$Z$-boson decays to a vector quarkonium plus a photon

We compute the decay rates for the processes $Z\to V+\gamma$, where $Z$ is the $Z$-boson, $\gamma$ is the photon, and $V$ is one of the vector quarkonia $J/\psi$ or $\mathrm{\Upsilon }(nS)$, with $n=1$, 2, or 3. Our computations include corrections through relative orders ${\alpha }_s$ and $v^2$ and resummations of logarithms of $m_Z^2/m_Q^2$, to all orders in ${\alpha }_s$, at next-to-leading-logarithmic accuracy. ($v$ is the velocity of the heavy quark $Q$ or the heavy antiquark $\overline{Q}$ in the quarkonium rest frame, and $m_Z$ and $m_Q$ are the masses of $Z$ and $Q$, respectively.) Our calculations are the first to include both the order-${\alpha }_s$ correction to the light-cone distributions amplitude and the resummation of logarithms of $m_Z^2/m_Q^2$ and are the first calculations for the $\mathrm{\Upsilon }(2S)$ and $\mathrm{\Upsilon }(3S)$ final states. The resummations of logarithms of $m_Z^2/m_Q^2$ that are associated with the order-${\alpha }_s$ and order-$v^2$ corrections are carried out by making use of the Abel-Padé method. We confirm the analytic result for the order-$v^2$ correction that was presented in a previous publication, and we correct the relative sign of the direct and indirect amplitudes and some choices of scales in that publication. Our branching fractions for $Z\to J/\psi +\gamma$ and $Z\to \mathrm{\Upsilon }(1S)+\gamma$ differ by $2.0\sigma$ and $-4.0\sigma$, respectively, from the branching fractions that are given in the most recent publication on this topic (in units of the uncertainties that are given in that publication). However, we argue that the uncertainties in the rates are underestimated in that publication.