A numerical evaluation of vacuum polarization tensor in constant external magnetic fields

TL;DR

This paper develops a numerical method to evaluate the photon vacuum polarization tensor in constant magnetic fields, focusing on Landau-level summation and error estimation relevant for heavy ion collision experiments.

Contribution

It introduces a numerical UV subtraction technique for evaluating the tensor's form factors and assesses the systematic errors from Landau-level truncation in realistic experimental conditions.

Findings

Error from Landau-level truncation is controllable at about 1% for electrons and muons.

The method enables accurate numerical evaluation of the polarization tensor in strong magnetic fields.

Results are relevant for interpreting magnetic field effects in heavy ion collision experiments.

Abstract

Hattori-Itakura have recently derived the full Landau-level summation form for the photon vacuum polarization tensor in constant external magnetic fields at the one-loop level. The Landau-level summation form is essential when the photon momentum exceeds the threshold of the pair creation of charged particles in a magnetic field stronger than the squared mass of the charged particle. The tensor has three different form factors depending on the tensor direction with respect to the external magnetic field. The renormalization is nontrivial because these form factors are expressed in terms of double or triple summation forms. We give a numerical UV subtraction method which can be applied to numerically evaluate the form factors in constant external magnetic fields. We numerically investigate the photon vacuum polarization tensor in the form of the Landau-level summation and estimate the systematic errors coming from truncation of the Landau-level summation in a parameter region realized in heavy ion collision experiments. We find that the error is practically controllable at an $O(10^{-2})$ level for electrons and muons in strong magnetic fields expected in heavy ion collisions in the experimentally feasible kinematic parameter regions.