Determination of crossing-symmetric $\pi\pi$ scattering amplitudes and the quark mass evolution of the $\sigma$ 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 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.