Asymptotically free four-Fermi theory in 4 dimensions at thez=3Lifshitz-like fixed point

Abstract

We show that a Nambu-Jona-Lasinio type four-fermion coupling at the $z=3$ Lifshitz-like fixed point in $3+1$ dimensions is asymptotically free and generates a mass scale dynamically. This result is nonperturbative in the limit of a large number of fermion species. The theory is ultraviolet complete and at low energies exhibits Lorentz invariance as an emergent spacetime symmetry. Many of our results generalize to $z=d$ in odd $d$ spatial dimensions; $z=d=1$ corresponds to the Gross-Neveu model. The above mechanism of mass generation has potential applications to the fermion mass problem and to dynamical electroweak symmetry breaking. We present a scenario in which a composite Higgs field arises from a condensate of these fermions, which then couples to quarks and leptons of the standard model. Such a scenario could eliminate the need for the Higgs potential and the associated hierarchy problem. We also show that the axial anomaly formula at $z=3$ coincides with the usual one in the relativistic domain.