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One-loop corrections to a scalar field during inflation

David Seery

TL;DR

<3-5 sentence high-level summary> This paper develops a formalism to compute leading one-loop quantum corrections to the power spectrum of a light scalar field during single-field slow-roll inflation, using Schwinger in-in techniques and a careful treatment of the interacting vacuum and derivative interactions that require ghost fields. By separating the calculation into a radiative correction to $\delta\phi$ near horizon exit and a subsequent $\delta N$-based assembly into the observable curvature perturbation $\zeta$, the work elucidates the infrared structure and renormalization of the loop corrections. The main result is a suppressed loop correction to the scalar power spectrum, with a characteristic logarithmic dependence on wavenumber, and a consistent renormalization framework that controls both ultraviolet and infrared divergences in an expanding spacetime.

Abstract

The leading quantum correction to the power spectrum of a gravitationally-coupled light scalar field is calculated, assuming that it is generated during a phase of single-field, slow-roll inflation.

One-loop corrections to a scalar field during inflation

TL;DR

<3-5 sentence high-level summary> This paper develops a formalism to compute leading one-loop quantum corrections to the power spectrum of a light scalar field during single-field slow-roll inflation, using Schwinger in-in techniques and a careful treatment of the interacting vacuum and derivative interactions that require ghost fields. By separating the calculation into a radiative correction to near horizon exit and a subsequent -based assembly into the observable curvature perturbation , the work elucidates the infrared structure and renormalization of the loop corrections. The main result is a suppressed loop correction to the scalar power spectrum, with a characteristic logarithmic dependence on wavenumber, and a consistent renormalization framework that controls both ultraviolet and infrared divergences in an expanding spacetime.

Abstract

The leading quantum correction to the power spectrum of a gravitationally-coupled light scalar field is calculated, assuming that it is generated during a phase of single-field, slow-roll inflation.

Paper Structure

This paper contains 32 sections, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Inflation from a single scalar field
  3. The background evolution
  4. Scalar perturbations
  5. Expectation values
  6. Quantum corrections
  7. Loop corrections from Schwinger integrals
  8. Schwinger's formalism.
  9. The interacting vacuum.
  10. Solution for propagators.
  11. Homogeneous equation.
  12. Propagator matrix.
  13. One-vertex, one-loop amplitudes.
  14. Theories with derivative interactions
  15. Feynman rules for the interacting scalar/ghost theory
  16. ...and 17 more sections

Figures (3)

  • Figure 1: Pure $\delta\phi$ vertices. A dot on a scalar line entering a vertex shows that a time derivative is applied to the field at the point of interaction. In terms of Eq. eq:derivative-action, the diagrams correspond to the vertices produced by ($a$) the potential $V$; ($b$) the $\lambda$ vertex; ($c$) the $\Gamma_1$ vertex; ($d$) the $\Gamma_2$ vertex; and ($e$) the $\omega$ vertex.
  • Figure 2: Scalar/ghost vertices. Solid lines represent the scalar field $\delta\phi$, whereas dashed lines represent the ghost. A dot on a scalar line entering a vertex shows that a time derivative is applied to the field at the point of interaction. Time derivatives are never applied to ghost fields. In terms of Eq. eq:ghost-action the diagrams correspond to the vertices produced by ($a$) the $\psi_1 \pi^2$ interaction; ($b$) the $\psi_2 \pi^2$ interaction; ($c$) the $\omega \dot{\theta} \pi \pi$ interaction; and ($d$) the $\omega \pi^3$ interaction.
  • Figure 3: Instability of the vacuum due to condensation. $\delta\phi$ particles emerge from the vacuum (represented by the hatched condensate) in a zero momentum state. The accumulation of such particles changes the homogeneous classical field configuration associated with the vacuum.