We study the $O\left(N\right)\mathrm{}$ linear sigma model with spontaneous symmetry breaking, using a Hartree-like Ansatz with a classical field and variational masses. We go beyond the Hartree approximation by including the two-loop contribution, the sunset diagram, using the 2PPI expansion. We compute numerically the effective potential at finite temperature. We find a phase transition of second order, while it is first order in the Hartree approximation. We also discuss some implications of the fact that, in this order, the decay of the sigma into two pions affects the thermal diagrams.

• Received 17 October 2002

DOI:https://doi.org/10.1103/PhysRevD.67.085006