Renormalization character and quantum S-matrix for a classically integrable theory

The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sine-Gordon model) and N = 2 (reduced O(4) σ-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the one-loop order. Its effect is like a local quartic coupling.