(3+1)-dimensional light-front model with spontaneous breaking of chiral symmetry

We investigate a (3+1)-dimensional toy model that exhibits spontaneous breakdown of chiral symmetry, both in a light-front (LF) Hamiltonian and in a Euclidean Schwinger-Dyson (SD) formulation. We show that both formulations are completely equivalent, provided the renormalization is properly done. The counterterm can be constructed explicitly by eliminating zero-mode degrees of freedom, giving rise to to an effective interaction: i.e., zero-mode dynamics, in the sense of an effective action, leads to a very simple set of modifications for the nonzero modes. We find that it is sufficient to renormalize terms that exist already in the canonical LF Hamiltonian independently. Chiral symmetry breaking is manifested via a ''kinetic mass'' counterterm, which is eventually responsible for the mass generation of the physical fermion of the model. The vertex mass in the LF calculation must be taken to be the same as the current quark mass in the SD calculation.