Renormalization group scaling in nonrelativistic QCD

TL;DR

The paper develops a nonrelativistic QCD (NRQCD) framework that remains compatible with mass-independent subtraction schemes by introducing a velocity-space renormalization group (VRG). By separating soft and ultrasoft modes and carefully counting powers in the velocity, the authors derive a consistent effective theory that reproduces the one-loop running of the static quark potential and the two-loop anomalous dimension of the heavy-quark production current. The VRG sums logarithms of the velocity by evolving a subtraction velocity nu from 1 to v, thereby simultaneously renormalizing soft and ultrasoft sectors. These results validate the NRQCD formulation and demonstrate its utility for threshold phenomena and quarkonium physics, with potential applicability to other nonrelativistic systems.

Abstract

We discuss the matching conditions and renormalization group evolution of non-relativistic QCD. A variant of the conventional MS-bar scheme is proposed in which a subtraction velocity nu is used rather than a subtraction scale mu. We derive a novel renormalization group equation in velocity space which can be used to sum logarithms of v in the effective theory. We apply our method to several examples. In particular we show that our formulation correctly reproduces the two-loop anomalous dimension of the heavy quark production current near threshold.

Figures (14)

• Figure 1: Fermion-antifermion propagation graph. The two $\otimes$ create and annihilate a fermion-antifermion pair.
• Figure 2: Plot of $\Lambda_{\rm QCD}/m\alpha_s^2(m v)$ as a function of $m/\Lambda_{\rm QCD}$, for $n_f=3$.
• Figure 3: Momentum space of size $mv$ is divided into boxes of size $mv^2$. A point in momentum space is labeled by $\mathbf p$ and $\mathbf k$.
• Figure 4: One loop vertex correction in the full theory and effective theory. The momenta are related by $P=p+k$, $P^\prime=p+k^\prime$.
• Figure 5: The Coulomb potential in the full theory, (a), is given by a local two-quark operator (b) in the effective theory.