  We study a recently proposed gauge invariant, non-local pure gauge
observable, which is equivalent to the gluon propagator in a certain gauge. The
correlator describes a gluon coupled to static sources and decays with
eigenvalues of the Hamiltonian, permitting a non-perturbative definition of a
gluonic parton mass. Detailed numerical tests of the observable are performed
in SU(2) gauge theories in 2+1 dimensions. In a Higgs model it reproduces the
physical W-boson mass, while in a confining theory its non-local nature results
in almost exclusive projection onto torelonic states. However, the gluon mass
can also be related to the mass difference between the lowest gluelumps, i.e.
gluon configurations bound to adjoint sources. Its value is computed in SU(2)
pure gauge theory in 2+1 dimensions to be m_A=0.360(19)g^2, which plays an
important role as "magnetic mass" of the four-dimensional theory at high
temperatures. The same quantity is found to determine the mass splitting
between vector and scalar mesons in the three-dimensional theory. In SU(3) pure
gauge theory in 3+1 dimensions the corresponding gluelump splitting yields
m_A\approx 370 MeV. Possible relations of this quantity with the QCD heavy
meson spectrum are discussed.
