Analytic results for planar three-loop integrals for massive form factors

TL;DR

This paper analytically evaluates all relevant planar three-loop Feynman integrals for massive form factors using differential equations, expressing results in terms of multiple polylogarithms for general and threshold cases.

Contribution

It provides explicit analytic solutions for all planar three-loop integrals for massive form factors, including at the threshold, using differential equations and polylogarithms.

Findings

Explicit expressions for 90 master integrals at general $q^2$

Explicit expressions for 51 master integrals at threshold $q^2=4m^2$

Results expressed in terms of multiple polylogarithms

Abstract

We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general $q^2$ are expressed in terms of multiple polylogarithms, and results for fiftyone master integrals at the threshold $q^2=4m^2$ are expressed in terms of multiple polylogarithms of argument one, with indices equal to zero or to a sixth root of unity.