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After the recent historical discovery of gravitational wave, it is curious tospeculate upon the detection prospect of the quantum graviton in theterrestrial accelerator-based experiment. We carefully investigate the "golden"channels, $J/\psi(\Upsilon)\to\gamma+\text{graviton}$, which can be pursued at\textsf{BESIII} and \textsf{Belle 2} experiments, by searching forsingle-photon plus missing energy events. Within the effective field theory(EFT) framework of General Relativity (GR) together with Nonrelativistic QCD(NRQCD), we are capable of making solid predictions for the corresponding decayrates. It is found that these extremely suppressed decays are completelyswamped by the Standard Model background events $J/\psi (\Upsilon)\to\gamma+\nu\bar{\nu}$. Meanwhile, we also study these rare decay processes inthe context of massive gravity, and find the respective decay rates in thelimit of vanishing graviton mass drastically differ from their counterparts inGR. Counterintuitive as the failure of smoothly recovering GR results may look,our finding is reminiscent of the van Dam-Veltman-Zakharov (vDVZ) discontinuitywidely known in classical gravity, which can be traced to the finitecontribution of the helicity-zero graviton in the massless limit. Nevertheless,at this stage we are not certain about the fate of the discontinuityencountered in this work, whether it is merely a pathology or not. If it couldbe endowed with some physical significance, the future observation of theserare decay channels, would, in principle, shed important light on the nature ofgravitation, whether the graviton is strictly massless, or bears a very smallbut nonzero mass.

Quarkonium decay into photon plus graviton: a golden channel to discriminate General Relativity from Massive Gravity?