Three-loop photon spectral density in QED

TL;DR

This paper computes the three-loop photon spectral density in QED with multiple electron species, expressing results through advanced polylogarithms and elliptic integrals, and verifies its accuracy via moments of the polarization operator.

Contribution

It provides a novel three-loop spectral density calculation in QED, utilizing complex iterated integrals and asymptotic analysis, advancing theoretical understanding of photon behavior.

Findings

Spectral density expressed in terms of Goncharov's polylogarithms and elliptic integrals.

Asymptotic behaviors at threshold and high energy are derived.

Spectral density accurately reproduces moments of the photon polarization operator.

Abstract

We calculate three-loop photon spectral density in QED with $N$ different species of electrons. The obtained results were expressed in terms of iterated integrals, which are either reduce to Goncharov's polylogarithms or can be written in terms of one-fold integrals of harmonic polylogarithms and complete elliptic integrals. In addition we provide threshold and high-energy asymptotics of the calculated spectral density. It is shown, that the use of the obtained spectral density correctly reproduces separately calculated moments of corresponding photon polarization operator.