altaffiliation[Also at]{ Departamento de F\'{\i}sica, Universidade Federal do
Par\'a, Caixa Postal 479, 66075-110, Bel\'em, PA, Brazil; e-mail:silperez@ufpa.br}
\author{A. J. da Silva}
\affiliation{Instituto de F\'{\i}sica, Universidade de S\~{a}o Paulo,
 Caixa Postal 66318, 05315-970, S\~{a}o Paulo - SP, Brazil}
\email{lcbrito,mgomes,silperez,
  ajsilva@fma.if.usp.br}

\begin{abstract}
The one-loop radiative correction to the Chern-Simons coefficient is
computed in noncommutative Chern-Simons-Higgs model at zero and at
finite temperature. At high temperature, we show that the static limit
of this correction is proportional to $TlogT$, in contrast with
the commutative case, where it goes like $T$. Our results are analytic
functions of the noncommutative parameter.

\end{abstract}
\maketitle

\section{Introduction}
    
The analysis of the radiative corrections to the Chern-Simons coefficient
has stimulated considerable interest in the recent years \cite{dunne_1}.  These studies
pointed out that the well known nonrenormalization theorem
\cite{coleman} may become invalid outside perturbation or even
perturbatively whenever infrared singularities occur.  Typical of the
last possibility are situations which potentially modify the long
distance behavior of the relevant models. Examples in this direction
are the breakdown of some continuous symmetry, thermal effects and
the possible noncommutativity of the underlying space. 
In this work
we are going to focus on the changes in the Chern-Simons coefficient
due to thermal effects and the noncommutativity of the space.
 
The appearance of noncommutative coordinates has an old history
\cite{jackiw} but gained impetus more recently, mainly due to its
connection with string theory \cite{witten}. One peculiar aspect of
these theories is the infrared/ultraviolet (IR/UV) mixing
\cite{seiberg}, i. e., the presence of infrared singularities even in
theories without massless fields. Such behavior may be ameliorated in
some supersymmetric models \cite{gomes}. The absence of IR/UV mixing
has also been verified for the pure $U(n)$ Chern-Simons model which
actually seems to be a free theory \cite{sheikh}.


At finite temperature, many features of noncommutativity have been
examined and applied to different field theoretical models
\cite{fischler}. One of the motivations for the studies in
noncommutative Quantum Field Theory (QFT) is to understand how the
thermal properties of physical systems are affected by the
noncommutativity.  At finite temperature, it is known that for small
momenta the amplitudes are, in general, not well behaved and this
feature is understood in terms of the new structure
introduced by the temperature, the velocity of the heat bath
\cite{bedaque}. On the other hand, in a noncommutative manifold, the
new structure is the $\theta^{\mu \nu}$ tensor, which measures the
noncommutativity of the space, and it also leads to an infrared
nonanalytic behavior, at least at zero temperature. As argued in
\cite{frenkel}, unless for some special limits, in noncommutative
finite temperature QFT there are many difficulties to evaluate
amplitudes in a closed form. Thus, to simplify matters, those authors
in particular have considered a generalization of the hard thermal
loop limit, which involves the noncommutative parameter.

    In a noncommutative manifold, it will appear trigonometric factors
  and, because of them, the amplitudes are separated in two parts, the
  planar and nonplanar ones. The first of them is proportional to the
  corresponding amplitude in the commutative case. The effects coming
  from the noncommutativity of the space can be extracted from the
  nonplanar contribution. In this paper we will study the corrections
  to the Chern Simons coefficient at finite temperature arising from
   the one-loop contributions to the gauge field two point
  function. To fix the notation, we first consider a commutative
  manifold. In \cite{parityvio}, it was shown that, in the
  Chern-Simons-Higgs model in the spontaneously broken phase,  the
  coefficient increases like $T$ in the static limit (in that work
  this result was extracted as a particular limit of the
  Maxwell-Chern-Simons-Higgs model). Here, we are going to extend this
  study to the noncommutative case and prove that it increases
  like $TlogT$ in this limit. As expected, although we are
  considering the Abelian model, because of the noncommutativity,
  there will be strong resemblances with the nonabelian theory.

The paper is organized as follows: In Section \ref{sec2} the
noncommutative version of the Chern-Simons-Higgs model in the broken
phase is presented. Section \ref{sec3} contains the one loop
corrections to the Chern-Simons coefficient in both commutative as
well as noncommutative case and at finite temperature in the imaginary time
formalism. The zero temperature results are obtained as consequence
of this evaluation. Finally, in Section \ref{sec4} the conclusions are
presented.


