Quadratic Abelian gauge fixing ($F[A]=\partial \cdot A+A^2$)

For $\delta A_{\mu }={\partial }_{\mu }\omega$, the real quadratic gauge functional varies by $(\square +2A\cdot \partial )\omega$. Its Faddeev-Popov operator depends on the gauge field, producing a ghost-gauge vertex even in an Abelian gauge theory. With $\Psi =\int \bar{c}(F[A]+\xi h/2)$, a consistent left BRST symmetry convention gives $s\Psi =hF+\xi h^2/2-\bar{c}(\square +2A\cdot \partial )c$. The auxiliary-field integral at $\xi =0$ imposes the exact condition; finite $\xi$ gives a weighted gauge condition. This real functional is distinct from a complex quadratic Abelian gauge condition.

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.