The Luttinger-Ward functional $\mathrm{\ensuremath{\Phi}}[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function
G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]\ensuremath{\propto}\ensuremath{\delta}\mathrm{\ensuremath{\Phi}}[\mathbf{G}]/\ensuremath{\delta}\mathbf{G}$ is not a single-valued functional of
G: in addition to the physical solution for $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]$, there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for $\mathbf{\ensuremath{\Sigma}}$ in terms of
G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the noninteracting Green's function
G0 converges to the correct physical branch of $\mathbf{\ensuremath{\Sigma}}$ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.
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 ${S}^{(\ensuremath{\xi})}$ depending analytically on an auxiliary complex parameter $\ensuremath{\xi}$, such that the Taylor expansion in $\ensuremath{\xi}$ 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 ${S}^{(\ensuremath{\xi})}$ is not granted, and we formulate a sufficient condition for converging to the correct result.