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 ${\mathrm{HD}}^{+}$ and ${\mathrm{H}}_2{}^{+}$), the transitions have involved the ground vibrational level $v=0$ and excited vibrational levels with quantum numbers up to $v^{\prime }=9$. 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 ${\mathrm{D}}_2{}^{+}$ 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.

Figure 1

Determination of the nuclear-electron mass ratio of ${\mathrm{HD}}^{+}$ from single transition frequencies. Shown are four already measured transitions $f_0, f_1, f_5$, and $f_9$ and two hot-band transitions considered for future measurement, $X:(v=9,N=1)\to (v^{\prime }=18,N^{\prime }=0)$ and $Y:(7,1)\to (15,0)$. Here $f_X$ (red) is a positive-mass-sensitivity transition and $f_Y$ (blue) is a suppressed-mass-sensitivity transition. Note that the band slope is similar for $f_0, f_1, f_5$, and $f_9$ because the abscissa is the normalized QED contribution ${\mathrm{\Delta }}_{\mathrm{nuc},\mathrm{QED}}/0.32$ [see Eq. (14)]. The width of each band takes into account (1) the uncertainty of the theoretical transition frequency due to the uncertainty of the Rydberg constant and the uncertainties of the rms charge radius of the proton and deuteron, (ii) the uncertainty of the experimentally measured frequency, and (iii) the uncertainty of the hyperfine structure theory. For the four measured transitions, the experimental results $f^{\left(\mathrm{expt}\right)}$ and theoretical results with CODATA 2018 constants $f_{2018}^{\left(\mathrm{theor}\right)}$ have been used in the plot. For the proposed transitions (red and blue bands) we have assumed a hypothetical perfect match $f^{\left(\mathrm{expt}\right)}=f_{2018}^{\left(\mathrm{theor}\right)}$, experimental uncertainties $u\left(f_X^{\left(\mathrm{expt}\right)}\right)=u\left(f_Y^{\left(\mathrm{expt}\right)}\right)=0.3\mathrm{kHz}$, and spin theory uncertainty 0.035 kHz. The yellow shaded range indicates the estimated normalized uncertainty of the unknown QED contribution to the theoretical transition frequency, stemming from the three contributions $u\left(\delta f_{\mathrm{QED},1}\right), u\left(\delta f_{\mathrm{QED},2}\right)$, and $u\left(\delta f_{\mathrm{QED},3}\right)$ (see the text for details). The horizontal black dashed lines indicate the $\pm 1$ standard uncertainty interval of the value of ${\mu }_{pd}/m_e$ according to CODATA 2018. The dark cyan dotted lines represent the $\pm 1$ standard uncertainty interval of ${\mu }_{pd}/m_e$ derived from Penning trap measurements. It is computed from the uncertainties of the atomic mass of the electron [8] (modified as in [14]), of the atomic mass of the deuteron [10], and of the proton-deuteron mass ratio [11].