  In response to the growing need for theoretical tools that can be used in QCD
to describe and understand the dynamics of gluons in hadrons in the Minkowski
space-time, the renormalization group procedure for effective particles (RGPEP)
is shown in the simplest available context of heavy quarkonia to exhibit a
welcome degree of universality in the first approximation it yields once one
assumes that beyond perturbation theory gluons obtain effective mass. Namely,
in the second-order terms, the Coulomb potential with Breit-Fermi spin
couplings in the effective quark-antiquark component of a heavy quarkonium, is
corrected in one-flavor QCD by a spin-independent harmonic oscillator term that
does not depend on the assumed effective gluon mass or the choice of the RGPEP
generator. The new generator we use here is much simpler than the ones used
before and has the advantage of being suitable for studies of the effective
gluon dynamics at higher orders than the second and beyond the perturbative
expansion.
