LHC Run-3, Dark Matter and Supersymmetric Spectra in the Supersymmetric Pati-Salam Model

Driven by the growing agreement between the experimentally measured muon anomalous magnetic moment and its SM prediction, we reexamine phenomenological consequences of the MSSM, which is embedded in the supersymmetric $SU(4)_C \times SU(2)_L \times SU(2)_R$ Pati-Salam model. In contrast to earlier studies that predominantly favored a specific sign for the Higgsino mass parameter, our analysis systematically explores both $μ> 0$, and $μ< 0$ scenarios in light of current collider, cosmological, and DM constraints. Within this framework, we identify viable parameter space regions where the observed DM relic density is reproduced through multiple mechanisms: co-annihilations involving sbottom-neutralino, gluino-neutralino, stop-neutralino, stau-neutralino, and chargino-neutralino coannihilation, as well as resonant s-annihilation channel via the pseudoscalar Higgs boson. We demonstrate that all such scenarios are consistent with present bounds from LHC supersymmetry searches, the Planck~2018 DM relic density bound, and current limits from DD DM searches. Our results reveal characteristic mass spectra associated with these mechanisms. In particular, sbottom-neutralino coannihilation typically requires sbottom masses near $2.8~\text{TeV}$, while gluino-neutralino and stop-neutralino coannihilation scenarios allow gluino masses in the range $1$--$3~\text{TeV}$ and stop masses between $1$ and $3.5~\text{TeV}$. In coannihilation-dominated regions, the stau and chargino masses may reach values as high as $3.8~\text{TeV}$, whereas viable $A$ resonance solutions are realized for pseudoscalar Higgs masses spanning approximately $1.6$--$3.8~\text{TeV}$. We anticipate that a portion of the parameter space will be accessible to supersymmetry searches in LHC Run-3 and future runs.