The relation between the gauge or global nature of a symmetry being broken and the order of the associated phase transition is clarified. First-order phase transitions in gauge theories are related to the asymptotic freedom of the Φ3 interaction in six dimensions. Spontaneous violation of lepton number in the standard electroweak theory provides a nontrivial illustration of our results.
'Order of Phase Transitions in a Field Theory,' V.P. Nair, Phys. Rev. D28, 409 (1983).