Compact $cs\bar{s}\bar{s}$ Tetraquarks in the Charm--Strange Sector: Mass Spectra, Rearrangement Decays and Regge Trajectories with $D_s$ Threshold Inputs

This work presents a spectroscopy-focused study of the compact open-charm, multi-strange tetraquark configuration $cs\bar{s}\bar{s}$, modeled as an axial diquark-antidiquark system $[cs][\bar{s}\bar{s}]$. The conventional $D_s$ meson spectrum is retained as a calibration sector for the model parameters and as a reference for the dominant two-meson thresholds; however, the primary emphasis is placed on the mass spectrum, threshold structure, rearrangement decay mechanisms, and Regge systematics of the $cs\bar{s}\bar{s}$ tetraquark. The spectrum is computed within a Cornell-potential framework using both semi-relativistic and non-relativistic treatments, with the $\bar{\mathbf{3}}-\mathbf{3}$ and $\mathbf{6}-\bar{\mathbf{6}}$ color configurations analyzed separately. Owing to the presence of two identical strange antiquarks, Pauli symmetry imposes restrictions on the $[\bar{s}\bar{s}]$ antidiquark, favoring axial-vector building blocks as the natural low-lying degrees of freedom for the $J^P=0^{+},1^{+},2^{+}$ tetraquark multiplet. The strong-decay sector is formulated in terms of the rearrangement topology $cs\bar{s}\bar{s}\to (c\bar{s})(s\bar{s})$, with the $D_s^{(\ast )}\eta$, $D_s^{(\ast )}{\eta }^{\mathrm{\prime }}$, and $D_s^{(\ast )}\phi$ modes identified through spin-color Fierz recoupling and normalized using two-point and three-point QCD-sum-rule amplitudes. Orbital and radial Regge trajectories are constructed to characterize the excitation patterns of the compact tetraquark and to benchmark them against the calibrated $D_s$ spectrum. This framework provides a dedicated phenomenological basis for identifying $cs\bar{s}\bar{s}$ candidates in hidden-strangeness open-charm final states.