Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Inflation with Non-minimal Coupling

Mark P. Hertzberg

TL;DR

This work analyzes inflation models with a non-minimal coupling $\xi$ to gravity, showing a stark split between single-field and multi-field cases. In the single-field scenario, gravity/kinetic sectors stay perturbative up to the Planck scale, while the potential can induce breakdown at high energies; in multi-field models, strong quantum corrections appear in the gravity/kinetic sectors and push the effective cutoff to $\Lambda \sim m_{\mathrm{Pl}}/\xi$, effectively ruling out Higgs-like inflaton scenarios. Curvature-squared inflation (Starobinsky) can be recast as a scalar-tensor theory and remains perturbatively well-behaved with a Planck-scale cutoff, in contrast to claims of early breakdown; these conclusions hold in both the Jordan and Einstein frames and are illuminated by a pion-Lagrangian analogy. The results imply limited prospects for embedding inflation in multi-field SM-like sectors while preserving EFT control, and they clarify the frame-independence of the conclusions.

Abstract

A simple realization of inflation consists of adding the following operators to the Einstein-Hilbert action: (partial phi)^2, lambda phi^4, and xi phi^2 R, with xi a large non-minimal coupling. Recently there has been much discussion as to whether such theories make sense quantum mechanically and if the inflaton phi can also be the Standard Model Higgs. In this note we answer these questions. Firstly, for a single scalar phi, we show that the quantum field theory is well behaved in the pure gravity and kinetic sectors, since the quantum generated corrections are small. However, the theory likely breaks down at ~ m_pl / xi due to scattering provided by the self-interacting potential lambda phi^4. Secondly, we show that the theory changes for multiple scalars phi with non-minimal coupling xi phi dot phi R, since this introduces qualitatively new interactions which manifestly generate large quantum corrections even in the gravity and kinetic sectors, spoiling the theory for energies > m_pl / xi. Since the Higgs doublet of the Standard Model includes the Higgs boson and 3 Goldstone bosons, it falls into the latter category and therefore its validity is manifestly spoiled. We show that these conclusions hold in both the Jordan and Einstein frames and describe an intuitive analogy in the form of the pion Lagrangian. We also examine the recent claim that curvature-squared inflation models fail quantum mechanically. Our work appears to go beyond the recent discussions.

On Inflation with Non-minimal Coupling

TL;DR

This work analyzes inflation models with a non-minimal coupling to gravity, showing a stark split between single-field and multi-field cases. In the single-field scenario, gravity/kinetic sectors stay perturbative up to the Planck scale, while the potential can induce breakdown at high energies; in multi-field models, strong quantum corrections appear in the gravity/kinetic sectors and push the effective cutoff to , effectively ruling out Higgs-like inflaton scenarios. Curvature-squared inflation (Starobinsky) can be recast as a scalar-tensor theory and remains perturbatively well-behaved with a Planck-scale cutoff, in contrast to claims of early breakdown; these conclusions hold in both the Jordan and Einstein frames and are illuminated by a pion-Lagrangian analogy. The results imply limited prospects for embedding inflation in multi-field SM-like sectors while preserving EFT control, and they clarify the frame-independence of the conclusions.

Abstract

A simple realization of inflation consists of adding the following operators to the Einstein-Hilbert action: (partial phi)^2, lambda phi^4, and xi phi^2 R, with xi a large non-minimal coupling. Recently there has been much discussion as to whether such theories make sense quantum mechanically and if the inflaton phi can also be the Standard Model Higgs. In this note we answer these questions. Firstly, for a single scalar phi, we show that the quantum field theory is well behaved in the pure gravity and kinetic sectors, since the quantum generated corrections are small. However, the theory likely breaks down at ~ m_pl / xi due to scattering provided by the self-interacting potential lambda phi^4. Secondly, we show that the theory changes for multiple scalars phi with non-minimal coupling xi phi dot phi R, since this introduces qualitatively new interactions which manifestly generate large quantum corrections even in the gravity and kinetic sectors, spoiling the theory for energies > m_pl / xi. Since the Higgs doublet of the Standard Model includes the Higgs boson and 3 Goldstone bosons, it falls into the latter category and therefore its validity is manifestly spoiled. We show that these conclusions hold in both the Jordan and Einstein frames and describe an intuitive analogy in the form of the pion Lagrangian. We also examine the recent claim that curvature-squared inflation models fail quantum mechanically. Our work appears to go beyond the recent discussions.

Paper Structure

This paper contains 9 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-minimally Coupled Models
  3. Quantum Corrections
  4. Single Field
  5. The Potential
  6. Multiple Fields
  7. An Analogy - the Pion Lagrangian
  8. Curvature-Squared Models
  9. Conclusions

Figures (2)

  • Figure 1: Tree-level scattering $2\phi\to2\phi$. Diagrams 1--3: In Jordan frame it is due to graviton exchange through t, u, and s-channels. Diagram 4: In Einstein frame it is due to a single 4-point vertex.
  • Figure 2: One-loop scattering $2\phi\to2\phi$. Diagrams 1--9 (left block): In Jordan frame it is due to graviton exchange. Diagrams 10--12 (right block): In Einstein frame it is due to a 4-point vertex. Top row: t-channel, middle row: u-channel, bottom row: s-channel.