Analytical approach to chiral symmetry breaking in Minkowsky space

The mass gap equation for spontaneous chiral symmetry breaking is studied directly in Minkowsky space. In hadronic physics, spontaneous chiral symmetry breaking is crucial to generate a constituent mass for the quarks, and to produce the Partially Conserved Axial Current theorems, including a small mass for the pion. Here a class of finite kernels is used, expanded in Yukawa interactions. The Schwinger-Dyson equation is solved with an analytical approach. This improves the state of the art of solving the mass gap equation, which is usually solved with the equal-time approximation or with the Euclidean approximation. The mapping from the Euclidean space to the Minkowsky space is also illustrated.