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Three-loop QCD+QED corrections to on-shell quark renormalization

Long Chen, Hong-Yang Han, Zhe Li, Marco Niggetiedt

TL;DR

This work computes the complete three-loop mixed QCD+QED corrections to the on-shell heavy-quark mass and wave-function renormalization constants, enabling a precise pole-versus-$\overline{\mathrm{MS}}$ mass conversion and the extraction of the quark-mass anomalous dimension in the presence of both interactions. It provides analytic expressions for $Z_{m}^{\text{os}}$, $Z_{\psi}^{\text{os}}$, and the $\overline{\mathrm{MS}}$–on-shell mass relation, including all pure QCD, pure QED, and mixed contributions with detailed color/flavor decompositions up to $\mathcal{O}(\alpha_s^m\alpha^n)$ where $m+n=3$. Additionally, the paper extends the trace-anomaly-subtracted $\sigma$-mass formalism to QCD+QED at three loops, delivering the $m_{\sigma}$–$m_{\mathrm{os}}$ and $m_{\sigma}$–$\overline{m}$ conversions. The results, obtained through a combination of advanced diagram generation, IBP reduction, master-integral evaluation, and cross-checks, underpin higher-precision predictions for heavy-quark processes at current and future colliders by reducing theoretical uncertainties in mass definitions and renormalization.”

Abstract

We present the three-loop QCD+QED mixed corrections to the on-shell heavy-quark mass and wave-function renormalization constants through orders $\mathcal{O}(α_s^mα^n)$ with $m+n=3$. From these results, we derive the three-loop relation between the pole mass and the $\overline{\text{MS}}$ mass, including the complete mixed QCD+QED contributions. The corresponding quark-mass anomalous dimension in the presence of both interactions is also extracted. Furthermore, we provide explicit conversion formulae up to the same perturbative order between the pole mass and the trace-anomaly subtracted $σ$-mass of heavy quark.

Three-loop QCD+QED corrections to on-shell quark renormalization

TL;DR

This work computes the complete three-loop mixed QCD+QED corrections to the on-shell heavy-quark mass and wave-function renormalization constants, enabling a precise pole-versus- mass conversion and the extraction of the quark-mass anomalous dimension in the presence of both interactions. It provides analytic expressions for , , and the –on-shell mass relation, including all pure QCD, pure QED, and mixed contributions with detailed color/flavor decompositions up to where . Additionally, the paper extends the trace-anomaly-subtracted -mass formalism to QCD+QED at three loops, delivering the – and – conversions. The results, obtained through a combination of advanced diagram generation, IBP reduction, master-integral evaluation, and cross-checks, underpin higher-precision predictions for heavy-quark processes at current and future colliders by reducing theoretical uncertainties in mass definitions and renormalization.”

Abstract

We present the three-loop QCD+QED mixed corrections to the on-shell heavy-quark mass and wave-function renormalization constants through orders with . From these results, we derive the three-loop relation between the pole mass and the mass, including the complete mixed QCD+QED contributions. The corresponding quark-mass anomalous dimension in the presence of both interactions is also extracted. Furthermore, we provide explicit conversion formulae up to the same perturbative order between the pole mass and the trace-anomaly subtracted -mass of heavy quark.
Paper Structure (14 sections, 61 equations, 2 figures)

This paper contains 14 sections, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and technicalities
  3. On-shell quark renormalization
  4. Computational setup
  5. Results up to three loops in mixed QCD+QED
  6. On-shell quark mass renormalization constant $Z_{m}^{\text{os}}$
  7. On-shell quark wave-function renormalization constant $Z_{\psi}^{\text{os}}$
  8. Mass relation between the $\overline{\mathrm{MS}}$ and the on-shell masses
  9. $\sigma$-mass for heavy quark in mixed QCD+QED
  10. Conclusion
  11. Explicit expressions of $\alpha_s$ and $\alpha_e$ up to two-loop order
  12. Explicit expressions of $Z_{m}^{\text{os}}$ at three-loop order
  13. Explicit expressions of $Z_{\psi}^{\text{os}}$ at three-loop order
  14. Explicit expressions of $Z_{\overline{m}}$ up to three-loop orders

Figures (2)

  • Figure 1: Representative Feynman diagrams contributing to the mixed QCD+QED corrections to the heavy quark self-energy up to two-loop order. The first column displays the diagrams at one-loop order, while the remaining columns show illustrative examples of the two-loop contribution.
  • Figure 2: Representative Feynman diagrams for the mixed QCD+QED corrections to the heavy quark self-energy at three-loop order.