Low energy scattering with a nontrivial pion

TL;DR

This paper investigates low energy pion scattering within a generalized linear sigma model, examining deviations from current algebra predictions when pion mass is introduced and how scalar mesons influence scattering lengths.

Contribution

It demonstrates how scalar mesons provide controlled corrections to the current algebra formula in a linear sigma model, extending understanding beyond the massless pion approximation.

Findings

The current algebra formula deviates slightly when pion mass is nonzero.

Scalar mesons contribute significant corrections to scattering lengths.

The model fits experimental data well with these corrections.

Abstract

An earlier calculation in a generalized linear sigma model showed that the well-known current algebra formula for low energy pion pion scattering held even though the massless Nambu Goldstone pion contained a small admixture of a two-quark two-antiquark field. Here we turn on the pion mass and note that the current algebra formula no longer holds exactly. We discuss this small deviation and also study the effects of an SU(3) symmetric quark mass type term on the masses and mixings of the eight SU(3) multiplets in the model. We calculate the s wave scattering lengths, including the beyond current algebra theorem corrections due to the scalar mesons, and observe that the model can fit the data well. In the process, we uncover the way in which linear sigma models give controlled corrections (due to the presence of scalar mesons) to the current algebra scattering formula. Such a feature is commonly thought to exist only in the non-linear sigma model approach.