Higher Legendre transforms and their relationship to Bethe–Salpeter kernels and r-field projectors

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Abstract
We analyze the structure of the higher Legendre transforms Γ(r){A}(r?1) of the generating functional G of the connected Green’s functions Gn in Euclidean boson field theories. In addition to the vertex functions, Γ(r) generates a variety of objects of interest for their r‐irreducibility in certain channels, e.g., r‐irreducible expectations, rth order Bethe–Salpeter kernels, and r‐field projectors. Our analysis is independent of perturbation theory, our definition of r‐irreducibility being based on Spencer’s idea of t‐lines. We derive formulas for ∂ntΓ(r){A;t} (in terms of either δnAΓ(r){A;t} or the Gn’s) to be used as input in the proofs of r‐irreducibility. For the case of the weakly coupled P(φ)2 model, we establish the existence of the moments δnAΓ(r){0;t} and their regularity in t.