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

Abstract

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 ${\mathcal{A}}_{\mathrm{eff}}[D_{\sigma },D_{\pi };S]$, depending on the propagators $D_{\sigma }$, $D_{\pi }$ of $\sigma -$ and $\pi -$mesons and on the full quark propagator $S$. The stationarity conditions $\delta {\mathcal{A}}_{\mathrm{eff}}/\delta S=0$, $\delta {\mathcal{A}}_{\mathrm{eff}}/\delta D_{\sigma }=0$, $\delta {\mathcal{A}}_{\mathrm{eff}}/\delta D_{\pi }=0$, 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.

Comments: 21 pages, 3 figures, minor text corrections, 2 references added, version published in Nucl. Phys. B