A note on theta dependence

The dependence on the topological θ angle term in quantum field theory is usually discussed in the context of instanton calculus. There the observables are 2π periodic, analytic functions of θ. However, in strongly coupled theories, the semi-classical instanton approximation can break down due to infrared divergences. Instances are indeed known where analyticity in θ can be lost, while the 2π periodicity is preserved. In this short Letter we exhibit a simple two-dimensional example where the 2π periodicity is lost. The observables remain periodic under the transformation θ↦θ+2kπ for some k⩾2. We also briefly discuss the case of four-dimensional N=2 supersymmetric gauge theories.