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The pairing symmetry of the newly proposed cobalt high temperature(high-$T_c$) superconductors formed by vertex shared cation-anion tetrahedralcomplexes is studied by the methods of mean field, random phase approximation(RPA) and functional renormalization group (FRG) analysis. The results of allthese methods show that the $d_{x^2-y^2}$ pairing symmetry is robustly favorednear half filling. The RPA and FRG methods, which are valid in weak interactionregions, predict that the superconducting state is also strongly orbitalselective, namely the $d_{x^2-y^2}$ orbital that has the largest density nearhalf filling among the three $t_{2g}$ orbitals dominates superconductingpairing. These results suggest that the new materials, if synthesized, canprovide indisputable test to high-$T_c$ pairing mechanism and the validity ofdifferent theoretical methods.

Robust d-wave pairing symmetry in multi-orbital cobalt high temperature superconductors