Mass-Varying Neutrino in sGB Gravity

• The mass-varying neutrino scenario is a model where neutrino masses arise dynamically from a scalar field coupled to both neutrinos and curvature, leading to environmental dependence.
• In the scalar–Gauss–Bonnet framework, the scalar field’s quadratic coupling to curvature creates position-dependent mass-squared splittings that significantly alter neutrino oscillation patterns.
• Solar neutrino experiments constrain the model parameters, offering empirical tests for deviations from standard oscillation models and insights into dark matter and dark energy links.

The mass-varying neutrino (MaVaN) scenario posits that neutrinos acquire their masses dynamically through coupling with a scalar field, such that neutrino masses become functions of environmental properties or cosmological time. This concept appears in several contexts, notably in attempts to unify dark matter and dark energy, in models of cosmic acceleration, and in exploring the links between neutrino properties and fundamental scalar fields. In the implementation within the Scalar–Gauss–Bonnet (sGB) gravity framework, the relevant scalar couples non-minimally both to the Gauss–Bonnet curvature invariant and directly to neutrinos, resulting in an effective neutrino mass that is sensitive to the local matter density via its influence on the scalar field profile. Experimental tests, particularly using solar neutrino data, provide constraints on the model parameters and probe the possible influence of gravitational phenomena on neutrino oscillations (Sadjad et al., 1 Oct 2025).

1. Scalar–Gauss–Bonnet Action and Coupling to Neutrinos

The theoretical structure is based on a scalar-tensor theory with the action: $$S = \int d^4x \sqrt{-g} \Bigg[ \frac{M_p^2}{2} R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - f(\phi) \mathcal{G} + \kappa_i \phi^2 (\bar{\nu}_i \nu_i) + ... \Bigg]$$ where:

• $M_p$ is the reduced Planck mass,
• $R$ is the Ricci scalar,
• $$\mathcal{G} = R^{2} - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$ is the Gauss–Bonnet invariant,
• $f(\phi) = \frac{1}{2}\xi \phi^2$ describes the quadratic coupling between the scalar and the GB invariant (with coupling parameter $\xi$ of dimension $[{\rm length}]^2$),
• $\kappa_i \phi^2$ yields a scalar-field dependent neutrino mass.

Variation of this action produces two crucial ingredients: modification of the field equation for $\phi$ via the coupling to curvature, and the dependence of the neutrino mass on the scalar field. This structure leads to environmental dependence of neutrino parameters as $\phi$ responds to local matter and curvature.

2. Environmental Dependence of the Scalar Field and Neutrino Mass

The scalar field satisfies the modified Klein–Gordon equation: $\Box \phi - \mathcal{G} f_{,\phi} = 0$ with $f_{,\phi} = \partial f / \partial \phi = \xi \phi$. The solution $\phi(r)$ depends on the background curvature profile, which is itself determined by the local matter density and geometry. In spherically symmetric static backgrounds (e.g., the Sun’s interior/exterior modeled as Schwarzschild), $\phi(r)$ is nearly constant and small in high-density environments, increasing to its asymptotic value outside.

The neutrino mass is set by a quadratic dependence: $m_{\nu, i}(r) = \sqrt{\kappa_i} \, \phi(r)$ Neutrino mass-squared differences acquire an explicit radial dependence: $$\Delta m^2_{21}(r) = \Delta \kappa'^{\,2}_{21}\, \hat{\phi}^4(r)$$ where $\hat{\phi}(r) = \phi(r)/\phi_{\mathrm{in}}(0)$ is the normalized scalar profile, and $\Delta \kappa'^{\,2}_{21}$ absorbs overall normalization and coupling strength. Thus, the MaVaN framework predicts variable neutrino oscillation parameters in regions with variable matter/curvature.

3. Implications for Neutrino Oscillation Phenomenology

Oscillations are governed by a space-dependent Hamiltonian. In the two-flavor ($\nu_e \leftrightarrow \nu_{\mu}$) basis, the vacuum oscillation term is

$$\mathscr{H}_{\mathrm{vac}}(r) = \frac{\Delta\kappa'^{\,2}_{21}\, \hat{\phi}^4(r)}{4E_{\nu}} \begin{pmatrix} -\cos 2\theta & \sin 2\theta \ \sin 2\theta & \cos 2\theta \end{pmatrix}$$

Matter effects are included as usual via the electron density-dependent potential $V_{\mathrm{matter}} = \sqrt{2} G_F N_e(r)$. The key distinction is that $\hat{\phi}(r)$ is small and nearly constant deep inside a dense object, suppressing the effective oscillation phase, and increases outside, restoring standard oscillation behavior. The phase accumulated is

$$\Phi_{21} = \int \frac{\Delta m^2_{21}(r)}{2E_{\nu}}\, dr = \int \frac{\Delta\kappa'^{\,2}_{21} \hat{\phi}^4(r)}{2E_{\nu}}\, dr$$

This predicts environmental suppression of flavor conversion in dense regions and restoration in lower-density regions, thereby modifying survival probabilities in a manner testable by solar neutrino experiments.

