Including lowest loop correction, the black-body radiation via the Born-Infeld theory of electrodynamics is obtained. The possible direct implications of the results on Cosmic Microwave Background map is argued.
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\begin{document}
\begin{center}
\textbf{Born-Infeld  Black-Body Radiation: Implications On CMB}

\vspace{0.7cm}
Maryam Hajirahimi$^{~1}$\footnote{email: maryam.hajirahimi@gmail.com}
~~~and~~~Amir H. Fatollahi$^{~2}$\footnote{email: ahfatol@gmail.com}
\vspace{0.7cm}

\textit{1) Islamic Azad University - South Tehran Branch \\ P. O. Box 11365, Tehran 4435, Iran \\
\vspace{0.5cm}
2)  Department of Physics, Alzahra University, \\ P. O. Box  19938, Tehran  91167, Iran}
\end{center}

\vspace{0.7cm}

\begin{abstract}
Including lowest loop correction, the black-body radiation via the
Born-Infeld theory of electrodynamics is obtained. The possible direct
implications of the results on Cosmic Microwave Background map is argued.
\end{abstract}
\vspace{1cm}

%\pacs{11.10.Nx}{Noncommutative field theory}%\pacs{11.10.Wx}{Finite-temperature field theory}
%\pacs{98.70.Vc}{Cosmic background radiation}

\newpage

The Born-Infeld theory of electrodynamics is given by the Lagrangian density \cite{bi}
\begin{equation}
\mathcal{L}=-\sqrt{|\det(\gamma\, \eta_{\mu\nu}+F_{\mu\nu})|} +\sqrt{-\det(\gamma\,\eta_{\mu\nu})}
\end{equation}
in which the second term is chosen so that the above Lagrangian reduces
to Maxwell one in the limit $\gamma\to\infty$. In above
$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$,
$\eta_{\mu\nu}=\mathrm{diag}\,(1,-1,-1,-1)$ as the spacetime
metric, and $\gamma$ is a new constant with dimension (energy)$^{2}$.
The original motivation to propose the non-linear theory above was to
overcome the problem of infinite field strength and energy
associated with the point charge source \cite{bi}. In particular,
the electrostatic field of a spherically symmetric solution of the
theory at the origin is finite and of order of $\gamma$.
It is shown that the Born-Infeld theory, though non-linear, possesses exact
plane-wave solutions \cite{boillat}.

The Born-Infeld theory has got a renewal interest since the discovery that
the low energy effective theories that govern the massless vector modes of open
string theories are of Born-Infeld type. In particular,
at tree level the effective Lagrangian which depends on the abelian field strength
$F_{\mu\nu}$ but not on its derivatives sum up to the Born-Infeld one \cite{tseytlin}.

The purpose of this note is to study the problem of black-body radiation
in the electrodynamics theory of Born-Infeld. Due to non-linear character
of the theory we expect interaction among the photons.
Including lowest loop correction, we compute the deviation from the
black-body radiation of Maxwell theory. The possible implications
of the results on Cosmic Microwave Background map is argued.

In $1+3$ spacetime dimensions the Lagrangian reduces to
\begin{equation}
\mathcal{L}= \sqrt{-\eta}~ \gamma^2
\bigg(1-\sqrt{1+\frac{2Q}{\gamma^2}-\frac{P^2}{\gamma^4}} \bigg)
\end{equation}
in which $Q$ and $P$ are given by
\begin{align}
Q&=\frac{1}{4}\, F_{\mu\nu}F^{\mu\nu} = \frac{1}{2}(\mathbf{B}^2 - \mathbf{E}^2) \\
P&=\frac{1}{4}\, {}^\ast F_{\mu\nu}F^{\mu\nu} =\mathbf{E}\cdot\mathbf{B}
\end{align}
with ${}^\ast F_{\mu\nu}=\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}/\,2$.
