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The renormalization procedure of the non-linear SU(2) sigma model in D=4proposed in hep-th/0504023 and hep-th/0506220 is here tested in a trulynon-trivial case where the non-linearity of the functional equation is crucial.The simplest example, where the non-linear term contributes, is given by thetwo-loop amplitude involving the insertion of two \phi_0 (the constraint of thenon-linear sigma model) and two flat connections. In this case we verify thevalidity of the renormalization procedure: the recursive subtraction of thepole parts at D=4 yields amplitudes that satisfy the defining functionalequation. As a by-product we give a formal proof that in D dimensions (withoutcounterterms) the Feynman rules provide a perturbative symmetric solution.Comment: Latex, 3 figures, 19 page

Renormalization of the non-linear sigma model in four dimensions. A two-loop example