QCD analysis of the \(P\)-wave charmonium electromagnetic Dalitz decays \(h_{c}\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-}\)

Abstract

The \(P\)-wave charmonium electromagnetic Dalitz decays \(h_{c}\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-}\) \((\ell=e, \mu)\) with large recoil momentum are investigated in the framework of perturbative QCD, and the soft contributions from the small recoil momentum region are described by the overlap of soft wave functions. The transition form factors \(f_{h_{c}\eta^{(\prime)}}(q^{2})\) and the normalized transition form factors \(F_{h_{c} \eta^{(\prime)}}(q^{2})\) in full kinematic region are derived for the first time. It is noticed that there are no extra IR divergences at the one-loop level and the tree level, and the transition form factors in which the relativistic corrections from the internal momentum of \(h_{c}\) are taken into account are insensitive to both the shapes of \(\eta^{(\prime)}\) distribution amplitudes and the invariant mass of the lepton pair in the large recoil momentum region. Furthermore, we find that the contributions from the soft mechanism and those from hard mechanism are comparable with each other in the branching ratios \(\mathcal{B}(h_{c}\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-})\). By employing the obtained \(F_{h_{c} \eta^{(\prime)}}(q^{2})\), we give the predictions of the branching ratios \(\mathcal{B}(h_{c}\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-})\), which may come within the range of measurement of present or near-future experiments.