Strong-Field QED and the Inverse Mellin Transform

TL;DR

This paper presents a novel application of the inverse Mellin transform to compute pair production widths and photon polarization functions in strong-field QED, enabling analysis across all energy ranges.

Contribution

The paper introduces the inverse Mellin transform technique to calculate photon polarization functions and pair production widths in strong magnetic fields, linking moments to derivatives at zero momentum.

Findings

Pair production width is proportional to derivatives of photon polarization at zero momentum.

The inverse Mellin transform allows calculation of the absorptive part of the photon polarization.

Dispersive parts can be obtained via Kramers-Kronig relations.

Abstract

We introduce the technique of inverse Mellin transform in a problem of strong-field QED. We show that the {\it moments} of pair production width in a uniform background magnetic field are proportional to the derivatives of photon polarization function at the zero momentum. Hence, the pair-production width or the absorptive part of the photon polarization function is calculable from the latter by the inverse Mellin transform. Using the {\it Kramers-Kronig} relation, the dispersive part of photon polarization function can be computed as well. Therefore the analytic property of the photon polarization function in all energy range is obtained. We also discuss briefly the possible extensions of this technique to other problems.