Non-metricity effects on electron scattering in bumblebee gravity

Abstract

We investigate non-metricity effects on electron scattering in metric-affine bumblebee gravity, where spontaneous Lorentz symmetry breaking is induced by a vector field acquiring a nonzero vacuum expectation value. Treating the affine connection as an independent variable and integrating it out leads to an effective description in which non-metricity modifies the dispersion relation of the bumblebee modes. From the full momentum-space propagator, we determine the pole structure that governs the interaction and construct the corresponding static Green function and interparticle potential. For a purely timelike background, the dispersion relation remains isotropic and produces a Coulomb potential with a uniformly rescaled effective coupling; consequently, the scattering amplitude preserves the Rutherford angular dependence, with the Lorentz-violating parameter entering only as an overall multiplicative factor. In contrast, a spacelike background induces anisotropy in the dispersion relation, leading to an orientation-dependent potential characterized by a quadrupolar modulation. This anisotropic structure propagates to the differential and integrated cross sections, introducing directional dependence while preserving the long-range character of the interaction. Finally, we consider phenomenological constraints from atomic physics. Hydrogen spectroscopy constrains the isotropic sector associated with the timelike configuration, whereas searches for anisotropies provide stronger limits on the quadrupolar contribution governed by $ξb^{2}$.

Figures (4)

• Figure 1: The interparticle potential $V(r)$ in the timelike configuration as a function of the radial distance $r$, for different values of the parameter $\xi b^{2}$.
• Figure 2: Radial and angular behavior of the modified potential $V(r,\hat{\alpha})$ in the spacelike configuration. The left panel shows the radial profile for $\hat{\alpha}=\pi/3$ and different values of $\xi b^{2}$, where the $1/r$ decay is preserved with a rescaled amplitude. The right panel presents the angular dependence at $r=1$.
• Figure 3: Differential cross section $\mathrm{d}\sigma/\mathrm{d}\Omega$ as a function of the scattering angle $\theta$ for different values of the parameter $\xi b^{2}$.
• Figure 4: Cross section in different parameter regimes. The upper--left panel shows $\sigma(x,y,\xi)$ as a function of $x$ for several values of $\xi$, with $m=1$ and $y=0.12$. The upper--right panel displays $\sigma(x,y,\xi)$ versus $x$ for different values of $y$, keeping $\xi b^{2}=0.5$ fixed. In the lower--left panel, $\sigma(\gamma_{\min},\xi)$ is plotted as a function of $\gamma_{\min}$ for distinct values of $\xi$, adopting $m=1$ and $\kappa=1$. The lower--right panel presents $\sigma(\gamma_{\min},\xi)$ as a function of $\xi b^{2}$ for several choices of $\gamma_{\min}$, with $m=\kappa=1$.