The large mass of the ninth pseudoscalar meson, the ${\eta }^{\prime }$, is believed to arise from the combined effects of the axial anomaly and the gauge field topology present in QCD. We report a realistic, $2+1$-flavor, lattice QCD calculation of the $\eta$ and ${\eta }^{\prime }$ masses and mixing which confirms this picture. The physical eigenstates show small octet-singlet mixing with a mixing angle of $\theta =-14.1(2.8)°$. Extrapolation to the physical light quark mass gives, with statistical errors only, $m_{\eta }=573(6)\mathrm{MeV}$ and $m_{{\eta }^{\prime }}=947(142)\mathrm{MeV}$, consistent with the experimental values of 548 and 958 MeV.

• Received 24 February 2010

DOI:https://doi.org/10.1103/PhysRevLett.105.241601