  Starting from the $0^{++}$ glueball mass and wave function computed from
lattice QCD, we compute the local potential between two constituent gluons.
Since the properties of constituent gluons are still a matter of research, we
allow for them to be either massless, or massive with a mass around 0.7 GeV.
Both pictures are actually used in the literature. When the gluons are
massless, the corresponding local potential is shown to be compatible with a
Cornell form, that is a linear confinement plus a short-range Coulomb part,
with standard values for the flux tube energy density and for the strong
coupling constant. When the gluons are massive, the confining potential is a
saturating one, commonly used to simulate string-breaking effects. These
results fill a gap between lattice QCD and phenomenological models: The picture
of the scalar glueball as a bound state of two constituent gluons interacting
via a phenomenological potential is shown to emerge from pure gauge lattice QCD
computations. Moreover, we show that the allowed potential shape is constrained
by the mass of the constituent gluons.

High Energy Physics - Phenomenology;High Energy Physics - Lattice