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Spin-orbit coupling (SOC) plays a crucial role in determining the spinstructure of an odd parity psedospin-triplet Cooper pairing state. Here, wepresent a thorough study of how SOC lifts the degeneracy among different p-wavepseudospin-triplet pairing states in a widely used microscopic model for$\mathrm{Sr_2 Ru O_4}$, combining a Ginzburg-Landau (GL) free energy expansion,a symmetry analysis of the model, and numerical weak-coupling renormalizationgroup (RG) and random phase approximation (RPA) calculations. These analysesare then used to critically re-examine previous numerical results on thestability of chiral p-wave pairing. The symmetry analysis can serve as a guidefor future studies, especially numerical calculations, on the pairinginstability in $\mathrm{Sr_2 Ru O_4}$ and can be useful for studying othermulti-band spin-triplet superconductors where SOC plays an important role.

Spin-orbit coupling and spin-triplet pairing symmetry in $\mathrm{Sr_2 Ru O_4}$