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Nonlocal Corrections to Scalar Field Effective Action in de Sitter spacetime

Will Cerne, Teruaki Suyama

TL;DR

This paper derives the one-loop effective action for a minimally coupled test scalar in a general FLRW background using the in-in formalism, revealing nonlocal memory effects and a stochastic noise term that arise at second order in the interaction. The authors perform a careful renormalization, separating Minkowski-like divergences and isolating finite, physically meaningful corrections to the local potential and to a nonlocal memory kernel, with a consistent renormalized equation of motion. In a local-approximation appropriate for slowly varying fields, the memory term reduces the drift friction via a negative contribution, while a concrete de Sitter analysis shows the coefficients A, B, C controlling these corrections; numerically, A scales as $H^4/m^2$ and the memory effects tend to suppress the infrared variance for a massive $\phi^4$ theory. The work also connects to stochastic inflation by mapping the results to a Langevin–Fokker–Planck framework with both white and colored noise, highlighting differences in the treatment of IR modes and offering a pathway to improved understanding of quantum corrections during inflation.

Abstract

We investigate the one-loop effective action for a test scalar field in a general Friedmann-Lemaître-Robertson-Walker (FLRW) background, specifically focusing on quantum corrections up to the second order in the interaction strength. By employing the Schwinger-Keldysh formalism, we derive the equation of motion for the field expectation value, which incorporates not only the standard local radiative corrections but also novel nonlocal features: a memory term and a stochastic noise term. We identify all ultraviolet divergent structures within these nonlocal terms and provide a consistent renormalization procedure. To analyze the physical impact of these terms, we apply a local approximation under the assumption of slowly-varying fields, by which the memory term acts as a negative contribution to the drift coefficient. As a concrete application, we consider a massive $φ^4$ theory and show that these one-loop corrections lead to a suppression of the field variance in the infrared regime compared to the tree-level results.

Nonlocal Corrections to Scalar Field Effective Action in de Sitter spacetime

TL;DR

This paper derives the one-loop effective action for a minimally coupled test scalar in a general FLRW background using the in-in formalism, revealing nonlocal memory effects and a stochastic noise term that arise at second order in the interaction. The authors perform a careful renormalization, separating Minkowski-like divergences and isolating finite, physically meaningful corrections to the local potential and to a nonlocal memory kernel, with a consistent renormalized equation of motion. In a local-approximation appropriate for slowly varying fields, the memory term reduces the drift friction via a negative contribution, while a concrete de Sitter analysis shows the coefficients A, B, C controlling these corrections; numerically, A scales as and the memory effects tend to suppress the infrared variance for a massive theory. The work also connects to stochastic inflation by mapping the results to a Langevin–Fokker–Planck framework with both white and colored noise, highlighting differences in the treatment of IR modes and offering a pathway to improved understanding of quantum corrections during inflation.

Abstract

We investigate the one-loop effective action for a test scalar field in a general Friedmann-Lemaître-Robertson-Walker (FLRW) background, specifically focusing on quantum corrections up to the second order in the interaction strength. By employing the Schwinger-Keldysh formalism, we derive the equation of motion for the field expectation value, which incorporates not only the standard local radiative corrections but also novel nonlocal features: a memory term and a stochastic noise term. We identify all ultraviolet divergent structures within these nonlocal terms and provide a consistent renormalization procedure. To analyze the physical impact of these terms, we apply a local approximation under the assumption of slowly-varying fields, by which the memory term acts as a negative contribution to the drift coefficient. As a concrete application, we consider a massive theory and show that these one-loop corrections lead to a suppression of the field variance in the infrared regime compared to the tree-level results.
Paper Structure (14 sections, 85 equations, 4 figures)

This paper contains 14 sections, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. One-Loop Corrections in the Effective Action of the Test Field: The General Case
  3. Renormalization
  4. Renormalization of the First Order Term
  5. Renormalization of the Second Order Term
  6. Renormalized equation of motion
  7. Discussion
  8. Connection to Stochastic Inflation
  9. Corrections to the Noise
  10. Conclusion
  11. Convergence Properties of $I(x)$
  12. Evaluation of the first term
  13. Evaluation of the second term
  14. Evaluation of the third term

Figures (4)

  • Figure 1: Coefficients $B$ and $C$ as functions of the mass parameter $m^2$ in units of $H^2$.
  • Figure 2: A comparison of the equilibrium probability distribution for three cases: tree level, the first order correction, and the full non-local corrections. $\Phi$ is given in units of $H$.
  • Figure 3: The relative difference of the variance from the tree level result as a function of $\frac{m^2}{H^2}$ for three different values of the coupling constant $\lambda$.
  • Figure 4: Expectation value $\frac{\langle \Phi^2 \rangle}{H^2}$ as a function of $\lambda$. Both panels show first-order and non-local corrections.