X(2370) glueball-like particle productions in $e^+e^-$ collisions at the BESIII energy and in pp collisions at the LHC energy with PACIAE model

Inspired by the BESIII newest observation of X(2370) glueball-like particle, we search its productions in both $e^+e^-$ collisions at $\sqrt{s}=$ 4.95 GeV and proton-proton (pp) collisions at $\sqrt{s}=$ 13 TeV with a parton and hadron cascade model PACIAE. In this model, the final partonic state (FPS) and the final hadronic state (FHS) are consecutively simulated and recorded. The X(2370) glueball- or tetraquark-state is then, respectively, recombined by two gluons or four quarks $ss\bar{s}\bar{s}$ in the FPS using the quantum statistical mechanics inspired dynamically constrained phase-space coalescence (DCPC) model. The X(2370) molecular-state is recombined by the baryon-antibaryon of $\Lambda$-$\bar{\Lambda}$ or $\Sigma$-$\bar{\Sigma}$, or by three mesons of $\pi^+\pi^{-}\eta'$, $K^+K^-\eta'$, or $K_S^0K_S^0\eta'$ in the FHS using DCPC model. In both $e^+e^-$ and pp collisions, significant discrepancies in the yields, the transverse momentum spectra and the rapidity distributions among the X(2370) glueball-, tetraquark-, and molecular-state are observed. These discrepancies are proposed as valuable criteria identifying the X(2370) different states from each other. Our results not only support the BESIII observation of glueball-like particle $\rm X(2370)$ production in $e^+e^-$ collisions, but also serve as a prediction for the $\rm X(2370)$ production in pp collisions. We strongly suggest the experimental measurement of the X(2370) glueball-like particle production in pp collisions at the LHC energies.