In section V ofthis paper, the author computes$\langle F_{\mu \nu} F^{\mu \nu} \rangle$. Using the definition of$F_{\mu \nu}$ it is not difficult to show that
$$\langle F_{\mu \nu} F^{\mu \nu}\rangle = 2 ( g^{\mu \lambda} g^{\nu \sigma} - g^{\mu \sigma} g^{\nu \lambda} ) T_{\mu \nu \lambda \sigma},$$
where$T_{\mu \nu \lambda \sigma} = \langle \nabla_{\mu} A_{\nu} \nabla_{\lambda} A_{\sigma} \rangle$. In order to evaluate$T_{\mu \nu \lambda \sigma}$, which is a divergent quantity, the author introduces a point-splitting regularization
$$ T_{\mu \nu \lambda \sigma} ( x^{\prime} ) = \frac{1}{2} \lim_{x^{\prime} \to x} [ \langle \nabla_{\mu} A_{\nu} ( x ) \nabla_{\lambda}^{\prime} A_{\sigma} ( x^{\prime} ) \rangle + \langle ( \nabla_{\lambda} A_{\sigma} ( x ) \nabla_{\mu}^{\prime} A_{\nu} ( x^{\prime} ) \rangle ] $$
which ensures that$T_{\mu \nu \lambda \sigma} = T_{\lambda \sigma \mu \nu}$ holds even after the regularization. The last equation can also be written in terms of the Green's function
$$ T_{\mu \nu \lambda \sigma} ( x^{\prime} ) = \frac{1}{2} \lim_{x^{\prime} \to x} [ \nabla_{\mu} \nabla_{\lambda}^{\prime} G_{\nu \sigma} ( x, x^{\prime} ) + \nabla_{\lambda} \nabla_{\mu}^{\prime} G_{\sigma \nu} ( x, x^{\prime} ) ]. $$
Now here is my question. Suppose I want to evaluate a similar object involving Faddeev–Popov ghost fields, such as$$\langle \nabla_{\mu} \bar{C}_{\nu} \nabla_{\lambda} C_{\sigma} \rangle,$$ where$\bar{C}_{\nu}$ and$C_{\sigma}$ are the vector ghost fields that appears, for instance, in quantum gravity. How should I perform the point-splitting in this case? Should I take the anticommuting nature of these fields into account and define something like
$$\frac{1}{2} [ \langle \nabla_{\mu} \bar{C}_{\nu} ( x ) \nabla_{\lambda}^{\prime} C_{\sigma} ( x^{\prime} ) \rangle - \langle \nabla_{\lambda} \bar{C}_{\sigma} ( x ) \nabla_{\mu}^{\prime} C_{\nu} ( x^{\prime} ) \rangle ]~?$$
