Non-abelian gauged NJL models on the lattice
  We use Monte Carlo simulation to probe the phase structure of a SU(2) gauge theory containing $N_f$ Dirac fermion flavors transforming in the fundamental representation of the group and interacting through an additional four fermion term. Pairs of physical flavors are implemented using the two tastes present in a reduced staggered fermion formulation of the theory. The resultant lattice theory is invariant under a set of shift symmetries which correspond to a discrete subgroup of the continuum chiral-flavor symmetry. The pseudoreal character of the representation guarantees that the theory has no sign problem. For the case of $N_f=4$ we observe a crossover in the behavior of the chiral condensate for strong four fermi coupling associated with the generation of a dynamical mass for the fermions. At weak gauge coupling this crossover is consistent with the usual continuous phase transition seen in the pure (ungauged) NJL model. However, if the gauge coupling is strong enough to cause confinement we observe a much more rapid crossover in the chiral condensate consistent with a first order phase transition 
Simon Catterall, Richard Galvez, Jay Hubisz, Dhagash Mehta, Aarti   Veernala
hep-lat hep-ph