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The symmetry of the superconducting condensate in Sr$_2$RuO$_4$ remainscontroversial after nuclear magnetic resonance (NMR) experiments recentlyoverturned the dominant chiral $p$-wave paradigm. Several theoretical proposalshave been put forward to account for existing experiments, including a$d+ig$-wave admixture, conjectured to be stabilized by longer-range Coulombinteractions. We perform a material-specific microscopic theoretical study ofpairing by spin- and charge-fluctuations in Sr$_2$RuO$_4$, including theeffects of spin-orbit coupling, and both local and longer-range Coulombrepulsion. The latter has important consequences for Sr$_2$RuO$_4$ due to thenear-degeneracy of symmetry-distinct pairing states in this material. We findthat both the $g$- and $d_{x^2-y^2}$-wave channels remain noncompetitivecompared to leading nodal $s'$, $d_{xy}$, and helical ($p$) solutions. Thissuggests nodal time-reversal symmetry broken $s'+id_{xy}$ or $s'+ip$ phases,promoted by longer-range Coulomb repulsion, as the most favorable candidatesfor Sr$_2$RuO$_4$. We analyse the properties of these states, and show that the$s'+id_{xy}$ solution agrees with the bulk of available experimental data,including recent discoveries from NMR, muon spin relaxation ($\mu$SR), andultrasound measurements.

Superconducting state of Sr$_2$RuO$_4$ in the presence of longer-range Coulomb interactions