Kinetic Heavy Quark Mass to Three Loops

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 $\overline{\mathrm{MS}}$ quark masses, we can establish precise relations between the kinetic and $\overline{\mathrm{MS}}$ charm and bottom masses. As compared to two loops, the precision is improved by a factor of 2 to 3. Our results constitute important ingredients for the precise determination of the Cabibbo–Kobayashi–Maskawa matrix element $|V_{cb}|$ at Belle II.

Figure 1

(a) Schematic Feynman diagram representing the scattering of an external current, represented by wavy lines and a heavy quark (solid line). The blob represents one-particle irreducible quantum corrections, which we consider up to three-loop order. (b) The most complicated master integral. Dotted lines represent massless relativistic propagators and solid lines stand for eikonal propagators with mass $y$.