Bound and Resonant States of Muonic Few-Body Coulomb Systems: Extended Stochastic Variational Approach

TL;DR

This work addresses the spectrum of bound and resonant states in hydrogen-like and molecular muonic Coulomb systems by developing an extended stochastic variational method (ESVM) combined with complex scaling (CSM) and explicitly correlated Gaussians (ECG). The approach delivers energy accuracies better than $0.1~\mathrm{eV}$ and maps complete spectra below the $X\mu(n=2)$ thresholds for three- and four-body systems, including many previously unresolved shallow resonances. By incorporating two-stage ESVM with scattering-like (molecular) configurations, the method robustly resolves near-threshold structures across diverse subchannels, as illustrated in the three- and four-body muonic ions $\mu\mu X$, $pp\mu$, $dd\mu$, $tt\mu$, $pd\mu$, $pt\mu$, $dt\mu$, and the four-body $\mu\mu pp$, $\mu\mu dd$, $\mu\mu tt$. The results provide a comprehensive reference for muon-catalyzed fusion physics and precision QED tests, and establish ESVM-CSM as a versatile tool for exploring exotic Coulombic few-body systems with dense continua.

Abstract

We compute the bound and resonant states of hydrogen-like muonic ions ($μμp$, $μμd$, $μμt$) and three-body muonic molecular ions ($ppμ$, $pdμ$, $ptμ$, $ddμ$, $dtμ$, $ttμ$), and the four-body double-muonic hydrogen molecule ($μμpp$) using an extended stochastic variational method combined with complex scaling. The approach provides a unified treatment of bound and quasibound states and achieves an energy accuracy better than $0.1~\mathrm{eV}$ across all systems studied. Complete spectra below the corresponding $n=2$ atomic thresholds are obtained, including several previously unresolved shallow resonances in both three- and four-body sectors.

Figures (6)

• Figure 1: Jacobi coordinate configurations of the $pp\mu$ system in the conventional GEM framework.
• Figure 2: Molecular configurations for the $dd\mu$ and $\mu\mu dd$ systems in extended SVM.
• Figure 3: Complex-energy spectra of the $\mu\mu pp$ system ($S_{12}=0$, $S_{34}=0$, $L=0$, $\theta=3^\circ$) obtained with SVM (top) and with ESVM (bottom).
• Figure 4: Complex-energy spectra of the $\mu\mu p$ system ($S_{12}=0$, $L=0$, $\theta=6^\circ$) .
• Figure 5: Complex-energy spectra of the $dt \mu$ system ($S_{12}=0$, $L=0$, $\theta=6^\circ$) .