Hyperspherical-coordinate approach to the spectra and decay widths of hybrid quarkonia

In this paper, we consider the possibility that Y(4260), Y(4360), \( \psi(4415) \) or X(4660) is the ground state or the first excited state of a hybrid charmonium and examine whether \( \Upsilon ( 10860) \) is the ground state or the first excited state of a hybrid bottomonium. Under a constituent quark model, we carry out numerical calculations to obtain the spectra of both electric gluon and magnetic gluon hybrid quarkonia. We see that the ground state of a magnetic gluon hybrid charmonium and the first excited state of an electric gluon hybrid are comparable in energy to \( \psi(4415) \) and that the first excited states of an electric gluon hybrid bottomonium and the ground state of a magnetic gluon hybrid appear a few hundred MeV above the mass of \( \Upsilon (10860 ) \). Also, we evaluate the widths of the decays of a hybrid charmonium into \(D^{(*)} \bar{D}^{(*) } \) and those of a hybrid bottomonium into \( B^{(*) } \bar{B}^{(* ) } \). We show that if the exotic meson candidates are hybrid charmonia, then they are magnetic gluon ones and argue that it seems unlikely that \( \Upsilon (10860) \) is the ground state of an electric gluon hybrid bottomonium.