The SU(2) Non-Linear Sigma-Model in 2+1 Dimensions: Perturbation Theory in a Polynomial Formulation

TL;DR

This paper develops a perturbation theory for the SU(2) non-linear sigma-model in 2+1 dimensions using a polynomial formulation, simplifying calculations and addressing one-loop infinities through normal ordering.

Contribution

It introduces a polynomial, first-order formulation of the SU(2) sigma-model, enabling straightforward perturbative calculations and handling divergences at one-loop order.

Findings

One-loop infinities are cured by normal ordering.

The polynomial formulation simplifies Feynman rules and calculations.

Scattering amplitudes relate directly to currents without group coordinates.

Abstract

We construct a perturbation theory for the SU(2) non-linear Sigma-model in 2+1 dimensions using a polynomial, first-order formulation, where the variables are a non-Abelian vector field L_mu (the left SU(2) current), and a non-Abelian pseudovector field $\theta_{\mu}$, which imposes the condition F_{mu nu}(L) = 0. The coordinates on the group do not appear in the Feynman rules, but their scattering amplitudes are easily related to those of the currents. We show that all the infinities affecting physical amplitudes at one-loop order can be cured by normal ordering, presenting the calculation of the full propagator as an example of an application.