Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals

TL;DR

This paper derives simplified analytical expressions for complex 3- and 4-loop Feynman integrals involving elliptic functions, advancing calculations relevant to quantum field theory and lattice models.

Contribution

It provides the first high-precision numerical identification of a compact analytical form for the 4-loop self-mass integral with equal masses, a key component in electron g-2 calculations.

Findings

Analytical expression for 3-loop self-mass integral simplified.

First high-precision numerical fit for 4-loop integral with equal masses.

Analytical forms for lattice integrals, including the Watson integral.

Abstract

In this paper we continue the work begun in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the 3-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the 4-loop on-shell self-mass integral with 4 equal masses, that is one of the master integrals of the 4-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the 4-dimensional generalized Watson integral.