$Z$ boson radiative decays to a $P$-wave quarkonium at NNLO and LL accuracy

TL;DR

This paper calculates the decay rates of Z bosons into P-wave quarkonium states with photons at high precision, including NNLO QCD corrections and leading logarithm resummation, revealing significant effects on certain channels.

Contribution

First comprehensive NNLO and LL resummation analysis of Z boson radiative decays into P-wave quarkonium within NRQCD, including validation of factorization at two-loop order.

Findings

Radiative corrections are significant for $ ext{chi}_{Q2}$ and $h_Q$ channels.

Leading logarithm resummation can alter decay width predictions by over 25%.

Effects of LL resummation are reduced at NNLO and NLO levels.

Abstract

In this work, we study the radiative decay of $Z$ boson to a $P$-wave quarkonium $H$ in association with a photon, where $H$ can be $\chi_{QJ}$, $h_Q$ with $Q=c,b$ and $J=0,1,2$. The helicity amplitudes and the unpolarized decay widths are evaluated up to QCD next-to-next-to-leading order (NNLO) within the framework of nonrelativistic QCD (NRQCD). For the first time, we check the NRQCD factorization for $h_Q$ exclusive production at two-loop order. The leading logarithms (LL) of $m_Z^2/m_Q^2$ in the leading-twist short-distance coefficients, which may potentially ruin the perturbative convergence, are resummed to all orders of $\alpha_s$ by employing the light-cone factorization. We find the radiative corrections are considerable for $\chi_{Q2}$ and $h_Q$ productions, while are moderate or even minor for other channels. We also notice that the LL resummation can change the leading-order predictions for decay widths by more than 25\% for $\chi_{c0,2}$ and $h_c$ productions, and by around 50\% for $\chi_{c1}$ production. However, effects of the LL resummation on the next-to-leading-order and NNLO predictions are notably mitigated. Some phenomenological explorations are also performed.