We discover new, hitherto unknown, shadow poles in Brune's alternative parametrization of $R$-matrix theory [Brune, Phys. Rev. C 66, 044611 (2002)]. Where these poles are, and how many there are, depends on how one continues $R$-matrix operators to complex wave numbers (especially the shift $S$ and penetration $P$ functions). This has little consequence for the exact $R$-matrix formalism (past the last energy threshold), as we show one can still always fully reconstruct the scattering matrix with only the previously known alternative parameters (poles and corresponding resonance widths), for which there were as many poles as the number of levels $N_{\lambda }$. However, we generalize the alternative parametrization to the Reich-Moore formalism and show that the choice of continuation is now critical as it changes the alternative parameters values (poles and residue widths are now complex). In order to establish nuclear libraries with alternative parameters, the nuclear community will thus have to decide what convention to adopt. We argue in favor of analytical continuation (against the legacy of Lane and Thomas approach) in the second article [P. Ducru, B. Forget, V. Sobes, G. Hale, and M. Paris, Phys. Rev. C 103, 064609 (2021)] of this trilogy on pole parametrizations of $R$-matrix theory, on which we build to establish the windowed multipole representation or $R$-matrix cross sections in the third and last article [P. Ducru, A. Alhajri, I. Meyer, B. Forget, V. Sobes, C. Josey, and J. Liang, Phys. Rev. C 103, 064610 (2021)]. We observe the first evidence of shadow poles in the alternative parametrization of $R$-matrix theory in the isotope ${}^{134}\mathrm{Xe}$ spin-parity group $J^{\pi }=1/2^{(-)}$ and show how they indeed depend on the choice of continuation to complex wave numbers.

中文翻译：

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R矩阵理论替代参数化中的阴影极点

我们在 Brune 的替代参数化中发现了新的、迄今为止未知的阴影极点 $\mathrm{电阻}$-矩阵理论 [Brune, Phys. 修订版 C 66 , 044611 (2002)]。这些极点在哪里，有多少，取决于人们如何继续$\mathrm{电阻}$- 复数波数的矩阵运算符（尤其是移位 $秒$ 和渗透 $磷$职能）。这对确切的影响不大$\mathrm{电阻}$-matrix 形式主义（超过最后一个能量阈值），正如我们所展示的，我们仍然可以仅使用先前已知的替代参数（极点和相应的共振宽度）完全重建散射矩阵，其中极点与级别数一样多 $N_{\lambda }$. 然而，我们将替代参数化推广到 Reich-Moore 形式主义，并表明延续的选择现在至关重要，因为它改变了替代参数值（极点和残差宽度现在很复杂）。为了建立具有替代参数的核库，核界因此必须决定采用什么公约。我们在第二篇文章中主张支持分析延续（反对 Lane 和 Thomas 方法的遗产）[P. Ducru, B. Forget, V. Sobes, G. Hale 和 M. Paris, Phys. Rev. C 103 , 064609 (2021)] 关于极点参数化的三部曲$\mathrm{电阻}$-矩阵理论，我们在此基础上建立窗口多极表示或$\mathrm{电阻}$-第三篇也是最后一篇文章中的矩阵横截面 [P. Ducru、A. Alhajri、I. Meyer、B. Forget、V. Sobes、C. Josey 和 J. Liang，Phys。修订版 C 103 , 064610 (2021)]。我们在替代参数化中观察到阴影极点的第一个证据$\mathrm{电阻}$-同位素中的矩阵理论 ${}^{134}氙$ 自旋奇偶群 $J^{\pi }=1/2^{(-)}$ 并展示它们确实如何依赖于复数波数的延续选择。