Bound and Resonant Spectra of Few-Lepton Coulomb Systems

Abstract

We present a unified calculation of bound and resonant states in purely leptonic Coulomb systems: $e^-e^-e^+$ ($\mathrm{Ps}^-$), $\mu^+e^-e^-$ ($\mathrm{Mu}^-$), $\mu^+\mu^+e^-$ ($\mathrm{Mu}_2^+$), $e^+e^+e^-e^-$ ($\mathrm{Ps}_2$), and $\mu^+\mu^+e^-e^-$ ($\mathrm{Mu}_2$). Using an extended stochastic variational method combined with the complex scaling method, we resolve the natural-parity $S$- and $P$-wave spectra. Near their respective $n=2$ thresholds, all three trilepton systems exhibit Gailitis--Damburg sequences generated by $2S$ and $2P$ Stark mixing and the resulting inverse-square attraction. Although microscopically distinct from the Efimov effect, this mechanism produces the same inverse-square asymptotics and geometric scaling. The $\mathrm{Ps}^-$ and $\mathrm{Mu}^-$ systems exhibit resonance sequences of comparable density, whereas the $\mathrm{Mu}_2^+$ produces a much denser spectrum, with 20 resolved members in the $^3P^o$ channel. In $\mathrm{Mu}_2^+$, the deeper states follow molecular Born--Oppenheimer configurations, while the near-threshold spectrum is governed by the atomic $\mathrm{Mu}(2)+\mu^+$ structure. In $\mathrm{Ps}_2$, coupling between threshold-degenerate configurations is essential for a near-threshold bound state. In $\mathrm{Mu}_2$, the Born--Oppenheimer organization of the resonance spectrum is channel dependent.