Consistency of EFT illuminated via relative entropy: A case study in scalar field theory

Relative entropy is a non-negative quantity and offers a powerful means of achieving a unified understanding of fundamental properties in physics, including the second law of thermodynamics and positivity bounds on effective field theories (EFTs). We analyze the relative entropy in scalar field theories and show that the non-negativity of relative entropy is potentially violated in perturbative calculations based on operator and loop expansions. Conversely, this suggests that the consistency of the EFT description in the scalar field theory can be identified by the sign of the relative entropy. In fact, we revisit an EFT of single-field inflation and present a relation between its non-linear parameter $f_{\rm NL}$ and the consistency condition of the EFT description derived from the relative entropy method. We find that interesting regions of $f_{\rm NL}$ that are observationally allowed can be constrained from the relative entropy by imposing the consistency of the EFT description when the EFT is generated via the interaction with heavy fields in UV theories.