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The energy level structure of ${}^{12}$C nucleus at a few MeV above thethree-$\alpha$ threshold is still unsatisfactory known. For instance, mostmicroscopic calculations predicted that there exist one $0^+$-state in thisenergy region besides the well known Hoyle state, while some experimental andtheoretical studies show the existing of two $0^+$-states.  In this paper, I will take a three-$\alpha$-boson (3$\alpha$) model for boundand 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 theelectric monopole ($E0$) operator. The strength distribution of the processwill be calculated as a function of $3\alpha$ energy using the Faddeevthree-body theory. The Hamiltonian for the $3\alpha$ system consists of two-and three-$\alpha$ potentials, and some three-$\alpha$ potentials withdifferent range parameters will be examined.  Results of the strength function show a double-peaked bump at low energyregion, which can be considered as two $0^+$-states. The peak at higher energymay originate from a 3$\alpha$ resonant state. However, it is unlikely that thepeak at the lower energy is related to a resonant state, which suggests that itmay be due to so called "ghost anomaly". Distributions of decaying particlesare also calculated.

Monopole transition strength function of ${}^{12}$C in three-$\alpha$ model