  Gauge-invariant non-local scalar and field strength operators have been
argued to have significance, e.g., as a way to determine the behaviour of the
screened static potential at large distances, as order parameters for
confinement, as input parameters in models of confinement, and as
gauge-invariant definitions of light constituent masses in bound state systems.
We measure such "correlators" in the 3d pure SU(2) and SU(2)+Higgs models on
the lattice. We extract the corresponding mass parameters and discuss their
scaling and physical interpretation. We find that the finite part of the MS-bar
scheme mass measured from the field strength correlator is large, more than
half the glueball mass. We also determine the non-perturbative contribution to
the Debye mass in the 4d finite T SU(2) gauge theory with a method due to
Arnold and Yaffe, finding $\delta m_D\approx 1.06(4)g^2T$.
