$0_^+ $ states of the ${ }^{12}$C nucleus: Faddeev calculation in configurations space

The $\alpha $-cluster model and Faddeev equations in configuration space are applied to study the ${ }^{12}$C nucleus. The model includes the Ali-Bodmer nuclear potential [1], attractive three-body potential, and takes into account the Coulomb interaction. An s-wave model [2] is adapted and parameters of the three-body potential are chosen to describe the first two 0$_^+ $ levels of ${ }^{12}$C. The value of the range parameter of the potential is adjusted to reproduce the position of diffraction minimum for the elastic form factor of ${ }^{12}$C. The model assumes a strong distortion of the charge density of $\alpha $ clusters inside the ${ }^{12}$C nucleus. It was found that the most probable configuration of the $\alpha $-clusters in the 0$_1^+ $state corresponds to an equilateral triangle with sides as large as 3.5 fm and in the 0$_2^+ $ state to a linear chain with the values of 2.9 fm and 13.1 fm for each link. Having calculated low-lying levels of ${ }^{12}$C, we found that the contributions of higher partial waves of nuclear interaction to the energy of 3$\alpha $-system are unnaturally large and some states turn to be overbound. Upon applying the method [3] based on the Pade approximation we got satisfactory description for the 0$_3^+ $ and 0$_4^+ $ states [4]. Additional $0^+$ broad resonance obtained in [3] was not found. 1. S. Ali, A. R. Bodmer, Nucl. Phys. 80, 99 (1966). 2. Z. Papp, et al. Few-Body Systems 30, 31 (2001). 3. C. Kurokawa and K. Kato, Phys. Rev. C76, 021301-1 (2005). 4. http://www.tunl.duke.edu/nucldata/.