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 is taken into account within the Debye-Hückel 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 shift does not depend on these models and follows 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.

\(^{12}\) C的第一个激发\(J^{\pi }=0^+\)态，即所谓的霍伊尔态，在三重\(\alpha \) ( \(^{4}\) He) 反应，这是在燃烧的恒星中合成\(^{12}\) C的主要贡献者。我们使用精确的三 - \(\alpha \)模型计算研究了库仑屏蔽对热等离子体环境中霍伊尔态能量转移的影响。在 Debye-Hückel 近似中考虑了\(\alpha \)簇之间的库仑筛选效应。为了概括我们的研究，我们使用两个标准\(\alpha \)-cluster 模型，它以不同的方式处理\(\alpha \)粒子之间的泡利原理。我们发现，当库仑屏蔽长度与由电子和\(\ alpha \)粒子。