Remark on pion scattering lengths

TL;DR

This paper derives an exact tree-level pion-pion scattering amplitude in the linear sigma model, revealing a geometric series pattern that aligns with chiral perturbation theory predictions up to high orders.

Contribution

It provides an exact expression for the pion scattering amplitude in the linear sigma model and connects it to chiral perturbation theory predictions, including higher-order terms.

Findings

Pattern of scattering lengths matches chiral perturbation theory predictions up to order p^6.

Derived explicit expressions for p^8 order terms.

Shows geometric series structure in the scattering amplitude.

Abstract

First it is shown that the tree amplitude for pion pion scattering in the minimal linear sigma model has an exact expression which is proportional to a geometric series in the quantity (s-$m_\pi^2$)/($m_B^2-m_\pi^2$), where $m_B$ is the sigma mass which appears in the Lagrangian and is the only a priori unknown parameter in the model. This induces an infinite series for every predicted scattering length in which each term corresponds to a given order in the chiral perturbation theory counting. It is noted that, perhaps surprisingly, the pattern, though not the exact values, of chiral perturbation theory predictions for both the isotopic spin 0 and isotopic spin 2 s-wave pion-pion scattering lengths to orders $p^2$, $p^4$ and $p^6$ seems to agree with this induced pattern. The values of the $p^8$ terms are also given for comparison with a possible future chiral perturation theory calculation. Further aspects of this approach and future directions are briefly discussed.