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 $m{\alpha }_s\gg {\Lambda }_{QCD}$. As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation $m{\alpha }_s^2\text{simg}{\Lambda }_{QCD}$. When ${\Lambda }_{QCD}\ll m{\alpha }_s^2$, this is equivalent to obtain the whole set of $O(m{\alpha }_s^{(n+4)}{\ln }^n{\alpha }_s)$ terms in the heavy quarkonium spectrum. The implications of our results in the non-perturbative situation $m{\alpha }_s\sim {\Lambda }_{QCD}$ are also mentioned.

Comments: 16 pages, LaTeX. Minor changes. Final version