Determination of Crossing-Symmetric Ππ Scattering Amplitudes and the Quark Mass Evolution of the Σ Constrained by Lattice QCD

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 parametrizations 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.