A 2 Loop 2PPI Analysis of λϕ⁴ at Finite Temperature

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We calculate the finite temperature effective potential of λϕ⁴ at the two loop order of the 2PPI expansion. This expansion contains all diagrams which remain connected when two lines meeting at the same point are cut and therefore sums systematically 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.