Nuclear Schwinger-Dyson formalism applied to finite baryon density. I. Formulation

The nuclear Schwinger-Dyson (NSD) formalism is presented for an application to nuclear matter. The NSD formalism consists of coupled Dyson equations of a nucleon and mesons. Because it includes meson self-energies in a nonperturbative way, higher order correlations beyond the Hartree-Fock approximation are taken into account. Some important differences between the NSD formalism for the system of finite baryon density and SD formalism of zero baryon density are shown. The main difference is due to the existence of condensed meson fields. By paying special attention to the treating of the condensed meson fields, the coupled Dyson equations of nucleon and mesons are derived based on a functional method. It is shown that this treating of the condensed fields naturally leads to two-tadpole energy, which cancels a half of the Hartree energy. A general representation of vector meson propagators is derived by using projection operators and by solving an inverse matrix problem. It is also shown that the NSD method is possible to be generalized to include sigma-omega meson mixings and new coupled meson propagators are obtained in a similar way to the nonmixed case. As a result, 5×5 components of the coupled meson propagator are expressed in terms of only four independent propagators. By using these meson propagators, explicit expressions of NSD coupled equations and the energy density of nuclear matter are derived for numerical calculations in a subsequent paper. Similarities and differences between NSD and traditional methods such as the mean-field theory or Hartree-Fock are discussed.