Algebraic approaches to hadrons and the identification ofι(1440)with glueball

The identifications of $\iota \mathrm{}\left(1440\right)\mathrm{}$ and $\theta \mathrm{}\left(1640\right)\mathrm{}$ with glueballs are difficult, if one relies on the popular simple quarkonium-glueball mass matrices. However, a different conclusion has been drawn from two distinct algebraic approaches. They are both based on QCD algebras and produce almost identical results for the ${\mathrm{O}}^{-+}$ mesons. In this paper, in the framework of chiral U(4)×U(4) QCD algebras, the problems of ${\mathrm{O}}^{-+}$ meson masses, mixings, decay constants, branching ratios of $\frac{J}{\psi }\to \iota \gamma$, ${\eta }^{\prime }\gamma$, and $\eta \gamma$, and the widths of the $\iota \to \rho \gamma$ and $2\gamma$ decays are discussed. It is found that the main features of the mixing parameters obtained previously in the U(3)×U(3) scheme remain intact and the $\iota \mathrm{}\left(1440\right)\mathrm{}$ can again be accommodated as a glueball which appreciably mixes with the ${\eta }^{\prime }$. It is also pointed out that the simple quarkonium-glueball mass matrices may fail to include the important effect of flavor-symmetry breaking and therefore are not very realistic. This is demonstrated by showing that the mass matrices can be reproduced in the present algebraic approach only if one is willing to take the symmetry limit for quantities which clearly involve the effect of symmetry breaking.