θ-vacuum -: Phase transitions and/or symmetry breaking at θ = π

Assuming that a quantum field theory with a theta-vacuum term in the action shows nontrivial theta-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 theta = pi or shows a singular behavior at some critical theta(c) between 0 and pi. 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.