Renormalization group improvement of the nonrelativistic QCD Lagrangian and heavy quarkonium spectrum

Abstract

We complete the leading-log renormalization group scaling of the nonrelativistic (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 $m{\ensuremath{\alpha}}_{s}\ensuremath{\gg}{\ensuremath{\Lambda}}_{\mathrm{QCD}}.$ As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation $m{\ensuremath{\alpha}}_{s}^{2}\ensuremath{\gtrsim}{\ensuremath{\Lambda}}_{\mathrm{QCD}}.$ When ${\ensuremath{\Lambda}}_{\mathrm{QCD}}\ensuremath{\ll}m{\ensuremath{\alpha}}_{s}^{2},$ this is equivalent to obtain the whole set of $O(m{\ensuremath{\alpha}}_{s}^{(n+4)}{\mathrm{ln}}^{n}{\ensuremath{\alpha}}_{s})$ terms in the heavy quarkonium spectrum. The implications of our results in the nonperturbative situation $m{\ensuremath{\alpha}}_{s}\ensuremath{\sim}{\ensuremath{\Lambda}}_{\mathrm{QCD}}$ are also mentioned.