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We generalize the recently introduced TRILEX approach (TRiply IrreducibleLocal EXpansion) to superconducting phases. The method treats simultaneouslyMott and spin-fluctuation physics using an Eliashberg theory supplemented bylocal vertex corrections determined by a self-consistent quantum impuritymodel. We show that, in the two-dimensional Hubbard model, at strong coupling,TRILEX yields a $d$-wave superconducting dome as a function of doping. Contraryto the standard cluster dynamical mean field theory (DMFT) approaches, TRILEXcan capture $d$-wave pairing using only a single-site effective impurity model.We also systematically explore the dependence of the superconductingtemperature on the bare dispersion at weak coupling, which shows a clear linkbetween strong antiferromagnetic (AF) correlations and the onset ofsuperconductivity. We identify a combination of hopping amplitudes particularlyfavorable to superconductivity at intermediate doping. Finally, we study within$GW$+EDMFT the low-temperature $d$-wave superconducting phase at strongcoupling in a region of parameter space with reduced AF fluctuations.

TRILEX and $GW$+EDMFT approach to $d$-wave superconductivity in the Hubbard model