Abstract
We compute the heavy quarkonium mass of $l\not= 0$ (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.
Highlights
The heavy quarkonium mass has been computed with increasing accuracy in the limit of very large mass over the years
The heavy quarkonium mass of the P-wave states has been computed in Ref. [1] to next-to-leading order (NLO), in Ref. [2] to NNLO, in Ref. [3] the ln αs term of the N3LO, in Refs. [4,5] with N3LO accuracy for the equal mass case and in Ref. [6] for the nonequal mass case
Once the spectrum has been obtained with N3LO accuracy, one can move to the step: the computation of the heavy quarkonium mass with N3LL accuracy by the resummation of the large logarithms
Summary
The heavy quarkonium mass has been computed with increasing accuracy in the limit of very large mass (i.e., in the strict weak-coupling approximation) over the years. Once the spectrum has been obtained with N3LO accuracy, one can move to the step: the computation of the heavy quarkonium mass with N3LL accuracy by the resummation of the large logarithms This is one of the main purposes of this paper, and we achieve this goal for arbitrary P-wave states. The applicability of a weak-coupling analysis to the first P-wave heavy quarkonium excited state (n 1⁄4 2, l 1⁄4 1) is an open issue Besides the aforementioned phenomenological analysis performed at strict weak coupling, we study the convergence of an alternative computational scheme that reorganizes the perturbative expansion of the weakcoupling computation This scheme is characterized by solving the Schrödinger equation including the static potential exactly (to the order it is known). 0, and for each individual potential: Vð1;0Þ, etc
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