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The $I=1$ $p$-wave and $I=2$ $s$-wave elastic $\pi$-$\pi$ scatteringamplitudes are calculated from a first-principles lattice QCD simulation usinga single ensemble of gauge field configurations with $N_{\mathrm{f}} = 2+1$dynamical flavors of anisotropic clover-improved Wilson fermions. This ensemblehas a large spatial volume $V=(3.7\mathrm{fm})^3$, pion mass $m_{\pi} =230\mathrm{MeV}$, and spatial lattice spacing $a_s = 0.11\mathrm{fm}$.Calculation of the necessary temporal correlation matrices is efficientlyperformed using the stochastic LapH method, while the large volume enables animproved energy resolution compared to previous work. For this single ensemblewe obtain $m_{\rho}/m_{\pi} = 3.350(24)$, $g_{\rho\pi\pi} = 5.99(26)$, and aclear signal for the $I=2$ $s$-wave. The success of the stochastic LapH methodin this proof-of-principle large-volume calculation paves the way forquantitative study of the lattice spacing effects and quark mass dependence ofscattering amplitudes using state-of-the-art ensembles.

$I=1$ and $I=2$ $\pi-\pi$ scattering phase shifts from $N_{\mathrm{f}} = 2+1$ lattice QCD