Finding the sigma pole by analytic extrapolation of $\pi\pi$ scattering data

TL;DR

This paper determines the sigma resonance pole from pi-pi scattering data using analytic extrapolation methods, incorporating recent experimental results and emphasizing the importance of Roy equations for precision.

Contribution

It introduces a model-independent analytic extrapolation approach to determine the sigma pole, highlighting the impact of parametrization choices on uncertainties.

Findings

The sigma pole position aligns with ChPT and Roy equation predictions.

Uncertainties increase due to sensitivity to parametrization.

Roy equations remain the most precise method for this determination.

Abstract

We investigate the determination of the $\sigma$ pole from $\pi\pi$ scattering data below the $K\bar{K}$ threshold, including the new precise results obtained from $K_{e4}$ decay by NA48/2 Collaboration. We discuss also the experimental status of the threshold parameters $a_0^0$ and $b_0^0$ and the phase shift $\delta_0^0$. In order to reduce the theoretical bias, we use a large class of analytic parametrizations of the isoscalar $S$-wave, based on expansions in powers of conformal variables. The $\sigma$ pole obtained with this method is consistent with the prediction based on ChPT and Roy equations. However, the theoretical uncertainties are now larger, reflecting the sensitivity of the pole position to the specific parametrizations valid in the physical region. We conclude that Roy equations offer the most precise method for the determination of the $\sigma$ pole from $\pi\pi$ elastic scattering.