A 3+1 Dimensional LF-Model with Spontaneous $χSB$

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. For the model considered, this means that if one uses the same transverse momentum cutoff on the SD and LF formulations then the vertex mass in the LF calculation must be taken to be the same as the current quark mass in the SD calculation. The kinetic mass term in the LF calculation is renormalized non-trivially, which is eventually responsible for the mass generation of the physical fermion of the model.