Effective three-body interactions in the alpha-cluster model for the   ^{12}C nucleus
  Properties of the lowest $0^{+}$ states of $^{12}\mathrm{C}$ are calculated to study the role of three-body interactions in the $\alpha$-cluster model. An additional short-range part of the local three-body potential is introduced to incorporate the effects beyond the $\alpha$-cluster model. There is enough freedom in this potential to reproduce the experimental values of the ground-state and excited-state energies and the ground-state root-mean-square radius. The calculations reveal two principal choices of the two-body and three-body potentials. Firstly, one can adjust the potentials to obtain the width of the excited $0_2^+$ state and the monopole $0_2^+ \to 0_1^+ $ transition matrix element in good agreement with the experimental data. In this case, the three-body potential has strong short-range attraction supporting a narrow resonance above the $0_2^+$ state, the excited-state wave function contains a significant short-range component, and the excited-state root-mean-square radius is comparable to that of the ground state. Next, rejecting the solutions with an additional narrow resonance, one finds that the excited-state width and the monopole transition matrix element are insensitive to the choice of the potentials and both values exceed the experimental ones. 
S. I. Fedotov, O. I. Kartavtsev, A. V. Malykh