Revisiting the pseudoscalar meson and glueball mixing and key issues in the search for pseudscalar glueball state

We revisit the mixing mechanism for pesudscalar mesons and glueball which is introduced by the axial vector anomaly. We demonstrate that the physical mass of the pseudoscalar glueball does not favor to be lower than 1.8 GeV if all the parameters are reasonably constrained. This conclusion, on the one hand, can accommodate the pseudoscalar glueball mass calculated by Lattice QCD, and on the other hand, is consistent with the high-statistics analyses at BESIII that all the available measurements do not support the presence of two closely overlapping pseudoscalar states in any exclusive channel. Such a result is in agreement with the recent claim that the slightly shifted peak positions for two possible states $\eta(1405)$ and $\eta(1475)$ observed in different channels are actually originated from one single state with the triangle singularity interferences. By resolving this long-standing paradox, one should pay more attention to higher mass region for the purpose of searching for the pseudoscalar glueball candidate.