Study of the O(N) linear sigma model at finite temperature using the   2PPI expansion
  We show that a new expansion which sums seagull and bubble graphs to all orders, can be applied to the O(N) linear sigma model at finite temperature. We prove that this expansion can be renormalised with the usual counterterms in a mass independent scheme and that Goldstone's theorem is satisfied at each order. At the one loop order of this expansion, the Hartree result for the effective potential (daisy and superdaisy graphs) is recovered. We show that at one loop 2PPI order, the self energy of the sigma meson can be calculated exactly and that diagrams are summed beyond the Hartree approximation. 
H. Verschelde and J. De Pessemier