The Pole Mass of The Heavy Quark. Perturbation Theory and Beyond

TL;DR

The paper argues that the heavy-quark pole mass cannot be defined precisely once nonperturbative QCD effects are included, due to an infrared renormalon that induces an $O(Λ_{QCD}/m_Q)$ ambiguity. Using Wilson's OPE, it shows how infrared effects are separated and absorbed into condensates, but the pole mass renormalon cannot be mapped to a local operator, leading to a fundamental limit on the pole mass. It demonstrates that inclusive widths are best described using a high-scale running mass $m_Q(μ)$ and a running $ar{Λ}(μ)$ in HQET, with IR effects controlled by the renormalization scale $μ$, thereby removing the pole mass from observable predictions. The authors conclude that the pole mass should be avoided in precision heavy-quark physics and emphasize the importance of the running mass for reliable extractions of heavy-quark parameters such as $|V_{cb}|$.

Abstract

The key quantity of the heavy quark theory is the quark mass $m_Q$. Since quarks are unobservable one can suggest different definitions of $m_Q$. 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 non-perturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order $\Lam /m_Q$. This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the $α_s$ 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 $\La \equiv M_{H_Q}-m_Q^{pole}$ can still be defined in Heavy Quark Effective Theory, but only at the price of introducing an explicit dependence on a normalization point $μ$: $\La (μ)$. Fortunately the pole mass $m_Q(0)$ {\em per se} does not appear in calculable observable quantities.