Endowing the nonlinear sigma model with a flat connection structure : a way to renormalization

We discuss the quantized theory of a pure-gauge non-abelian vector field (flat connection) as it would appear in a mass term a la Stueckelberg. However the paper is limited to the case where only the flat connection is present (no field strength term). The perturbative solution is constructed by using only the functional equations and by expanding in the number of loops. In particular we do not use a perturbative approach based on the path integral or on a canonical quantization. It is shown that there is no solution with trivial S-matrix. Then the model is embedded in a nonlinear sigma model. The solution is constructed by exploiting a natural hierarchy in the functional equations given by the number of insertions of the flat connection and of the constraint component of the sigma field. The amplitudes with the sigma field are simply derived from those of the flat connection and of the constraint component. Unitarity is enforced by hand by using Feynman rules. We demonstrate the remarkable fact that in generic dimensions the naive Feynman rules yield amplitudes that satisfy the functional equations. This allows a dimensional renormalization of the theory in D = 4 by recursive subtractions of the poles in the Laurent expansion. Thus one gets a finite theory depending only on two parameters. The novelty of the paper is the use of the functional equation associated to the local left multiplication introduced by Faddeev and Slavnov, here improved by adding the external source coupled to the constrained component. It gives a powerful tool to renormalize the nonlinear sigma model.