2PI Effective Action from Double Legendre Transformation

Question

This answer (https://physics.stackexchange.com/q/348673) provides good intuition for why Legendre transformation induces 1-particle irreducible graphs: It mainly tries to convey the idea that the condition for tree level functional expansion for the effective action is equivalent to a Legendre transformation, and due to the uniqueness for decompositions of connected graphs into trees with 1PI vertices, Legendre transformation should leads to one-particle irreducible graphs.

Is there a similar argument for 2PI effective actions? Or a derivation of 2PI skeleton expansion to all orders?

For the moment, I would like to consider the Dyson equation as an extra consistency condition between 1PI and 2PI, such that the functional derivative give rise to the 1PI self-energy: \begin{equation} \frac{\delta \Gamma_2[\phi,G]}{\delta G} = \Sigma_{\text{1PI}}, \end{equation} which means that $\Gamma[\phi,G]$ must only contain two-particle irreducible graphs (Because functional derivative w.r.t. $G$ can be represented graphically as cutting one line.). The Dyson equation looks like following: \begin{equation} \frac{\delta[S[\phi]+\imath\text{Tr}\ln(G^{-1})+\imath\text{Tr}G_0^{-1}G+ \Gamma_2[\phi,G]]}{\delta G}=0=-\imath G^{-1}+\imath G_0^{-1}+\Sigma_{\text{1PI}}. \end{equation} The expression on the left hand side come from the double Legendre transformation.

Answer 1

It is difficult to find an explicit reference that really shows this property, but one can convince oneself from eq. (2.17) in PhysRevD.10.2428. Note here that the constant is the log of the free Green function.

At the physical point, we can write the expression as

$ \Gamma^2 = \Gamma^1 - \frac{i \hbar}{2} \text{tr}\left(G_0^{-1} G \right) + \frac{i\hbar}{2} \text{tr} \log \left( 1 - \Sigma G \right) \qquad \text{with} \quad \Sigma \propto \frac{\delta \Gamma^2}{\delta G} \, . $

$\Gamma^1 $ are all 1-PI diagrams. The trick is now that we know what all 1-PI diagrams, which are 2-PR, look like. They must fall apart when cutting two lines. So they must be loops of the type $\text{tr} G \Sigma G \Sigma ....$, if $\Sigma$ is 1PI with external legs. If one now starts to expand $\text{tr} \log \left( 1 - \Sigma G \right)$, one finds all these contributions. These are now the diagrams, which are deleted from $\Gamma^1$ and $\Gamma^2$ has only 2PI diagrams remaining.

If you want a different perspective on this you can look into "Functional Methods in Quantum Field Theory and Statistical Physics" from Vasiliev. There, the diagrammatic parts are derived by recursion relations. (Note: this book is not the easiest to read, but the most complete reference on effective actions I could find till now.)