Correlation functions in the non-commutative Wess–Zumino–Witten model

We develop a systematic perturbative expansion and compute the one-loop two-points, three-points and four-points correlation functions in a non-commutative version of the U(N) Wess–Zumino–Witten model in different regimes of the θ-parameter showing in the first case a kind of phase transition around the value θc=p2+4m2/(Λ2p), where Λ is a ultraviolet cut-off in a Schwinger regularization scheme. As a by-product we obtain the functions of the renormalization group, showing they are essentially the same as in the commutative case but applied to the whole U(N) fields; in particular there exists a critical point where they are null, in agreement with a recent background field computation of the beta-function, and the anomalous dimension of the Lie algebra-valued field operator agrees with the current algebra prediction. The non-renormalization of the level k is explicitly verified from the four-points correlator, where a left-right non-invariant counter-term is needed to render finite the theory, that it is however null on-shell. These results give support to the equivalence of this model with the commutative one.