\section{Noncommutative Chern-Simons-Higgs model}
\label{sec2}

Noncommutative quantum field theories are defined in a manifold where
the coordinates do not commute among themselves. Rather, the
commutator between two position operators is postulated to be

\begin{equation}
[\hat x^{\mu},\, \hat x^{\nu}] = i \theta^{\mu \nu},
\end{equation}

\noindent where $\theta^{\mu \nu}$ is an antisymmetric matrix, which
commutes with the $\hat x$'s. The algebra of operators in such a kind
of manifold has extensively been studied \cite{gronewold}, and many
properties are known (see \cite{douglas} for a review). For example,
due to the Wigner-Moyal correspondence, instead of working with
functions of the above noncommutative coordinates, one may use
ordinary functions of commutative variables embodied with the so
called Moyal product, defined as

\begin{equation}
f(x) \ast g(x) = \left[e^{(i/2) \theta^{\mu \nu}
\partial^{(\zeta)}_{\mu} \partial^{(\eta)}_{\nu}} f(x + \zeta) g(x +
\eta) \right]_{\zeta=0=\eta}.
\end{equation}


Using this definition, one can   study quantum field theories in
 a noncommutative space, by replacing the standard pointwise product 
of fields by the Moyal one. For simplicity, in this
paper we shall keep $\theta^{0i}=0$.

 Let us consider the Chern-Simons-Higgs model in a noncommutative
space defined by the Lagrangian density

\begin{eqnarray}\label{lagrangian}
{\cal{L}} &=& \frac{\kappa}{2} \epsilon^{\mu \nu \lambda} \left[A_{\mu}
\ast \partial_{\nu} A_{\lambda} + \frac{2 i}{3} A_{\mu}\ast
A_{\nu} \ast A_{\lambda} \right] \nonumber \\
&+& (D_{\mu} \Phi) \ast (D^{\mu} \Phi)^{\dag} -
\frac{\lambda}{4}\left[\Phi \ast \Phi^{\dag} - v^2\right]_{\ast}^2,
\end{eqnarray}

\noindent with $D_{\mu} \Phi = \partial_{\mu} \Phi - i \Phi \ast
A_{\mu}$. The above Lagrangian is invariant under the infinitesimal
 gauge transformation


\[
\Phi \rightarrow \Phi  - i \Phi \ast \alpha,
\]

\begin{equation}
A_{\mu} \rightarrow A_{\mu} - \partial_{\mu} \alpha + i \alpha \ast
A_{\mu} - i A_{\mu} \ast \alpha.
\end{equation}

In the spontaneously broken phase, $\langle \Phi \rangle \equiv
v\ne 0$, one can choose the decomposition $\Phi = v +
\frac{1}{\sqrt{2}}(\sigma + i \chi)$ and rewrite
Eq. (\ref{lagrangian}) as

\begin{eqnarray}\label{lagrang2}
{\cal{L}} &=& \frac{\kappa}{2} \epsilon^{\mu \nu \lambda} \left[A_{\mu}
\ast \partial_{\nu} A_{\lambda} + \frac{2 i}{3} A_{\mu}\ast
A_{\nu} \ast A_{\lambda} \right] + \frac{m^2}{2} A_{\mu} \ast
A^{\mu} - \frac{1}{2 \xi} (\partial_{\mu}
A^{\mu}){\ast}(\partial_{\nu} A^{\nu})  \nonumber \\ 
&+& \frac{1}{2}(\partial_{\mu}{\sigma})\ast(\partial^{\mu}{\sigma}) -
\frac{m_{\sigma}^2}{2} \sigma \ast \sigma +
\frac{1}{2}(\partial_{\mu}{\chi})\ast(\partial^{\mu}{\chi}) -
\frac{m_{\chi}^2}{2} \chi\ast \chi \nonumber \\ 
&-&  \frac{1}{2} A_{\mu} \ast \left(\sigma \ast  
{\buildrel\,\leftrightarrow\over{\partial^{\mu}}\!\!}
\, \, \chi  - \chi \ast  
{\buildrel\,\leftrightarrow\over{\partial^{\mu}}\!\!} \,\,
\sigma  - i [\sigma ,\partial^{\mu}\sigma]_{\ast} - i [\chi
,\partial^{\mu}\,\chi]_{\ast} \right) \nonumber \\ 
&+& \frac{1}{2}  A_{\mu} \ast A^{\mu} \ast\left(\sigma \ast \sigma
+ \chi \ast \chi + 2 \sqrt{2} v
\sigma + i [\sigma,\chi]_{\ast} \right) \nonumber \\ 
&-& \frac{\lambda}{2 \sqrt{2}} v \left\{ \sigma,
\frac{(\sigma \ast \sigma + \chi \ast \chi)}{2} +
\frac{i}{2}[\chi,\sigma]_{\ast}\right\}_{\ast} -
\frac{\lambda}{16}\left(\sigma \ast \sigma + \chi \ast \sigma + i
[\chi,\sigma]_{\ast}\right)_{\ast}^2,
\end{eqnarray}



\noindent where we have chosen the $R_{\xi}$ gauge, specified by the gauge
fixing Lagrangian

\begin{equation}
{\cal{L}}_{\mbox{GF}} = - \frac{1}{2\xi} \left(\partial_{\mu} A^{\mu}
+ \xi \sqrt{2}v \chi\right)_{\ast}^2,
\end{equation}

\noindent which has the merit of canceling the nondiagonal terms in
the quadratic part of the model. We have also defined


\begin{equation}
m^2= 2 v^2, \qquad m_{\sigma}^2= \lambda v^2, \qquad m_{\chi}^2=
\xi m^2.
\end{equation}

\noindent Furthermore, the notations $[ \, , \, ]_{\ast}$ and $\{ \, ,
\, \}_{\ast}$
stand, respectively, for the commutator and anticommutator using the
Moyal product.

