# On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguity of the Pole Mass

## TL;DR

This paper develops a renormalization group formalism to analyze the effects of lighter massive quarks on the pole mass of heavy quarks, providing precise mass difference calculations and exploring the ambiguity in the pole mass due to large order asymptotics.

## Contribution

It introduces a systematic formalism for mass effects in heavy quark pole mass relations, enabling summation of logarithms and precise ambiguity estimates.

## Key findings

- Calculated pole mass differences for top, bottom, and charm quarks with ~20 MeV precision.
- Predicted ${
m O}( ext{alpha}_s^4)$ virtual quark mass corrections.
- Found the pole mass ambiguity to be around 250 MeV, larger than experimental projections.

## Abstract

We provide a systematic renormalization group formalism for the mass effects in the relation of the pole mass $m_Q^{\rm pole}$ and short-distance masses such as the $\overline{\rm MS}$ mass $\overline{m}_Q$ of a heavy quark $Q$, coming from virtual loop insertions of massive quarks lighter than $Q$. The formalism reflects the constraints from heavy quark symmetry and entails a combined matching and evolution procedure that allows to disentangle and successively integrate out the corrections coming from the lighter massive quarks and the momentum regions between them and to precisely control the large order asymptotic behavior. With the formalism we systematically sum logarithms of ratios of the lighter quark masses and $m_Q$, relate the QCD corrections for different external heavy quarks to each other, predict the ${\cal O}(\alpha_s^4)$ virtual quark mass corrections in the pole-$\overline{\rm MS}$ mass relation, calculate the pole mass differences for the top, bottom and charm quarks with a precision of around $20$ MeV and analyze the decoupling of the lighter massive quark flavors at large orders. The summation of logarithms is most relevant for the top quark pole mass $m_t^{\rm pole}$, where the hierarchy to the bottom and charm quarks is large. We determine the ambiguity of the pole mass for top, bottom and charm quarks in different scenarios with massive or massless bottom and charm quarks in a way consistent with heavy quark symmetry, and we find that it is $250$ MeV. The ambiguity is larger than current projections for the precision of top quark mass measurements in the high-luminosity phase of the LHC.