A Weak Power-Counting Theorem for the Renormalization of the Non-Linear Sigma Model in Four Dimensions

TL;DR

This paper proves a weak power-counting theorem for the four-dimensional non-linear sigma model, showing that only a finite set of divergent amplitudes need renormalization despite infinitely many divergences.

Contribution

It introduces a weak power-counting theorem for the non-linear sigma model, demonstrating finite renormalization requirements at one-loop level in four dimensions.

Findings

Finite number of divergent amplitudes require renormalization.

Counterterms derived for four-point amplitudes.

Hierarchy of amplitudes based on the functional equation.

Abstract

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.