[Paper Review] Emergent Electroweak Gravity

Bob McElrath|ArXiv.org|Dec 15, 2008

Particle physics theoretical and experimental studies9 references3 citations

TL;DR

This paper proposes that massive cosmological relics, such as weakly interacting massive particles (WIMPs) or cosmic neutrinos, form a superfluid state due to wave packet broadening and negligible elastic scattering, leading to emergent gravity via a Goldstone mechanism. The condensate of fermions with repulsive interactions (e.g., via Z⁰ exchange) realizes the vierbein of general relativity, and long-wavelength fluctuations yield a spin-2 graviton, resulting in a low-energy gravitational theory with Newtonian strength arising from the Standard Model’s electroweak sector.

ABSTRACT

We show that any massive cosmological relic particle with small self-interactions is a super-fluid today, due to the broadening of its wave packet, and lack of any elastic scattering. The WIMP dark matter picture is only consistent its mass $M \gg M_{ m Pl}$ in order to maintain classicality. The dynamics of a super-fluid are given by the excitation spectrum of bound state quasi-particles, rather than the center of mass motion of constituent particles. If this relic is a fermion with a repulsive interaction mediated by a heavy boson, such as neutrinos interacting via the $Z^0$, the condensate has the same quantum numbers as the vierbein of General Relativity. Because there exists an enhanced global symmetry $SO(3,1)_{space} imes SO(3,1)_{spin}$ among the fermion's self-interactions broken only by its kinetic term, the long wavelength fluctuation around this condensate is a Goldstone graviton. A gravitational theory exists in the low energy limit of the Standard Model's Electroweak sector below the weak scale, with a strength that is parametrically similar to $G_N$.

Motivation & Objective

• To resolve the inconsistency of WIMP dark matter scenarios by showing that WIMPs cannot remain both localized and decoupled over cosmological timescales.
• To demonstrate that massive, weakly interacting relics naturally form a superfluid due to wave packet broadening and lack of scattering.
• To establish that the electroweak sector of the Standard Model can give rise to emergent gravity through spontaneous symmetry breaking of an enhanced SO(3,1)×SO(3,1) global symmetry.
• To show that the long-wavelength fluctuations of the superfluid condensate yield a massless spin-2 Goldstone boson identified as a graviton, leading to a low-energy gravitational theory.
• To argue that this emergent gravity is consistent with the Weinberg-Witten theorem due to the presence of a physical background and broken diffeomorphism invariance.

Proposed method

• Modeling relic particles as massive fermions with small self-interactions, showing that wave packet broadening dominates over scattering in the absence of elastic collisions.
• Applying the quantum liquid condition Δx ≫ n⁻¹ᐟ³ to determine when collective behavior and superfluidity emerge, using the time evolution of free-particle wave packets.
• Identifying the condensate of fermions with repulsive Z⁰- mediated interactions as a state with the quantum numbers of the vierbein (tetrad) field in general relativity.
• Analyzing the enhanced global symmetry SO(3,1)ₐₗₐcₑ × SO(3,1)ₛₚᵢₙ, broken only by the kinetic term, leading to a Goldstone mechanism for the emergence of a spin-2 graviton.
• Deriving the effective action for the condensate field Ẽᵃᵤ, showing it acts as the order parameter for symmetry breaking and gives rise to the metric gᵤᵥ = ẼᵃᵤẼᵇᵥηₐ_b.
• Establishing that the emergent gravity is not diffeomorphism-invariant due to the physical background, thus evading the Weinberg-Witten theorem.

Experimental results

Research questions

• RQ1 Can massive cosmological relics like WIMPs or neutrinos form a superfluid state due to wave packet broadening and negligible scattering?
• RQ2 Does the electroweak sector of the Standard Model, when extended to include such relics, give rise to an emergent gravitational theory?
• RQ3 Can the condensate of fermions with Z⁰-mediated repulsive interactions realize the vierbein of general relativity?
• RQ4 What is the role of the enhanced SO(3,1)×SO(3,1) global symmetry in the emergence of a Goldstone graviton?
• RQ5 How does the emergent gravity in this model evade the Weinberg-Witten theorem despite being a spin-2 theory?

Key findings

• Cosmological relics with M ≫ M_Pl are required for classicality; otherwise, wave packet broadening leads to superfluidity, invalidating the WIMP dark matter picture.
• The quantum liquid condition Δx ≫ n⁻¹ᐟ³ is satisfied for decoupled relics when the cross section σ < λ(1−v²)/n²ᐟ³, which holds for weakly interacting particles.
• The fermion condensate acquires the quantum numbers of the vierbein, with Ẽᵃᵤ = ⟨ψ̄γᵃσᵘψ⟩ forming the order parameter for SO(3,1)×SO(3,1) symmetry breaking.
• The long-wavelength fluctuations of the condensate produce a massless spin-2 Goldstone boson gᵤᵥ = ẼᵃᵤẼᵇᵥηₐ_b, identified as the graviton.
• The emergent gravitational theory has a strength parametrically similar to Newton’s constant G_N, arising from the electroweak sector without new physics.
• The theory is not diffeomorphism-invariant due to the physical background and nonlocality of the vierbein, thus avoiding the Weinberg-Witten theorem.