Pion pair condensed nuclear matter without a σ meson

The nonlinear σ model that includes vector meson contributions is utilized for the study of nuclear matter. Here, the σ meson degree of freedom is eliminated by the condition that σ2+π2 is invariant under both chiral transformation and isospin rotation. The nuclear matter consists of nucleons, pions, and vector (ω) mesons. In the model, there are only three free parameters, and the minimum energy and desired incompressibility at the normal nuclear density for the pion pair condensed nuclear matter can be obtained by adjusting these parameters.