Proximity of $f_0(1500)$ and $f_0(1710)$ to the scalar glueball

Abstract

Within a nonlinear chiral Lagrangian framework, the underlying mixings among quark-antiquark, four-quark and glue components of $f_0(1500)$ and $f_0(1710)$ are studied in a global picture that includes all isosinglet scalar mesons below 2 GeV. The quark components are introduced in the Lagrangian in terms of two separate nonets (a quark-antiquark nonet and a four-quark nonet) which can mix with each other and with a scalar glueball. The free parameters of the Lagrangian are studied by a simultaneous fit to more than 20 experimental data and constraints on the mass spectrum, decay widths, and decay ratios of the isosinglet scalars below 2 GeV. Moreover, constraints on the mass spectrum and decay widths of isodoublet and isovector scalars below 2 GeV as well as pion-pion scattering amplitude are also taken into account. The insights gained in this global picture, due to the complexities of the mixings as well as the experimental uncertainties, are mainly qualitative but are relatively robust, and reveal that the lowest scalar glueball hides between $f_0(1500)$ and $f_0(1710)$, resulting in a considerable mixing with various quark components of these two states. The overall current experimental and theoretical uncertainties do not allow to pin down the exact glue components of isosinglet states, nevertheless it is shown that the $f_0(1500)$ and $f_0(1710)$ have the highest glue component. While this global study does not allow precision predictions for each individual state, it provides useful "family" correlations among the isosinglet states that are found insightful in probing the substructure of all scalars, in general, and the isosinglets, in particular. The overall estimate of the scalar glueball mass is found to be $1.58 \pm 0.18$ GeV.