# Spontaneous Lorentz Violation: Non-Abelian Gauge Fields as Pseudo-Goldstone Vector Bosons

## TL;DR

This paper proposes that non-Abelian gauge fields can be viewed as pseudo-Goldstone bosons resulting from spontaneous Lorentz invariance violation, with potential observable effects at high energy scales near the Planck scale.

## Contribution

It introduces a novel interpretation of non-Abelian gauge fields as pseudo-Goldstone bosons from Lorentz violation, incorporating a nonlinear vector field constraint in Yang-Mills theories.

## Key findings

- Massless pseudo-Goldstone vector bosons protected by gauge invariance.
- Lorentz and CPT violating couplings cancel out in lowest order processes.
- Physical Lorentz violation may emerge if gauge invariance is slightly broken at small scales.

## Abstract

We argue that non-Abelian gauge fields can be treated as the pseudo-Goldstone vector bosons caused by spontaneous Lorentz invariance violation (SLIV). To this end, the SLIV which evolves in a general Yang-Mills type theory with the nonlinear vector field constraint $Tr(% \boldsymbol{A}_{\mu }\boldsymbol{A}^{\mu})=\pm M^{2}$ ($M$ is a proposed SLIV scale) imposed is considered in detail. With an internal symmetry group $G$ having $D$ generators not only the pure Lorentz symmetry SO(1,3), but the larger accidental symmetry $SO(D,3D)$ of the SLIV constraint in itself appears to be spontaneously broken as well. As a result, while the pure Lorentz violation still generates only one genuine Goldstone vector boson, the accompanying pseudo-Goldstone vector bosons related to the $SO(D,3D)$ breaking also come into play in the final arrangement of the entire Goldstone vector field multiplet. Remarkably, they remain strictly massless, being protected by gauge invariance of the Yang-Mills theory involved. We show that, although this theory contains a plethora of Lorentz and $CPT$ violating couplings, they do not lead to physical SLIV effects which turn out to be strictly cancelled in all the lowest order processes considered. However, the physical Lorentz violation could appear if the internal gauge invariance were slightly broken at very small distances influenced by gravity. For the SLIV scale comparable with the Planck one the Lorentz violation could become directly observable at low energies.