  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.
