Lorentz-violating alternative to the Higgs mechanism?

We consider a four-dimensional field-theory model with two massless fermions, coupled to an Abelian vector field without flavor mixing, and to another Abelian vector field with flavor mixing. Both Abelian vectors have a Lorentz-violating kinetic term, introducing a Lorentz-violation mass scale $M$, from which fermions and the flavor-mixing vector get their dynamical masses, whereas the vector coupled without flavor 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, while the vector mass is of the order of the heavy fermion mass. 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.

Figure 1

The decomposition (26) of the $B$-meson-fermion vertex function into a regular and a pole (singular) part; the pole is attributed to the exchange of a massless Goldstone spin-zero excitation which though is absent from the physical spectrum (scattering amplitude) due to delicate cancellations between singular vertex and propagation poles. The notation “R” stands for regular (nonsingular) parts and “P” for pole (singular) graphs.

Figure 2

The upper figure shows the quantum corrections to the $B$-meson fermion vertex function, including the one-vector-meson-irreducible parts $T^{\prime }$. The middle figure shows the decomposition of $T^{\prime }$ into a regular (R) and a pole (P) part, due to the Goldstone scalar exchange (denoted by a dashed line, corresponding to the $1/k^2$ pole structure). The lower figure gives the total scattering amplitude $T$ of two fermions in terms of the $T^{\prime }$ part and the one $B$-vector-meson exchange.

Figure 3

The upper figure (a) shows the complete scattering amplitude for two fermion scattering $T$, while the lower figure (b) details its pole structure, using the vertex decomposition (26), cf. Fig. 1.