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Based on the NRQCD factorization formalism, we calculate thenext-to-next-to-leading order QCD corrections to the heavy quarkonium $\eta_c(\eta_b)$ production associated with a photon at electron-positron colliders.By matching the amplitudes calculated in full QCD theory to a series ofoperators in NRQCD, the short-distance coefficients up to NNLO QCD radiativecorrections are determined. It turns out that the full set of master integralsthat we obtained could be analytically expressed in terms of GoncharovPolylogarithms, Chen's iterated integrals, and elliptic functions, which mostlydo not exist in the literature and could be employed in the analyses of otherphysical processes. In phenomenology, numerical calculations of NNLO K-factorsand cross sections of $e^+e^-\rightarrow \gamma + \eta_c(\eta_b)$ processes inBESIII and B-factory experiments are performed, which may stand as a test ofthe NRQCD higher order calculation while confronting to the data.

NNLO QCD Corrections to $\gamma + \eta_c(\eta_b)$ Exclusive Production in Electron-Positron Collision