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Heavy Quarkonium Spectrum at ${\cal O}(α_s^5m_q)$ and Bottom/Top Quark Mass Determination

A. A. Penin, M. Steinhauser

TL;DR

The study addresses precise determination of heavy-quark masses by exploiting perturbative heavy-quarkonium dynamics near threshold. It uses the nonrelativistic effective theory pNRQCD to compute the complete O(α_s^5 m_q) ground-state energy, including beta-function driven contributions and ultrasoft effects, and applies the result to extract the MSbar bottom mass from the Υ(1S) and to formulate a threshold-energy relation for the top quark. Key contributions include the explicit decomposition of the N^3LO energy correction, the handling of IR/UV cancellations, and the derivation of a universal E_res–m_t relation with a quantified perturbative uncertainty. The findings yield m_bbar(m_b) = 4.346 ± 0.070 GeV and a robust top-threshold formula, enabling future precision extractions of m_t from threshold scans with about 80 MeV theoretical precision. Overall, the work strengthens perturbative control over heavy-quark masses and provides practical formulas for phenomenology at high-energy colliders.

Abstract

We present the next-to-next-to-next-to-leading ${\cal O}(α_s^5m_q)$ result for the ground state energy of a heavy quarkonium system. On the basis of this result we determine the bottom quark mass from $Υ(1S)$ resonance and provide an explicit formula relating the top quark mass to the resonance energy in $t\bar t$ threshold production.

Heavy Quarkonium Spectrum at ${\cal O}(α_s^5m_q)$ and Bottom/Top Quark Mass Determination

TL;DR

The study addresses precise determination of heavy-quark masses by exploiting perturbative heavy-quarkonium dynamics near threshold. It uses the nonrelativistic effective theory pNRQCD to compute the complete O(α_s^5 m_q) ground-state energy, including beta-function driven contributions and ultrasoft effects, and applies the result to extract the MSbar bottom mass from the Υ(1S) and to formulate a threshold-energy relation for the top quark. Key contributions include the explicit decomposition of the N^3LO energy correction, the handling of IR/UV cancellations, and the derivation of a universal E_res–m_t relation with a quantified perturbative uncertainty. The findings yield m_bbar(m_b) = 4.346 ± 0.070 GeV and a robust top-threshold formula, enabling future precision extractions of m_t from threshold scans with about 80 MeV theoretical precision. Overall, the work strengthens perturbative control over heavy-quark masses and provides practical formulas for phenomenology at high-energy colliders.

Abstract

We present the next-to-next-to-next-to-leading result for the ground state energy of a heavy quarkonium system. On the basis of this result we determine the bottom quark mass from resonance and provide an explicit formula relating the top quark mass to the resonance energy in threshold production.

Paper Structure

This paper contains 4 sections, 19 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Perturbative heavy quarkonium ground state energy at ${\cal O}({\alpha_s^5m_q})$
  3. Bottom and top quark mass determination
  4. Conclusions

Figures (2)

  • Figure 1: (a) Bottom quark pole mass, $m_b$, as a function of the renormalization scale $\mu$, used for the numerical evaluation of $E_1$ where the short-dashed, long-dashed and solid line corresponds to the NLO, NNLO and N$^3$LO approximations. (b) $\overline{\rm MS}$ bottom quark mass $\overline{m}_b(\overline{m}_b)$, as a function of the renormalization scale $\mu$, which is used for the extraction of the pole mass $m_b$ (cf. (a)). For the conversion from the pole to the $\overline{\rm MS}$ mass the method described around Eq. (\ref{['mustar']}) is used. (c) like (b), however, the upsilon-expansion is used.
  • Figure 2: Ground state energy $E^{\rm p.t.}_1$ of the $t\bar{t}$ bound state in the zero-width approximation as a function of the renormalization scale $\mu$ for $S=0$ (a) and $S=1$ (b). The short-dashed, long-dashed and solid line corresponds to the NLO, NNLO and N$^3$LO approximations.