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.

• Received 17 September 2013

DOI:https://doi.org/10.1103/PhysRevD.89.025008