 To complete the Lagrangian of the model, one has to add to
  Eq. (\ref{lagrang2}) the contribution of the Faddeev-Popov  ghost
  field  given by

\begin{equation}
{\cal L}_{\mbox{FP}} = \frac{1}{2 \xi} \bar{c} \,\partial^2 \, c +
\frac{i}{4 \xi} \left(\bar{c} \ast A_{\mu} \ast \partial^{\mu} c -
A_{\mu} \ast \partial ^{\mu} c \ast \bar{c} \right) +i\xi v \bar{c}*\chi*c,
\end{equation}

\noindent where $\bar{c}$ and $c$ are the ghost fields.


 
The {\bf propagator} for the gauge field is 

\begin{equation}
D_{\mu \nu} (p) = \frac{i}{p^2-M^2} \left[ -m^2 g_{\mu \nu}
+ p_{\mu} p_{\nu} \frac{m^2 - \xi \kappa^2}{p^2 + \xi m^2} + i
\kappa \epsilon_{\mu \nu \lambda} p^{\lambda} \right]\frac{1}{\kappa^2} ,
\end{equation}

\noindent 
 where $M=\frac{m^2}{\kappa}$.  The propagators for the other  fields
 are the standard ones ($D_{\sigma}(p)=
i/(p^2 - m_{\sigma}^2)$, $D_{\chi} = i/(p^2- m_{\chi}^2)$  and $D_{c}=\frac{i}{p^2}$). These
propagators are not affected by the noncommutativity.

At finite temperature and using the imaginary time formalism, the
gauge propagator is ($\kappa \rightarrow i \kappa$ in the Euclidean
space)

\begin{equation}
D_{\mu \nu} (p) =  \frac{1}{p^2+M^2} \left[ m^2
\delta_{\mu \nu} - p_{\mu} p_{\nu} \frac{m^2 - \xi \kappa^2}{p^2 + \xi
m^2} -  \kappa \epsilon_{\mu \nu \lambda} p^{\lambda} \right]\frac{1}{\kappa^2}.\end{equation}

\noindent with $p^{\mu}\equiv (p^0,\vec{p})=(2 \pi n T, \vec{p})$. 

\section{One-loop results}
\label{sec3}

In this section we will compute the one-loop radiative correction to
the Chern-Simons coefficient of the above model in both commutative
and noncommutative cases  at finite temperature in the imaginary
time formalism.


\subsection{Commutative case}

 Let us start by evaluating the parity violating part of the
polarization tensor $\Pi_{\mu \nu}$ in the commutative case. There is
only one diagram,  Fig. 1$a$, to evaluate.  At one loop, the ghost
field does not contribute to the parity violating part,
 as can be seen from ${\cal L}_{\mbox{FP}}$. Thus, we have



\begin{equation}\label{eq.01}
\Pi_{\mu \nu}^{odd} (p)\equiv {\cal{\pi}}_{\mu \nu} (p) =\frac{8 v^2}{\kappa} T \sum_{n} \int \frac{d^2
k}{(2 \pi)^2} \frac{\epsilon_{\mu \nu\lambda} k^{\lambda}}{(k^2 +
M^2)[(p-k)^2 + m^2_{\sigma}]}. 
\end{equation}

\noindent  The consideration of the static
limit, where $p_0=0$ and $\mid \vec{p} \mid$ is small, yields

\begin{equation}\label{tensorI}
{\cal{\pi}}_{0i}(p_0=0)= \frac{8 v^2}{\kappa}\int
\frac{d^2k}{(2 \pi)^2} \epsilon_{0ij} k^j T \sum_{n}\frac{1}{k_0^2 +
w_M^2} \frac{1}{k_0^2 + w_{\sigma}^2(p)}.
\end{equation}

\noindent Note that, in the static limit, ${\cal{\pi}}_{ij}$ vanishes
due to the antisymmetry of the integrand in
Eq.~(\ref{eq.01}). We have also introduced the notation
$w_M^2=\vec{k}^2 + M^2$ and $w_{\sigma}^2(p)=(\vec{p}-\vec{k})^2 +
m_{\sigma}^2$. Using that

\begin{eqnarray}\label{expansion}
\frac{1}{k_0^2 + (\vec{p}-\vec{k})^2 + m_{\sigma}^2} &=& \frac{1}{k_0^2 +
w_{\sigma}^2} + \frac{2 \vec{k} \cdot \vec{p} }{(k_0^2 +
w_{\sigma}^2)^2} + {\cal{O}}(\vec{p}^{\phantom a 2})\nonumber \\
&=& \left(1 - 2 \vec{k} \cdot \vec{p} \frac{\partial}{\partial
m_{\sigma}^2} \right) \frac{1}{k_0^2 +
w_{\sigma}^2} + {\cal{O}}(\vec{p}^{\phantom a 2}),
\end{eqnarray}
 
