Radiative decay of muonic molecules in resonance states

In this study, we theoretically investigated x-ray spectra from the radiative decay of muonic deuterium molecules in resonance states $\mathrm{dd}{\ensuremath{\mu}}^{*}$, which plays an important role in a new kinetic model of muon catalyzed fusion ($\ensuremath{\mu}\mathrm{CF}$). The resonance states are Feshbach resonances located below the $\mathrm{d}\ensuremath{\mu}(n=2$) $+$ d threshold energy and radiatively decay into the continuum or bound states. The x-ray spectra having characteristic shapes according to the radial distribution of the two nuclei are obtained using precise three-body wave functions. We carefully examined the convergence of the x-ray spectra and achieved agreements between the length- and velocity-gauge calculations. We revealed a nonadiabatic kinetic energy distribution of the decay fragments, indicating that the radiative decay becomes a heating source of muonic atoms. We also investigated the decay branch that directly results in bound-state muonic molecules. Some resonance states $\mathrm{dd}{\ensuremath{\mu}}^{*}$ and $\mathrm{dt}{\ensuremath{\mu}}^{*}$ are found to have high branching ratios to the bound state where intramolecular nuclear fusion occurs. The formation of the muonic molecules in the bound states from metastable muonic atoms can be a fast track in the $\ensuremath{\mu}\mathrm{CF}$ cycle, which skips a slow path to form the bound state from the ground-state muonic atoms and increases the $\ensuremath{\mu}\mathrm{CF}$ cycle rate. Although the spectra from the radiative decays are located in the energy range of 1.5--1.997 keV, which is close to the $\mathrm{K}\ensuremath{\alpha}$ x-ray of 1.997 keV from muonic deuterium atoms, state-of-the-art microcalorimeters can distinguish them.