The pion-pion scattering amplitude. III: Improving the analysis with forward dispersion relations and Roy equations

TL;DR

This paper refines pion-pion scattering amplitude analysis by incorporating forward dispersion relations and Roy equations, ensuring analyticity and consistency with experimental data, leading to more reliable low-energy parameters and physical predictions.

Contribution

The paper introduces improved fits to pion-pion scattering amplitudes that satisfy analyticity constraints using forward dispersion relations and Roy equations, enhancing the reliability of the amplitude parametrizations.

Findings

Amplitudes satisfy analyticity and dispersion relations within errors.

Central values of amplitudes are consistent with constraints, with minimal changes.

Precise low-energy parameters and phase shifts are determined.

Abstract

We complete and improve the fits to experimental $\pi\pi$ scattering amplitudes, both at low and high energies, that we performed in the previous papers of this series. We then verify that the corresponding amplitudes satisfy analyticity requirements, in the form of partial wave analyticity at low energies, forward dispersion relations (FDR) at all energies, and Roy equations below$\bar{K}K$ threshold; the first by construction, the last two, inside experimental errors. Then we repeat the fits including as constraints FDR and Roy equations. The ensuing central values of the various scattering amplitudes verify very accurately FDR and, especially, Roy equations, and change very little from what we found by just fitting data, with the exception of the D2 wave phase shift, for which one parameter moves by $1.5 \sigma$. These improved parametrizations therefore provide a reliable representation of pion-pion amplitudes with which one can test various physical relations. We also present a list of low energy parameters and other observables. In particular, we find $a_0^{(0)}=0.223\pm0.009 M^{-1}_\pi$, $a_0^{(2)}=-0.0444\pm0.0045 M^{-1}_\pi$ and $\delta_0^{(0)}(m^2_K)-\delta_0^{(2)}(m^2_K)=50.9\pm1.2^{\rm o}$.