  We expand Hilbert series technologies in effective field theory for the
inclusion of massive particles, enabling, among other things, the enumeration
of operator bases for non-linearly realized gauge theories. We find that the
Higgs mechanism is manifest at the level of the Hilbert series, as expected for
the partition function of an $S$-matrix that is subject to the Goldstone
equivalence theorem. In addition to massive vectors, we detail how other
massive, spinning particles can be studied with Hilbert series; in particular,
we spell out the ingredients for massive gravity in general spacetime
dimensions. Further methodology is introduced to enable Hilbert series to
capture the effect of spurion fields acquiring vevs. We apply the techniques to
the Higgs Effective Field Theory (HEFT), providing a systematic enumeration of
its operator basis. This is achieved both from a direct and a custodial
symmetry spurion-based approach; we compare and contrast the two approaches,
and our results to those appearing in previous literature.
