We present high-precision spectroscopy of the ground-state hyperfine structure of HD$^+$ at 4~T. We determine the bound-electron $g$ factor, $g_{e,\mathrm{bound}} = -2.002\,278\,540\,96(40)$, to a relative uncertainty of $2\times$10$^{-10}$, the most precise determination of a bound-electron $g$ factor of a molecular ion to date. The experimental value agrees with recently developed ab initio theory that now includes quantum-electrodynamical effects up to order $α^5$ and has reduced the theoretical uncertainty by three orders of magnitude [O. Kullie \textit{et al.}, Phys. Rev. A 112 052813 (2025)]. In addition, we extract the scalar spin-spin interaction coefficients $E_4$~=~925\,395.758(41)$\,$kHz (electron-proton) and $E_5$~=~142\,287.821(22)$\,$kHz (electron-deuteron), which show a moderate tension with another state-of-the-art theoretical prediction [M. Haidar \textit{et al.}, Phys. Rev. A 106 042815 (2022)].
The search for physics beyond the Standard Model requires isolating potentially minute deviations from the overwhelmingly dominant effects of known physics. This is certainly the case for precision experiments at low energy that compare measured observables to predicted values from fundamental theory, in particular quantum electrodynamics (QED), and independently determined fundamental constants. By choosing systems that are sensitive to particular interactions, such experiments become a search platform that is competitive with, and complementary to, accelerator-based high-energy experiments [1]. Alternatively, under the assumption that the Standard Model is correct, precise measurements and theory can be combined to determine fundamental constants to high precision [2][3][4][5].
Similar to one of the simplest and most studied atomic systems, hydrogen, diatomic molecular hydrogen ions (MHI) contain only a single electron and are exceptionally well-described by quantum electrodynamics (QED) theory [6,7]. However, as molecular systems, MHI possess additional rotational and vibrational degrees of freedom that give rise to a rich energy spectrum comprising hundreds of rovibrational states in the electronic ground state. These energies depend on fundamental constants, such as the proton-to-electron mass ratio mp /me, the proton charge radius r p , the Rydberg constant R ∞ , and the fine-structure constant α, with well-understood scaling [7]. As a result, high-precision spectroscopy of MHI in conjunction with ab initio theory can be used to determine fundamental constants, test molecular QED, and probe beyond-Standard-Model (BSM) forces [5,[8][9][10][11].
Furthermore, measurements of the rovibrational transition frequencies of H + 2 and its antimatter counterpart H -2 are also very attractive for a future test of CPT invariance [12][13][14][15][16][17] that would provide a unique opportunity to compare the properties of the proton and antiproton along with proton-proton (p-p) and antiprotonantiproton (p-p) interactions. Compared to atomic (anti)hydrogen, such a test would be three orders of magnitude more sensitive to the proton/electron vs. antiproton/positron mass ratio difference mp /me -mp /mē. While production of H -2 remains a challenge [18][19][20], recent years have seen considerable development in precision spectroscopy of H + 2 and HD + which now enables comparisons of experiment and theory at the level of 1×10 -11 relative uncertainty [9][10][11]21].
In this work we aim to use the techniques of singleion precision Penning-trap experiments to perform highprecision measurements on HD + and extend the reach of MHI as a platform for fundamental physics studies. These techniques, namely electron spin resonance spec-troscopy (ESR) enabled by image current detection and the continuous Stern-Gerlach effect (CSGE) [22], have recently been applied to HD + and provide deterministic and non-destructive state preparation and detection [17]. This experimental platform presents an opportunity for a strong improvement in experimental resolution. For context, the only previous ESR experiment [23] on MHI was performed nearly 40 years ago on an ensemble of ions and determined bound-electron g factors to a relative precision of around 1 × 10 -6 . Today, experiments using single atomic ions routinely perform measurements of boundelectron g factors to sub-ppb or even lower uncertainty [24][25][26].
This perspective of greatly improved experimental accuracy triggered a new ab initio calculation of the boundelectron g factor that improved the only, more-than-40year-old previous calculation [27] by more than three orders of magnitude and allows us to now perform a highprecision comparison of theory and experiment. Such comparisons, and, more generally, accurate determinations of the hyperfine structure (HFS), are crucial as HFS affects all rovibrational transition measurements in MHI and are required to extract spin-averaged frequencies from measured transition frequencies. Currently, the extraction of fundamental constants from MHI spectroscopy is limited by the theoretical uncertainty of the spin-averaged frequencies. However, once these theoretical calculations improve, precise knowledge of the HFS will be required for improved determinations of these fundamental constants. Moreover, while theory [28] and experiment [11,29,30] on the HFS of H + 2 currently agree, discrepancies have been observed in HD + [31] that call for a resolution.
Here, we present the results of a new determination of the ground-state HFS of HD + using ESR in a precision Penning-trap experiment and measure the boundelectron g factor with orders of magnitude improved precision. To perform these measurements, we directly drive magnetic dipole transitions between the electron spin states of a single molecular ion in the Alphatrap cryogenic Penning-trap apparatus [32]. In the strong 4-T magnetic field used to trap the ion, these transition frequencie
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