Renormalization group improvement of the NRQCD Lagrangian and heavy quarkonium spectrum

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.