Order-v^4 Corrections to S-wave Quarkonium Decay

TL;DR

This work advances NRQCD-based calculations of heavy-quarkonium decay by deriving relativistic corrections up to order $v^4$ for all relevant S-wave channels. It provides explicit short-distance coefficients (F, G, H) for decays ${}^1S_0 o LH$, ${}^1S_0 o wo ext{photons}$, ${}^3S_1 o e^+e^-$, and ${}^3S_1 o LH$, including the treatment of infrared divergences in the ${}^3S_1 o LH$ three-gluon channel and the necessary operator matching to color-octet matrix elements. The results confirm known $v^2$ contributions, reveal channel-dependent convergence of the $v$-expansion (good for some decays, problematic for ${}^3S_1$-LH at $v^2$ but improved at $v^4$), and provide new $v^4$-level combinations such as $H^1+H^2$ across several channels. The analysis emphasizes the bound of unknown operator matrix elements and suggests that spin symmetry and lattice inputs will be essential to achieving precise phenomenology for quarkonium decays.

Abstract

We compute corrections of relative order v^4 to the rates for the decays of ^1S_0 heavy quarkonium into two photons and into light hadrons and for the decays of ^3S_1 heavy quarkonium into a lepton pair and into light hadrons. In particular, we compute the coefficients of the decay operators that have the same quantum numbers as the heavy quarkonium. We also confirm previous calculations of the order-v^2 corrections to these rates. We find that the v expansion converges well for the decays of ^1S_0 heavy quarkonium and for the decay of ^3S_1 heavy quarkonium into a lepton pair. Large higher-order-in-v corrections appear in the decay of ^3S_1 heavy quarkonium into light hadrons. However, we find that the series of coefficients of operators with ^3S_1 quantum numbers, which yields a large correction in order v^2, yields a smaller correction in order v^4.