Vertex Function in a Fermi Liquid near a q=0 Charge Quantum Critical Point

We analyze the vertex function for a two-dimensional Fermi liquid close to a q=0 charge quantum critical point (QCP). We show that there is an inconsistency in the calculation for the quasiparticle residue (Z) if one extracts Z from the one-loop self-energy and from the Ward identity, when one computes the vertex function within RPA. This inconsistency can be resolved if one includes an infinite series of ladder diagrams in the calculation of the vertex function. These diagrams are treated as zero within a conventional Fermi liquid theory with static interaction but are finite when using the renormalized, frequency and momentum dependent interaction. We demonstrate explicitly that including these terms changes the vertex function by a factor of Z^-1, which resolves the discrepancy between the two calculations.