Quasi-doublets in the $3\alpha$ Spectrum of $^{12}$C

Abstract Within the three-$\alpha$ cluster model of $^{12}$C, we perform configuration-space Faddeev calculations of the low-lying $0^+$ and $2^+$ states. Resonance states are treated using a variant of the analytic continuation in the coupling constant (ACCC) method. The phenomenological Hamiltonian includes both two-body and three-body interactions. The recently proposed effective-field-theory-motivated leading-order (LO) $\alpha\alpha$ interaction is approximated by a Gaussian-type (LO-simulating) potential to reproduce the experimental $\alpha$--$\alpha$ scattering phase shifts. The parameters $(V,\beta)$ of the three-body force, $ V_{3\mathrm{bf}}(\rho) = V \exp\!\left[ -\frac{1}{2} \left(\frac{\rho}{\beta}\right)^2 \right], $ are determined by fitting the experimental spectrum of $^{12}$C. We investigate the possibility of a universal set of two- and three-body interactions capable of simultaneously reproducing the energies of lowest $0^+$ or $2^+$ states. For the $2^+$ levels, we find that the condition $\beta > 3$ fm must be satisfied for such universality. The ACCC method, combined with the effective-mass approach, reveals level anticrossing and demonstrates coupling between the $0_1^+$ and $0_2^+$ states, as well as between the $2_1^+$ and $2_2^+$ states, which can be interpreted as quasidoublet structures in the $3\alpha$ spectrum. The calculated results are compared with those obtained using the Ali--Bodmer potentials and the next-to-next-to-leading-order (NNLO) interaction. We find that the LO-simulating potential reduces the strength of the three-body force required to reproduce the low-lying spectrum.