Two-dimensional classically solvable model with nontrivial scattering

A study is made of the model $({:}j^{\mu}j_{\mu}\varphi{:})_2$, which describes the interaction of a massive scalar field $\varphi$ with a massless spinor field. It is shown that in the classical case this model has an explicit solution, while in the quantum case it leads to nontrivial transitions between the boson and fermion particles.