The Kinetic Heavy Quark Mass to Three Loops

Abstract

We compute three-loop corrections to the relation between the heavy quark masses defined in the pole and kinetic schemes. Using known relations between the pole and $\overline{\rm MS}$ quark masses we can establish precise relations between the kinetic and $\overline{\rm MS}$ charm and bottom masses. As compared to two loops, the precision is improved by a factor two to three. Our results constitute important ingredients for the precise determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ at Belle~II.