Full time-dependent Hartree-Fock solution of largeNGross-Neveu models

Abstract

We find the general solution to the time-dependent Hartree-Fock problem for scattering solutions of the Gross--Neveu models, with both discrete (${\mathrm{GN}}_{2}$) and continuous (${\mathrm{NJL}}_{2}$) chiral symmetry. We find new multibreather solutions both for the ${\mathrm{GN}}_{2}$ model, generalizing the Dashen--Hasslacher--Neveu breather solution, and also new twisted breathers for the ${\mathrm{NJL}}_{2}$ model. These solutions satisfy the full time-dependent Hartree-Fock consistency conditions, and only in the special cases of ${\mathrm{GN}}_{2}$ kink scattering do these conditions reduce to the integrable Sinh--Gordon equation. We also show that all baryons and breathers are composed of constituent twisted kinks of the ${\mathrm{NJL}}_{2}$ model. Our solution depends crucially on a general class of transparent, time-dependent Dirac potentials found recently by algebraic methods.