\noindent 
where $w_\sigma\equiv w_\sigma(0)$, we write Eq.  (\ref{tensorI})
for small external momentum as

\begin{equation}
{\cal{\pi}}_{0i}(p_0=0) = \frac{8 v^2}{\kappa}\int \frac{d^2k}{(2
\pi)^2} \left(1 - 2 \vec{k} \cdot \vec{p} \frac{\partial}{\partial
m_{\sigma}^2} \right) \epsilon_{0ij} k^j T \sum_{n}\frac{1}{k_0^2 +
w_M^2} \frac{1}{k_0^2 + w_{\sigma}^2} + {\cal{O}} (\vec{p}^{\phantom a
  2}).
\end{equation}

After that, we evaluate the sum in $k_0$ through the use of

\begin{equation}\label{sum}
\sum_{n=-\infty}^{n=+\infty} f(n) = - \pi \sum'
[f(z)\cot(\pi z)],
\end{equation}

\noindent where the prime in the sum indicates that it runs only over the residues of the
poles of $f(z)$. Thus, we find

\begin{eqnarray}\label{planar}
{\cal{\pi}}_{0i}(p_0=0) &=&  \frac{4 v^2}{\kappa} \int
\frac{d^2k}{(2 \pi)^2} \epsilon_{0ij} k^j \nonumber \\
&\times& \left(1 - 2 \vec{k} \cdot \vec{p} \frac{\partial}{\partial
m_{\sigma}^2}\right) \left\{ \frac{1}{m_{\sigma}^2 - M^2} 
\left[ \frac{\coth{(\beta w_M/2)}}{w_M} -
\frac{\coth{(\beta w_{\sigma}/2)}}{w_{\sigma}}\right] \right\} . 
\end{eqnarray}

Using that $\coth{(\beta x/2)}=1 + \frac{2}{e^x-1}$, we can separate
 the zero temperature from the finite temperature part. The first
one can be computed straightforwardly, giving

\begin{eqnarray}\label{planar0}
{\cal{\pi}}_{0i}(p_0=0;T=0) &=& \frac{4 v^2}{\kappa} \int
\frac{d^2k}{(2 \pi)^2} \epsilon_{0ij} k^j  \left(1 - 2 \vec{k} \cdot \vec{p} \frac{\partial}{\partial
m_{\sigma}^2}\right) \left[ \left( \frac{1}{m_{\sigma}^2 - M^2}\right) 
\left( \frac{1}{w_M} -
\frac{1}{w_{\sigma}}\right) \right] \nonumber \\
&=& \frac{2}{3 \pi}
\frac{|\kappa|}{\kappa} \frac{(1 + \frac{1}{2}
\frac{m_{\sigma}}{M})}{(1 + \frac{m_{\sigma}}{M})^2}\epsilon_{0ij}p^j.
\end{eqnarray}

The finite temperature part of Eq. (\ref{planar}) is more complicated
but can be expressed in terms of polylogarithm functions as

\begin{eqnarray}
{\cal{\pi}}_{0i}(p_0=0;T)&=& - \frac{4 v^2}{\pi \kappa} \epsilon_{0ij}
p^j \frac{\partial}{\partial m_{\sigma}^2} \left(\frac{f(M,m_{\sigma},T)}{m_{\sigma}^2 -
M^2}\right) 
\end{eqnarray}
 
\noindent  where \cite{leon}  

\begin{equation}\label{function1}
f(M,m_{\sigma},T)=\left[ T^3 PolyLog(3,e^{-\beta M}) + T^2 M
PolyLog(2,e^{-\beta M})\right] - [ M\rightarrow m_{\sigma}]
\end{equation}

\noindent  
and

\begin{equation}\label{poly}
PolyLog(b, a) \equiv \sum_{n=1}^{\infty} \frac{a^n}{n^b}.
\end{equation}

In the high temperature limit \cite{kapusta}, the above expression
furnishes 

\begin{equation}\label{planarT}
{\cal{\pi}}_{0i} (p_0=0;T)=-\frac{4 v^2}{\pi \kappa} \epsilon_{0ij}p^j
T \frac{\partial}{\partial m_{\sigma}^2} \left[\frac{m_{\sigma}^2 \log
(M/{m_{\sigma}})}{m_{\sigma}^2 - M^2} \right].
\end{equation}

The results (\ref{planar0}) and (\ref{planarT}) agree with the
corresponding ones in the Chern-Simons-Higgs limit of the
Maxwell-Chern-Simons-Higgs model calculated in \cite{parityvio},
as commented in the introduction.

\subsection{Noncommutative case}

Next, let us determine the parity violating part of the polarization
tensor in the noncommutative Chern-Simons-Higgs model. In this case,
 both diagrams in Fig. 1 contribute. As in the commutative case,
at one-loop the ghost field does not give a parity violating part
correction to the polarization tensor. The diagram in Fig.  1$a$  gives

\begin{equation}
\Pi^{odd}_{a,{\mu\nu}} \equiv {\cal{\pi}}_{\mbox{a},\mu \nu}
(p)=\frac{8 v^2}{\kappa} \sum_{n} \int \frac{d^2 k}{(2 \pi)^2}
\frac{\epsilon_{\mu \nu\lambda} k^{\lambda} cos^2(k \wedge p)}{(k^2 +
M^2)[(p-k)^2 + m^2_{\sigma}]},
\end{equation}
 