Evidently the Lagrangian is invariant under the abelian gauge transformation
$A\to A+\partial \Lambda$, for any differentiable function $\Lambda(x)$.

The Lagrangian might be expanded for small values of $\gamma^{-1}$, for which, by
keeping the lowest orders, one gets
\begin{equation}
\mathcal{L} = -\frac{1}{4}\, F_{\mu\nu}F^{\mu\nu} + \frac{1}{2 \gamma^2}
(Q^2+P^2) + O(\gamma^{-4})
\end{equation}
The first term above is simply the Maxwell term, by which the free theory is presented.
The other terms introduce interaction among the photons. By the lowest order kept,
in the language of Feynman diagrams, we have 4-leg vertex which is given by
\begin{align}
 V_4=& \frac{-1}{\gamma^2} \bigg[
 (p\cdot q) (r\cdot s)\, \eta^{\sigma\rho}\eta^{\mu\nu} +
(p\cdot r) (q\cdot s)\, \eta^{\mu\sigma}\eta^{\nu\rho} +
(p\cdot s) (q\cdot r)\, \eta^{\mu\rho}\eta^{\nu\sigma}  \cr
& +p^\nu q^\mu r^\rho s^\sigma + p^\sigma q^\rho r^\mu s^\nu + p^\rho q^\sigma r^\nu s^\mu
- (s\cdot r) p^\nu q^\mu \eta^{\sigma \rho} -(s\cdot p) q^\sigma r^\nu \eta^{\mu\rho} \cr
& - (s\cdot q) p^\sigma r^\mu \eta^{\nu\rho} -(q\cdot r) p^\rho s^\mu \eta^{\nu\sigma}
- (p\cdot r) q^\rho s^\nu \eta^{\mu\sigma} - (q\cdot p) r^\rho s^\sigma \eta^{\mu\nu}\cr
&+ (s^\alpha q^\beta \varepsilon_{\alpha\beta}^{~~~\rho\nu})(r^\kappa p^\lambda \varepsilon_{\kappa\lambda}^{~~~\sigma\mu})+
 (s^\alpha p^\beta \varepsilon_{\alpha\beta}^{~~~\rho\mu})(r^\kappa q^\lambda \varepsilon_{\kappa\lambda}^{~~~\sigma\nu})\cr
&+ (s^\alpha r^\beta \varepsilon_{\alpha\beta}^{~~~\rho\sigma})(q^\kappa p^\lambda \varepsilon_{\kappa\lambda}^{~~~\nu\mu})\bigg]
\end{align}
where the momenta and indices are introduced in Fig.~1.
Also, the full quantum theory is given after adding gauge fixing term,
which may be taken as usual $-\frac{1}{2}(\partial\cdot A)^2$, and the corresponding
ghost sector. Since in the present theory the gauge group is abelian, the ghosts
do not interact with the photons, hence the ghost sector simply can be integrated out.

\begin{figure}[t]\begin{center}
\includegraphics[scale=1]{fig_1.eps}
\caption{The 4-leg photon vertex.}
\end{center}\end{figure}

Since the coupling constant $\gamma^{-2}$ has negative mass dimension, the present
theory, as a quantum field theory, is not renormalizable. This suggests
that the proper interpretation of the Born-Infeld theory would be as an
effective theory of an underlying finite or renormalizable theory, a picture
that is supported by the string theory calculations \cite{tseytlin}. In lack
of the full underlying theory one might try to explore implications of the
Born-Infeld theory at higher orders of perturbation, for which one has to
give a recipe for the problem of infinities.
As mentioned, here we consider the black-body radiation problem, and the natural
framework for dealing with the interaction contribution
is finite temperature field theory \cite{fin-temp-book, kapusta-book}. As known, and
we will see in more detail below, the problem of infinities finds a natural solution
in this framework \cite{fin-temp-book, kapusta-book}.
