In this work, we present a phenomenological study of the complete analytical solution to the bound eigenstates and eigenvalues of the Yukawa potential obtained previously using the hidden supersymmetry of the system and a systematic expansion of the Yukawa potential in terms of δ = a 0 ∕D, where a 0 is the Bohr radius and D is the screening length. The eigenvalues, ϵnl(δ), are given in the form of Taylor series in δ which can be systematically calculated to the desired order δk. Coulomb l-degeneracy is broken by the screening effects and, for a given n, ϵnl(δ) is larger for higher values of l which causes the crossing of levels for n ≥ 4. The convergence radius of the Taylor series can be enlarged up to the critical values using the Padé approximants technique which allows us to calculate the eigenvalues with high precision in the whole rage of values of δ where bound states exist, and to reach a precise determination of the critical screening lengths, δnl. Eigenstates have a form similar to the solutions of the Coulomb potential, with the associated Laguerre polynomials replaced by new polynomials of order δk with r-dependent coefficients which, in turn, are polynomials in r. In general we find sizable deviations from the Coulomb radial probabilities only for screening lengths close to their critical values. We use these solutions to find the squared absolute value at the origin of the wave function for l = 0, and their derivatives for l = 1, for the lowest states, as functions of δ, which enter the phenomenology of dark matter bound states in dark gauge theories with a light dark mediator.

中文翻译：

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来自隐藏超对称的 Yukawa 势的束缚态

在这项工作中，我们对先前使用系统的隐藏超对称性和 Yukawa 势在 δ = a 0 ∕D 方面的系统扩展获得的 Yukawa 势的束缚本征态和本征值的完整解析解进行了唯象学研究，其中a 0 是玻尔半径，D 是屏蔽长度。特征值 εnl(δ) 以 δ 中的泰勒级数的形式给出，可以系统地计算到所需的阶 δk。库仑 l 简并性被筛选效应打破，对于给定的 n，εnl(δ) 在 l 的较高值时较大，这会导致 n ≥ 4 的能级交叉。泰勒级数的收敛半径可以使用 Padé 逼近技术扩大到临界值，这使我们能够在存在束缚态的整个 δ 值范围内高精度地计算特征值，并达到精确确定临界筛选长度，δnl。本征态具有类似于库仑势解的形式，相关的拉盖尔多项式被新的 δk 阶多项式取代，其系数与 r 相关，而这些多项式又是 r 中的多项式。一般来说，我们发现只有在接近其临界值的筛选长度时才会与库仑径向概率有相当大的偏差。我们使用这些解决方案来找到 l = 0 时波函数原点处的平方绝对值，以及它们对于 l = 1 的导数，对于最低状态，