Relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks

We compute the relation between the pole mass and the kinetic mass of a heavy quark to three loops. Using the known relation between the pole and the $\overline{\mathrm{MS}}$ mass we obtain precise conversion relations between the $\overline{\mathrm{MS}}$ and kinetic masses. The kinetic mass is defined via the moments of the spectral function for the scattering involving a heavy quark close to threshold. This requires the computation of the imaginary part of a forward-scattering amplitude up to three-loop order. We discuss in detail the expansion procedure and the reduction to master integrals. For the latter analytic results are provided. We apply our result both to charm and bottom quark masses. In the latter case we compute and include finite charm quark mass effects. Furthermore, we determine the large-${\beta }_0$ result for the conversion formula at four-loop order. For the bottom quark we estimate the uncertainty in the conversion between the $\overline{\mathrm{MS}}$ and kinetic masses to about 15 MeV which is an improvement by a factor of 2–3 as compared to the two-loop formula. The improved precision is crucial for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ at Belle II.

Figure 1

Sample Feynman diagrams for the scattering process of an external current (wavy line) and a heavy quark (solid line). Gluons are represented by curly lines.

Figure 2

Two possible momentum routings of a two-loop diagram. The naive regions in the first case (a) do not correspond to all regions while the routing in (b) correctly reveals all regions.

Figure 3

Sample singlet-type Feynman diagrams. The external currents are drawn with wavy lines, heavy quarks with solid lines and gluons are represented by curly lines. Diagrams (a), (c) and (d) are zero for an external vector current but not for an external scalar current.

Figure 4

The one- and two-loop master integrals for the double ultrasoft region. Dashed lines represent massless propagators, while solid lines and double lines represent the linear massive ($2p·k_i-y$) and massless ($2p·k_i$) HQET-like propagators, respectively, with $p^2=m_b^2$ and the loop momentum $k_i$ ($i=1$, 2).

Figure 5

The three-loop master integrals for the triple ultrasoft region. The same notation as in Fig. 4 is used.

Figure 6

Sample Feynman diagrams containing closed loops with charm quarks. The same notation as in Fig. 1 is used.

Figure 7

${\overline{m}}_b^{\mathrm{kin}}$ computed from ${\overline{m}}_b({\mu }_s)$ using one-, two- and three-loop (blue, red, and black lines, respectively) accuracy as a function of ${\mu }_s$. At each loop order, four lines are shown, one for each of schemes A–D.

Figure 8

${\overline{m}}_b({\overline{m}}_b)$ computed from $m^{\mathrm{kin}}$ using one-, two- and three-loop (blue, red, and black lines, respectively) accuracy as a function of ${\mu }_s$. At each loop order, four lines are shown, one for each of the schemes A–D.