Relativistic corrections to heavy quarkonium $+\gamma$ production at super Z factory

Abstract

In the framework of nonrelativistic quantum chromodynamics, we study the $\mathcal{O}(v^2)$ relativistic corrections to heavy-quarkonium production associated with a photon at future electron-positron colliders operated in the Z-factory mode. S-, P-, and D-wave heavy quarkonia as well as the mixed state $\psi(3770)$ are considered. We comprehensively investigate tree-level $s$-channel processes for all states, together with $t$-channel contributions that apply only to ${}^3L_1$ states. It is found that the relativistic corrections substantially reduce the LO cross sections for charmonium states by roughly $20\%$-$70\%$, while the reduction is milder for bottomonium, at the level of $10\%$-$30\%$. Event yields for all studied channels are calculated, and sizable event yields are expected for $\eta_c$, $J/\psi$, $\psi(2S)$, $\Upsilon$, and $\psi(3770)$, especially at FCC-ee.