[en] 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. (author)