Theta-vacuum: Phase Transitions and/or Symmetry Breaking at $θ= π$

Abstract

Assuming that a quantum field theory with a $θ$-vacuum term in the action shows non-trivial $θ$-dependence and provided that some reasonable properties of the probability distribution function of the order parameter hold, we argue that the theory either breaks spontaneously CP at $θ= π$ or shows a singular behavior at some critical $θ_c$ between 0 and $π$. This result, which applies to any model with a pure imaginary contribution to the euclidean action consisting in a quantized charge coupled to a phase, as QCD, is illustrated with two simple examples; one of them intimately related to Witten's result on SU(N) in the large $N$ limit.