We calculate the θ dependence in a cousin of QCD, where the vacuum structure can be analyzed exactly. The theory is N=2 SU (2) gauge theory with NF=0, 1, 2, 3 flavors of fundamentals, explicitly broken to N=1 via an adjoint superpotential, and coupled to anomaly mediated supersymmetry breaking (AMSB). The hierarchy m₀₌ₒ₁ μ₍=₁ Λ ensures the validity of our IR analysis. As expected from ordinary QCD, the vacuum energy is a function of θ which undergoes 1st order phase transitions between different vacua where the various dyons condense. For NF=0 we find the expected phase transition at θ=π, while for NF=1, 2, 3 we find phase transitions at fractional values of π.
Csáki et al. (Tue,) studied this question.