Abstract
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 $\overline{\rm MS}$ mass we obtain precise conversion relations between the $\overline{\rm MS}$ 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 $\overline{\rm MS}$ and kinetic masses to about 15 MeV which is an improvement by a factor two to three 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.
Highlights
Quark masses enter the QCD Lagrangian density as free parameters and as such they have to be renormalized once higher-order corrections are considered
The pole mass scheme (OS) has the advantage that it is based on a physical definition since it requires that, order by order in perturbation theory, the inverse heavy-quark propagator has a zero at the position of the pole mass
The minimal subtraction scheme only subtracts the divergent parts of the quantum corrections to the quark twopoint function and combines them with the bare mass to arrive at the renormalized MS quark mass
Summary
Quark masses enter the QCD Lagrangian density as free parameters and as such they have to be renormalized once higher-order corrections are considered. The pole mass, on the other hand, suffers from renormalon ambiguities, which manifest themselves through an ill-behaved perturbative series This can already be seen in the relation between the on-shell and MS mass which suffers from large. The four-loop term in the mass relation amounts to about 100 MeV for bottom quarks [4,5], which is much larger than the current uncertainty of the MS mass as, e.g., extracted from lattice calculations or low-moment sum rules The various mass definitions can be converted into each other within perturbation theory Such a conversion is frequently needed in practical calculations as well as in the extraction of mass values from experiments. In Appendix A we report the analytic expressions up to Oðα3sÞ of the heavy-quark effective theory (HQET) parameters Λ , μπ, ρD and rE computed in perturbative QCD. In Appendix B we discuss in detail the calculation of the most difficult master integral while in Appendix C we provide analytic results of auxiliary integrals which were useful in the course of our calculation
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