Glueball states in a constituent gluon model

In a model with a dynamical gluon mass, we investigate the bound states of two and three gluons via a Schrödinger equation. The short distance potential is approximated by one-gluon exchange while the long distance part is assumed to be of a breakable string. We estimate the masses and in particular the sizes of low-lying bound states with no orbital angular momentum. By considering quantum-mechanical smearing of the gluon fields and normalizing to lattice results on $M_{0^{++}}$ and $M_{2^{++}},$ we find that the $0^{++}$ glueball is rather small in size compared with the others. The fitted gluon mass is of order 600 to 700 MeV, which is reasonable. The $0^{-+},\hspace{0.5em}{1}^{--},$ and $3^{--}$ three gluon glueball states are nearly degenerate, and their mass ratio with $2^{++}$ is largely independent of all the parameters and consistent with lattice calculations. We estimate the mass of the $1^{--}$ glueball to be around $3.1–3.7\hspace{0.5em}\mathrm{GeV},$ which is close to the mass of $J/\psi$ and ${\psi }^{\prime }.$