Two-loop two-particle point irreducible analysis ofλφ4theory at finite temperature

We calculate the finite temperature effective potential of $\lambda {\phi }^4$ at the two-loop order of the two-particle point irreducible (2PPI) expansion. This expansion contains all diagrams that remain connected when two lines meeting at the same point are cut and therefore the expansion systematically sums the bubble graphs. At one loop in the 2PPI expansion, the symmetry restoring phase transition is first order. At two loops, we find a second order phase transition with mean field critical exponents.