Resummation of relativistic corrections to heavy quarkonium $+\gamma$ production at $Z$ factory

Cross sections of the heavy quarkonium production process $e^{+}{e}^{-}\to H+\gamma$ at the c.m. energy of $Z^0$-boson resonance are calculated at leading order in ${\alpha }_s$ with relativistic corrections resummed within the framework of nonrelativistic QCD (NRQCD). $S$- and $P$-wave charmonia and bottomonia are considered. For $J\mathrm{/}\psi (\mathrm{{\rm Y}})$ production, the result of vector-meson-dominance (VMD) approach is also given. Relativistic contributions to charmonium production are of the same magnitude as next-to-leading order QCD corrections, decreasing cross section 20-30%. Relativistic corrections to bottomonium production are negligible, not exceeding 2%.