Cosmon–Higgs Coupling in Scalar-Tensor Gravity

• Cosmon-Higgs coupling is a nonminimal interaction between the Higgs doublet and a JBD-type scalar field that dynamically modulates gravity and electroweak symmetry breaking.
• The model employs a curvature-dependent effective potential to adjust the Higgs vacuum expectation value and mass, linking cosmic curvature with particle physics.
• Cosmologically, the framework predicts novel phases including a time-varying, sign-changing gravitational constant, with epochs of repulsive gravity and multiple Big Bang singularities.

The cosmon-Higgs coupling refers to a nonminimal interaction between the Standard Model Higgs sector and gravity, modeled by incorporating the Higgs doublet with a Jordan-Brans-Dicke (JBD)–type scalar field, leading to a scalar-tensor theory where the Higgs field dynamics directly influence the gravitational sector. In this framework, the Higgs field acts as the “cosmon”—a dynamical scalar field modulating the effective gravitational coupling and symmetry breaking in the early universe. This coupling gives rise to a time-varying and even sign-changing effective gravitational constant, modifies electroweak symmetry breaking, and produces a nontrivial cosmological history, including distinct phases with repulsive gravity and multiple Big Bang singularities (Arik et al., 18 Mar 2025).

1. Action and Lagrangian Structure

The cosmon-Higgs coupling is formulated by extending the Einstein-Hilbert action to include both a Brans-Dicke–type scalar and the Standard Model Higgs doublet. In the unitary gauge, the Higgs field $H$ is represented as $H = (0,\,\phi/\sqrt{2})^T$, with $\phi$ both the physical Higgs and the Brans-Dicke scalar. The Lagrangian density becomes

$$\mathcal{L} = \sqrt{-g} \left\{ \frac{1}{2} \left( M_P^2 + \frac{\phi^2}{2\omega} \right) R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi\, \partial_\nu \phi - \frac{m^2}{2} \phi^2 - \frac{\lambda}{4} \phi^4 - A \right\},$$

where $M_P$ is the reduced Planck mass, $R$ the Ricci scalar, $\omega$ the Brans-Dicke parameter, $m$ and $\lambda$ the Higgs mass and quartic couplings, and $A$ the cosmological constant term. The term $\phi^2 R/(2\omega)$ constitutes the essential cosmon-Higgs (nonminimal) coupling that distinguishes this model from canonical scalar-tensor or Higgs cosmologies (Arik et al., 18 Mar 2025).

2. Curvature-Dependent Effective Potential

The model yields a curvature-dependent effective potential for the scalar field:

$$V_{\rm eff}(\phi, R) = \left( \frac{\omega}{2} + 1 \right) R \phi^2 + A - m^2 \phi^2 + \lambda \phi^4.$$

Here, the Brans-Dicke parameter $\omega$ determines the coupling between the Ricci curvature and the quadratic Higgs term, leading to a dynamical mass shift for the Higgs field as the curvature evolves. The effective potential encodes how cosmic expansion and local curvature directly alter symmetry-breaking dynamics and scalar sector spectra (Arik et al., 18 Mar 2025).

3. Spontaneous Symmetry Breaking and Higgs Sector Dynamics

Minimization of $V_{\rm eff}(\phi, R)$ yields a curvature- and $\omega$-dependent vacuum expectation value (VEV) and mass for the Higgs mode:

$$\langle H \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \ v \end{pmatrix}, \qquad v = \sqrt{ \frac{m^2 - (\frac{\omega}{2} + 1) R } {2\lambda} },$$

with physical Higgs mass

$$m_H(R, \omega) = 2 \sqrt{ m^2 - (\tfrac{\omega}{2} + 1) R }.$$

Thus, both the Higgs VEV and its mass are dynamical, functions of spacetime curvature and the Brans-Dicke parameter, tying the onset and strength of electroweak symmetry breaking directly to the geometric and scalar structure of the universe. This interdependence is not present in Standard Model cosmology or minimal scalar-tensor gravity models.

4. Cosmological Consequences: Two Big Bangs and $G_{\rm eff}$ Evolution

The nonminimal scalar-gravity coupling leads to a generalized Friedmann equation with a dynamical gravitational “constant”:

$$G_{\rm eff} = \left[ \frac{1}{G_N} + \frac{2 \phi^2}{\omega} \right]^{-1},$$

with $G_N$ the usual Newton’s constant. The cosmological evolution in this framework can be characterized as follows for negative $\omega$:

• For vanishing scale factor ($a \to 0$, $\phi \to 0$): $G_{\rm eff} \to 0$—the first Big Bang, with vanishing strength of gravity.
• For $0 < \phi < \phi_c$ ($\phi_c^2 = -2\omega M_P^2$): $G_{\rm eff} < 0$—the universe enters a repulsive-gravity epoch.
• As $\phi \to \phi_c$: The denominator crosses zero, so $G_{\rm eff} \to \pm\infty$—interpreted as a second singularity, or “second Big Bang.”
• For $\phi > \phi_c$: $G_{\rm eff} > 0$—standard gravitational dynamics resume.

Explicit solutions using ansätze such as $H^2(a) = h_1 a^{-4} + h_2 a^{-3} + h_3$ and $\phi(a) = f_1 + f_2 a$ demonstrate this sequence in open FLRW universes (forcing $\omega = -3/2$), but not in closed or flat cases, where energy conditions or signature requirements rule out analogous solutions (Arik et al., 18 Mar 2025).

5. Physical Implications and Interpretive Context

The cosmon-Higgs coupling framework introduces several novel cosmological phenomena:

• A time-varying, sign-changing effective Newton constant ($G_{\rm eff}$) supports a transient repulsive phase, conducive to homogenizing the early universe.
• The Higgs VEV and mass become dynamical, curvature-dependent quantities, intimately connecting cosmic geometry with electroweak-scale physics.
• At late times (large $a$, large $\phi$), the nonminimal coupling dilutes, and ordinary Einstein gravity and Standard Model Higgs physics are recovered.

A plausible implication is that the cosmon-Higgs model realizes a unification of gravitational-coupling evolution, symmetry breaking, and cosmic acceleration in a single scalar sector, and that the Higgs field itself may serve as the archetypal “cosmon” field relevant for broader quintessence and modified gravity paradigms.

6. Relation to Broader Scalar-Tensor and Quintessence Models

In the broader context, the cosmon-Higgs coupling is a particular realization of scalar-tensor gravity theories wherein the scalar is not an additional field but the Standard Model Higgs itself. This approach provides a natural mechanism for time-varying gravitational coupling and curvature-driven symmetry breaking. It differs from conventional quintessence mechanisms, where an independent light scalar is introduced, by identifying this field with a known Standard Model degree of freedom and placing strong model-dependent constraints on the coupling and potential structure via Standard Model parameters and observed Higgs properties (Arik et al., 18 Mar 2025).