$\eta$-glueball mixing from $N_f=2$ lattice QCD

We perform the first lattice study on the mixing of the isoscalar pseudoscalar meson $\eta$ and the pseudoscalar glueball $G$ in the $N_f=2$ QCD at the pion mass $m_\pi\approx 350$ MeV. The $\eta$ mass is determined to be $m_\eta=714.1(5.4)$ MeV. Through the Witten-Veneziano relation, this value can be matched to a mass value of $\sim 935$ MeV for the $SU(3)$ counterpart of $\eta$. Based on a large gauge ensemble, the $\eta-G$ mixing energy and the mixing angle are determined to be $|x|=107(14)$ MeV and $|\theta|=3.47(46)^\circ$ from the $\eta-G$ correlators that are calculated using the distillation method. It is also found that $m_\eta$ can be correctly derived from $\eta-\eta_c$ correlators with $\eta_c$ being the pseudoscalar charmonium. We conclude that the $\eta-G$ mixing is tiny and the topology induced interaction contributes most of $\eta$ mass owing to the QCD $U_A(1)$ anomaly.