Annihilation of S-wave quarkonia and the measurement of alpha_s

TL;DR

This paper analyzes relativistic corrections to S-wave quarkonium annihilation within NRQCD, showing that order $v^2$ corrections can be expressed in terms of the heavy-quark pole mass and the quarkonium mass, enabling predictions for the ratio of hadronic to radiative widths for $\eta_c$ and $\eta_b$. It demonstrates that color-octet contributions to hadronic decays of spin-triplet states are significant despite arising at $\mathcal{O}(v^4)$, and provides rough RG-based estimates to gauge their size. By applying these results to $\Upsilon$ decays, the authors extract $\alpha_s$ with a range $0.097-0.117$ at the $Z$-boson scale, limited mainly by the uncertainty in the $b$-quark pole mass and the crude octet estimates. The work emphasizes that improving the pole-mass determination and performing NNLO calculations are essential for a more precise quarkonium-based determination of $\alpha_s$, and it outlines how future measurements of quarkonium decays could sharpen these constraints.

Abstract

We analyze the relativistic corrections to annihilation rates of S-wave quarkonia within the framework of NRQCD. We show that order v^2 corrections can be expressed in terms of the heavy quark pole mass and the quarkonium mass. The ratio of hadronic to radiative annihilation rates for eta_b and eta_c can therefore be predicted accurately. The contributions of color-octet operators to the hadronic decay rates of spin-triplet quarkonia are shown to be significant, even though they arise at order v^4 in the velocity expansion. We provide a rough estimate of the color-octet contributions and extract the value of alpha_s from the experimental data on Upsilon decays.

Figures (1)

• Figure 1: Contributions of higher Fock states to the hadronic annihilation of spin-1 S-wave quarkonia. All diagrams shown here contribute to the rate at order $\alpha_s^2 v^4$.