  If scale invariance is exact, unparticles are unlikely to be probed in
colliders since there are stringent constraints from astrophysics and
cosmology. However these constraints are inapplicable if scale invariance is
broken at a scale mu >~ 1 GeV. The case 1 GeV <~ mu < M_Z is particularly
interesting since it allows unparticles to be probed at and below the Z pole.
We show that mu can naturally be in this range if only vector unparticles
exist, and briefly remark on implications for Higgs phenomenology. We then
obtain constraints on unparticle parameters from e+ e- -> mu+ mu- cross-section
and forward-backward asymmetry data, and compare with the constraints from
mono-photon production and the Z hadronic width.