According to the recipe, the regulated finite temperature
expressions are given simply after the subtraction of the
zero temperature counterpart expressions \cite{kapusta-book}. By this,
the study of thermodynamical properties may give a basis to explore
the implications of the effective theory at higher order of perturbations.

At 1-loop order (no vertex), the free Maxwell theory, plus the contribution
from the ghost sector, simply give the standard Planck expression
for black-body radiation. The first contribution from interactions
comes at 2-loop (one vertex). As mentioned, to get the relevant
regulated contributions one subtracts an altered form of the
diagrams, the so-called parenthesized diagrams, from the original
ones. At 2-loop order by ``parenthesized" we simply mean that in
one of the loops we send temperature to zero
\cite{kapusta-book,kapusta}. In the present case we have the
contribution from the following diagrams \cite{kapusta-book,kapusta}

\begin{figure}[h]\begin{center}
\includegraphics[scale=1]{fig_2.eps}
\caption{a) the original 2-loop diagram. b) the parenthesized version of a).}
\end{center}\end{figure}

Using the identity
\begin{equation}
\varepsilon^{\mu\nu\rho\sigma} \varepsilon_{\mu\nu}^{~~~\kappa\lambda}
= -2(\eta^{\rho\kappa}\eta^{\sigma\lambda}-\eta^{\rho\lambda}\eta^{\sigma\kappa})
\end{equation}
the original diagram (Fig.~2a), in units $c=\hbar=k_\mathrm{B}=1$, is associated with the expression
\begin{align}
\mathcal{F}_\mathrm{2-loop}^\mathrm{orig.}=\frac{1}{8\gamma^2}
\int \frac{\d^3p}{(2\pi)^3}\int\frac{\d^3q}{(2\pi)^3}\; \frac{1}{\beta} \sum_{n_p} \frac{1}{\beta} \sum_{n_q}
\bigg(\frac{8(p\cdot q)^2}{p^2 q^2}+7\bigg)
\end{align}
in which $\beta=T^{-1}$ is the inverse of the temperature, and 4-vectors $p$ and $q$ have the general
form of $k=(2\pi i n_k T, \mathbf{k})$, with $n_k$ taking all integer values. In the parenthesized form
of the diagram one takes the limit $\beta\to \infty$ in one of the summations of the above expression,
turning the integer variable to a continuous one \cite{kapusta}.
Following \cite{kapusta}, one can transform the summations to integrals in the complex-plane
\begin{align}
T\sum_{n=-\infty}^\infty f(k_0=2\pi i n  T)  = & \frac{1}{2\pi i} \int_{-i\infty +\epsilon}^{i\infty +\epsilon}
\d k_0 \frac{1}{\exp(\beta k_0)-1} \big[f(k_0) + f(-k_0)] \cr
& + \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \d k_0 f(k_0), ~~~~~~~\mathrm{with}~~\epsilon\to 0^+
\end{align}
We mention that the second term is independent of temperature, and survives in the limit $\beta\to\infty$.
The first term can be calculated by the residues at the poles of the function $f(k_0)$ in the right half-plane.

By using above one finds for the original diagram
\begin{align}
\mathcal{F}_\mathrm{2-loop}^\mathrm{orig.}=\frac{1}{8\gamma^2}& \int \frac{\d^3p}{(2\pi)^3}\int\frac{\d^3q}{(2\pi)^3}\cr
&\bigg\{\frac{4N_p N_q} {\omega_p \omega_q} \Big[(\omega_p\omega_q - \mathbf{p}\cdot\mathbf{q})^2
+ (\omega_p\omega_q + \mathbf{p}\cdot\mathbf{q})^2 \Big] \cr
&+2\frac{N_p+ N_q} {\omega_p \omega_q} \Big[(\omega_p\omega_q - \mathbf{p}\cdot\mathbf{q})^2
+ (\omega_p\omega_q + \mathbf{p}\cdot\mathbf{q})^2 \Big] \bigg\}\cr
&+ \infty\mathrm{~terms~independent~of~}T
\end{align}
in which $N_k=\big(\exp(\beta \omega_k) -1\big)^{-1}$, with $\omega_k=|\mathbf{k}|$.
