Renormalization group improvement of the NRQCD Lagrangian and heavy quarkonium spectrum

We complete the leading-log renormalization group scaling of the NRQCD Lagrangian at $O(1\mathrm{/}{m}^2)$. 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 $m{\alpha }_s\gg {\mathrm{\Lambda }}_{QCD}$. As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation $m{\alpha }_s^2\text{simg}{\mathrm{\Lambda }}_{QCD}$. When $\Lambda_{QCD} \ll m\alpha_s^2$,thisisequivalenttoobtainthewholesetof$O(m\alpha_s^{(n+4)} \ln^n \alpha_s)$ 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.