  We perform a systematic analysis of an extension of the Standard Model that
includes a complex singlet scalar field and is scale invariant at the tree
level. We call such a model the Minimal Scale Invariant extension of the
Standard Model (MSISM). The tree-level scale invariance of the model is
explicitly broken by quantum corrections, which can trigger electroweak
symmetry breaking and potentially provide a mechanism for solving the gauge
hierarchy problem. Even though the scale invariant Standard Model is not a
realistic scenario, the addition of a complex singlet scalar field may result
in a perturbative and phenomenologically viable theory. We present a complete
classification of the flat directions which may occur in the classical scalar
potential of the MSISM. After calculating the one-loop effective potential of
the MSISM, we investigate a number of representative scenarios and determine
their scalar boson mass spectra, as well as their perturbatively allowed
parameter space compatible with electroweak precision data. We discuss the
phenomenological implications of these scenarios, in particular, whether they
realize explicit or spontaneous CP violation, neutrino masses or provide dark
matter candidates. In particular, we find a new minimal scale-invariant model
of maximal spontaneous CP violation which can stay perturbative up to
Planck-mass energy scales, without introducing an unnaturally large hierarchy
in the scalar-potential couplings.
