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Heavy Quark Masses from the $Q\bar Q$ Threshold and the Upsilon Expansion

A. H. Hoang

TL;DR

The paper argues that heavy-quark masses defined in the 1S scheme, based on half the perturbative energy of the ground quarkonium state, provide a short-distance, low-scale mass with superior perturbative convergence for threshold and nonrelativistic QCD problems. It introduces the upsilon expansion to consistently apply the 1S mass to non-Coulombic observables, demonstrating improved stability in $tar{t}$ threshold production, $$ sum rules for $b$ quarks, and inclusive $B$ decays, and shows how to extract accurate $ar{ ext{MS}}$ masses from the 1S mass with three-loop relations. The results yield precise determinations: $M_b^{1S}$ around 4.71–4.73 GeV and $ar{m}_b(ar{m}_b)$ near 4.21 GeV, with top-quark implications suggesting sub-GeV-level precision in future threshold scans. The approach provides a coherent framework linking low-scale quark masses to high-energy observables while highlighting the interplay with $ar{ ext{MS}}$ masses and $eta$-function inputs.

Abstract

Recent results from studies using half the perturbative mass of heavy quark-antiquark n=1, ${}^3S_1$ quarkonium as a new heavy quark mass definition for problems where the characteristic scale is smaller than or of the same order as the heavy quark mass are reviewed. In this new scheme, called the 1S mass scheme, the heavy quark mass can be determined very accurately, and many observables like inclusive B decays show nicely converging perturbative expansions. Updates on results using the 1S scheme due to new higher order calculations are presented.

Heavy Quark Masses from the $Q\bar Q$ Threshold and the Upsilon Expansion

TL;DR

The paper argues that heavy-quark masses defined in the 1S scheme, based on half the perturbative energy of the ground quarkonium state, provide a short-distance, low-scale mass with superior perturbative convergence for threshold and nonrelativistic QCD problems. It introduces the upsilon expansion to consistently apply the 1S mass to non-Coulombic observables, demonstrating improved stability in threshold production, sum rules for quarks, and inclusive decays, and shows how to extract accurate masses from the 1S mass with three-loop relations. The results yield precise determinations: around 4.71–4.73 GeV and near 4.21 GeV, with top-quark implications suggesting sub-GeV-level precision in future threshold scans. The approach provides a coherent framework linking low-scale quark masses to high-energy observables while highlighting the interplay with masses and -function inputs.

Abstract

Recent results from studies using half the perturbative mass of heavy quark-antiquark n=1, quarkonium as a new heavy quark mass definition for problems where the characteristic scale is smaller than or of the same order as the heavy quark mass are reviewed. In this new scheme, called the 1S mass scheme, the heavy quark mass can be determined very accurately, and many observables like inclusive B decays show nicely converging perturbative expansions. Updates on results using the 1S scheme due to new higher order calculations are presented.

Paper Structure

This paper contains 12 sections, 16 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. THE $1S$ MASS
  3. APPLICATION TO $Q\bar{Q}$ SYSTEMS -- $1S$ MASS DETERMINATION
  4. $t\bar{t}$ production at threshold
  5. $\Upsilon$ mesons
  6. $1S$ MASS AND NON-$Q\bar{Q}$ SYSTEMS
  7. The upsilon expansion
  8. Inclusive B decays
  9. The top width and $\Delta \rho$
  10. Determination of $\overline{\hbox{MS}}$ masses
  11. OTHER MASS DEFINITIONS
  12. ACKNOWLEDGEMENTS

Figures (2)

  • Figure 1: The total normalised vector-current-induced $t\bar{t}$ cross section at the LC versus the c.m. energy in the threshold regime at LO (dotted curves), NLO (dashed) and NNLO (solid) in the pole (upper figure), $\overline{\hbox{MS}}$ (middle) and $1S$ (lower) mass schemes for $\alpha_s(M_Z)=0.118$ and $\mu=15$, $30$, $60$ GeV. The plots have been generated from results obtained in Hoang1Hoang2. NNLO calculations for the cross section have also been carried out in Melnikov2Yakovlev1Beneke1Sumino1Penin1.
  • Figure 2: The dark regions show the allowed bottom mass values as a function of $\alpha_s$ using NNLO $\Upsilon$ sum rules in the pole (upper figure), $\overline{\hbox{MS}}$ (middle) and $1S$ (lower figure) mass schemes. The diagrams have been generated from results obtained in Hoang3Hoang4. Mass extractions for $\alpha_s(M_Z)=0.118\pm 0.004$ are indicated. Sum rule analyses at NNLO have also been carried out in Melnikov1Penin2Beneke2.