The neutrality effects on the phase structure of the linear sigma model with the nonstandard symmetry breaking term is systematically studied by means of the Cornwall-JackiwTomboulis effective potential. The latter quantity is calculated in the improved Hartree-Fock approximation which preserves the Goldstone theorem .and the thermodynamic consistency. Its results that in the region µ > mπ, as a function of the temperature T , the pion condensate undergoes a second order phase transition, as a function of the isospin chemical µ, it undergoes a first order phase transition. In the region µ mπ , as a function of the temperature T , the pion condensate undergoes a second order phase transition, as a function of the isospin chemical µ, it undergoes a first order phase transition. In the region µ 0, λ2 = σ2 > 0, b = π , 2 2fπ 2 (5) in which mσ , mπ are respectively the masses of sigma and pion mesons in vacuum, and fπ is the pion decay constant in vacuum. As was shown in [21] the field operators σ and πi , (i = 1, 2, 3) develop the following expectation values in the ground state hσi = u = fπ , hπi i = v = 0, NEUTRALITY EFFECTS ON THE PHASE STRUCTURE OF. 215 for µ mπ . In the tree approximation u and v are determined from the minimization of the potential energy of (4), namely, they fulfill the equations −m2 + λ2 (u2 + v 2 ) u = 0, (6) 2 2 2 2 2 2 −µ − m + λ (u + v ) + mπ v = 0, (7) which yield u = fπ , v = 0, for µ mπ . Realizing the shifts on field operators σ → u + σ, π1 → v + π1 , π2,3 → π2,3 , (8) and then inserting (8) into (4) it is derived the interaction Lagrangian LI = m2 u − λ2 u(u2 + v 2 ) σ + m2 v + µ2 v − λ2 v(u2 + v 2 ) − 2bv π1 −λ2 (σ 2 + ~π 2 )(uσ + vπ1 ) − λ2 2 (σ + ~π 2 )2 , 4 (9) and the inverse propagators iD0−1 (k; v, .
