Study of $χ_{c1,2}$ decays to $PPP$ via light meson resonances

The three-body decays of charmonium states provide a valuable laboratory for probing non-perturbative QCD dynamics and investigating light meson resonances. Three-body decays $\chi _{c1,2} \to P_{1}P_{2}P_{3}$ are studied in this work ($P$ denotes the light pseudoscalar meson). Branching ratio analysis is performed for the $\chi _{c1,2} \to P_{1}P_{2}P_{3}$ decays incorporating light vector ($V$) meson resonance, light tensor ($T$) meson resonance and light scalar ($S$) meson resonance contributions. Based on the SU(3) flavor symmetry approach, we calculate the branching ratios under both the narrow width approximation and the finite width effect, and provide the predict results in the $\chi _{c1,2}\to (V/T)P_{3} \to P_{1}P_{2}P_{3}$ processes. Furthermore, based on the significant $a^{+}_{0}(980)$ resonance signal in $\chi _{c1} \to \pi^{+ }\pi^{- }\eta$ decay, the predict results for $\chi _{c1}\to SP_{3} \to P_{1}P_{2}P_{3}$ are also obtained. While the resonance signals observed in current experiments are mainly concentrated in the $\chi_{c1} \to \pi^+ \pi^- \eta$ decay, our results suggest potential resonance contributions in other channels. Especially in $\chi _{c1,2}\to KK\pi$ channels, the predicted results show the significant resonance contributions from $V$ mesons and $T$ mesons, which is worth further experimental verification and exploration.