  The standard model can be extended to include weakly-interacting light
particle (WILP): real or complex singlet scalar, Majorana or Dirac neutral
fermion, neutral or hidden-charged vector boson, etc. Imposing the $Z_2$
symmetry, these particles can be lifted as the weakly-interacting massive
particle (WIMP), the candidate of dark matter. Instead, imposing the shift
symmetry on the scalar components gives rise to the axion-like particle, dark
photon, etc. Utilizing these light degree of freedom along with the standard
model particles and imposing different symmetries, we construct the complete
and independent sets of effective operators up to dimension eight with the
Young tensor technique, consistent with counting from the Hilbert series.
