  The vacuum angle $\theta$ renormalization is studied for a toy model of a
quantum particle moving around a ring, threaded by a magnetic flux $\theta$.
Different renormalization group (RG) procedures lead to the same generic RG
flow diagram, similar to that of the quantum Hall effect. We argue that the
renormalized value of the vacuum angle may be observed if the particle's
position is measured with finite accuracy or coupled to additional slow
variable, which can be viewed as a coordinate of a second (heavy) particle on
the ring. In this case the renormalized $\theta$ appears as a magnetic flux
this heavy particle sees, or the Berry phase, associated with its slow
rotation.
