Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

TL;DR

This paper computes a complex four-loop conformal Feynman integral using differential equations and a transcendental basis, providing explicit solutions in terms of polylogarithms.

Contribution

It introduces a novel method for evaluating four-loop conformal integrals by applying differential equations and a transcendental basis, yielding explicit analytical results.

Findings

Solution up to weight eight in multiple polylogarithms

Analytical expression in terms of single-valued harmonic polylogarithms

Results for all 213 master integrals in dimensional regularization

Abstract

We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.