Legendre transforms and r-particle irreducibility in quantum field theory: The formal power series framework

We study the higher Legendre transforms Γ^(r) {A } of the generating functional G for connected Green's functions in Euclidean boson field theories. To analyze Γ^(r) {A } rigorously even when it does not make sense as an ordinary functional, we develop the framework of formal power series in A. For r = 1, 2 we isolate regularity conditions on G that ensure the existence of Γ^(r) as a formal power series and we verify these conditions for the weakly coupled P(φ)₂ model. We also establish the improved regularity of the functional Φ^(r) obtained by subtracting from Γ^(r) its trivial singular part.