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 \(cs\bar{s}\bar{s}\rightarrow(c\bar{s})(s\bar{s})\), with the \(D_s^{(*)}\eta\), \(D_s^{(*)}\eta'\), and \(D_s^{(*)}\phi\) 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.