After subtracting parenthesized diagram, the contribution from 2-loop order to the
free-energy per unit volume is found
\begin{align}
\mathcal{F}_\mathrm{2-loop} = & \frac{1}{\gamma^2} \int
\frac{\d^3p}{(2\pi)^3}\int\frac{\d^3q}{(2\pi)^3}\; N_p N_q
\frac{\omega_p^2\omega_q^2 +
(\mathbf{p}\cdot\mathbf{q})^2}{\omega_p \omega_q}\cr
& + \infty\mathrm{~terms~independent~of~}T
\end{align}
Using
\begin{equation}\label{12}
\int_0^\infty\frac{s^{2m +1}\d s}{\e^{s}-1}=(2m +1)!\;\zeta(2m +2)
\end{equation}
with $\zeta(t)$ as the Riemann zeta-function, one can do the integrations over
$\d^3q=\omega_q^2\d\omega_q\d\Omega_q$ and $\d\Omega_p$ to get
\begin{equation}
\mathcal{F}_\mathrm{2-loop} = \frac{T^4}{45\,\gamma^2}
\int_0^\infty \frac{\omega^3 \d\omega}{ \exp(\beta\omega)-1 }
\end{equation}
The internal energy per unit volume is then given by
\begin{align}
U&=\mathcal{F} +\beta \partial_\beta \mathcal{F} \\
&=\int_0^\infty\d\omega ~U(\omega)
\end{align}
for which we find
\begin{equation}
U_\mathrm{2-loop}(\omega) = - \frac{T^4 }{45\,\gamma^2}
\omega^3 \left[  \frac{3 }{\exp(\beta \omega)-1}
+ \frac{\beta\omega \exp(\beta\omega)}{(\exp(\beta \omega)-1)^2}\right]
\end{equation}
as the energy per unit volume in the frequency interval $(\omega,\omega+\d\omega)$.
In fact this expression gives the correction to the standard Planck expression
\begin{equation}
U_\mathrm{Planck} (\omega)= \frac{1}{\pi^2}\frac{\omega^3}{\exp(\beta\omega)-1}
\end{equation}
Using (\ref{12}) and
\begin{equation}
\int_0^\infty\frac{s^{2m +2}\e^{s}\d s}{(\e^{s}-1)^2}=(2m +2)!\;\zeta(2m +2)
\end{equation}
one finds the correction to Stefan's law
\begin{equation}
U_\mathrm{2-loop}=-\frac{7\pi^4}{675 \gamma^2} \; T^8
\end{equation}

Although it is hard to imagine that the deviations
from Maxwell theory can be detected in a laboratory black-body radiation,
one may look for an indication of deviations in the signals we are getting
from the extremely hot seconds of early universe. In fact, the energy scale
that one expects for relevance of possible deviations is as much as high that
suggests maybe it has been available for particles only in the early universe.
So an excellent way to test the current theories would be the study of what are
left for us as early universe's heir, the most important among them the Cosmic
Microwave Background radiation. The reason is, CMB map is just a tableau of
events which happened at the first epochs of universe, at the decoupling era
or much earlier, when the energies were sufficiently high to make relevant
possible deviations.
Based on this way of reasoning, there have been several proposals for this kind
of studies, among them one may call, 1) exploring the consequences of space-time
uncertainty relations of the form $\Delta t \Delta x\geq l_s^2$ in the
context of inflation theory and its implications on CMB data \cite{tx-uncer},
2) the efforts to formulate and study the noncommutative versions of inflation
theory \cite{nc-cosmos, fuzzy-sphere}, 3) the direct consequences of
space noncommutativity on CMB anisotropy \cite{nc-cmb}, 4) the study of theories
with noncommutative electromagnetic fields - that is considering the fields, rather
than the coordinates, noncommutative - which may cause anisotropy on CMB, as well
as the pattern we expect to see in polarized CMB data \cite{nc-field}.