\noindent where $k \wedge p = \frac{1}{2} \theta^{\mu \nu} k_{\mu}
p_{\nu}$.  Again, we will consider only the static limit. Therefore,
we have

\begin{eqnarray}
{\cal{\pi}}_{\mbox{a},0i}(p_0=0) &=& \frac{4 v^2}{\kappa}\int
\frac{d^2k}{(2 \pi)^2} \left[ 1 + \cos(2 \vec{k} \wedge \vec{p})
\right] \epsilon_{0ij} k^j \nonumber \\  &\times& T
\sum_{n}\frac{1}{k_0^2 + w_M^2} \frac{1}{k_0^2 + w_{\sigma}^2(p)}.
\end{eqnarray}

To compute this integral, let us note that, while the denominator has
only sums of $\vec{k}$ and $\vec{p}$ and so can be expanded for small
values of $\vec{p}$, as we have done in the commutative case, the
numerator has the product $\theta^{ij} k^i p^j$ which is not
necessarily small for small $\vec{p}$. To avoid possible problems, we
will proceed as follows. We define a new variable $\tilde{p}^i=
\theta^{ij} p^j$ such that $p \cdot\tilde{p} =0$. Afterwards, we
evaluate the sum, by expanding the denominator for small $p$ as in
Eq. (\ref{expansion}) and then  use Eq. (\ref{sum}), yielding


\begin{eqnarray}\label{noncommut}
{\cal{\pi}}_{\mbox{a},0i} (p_0=0) &=&  \frac{2 v^2}{\kappa} \int
\frac{d^2k}{(2 \pi)^2} \left[1 + \cos(2 \vec{k} \cdot \tilde{\vec{p}})
\right] \epsilon_{0ij} k^j \nonumber \\ &\times& \left(1 - 2 \vec{k}
\cdot \vec{p} \frac{\partial}{\partial m_{\sigma}^2} \right) \left\{
\frac{1}{m_{\sigma}^2 - M^2} \left[\frac{\coth{(\beta w_M/2)}}{w_M} -
\frac{\coth(\beta w_{\sigma}/2)}{w_{\sigma}} \right] \right\} \nonumber
\\ &\equiv& A_{0i} + B_{0i}
\end{eqnarray}

\noindent where $A_{0i}$ and $B_{0i}$ are respectively the planar and
non-planar contributions to ${\cal{\pi}}_{\mbox{a},0i}(p_0=0)$; 
the $\tilde{p}\equiv\mid \vec{\tilde{p}} \mid\rightarrow 0$ limit will be taken shortly.

The planar part is exactly one half of Eq. (\ref{planar}). Thus, going
to the non-planar contribution,  from Eq. (\ref{noncommut}) 
we have

\begin{eqnarray}
B_{0i} &=& \frac{2 v^2}{\kappa} \int \frac{d^2k}{(2 \pi)^2} \cos(2
 \vec{k} \cdot \tilde{\vec{p}}) \epsilon_{0ij} k^j \nonumber \\
 &\times & \left(1 - 2 \vec{k} \cdot \vec{p} \frac{\partial}{\partial
 m_{\sigma}^2} \right) \left\{ \frac{1}{m_{\sigma}^2 - M^2}
 \left[\frac{\coth{(\beta w_M/2)}}{w_M} - \frac{\coth(\beta
 w_{\sigma}/2)}{w_{\sigma}} \right] \right\} \nonumber \\  &=& -
 \frac{4 v^2}{\kappa} \int \frac{d^2k}{(2 \pi)^2} \cos(\vec{k} \cdot
 \vec{\tilde{p}}) \epsilon_{0ij} k^j \vec{k} \cdot \vec{p} \nonumber
 \\  &\times& \frac{\partial}{\partial m_{\sigma}^2} \left\{
 \frac{1}{m_{\sigma}^2 - M^2}\left[\frac{\coth{(\beta w_M/2)}}{w_M} -
 \frac{\coth(\beta w_{\sigma}/2)}{w_{\sigma}} \right] \right\}
 \nonumber \\  &=& - \frac{4 v^2}{\kappa} \frac{\epsilon_{0ij} p^j}
 {(2 \pi)^2} \int d^2 k \frac{\cos(\vec{k} \cdot \vec{\tilde{p}})
 (\vec{k} \cdot \vec{p})^2}{|\vec{p}|^2} \nonumber \\  &\times&
 \frac{\partial}{\partial m_{\sigma}^2} \left\{ \frac{1}{m_{\sigma}^2
 - M^2} \left[\frac{\coth{(\beta w_M/2)}}{w_M} - \frac{\coth(\beta
 w_{\sigma}/2)}{w_{\sigma}} \right] \right\},
\end{eqnarray}

\noindent where we have used polar coordinates such that

\begin{equation}
\vec{k}\equiv \frac{(\vec{k} \cdot \vec{p}) \vec{p}}{|\vec{p}|^2} +
\frac{(\vec{k} \cdot \vec{\tilde{p}})
\vec{\tilde{p}}}{|\vec{\tilde{p}}|^2}.
\end{equation}
 
