$η-η^{\prime}$ mixing

The $η-η^{\prime}$ 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 $η^{\prime}$, but is suppressed in the $η$ amplitude by a factor $m_η^{2}/m_{η^{\prime}}^{2}$. 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 $η-η^{\prime}$ system could be described by a meson field renormalization and a new mixing angle $θ$ which differs from the usual mixing angle $θ_{P}$ 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 $η-η^{\prime}$ mixing angle and the momentum-dependent mixing term. Assuming nonet symmetry for the $η_{0}$ singlet amplitude, from the sum rules relating $θ$ and $d$ to the measured vector meson radiative decays amplitudes, we obtain consistent solutions with $θ=-(13.99\pm 3.1)^{\circ}$, $d=0.12\pm 0.03$ from $ρ\toηγ$ and $η^{\prime}\toργ$ decays, for $ω$ , $θ=-(15.47\pm 3.1)^{\circ}$, $d=0.11\pm 0.03$, and for $ϕ$, $θ=-(12.66\pm 2.1)^{\circ}$, $d=0.10\pm 0.03$. It seems that vector meson radiative decays would favor a small $η-η^{\prime}$ mixing angle and a small momentum-dependent mixing term.

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