A Lorentz-Violating Alternative to Higgs Mechanism?

We consider a four-dimensional field-theory model with two massless fermions, coupled to an Abelian vector field without flavour mixing, and to another Abelian vector field with flavour mixing. Both Abelian vectors have a Lorentz-violating kinetic term, introducing a Lorentz-violation mass scale $M$, from which fermions and the flavour-mixing vector get their dynamical masses, whereas the vector coupled without flavour mixing remains massless. When the two coupling constants have similar values in order of magnitude, a mass hierarchy pattern emerges, in which one fermion is very light compared to the other, whilst the vector mass is larger than the mass of the heavy fermion. The work presented here may be considered as a Lorentz-symmetry-Violating alternative to the Higgs mechanism, in the sense that no scalar particle (fundamental or composite) is necessary for the generation of the vector-meson mass. However, the model is not realistic given that, as a result of Lorentz Violation, the maximal (light-cone) speed seen by the fermions is smaller than that of the massless gauge boson (which equals the speed of light in vacuo) by an amount which is unacceptably large to be compatible with the current tests of Lorentz Invariance, unless the gauge couplings assume unnaturally small values. Possible ways out of this phenomenological drawback are briefly discussed, postponing a detailed construction of more realistic models for future work.