  The energy level structure of ${}^{12}$C nucleus at a few MeV above the
three-$\alpha$ threshold is still unsatisfactory known. For instance, most
microscopic calculations predicted that there exist one $0^+$-state in this
energy region besides the well known Hoyle state, while some experimental and
theoretical studies show the existing of two $0^+$-states.
  In this paper, I will take a three-$\alpha$-boson (3$\alpha$) model for bound
and continuum states in ${}^{12}$C, and study a transition process from the
${}^{12}$C($0_1^+$) ground state to 3$\alpha$ $0^+$ continuum states by the
electric monopole ($E0$) operator. The strength distribution of the process
will be calculated as a function of $3\alpha$ energy using the Faddeev
three-body theory. The Hamiltonian for the $3\alpha$ system consists of two-
and three-$\alpha$ potentials, and some three-$\alpha$ potentials with
different range parameters will be examined.
  Results of the strength function show a double-peaked bump at low energy
region, which can be considered as two $0^+$-states. The peak at higher energy
may originate from a 3$\alpha$ resonant state. However, it is unlikely that the
peak at the lower energy is related to a resonant state, which suggests that it
may be due to so called "ghost anomaly". Distributions of decaying particles
are also calculated.