Much efforts are currently being devoted to examining the CMB temperature pattern
measured with the Wilkinson Microwave Anisotropy Probe (WMAP) \cite{wmap} as
well as the new launched Planck Observatory \cite{planck}.
It would be extremely important if the present and forthcoming
data indicate any significant evidence for deviations from Maxwell theory
of electromagnetism. In particular, due to the mentioned relation between
Born-Infeld type theories and string theory calculations \cite{tseytlin},
comparison with the expectations from Born-Infeld theory may provide
a basis for an indirect test of string theory.

\vspace{0.5cm}
\noindent\textbf{Acknowledgments:}

The work by A. H. F. is partially supported by the Research Council of Alzahra University.

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\end{document}

0909.2620.extracted_bib
\begin{thebibliography}{99}

\bibitem{bi} M. Born and L. Infeld, ``Foundations Of The New Field Theory,"
Nature \textbf{132} (1933) 1004; Proc. Roy. Soc. (London) \textbf{144} (1934) 425.

\bibitem{boillat} G. Boillat, ``Exact Plane-Wave Solution Of Born-Infeld Electrodynamics,"
Lett. Nuovo Cimento \textbf{4} (1972) 274.

\bibitem{tseytlin} E.~S. Fradkin and A.~A. Tseytlin,
`` Nonlinear Electrodynamics From Quantized Strings," Phys. Lett. B \textbf{163} (1985) 123;
A.~A.~Tseytlin, ``Vector Field Effective Action In The Open Superstring Theory,"
Nucl. Phys. B \textbf{276} (1986) 391; Err.-ibid. \textbf{291} (1987) 876.

\bibitem{fin-temp-book} M. Le Bellac, ``Thermal Field Theory," Cambridge Univ. Press, 1996;
A.~Das, ``Finite Temperature Field Theory," World Scientific, 1997.

\bibitem{kapusta-book} J. I. Kapusta, ``Finite Temperature Field Theory," Cambridge Univ. Press, 1989.

\bibitem{kapusta} J. I. Kapusta, ``Quantum Chromodynamics At High Temperature,"
Nucl. Phys. B {\bf 148} (1979) 461.

\bibitem{tx-uncer} S. Tsujikawa, R. Maartens, and R. Brandenberger,
``Non-Commutative Inflation And The CMB," Phys. Lett. B \textbf{574}
(2003) 141, astro-ph/0308169; Q. G. Huang and M. Li,
``Noncommutative Inflation And The CMB Multipoles," JCAP
\textbf{0311} (2003) 001, astro-ph/0308458;  G. Calcagni and S.
Tsujikawa, ``Observational Constrains On Patch Inflation In
Noncommutative Spacetime," Phys. Rev. D \textbf{70} (2004) 103514,
astro-ph/0407543; G. Calcagni, ``Noncommutative Models In Patch
Cosmology," Phys. Rev. D \textbf{70} (2004) 103525, hep-th/0406006; Y. S.
Myung, ``Cosmological Parameters In Noncommutative Inflation,"
Phys. Lett. B \textbf{601} (2004) 1, hep-th/0407066; Q. G. Huang and M.
Li, ``Power Spectra In Spacetime Noncommutative Inflation,"
Nucl. Phys. B \textbf{713} (2005) 219, astro-ph/0311378; H. Kim, G. S.
Lee, H. W. Lee, and Y. S. Myung, ``Second-Order Corrections To
Noncommutative Spacetime Inflation," Phys. Rev. D \textbf{70} (2004) 043521,
hep-th/0402198; H. Kim, G. S. Lee, and Y. S. Myung,
``Noncommutative Spacetime Effect On The Slow-Roll Period Of
Inflation," Mod. Phys. Lett. A \textbf{20} (2005) 271, hep-th/0402018.

