  We propose the minimal, lepton-number conserving, SU(3)xSU(2)xU(1)
gauge-singlet, or phantom, extension of the Standard Model. The extension is
natural in the sense that all couplings are of O(1) or forbidden due to a
phantom sector global U(1)_D symmetry, and basically imitates the standard
Majorana see-saw mechanism. Spontaneous breaking of the U(1)_D symmetry
triggers consistent electroweak gauge symmetry breaking only if it occurs at a
scale compatible with small Dirac neutrino masses and baryogenesis through
Dirac leptogenesis. Dirac leptogenesis proceeds through the usual
out-of-equilibrium decay scenario, leading to left and right-handed neutrino
asymmetries that do not fully equilibrate after they are produced. The model
contains two physical Higgs bosons and a massless Goldstone boson. The
existence of the Goldstone boson suppresses the Higgs to bb branching ratio and
instead the Higgs bosons will mainly decay to invisible Goldstone and/or to
visible vector boson pairs. In a representative scenario, we estimate that with
30 fb^-1 integrated luminosity, the LHC could discover this invisibly decaying
Higgs, with mass ~120 GeV. At the same time a significantly heavier, partner
Higgs boson with mass ~210 GeV could be found through its vector boson decays.
Electroweak constraints as well as astrophysical and cosmological implications
are analysed and discussed.
