$\eta-\eta^{\prime}$ mixing

The $\eta -\eta^{\prime}$ mixing mass term due to the derivative coupling $SU(3)\times SU(3)$ symmetry breaking term, produces an additional momentum-dependent pole term for processes with $\eta^{\prime}$, but is suppressed in the $\eta$ amplitude by a factor $m_{\eta}^{2}/m_{\eta^{\prime}}^{2}$. This seems to be the origin of the two-angle description of the pseudo-scalar decay constants used in the literature. In this paper, by diagonalizing both the mixing mass term and the momentum-dependent mixing term, we show that the $\eta -\eta^{\prime}$ system could be described by a meson field renormalization and a new mixing angle $\theta$ which differs from the usual mixing angle $\theta_{P}$ by a small momentum-dependent mixing $d$ term. This new mixing scheme with exact treatment of the momentum-dependent mixing term, is actually simpler than the perturbation treatment and should be used in any determination of the $\eta -\eta^{\prime}$ mixing angle and the momentum-dependent mixing term. Assuming nonet symmetry for the $\eta_{0}$ singlet amplitude, from the sum rules relating $\theta$ and $d$ to the measured vector meson radiative decays amplitudes, we obtain consistent solutions with $\theta=-(13.99\pm 3.1)^{\circ}$, $d=0.12\pm 0.03$ from $\rho\to\eta\gamma$ and $\eta^{\prime}\to\rho\gamma$ decays, for $\omega$ , $\theta=-(15.47\pm 3.1)^{\circ}$, $d=0.11\pm 0.03$, and for $\phi$, $\theta=-(12.66\pm 2.1)^{\circ}$, $d=0.10\pm 0.03$. It seems that vector meson radiative decays would favor a small $\eta-\eta^{\prime}$ mixing angle and a small momentum-dependent mixing term.