Abstract
Abstract 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.
Highlights
The Yukawa potential is the effective non-relativistic description of the interaction of two particles due to the exchange of a massive particle of mass M
The value of the critical screening lengths are important for some applications, specially the ground state critical screening, and have been calculated using non-perturbative methods [29] and numerically solving the Yukawa potential for n = 0 to n = 9 [31]
Self-interacting dark matter has been proposed as a possibility to solve this problem [37] but there is some tension with data [38][39], which is alleviated if we consider that the self-interaction is induced by the exchange of a massive mediator which, for non-relativistic dark matter, yields a Yukawa potential [40][41]
Summary
The Yukawa potential is the effective non-relativistic description of the interaction of two particles due to the exchange of a massive particle of mass M. The value of the critical screening lengths are important for some applications, specially the ground state critical screening, and have been calculated using non-perturbative methods [29] and numerically solving the Yukawa potential for n = 0 to n = 9 [31]. Self-interacting dark matter has been proposed as a possibility to solve this problem [37] but there is some tension with data [38][39], which is alleviated if we consider that the self-interaction is induced by the exchange of a massive mediator which, for non-relativistic dark matter, yields a Yukawa potential [40][41]. Yukawa interaction between two dark matter particles can give rise to dark matter bound states (darkonium) which drives us to study of the Yukawa potential at the quantum level.
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