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-H\"uckel 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.