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Sub-Leading Logarithms for Scalar Potential Models on de Sitter

S. P. Miao, N. C. Tsamis, R. P. Woodard

TL;DR

This paper addresses how the first sub-leading infrared logarithm emerges in de Sitter scalar potential models. By combining a carefully renormalized 1-loop effective potential with Starobinsky’s stochastic Langevin framework, the authors identify the specific part of $V'_{\rm eff}$ that yields sub-leading $\ln[a(t)]$ terms and derive explicit expressions for $V'_{\rm eff,1}$, then predict the order-$\lambda$ correction to $\langle \phi^2 \rangle$ within the stochastic approach. The authors verify these stochastic predictions against a fully dimensionally regulated quantum field theory computation of $\langle \phi^2 \rangle$ at order $\lambda$, finding agreement up to a small shift in the stochastic mode-lower limit. The work clarifies how sub-leading infrared logarithms arise from the 1-loop effective potential, discusses ultraviolet sensitivity, and points toward possible RG-improvement to handle derivative interactions in more general theories.

Abstract

The continual production of long wavelength scalars and gravitons during inflation injects secular growth into loop corrections which would be constant in flat space. One typically finds that each additional factor of the loop counting parameter can induce up to a certain number of logarithms of the scale factor. Loop corrections that attain this number are known as ``leading logarithms''; those with fewer are sub-leading. Starobinsky's stochastic formalism has long been known to reproduce the leading logarithms of scalar potential models. We show that the first sub-leading logarithm is captured by applying the stochastic formalism to a certain part of the 1-loop effective potential. This is checked at 2-loops for a massless, minimally coupled scalar with a quartic self-interaction on de Sitter background.

Sub-Leading Logarithms for Scalar Potential Models on de Sitter

TL;DR

This paper addresses how the first sub-leading infrared logarithm emerges in de Sitter scalar potential models. By combining a carefully renormalized 1-loop effective potential with Starobinsky’s stochastic Langevin framework, the authors identify the specific part of that yields sub-leading terms and derive explicit expressions for , then predict the order- correction to within the stochastic approach. The authors verify these stochastic predictions against a fully dimensionally regulated quantum field theory computation of at order , finding agreement up to a small shift in the stochastic mode-lower limit. The work clarifies how sub-leading infrared logarithms arise from the 1-loop effective potential, discusses ultraviolet sensitivity, and points toward possible RG-improvement to handle derivative interactions in more general theories.

Abstract

The continual production of long wavelength scalars and gravitons during inflation injects secular growth into loop corrections which would be constant in flat space. One typically finds that each additional factor of the loop counting parameter can induce up to a certain number of logarithms of the scale factor. Loop corrections that attain this number are known as ``leading logarithms''; those with fewer are sub-leading. Starobinsky's stochastic formalism has long been known to reproduce the leading logarithms of scalar potential models. We show that the first sub-leading logarithm is captured by applying the stochastic formalism to a certain part of the 1-loop effective potential. This is checked at 2-loops for a massless, minimally coupled scalar with a quartic self-interaction on de Sitter background.
Paper Structure (14 sections, 49 equations, 4 figures)

This paper contains 14 sections, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Counterterms
  3. Feynman Rules
  4. The Conformal Counterterm
  5. The Vertex Counterterm
  6. $V_{\rm eff}(\phi)$ at 1-Loop
  7. The Propagator in a Constant Background
  8. 1-Loop Effective Potential
  9. First Sub-Leading Logarithm Contribution
  10. Stochastic Predictions at Order $\lambda$
  11. $\langle \Omega \vert \phi^2(t,\vec{x}) \vert \Omega \rangle$ at Order $\lambda$
  12. The Schwinger-Keldysh Formalism
  13. Dimensionally Regulated Computation
  14. Conclusions

Figures (4)

  • Figure 1: 1-loop contributions to the scalar self-mass. The leftmost diagram is the primitive contribution and the rightmost one is the conformal counterterm.
  • Figure 2: 1-loop contributions to the $s$-channel vertex function. The leftmost diagram is the primitive contribution and the rightmost one is the vertex counterterm.
  • Figure 3: One term in the sum over 1PI vacuum graphs that contributes to the 1-loop effective potential. Each vertex contributes a factor of $\frac{\lambda}{2} \phi^2$.
  • Figure 4: Order $\lambda$ contributions to $\langle \Omega \vert \phi^2(x) \vert \Omega \rangle$. The rightmost diagram is the primitive contribution and the leftmost one is from the conformal counterterm.