 Two-photon scalar Ward identity

In a two-photon process involving a charged scalar field, the full amplitude obeys a Ward identity under replacement of either external photon polarization vector by that photon's four-momentum. At tree level in scalar quantum electrodynamics, both orders of single-photon attachment and the two-photon contact diagram contribute.

Contracting a derivative vertex converts its momentum factor into a difference of inverse scalar propagators. Intermediate terms cancel between the attachment orders and the contact interaction; external inverse propagators vanish for on-shell external states. Omitting the contact term generally leaves a nonzero result, even when both exchange diagrams are present. The cancellation is a practical test of the amplitude's gauge consistency.^[ref]