$Z$ boson radiative decays to a $S$-wave quarkonium at NNLO and NLL accuracy

TL;DR

This paper calculates high-precision QCD corrections to Z boson radiative decays into S-wave quarkonium states, combining NNLO and NLL resummation techniques to improve theoretical predictions.

Contribution

It provides the first analytic expressions of NLL resummation at (^2) order for Z boson decays into quarkonium within the NRQCD framework.

Findings

() corrections are about 10% for _c and _ productions.

(^2) corrections are moderate for charmonium and very small for bottomonium.

NLL resummation significantly impacts the NRQCD predictions, especially for J// production.

Abstract

Within the framework of nonrelativistic QCD (NRQCD) factorization formalism, we compute QCD next-to-next-to-leading order (NNLO) corrections to the helicity amplitudes as well as the decay width of $Z\to H+\gamma$, where $H$ can be $\eta_Q (Q=c,b), J/\psi$, or $\Upsilon$. In addition, we resum the next-to-leading logarithms (NLL) of ${m_Z^2}/{m_Q^2}$ to all orders of $\alpha_s$ for the leading-twist helicity amplitude by employing the light-cone factorization approach. It is worth mentioning that we obtain the analytic expressions of the truncated NLL at $\alpha_s^2$. We find that the $\mathcal{O}(\alpha_s)$ corrections are around 10\% for $\eta_c$ and $\Upsilon$ productions, however insignificant for $J/\psi$ and $\eta_b$ productions. The $\mathcal{O}(\alpha_s^2)$ corrections are moderate for charmonium production, while very small for bottomonium production. Moreover, it is found that the NLL resummation can considerably alter the NRQCD prediction, especially for $J/\psi$ production. Combining the NRQCD and light-cone computation, we make phenomenological predictions on the decay widths and branching fractions. In addition, we investigate the dependence of the theoretical results on the heavy quark mass, and find the branching fraction of $Z\to H+\gamma$ monotonically decreases as $m_Q$ increases.