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.