Small vacuum energy from small equivalence violation in scalar gravity

TL;DR

The paper gives a six-dimensional gauge-theory origin for the small shift-symmetry violation needed in a scalar-gravity solution to the cosmological constant problem, where the Goldberger-Wise field is a pseudo-Nambu-Goldstone boson. It argues that the equivalence-principle violation and the vacuum energy seen by scalar gravity can naturally be exponentially small.

Key findings

• Scalar gravity, the Nordström/Einstein-Fokker theory, faithfully reproduces qualitative features of the cosmological constant problem, and the Weinberg no-go theorem extends to it; a proposed solution uses a very small violation of the scalar equivalence principle via a CFT deformation and Goldberger-Wise radion stabilization with a pNGB.
• In the 6D construction, the Goldberger-Wise field is identified with the sixth component of a gauge field, \(A_6\), and its bulk potential arises from non-local Aharonov-Bohm phases: \(V_{\rm bulk}(\rho,\omega) \simeq \epsilon M_6^2 \cos(2\pi e\omega)\), with \(\epsilon = \frac{1}{2\pi\rho M_6^6}\frac{m_\xi}{4\pi^2\rho^{5/2}} e^{-2\pi\rho m_\xi}\), exponentially suppressed when \(2\pi\rho m_\xi \gg 1\).
• Dimensional reduction gives the desired 5D effective action with an approximate shift symmetry; higher-dimensional operators and corrections are suppressed by \(\epsilon\), so the dilaton potential remains of the form \(\kappa(g(\phi))\phi^4\) and the \(\beta\)-function stays parametrically small.
• The paper notes that scalar gravity replicates the hierarchy \(\lambda \sim 10^{-120}\), and that the 6D parameters can yield \(\epsilon \sim 10^{-120}\) with \(N=\rho m_\xi \sim 100\), under mild hierarchies \(M_5 > M_6 > m_\xi > 1/\rho > k_5 \gg M_{\rm pl}\).

Methods

• Theoretical effective-field-theory construction: review of the 4D dilaton potential and CFT deformation, the 5D AdS/CFT dual of RS1 without a UV brane, and a 6D warped effective theory with a compact sixth dimension, gauge field \(A_6\), heavy charged scalar, and axion.
• Derives equations of motion and junction conditions, solves perturbatively in the backreaction parameter \(\eta\) and small explicit breaking \(\epsilon\), and matches the 6D solution to the 5D effective action and radion/dilaton potential.
• No experimental data or numerical simulation; the analysis is analytic and based on effective field theory.

Limitations

• Stated: a full solution must also address cosmological evolution including obtaining a hot big bang, left for future work; embedding in a full string theory UV completion is beyond scope; translating the \(A_6\) mechanism into 4D QFT language is left for future work; scalar gravity itself is experimentally ruled out, e.g. light does not bend in a scalar gravity field.
• Stated/evident: the 5D analysis assumes small backreaction, small \(\omega_*\), and neglects higher-order potential terms; perturbation in \(\eta\) requires a shift symmetry to ensure higher-order corrections remain higher order in \(\epsilon\); quantum-gravity breaking of global symmetries motivates the 6D gauge construction, but a full quantum-gravity completion is not provided.
• Evident: no observational tests or quantitative cosmological predictions are presented; the mechanism addresses particle-physics fine-tuning rather than cosmological dynamics.