Realistic SM extensions valid up to infinite energy

In the last years the colliders could explore a wider range of energies, but the absence of new particles close enough to the weak scale makes the SM more and more un-natural. In a framework where only the physical corrections to the dimensionful parameters are considered and the gravitational interactions are supposed to be weak at all scales, we want to extend the SM with new degrees of freedom at the TeV scale, such that the behavior of the coupling constants at high energy has no Landau poles and it is under perturbative control. We analyzed the Trinification models, that extend the SM gauge group to SU(3)_L x SU(3)_R x SU(3)_c, to find a model where all the couplings go to zero at infinite energy (Totally Asymptotically Free model). We studied the possible signals of the newly introduced particles that are measurable at colliders. We considered a toy model with gauge group SU(N) featuring vector-like fermions and charged scalars: all the couplings approach an interacting fixed point in the Veneziano limit (Totally Asymptotically Safe behavior). We considered the addition of $N_F$ vector-like fermions to the SM: the complete resummation of the corrections at any order in perturbation theory but at leading order in 1/N_F is known. Considering such contributions in the Renormalization Group equations allows the taming of the Landau poles for all the SM couplings while an interacting fixed point arises and the model becomes Totally Asymptotically Safe.