4. Constraints from Solar Neutrino Observations

By confronting the model with solar neutrino data (including measurements from Kamiokande, Super-Kamiokande, SNO, Borexino, etc.), best-fit values for the model parameters are extracted. The analysis employs a least-squares fit to the electron neutrino survival probability as a function of energy,

$$P_{ee}(E_\nu) = \frac{1}{2}\left[1 + \cos 2\theta_M\, \cos 2\theta\right]$$

where the effective mixing angle in matter is

$$\tan 2\theta_M = \frac{\sin 2\theta}{\cos 2\theta - A}$$

with

$$A = \frac{2\sqrt{2} G_F N_e E_\nu}{\Delta\kappa'^{\,2}_{21} \hat{\phi}^4}$$

The reliable fit to data places constraints on both $\xi$ (rescaled: $\xi' = \xi/R_\odot^2$) and $\Delta\kappa'^{\,2}_{21}$, e.g., $\xi' \sim 7.8\times10^{11}$, and $$\Delta\kappa'^{\,2}_{21} \sim 1\times10^{-23}\,{\rm eV}^2$$, while the condition $\phi_{\mathrm{in}}(0) \ll 10^{-8} M_p$ ensures the approximate validity of the Schwarzschild solution for the background metric. The sGB-based MaVaN scenario remains fully consistent with the Large Mixing Angle–MSW paradigm, with deviations potentially accessible to future high-precision studies.

5. Comparison to Standard Mass-Varying Neutrino and Oscillation Models

Unlike standard MaVaN scenarios—where a scalar field is coupled directly to neutrinos, but not to curvature—the sGB framework introduces implicit environmental dependence through the non-minimal curvature coupling. The result is that neutrino masses can adjust to the local gravitational and matter content, providing a richer structure for parameter space searches. The quadratic dependence on the scalar and resulting $\hat{\phi}^4$ dependence for mass-squared splittings create distinctive oscillation behaviors as functions of local density, distinguishing sGB MaVaNs both from constant-mass and generic MaVaN models.

6. Key Formulas and Theoretical Summary

Formula	Meaning	Context
$$\mathcal{G} = R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$	Gauss–Bonnet invariant	Scalar-curvature coupling
$f(\phi) = \frac{1}{2} \xi \phi^2$	Quadratic scalar–GB coupling function	Action, sGB term
$m_{\nu, i}(r) = \sqrt{\kappa_i} \phi(r)$	Scalar field–dependent neutrino mass	Neutrino sector term
$$\Delta m^2_{21}(r) = \Delta\kappa'^{\,2}_{21}\, \hat{\phi}^4(r)$$	Position-dependent mass-squared splitting	Oscillation phenomenology
$$\Phi_{21} = \int \frac{\Delta m^2_{21}(r)}{2E_\nu} dr$$	Oscillation phase incorporating scalar dependence	Observable phase difference
$\tan 2\theta_M = \sin 2\theta/(\cos 2\theta - A)$	Effective mixing angle in matter, with $A$ containing $\hat{\phi}$-dependent splitting	Survival probability

These formulas collectively illustrate the structure and consequences of MaVaN models in sGB gravity.

7. Implications, Prospects, and Significance

This extension of MaVaN physics to scalar–Gauss–Bonnet gravity links neutrino properties to gravitational curvature, offering new probes of environmental and cosmological neutrino physics. The model’s successful reproduction of standard oscillation phenomenology in appropriate limits, combined with new testable deviations, demonstrates its compatibility with current data and potential for future discovery. Constraints derived from solar neutrino experiments serve both as checks for the sGB framework and as ways to probe possible gravitationally induced neutrino mass variation, providing a nontrivial interface between neutrino phenomenology and modified gravity frameworks (Sadjad et al., 1 Oct 2025).