Is there gluonium in the $η^\prime$? The flavour structure of $η$ and $η^\prime$ mixing in radiative $V\to Pγ$ and $P\to Vγ$ decays

Abstract

Radiative $V\to Pγ$ and $P\to Vγ$ decays are conventionally described using constituent-quark mass differences and phenomenological wavefunction-overlap factors as sources of flavour-symmetry breaking. We investigate whether the overlap factors can instead be understood as the effect of mixing of the $η$ and $η^\prime$ with an additional pseudoscalar state, described by an $SO(4)$ rotation among four basis states. A global fit to current data yields a significantly better description than the corresponding three-state limit without overlap factors, while preserving the established $η$-$η^\prime$ and vector-meson mixing angles, and shows that the additional state mixes predominantly with the non-strange component. The conventional overlap-factor description is recovered exactly as the special case in which this mixing is purely non-strange. The non-strange pattern disfavours both a flavour-singlet admixture, as expected from mixing with a pseudoscalar glueball, and the gluonium scenario considered in previous analyses. Radiative decays therefore provide no compelling evidence for gluonium in the $η^\prime$, and the overlap factors are better interpreted in terms of a radially excited non-strange state.