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 quark 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 pseudoscalar 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.

我们在 SU(3) 风味极限中的广义线性 sigma 模型的框架内详细研究了标量和伪标量介子和胶球之间的相互作用。我们方法的基础是发展对光标量和伪标量介子（高达约 2 GeV）的全局理解，从而探索复合夸克物质场和胶球之间的潜在混合。拉格朗日的手征扇区由代表夸克-反夸克和四夸克的两个手性九线组成（按领先顺序，该扇区包含具有八个或更少数量的夸克或反夸克线的项）。拉格朗日函数还包含一个代表标量和伪标量胶球及其与物质手征场相互作用的扇区，以精确实现 QCD 的轴向和迹异常的方式。在这个构造中，模型有两个标量八位组和两个伪标量八位组（每个都是两个和四夸克分量的线性组合），以及三个标量 SU(3) 单线态和三个伪标量 SU(3) 单线态（每个二夸克和四夸克分量以及胶球分量的线性组合）。随着实验群众的投入${\mathrm{一种}}_0(980)$, ${\mathrm{一种}}_0(1450)$，以及广大 $\pi (137)$, $\pi (1300)$和它们的衰减常数，我们进行了广泛的数值模拟来确定模型参数空间的边界。我们进一步结合了关于 eta 态的质谱以及几个衰变宽度和衰变比的实验数据。$F_0$状态以放大参数空间并对伪标量和标量 SU(3) 八位字节和单线态的子结构以及伪标量和标量胶球质量进行预测。我们发现标量和伪标量胶球质量分别约为 1.6 和 2.0 GeV。