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.Comment: 33 pages, 40 eps figures, based on PhD thesis submitted to Manchester
University in Oct 200