QCD Corrections to QED Vacuum Polarization
Castelo Ferreira, P. and Dias de Deus, J. (2008), "QCD Corrections to QED Vacuum Polarization", Eur. Phys. J., C54: 539.

Abstract:
We compute QCD corrections to QED calculations for vacuum polarization in background magnetic fields. Formally, the diagram for virtual $e bar{e}$ loops is identical to the one for virtual $q bar{q}$ loops. However due to confinement, or to the growth of $alpha_s$ as $p^2$ decreases, a direct calculation of the diagram is not allowed. At large $p^2$ we consider the virtual $q bar{q}$ diagram, in the intermediate region we discuss the role of the contribution of quark condensates $< q bar{q} >$ and at the low-energy limit we consider the $pi^0$, as well as charged pion $pi^+ pi^-$ loops. Although these effects seem to be out of the measurement accuracy of photon-photon laboratory experiments they may be relevant for $gamma$-ray burst propagation. In particular, for emissions from the center of the galaxy (8.5 kpc), we show that the mixing between the neutral pseudo-scalar pion $pi_0$ and photons renders a deviation from the power-law spectrum in the $TeV$ range. As for scalar quark condensates $< q bar{q} >$ and virtual $q bar{q}$ loops are relevant only for very high radiation density $sim 300 MeV/fm^3$ and very strong magnetic fields of order $sim 10^{14} T$.

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