Understanding the non-trivial isoscalar pseudoscalar structures in the KS KSπ⁰ spectra in the J/ψ radiative decay

Initiated by the recent observation of a flattened lineshape of IJPC=00⁻⁺ around 1.4~ 1.5 GeV in the KS KS π⁰ invariant mass spectrum by BESIII, we make a systematic partial wave analysis of J/ψ →γηX→ γ KK̄π based on an isobaric approach. We demonstrate that in the scenario of the first radial excitations of the isoscalar pseudoscalar from the KK̄π threshold to about 1.6 GeV the non-trivial KS KS π⁰ invariant mass spectrum can be explained by the coupled-channel effects with the presence of the triangle singularity mechanism. It shows that a combined fit of the three-body and two-body spectra can be achieved which suggests that the one-state solution around 1.4~ 1.5 GeV proposed before still holds well. In particular, we show that the coupled-channel effects between the two most important quasi-two-body decay channels, K^*K̄+c.c. and a₀(980)π, can be well described by taking into account the one-loop corrections in the isobaric approach. This is because the isoscalar pseudoscalar states are coupled to the K^*K̄+c.c. and a₀(980)π channels via the P and S waves, respectively. As a consequence, the coupled-channel effects can be largely absorbed into the redefinition of the tree-level effective couplings with the transition amplitudes computed to the order of one-loop corrections. Then, the coupled-channel effects can be estimated by the contributions from the one-loop rescattering amplitudes in comparison with the tree-level ones, where we find that the rescattering contributions from the P-wave into the S-wave, or vice verse, are apparently suppressed in the kinematic region near threshold.