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We calculate the two-particle irreducible (2PI) effective potential of theO(N) linear sigma model in 1+1 dimensions. The approximations we use are thenext-to-leading order of a 1/N expansion (for arbitrary N) and a kind of"resummed loop approximation" for N=1. We show that the effective potential ofthe 1/N expansion is convex for N=4 and N=10 whereas it is not for the "loop"expansion and the case N=1 of the 1/N expansion.Comment: LaTeX, 5 pages. Contribution to the Proceedings of 6th Conference on  Strong and Electroweak Matter 2004 (SEWM04), Helsinki, Finland, 16-19 Jun  200

SCALAR O(N) MODEL AT FINITE TEMPERATURE— 2PI EFFECTIVE POTENTIAL IN DIFFERENT APPROXIMATIONS