The MSR mass and the $$ \mathcal{O}\left({\Lambda}_{\mathrm{QCD}}\right) $$ renormalon sum rule

Abstract

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 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 MS mass concept to renormalization scales 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 Revolution. Using Revolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the O(Λ QCD) renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the O(Λ 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.

On the light massive flavor dependence of the large order asymptotic behavior and the ambiguity of the pole mass

“…(1.1)-(1.3) agree remarkably well with the corresponding large order asymptotic behavior already beyond the terms of O(α s ) (so that the terms of the series are known quite precisely to all orders) and because even for orders where the QCD corrections still decrease with order they can be very large numerically and make phenomenological applications difficult. The pole mass scheme has therefore been abandoned in high precision top, bottom and charm quark mass analyses in favor of quark mass schemes such as MS or low-scale short distance masses such as the kinetic mass [15], the potential-subtracted (PS) mass [16], the 1S mass [17][18][19], the renormalon-subtracted (RS) mass [20], the jet mass [21,22] or the MSR mass [23,24]. These mass schemes do not have an O(Λ QCD ) renormalon and are called short-distance masses.…”

“…the recent work of refs. [24,[27][28][29]). However, when quoting a concrete numerical size of the ambiguity, criteria common for converging series cannot be applied, and it is instrumental to consider more global aspects of the series and the quantity it describes.…”

“…The essential technical tool to set up the formalism is the MSR mass m MSR Q (R) [23,24]. Like the perturbative series for the pole-MS mass relation, the pole-MSR mass relation is calculated from on-shell heavy quark self energy diagrams, but has also linear dependence on R. It is the basis of the RG formalism we propose, allows to precisely capture the QCD corrections from the different quark mass scales and, in particular, to encode and study issue (1) coming from HQS.…”

“…Like the perturbative series for the pole-MS mass relation, the pole-MSR mass relation is calculated from on-shell heavy quark self energy diagrams, but has also linear dependence on R. It is the basis of the RG formalism we propose, allows to precisely capture the QCD corrections from the different quark mass scales and, in particular, to encode and study issue (1) coming from HQS. The renormalization group evolution in the scale R is described by R-evolution [23,24], which is free of the O(Λ QCD ) renormalon, and allows to sum large logarithms of ratios of the quark masses in the evolution between the quark mass scales. Using the concepts of the MSR mass and the R-evolution it is then possible to relate the pole-MS masses of the top, bottom and charm quarks to each other.…”

“…The definition generalizes the one already provided in ref. [24], which only considered n Q massless quarks. The notation used for the virtual quark mass corrections involving the functions δ (q,q ,... ) Q (r qQ , r q Q , .…”