The angular part of the above integral  can be expressed in terms of
Bessel functions, as follows

\begin{equation}\label{integral}
B_{0i} = - \frac{2 v^2}{\kappa} \frac{\epsilon_{0ij} p^j}{\pi}
\frac{\partial}{\partial m_{\sigma}^2} \left\{ \frac{1}{m_{\sigma}^2 -
M^2}\int_0^{\infty} dk \frac{k^2}{\tilde p} J_1 (k \tilde p )
\left[\frac{\coth{(\beta w_M/2)}}{w_M} - \frac{\coth(\beta
w_{\sigma}/2)}{w_{\sigma}} \right]\right\}.
\end{equation}

\noindent 
These integrals
 are evaluated in the appendix \ref{a1}. Here, we will only write the
 final results. So, we have:

\begin{equation}\label{nonplanar0}
B_{0i}(T=0)= - \frac{2 v^2}{ \pi \kappa} \epsilon_{0ij} p^j
\frac{\partial}{\partial m_{\sigma}^2} \left\{ \frac{1}{m_{\sigma}^2 -
M^2} \left[\left(M \frac{e^{- \tilde p M}}{(\tilde p )^2} + \frac{e^{-
\tilde p M}}{(\tilde p )^3}\right) - (M \rightarrow m_{\sigma}
)\right] \right\}
\end{equation}

\noindent and

\begin{eqnarray}\label{nonplanarT}
B_{0i}(T)&=&- \frac{4 v^2}{\pi \kappa} \epsilon_{0ij} p^j 
\frac{\partial}{\partial m_{\sigma}^2} \left\{  \frac{1}{m_{\sigma}^2
- M^2}\right. \nonumber \\ 
&\times&\left. \left[ \left(M T^2 \sum_{n=1}^{\infty} \frac{e^{- \beta M
\sqrt{n^2+\tau^2}}} {n^2 + \tau^2} + T^3 \sum_{n=1}^{\infty}
\frac{e^{- \beta M \sqrt{n^2+\tau^2}}} {(n^2 + \tau^2)^{3/2}}\right) -
\left( M \rightarrow m_{\sigma}\right) \right] \right\}, 
\end{eqnarray} 


\noindent where $\tau= \tilde p T$. Note the apparent singularity
in Eq. (\ref{nonplanar0}) at $\tilde{p} =0$; this kind of
structure is the well known infrared singularity, characteristic of
noncommutative field theories \cite{seiberg}.  Here, however, as
 we will shortly see, this singularity cancels in the final
result. Next, let us check if these results are consistent with the
 $\theta=0$ limit, namely, if in this limit we obtain the 
other half of Eqs. (\ref{planar0}) and (\ref{planarT}), so that the
 commutative result is recovered. For the zero temperature part,
expanding Eq. (\ref{nonplanar0}) for small values of $\tilde{p} $, we
 get

\begin{equation}\label{eq.03}
B_{0i}(T=0)= \epsilon_{0ij} p^j \left( \frac{1}{3 \pi}
\frac{|\kappa|}{\kappa} \frac{(1 + \frac{1}{2}
\frac{m_{\sigma}}{M})}{(1 + \frac{m_{\sigma}}{M})^2} - \frac{v^2}{4
\pi \kappa } \tilde p  \right) + {\cal{O}}(\tilde{p}^{\phantom a 2}).
\end{equation}

\noindent When $\theta \rightarrow 0$, we obtain one half of
Eq. (\ref{planar0}), completing the expected result for the
commutative case.

Now, looking at the temperature dependent part,
Eq. (\ref{nonplanarT}), when $\theta=0$, we found that the result is
again proportional to function defined in
Eq. (\ref{function1}):

\begin{eqnarray}\label{eq.02}
B_{0i}(T) &=&  - \frac{4 v^2}{\pi \kappa} \epsilon_{0ij} p^j
 \frac{\partial}{\partial m_{\sigma}^2} \left(\frac{f(M,m_{\sigma},T)}{m_{\sigma}^2 - M^2}\right).
\end{eqnarray} 

From the asymptotic behavior of the polylogarithm functions
\cite{leon}, we found that in the high temperature limit,
Eq. (\ref{eq.02}) behaves as

\begin{equation}
B_{0i}(T) \rightarrow \frac{-2 v^2}{\pi \kappa} \epsilon_{0ij} p^j T
 \frac{\partial}{\partial m_{\sigma}^2}\left[\frac{m_{\sigma}^2
 \log(M/m_{\sigma})}{m_{\sigma}^2 - M^2}\right] \equiv \frac{1}{2}
 {\cal{\pi}}_{0i} (p_0=0,T).
\end{equation}

Once we have obtained the commutative limit, we consider next the first
correction,  which in this case is proportional to $\tau^2$. The
computations for this term are similar to the ones that we have done 
 so far, and finally, we can write the result, up to $\tau^4$, as


\begin{eqnarray}\label{eq.04}
B_{0i}(T)&=&
- \frac{2 v^2}{\pi \kappa} \epsilon_{0ij} p^j  T 
\frac{\partial}{\partial m_{\sigma}^2}\left\{ \frac{1}{m_{\sigma}^2 -
M^2}\right. \nonumber \\ 
&\times& \left. \left[ m^2_{\sigma} \log(M/m_{\sigma}) + \tau^2\frac{1
}{8 T^2}\left(M^4 \log(M/T) - m_{\sigma}^4 \log(m_{\sigma}/T)
\right)\right] \right\} + {\cal{O}} (\tau^4).
\end{eqnarray}
 
Before proceeding to the computation of the remaining diagram, it is
worth noting that both the temperature independent as well as
temperature dependent parts of $B_{0i}$, Eqs. (\ref{eq.03}) and
(\ref{eq.04}), are analytic functions of $\tilde{p}$: no infrared
singularities shows up.

