QCD Corrections to QED Vacuum Polarization

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 \left< q\bar{q} right> and virtual $q\bar{q}$ loops are relevant only for very high radiation density $\sim 300MeV/f{m}^3$ and very strong magnetic fields of order $\sim 10^{14}T$.

Comments: 15 pages, 4 figures; Final version