  Assuming that a quantum field theory with a $\theta$-vacuum term in the
action shows non-trivial $\theta$-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.
