Abstract
We obtain the potential nonrelativistic quantum chromodynamics Lagrangian relevant for $S$-wave states with next-to-next-to-next-to-leading logarithmic accuracy. We compute the heavy quarkonium mass of spin-averaged $l=0$ (angular momentum) states, with otherwise arbitrary quantum numbers, with next-to-next-to-next-to-leading logarithmic accuracy. These results are complete up to a missing contribution of the two-loop soft running.
Highlights
High order perturbative computations in heavy quarkonium require the use of effective field theories (EFTs), as they efficiently deal with the different scales of the system
By incorporating the heavy quark effective theory (HQET) Wilson coefficients with LL accuracy2 in the α=m4 and α2=m3 spin-independent potentials, the divergent structure of their expectation value determines the piece associated to these potentials of the renormalization group (RG) equation of the spin-independent delta potential with N3LL precision
Combined with the previous results we solve this equation and obtain the complete NLL running of the delta potential
Summary
High order perturbative computations in heavy quarkonium require the use of effective field theories (EFTs), as they efficiently deal with the different scales of the system. Since some expectation values are divergent, some of these energy shifts are logarithmic enhanced, i.e., of order Oðmα lnðmναÞÞ Such corrections contribute to the heavy quarkonium mass at N3LL. By incorporating the heavy quark effective theory (HQET) Wilson coefficients with LL accuracy in the α=m4 and α2=m3 spin-independent potentials, the divergent structure of their expectation value (tantamount to computing potential loops) determines the piece associated to these potentials of the renormalization group (RG) equation of the spin-independent delta potential with N3LL precision. The reason is the generation of singular potentials through divergent ultrasoft loops Note that with the precision achieved in this paper we need in some cases the two-loop running of the coupling when solving the RG equations
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