Coupled $\pi \pi ,K\bar{K}$ scattering in $P$-wave and the $\rho$ resonance from lattice QCD

We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson scattering in $P$-wave, by calculating correlation functions using lattice QCD with light quark masses such that $m_{\pi }=236$ MeV in a cubic volume of $\sim (4 ,$\mathrm{f}\mathrm{m}$)^3$. Variational analyses of large matrices of correlation functions computed using operator constructions resembling $\pi \pi$, $K\bar{K}$ and $q\bar{q}$, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic $\pi \pi$ scattering region we obtain a detailed energy-dependence for the phase-shift, corresponding to a $\rho$ resonance, and we extend the analysis into the coupled-channel $K\bar{K}$ region for the first time, finding a small coupling between the channels.