arXiv 2102.071602021-02hep-phphysics.plasm-phquant-ph

隐超对称性中Yukawa势的束缚态

Bound states of the Yukawa potential from hidden supersymmetry

M. Napsuciale, S. Rodríguez

备注：: References added, typos fixed, Mathematica code for the calculations released. Version to appear in Progress of Theoretical and Experimental Physics

摘要：

在这项工作中，我们提出了一个唯象研究的完整的解析解的束缚本征态和本征值的Yukawa势先前获得使用隐藏的超对称性的系统和系统的展开的Yukawa势的$δ=a_{0}/D$，其中$a_{0}$是玻尔半径和$D$是屏蔽长度。本征值$ε_{nl}(δ)$以$δ$中的泰勒级数的形式给出，其可以被系统地计算到期望的阶数$δ^{k}$。库仑$l$ -简并性被屏蔽效应破坏，对于给定的$n$，$l$值越高，$ε_{nl}(δ)$越大，导致$n\ge4$的能级交叉。采用Padé近似技术，泰勒级数的收敛半径可以扩大到临界值，从而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本征值，并精确地确定临界屏蔽长度$δ_{nl}$ .本征态具有类似于库仑势的解的形式，其中相关联的拉盖尔多项式被具有$r$相关系数的$δ^{k}$阶的新多项式代替，所述系数又是$r$中的多项式。在一般情况下，我们发现相当大的偏离库仑径向概率的屏蔽长度接近其临界值。我们使用这些解决方案，以找到在原点的波函数的绝对值的平方$l=0$，和他们的衍生物为$l=1$，为最低的状态，作为$δ$的功能，进入暗物质的束缚态的暗规范理论与光暗介质的现象。

英文摘要：

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, $ε_{nl}(δ)$, are given in the form of Taylor series in $δ$ which can be systematically calculated to the desired order $δ^{k}$. Coulomb $l$-degeneracy is broken by the screening effects and, for a given $n$, $ε_{nl}(δ)$ is larger for higher values of $l$ which causes the crossing of levels for $n\ge4$. 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, $δ_{nl}$. Eigenstates have a form similar to the solutions of the Coulomb potential, with the associated Laguerre polynomials replaced by new polynomials of order $δ^{k}$ 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.

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摘要：

英文摘要：

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