Two-photon vertex of a scalar field-strength interaction ($\mathcal{A}=\lambda [(q_1\cdot q_2)({ϵ}_1\cdot {ϵ}_2)-(q_1\cdot {ϵ}_2)(q_2\cdot {ϵ}_1)]$)

The interaction $\lambda \Phi F_{\mu \nu }{F}^{\mu \nu }/4$ gives the displayed on-shell scattering amplitude, up to an overall phase, for a scalar field decaying to two photons. Substituting either photon four-momentum for its polarization vector makes the expression vanish, verifying its Ward identity. In the parent rest frame, summing the two physical transverse photon polarizations gives $\sum |\mathcal{A}{|}^2=|\lambda {|}^2{M}^4/2$. Including the identical-particle factor in a final-state phase-space integral gives $\Gamma =|\lambda {|}^2{M}^3/(64\pi )$.