The Dispersive Approach to Electroweak Processes in the Background Magnetic Field

The electroweak phenomena associated with an intensive background magnetic field are rather rich. Under a background magnetic field, a physical photon can decay into an e+e− pair or split into two photons. Such processes are relevant to the attenuation of gamma-rays from pulsars. Similarly, with a sufficient energy, a neutrino can go through the decays ν → νe+e−3 and ν → νγ. For processes without charged fermions in the final state, such as γ → γγ or ν → νγ, their decay widths can be expressed as asymptotic series in B for B < Bc. However, such asymptotic expansions are not possible for ν → νe+e− or γ → e+e− since the wave functions of final-state fermions are non-analytic with respect to the magnetic field strength at B = 0. Previously, the photon absorption coefficients due to γ → e+e− were computed in two different ways. One either directly squares the γ → e+e− amplitude using the exact electron (positron) wave function in the background magnetic field, or applies the optical theorem on the photon polarization function Πμν. In both approaches, the results are valid only for B < Bc and ω sin θ 2me, where ω is the photon energy and θ is the angle between the magnetic-field direction and the direction of photon propagation. It has been pointed out that a correct description of γ → e+e− near the pair-production threshold ω sin θ ≈ 2me is crucial for astrophysical applications. A numerical study taking into account the threshold behavior of γ → e+e− was also carried out. In this work, we re-examine the previous analytic approaches to γ → e+e− , clarifying their implicit assumptions which lead to incorrect threshold behavior for the above decay. Let us follow Ref. 6 which begins with the proper-time representation of photon polarization function Πμν in the background magnetic field: