The resonating mean-field theoretical approach to the Nambu-Jona-Lasinio model -$σ−π$ sector-

Using the Nambu-Jona-Lasinio (NJL) model, dynamical chiral-symmetry breaking, light-mesons spectra and properties of the mesons have been investigated by a conventional Hartree-Fock approach. To show the advantage of the resonating (Res) mean-field theory for a fermion system with large quantum fluctuations over the usual mean-field theory, we have applied it successfully to the NJL model to describe phenomena associated with a pionic excitation. Further we use it to describe precisely the phenomena including a sigma mesonic excitation. A state with large quantum fluctuations is approximated by superposition of three Dirac seas, non-orthogonal Slater determinants (S-dets) with different correlation structures. Suppose the Dirac seas are composed of “constituent quark masses”. We make a direct optimization of Res-mean-field energy functional, i.e., a variation of Res-mean-field ground-state energy with respect to Res-mean-field parameters, the “constituent quark masses”. The Res-mean-field ground and excited states generated with the S-dets explain most of mass spectra and associated properties of the pion and sigma meson.