  Phenomenological implications of a minimal extension to the Standard Model
are considered, in which a Nambu-Goldstone boson emerges from the spontaneous
breaking of a global U(1) symmetry. This is felt only by a scalar field which
is a singlet under all Standard Model symmetries, and possibly by neutrinos.
Mixing between the Standard Model Higgs boson field and the new singlet field
may lead to predominantly invisible Higgs boson decays. The "natural" region in
the Higgs boson mass spectrum is determined, where this minimally extended
Standard Model is a valid theory up to a high scale related with the smallness
of neutrino masses. Surprisingly, this region may coincide with low visibility
of all Higgs bosons at the LHC. Monte-Carlo simulation studies of this
"nightmare" situation are performed and strategies to search for such Higgs
boson to invisible (Nambu-Goldstone boson) decays are discussed. It is possible
to improve the signal-to-background ratio by looking at the distribution of
either the total transverse momentum of the leptons and the missing transverse
momentum, or by looking at the distribution of the azimuthal angle between the
missing transverse momentum and the momentum of the lepton pair for the Z- and
Higgs-boson associated production. We also study variations of the model with
non-Abelian symmetries and present approximate formulae for Higgs boson decay
rates. Searching for Higgs bosons in such a scenario at the LHC would most
likely be solely based on Higgs to "invisible" decays.