Evaluating the  graph in Fig. 1.$b$, we have

\begin{equation}
\Pi_{\mbox{b},\mu \nu} (p)=2 \kappa^2 \int \frac{d^3
k}{(2 \pi)^2} \epsilon_{\mu \alpha \beta} \epsilon_{\sigma \rho \nu}
\sin^2(k \wedge p) D^{\alpha \rho} (k+p) D^{\sigma \beta} (k)
\end{equation}

\noindent and considering only the parity violating part of this diagram,
we obtain

\begin{eqnarray}
{\cal{\pi}}_{\mbox{b},\mu \nu}(p) &=&2 \kappa^2\int \frac{d^3
k}{(2 \pi)^3} \frac{\sin^2(k \wedge p) \epsilon_{\mu \nu \lambda}}{(m^4
+ \kappa^2 k^2) [m^4 + \kappa^2 (k+p)^2]} \nonumber \\
&\times& \left\{ \kappa m^2 p^{\lambda} - \kappa (m^2 - \xi \kappa^2) k
\cdot (k+p) \left[ \frac{k^{\lambda}}{k^2 + \xi m^2} -
\frac{(k+p)^{\lambda}}{(k+p)^2 + \xi m^2} \right]\right\}
\end{eqnarray}


The integrals that appear in this expression are similar to the
ones we have computed before. So, without going into details, for
  $p \rightarrow 0$ and $\tilde{p}\rightarrow 0$,  the
calculation gives

\begin{equation}
{\cal{\pi}}_{\mbox{b}, 0i} (p_0=0;T=0) = \frac{1}{16 \pi} \frac{3
m^2 - \xi \kappa^2}{\kappa} \tilde{p}\, \epsilon_{0ij} p^j
\end{equation}

\noindent and, for high temperature limit and also small
$\tilde{p}$ and $p$
\begin{equation}
{\cal{\pi}}_{\mbox{b}, 0i} (p_0=0;T) = -
\frac{\epsilon_{0ij}p^j}{16 \pi \kappa} \frac{\tau^2}{T}\left[2 m^2
\log(M/T) + (m^2 - \xi \kappa^2)F\right]
\end{equation}

\noindent where

\begin{equation}
F\equiv \frac{\xi^2 m^4(\xi m^2 + M^2)}{(\xi m^2 -
M^2)^3}\log(\sqrt{\xi}m/T) - \frac{M^2(M^4 + 4 \xi^2 m^4 - 3 \xi m^2
M^2)}{(\xi m^2 - M^2)^3} \log(M/T).
\end{equation}

It is worth noting here the gauge dependence of
${\cal{\pi}}_{b}$. Naively, we could expect that there would be no
gauge dependence at this point of the calculation, as it happens
in the commutative
case. But, this graph gives a purely noncommutative contribution to
$\Pi_{\mu \nu}$, which will disappear when $\theta$ goes to
zero. Therefore, there is no relation between it and what
is found in the commutative case. We recall that a dependence on the gauge
parameter was also  obtained in the commutative studies of the
nonabelian gauge models \cite{pisarski}.


\section{Conclusions}
\label{sec4}

The graphs computed in section \ref{sec3} suffice to obtain the
one-loop contribution to the parity violating part of the polarization
tensor and so, to the Chern-Simons coefficient.  Collecting the
results coming from ${\cal{\pi}}_a$ and ${\cal{\pi}}_{b}$, we obtain
the zero and finite temperature contributions to
${\cal{\pi}}^{NC}$. So,


\begin{equation}\label{res0}
{\cal{\pi}}_{0i}^{NC} (p_0=0;T=0) = \frac{1}{\pi \kappa} \epsilon_{0ij} p^j
\left[\frac{2 |\kappa|}{3} \frac{ \frac{1}{2}
\frac{m_{\sigma}}{M}}{(1 + \frac{m_{\sigma}}{M})^2} +
\frac{\tilde{p}}{4} (m^2 - \frac{3}{4}\xi \kappa^2)\right]
\end{equation}

\noindent and

\begin{eqnarray}\label{resT}
{\cal{\pi}}_{0i}^{NC} (p_0=0;T) &=& - \frac{1}{\pi \kappa}
\epsilon_{0ij} p^j T \nonumber \\ 
&\times& \left[ 2 v^2 \frac{\partial}{\partial m_{\sigma}^2}\left(
\frac{2m^2_{\sigma}\log(M/m_{\sigma}) + \frac{\tilde{p}^2}{8}(M^4
\log(M/T) - m^4_{\sigma}\log(m_{\sigma}/T))}{m_{\sigma}^2 - M^2}\right)
\nonumber \right. \\
&+& \left. \frac{\tilde{p}}{16} (2 m^2 \log(M/T)) + (m^2 - \xi \kappa^2) F
  \right] \\ \nonumber
\end{eqnarray} 



 From Eqs. (\ref{res0})
and (\ref{resT}) we can see that the commutativity can
be recovered straightforwardly, by considering the limit $\tilde{p}
\rightarrow 0$. In other words, there are no Seiberg poles appearing
in this limit.

