 Solution

The ghost kinetic term as printed is $-{\partial }^{\mu }{\bar{c}}^a{\partial }_{\mu }{c}^a$. Integration by parts gives ${\bar{c}}^a\square c^a$, so its Fourier-space quadratic kernel is $K_c^{ab}(p)=-p^2{\delta }^{ab}$. Completing the Grassmann square in the same generating functional givesKeep the source order fixed, using a left derivative for $\bar{\eta }$ and a right derivative for $\eta$. Including the two insertion factors produces $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$, just as in the Gaussian generating functional for a Dirac field. ThereforeThe minus sign follows from the particular ghost Lagrangian in this paper. A convention with the opposite ghost kinetic sign has the opposite $c\bar{c}$ propagator; redefining the Faddeev-Popov antighost field changes its interaction and source signs at the same time. Mixing propagator and vertex conventions would give an erroneous loop sign. This is the ghost propagator for a negative derivative kinetic term.