Inter-orbital p- and d-wave pairings between $d_{xz\mathrm{/}yz}$ and $d_{xy}$ orbitals in Sr${}_2$RuO${}_4$

We study the pairing symmetry of Sr${}_2$RuO${}_4$ through the group-theoretical approach. We emphasize the role of pairing interaction between the quasi-one-dimensional(Q1D) $d_{xz\mathrm{/}yz}$ and quasi-two-dimensional(Q2D) $d_{xy}$ orbitals. It is found that two degenerate inter-orbital time-reversal-invariant(TRI) p-wave pairings, one is spin-singlet and the other spin-triplet with out-of-plane $\mathit{d}$-vector, could be the most promising candidates. Several important physical quantities are presented, including the near-nodal gap structure, the unchanged out-of-plane Knight shift, and no split transition under strain, which are consistent with the experiments. In addition, these p-wave pairings shed light on resolving the contradiction between the time-reversal breaking and reduced in-plane Knight shift measurements. As the system reaches the Van Hove singularity under applied strain, the pairing symmetry would become a d-wave pairing mainly consisting of inter-orbital components, which could be responsible for the strained 3$K$ phase.