Confronting infrared divergences in de Sitter: loops, logarithms and the stochastic formalism

Abstract: A well-established result in quantum field theory in four-dimensional de Sitter space is that the vacuum state of a massless scalar field breaks the de Sitter isometry group, leading to time-dependent (secular) growth in correlation functions computed in inflationary coordinates. This behavior is widely believed to extend to more general theories involving light scalar fields with weak non-derivative interactions. In such cases, secular growth is thought to be further amplified by loop corrections, and the stochastic formalism is often regarded as the appropriate framework to resum these infrared effects. In this article we challenge this prevailing view. A crucial distinction must be made between two cases: a massless scalar field protected by a shift symmetry, and a light scalar without such a symmetry. In the former, the shift symmetry enforces derivative interactions, yielding observables in which secular growth plays no physical role. In the latter, although correlation functions develop infrared divergences in the massless limit, they remain fully invariant under the de Sitter isometry group. We analyze the structure of these divergences arising from loop integrals and show that, in the soft-momentum limit, they do not alter the time dependence of tree-level correlators. In fact, using a de Sitter-invariant renormalization scheme based on Wilson's axioms for integration, these divergences can be systematically removed order by order. We therefore conclude that neither massless nor light scalar fields in de Sitter space exhibit genuine secular growth. We further discuss the implications of these findings for the validity and scope of the stochastic approach to inflation.