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

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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 {bold Q}=({pi},{pi}) 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 d{sub x{sup 2}{minus}y{sup 2}} gap is large. These higher-order corrections also result in a stronger pairing potential in the d{sub x{sup 2}{minus}y{sup 2}} channel relative to the one-loop Eliashberg contribution. {copyright} {ital 1997} {ital The American Physical Society}