Vertex corrections and two-loop pairing potential in nearly antiferromagnetic Fermi liquids

We have calculated, within perturbation theory, the correction to the Eliashberg approximation for nearly antiferromagnetic Fermi liquids. The vertex corrections to the effective interaction that determines the normal-state properties of the model result in a increase of the quasiparticle spin-fluctuation coupling for spin-fluctuation momenta near the antiferromagnetic wave vector Q=(π,π) for a large Fermi surface. This increase gets larger as the antiferromagnetic correlation length increases and is very anisotropic. It is most important for points on the Fermi surface where the ${\mathrm{d}}_{{\mathrm{x}}^2\mathrm{-}{\mathrm{y}}^2}$ gap is large. These higher-order corrections also result in a stronger pairing potential in the ${\mathrm{d}}_{{\mathrm{x}}^2\mathrm{-}{\mathrm{y}}^2}$ channel relative to the one-loop Eliashberg contribution.