Glueball States in a Constituent Gluon Model

In a model with dynamical gluon mass, we investigate the bound states of two and three gluons via a Schr\"odinger 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 {\it 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 3-gluon glueballs $0^{-+}$, $1^{--}$ and $3^{--}$ states are nearly degenerate, and their mass ratio with $2^{++}$ is largely independent of all parameters and consistent with lattice calculations. We estimate the mass of $1^{--}$ glueball to be $3.1-3.7$ GeV, which is close to the mass of $J/\psi$ and $\psi^\prime$.