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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>σ</a:mi></a:math>, 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi><c:mi>π</c:mi></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>σ</e:mi></e:math> pole position is determined with minimal systematic uncertainty. The evolution of the now well-constrained <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>σ</g:mi></g:math> pole with varying light quark mass is presented, showing how it transitions from a bound-state to a broad resonance.