A Critical Surface of Chiral-invariant System with Gauge Boson and   Fermions
  In the chirally-invariant context of the $U_{em}(1)$ gauge interaction and four-fermion interactions for ordinary and mirror fermions, the Schwinger-Dyson equation for the fermion self-energy function is studied on a lattice. We find that a sensible infrared limit can be defined on a critical surface, which is consistent with the critical line found in the continuum theory. 
She-Sheng XUE