We perform a coupled channel study of the meson-meson $S$ waves with isospin ($I$) 0 and $1/2$ up to 2 GeV. A new approach is derived that allows one to include the many channels $\pi \pi$, $K\overline{K}$, $\eta \eta$, $\sigma \sigma$, $\eta {\eta }^{\prime }$, ${\eta }^{\prime }{\eta }^{\prime }$, $\rho \rho$, $\omega \omega$, $\omega \varphi$, $\varphi \varphi$, $a_1\pi$, and ${\pi }^{*}\pi$ with still few free parameters. It follows that coupled channel dynamics is strong and cannot be neglected in order to study resonance properties in the region 1.4–1.6 GeV. All the resonances with masses below 2 GeV and $I=0$ and $1/2$ are generated. We identify the $f_0(1710)$ and an important contribution to the $f_0(1500)$ as an unmixed glueball. This is based on an accurate agreement of our results with predictions of lattice QCD and the chiral suppression of the coupling of a scalar glueball to $\overline{q}q$. Another pole, mainly corresponding to the $f_0(1370)$, is a pure octet state.

• Received 31 January 2008

DOI:https://doi.org/10.1103/PhysRevLett.101.252002