I have a long Lagrangian when I apply the Slavnov operator all terms cancel except for the Gauge fixing term and the ghost term. I am using an unusual gauge fixing condition, $$F=(\partial_\mu + \frac{\lambda}{2}A_\mu) A^\mu.$$ My question is should the Slavnov operator always cancel out no matter what gauge condition is used?
$$\mathcal{L}_{eff} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} -\frac{1}{2\zeta} \left( \partial_\mu A^\mu + \frac{\lambda}{2} A_\mu A^\mu \right)^{2} + \overline{\omega} \left(\partial_\mu \partial^\mu + \lambda A_\mu \partial^\mu \right) \omega.$$
Where I have used $sA_\mu = \partial_\mu\omega$, $s\overline{\omega}=\dfrac{\partial_\mu A^\mu + \frac{\lambda}{2} A_\mu A^\mu}{\zeta}$ and $s\omega = 0$.
When I apply $s\mathcal{L}_{eff}$ I should expect to obtain $0$, right?
But I obtain
$$s\mathcal{L}_{eff}= -\frac{1}{\zeta} F\left( \partial_\mu \partial^\mu + \lambda A_\mu \partial^\mu \right) + \frac{F}{\zeta}\left(\partial_\mu \partial^\mu + \lambda A_\mu \partial^\mu \right)\omega +\overline{\omega}\lambda \partial_\mu \omega \partial^\mu \omega = \overline{\omega}\lambda \partial_\mu \omega \partial^\mu \omega$$
Any ideas?
