Renormalization group improvement of the NRQCD Lagrangian and heavy quarkonium spectrum

TL;DR

The paper develops a renormalization-group framework for heavy-quark bound states by performing LL RG improvement of the NRQCD Lagrangian at $O(1/m^2)$ and NNLL RG improvement of the singlet sector of pNRQCD when $m\alpha_s \gg \Lambda_{\rm QCD}$. It derives the complete LL running of four-heavy-fermion operators in NRQCD, then propagates to the pNRQCD potentials, providing explicit NNLL running for singlet potentials and related coefficients via a set of RG equations with initial conditions at the matching scales. Using these results, the authors obtain the heavy-quarkonium spectrum with the same accuracy, including both potential and ultrasoft contributions, and discuss how nonperturbative effects enter in regimes $\Lambda_{\rm QCD}\sim m\alpha_s^2$ or $\Lambda_{\rm QCD}\ll m\alpha_s^2$. The work clarifies the perturbative structure of heavy quarkonia, compares with previous formalisms (e.g., vNRQCD), and sets the stage for lattice and phenomenological applications by expressing potentials in terms of Wilson loops and NRQCD matching coefficients.

Abstract

We complete the leading-log renormalization group scaling of the NRQCD Lagrangian at $O(1/m^2)$. The next-to-next-to-leading-log renormalization group scaling of the potential NRQCD Lagrangian (as far as the singlet is concerned) is also obtained in the situation $mα_s \gg Λ_{QCD}$. As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation $mα_s^2 \simg Λ_{QCD}$. When $Λ_{QCD} \ll mα_s^2$, this is equivalent to obtain the whole set of $O(mα_s^{(n+4)} \ln^n α_s)$ terms in the heavy quarkonium spectrum. The implications of our results in the non-perturbative situation $mα_s \sim Λ_{QCD}$ are also mentioned.