Determination of crossing-symmetric $\pi\pi$ scattering amplitudes and the quark mass evolution of the $\sigma$ constrained by lattice QCD

TL;DR

This paper combines lattice QCD data and dispersion relations to accurately determine the $\sigma$ resonance pole position and its evolution with quark mass, reducing systematic uncertainties in $\pi\pi$ scattering amplitudes.

Contribution

It introduces a method to incorporate crossing symmetry and analyticity constraints into lattice QCD analyses of $\pi\pi$ scattering, improving the determination of the $\sigma$ resonance.

Findings

The $\sigma$ pole position can be determined with minimal systematic uncertainty.

The $\sigma$ transitions from a bound state to a broad resonance with increasing quark mass.

Certain amplitude parameterizations satisfy all physical constraints.

Abstract

Lattice QCD spectra can be used to constrain partial-wave scattering amplitudes that, while satisfying unitarity, do not have to respect crossing symmetry and analyticity. This becomes a particular problem when extrapolated far from real energies, e.g. in the case of broad resonances like the $\sigma$, leading to large systematic uncertainties in the pole position. In this manuscript, we will show how dispersion relations can implement the additional constraints, using as input lattice--determined $\pi\pi$ partial-wave scattering amplitudes with isospin--0,1,2. We will show that only certain combinations of amplitude parameterizations satisfy all constraints, and that when we restrict to these, the $\sigma$ pole position is determined with minimal systematic uncertainty. The evolution of the now well-constrained $\sigma$ pole with varying light quark mass is presented, showing how it transitions from a bound-state to a broad resonance.