Heavy flavored hydrogen molecule systems

This study provides a comprehensive analysis of $S$-wave exotic hydrogen-like three-body systems ($pp\mu^-$, $pp\tau^-$, $\mu^-\mu^-p$, $\tau^-\tau^-p$, $p\mu^-\tau^-$) with spin-parity $J^P = 1/2^+$ and $3/2^+$, and four-body systems ($pp\mu^-\mu^-$, $pp\tau^-\tau^-$) with $J^P = 0^+$, $1^+$, and $2^+$. We use complex scaling and Gaussian expansion methods to solve the complex-scaled Schr\"{o}dinger equation and obtain possible bound and quasi-bound states. The resulting binding energies range from $-33.8$~keV to $-340$~eV. Notably, we present the first theoretical estimation of the bound-state energy levels of $pp\mu^-\mu^-$ and $pp\tau^-\tau^-$, which is of significant importance for understanding exotic few-body Coulomb systems. We further analyze spin configurations and root-mean-square radii to elucidate the spatial structure of these bound and quasi-bound states. Our results reveal that $K$-type spatial configurations play a crucial role in accurately describing bound and quasi-bound states in the hydrogen-molecule-like systems $pp\mu^-\mu^-$ and $pp\tau^-\tau^-$. Incorporating $K$-type configurations significantly alters the mass spectra of these states. Future muon colliders and muon facilities may offer promising platforms for the possible copious production of such heavy flavored hydrogen molecules and molecular ions. For instance, scattering processes such as $2\mu^- + \mathrm{H_2} \to \mathrm{H_{2\mu}} + 2e^-$, $\mu^- + \mathrm{H_2} \to \mathrm{H_{\mu e}} + e^-$, and $\mu^- + \mathrm{H_2^+} \to \mathrm{H_{2\mu}^+} + e^-$ could be utilized, facilitating detailed studies of intriguing states such as $\mathrm{H_{2\mu}}$, $\mathrm{H_{\mu e}}$, and $\mathrm{H_{2\mu}^+}$.