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

Abstract

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 $\varphi$, $\theta =-(12.66\pm 2.1{)}^{\circ }$, $d=0.10\pm 0.03$. It seems that vector meson radiative decays would favor a small $\eta -{\eta }^{\prime }$ mixing angle and a small momentum-dependent mixing term.

Comments: LaTeX, 11 pages, v2, text and references added, to appear in PRD