Understand
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 $\Upsilon(nS)$, 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 NLL accuracy.
- ($v$ is the velocity of the heavy quark $Q$ or the heavy antiquark $\bar{Q}$ 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 $\Upsilon(2S)$ and $\Upsilon(3S)$ 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-Pad\'e method.