The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

TL;DR

This paper investigates the connection between the linear and nonlinear sigma models in four dimensions, focusing on the large mass limit and employing a hierarchy principle to unify different limits at one-loop perturbation theory.

Contribution

It extends the hierarchy principle to match linear and nonlinear sigma models in both chiral and strong coupling limits at one loop, providing a systematic approach for Green functions with many pion legs.

Findings

Successful matching in the chiral limit is straightforward.

Matching in the strong coupling limit requires careful normalization and functional equation use.

Hierarchy principle effectively unifies different limits at one loop.

Abstract

In perturbation theory we study the matching in four dimensions between the linear sigma model in the large mass limit and the renormalized nonlinear sigma model in the recently proposed flat connection formalism. We consider both the chiral limit and the strong coupling limit of the linear sigma model. Our formalism extends to Green functions with an arbitrary number of pion legs,at one loop level,on the basis of the hierarchy as an efficient unifying principle that governs both limits. While the chiral limit is straightforward, the matching in the strong coupling limit requires careful use of the normalization conditions of the linear theory, in order to exploit the functional equation and the complete set of local solutions of its linearized form.