Hedin's equations and the ground state energy

Hedin’s equations are an iterative scheme to calculate the Green’s function $G$, the self-energy $Sigma$, the vertex $Gamma$, the polarizability $chi$, and the screened interaction $W$.

However, is there a nice identity that gives me the ground state energy only in terms of $G$, $Sigma$, $Gamma$, $chi$, and/or $W$?

I know that there are approximations to Hedin’s equations, like the GW approximation or RPA. There we can write the ground state energy as:
$$
E_0^{text{RPA}} = frac{1}{2pi}int text d omega ; text{Tr} lnbig(1-chi(text i omega) Vbig),
$$
where $V$ is the Coulomb operator.

What if I don’t make any approximations but solve Hedin’s equations such that I have all five quantities $G$, $Sigma$, $Gamma$, $chi$, and $W$ (I know that this is not possible in real life), how do I find the ground state energy?

Just from the diagrammatics I guess that it must be something like
$$
E_0 = GSigma
$$
or so… . Can someone help me out?

PS: I know that there is the Galitskii-Migdal formula such that I can write the ground state energy in terms of the Green’s function. But there are operators in the integrand thus the equation is not practical for an implementation into a computer code.