Spontaneous Lorentz violation: Non-Abelian gauge fields as pseudo-Goldstone vector bosons

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(AμAμ)=±M2 (M is a proposed SLIV scale) imposed is considered in detail. Specifically, we show that in a theory 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, although the pure Lorentz violation on its own 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 properly completing the whole gauge multiplet of the internal symmetry group G taken. Remarkably, they appear to be strictly massless as well, being protected by the starting non-Abelian gauge invariance of the Yang-Mills theory involved. When expressed in terms of the pure Goldstone vector modes, this theory look essentially nonlinear and contains a plethora of Lorentz and CPT violating couplings. However, they do not lead to physical SLIV effects which turn out to be strictly cancelled in all the lowest order processes considered.