The MSR Mass and the ${\cal O}(\Lambda_{\rm QCD})$ Renormalon Sum Rule

We provide a detailed description and analysis of a low-scale short-distance mass scheme, called the MSR mass, that is useful for high-precision top quark mass determinations, but can be applied for any heavy quark $Q$. In contrast to earlier low-scale short-distance mass schemes, the MSR scheme has a direct connection to the well known $\overline{\rm MS}$ mass commonly used for high-energy applications, and is determined by heavy quark on-shell self-energy Feynman diagrams. Indeed, the MSR mass scheme can be viewed as the simplest extension of the $\overline{\rm MS}$ mass concept to renormalization scales $\ll m_Q$. The MSR mass depends on a scale $R$ that can be chosen freely, and its renormalization group evolution has a linear dependence on $R$, which is known as R-evolution. Using R-evolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the ${\cal O}(\Lambda_{\rm QCD})$ renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the ${\cal O}(\Lambda_{\rm QCD})$ renormalon sum rule, and can be applied to any perturbative series. The relations of the MSR mass scheme to other low-scale short-distance masses are analyzed as well.