The bumblebee field excitations in a cosmological braneworld

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 $Ad{S}_5$ 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 $Ad{S}_5$ 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 $z^{-\lambda b^2{l}^2}$.

15 pages, no figure