VERTEX CORRECTION TO PAIRING AT A VAN-HOVE SINGULARITY

We calculate in a k-averaging approximation, the vertex corrections fo r pairing at a Van Hove singularity mediated by an Einstein phonon. Th e vertex correction leads to a modest reduction in T-c, which can stil l reach the experimental values for realistic phonon couplings, and to a significant deepening of the dip in the isotope shift at the T-c ma ximum. Interference between first- and second-order vertex graphs dimi nishes the effect of the Coulomb interaction on T-c. The isotope-shift dip is shallower for repulsive Coulomb interaction, and deeper in the case of an attractive interaction introduced to model an excitonic co ntribution to pairing.