A 3+1 Dimensional LF-Model with Spontaneous $\chi SB$

TL;DR

This paper demonstrates the equivalence of light-front and Euclidean Schwinger-Dyson formulations in a 3+1 dimensional toy model exhibiting spontaneous chiral symmetry breaking, highlighting the importance of proper renormalization.

Contribution

It establishes the equivalence between LF and SD formulations for a chiral symmetry breaking model with proper renormalization, emphasizing the non-trivial renormalization of the kinetic mass.

Findings

LF and SD formulations are equivalent with proper renormalization

Mass generation is linked to non-trivial kinetic mass renormalization

Vertex mass matches current quark mass when cutoffs are aligned

Abstract

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.