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Matching at one loop for the four-quark operators in NRQCD

A. Pineda, J. Soto

TL;DR

The paper performs a one-loop matching of NRQCD/NRQED four-quark operators at O(1/m^2) for both unequal and equal heavy-quark masses using dimensional regularization to regulate UV and IR divergences, carefully treating Coulomb infrared behavior. It derives explicit expressions for the Wilson coefficients d_ss, d_sv, d_vs, d_vv in both mass scenarios, and discusses how the strong coupling α_s should be evaluated at the heavy-quark mass scale for the Wilson coefficients versus the scale for gluon fields. The results clarify flavor dependence and scale-setting in matching, and include consistency checks with known spin-dependent results and the electromagnetic current. These coefficients enable improved perturbative predictions for heavy-quarkonium spectra and NRQED bound states and have potential impact on future high-precision calculations such as O(m α_s^5) in heavy-quark systems and precision recoil studies in atomic systems.

Abstract

The matching coefficients for the four-quark operators in NRQCD (NRQED) are calculated at one loop using dimensional regularization for ultraviolet and infrared divergences. The matching for the electromagnetic current follows easily from our results. Both the unequal and equal mass cases are considered. The role played by the Coulomb infrared singularities is explained in detail.

Matching at one loop for the four-quark operators in NRQCD

TL;DR

The paper performs a one-loop matching of NRQCD/NRQED four-quark operators at O(1/m^2) for both unequal and equal heavy-quark masses using dimensional regularization to regulate UV and IR divergences, carefully treating Coulomb infrared behavior. It derives explicit expressions for the Wilson coefficients d_ss, d_sv, d_vs, d_vv in both mass scenarios, and discusses how the strong coupling α_s should be evaluated at the heavy-quark mass scale for the Wilson coefficients versus the scale for gluon fields. The results clarify flavor dependence and scale-setting in matching, and include consistency checks with known spin-dependent results and the electromagnetic current. These coefficients enable improved perturbative predictions for heavy-quarkonium spectra and NRQED bound states and have potential impact on future high-precision calculations such as O(m α_s^5) in heavy-quark systems and precision recoil studies in atomic systems.

Abstract

The matching coefficients for the four-quark operators in NRQCD (NRQED) are calculated at one loop using dimensional regularization for ultraviolet and infrared divergences. The matching for the electromagnetic current follows easily from our results. Both the unequal and equal mass cases are considered. The role played by the Coulomb infrared singularities is explained in detail.

Paper Structure

This paper contains 7 sections, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting the matching
  3. Unequal mass case
  4. Equal mass case
  5. Discussion
  6. Conclusions
  7. Coulomb singularity

Figures (2)

  • Figure 1: We show the relevant diagrams for the matching of the four-fermion operators at order $O(1/m^2)$ and one loop for the unequal mass case. The incoming and outcoming particles are on-shell and exactly at rest.
  • Figure 2: We show the relevant diagrams to the matching for the four-fermion operators at order $O(1/m^2)$ and one loop that only appear for the equal mass case. The incoming and outcoming particles are on-shell and exactly at rest.