Coulomb screening effect on the Hoyle state energy in thermal plasmas

The first excited $J^pi=0^+$ state of $^{12}$C, the so-called Hoyle state, plays an essential role in a triple-$alpha$ ($^4$He) reaction, which is a main contributor to the synthesis of $^{12}$C in a burning star. We investigate the Coulomb screening effects on the energy shift of the Hoyle state in a thermal plasma environment using precise three-$alpha$ model calculations. The Coulomb screening effect between $alpha$ clusters are taken into account within the Debye-Huckel approximation. To generalize our study, we utilize two standard $alpha$-cluster models, which treat the Pauli principle between the $alpha$ particles differently. We find that the energy shifts do not depend on these models and follow a simple estimation in the zero-size limit of the Hoyle state when the Coulomb screening length is as large as a value typical of such a plasma consisting of electrons and $alpha$ particles.