  We analyze one-loop vacuum stability, perturbativity, and phenomenological
constraints on a complex singlet extension of the Standard Model (SM) scalar
sector containing a scalar dark matter candidate. We study vacuum stability
considerations using a gauge-invariant approach and compare with the
conventional gauge-dependent procedure. We show that, if new physics exists at
the TeV scale, the vacuum stability analysis and experimental constraints from
the dark matter sector, electroweak precision data, and LEP allow both a
Higgs-like scalar in the mass range allowed by the latest results from CMS and
ATLAS and a lighter singlet-like scalar with weak couplings to SM particles. If
instead no new physics appears until higher energy scales, there may be
significant tension between the vacuum stability analysis and phenomenological
constraints (in particular electroweak precision data) to the extent that the
complex singlet extension with light Higgs and singlet masses would be ruled
out. We comment on the possible implications of a scalar with ~125 GeV mass and
future ATLAS invisible decay searches.
