$\eta$-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 $m_\eta=714(6)(16)$ MeV. Through the Witten-Veneziano relation, this value can be matched to a mass value of $\sim 981$ MeV for the $\mathrm{SU(3)}$ counterpart of $\eta$. Based on a large gauge ensemble, the $\eta-G$ mixing energy and the mixing angle are determined to be $|x|=107(15)(2)$ MeV and $|\theta|=3.46(46)^\circ$ from the $\eta-G$ correlators that are calculated using the distillation method. We conclude that the $\eta-G$ mixing is tiny and the topology induced interaction contributes most of $\eta$ mass owing to the QCD $\mathrm{U_A(1)}$ anomaly.