P -wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy

Abstract

We compute the heavy quarkonium mass of l = 0 (angular momentum) states, with otherwise arbitrary quantum numbers, with next-next-to-next-to-leading logarithmic (N 3 LL) accuracy. This constitutes the first observable in heavy quarkonium for which two orders of the weak-coupling expansion sensitive to the ultrasoft scale are known and the resummation of ultrasoft logarithms is made. We also obtain, for the first time, resummed N 3 LL expressions for the different fine and hyperfine energy splittings of these states, which are not sensitive to the ultrasoft scale but still require resummation of (hard) logarithms. We do this analysis for the equal and non-equal mass cases. We also study an alternative computational scheme that treats the static potential exactly.We then perform a comprehensive phenomenological analysis: we apply these results to the n = 2, l = 1 bottomonium, B c and charmonium systems and study their convergence.

Electric dipole transitions of 1P bottomonia

“…In Refs. [20,23] the effect of taking ν f = 0.7 GeV has been considered. The impact on the bottomonium mass is at most 1%.…”

“…There seems to be a growing consensus in the literature that the weak-coupling regime mv 2 Λ QCD may indeed be applied to many physical observables in the bottomonium sector including n = 2 bottomonium states (for early work see [16][17][18], for reviews see [8,9,13], for recent work see [19,20]). In order to reach this conclusion, it is crucial, however, to have a proper treatment for the large terms appearing in the perturbative expansion.…”

“…The authors obtain agreement between theory and experiment for the case of the charmonium and bottomonium ground states and for the n = 2 excitations of the bottomonium. Very recently, the same scheme has been applied to the spectrum of n = 2, l = 1 quarkonium states in [20]. Hence, another motivation for the present study is to probe weakly coupled pNRQCD in the context of electric dipole transitions from the spin-triplet and spin-singlet lowest bottomonium P -wave states.…”

Matching coefficients in nonrelativistic QCD to two-loop accuracy

“…The superscript "c" in Eq. (16) denotes that the result is matched to the Lagrange density (7), whereas in the scattering channel we match our expressions to Eq. (6).…”

“…Note that for P -wave states the N 3 LL pNRQCD Lagrange density is complete and can be found in Ref. [16].…”