Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Loop Scalar Self-Mass during Inflation

T. Brunier, V. K. Onemli, R. P. Woodard

TL;DR

The paper performs a complete two-loop computation of the fully renormalized scalar self-mass-squared for a massless, minimally coupled $\varphi^4$ field during inflation in a locally de Sitter background, employing the Schwinger-Keldysh formalism and dimensional regularization. It shows that curvature induces a nonzero mass renormalization and a time-dependent finite piece at one loop, and reveals a rich, causal nonlocal structure at two loops with overlapping divergences canceled by counterterms $\delta m^2$, $\delta \lambda$, and $\delta Z$; the final two-loop self-mass-squared comprises nonlocal derivative kernels and a local $a^4\delta^D(x-x')$ part with $\ln(a)$-dependent contributions. The work analyzes stability and links to stochastic inflation, suggesting possible initial-state modifications to account for exponentially decaying terms, and sets the stage for applying these results to the quantum-corrected field equations governing inflationary dynamics. Overall, the results demonstrate how inflationary expansion generates and regulates quantum corrections to scalar dynamics in a cosmological setting, with implications for early-universe phenomenology and stochastic approaches.

Abstract

We work in the locally de Sitter background of an inflating universe and consider a massless, minimally coupled scalar with a quartic self-interaction. We use dimensional regularization to compute the fully renormalized scalar self-mass-squared at one and two loop order for a state which is released in Bunch-Davies vacuum at t=0. Although the field strength and coupling constant renormalizations are identical to those of lfat space, the geometry induces a non-zero mass renormalization. The finite part also shows a sort of growing mass that competes with the classical force in eventually turning off this system's super-acceleration.

Two Loop Scalar Self-Mass during Inflation

TL;DR

The paper performs a complete two-loop computation of the fully renormalized scalar self-mass-squared for a massless, minimally coupled field during inflation in a locally de Sitter background, employing the Schwinger-Keldysh formalism and dimensional regularization. It shows that curvature induces a nonzero mass renormalization and a time-dependent finite piece at one loop, and reveals a rich, causal nonlocal structure at two loops with overlapping divergences canceled by counterterms , , and ; the final two-loop self-mass-squared comprises nonlocal derivative kernels and a local part with -dependent contributions. The work analyzes stability and links to stochastic inflation, suggesting possible initial-state modifications to account for exponentially decaying terms, and sets the stage for applying these results to the quantum-corrected field equations governing inflationary dynamics. Overall, the results demonstrate how inflationary expansion generates and regulates quantum corrections to scalar dynamics in a cosmological setting, with implications for early-universe phenomenology and stochastic approaches.

Abstract

We work in the locally de Sitter background of an inflating universe and consider a massless, minimally coupled scalar with a quartic self-interaction. We use dimensional regularization to compute the fully renormalized scalar self-mass-squared at one and two loop order for a state which is released in Bunch-Davies vacuum at t=0. Although the field strength and coupling constant renormalizations are identical to those of lfat space, the geometry induces a non-zero mass renormalization. The finite part also shows a sort of growing mass that competes with the classical force in eventually turning off this system's super-acceleration.

Paper Structure

This paper contains 13 sections, 103 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Feynman Rules in the Schwinger-Keldysh Formalism
  3. One Loop Counterterms
  4. The Sunset Diagram
  5. The Snowman Diagram
  6. The Counter Diagrams
  7. The Fully Renormalized Result
  8. Discussion
  9. Appendix
  10. Partial Integration Identities
  11. The Square of a Propagator
  12. The Cube of a Propagator
  13. Evaluating the Integrals for the Snowman Diagram

Figures (5)

  • Figure 1: Generic one loop self-mass-squared.
  • Figure 2: Generic tree and one loop vertex function.
  • Figure 3: Generic Sunset Diagram.
  • Figure 4: Generic Snowman Diagram with mass counterterm.
  • Figure 5: Generic Counter Diagrams. The one on the left is the same as Fig. \ref{['fig1']} but with $\delta \lambda$ in place of $\lambda$. The one on the right is the field strength renormalization.