Multiloop contributions to the on-shell-$ \overline{\rm{MS}}$ heavy quark mass relation in QCD and the asymptotic structure of the corresponding series: the updated consideration

The asymptotic structure of the QCD perturbative relation between the on-shell and $\overline{\rm{MS}}$ heavy quark masses is studied. We estimate the five and six-loop contributions to this relation by three different techniques. First, the effective charges motivated approach in two variants is used. Second, the results following from the large-$\beta_0$ approximation are analyzed. Finally, the consequences of applying the asymptotic renormalon-based formula are investigated. We show that all approaches lead to corrections which are qualitatively consistent in order of magnitude. Their sign-alternating character in powers of the number of massless quarks is demonstrated. We emphasize that there is no contradiction in the behavior of the fine structure of the renormalon-based estimates with other approaches if one use the detailed information about the normalization factor included in the renormalon asymptotic formula. The obtained five- and six-loop estimates indicate that in the case of the $b$-quark the asymptotic character of the studied relation manifests itself above the fourth order of PT, whereas for the $t$-quark it starts to reveal itself after the seventh order. This allows to conclude that like the running masses, the pole masses of the $b$ and especially $t$-quark in principle may be used in the phenomenologically-oriented studies.