Isoscalar $\pi \pi$ scattering and the $\sigma$ meson resonance from QCD

We present for the first time a determination of the energy dependence of the isoscalar $\pi \pi$ elastic scattering phase-shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum we obtain the $S$-wave phase-shift up to the $K\bar{K}$ threshold. Calculations are performed at two values of the $u,d$ quark mass corresponding to $m_{\pi }=236,391$ MeV and the resulting amplitudes are described in terms of a $\sigma$ meson which evolves from a bound-state below $\pi \pi$ threshold at the heavier quark mass, to a broad resonance at the lighter quark mass.