Shifted-action expansion and applicability of dressed diagrammatic schemes

While bare diagrammatic series are merely Taylor expansions in powers of interaction strength, dressed diagrammatic series, built on fully or partially dressed lines and vertices, are usually constructed by reordering the bare diagrams, which is an a priori unjustified manipulation, and can even lead to convergence to an unphysical result [E. Kozik, M. Ferrero, and A. Georges, Phys. Rev. Lett. 114, 156402 (2015)]. Here we show that for a broad class of partially dressed diagrammatic schemes, there exists an action $S^{\left(\xi \right)}$ depending analytically on an auxiliary complex parameter $\xi$, such that the Taylor expansion in $\xi$ of correlation functions reproduces the original diagrammatic series. The resulting applicability conditions are similar to the bare case. For fully dressed skeleton diagrammatics, analyticity of $S^{\left(\xi \right)}$ is not granted, and we formulate a sufficient condition for converging to the correct result.