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Using a non-perturbative approximation for the partition function of acomplex scalar model, which features spontaneous symmetry breaking, weexplicitly derive the flattening of the effective potential in the regionlimited by the minima of the bare potential. This flattening occurs in thelimit of infinite volume, and is a consequence of the summation over thecontinuous set of saddle points which dominate the partition function. We alsoprove the convexity of the effective potential and generalize the MaxwellConstruction for scalar theories with O(N) symmetry. Finally, we discuss whythe flattening of the effective potential cannot occur in the Abelian Higgstheory.

Maxwell Construction for Scalar Field Theories with Spontaneous Symmetry Breaking