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 $\mathscr{A}_{\rm eff}[D_{\sigma},D_{\pi}; S]$, depending on the propagators $D_\sigma$, $D_\pi$ of $\sigma-$ and $\pi-$mesons and on the full quark propagator $S$. The stationarity conditions $\delta \mathscr{A}_{\rm eff}/ \delta S = 0$, $\delta \mathscr{A}_{\rm eff}/ \delta D_\sigma = 0$, $\delta \mathscr{A}_{\rm eff}/ \delta D_\pi = 0$, 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 "$\Phi-$derivable" approach and provide a functional formulation for diagrammatic resummations of $1/N_c-$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.