The excited state of the $\alpha$-particle: a benchmark study

A benchmark study is performed for the excited state of ${}^4$He. When the Coulomb interaction is switched off, the ${}^4$He nucleus exhibits a bound excited state in the vicinity of $p-{}^3$H threshold. As the Coulomb interaction is gradually introduced, the excited state crosses the threshold and eventually becomes a resonant state. Using three numerical methods, we track the evolution of this excited state and determine the resonance energy and width. Comparisons of the theoretical predictions reveal a significant discrepancy with commonly used $R$-matrix values based on the analysis of the experimental data. We explain the origin for this discrepancy. Additionally, the two-level energy spectrum of ${}^4$He in the absence of the Coulomb force exhibits characteristics linked to Efimov physics, suggesting a reduced sensitivity to interaction details.