Prospects for the determination of fundamental constants with beyond-state-of-the-art uncertainty using molecular hydrogen ion spectroscopy

The proton, deuteron, and triton masses can be determined relative to the electron mass via rovibrational spectroscopy of molecular hydrogen ions. This has to occur via comparison of the experimentally measured transition frequencies and the ab initio calculated frequencies, whose dependence on the mass ratios can be calculated precisely. In precision experiments to date (on HD+ and H2+), the transitions have involved the ground vibrational level v = 0 and excited vibrational levels with quantum numbers up to v' = 9. For these transitions, the sensitivity of the ab initio frequency to the high-order QED contributions is correlated with that to the mass ratios. This prevents an efficient simultaneous determination of these quantities from experimental data, so the accuracy of the mass ratios is essentially limited by the theoretical uncertainty. Here we analyze how the accuracy of mass ratios may be improved by providing experimental transition frequencies between levels with larger quantum numbers, whose sensitivity to the mass ratio is positive rather than negative, or close to zero. This allows the unknown QED contributions and involved fundamental constants to be much more efficiently determined from a joint analysis of several measurements. We also consider scenarios where transitions of D2+ are included. We find these to be powerful approaches, allowing us in principle to reach uncertainties for the mass ratios approximately three orders smaller than reported by CODATA 2018. Improvements by a factor of 3.5 for the Rydberg constant, and 11 (14) for the proton (deuteron) charge radius, are also projected.