Mean-field theory for self-interacting relativistic Luttinger fermions

We investigate a class of quantum field theories with relativistic Luttinger fermions and local self-interaction in scalar channels. For an understanding of possible low-energy phases, we first classify the set of mass terms arising from scalar fermion bilinears. For large flavor numbers, we show that each of our models features a coupling branch in which the theory is asymptotically free. In order to address the long-range behavior, we use mean-field theory which is exact in the limit of large flavor numbers. We identify two models which undergo dimensional transmutation, interconnecting the asymptotically free high-energy regime with an ordered low-energy phase sustaining a vacuum condensate. We also study the analytic structure of the Luttinger-fermionic propagator in the various possible gapped phases.