Linear sigma model at finite temperature
  The chiral phase transition is investigated within the framework of thermal field theory using the O(N) linear sigma model as an effective theory. We calculate the thermal effective potential by using the Cornwall-Jackiw-Tomboulis formalism of composite operators. The thermal effective potential is calculated for N=4 involving as usual the sigma and the three pions, and in the large-N approximation involving N-1 pion fields. In both cases, the system of the resulting gap equations for the thermal effective masses of the particles has been solved 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, while in the large N approximation the phase transition appears as second-order. In this analysis, we have ignored quantum fluctuations and have used the imaginary time formalism for calculations. We have extended our calculation in order to include the full effect of two loops in the calculation of the effective potential. In this case, the effective masses are momentum dependent. In order to perform the calculations, we found the real time formalism to be convenient. We have calculated the effective masses of pions at the low-temperature phase and we found a quadratic dependence on temperature, in contrast to the Hartree case, where the mass is proportional to temperature. The sigma mass was investigated in the presence of massive pions, and we found a small deviation compared to the Hartree case. In all cases, the system approaches the behaviour of the ideal gas at the high temperature limit. 
Nicholas Petropoulos