Pion condensation and realistic interactions

The static pion $p$-wave self-energy in symmetric nuclear matter is calculated, taking into account the effects of nucleon particle-hole (ph) and $\Delta$-isobar nucleon-hole ($\Delta \mathrm{h}$) states. The residual interaction between ph states is derived from a Brueckner $G$ matrix which depends on the starting energy and three momentum variables. The coupling of ph and $\Delta \mathrm{h}$ states and the $\Delta \mathrm{h}-\Delta \mathrm{h}$ interaction is described by $\pi$ and $\rho$ exchange taking into account the effects of $\mathrm{NN}$ correlations. The resulting self-energies are analyzed in the model of constant interaction strengths which is commonly used. This analysis yields effective interaction strengths weakly depending on the momentum of the pion field as well as on the nuclear density. Only in the limit of vanishing pion momentum the strength of the ph interaction is connected with the Landau parameter $G_0^{\prime }$. The present calculation yields pion condensation at twice the empirical nuclear matter density when isobars are included.