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Gravity and the Hierarchy Problem

Thede de Boer, Jisuke Kubo, Manfred Lindner, Markus Reinig

TL;DR

The paper addresses dynamically generating the Planck mass and solving the Higgs hierarchy problem within a scale-invariant gravity framework. A Standard Model conformal sector is paired with a non-Abelian G-sector and a quadratic gravity term, such that the Einstein-Hilbert action and a gravitationally induced Higgs mass arise through dimensional transmutation and loop-level couplings, without a fundamental portal between sectors. Inflation is Starobinsky-like in a semi-conformal regime, with gamma fixed by the CMB amplitude, while a Weyl-squared term introduces a spin-two ghost whose unitarity is discussed and mitigated by various proposals; the G-sector also provides massless Nambu-Goldstone bosons that behave as dark radiation. The mechanism offers a concrete link between Planck-scale emergence, inflation, and Higgs naturalness, with potential generalizations to broader BSM scenarios and inviting further non-perturbative studies.

Abstract

We propose a mechanism where the dynamical generation of the Planck mass in scale invariant gravity leads to Einstein gravity, successful inflation and an explanation of the hierarchy problem of the Standard Model. We will discuss the scale generation by dynamical symmetry breaking and phenomenological consequences.

Gravity and the Hierarchy Problem

TL;DR

The paper addresses dynamically generating the Planck mass and solving the Higgs hierarchy problem within a scale-invariant gravity framework. A Standard Model conformal sector is paired with a non-Abelian G-sector and a quadratic gravity term, such that the Einstein-Hilbert action and a gravitationally induced Higgs mass arise through dimensional transmutation and loop-level couplings, without a fundamental portal between sectors. Inflation is Starobinsky-like in a semi-conformal regime, with gamma fixed by the CMB amplitude, while a Weyl-squared term introduces a spin-two ghost whose unitarity is discussed and mitigated by various proposals; the G-sector also provides massless Nambu-Goldstone bosons that behave as dark radiation. The mechanism offers a concrete link between Planck-scale emergence, inflation, and Higgs naturalness, with potential generalizations to broader BSM scenarios and inviting further non-perturbative studies.

Abstract

We propose a mechanism where the dynamical generation of the Planck mass in scale invariant gravity leads to Einstein gravity, successful inflation and an explanation of the hierarchy problem of the Standard Model. We will discuss the scale generation by dynamical symmetry breaking and phenomenological consequences.
Paper Structure (8 sections, 31 equations, 4 figures)

This paper contains 8 sections, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Generating the Planck mass and inducing the Higgs mass
  4. Inflation and phenomenological consequences
  5. The coefficient gamma of the squared Ricci scalar
  6. The coefficient kappa of the squared Weyl tensor
  7. The Nambu-Goldstone bosons
  8. Summary and conclusions

Figures (4)

  • Figure 1: Generating the Einstein-Hilbert term, where $g_r$ is graviton line. The non-perturbative effect is consolidated by the grey disc, and $\cdots$ stand for higher order terms containing more that three derivatives.
  • Figure 2: Inducing the Higgs mass term. The Einstein-Hilbert term is first formally replaced by $-\frac{\xi_f}{2}f^2 R$, where $f$ is a Froggatt-Nielsen field. The diagram shows schematically the gravitational linking between $H$ and $f$ at one-loop, which induces the Higgs mass term once $\xi_f f^2$ is replaced by $M_\text{Pl}^2$.
  • Figure 3: $\gamma$ vs $N_e$.
  • Figure 4: $T_\text{RH}$ and $E_\text{max}$ vs $N_e$ (left), and $m_\text{min}$ vs $N_e$ (right) with ${\cal H}_0= 67.66$ km s$^{-1}$ Mpc$^{-1}$, $k_*=0.002~\text{Mpc}^{-1}$ and $g_\text{RH}=106.75$, where $m_\text{min}$ is the minimum of the induced Higgs mass $(-2 \mu_H^2)^{1/2}$ (for ${\cal C}=1$) and is calculated according to the chain (\ref{['chain']}). $m_\text{mim}$ does not change when we use ${\cal H}_0= (73.0\pm 1.0)~$ km s$^{-1}$ Mpc$^{-1}$ of Riess:2021jrx, because $66.89$ in (\ref{['Ne']}) should be replaced by $66.81$. The blue line denotes the Higgs mass $125$ GeV.