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1. 3 discusses the concept of the effective electric charge of a neutrino from a quantum field theoretical point of view. As neutrinos do not couple to photons in the tree level in the standard model so naturally if the neutrinos acquire some electric charge it must be an effective one. 3. 4 describes the methods for handling ultraviolet divergences in calculations involving background magnetic fields. This chapter concludes with a general discussion on the various ideas exposed in the following sections.

Such calculations for γγ → νν were done some time ago [87, 55]. One of the important features of this calculation is that in the 4-fermi limit, the diagram contains three electron propagators. In such situations, the phase factor Ψ(x, x′ ) appearing in the Schwinger propagator of Eq. 2) cannot be disregarded6 . In the calculation, only the linear term in B was retained in the amplitude so that the results are valid only for small magnetic fields. However, since no effective Lagrangian was used, the results are valid even when the energies of the neutrinos and/or the photons are comparable to, or greater than, the electron mass.

In presence of a medium without any magnetic field the most general expression of the polarization tensor can be written in terms of form factors along with the new tensors constructed out of uµ and the ones we already had in absence of a medium as [88], Πµν (K, ω) = ΠT Tµν + ΠL Lµν + ΠP Pµν . 4: One-loop diagram for vacuum polarization. 30) and gµρ = gµρ − kµ kρ . 31) In Eq. 26) all form-factors are functions of ω, K and other parameters as chemical potential µ and inverse temperature β. In Eq.