Algebraic approach to boson spectroscopy—study of theJPC=1++nonet and theD-meson decays

An approach to boson spectroscopy without assuming SU(6) is discussed in the theoretical framework of (a) chiral $\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ charge algebra supplemented by the presence of exotic commutation relations which characterize the symmetry breaking, (b) asymptotic SU(3), and (c) the hypothesis of asymptotic algebraic realization of SU(3) in the algebra $[A_i,A_j]=i{f}_{\mathrm{ijk}}{V}_k$. The constraints obtained always involve the ideal or the SU(6) nonet scheme of bosons as a special case. However, the deviation from this ideal nonet scheme is completely prescribed by the constraints. As an illustration, the question whether the $1^{++}$ nonet can form an almost ideal nonet similar to the $1^{--}$ is studied in some detail. The $D$-meson decays, $D\to K\overline{K}\pi$ and $\eta \pi \pi$, are especially studied. It is argued that the present experiment can be fitted to an almost ideal nonet structure of the $1^{++}$ nonet, if the $A_1$ mass lies around 1070-1100 MeV.