  We investigate the continuum limit of the step scaling function in the 2-d
O(3) model with different theta-vacua. Since we find a different continuum
value of the step scaling function for each value of theta, we can conclude
that theta indeed is a relevant parameter of the theory and does not get
renormalized non-perturbatively. Furthermore, we confirm the result of the
conjectured exact S-matrix theory, which predicts the continuum value at theta
= pi. To obtain high precision data, we use a modified Hasenbusch improved
estimator and an action with an optimized constraint, which has very small
cut-off effects. The optimized constraint action combines the standard action
of the 2-d O(3) model with a topological action. The topological action
constrains the angle between neighboring spins and is therefore invariant
against small deformations of the field.
