Extraction of isoscalar $\pi \pi$ phase-shifts from lattice QCD

We conduct a two-flavor ($N_f=2$) lattice QCD calculation of the elastic phase-shifts for pion-pion scattering in the scalar, isoscalar channel (the $\sigma$-meson). The calculation is performed for two quark masses corresponding to a pion mass of $315\text{ MeV}$ and $227\text{ MeV}$. The $\sigma$-meson parameters are extracted using various parametrizations of the scattering amplitude. The results obtained from a chiral unitary parametrization are extrapolated to the physical point and read $M_\sigma = (440^{+10}_{-16}(50) - i,240(20)(25))$\text{ MeV}$$, where the uncertainties in the parentheses denote the stochastic and systematic ones. The behavior of the $\sigma$-meson parameters with increasing pion mass is discussed as well.