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 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 IR divergences at one-loop level, and the transition form factors with the relativistic corrections from the internal momentum of $h_{c}$ 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. Intriguingly, unlike the situation in the $S$-wave charmonium decays $J/\psi\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-}$, we find the contributions from the small recoil momentum region are comparable with those from the large recoil momentum region in the $P$-wave charmonium decays $h_{c}\rightarrow\eta^{(\prime)}\ell^{+}\ell^{-}$. 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.