The bumblebee field excitations in a cosmological braneworld

TL;DR

This paper studies how spacetime curvature and extra dimensions influence the excitations of the bumblebee field, revealing mass generation, mode behavior, and decay properties in a cosmological braneworld context.

Contribution

It introduces a model of the bumblebee field in an AdS_5 bulk, analyzing its excitations, mass spectrum, and decay in a cosmological braneworld setting, highlighting effects of curvature and Lorentz violation.

Findings

Transverse modes acquire massive Kaluza-Klein towers.

Longitudinal mode gains Lorentz-violating mass and decays over time.

Cosmological expansion induces decay of the longitudinal mode.

Abstract

We investigated the effects of the spacetime curvature and extra dimensions on the excitations of a self-interacting vector field known as the bumblebee field. The self-interacting quadratic potential breaks the gauge invariance and the vacuum expectation value (VEV) of the bumblebee field $b_M$ violates the local particle Lorentz symmetry. By assuming the bumblebee field living in a $AdS_{5}$ bulk, we found an exponential suppression of the self-interacting constant $\lambda$ and the bumblebee VEV along the extra dimension. The fluctuations of the bumblebee upon the VEV can be decomposed into transverse and longitudinal modes with respect to $b_{M}$. Despite the curvature, the transverse mode acquires massive Kaluza-Klein towers, while the longitudinal mode acquires LV mass $\lambda b^{2}$. On the other hand, the current conservation law prevents massive Kaluza-Klein modes for the longitudinal mode. For a spacelike $b_{M}$ along the extra dimension and assuming a FRW 3-brane embedded in the $AdS_{5}$ yields to an additional dissipative term to the longitudinal mode. The cosmological expansion leads to decay of the longitudinal mode in a time $\Delta t \approx H^{-1}$, where $H=\dot{a}/a$ is the Hubble parameter and $a(t)$ is the scale factor. For a timelike $b_{M}$, the longitudinal mode does not propagate on the brane and its amplitude decays in time with $a^{-3}$ and in the extra dimension with $z^{-\lambda b^{2}l^{2}}$.