A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the MS-pole quark mass relation

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 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.

Automated summary

The described method allows for numerical computation of multi-loop integrals in the entire kinematic range of the dimensionless parameter x, without the need for any special form, providing results suitable for fast numerical evaluation and accurate enough for phenomenological applications. It was applied to four-loop on-shell integrals, computing coefficient functions in the mass relation with eight color structures. Analytic results were obtained for these coefficient functions using harmonic polylogarithms and iterated integrals, enabling validation of numerical accuracy.