Pseudoscalar glueball, the axial-vector anomaly, and the mixing problem for pseudoscalar mesons

If the $G\left(1440\right)\mathrm{}$ observed in $\psi \to \gamma G$ is a pseudoscalar glueball its relationship with other pseudoscalar mesons must be understood. We present a simple, unified picture of these mesons in which there must be mixing between glue matter and quark matter. Our model, an extension of an effective Lagrangian which solved the U(1) problem by incorporating the axial-vector anomaly, dictates a relationship between ${\eta }^{\prime }$ and $G$. We are readily able to explain why the quark-matter meson ${\eta }^{\prime }$ is at least as prominent as the glueball $G$ in the gluon-dominated reaction $\psi \to \gamma X$.