Understanding the non-trivial isoscalar pseudoscalar structures in the $K_S K_S\pi^0$ spectra in the $J/\psi$ radiative decay

Abstract

Initiated by the recent observation of a flattened lineshape of $IJ^{PC}=00^{-+}$ around $1.4\sim 1.5$ GeV in the $K_S K_S \pi^0$ invariant mass spectrum by BESIII, we make a systematic partial wave analysis of $J/\psi \to\gamma\eta_X\to \gamma K\bar{K}\pi$ based on an isobaric approach. We demonstrate that in the scenario of the first radial excitations of the isoscalar pseudoscalar from the $K\bar{K}\pi$ threshold to about 1.6 GeV the non-trivial $K_S K_S \pi^0$ 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 Dalitz plots, three-body and two-body spectra can be achieved which suggests that the one-state solution around $1.4\sim 1.5$ GeV proposed before still holds well. In particular, we show that the coupled-channel effects between the three important quasi-two-body decay channels, $K^*\bar{K}+c.c.$, $\kappa \bar{K}+c.c.$ and $a_0(980)\pi$, 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 $K^*\bar{K}+c.c.$ and $a_0(980)\pi$ ($\kappa \bar{K}+c.c.$) 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.