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

We complete the leading-log renormalization group scaling of the nonrelativistic (NRQCD) Lagrangian at ${O(1/m}^2).\mathrm{}$ 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 $m{\alpha }_s\gg {\Lambda }_{\mathrm{QCD}}.$ As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation $m{\alpha }_s^2\gtrsim {\Lambda }_{\mathrm{QCD}}.$ When ${\Lambda }_{\mathrm{QCD}}\ll m{\alpha }_s^2,$ this is equivalent to obtain the whole set of $O\left(m{\alpha }_s^{\mathrm{}(n+4\mathrm{})\mathrm{}}{\ln }^n{\alpha }_s\right)$ terms in the heavy quarkonium spectrum. The implications of our results in the nonperturbative situation $m{\alpha }_s\sim {\Lambda }_{\mathrm{QCD}}$ are also mentioned.