\bibitem{nc-cosmos} F. Lizzi, G. Mangano, G. Miele, and M. Peloso,
``Cosmological Perturbations And Short Distance Physics From
Noncommutative Geometry," JHEP \textbf{0206} (2002) 049, hep-th/0203099;
F. Nasseri and S. A. Alavi, ``Noncommutative Decrumpling Inflation
And Running Of The Spectral Index," Int. J. Mod. Phys. D \textbf{14} (2005)
621, hep-th/0410259; S. A. Alavi and F. Nasseri, ``Running Of
The Spectral Index In Noncommutative Inflation," Int. J. Mod. Phys.
A \textbf{20} (2005) 4941, astro-ph/0406477; C. Chu, B. R. Greene, and
G. Shiu, ``Remarks On Inflation And Noncommutative Geometry,"
Mod. Phys. Lett. A \textbf{16} (2001) 2231-2240, hep-th/0011241; G.
Amelino-Camelia and S. Majid, ``Waves On Noncommutative
Spacetime And Gamma-Ray Bursts," Int. J. Mod. Phys. A \textbf{15} (2000)
4301, hep-th/9907110; G. D. Barbosa and N. Pinto-Neto, ``Noncommutative Geometry And Cosmology,"
Phys. Rev. D \textbf{70} (2004) 103512, hep-th/0407111;
R. H. Brandenberger and J. Martin, ``On Signatures Of Short Distance Physics In The Cosmic Microwave Background,"
Int. J. Mod. Phys. A \textbf{17} (2002) 3663, hep-th/0202142;
S. Alexander, R. Brandenberger, and J. Magueijo, ``Non-Commutative Inflation,"
Phys. Rev. D \textbf{67} (2003) 081301, hep-th/0108190.

\bibitem{fuzzy-sphere} M. Fukuma, Y. Kono, and A. Miwa, ``Effects Of Space-Time Noncommutativity
On The Angular Power Spectrum Of The CMB," Nucl. Phys. B \textbf{682} (2004)
377, hep-th/0307029; M. Fukuma, Y. Kono, and A. Miwa, ``A
Mechanism Of The Large-Scale Damping In The CMB Anisotropy,"
Nucl. Phys. B \textbf{703} (2004) 293, hep-th/0312298; M. Fukuma, Y.
Kono, and A. Miwa, ``Noncommutative Inflation And The Large-Scale
Damping In The CMB Anisotropy," hep-th/0401153.

\bibitem{nc-cmb} A. H. Fatollahi and M. Hajirahimi,
``Noncommutative Black-Body Radiation: Implications On Cosmic Microwave Background,"
Europhys. Lett. \textbf{75} (2006) 542-547, astro-ph/0607257;
H.-C. Kim, C. Rim, J. H. Yee, ``Blackbody Radiation In $\kappa$-Minkowski Spacetime,"
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E. Akofor, A. P. Balachandran, S. G. Jo, A. Joseph, and B. A. Qureshi,
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JHEP \textbf{0805} (2008) 092, arXiv:0710.5897.

\bibitem{nc-field} J. Gamboa, J. Lopez-Sarrion, and A. P. Polychronakos, ``Ultraviolet Modified Photons
And Anisotropies In The Cosmic Microwave Background Radiation," Phys. Lett. B
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A. H. Fatollahi, and  M. Hajirahimi, ``Black-Body Radiation Of Noncommutative Gauge Fields,"
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\bibitem{wmap} G. Hinshaw, et al.,
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arXiv:0803.0732; for a complete bibliographys of the WMAP project see:\\
http://lambda.gsfc.nasa.gov/product/map/current/map\_bibliography.cfm

\bibitem{planck} http://www.esa.int/esaMI/Planck/

\end{thebibliography}