• Lorentz violation
• Weighted power counting

Lorentz symmetry is one of the best tested symmetries in nature. Usually it is considered to hold at arbitrarily high energies and large distances. However, in the last years a great amount of activity has been devoted to establish the limits of validity of this symmetry and predict the consequences of its hypothetical violation.
In a Lorentz-Violating theory, the UV behavior of the propagator can be improved adding terms containing higher space derivatives. The class of renormalizable theories can be significantly enlarged, while the absence of higher time derivatives ensures the unitarity of the S-matrix. If the additional terms satisfy a modified version of power counting (“weighted power counting”), in which the dimensions of the vertices are substituted by a weighted dimensions, then no higher-time derivatives are generated by renormalization.
In flat space and in the realm of perturbation theory, it is possible to construct theories that include scalars, fermions and gauge fields: it is possible to construct also Lorentz-Violating Standard Model Extensions (LVSMEs) without violating physical principles.
In some of these extensions, the four fermion interactions are renormalizable; they can trigger a Nambu--Jona-Lasinio mechanism that, due to a dynamical symmetry breaking, can generate a fermion condensate and give masses to the particles, also if the elementary scalars are suppressed. The Higgs bosons emerge at low-energy as a composite scalar field, together with the Goldstone bosons.
In this thesis we study some models of scalarless LVSMEs (i.e. no elementary scalar fields are present), in which the particles gain mass by means of the Nambu--Jona-Lasinio mechanism. First we calculate the effective potential, in the leading order of large N approximation. We show that there is a Lorentz-invariant minimum, and that, considering more generations of fermions, the CKM matrix can emerge. We compute the low-energy effective action and study the phenomenology of our model, in the leading order of large N limit.
At low-energy, one or more doublets of Higgs bosons can emerge as composite fields, with masses within the present experimental bounds. In this minimal LVSME, the neutrinos remain massless, but we show that the Lorentz-violating coefficients can also explain neutrino oscillations.
We examine also the high-energy properties of this model. Since the gauge couplings are super-renormalizable, they decouple and only the four fermion vertices are relevant. We study the most general model containing only four fermion interactions: we perform the one-loop renormalization and calculate the beta functions. We present a technique, inspired by Zimmermann's reduction of couplings, for the determination of the domain of asymptotic freedom in the case of several coupling constants, showing that this property is compatible with the dynamical symmetry breaking.