Decays of the vector glueball

We calculate two- and three-body decays of the (lightest) vector glueball into (pseudo)scalar, (axial-)vector, as well as pseudovector and excited vector mesons\textbf{ }in the framework of a model of QCD. While absolute values of widths cannot be predicted because the corresponding coupling constants are unknown, some interesting branching ratios can be evaluated\textbf{ }by setting the mass of the yet hypothetical vector glueball to $3.8$ GeV as predicted by quenched Lattice QCD\textbf{.} We find that the decay mode $\omega\pi\pi$ should be one of the largest (both through the decay chain $\mathcal{O}\rightarrow b_{1}\pi\rightarrow$ $\omega\pi\pi$ and through the direct coupling $\mathcal{O}\rightarrow\omega\pi\pi$)$.$ Similarly, the (direct and indirect) decay into $\pi KK^{\ast}(892)$ is sizable. Moreover, the decays into $\rho\pi$ and $K^{\ast}(892)K$ are, although subleading, possible and could play a role in explaining the $\rho\pi$ puzzle of the charmonium state $\psi(2S)$ thank to a (small) mixing with the vector glueball. The vector glueball can be directly formed at the ongoing BESIII experiment as well as at the future PANDA experiment at the FAIR facility. If the width is sufficiently small ($\lesssim100$ MeV) it should not escape future detection. It should be stressed that the employed model is based on some inputs and simplifying assumptions: the value of glueball mass (at present, the quenched lattice value is used), the lack of mixing of the glueball with other quarkonium states, and the use of few interaction terms. It then represents a first step toward the identification of the main decay channels of the vector glueball, but shall be improved when corresponding experimental candidates and/or new lattice results will be available.