Spinless mesons and glueballs mixing patterns in SU(3) flavor limit

We present a detailed study of the interactions among scalar and pseudoscalar mesons and glueballs within the framework of the generalized linear sigma model in the SU(3) flavor limit. The basis of our approach is to develop a global understanding of light scalar and pseudoscalar mesons (up to around 2 GeV) and thereby explore the underlying mixings among composite quark matter fields and glueballs. The chiral sector of the Lagrangian is formulated in terms of two chiral nonets representing quark-antiquarks and tetraquarks (in the leading order, this sector contains terms with eight or fewer number of quak or antiquark lines). The Lagrangian also contains a sector that represents scalar and pseudoscalar glueballs and their interactions with the matter chiral fields, in a manner that the axial and trace anomalies of QCD are exactly realized. In this construct, the model has two scalar octets and two pdeudoscalar octets (each a linear combination of two- and four-quark components), as well as three scalar SU(3) singlets and three pseudoscalar SU(3) singlets (each a linear combination of two- and four-quark components as well as a glueball component). With the inputs of the experimental masses of $a_0(980)$, $a_0(1450)$, and the masses of $\pi(137)$, $\pi(1300)$ and their decay constants, we perform an extensive numerical simulation to determine the boundaries of the parameter space of the model. We further incorporate experimental data on the mass spectrum of eta states as well as on several decay widths and decay ratios of $f_0$ states to zoom in on the parameter space and make predictions for the substructure of pseudoscalar and scalar SU(3) octets and singlets as well as for the pseudoscalar and scalar glueball masses. We find the scalar and pseudoscalar glueball masses around 1.6 and 2.0 GeV, respectively.