Heavy neutral bosons and dark matter in the 3-3-1 model with axionlike particle

We consider heavy neutral bosons in the 3-3-1 model with axionlike particles (331ALP), including the Higgs boson and the $Z^{\mathrm{\prime }}$ boson which are outside the standard model (SM). Based on gluon-gluon fusion at the LHC, we investigate the signals of cross-sections in the parameter space region satisfying the current experimental limits of lepton flavor violating decay, including processes involving both charged leptons and Higgs boson, and provide predictions of $m_{h_2}\ge 600\mathrm{G}\mathrm{e}\mathrm{V}$. A new gauge boson, labeled as $Z^{\mathrm{\prime }}$, is predicted $m_{Z^{\mathrm{\prime }}}\ge 5.1\mathrm{T}\mathrm{e}\mathrm{V}$ based on the search for high-mass dilepton resonances at ATLAS and CMS. We consider the stability of odd-$Z_2$ particles, with $Z_2$ is assumed a residual symmetry after spontaneous symmetry breaking stages, to point out dark matter candidates in the model. Investigating the relic density of dark matter within experimentally permissible limits, we established a relationship between the mass of dark matter and the breaking scale of axion.