$λφ^4$ as an Effective Theory in de Sitter

TL;DR

This work develops an EFT framework for a light scalar in de Sitter by integrating out a heavy sector and tracking the resulting density-matrix dynamics on cosmological scales. The authors perform tree- and loop-level matching to two UV realizations, revealing a unitary, local EFT when the heavy mass satisfies $m^2/H^2\gg1$, and a non-unitary, diffusion-dominated regime when $m^2/H^2\ll1$, with decoherence encoded by off-diagonal density-matrix terms. A key finding is the sign change of the effective quartic coupling $\lambda_{ m eff}$ as the cutoff crosses the horizon, aligning loop-level results with stochastic-inflation expectations in the appropriate regime. The study connects Wilsonian EFTs to stochastic descriptions, clarifying regimes of validity for cosmological EFTs and highlighting how non-unitary effects emerge and compete with perturbative control. The results provide a concrete link between the non-perturbative stochastic framework and a structured EFT description, with implications for the treatment of IR divergences, decoherence, and the late-time behavior of correlators in de Sitter space.

Abstract

Effective field theories (EFTs) provide a powerful framework to parametrise unknown aspects of possible ultraviolet (UV) physics. For scalar fields in de Sitter space, however, new emergent phenomena can arise when the cut-off scale of the theory lies below the horizon scale $H$, as seen in the stochastic formalism of inflation. In this work, we study EFTs that, at leading order, reproduce the standard quartic theory in de Sitter, but with a variable cut-off identified with the mass of an integrated-out hidden sector. We perform the complete analytic computation for the tree- and loop-level matching between the effective $λφ^4$ theory and two possible UV realisations. We find that when the cut-off is much larger than the horizon, the theory admits a unitary description, up to exponentially suppressed corrections. In contrast, when the cut-off is lowered below $H$, the system evolves into a mixed state and diffusive effects emerge. Nevertheless, at leading order, the EFT remains local and reproduces the same effective quartic coefficient as in the unitary regime. Furthermore, for the EFT matching at the loop-level, the effective quartic coupling changes sign and becomes negative as the cut-off decreases, in agreement with the result obtained from the stochastic formalism. In general, for cosmological EFTs, our findings highlight the role of non-unitary effects and illustrate their regimes of validity, within and beyond perturbation theory.

Figures (5)

• Figure 1: Quartic vertices appearing at the tree level: Dashed lines represent $\phi_a$ fields and solid black lines represent $\phi_r$ fields. A contraction between two $\phi_r$ fields corresponds to a Keldysh propagator $G_K$, while a contraction between $\phi_r$ and $\phi_a$ corresponds to a retarded or advanced propagator, $G_{R}$ or $G_{A}$.
• Figure 2: Quartic vertices appearing at 1-loop. Lines in red are $\sigma$ fields and in black $\phi$ fields. (a) and (b) are the diagrams from the unitary theory while (c) is the non-unitary vertex that leads to decoherence.
• Figure 3: Loop diagrams contributing at 1-loop to the four-point function. Lines in red are $\sigma$ fields and in black $\phi$ fields (a) contribution to the $\phi_r^4$ vanishes since it does not have support for any time, (b) contribution to the $\phi_a^4$ has the same propagator structure inside the loop and thus vanishes.
• Figure 4: The effective quartic coupling $\lambda_{\mathrm{eff}}$ obtained by matching the four-point function in Eq. \ref{['eq_loopIR']}, with $\mu_R$ fixed to be $H$. The blue solid line represents the exact result, the green dashed line corresponds to the coefficient derived from the stochastic description in Eq. \ref{['eq:stochasticquarticcoupling']}, and the pink dashed line illustrates the interpolation valid in the large-mass limit.
• Figure 5: The probability distributions of $\phi$ by tracing out $\sigma$: a) heavy tail \ref{['PDF:quartic']} in the small mass limit; b) illustration of the non-Gaussian tail in the large mass limit where the effective $\lambda\phi^4$ breaks down and UV physics becomes relevant. The dashed gray lines are the Gaussian distributions.