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We present the first lattice QCD study of coupled isoscalar$\pi\pi,K\overline{K},\eta\eta$ $S$- and $D$-wave scattering extracted fromdiscrete finite-volume spectra computed on lattices which have a value of thequark mass corresponding to $m_\pi\sim391$ MeV. In the $J^P=0^+$ sector we findanalogues of the experimental $\sigma$ and $f_0(980)$ states, where the$\sigma$ appears as a stable bound-state below $\pi\pi$ threshold, and, similarto what is seen in experiment, the $f_0(980)$ manifests itself as a dip in the$\pi\pi$ cross section in the vicinity of the $K\overline{K}$ threshold. For$J^P=2^+$ we find two states resembling the $f_2(1270)$ and $f_2'(1525)$,observed as narrow peaks, with the lighter state dominantly decaying to$\pi\pi$ and the heavier state to $K\overline{K}$. The presence of all thesestates is determined rigorously by finding the pole singularity content ofscattering amplitudes, and their couplings to decay channels are establishedusing the residues of the poles.

Isoscalar $\pi\pi, K\overline{K}, \eta\eta$ scattering and the $\sigma, f_0, f_2$ mesons from QCD