Next-to-leading-order QCD corrections to $S$- and $P$-wave heavy quarkonium decay to $l^{+}l^{-} \gamma$
In this work, we comprehensively study the total and differential decay widths of the radiative Dalitz decays $H \to l^+l^-\gamma$ up to QCD next-to-leading order (NLO) accuracy within the framework of NRQCD factorization. Our calculation includes the decays of $S$-wave states ($\eta_c, \eta_b$) and the $P$-wave triplets ($\chi_{cJ}, \chi_{bJ}$ for $J=0,1,2$) to both electron ($l=e$) and muon ($l=\mu$) final states. To match realistic experimental detection thresholds, systematic kinematic cuts are implemented on the final-state photon energy. Our analysis of the lepton-pair invariant mass distribution shows distinct singular behaviors across the multiplets originating from the lepton-pair threshold region, which is regularized by the lepton mass, and the soft-photon region, respectively. For the $S$-wave states, there is only one peak near the lepton-pair threshold region, while for the $J=0$ and $J=2$ $P$-wave states, the peaks show up in both regions. However, for the $J=1$ $P$-wave states, the peak only appears in the soft-photon region. Such features provide a rich venue to probe the $\gamma^{\ast}\to l^{+}l^{-}$ form factor in heavy quarkonium decay. Integrating over the bounded phase spaces reveals a distinct hierarchy among the $\chi_{QJ}$ states in the sensitivity of our theoretical predictions to the soft-photon energy cuts, ordered as $\chi_{Q1} >\chi_{Q2}>\chi_{Q0}$. In the $\chi_{cJ}\to l^{+}l^{-}\gamma$ cases, as the energy cut increases from 100 $\mathrm{MeV}$, to 500 $\mathrm{MeV}$ the theoretical predictions at QCD NLO are reduced by $6\%(\chi_{c0})$,$66\%(\chi_{c1})$, and $21\%(\chi_{c2})$ for $l=e$, and by $17\%(\chi_{c0})$,$66\%(\chi_{c1})$, and $39\%(\chi_{c2})$ for $l=\mu$. Comparing our predictions with the upcoming high-precision experimental tests at BESIII will definitely deepen our understanding of the predictive power of perturbative calculations.