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Recently proposed de Sitter Swampland conjectures imply non-trivialconstraints on a scalar field potential in any effective field theory thatadmits a quantum gravity completion. The original conjecture apparentlyexcludes many phenomenologically motivated scalar potentials with de Sitterextrema, such as the perturbative Standard Model Higgs potential and the QCDaxion potential, as the viable low-energy theories. Subsequently, the refined,weaker conjecture was proposed. However, the full effective potential, havingbeen defined as a Legendre transform, is necessarily convex. This ensures thatthe original Swampland conjecture can actually be satisfied inphenomenologically relevant models, providing no de Sitter vacua exist.

A brief remark on convexity of effective potentials and de Sitter Swampland conjectures