Scalar quantum electrodynamics Ward identity

The scalar Ward identity relates the charged-scalar photon vertex to differences of inverse scalar propagators. At tree level, contraction with photon momentum $q=p^{\prime }-p$ usesIncluding the appropriate charge and vertex conventions makes the right side the difference of inverse free propagators. The full identity uses the dressed vertex and full inverse propagators instead.

Scalar quantum electrodynamics also contains a two-photon contact interaction from the square of the gauge covariant derivative. Its coefficient is tied to the one-photon and kinetic terms by gauge invariance. Consequently, checks of scalar scattering must include both sequential one-photon attachments and the contact term. Renormalization conventions preserving the symmetry retain these relations between the coefficients.

In $D=4-ϵ$, the two surviving zero-momentum seagull bubbles contribute $-4e_D^4{\int }_l{P}^{\mu \nu }(l)/[l^2(l^2-m^2)]+(\mu \leftrightarrow \nu )$. Tensor averaging gives $(D-1)/D$, and the scalar integral has pole $i/(8{\pi }^2ϵ)$. Cancellation against $2i{e}_D^2(Z_4-1)g^{\mu \nu }$ gives the displayed result. A positive scalar mass separates the ultraviolet pole from the scaleless massless limit. The scalar QED Ward identity gives $Z_4=Z_2$ in this convention.

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]