[Paper Review] Inertial Symmetry Breaking

Christopher T. Hill|arXiv (Cornell University)|Mar 19, 2018

Quantum chaos and dynamical systems5 citations

TL;DR

This paper proposes 'inertial symmetry breaking'—a mechanism in Weyl-invariant theories where scale and gauge symmetries break dynamically due to the universe's expansion, without requiring a potential. The Weyl current's kernel, $ K $, evolves to a constant during pre-inflationary expansion, generating the Planck mass $ M_{\text{Pl}} \propto \sqrt{\overline{K}} $, which acts as the order parameter, enabling spontaneous breaking of scale and $ U(1) $ symmetries while preserving Weyl invariance in the quantum theory via field VEV-dependent renormalization.

ABSTRACT

We review and expand upon recent work demonstrating that Weyl invariant theories can be broken "inertially," which does not depend upon a potential. This can be understood in a general way by the "current algebra" of these theories, independently of specific Lagrangians. Maintaining the exact Weyl invariance in a renormalized quantum theory can be accomplished by renormalization conditions that refer back to the VEV's of fields in the action. We illustrate the computation of a Weyl invariant Coleman-Weinberg potential that breaks a U(1) symmetry together,with scale invariance.

Motivation & Objective

• To propose a new mechanism for spontaneous symmetry breaking in Weyl-invariant theories that does not rely on scalar potentials.
• To identify the kernel $ K $ of the Weyl current as the dynamical order parameter for scale symmetry breaking.
• To demonstrate how the Planck mass emerges naturally from the time evolution of $ K $ during cosmic expansion.
• To show that $ U(1) $ symmetry breaking can occur via this inertial mechanism, even in the absence of a potential.
• To develop a Weyl-invariant Coleman-Weinberg potential using field VEVs as renormalization scales.

Proposed method

• The Weyl current $ K_{\mu} = \partial_{\mu}K $ is analyzed in the Jordan frame, where Weyl invariance is manifest.
• The time evolution of $ K_{0} \propto a(t)^{-3} $ is shown to dilute to zero during pre-inflationary expansion, driving $ K \to \overline{K} $, a constant.
• The constant $ \overline{K} $ is identified as the order parameter for scale symmetry breaking, with $ M_{\text{Pl}}^2 \propto \overline{K} $.
• For multi-scalar theories, $ K = \sum_i (1 - \alpha_i)\phi_i^2 / 2 $, leading to an ellipsoidal constraint on VEVs and a dynamically generated Planck scale.
• A Weyl-invariant Coleman-Weinberg potential is computed using field VEVs as renormalization scales, ensuring consistency with Weyl invariance.
• The mechanism is applied to a $ U(1) $-gauge theory, showing that the Higgs field becomes the massless dilaton, while the $ U(1) $ symmetry is broken inertially.

Experimental results

Research questions

• RQ1 How can Weyl symmetry be spontaneously broken without a scalar potential?
• RQ2 What is the dynamical order parameter for scale symmetry breaking in a Weyl-invariant quantum field theory?
• RQ3 How does the Planck mass emerge dynamically from cosmological expansion in the Jordan frame?
• RQ4 Can gauge symmetries like $ U(1) $ be broken via inertial symmetry breaking, independent of potentials?
• RQ5 How can Weyl-invariant quantum field theories be consistently renormalized when mass scales arise dynamically?

Key findings

• The kernel $ K $ of the Weyl current evolves to a constant $ \overline{K} $ during pre-inflationary expansion, which directly determines the Planck mass via $ M_{\text{Pl}}^2 \propto \overline{K} $.
• Inertial symmetry breaking occurs without any potential, relying solely on the dilution of the Weyl current density $ K_0 \sim a(t)^{-3} $.
• For a single scalar field with non-minimal coupling $ \alpha < 0 $, the mechanism generates a positive Planck mass and leads to eternal inflation.
• In multi-scalar theories, the VEVs of scalar fields are constrained to an ellipsoid defined by $ \sum_i (1 - \alpha_i)\phi_i^2 = 2\overline{K} $, ensuring a consistent Planck scale.
• A Weyl-invariant Coleman-Weinberg potential is constructed by using the VEVs of fields as renormalization scales, with $ f^2 = 2\overline{K} $ as the natural renormalization scale.
• The $ U(1) $ symmetry is broken solely by the inertial mechanism, with the Higgs boson becoming the massless dilaton, and the cosmological constant vanishes due to flat directions.