Power Counting in Dimensionally Regularized NRQCD

TL;DR

The paper addresses the challenge of manifest velocity power counting in NRQCD by introducing two distinct gluon fields to separately capture instantaneous potential interactions and on-shell radiation. Through a Yukawa-theory warmup and a detailed NRQCD current-matching calculation to O(g^2 v^2), it demonstrates that including both gluon sectors yields consistent infrared behavior and correct subleading matching coefficients. The work provides a concrete, gauge-invariant framework for systematic subleading corrections in heavy-quark dynamics near threshold, with clear implications for quarkonium production and sum rules. Overall, it establishes a robust method for deriving and applying velocity-based power counting in dimensionally regulated nonrelativistic effective theories.

Abstract

We present a scheme for calculating in NRQCD with consistent power counting in the heavy quark velocity v. As an example, we perform the systematic matching of an external current onto NRQCD at subleading order in v, a calculation relevant for the process e^+e^- -> hadrons near threshold. Consistent velocity power counting in dimensional regularization is achieved by including two distinct gluon fields, one corresponding to gluon radiation and one corresponding to an instantaneous potential. In this scheme power counting is manifest in any gauge, and also holds for non-gauge interactions. The matching conditions for an external vector current in NRQCD are calculated to O(g^2 v^2) and the cancellation of infrared divergences in the matching conditions is shown to require both gluon fields. Some subtleties arising in the matching conditions at subleading order are addressed.

Figures (5)

• Figure 1: Two of the one loop contributions to the matching of an external current in HQET.
• Figure 2: One loop diagrams in Yukawa theory.
• Figure 3: Infrared divergent one loop diagrams in NRY. The dashed line corresponds to the potential scalar while the solid line is the radiation scalar.
• Figure 4: One loop contributions to quark-antiquark production in QCD.
• Figure 5: One loop contributions to quark-antiquark production in NRQCD. The dashed line corresponds to a potential $A_0$ gluon, the dashed gluon line to a potential $A_i$ gluon. The shaded circles represent (b) the ${\bf p}\cdot {\bf A}$ vertex, (c) the Fermi vertex, (d) the Darwin vertex, (e) the relativistic kinematic correction to the fermion leg, and (f) ${\bf O}_2$. Implicit in both (d) and (e) are graphs with the same operator insertion on the antiquark line. The wavefunction graphs vanish.