Linear sigma model and chiral symmetry at finite temperature

The chiral phase transition is investigated within the framework of the linear sigma model at finite temperature. We concentrate on the meson sector of the model and calculate the finite temperature effective potential in the Hartree approximation by using the Cornwall-Jackiw-Tomboulis formalism of composite operators. The effective potential is calculated for N=4 involving the usual sigma and three pions and in the large N approximation involving N-1 pion fields. In the N=4 case we have examined the theory both in the chiral limit and with the presence of a symmetry breaking term which generates the pion masses. In both cases we have solved the system of the resulting gap equations for the thermal effective masses of the particles numerically and we have investigated the evolution of the effective potential. In the N=4 case there is indication of a first order phase transition and the Goldstone theorem is not satisfied. The situation is different in the general case using the large $N$ approximation, the Goldstone theorem is satisfied and the phase transition is of the second order. For this analysis we have ignored quantum effects and we used the imaginary time formalism for calculations.