Ward identities for strongly coupled Eliashberg theories
  We discuss Ward identities for strongly interacting fermion systems described by Eliashberg-type theories. We show that Ward identities are not in conflict with Migdal theorem. We derive diagrammatically Ward identity for a charge vertex in a Fermi liquid, and when a   Fermi liquid is destroyed at quantum criticality. We argue that   Ward identity for a spin vertex cannot be obtained within   Eliashberg theory. 
Andrey V. Chubukov