A new approach to nonrenormalizable theories

The problem of constructing a sensible physical theory out of a nonrenormalizable classical action is discussed and instanced in the paradigmatic case of the nonlinear sigma model. At variance with the approach of algebraic renormalization, the perturbative expansion in $\hbar$ is not constructed on the most general classical action, where each new independent divergent amplitude induces a novel parameter. Instead a solution is searched where the number of free parameters is fixed and all the requirements associated to symmetry properties, defining equations, locality, physical unitarity are met. Once these conditions are obeyed, the theory can be tested experimentally. This program can be realized in the case of the nonlinear sigma model. This is achieved if one looks at the fields as parameters of a gauge field of zero strength (flat connection). In this important example, some technical items are developed which will be useful in other more complex models (e.g. massive Yang-Mills theory) as local functional equations, hierarchy, weak power-counting. The final result is a model depending on two parameters: the v.e.v. of the spontaneous breakdown of symmetry and the scale of the radiative corrections. The subtraction of the divergences is performed in dimensional regularization by minimal subtraction on the properly normalized 1-PI amplitudes.