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The cosmon-Higgs coupling pertains to theoretical frameworks in which the Standard Model Higgs field interacts nonminimally with spacetime curvature, typically via an additional Jordan–Brans–Dicke (JBD)–type term. In these models, the Higgs doublet field is promoted to the role of a dynamical scalar that modulates the effective gravitational constant and participates in cosmic evolution. The hybrid JBD-Higgs scenario leads to curvature-dependent symmetry breaking scales, Higgs mass variations, and novel cosmological dynamics, including a phase where gravity becomes repulsive. Detailed realization of these phenomena is found in the model of Arık & Tok ("Higgs mode of modified cosmology"), which integrates the Standard Model with a JBD scalar sector, generating a dynamic and curvature-sensitive interplay between the Higgs field and gravitation (Arik et al., 18 Mar 2025).

1. Lagrangian and Action Structure

The foundational action for the cosmon-Higgs system in the JBD-Higgs framework comprises the Einstein–Hilbert term, the nonminimal JBD–type coupling, Higgs sector dynamics, and the Standard Model content. In natural units ($c = \hbar = 1$), the key fields are the metric $g_{\mu\nu}$, Planck mass $M_P^2 = 1/(8\pi G_N)$, the Brans–Dicke scalar $\phi$, and the Higgs doublet $H$.

In unitary gauge ($H^\dagger H = \phi^2/2$), the Lagrangian density is: $$\mathcal{L} = \sqrt{-g} \left\{ \frac{1}{2} \left( M_P^2 + \frac{\phi^2}{2\omega} \right) R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{m^2}{2} \phi^2 - \frac{\lambda}{4} \phi^4 - A \right\}$$ Here, $\omega$ is the Brans–Dicke parameter, $R$ the Ricci scalar, $m$ and $\lambda$ are the Higgs mass and self-coupling, and $A$ a cosmological constant component. The $\phi^2 R$ term realizes the cosmon-Higgs coupling, making the effective gravitational sector scalar-dependent.

2. Effective Potential and Curvature Dependence

The effective potential in this scheme is explicitly curvature dependent and governed by both the Ricci scalar $R$ and the parameter $\omega$. Extracting the relevant terms, the effective potential is: $$V_{\text{eff}}(\phi, R) = \Bigl( \frac{\omega}{2} + 1 \Bigr) R \phi^2 + A - m^2 \phi^2 + \lambda \phi^4$$ This structure indicates that the vacuum expectation value (VEV) and mass of the Higgs are not universal constants but instead are dynamical functions sensitive to both large-scale geometry and scalar sector parameters. The Brans–Dicke parameter $\omega$ modulates the curvature-induced modifications to the Higgs mass threshold, directly influencing symmetry-breaking conditions.

3. Symmetry Breaking, Higgs Mass, and Vacuum Expectation Value

Minimization of the effective potential with respect to $\phi$ leads to the vacuum condition: $$\frac{\partial V_{\text{eff}}}{\partial\phi} = 0 \;\;\Longrightarrow\;\; 2\bigl((\tfrac{\omega}{2}+1)R-m^2\bigr)\,\phi + 4\lambda\,\phi^3 = 0$$ Solving for the nontrivial vacuum yields

$$\phi^2 = \frac{m^2 - (\frac{\omega}{2} + 1) R}{2\lambda}$$

Hence, the electroweak VEV is

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

The physical Higgs mass becomes curvature and parameter dependent: $$m_H(R, \omega) = 2\sqrt{m^2 - (\frac{\omega}{2} + 1)R}$$ Thus, both the scale of symmetry breaking and the Higgs mass are dynamical quantities dictated by cosmic evolution.

4. Cosmological Solutions and Dynamical Newton’s Constant

Variation of the action with respect to the metric yields a modified Friedmann equation where the effective Newton’s constant is dynamically set by the scalar field: $$G_{\text{eff}}(\phi) = \left(M_P^2 + \frac{\phi^2}{2\omega}\right)^{-1}$$ or, in terms of $G_N$,

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

For negative $\omega$, this induces a notable cosmological scenario:

• As the universe emerges from the initial singularity ($a=0$, $\phi \to 0$): $G_{\text{eff}} \to 0$ (vanishing gravitational strength; “first Big Bang”).
• As $\phi$ increases to $\phi_c^2 = -2\omega M_P^2$: $G_{\text{eff}} < 0$ (a repulsive gravity era).
• At $\phi = \phi_c$, $G_{\text{eff}} \to \pm\infty$ (“second Big Bang”).
• For $\phi > \phi_c$, standard gravity ($G_{\text{eff}} > 0$) is restored.

Open FLRW models with negative $\omega$ (specifically $\omega = -3/2$) realize a two–Big–Bang sequence, while closed and flat universes favor other behaviors or parameter constraints.

5. Physical Implications and Dynamical Regimes

The cosmon-Higgs coupling yields several interlinked physical consequences:

• The effective gravitational coupling is rendered temporally variable and sign-changing, introducing a natural repulsive phase that may contribute to early-universe homogenization and isotropization.
• There is an intrinsic link between large-scale geometry and microscopic electroweak properties: both the Higgs VEV and mass are curvature- and $\omega$-dependent.
• At late cosmological times (large scale factor $a$, large $\phi$), the nonminimal contribution ($\phi^2/(2\omega)$) becomes negligible, and standard Einstein gravity with fixed Newton’s constant and conventional Higgs physics is asymptotically recovered.
• The model exemplifies settings where the Higgs field serves the dual role of the “cosmon” in cosmological scalar field models, thereby unifying gravitational-coupling evolution, symmetry breaking, and cosmic acceleration within a single sector (Arik et al., 18 Mar 2025).

6. Broader Context and Theoretical Significance

The cosmon-Higgs mechanism, as realized in the JBD-Higgs hybrid, directly incorporates scalar-tensor modifications of gravity into the Higgs sector, allowing for dynamical interplay between fundamental constants and cosmological evolution. It bridges ideas from quintessence, gravitational scalarization, and standard model Higgs physics. The curvature-induced transitions and temporally varying Newton’s constant are distinctive, offering unique testable implications for early-universe physics, cosmic inflation, and potential signals in precision gravitational and cosmological observations.

A plausible implication is that such frameworks provide a robust setting for revisiting signatures of non-standard gravitational regimes in the pre-inflationary universe, as well as for examining how cosmic history may imprint itself on low-energy parameters of the Standard Model through dynamical symmetry breaking coupled to spacetime curvature (Arik et al., 18 Mar 2025).