Convexity of the effective potential

Abstract

The authors show that an interpolated loop expansion produces a convex effective potential for Higgs fields in the vector representations of SU(N) and SO(N) and the adjoint representation of any simple Lie group, provided one considers the Higgs fields as a sector of a gauge theory and use the gauge fixing freedom to choose a 't Hooft-type gauge fixing term. The adjoint case is considered in some detail, exploiting the correspondence between the Dynkin diagrams used to classify the symmetry breaking and the Coexeter graphs that describe the symmetries of regular polytopes to determine the shape of the linearly interpolated regions that appear in the true, convex effective potential.