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

Understand

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.