O(alpha_s^5 m) quarkonium 1S spectrum in large-beta_0 approximation and renormalon cancellation

TL;DR

The work addresses precision predictions for the heavy-quarkonium $1S$ spectrum and shows that achieving $O(\alpha_S^4 m)$ accuracy requires (i) the $O(\alpha_S^4 m)$ MSbar-pole mass relation and (ii) the $O(\alpha_S^5 m)$ binding energy in the large-$\beta_0$ limit. The authors compute the latter analytically, obtaining the leading $c_3$ coefficient in the large-$\beta_0$ regime and demonstrating renormalon cancellation when the spectrum is expressed in terms of the MSbar mass. They carefully analyze the phenomenology for bottomonium and toponium, finding improved convergence when using the MSbar basis and highlighting the scale sensitivity and the anticipated size of remaining higher-order terms. The results indicate that the $1S$ spectrum can be predicted at genuine $O(\alpha_S^4 m)$ accuracy within this approximation, contingent on the four-loop mass relation, and they discuss potential nonperturbative uncertainties and next-to-leading renormalon effects. Overall, the study clarifies the role of renormalons in heavy-quark bound states and informs precise determinations of heavy-quark masses.

Abstract

Presently the quarkonium spectrum, written in terms of the quark MSbar-mass, is known at O(alpha_s^3 m) accuracy. We point out that in order to achieve O(alpha_s^4 m) accuracy it is sufficient to calculate further (I) the O(alpha_s^4 m) relation between the MSbar-mass and the pole-mass, and (II) the binding energy at O(alpha_s^5 m) in the large-beta_0 approximation. We calculate the latter correction analytically for the 1S-state and study its phenomenological implications.