Higher Order Corrections to the Cross Section of e+e- to Quarkonium plus gamma

The production cross sections of $e^+e^- $ to a heavy quarkonium with $C$-parity even $S-$wave and $P-$wave associated with a photon are analyzed in the framework of non-relativistic Quantum Chromodynamics(NRQCD) factorization formalism. The short-distance coefficients are determined up to the next-to-leading order(NLO) QCD radiative corrections for $S-$ and $P-$wave and NLO relativistic corrections for $S-$wave in analytical forms. Applying the results to estimate the cross sections we find that both the radiative and relativistic corrections are considerable.