We consider theories with a large number N F of charged fermions and compute the renormalisation group equations for the gauge, Yukawa and quartic couplings resummed at leading order in 1/N F . We construct extensions of the Standard Model where SU(2) and/or SU(3) are asymptotically safe. When the same procedure is applied to the Abelian U(1) factor, we find that the Higgs quartic can not be made asymptotically safe and stay perturbative at the same time.

Motivated by new ideas about the Higgs mass naturalness problem, we present realistic TeV-scale extensions of the Standard Model, into the gauge group SU(3) L ⊗ SU(3) R ⊗ SU(3) c , such that all gauge, Yukawa and quartic couplings can be extrapolated up to infinite energy. Three generations of chiral fermions and Higgses are needed, as well as some extra fermion.

We extend the list of theories featuring a rigorous interacting ultraviolet fixed point by constructing the first theory featuring a Higgs-like scalar with gauge, Yukawa and quartic interactions. We show that the theory enters a perturbative asymptotically safe regime at energies above a physical scale . We determine the salient properties of the theory and use it as a concrete example to test whether scalars masses unavoidably receive quantum correction of order . Having at our dispose a calculable model allowing us to precisely relate the IR and UV of the theory we demonstrate that the scalars can be lighter than . Although we do not have an answer to whether the Standard Model hypercharge coupling growth toward a Landau pole at around ∼ 10 40 GeV can be tamed by non-perturbative asymptotic safety, our results indicate that such a possibility is worth exploring. In fact, if successful, it might also offer an explanation for the unbearable lightness of the Higgs.