The Heavy Quarkonium Spectrum at Order $mα_s^5 \ln α_s$

TL;DR

The paper develops the NRQCD→pNRQCD effective-field-theory framework to study heavy quarkonia near threshold, exploiting the scale hierarchy ($m$, $mv$, $mv^2$) to systematically compute leading-log corrections. It provides explicit matching coefficients up to $O(1/m^2)$, derives the singlet/octal dynamics in pNRQCD, and obtains the leading-log corrections to the heavy quarkonium spectrum at $O(m α_s^5)$, including ultrasoft contributions and μ-dependence cancellation. The results establish the groundwork for a complete NNNLO analysis and offer a path to precision predictions for observables such as $t\bar t$ threshold production and Υ spectroscopy, while outlining necessary higher-loop inputs and nonperturbative considerations. This framework helps quantify theoretical uncertainties and guides future refinements toward full NNNLO accuracy.

Abstract

We compute the complete leading-log terms of the next-to-next-to-next-to-leading-order corrections to potential NRQCD. As a by-product we obtain the leading logs at $O(mα_s^5)$ in the heavy quarkonium spectrum. These leading logs, when $Λ_{QCD} \ll mα_s^2$, give the complete $O(mα_s^5 \ln α_s)$ corrections to the heavy quarkonium spectrum.

Figures (1)

• Figure 1: The UV divergences of this diagram in pNRQCD fix the $\mu$ dependence of $\alpha_{V_s}$, $D_s^{(1)}$, $D_s^{(2)}$, $Z_s^{(0)}$, $Z_s^{(1)}$ and $Z_s^{(2)}$.