  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.
