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Let $p_1,p_2$ be the incoming scalar and antiscalar momenta and $k_1,k_2$ the outgoing photon momenta, so $p_1+p_2=k_1+k_2$. Up to the common overall phase fixed by the $S$-matrix convention, the connected leading amplitude iswhereOn shell, the denominators are $-2p_1\cdot k_1=-2p_2\cdot k_2$ and $-2p_1\cdot k_2=-2p_2\cdot k_1$. Contracting with $k_{1\mu }$ therefore givesusing momentum conservation and $k_2\cdot {ϵ}_2=0$. The same calculation with the photons interchanged gives $k_{2\nu }{\mathcal{M}}^{\mu \nu }{ϵ}_{1\mu }^{\ast }=0$. The two exchange diagrams and the seagull term are all required for this Ward identity.

The leading contributions to ${\delta }_2$ and ${\delta }_m$ come from the one-loop electron self-energy: one internal electron line and one internal photon line join two electron-photon vertices. Its coefficients of ${\gamma }^{\mu }{p}_{\mu }$ and $m$ determine the field-strength and mass counterterms. The leading contribution to ${\delta }_1$ comes from the one-loop electron-photon vertex correction, with two internal electron propagators and one internal photon propagator, together with the corresponding local counterterm vertices. The Ward identity gives ${\delta }_1={\delta }_2$ in a gauge-invariant scheme.

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

For a real on shell photon of null momentum $k$, a physical polarization satisfies $k\cdot ϵ=0$ and is defined modulo $ϵ\mapsto ϵ+\alpha k$. The two-dimensional quotient carries the two physical photon helicities. In a transverse gauge choose ${ϵ}^0=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. A Ward identity ensures that replacing an external polarization by its momentum gives zero in the summed physical amplitude.

Define the proper electromagnetic vertex with the electric charge removed, and define $S^{-1}$ without the propagator's overall factor $i$. The displayed off-shell identity follows by localizing the vector symmetry in a time-ordered correlation function, including the contact terms at the charged-field insertions, and amputating. Its on-shell form gives the Ward identity for physical photon amplitudes. At zero photon momentum it identifies the vertex with the momentum derivative of the inverse fermion propagator.

An external source is a prescribed field coupled to a dynamical field or operator in the action, for example $S_J=S_0+\int J_i(x){\varphi }_i(x)d^{d}x$. Functional derivatives of the generating functional with respect to $J$ insert the corresponding fields into correlation functions. The source is varied to generate these insertions, rather than integrated over as a dynamical degree of freedom. Transforming it in the appropriate representation maintains an invariant source coupling when deriving a Ward identity.

For tree Electron-Positron annihilation into three photons, all six orders along the fermion line must be summed. Write $D(r)=/r-m$ and $S(r)=D(r{)}^{-1}=(/r+m)/(r^2-m^2)$. Contracting an internal photon with its momentum gives $S(R)/qS(r)=S(R)-S(r)$ when $q=r-R$. Contracting an end photon removes its adjacent Dirac propagator by the external Dirac equation, with a minus sign at the electron end and a plus sign at the positron end. For either fixed order of the two uncontracted photons, the three insertion positions therefore telescope to zero. Summing their two orders proves the Ward identity. The mass terms in both remaining Dirac propagators are essential when $m\ne 0$. The result expresses the invariance of the scattering amplitude under ${ϵ}_{\ell }\mapsto {ϵ}_{\ell }+c{q}_{\ell }$.