We study the O(3) sigma model in $D=2$ on the lattice with a Boltzmann weight linearized in $\beta$ on each link.
While the spin formulation now suffers from a sign-problem the equivalent loop model remains positive and
becomes particularly simple. By studying the transfer matrix and by performing Monte Carlo simulations in the loop form
we study the mass gap coupling in a step scaling analysis. The question addressed is, whether or not such a
simplified action still has the right universal continuum limit. If the answer is affirmative this would be
helpful in widening the applicability of worm algorithm methods.