A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $\overline{\rm MS}$-pole quark mass relation

TL;DR

The paper introduces a semi-analytic numerical method for computing multi-loop Feynman integrals across the entire kinematic range, applied to four-loop quark mass relations, providing both numerical and analytic results.

Contribution

It presents a versatile differential equation-based approach for multi-loop integrals that does not rely on special forms, enabling precise numerical and analytic results for complex Feynman integrals.

Findings

Successfully computed four-loop on-shell integrals for heavy quark mass relations.

Derived analytic expressions in harmonic polylogarithms for eight coefficient functions.

Validated numerical results with analytic expressions, ensuring high precision.

Abstract

We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter $x$ and the dimension $d$, in the whole kinematic range of $x$. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the $\overline{\rm MS}$ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.