Black-body radiation induced photodissociation and population redistribution of weakly bound states in H$_2^+$

Molecular hydrogen ions in weakly bound states close to the first dissociation threshold are attractive quantum sensors for measuring the proton-to-electron mass ratio and hyperfine-induced ortho-para mixing. The experimental accuracy of previous spectroscopic studies relying on fast ion beams could be improved by using state-of-the-art ion trap setups. With the electric dipole moment vanishing in H$_2^+$ and preventing fast spontaneous emission, radiative lifetimes of the order of weeks are found. We include the effect of black-body radiation that can lead to photodissociation and rovibronic state redistribution to obtain effective lifetimes for trapped ion experiments. Rate coefficients for bound-bound and bound-continuum processes were calculated using adiabatic nuclear wave functions and nonadiabatic energies, including relativistic and radiative corrections. Effective lifetimes for the weakly bound states were obtained by solving a rate equation model and lifetimes in the range of 4 to 523~ms and $>$215~ms were found at room temperature and liquid nitrogen temperature, respectively. Black-body induced photodissociation was identified as the lifetime-limiting effect, which guarantees the purity of state-selectively generated molecular ion ensembles. The role of hyperfine-induced $g/u$-mixing, which allows pure rovibrational transitions, was found to be negligible.