Which gauge fixing conditions are allowed to choose for this approach?
For example (following the steps of Peskin in 9.4) I can choose a "modified" lorenz gauge ( for a abelian theory):
$$ (\partial_\mu \hat{A}^\mu)^² = 0 $$
With $ \hat{A}^\mu = A^\mu + \partial_\mu \alpha $ is the gauge transformed of $ A $
For the ghosts I end up with: $$ L= c(\partial_\mu \hat{A}^\mu) \partial^2 \overline{c}$$ or $$ L= c(\partial_\mu A^\mu + \partial^2 \alpha) \partial^2 \overline{c}$$
I assumed to find an equal result as for the very similar Lorenz condition. Or if not I thought the formalism would introduces ghost's to keep the theory stable. But even then I can not interpret this Lagrangian. The ghosts have no kinetic term and the interaction vertex with the gauge field depends on moment of the gauge field and the $ p^2 $ of the ghosts. But the ghost are massless here so $ p^2 = m^2 =0$. What is wrong with this gauge fixing? What says this Lagrangian about the interactions and "propagator" of the ghost fields?