Is the 2 D O ( 3 ) Nonlinear σ Model Asymptotically Free ?

We report the results of a Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear σ model. The notable finding is that it agrees very well with both the prediction inspired by Zamolodchikovs’ S-matrix ansatz and with the continuum limit of the dodecahedron spin model. The latter finding renders the existence of asymptotic freedom in the O(3) model rather unlikely. Asymptotic freedom is one of the corner stones of modern particle physics. This special property of certain non-Abelian models is supposed to explain many remarkable properties of quantum field theory, such as the existence of a non-trivial continuum limit, the possibility of grand unification of all interactions, etc. This property was discovered in 1973 in perturbation theory (PT) [1, 2]. However the correctness of this technique within a nonperturbative setting, such as the one offered by lattice quantuum field theory, remains mathematically unsettled. In the past, we have repeatedly pointed