Furthermore, looking at the finite temperature result,
Eq. (\ref{resT}), in the static limit we can extract the leading
behavior in the high temperature regime and first order correction in
the noncommutative parameter as being proportional to $T
logT$. So, our calculation provides a logarithm correction to
the result obtained in \cite{parityvio} for the commutative version of
the same model. 

Concerning gauge invariance it should be pointed out that
previous studies of noncommutative gauge theories have shown that, 
as it happens in the commutative non-Abelian CS model,
invariance under large gauge transformation requires that the CS coefficient
be quantized \cite{quantization}. At finite temperature, the 
verification of this property for the effective action is a highly non
trivial task which probably involves all orders of perturbation and
also other (nonlocal) interactions. Even in the
commutative setting, invariance under large non-Abelian gauge
transformation was verified only in simplified situations
\cite{seminara}. In our calculations large gauge transformation is
certainly partially  broken. In spite of that, our result should still
be a good approximation for small couplings.


Possible extensions of this work are the analysis of the parity even
part of the gauge field two point function and the investigation of
the properties of its effective action under small gauge
transformation. This last study requires the calculation of the gauge
field three and four point functions.  Such work is in progress.

After the completion of this work, we became aware of reference
\cite{chandrasekhar}, where similar studies were made for Maxwell gauge field
coupled with Majorana f\'ermions.

\section{Acknowledgments}

This work was partially supported by  Conselho
Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq) and
 Funda\c{c}\~ao de Amparo a Pesquisa do Estado de S\~ao Paulo (FAPESP).


\begin{appendix}
\section{Useful Integrals.}
\label{a1}
In this appendix, we will compute the integrals that appear in
Eq. (\ref{integral}). Let us define

\begin{eqnarray}
I(M) &\equiv& \int_0^{\infty} k^2 dk J_1 (k \tilde{p})
\frac{\coth{(\beta w_M/2)}}{w_M}. \nonumber \\
&=& \int_0^{\infty} k^2 dk J_1 (k \tilde{p})
\frac{1}{w_M} + 2 \int_0^{\infty} k^2 dk J_1 (k \tilde{p})
\frac{n_B(w_M)}{w_M}.
\end{eqnarray}

Using that 

\begin{equation}
\frac{1}{e^x-1}= \sum_{n=1}^{\infty} e^{-n x}
\end{equation}

\noindent we can rewrite $I(M)$ as

\begin{eqnarray}
I(M)&=& \int_M^{\infty} dx \sqrt{x^2 - M^2} J_1[\tilde{p}\sqrt{x^2 -
M^2}] \nonumber \\
&+& T^2 \sum_0^{\infty} \int_{\beta M}^{\infty} dx \sqrt{x^2 - \beta
M^2} J_1[\tau \sqrt{}x^2 - b^2 M^2] e^{-n x} \nonumber \\
&=&M^2 \int_1^{\infty} dy \sqrt{y^2 -1} J_1[\tilde{p} M \sqrt{y^2 -1}]
\nonumber \\
&+& M^2 \sum_0^{\infty} \int_1^{\infty} dy \sqrt{y^2 - 1} J_1[\tau
\beta M \sqrt{y^2 -1}] e^{-\beta M y n}
\end{eqnarray}

 These integrals can be evaluated using the standard result \cite{gradshteyn}

\begin{equation}
\int_1^{\infty} dx (x^2 -1)^{\nu/2} e^{- \alpha x} J_{\nu}[\beta
\sqrt{x^2 -1}] =\sqrt{\frac{2}{\pi}}\beta^{\nu} (\alpha^2 +
\beta^2)^{-\nu/2  - 1/4} K_{\nu + 1/2} (\sqrt{\alpha^2 + \beta^2}).
\end{equation}

\noindent  Then, it is straightforward to verify that

\begin{eqnarray}
I(M) &=& M^2 e^{- \tilde p  M} \left[\frac{1}{\tilde p  M} + \frac{1}{(\tilde p
 M)^2}\right] \nonumber \\
 &+& \tilde p  M T^2 \sum_{n=1}^{\infty}
\frac{e^{- \beta M \sqrt{n^2 + \tau^2}}}{n^2 +\tau^2} +\tilde  p  T^3
\sum_{n=1}^{\infty} \frac{e^{- \beta M \sqrt{n^2 + \tau^2}}}{(n^2
+\tau^2)^{3/2}}.
\end{eqnarray}

Note that here we have computed both zero temperature and finite
temperature parts together,  although in sections \ref{sec2} and
\ref{sec3} they occur separately.


\end{appendix}

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\newpage

\begin{figure*}\
\includegraphics{parity_nc.eps}
\caption{One loop graphs contributing to the parity violating part of the
$A_\mu$ two point function.} 
\end{figure*}


\end{document}







