Exploring the Singlino-dominated Thermal Neutralino Dark Matter in the $Z_3$ invariant NMSSM

We examine the parameter space of the Next to Minimal Supersymmetric Standard Model (NMSSM) with Singlino-dominated neutralino $\widetilde\chi_1^0$ as the lightest supersymmetric particle (LSP). Our study focuses on identifying the regions within this parameter space that produce a thermal relic abundance of $\widetilde\chi_1^0$ smaller than the observed cold dark matter relic density while remaining consistent with constraints from LEP measurements, low-energy experiments, Higgs measurements, LHC data, and dark matter direct detection experiments. We identify the dominant annihilation modes of the LSP neutralino across varying LSP mass ranges $\sim \mathcal{O}(1)-\mathcal{O}(10^{3})~$GeV. Furthermore, we conduct a benchmark study to assess the production rates of triple-boson final states emerging from direct electroweakino pair production at the LHC. Drawing insights from these findings, we perform a detailed collider analysis to explore the future potential of probing the triple-boson final states involving a light Higgs boson at the high-luminosity LHC (HL-LHC).