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

We present the first lattice QCD study of coupled isoscalar $\pi \pi ,K\bar{K},\eta \eta$ $S$- and $D$-wave scattering extracted from discrete finite-volume spectra computed on lattices which have a value of the quark mass corresponding to $m_{\pi }\sim 391$ MeV. In the $J^P=0^{+}$ sector we find analogues of the experimental $\sigma$ and $f_0(980)$ states, where the $\sigma$ appears as a stable bound-state below $\pi \pi$ threshold, and, similar to 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 $K\bar{K}$ threshold. For $J^P=2^{+}$ we find two states resembling the $f_2(1270)$ and $f_2^{\mathrm{\prime }}(1525)$, observed as narrow peaks, with the lighter state dominantly decaying to $\pi \pi$ and the heavier state to $K\bar{K}$. The presence of all these states is determined rigorously by finding the pole singularity content of scattering amplitudes, and their couplings to decay channels are established using the residues of the poles.