  A toy model is proposed for four dimensional non-abelian gauge theories
coupled to a large number of fermionic degrees of freedom. As the number of
flavors is varied the gauge theory may be confining, walking or conformal. The
toy model mimicking this feature is the two dimensional O(3) sigma model with a
theta-term. For all theta the model is asymptotically free. For small theta the
model is confining in the infra red, for theta = pi the model has a non-trivial
infra red fixed point and consequently for theta slightly below pi the coupling
walks. The first step in investigating the notoriously difficult systematic
effects of the gauge theory in the toy model is to establish non-perturbatively
that the theta parameter is actually a relevant coupling. This is done by
showing that there exist quantities that are entirely given by the total
topological charge and are well defined in the continuum limit and are
non-zero, despite the fact that the topological susceptibility is divergent.
More precisely it is established that the differences of connected correlation
functions of the topological charge (the cumulants) are finite and non-zero and
consequently there is only a single divergent parameter in Z(theta) but
otherwise it is finite. This divergent constant can be removed by an
appropriate counter term rendering the theory completely finite even at theta >
0.
