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)). 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 malpha(s)>Lambda(QCD). As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation malpha(s)(2)greater than or similar toLambda(QCD). When Lambda(QCD)<malpha(s)(2), this is equivalent to obtain the whole set of O(malpha(s)((n+4))ln(n)alpha(s)) terms in the heavy quarkonium spectrum. The implications of our results in the nonperturbative situation malpha(s)similar toLambda(QCD) are also mentioned.