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

Abstract

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 Stückelberg. 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 constrained 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 naïve 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. One loop counterterms for

Path-integral over non-linearly realized groups and hierarchy solutions

“…It has been recently pointed out [1]- [3] that such a definition can be implemented through the local functional equation which expresses the invariance of the Haar measure under local left group multiplication. The subtraction procedure is required to be symmetric, thus preserving the validity of the local functional equation to all orders in the loop expansion [3].…”

“…This goes under the name of hierarchy principle [1]. Moreover there is only a finite number of divergent ancestor amplitudes at every loop order (weak power-counting theorem [2], [3]).…”

Managing γ 5 in Dimensional Regularization II: the Trace with more γ 5’s

“…The ABJ anomaly [29]- [32] and the LFE (Local Functional Equation) [33]- [36] associated to the abelian local chiral transformations have been verified by explicit calculations.…”

Massive Yang-Mills theory based on the nonlinearly realized gauge group

“…It is not invariant under local SU (2) R symmetry, since the latter forbids the Stückelberg mass term because of the transformation property given in eq. (10).…”

“…In the nonlinear sigma model the way out is to make use of the hierarchy principle [10] for the vertex functional, i.e. to fix the φ-amplitudes in terms of ancestor amplitudes involving only the insertion of the flat connection and the nonlinear sigma model constraint.…”