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Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD

K. G. Chetyrkin, J. H. Kühn, C. Sturm

TL;DR

The paper computes four-loop (alpha_s^3) corrections to the first two Taylor coefficients of the heavy-quark vacuum polarization function, focusing on non-singlet diagrams. It uses integration-by-parts and Laporta's algorithm to reduce thousands of integrals to 13 master integrals, with a mix of analytic and high-precision numerical evaluations. The authors provide explicit expressions for the coefficients C_0^{(3)} and C_1^{(3)} and demonstrate their impact on the charm- and bottom-quark masses via QCD sum-rule moments, reporting m_c(3 GeV) ≈ 1.023 GeV and m_b(10 GeV) ≈ 3.665 GeV with substantially reduced theoretical uncertainties. An independent confirmation and note on reduced scale dependence corroborate the robustness of these four-loop results for precise quark-mass determinations.

Abstract

New results at four-loop order in perturbative QCD for the first two Taylor coefficients of the heavy quark vacuum polarization function are presented. They can be used to perform a precise determination of the charm- and bottom-quark mass. Implications for the value of the quark masses are briefly discussed.

Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD

TL;DR

The paper computes four-loop (alpha_s^3) corrections to the first two Taylor coefficients of the heavy-quark vacuum polarization function, focusing on non-singlet diagrams. It uses integration-by-parts and Laporta's algorithm to reduce thousands of integrals to 13 master integrals, with a mix of analytic and high-precision numerical evaluations. The authors provide explicit expressions for the coefficients C_0^{(3)} and C_1^{(3)} and demonstrate their impact on the charm- and bottom-quark masses via QCD sum-rule moments, reporting m_c(3 GeV) ≈ 1.023 GeV and m_b(10 GeV) ≈ 3.665 GeV with substantially reduced theoretical uncertainties. An independent confirmation and note on reduced scale dependence corroborate the robustness of these four-loop results for precise quark-mass determinations.

Abstract

New results at four-loop order in perturbative QCD for the first two Taylor coefficients of the heavy quark vacuum polarization function are presented. They can be used to perform a precise determination of the charm- and bottom-quark mass. Implications for the value of the quark masses are briefly discussed.

Paper Structure

This paper contains 4 sections, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Generalities
  3. Calculations and Results
  4. Summary and Conclusion

Figures (2)

  • Figure 1: Factorized or analytically known master integrals. The solid (dashed) lines denote massive (massless) propagators.
  • Figure 2: Master integrals where only a few terms of their $\varepsilon$-expansion are known analytically. The solid (dashed) lines denote massive (massless) propagators.