Renormalization of the Non-Linear Sigma Model in Four Dimensions. A two-loop example

TL;DR

This paper tests a renormalization method for the non-linear SU(2) sigma model in four dimensions using a complex two-loop example, confirming the method's validity and providing a formal proof of symmetry in Feynman rules.

Contribution

It demonstrates the effectiveness of a renormalization procedure in a non-trivial two-loop case of the non-linear sigma model, validating the approach and offering a formal proof of symmetry.

Findings

Renormalization procedure successfully applied at two-loop level.

Amplitudes satisfy the functional equation after subtraction.

Feynman rules yield a symmetric solution in D dimensions.

Abstract

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.