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The Luttinger-Ward functional was proposed more than five decades ago toprovide a link between static and dynamic quantities in a quantum many-bodysystem. Despite its widespread usage, the derivation of the Luttinger-Wardfunctional remains valid only in the formal sense, and even the very existenceof this functional has been challenged by recent numerical evidence. In asimpler and yet highly relevant regime, namely the Euclidean lattice fieldtheory, we rigorously prove that the Luttinger-Ward functional is awell-defined universal functional over all physical Green's functions. Usingthe Luttinger-Ward functional, the free energy can be variationally minimizedwith respect to Green's functions in its domain. We then derive the widely usedbold diagrammatic expansion rigorously, without relying on formal argumentssuch as partial resummation of bare diagrams to infinite order.

Variational structure of Luttinger-Ward formalism and bold diagrammatic expansion for Euclidean lattice field theory