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Spontaneous Symmetry Breakings in the Linear Sigma Model at finite baryonic density

The linear sigma model with pions and sigma coupled to baryons and a (massive) gauge vector meson is considered for the description of a finite baryonic density system. The stability equation for the bound system is studied in the homogeneous limit considering quantum fluctuations for the scalar and pseudo scalar fields within a variational method with truncations. Nearly exact solutions for the stability equation which also solve the field equations are sought semi-analytically. Bosonic fields are found to have non zero expected classical values at finite density corresponding to condensates of dynamical symmetry breakings. The two components of baryon field, neutron and proton for example, acquire different masses due to the isospin symmetry breaking and these states may even oscillate in the medium. There is a non trivial solution for the (massive) vector meson field indicating the occurrence of a gauge dynamical symmetry breaking at finite density typical of a superconductor.