Pole mass of the heavy quark: Perturbation theory and beyond

The key quantity of the heavy quark theory is the quark mass ${\mathit{m}}_{\mathit{Q}}$. Since quarks are unobservable one can suggest different definitions of ${\mathit{m}}_{\mathit{Q}}$. One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. We show that no precise definition of the pole mass can be given in the full theory once nonperturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order ${\mathrm{\Lambda }}_{\mathrm{QCD}}$/${\mathit{m}}_{\mathit{Q}}$. This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the ${\mathrm{\alpha }}_{\mathit{s}}$ series; the corresponding singularity in the Borel plane is situated at 2π/b. A peculiar feature is that this renormalon is not associated with the matrix element of a local operator. The difference Λ¯≡${\mathit{M}}_{\mathit{H}\mathit{Q}}$-${\mathit{m}}_{\mathit{Q}}^{\mathrm{pole}}$ can still be defined by heavy quark effective theory, but only at the price of introducing an explicit dependence on a normalization point μ: Λ¯(μ). Fortunately the pole mass ${\mathit{m}}_{\mathit{Q}}$(0) per se does not appear in calculable observable quantities.