= Solution
The ghost kinetic term as printed is $-\partial^\mu\bar c^a\partial_\mu c^a$. Integration by parts gives $\bar c^a\Box c^a$, so its Fourier-space quadratic kernel is $K_c^{ab}(p)=-p^2\delta^{ab}$. Completing the Grassmann square in the same generating functional gives
$$
Z_0[\bar\eta,\eta]=\exp[-i\bar\eta K_{c,F}^{-1}\eta].
$$
Keep the source order fixed, using a left derivative for $\bar\eta$ and a right derivative for $\eta$. Including the two insertion factors produces $\langle Tc\bar c\rangle=iK_{c,F}^{-1}$, just as in the <Gaussian generating functional for a Dirac field>. Therefore
$$
\boxed{\langle\Omega|T c^a(x)\bar c^b(y)|\Omega\rangle=\delta^{ab}\int\frac{d^4p}{(2\pi)^4}e^{-ip(x-y)}\frac{-i}{p^2+i0}.}
$$
The 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>.
Back to article page
