Theta-vacuum: Phase Transitions and/or Symmetry Breaking at $\theta =\pi$

Abstract

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.

Comments: 9 pages, 2 figures, 2 references added, final version to appear in Progr. Theor. Phys