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Starting from the $0^{++}$ glueball mass and wave function computed fromlattice QCD, we compute the local potential between two constituent gluons.Since the properties of constituent gluons are still a matter of research, weallow 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 aremassless, the corresponding local potential is shown to be compatible with aCornell form, that is a linear confinement plus a short-range Coulomb part,with standard values for the flux tube energy density and for the strongcoupling constant. When the gluons are massive, the confining potential is asaturating one, commonly used to simulate string-breaking effects. Theseresults fill a gap between lattice QCD and phenomenological models: The pictureof the scalar glueball as a bound state of two constituent gluons interactingvia a phenomenological potential is shown to emerge from pure gauge lattice QCDcomputations. Moreover, we show that the allowed potential shape is constrainedby the mass of the constituent gluons.Comment: 4 figures; Comments added, one typo corrected in v2. V3 accepted for  publication in EPJA : major changes, content enlarged, inclusion of massive  gluon

Effective potential between two gluons from the scalar glueball