Renormalization group improvement of the nonrelativistic RQCD 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 m α s ≫ Λ QCD . As a by-product, we obtain the heavy quarkonium spectrum with the same accuracy in the situation m α 2 s ≳ Λ QCD . When Λ QCD ≪ m α 2 s , this is equivalent to obtain the whole set of O ( m α ( n + 4 ) s ln n α s ) terms in the heavy quarkonium spectrum. The implications of our results in the nonperturbative situation m α s ∼ Λ QCD are also mentioned