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$η-η^{\prime}$ Mixing Angle from Vector Meson Radiative Decays

The octet-singlet $η-η^{\prime}$ mixing mass term could have a derivative $O(p^{2})$ term as found in recent analysis of the $η-η^{\prime}$ system. This term gives rise to an additional momentum-dependent pole contribution which is suppressed by a factor $m_η^{2}/m_{η^{\prime}}^{2}$ for $η$ relative to the $η^{\prime}$ amplitude. The processes with $η$ meson can then be described, to a good approximation, by the momentum-independent mixing mass term which gives rise to a new $η-η^{\prime}$ mixing angle $θ_{P}$, like the old $η-η^{\prime}$ mixing angle used in the past, but a momentum-dependent mixing term $d$, like $\sin(θ_{0}-θ_{8})$ in the two-angle mixing scheme used in the parametrization of the pseudo-scalar meson decay constants in the current literature, is needed to describe the amplitudes with $η^{\prime}$. In this paper, we obtain sum rules relating $θ_{P}$ and $d$ to the physical vector meson radiative decays with $η$ and $η^{\prime}$, as done in our previous work for $η$ meson two-photon decay, and with nonet symmetry for the $η^{\prime}$ amplitude, we obtain a mixing angle $θ_{P}=-(18.76\pm 3.4)^{\circ}$, $d=0.10\pm 0.03$ from $ρ\toηγ$ and $η^{\prime}\toργ$ decays, for $ω$, $θ_{P}=-(15.81\pm 3.1)^{\circ}$, $d=0.02\pm 0.03$, and for $ϕ$, $θ_{P}=-(13.83\pm 2.1)^{\circ}$, $d=0.08\pm 0.03$. A larger value of $0.06\pm 0.02$ for $d$ is obtained directly from the nonet symmetry expression for the $η^{\prime}\toωγ$ amplitude. This indicates that more precise vector meson radiative decay measured branching ratios and higher order SU(3) breaking effects could bring these values for $θ_{P}$ closer and allows a better determination of $d$.