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

TL;DR

This paper discusses the implications of de Sitter Swampland conjectures on scalar potentials, emphasizing the convexity of the effective potential and its impact on the existence of de Sitter vacua in phenomenological models.

Contribution

It clarifies how the convexity of the effective potential affects the validity of Swampland conjectures, reconciling them with phenomenologically relevant scalar potentials.

Findings

Convexity of the effective potential ensures the original Swampland conjecture can be satisfied.

Many scalar potentials with de Sitter extrema are excluded by the conjecture due to convexity.

No de Sitter vacua exist in models where the effective potential is convex.

Abstract

Recently proposed de Sitter Swampland conjectures imply non-trivial constraints on a scalar field potential in any effective field theory that admits a quantum gravity completion. The original conjecture apparently excludes many phenomenologically motivated scalar potentials with de Sitter extrema, such as the perturbative Standard Model Higgs potential and the QCD axion potential, as the viable low-energy theories. Subsequently, the refined, weaker conjecture was proposed. However, the full effective potential, having been defined as a Legendre transform, is necessarily convex. This ensures that the original Swampland conjecture can actually be satisfied in phenomenologically relevant models, providing no de Sitter vacua exist.