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Analyses of the inelastic $\alpha$+$^{12}$C scattering at medium energieshave indicated that the strength of the Hoyle state (the isoscalar 0$^+_2$excitation at 7.65 MeV in $^{12}$C) seems to exhaust only 7 to 9% of themonopole energy weighted sum rule (EWSR), compared to about 15% of the EWSRextracted from inelastic electron scattering data. The full monopole transitionstrength predicted by realistic microscopic $\alpha$-cluster models of theHoyle state can be shown to exhaust up to 22% of the EWSR. To explore themissing monopole strength in the inelastic $\alpha$+$^{12}$C scattering, wehave performed a fully microscopic folding model analysis of the inelastic$\alpha$+$^{12}$C scattering at $E_{\rm lab}=104$ to 240 MeV using the3-$\alpha$ resonating group wave function of the Hoyle state obtained byKamimura, and a complex density-dependent M3Y interaction newly parametrizedbased on the Brueckner Hartree Fock results for nuclear matter. Our foldingmodel analysis has shown consistently that the missing monopole strength of theHoyle state is not associated with the uncertainties in the analysis of the$\alpha$+$^{12}$C scattering, but is most likely due to the short lifetime andweakly bound structure of this state which significantly enhances absorption inthe exit $\alpha$+$^{12}$C$^*(0^+_2)$ channel.

Missing monopole strength of the Hoyle state in the inelastic $\alpha$+$^{12}$C scattering