Tree photon-scattering cancellation in a quadratic Abelian gauge (${\mathcal{M}}_{\gamma \gamma \to \gamma \gamma }^{\mathrm{tree}}=0$)

The nonlinear gauge fixing $\partial \cdot A+\lambda A^2$ produces cubic and quartic photon Feynman vertices. For physical photon polarization vectors, the two-external-leg cubic vertex reduces to a longitudinal internal momentum times the scalar product of the external polarizations. Its exchanged photon propagator cancels that momentum squared. Each of the three exchange graphs contributes $+4i{\lambda }^2({\epsilon }_i\cdot {\epsilon }_j)({\epsilon }_k\cdot {\epsilon }_l)$, cancelling the corresponding term of the quartic vertex. This vanishing physical tree-level Feynman diagram sum does not assert that the off shell four-point function vanishes.