Mass-Energy Compensation Effect of 3$$\alpha$$ Hamiltonian

The $$3\alpha$$ phenomenological model describes the structure of the carbon-12 nucleus as a cluster of three alpha particles. This model includes a pairwise $$\alpha{-}\alpha$$ interaction and a three-body force. To fit the three-body potential, the $${}^{12}$$ C data are used, while ensuring that the pair potential reproduces the $$\alpha{-}\alpha$$ scattering data. Alternatively, the mass–energy compensation (MEC) effect can be used to simulate the effect of the three-body potential by adjusting the mass of the $$\alpha$$ particle within the effective-mass approach. We demonstrate the MEC effect for the $$3\alpha$$ ground state by numerically solving the differential Faddeev equation, in which the $$\alpha{-}\alpha$$ interaction is described by the Ali–Bodmer potential. The effective masses of $$\alpha$$ particles are evaluated for the ground and excited $$0^{+}$$ and bound $$2^{+}$$ states. We demonstrate a coupling between the ground and first excited $$0^{+}$$ states, indicated by an anti-crossing of these energy levels in the energy–mass coordinates. A correspondence between the effective mass and a three-body potential is demonstrated. We discuss the results of the $$0^{+}_{2}$$ calculations for various models of the $$\alpha{-}\alpha$$ interaction.