Detecting the pure triangle sigularity effect through the $psi(2S) to p bar{p} eta / p bar{p} pi^0$ process

In this work, the triangle singularity mechanism is investigated in the $psi(2S) to p bar{p} eta / p bar{p} pi^0$ process. The triangle loop composed by $J/psi$, $eta$ and $p$ has a singularity in the physical kinematic range for the $psi(2S) to p bar{p} eta / p bar{p} pi^0$ process, and it would generate a very narrow peak in the invariant mass spectrum of $peta (pi)$ around $1.56387$ GeV, which is far away from both the threshold and relative resonances. In these processes, all the involved vertices are constrained by the experimental data. Thus, we can make a precise model independent prediction here. It turns out that the peak in the $peta$ invariant mass spectrum is visible, while it is very small in the $ppi^0$ invariant mass spectrum. We expect this effect shown in $p bar{p} eta$ final state can be observed by the Beijing Spectrometer (BESIII) and Super Tau-Charm Facility (STCF) in the future.