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Register a new userA Weak Power-Counting Theorem for the Renormalization of the Non-Linear Sigma Model in Four Dimensions
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The formulation of the non-linear sigma model in terms of flat connection allows the construction of a perturbative solution of a local functional equation encoding the underlying gauge symmetry. In this paper we discuss some properties of the solution at the one-loop level in D=4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of divergent amplitudes have to be renormalized. The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counteterms are given in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. The latter contain only insertions of the composite operators $phi_0$ (the constraint of the non-linear sigma model) and $F_mu$ (the flat connection). These amplitudes are at the top of a hierarchy implicit in the functional equation. As an example we derive the counterterms for the four-point amplitudes.
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The renormalization procedure of the non-linear SU(2) sigma model in D=4 proposed in hep-th/0504023 and hep-th/0506220 is here tested in a truly non-trivial case where the non-linearity of the functional equation is crucial. The simplest example, where the non-linear term contributes, is given by the two-loop amplitude involving the insertion of two phi_0 (the constraint of the non-linear sigma model) and two flat connections. In this case we verify the validity of the renormalization procedure: the recursive subtraction of the pole parts at D=4 yields amplitudes that satisfy the defining functional equation. As a by-product we give a formal proof that in D dimensions (without counterterms) the Feynman rules provide a perturbative symmetric solution.
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