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

TL;DR

This paper calculates the heavy quarkonium spectrum for states with non-zero angular momentum at N$^3$LL accuracy, including resummation of ultrasoft logarithms, providing new precise theoretical predictions for various heavy quarkonium systems.

Contribution

It presents the first N$^3$LL resummed calculations of heavy quarkonium masses and fine/hyperfine splittings, including ultrasoft logarithm resummation and an alternative potential approach.

Findings

First N$^3$LL accurate spectrum for $l ot= 0$ states.

Resummed expressions for fine and hyperfine splittings.

Application to bottomonium, $B_c$, and charmonium systems.

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.