Variational path-integral approach to back-reactions of composite mesons in the Nambu-Jona-Lasinio model

For the investigation of back-reactions of composite mesons in the NJL model, a variational path-integral treatment is formulated which yields an effective action Aeff[Dσ,Dπ;S], depending on the propagators Dσ, Dπ of σ− and π−mesons and on the full quark propagator S. The stationarity conditions δAeff/δS=0, δAeff/δDσ=0, δAeff/δDπ=0, then lead to coupled Schwinger-Dyson (SD) equations for the quark self-energy and the meson polarization functions. These results reproduce and extend results of the so-called “Φ−derivable” approach and provide a functional formulation for diagrammatic resummations of 1/Nc−corrections in the NJL model. Finally, we perform a low-momentum estimate of the quark and meson loop contributions to the polarization function of the pion and on this basis discuss the Goldstone theorem.