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While the Hoyle state (the isoscalar $0^+_2$ excitation at 7.65 MeV in$^{12}$C) has been observed in almost all the electron and $\alpha$ inelasticscattering experiments, the second $2^+$ excited state of $^{12}$C at $E_{\rmx}\approx 10$ MeV, believed to be an excitation of the Hoyle state, has notbeen clearly observed in these measurements excepting the high-precision \aapexperiments at $E_\alpha=240$ and 386 MeV. Given the (spin and isospin zero)$\alpha$-particle as a good probe for the nuclear isoscalar excitations, itremains a puzzle why the peak of the $2^+_2$ state could not be clearlyidentified in the measured \aap spectra. To investigate this effect, we haveperformed a microscopic folding model analysis of the \ac scattering data at240 and 386 MeV in both the Distorted Wave Born Approximation (DWBA) andcoupled-channel (CC) formalism, using the nuclear transition densities given bythe antisymmetrized molecular dynamics (AMD) approach and a complex CDM3Y6density dependent interaction. Although AMD predicts a very weak transitionstrength for the direct $(0^+_1\to 2^+_2)$ excitation, our detailed analysishas shown evidence that a weak \emph{ghost} of the $2^+_2$ state could beidentified in the 240 MeV \aap data for the $0^+_3$ state at 10.3 MeV, when theCC effects by the indirect excitation of the $2^+_2$ state are taken intoaccount. Based on the same AMD structure input and preliminary \aap data at 386MeV, we have estimated relative contributions from the $2^+_2$ and $0^+_3$states to the excitation of $^{12}$C at $E_{\rm x}\approx 10$ MeV as well aspossible contamination by $3^-_1$ state.

Hindrance of the excitation of the Hoyle state and the ghost of the $2^+_2$ state in $^{12}$C