## Abstract

Molecular hydrogen ions are of high interest in fundamental physics and metrology. In particular, the long-range baryon–baryon interaction and the degrees of freedom of baryon rotational and vibrational motion are key features of molecular hydrogen ions. Here we present the measurement of a one-photon transition of HD^{+}, a fourth overtone vibrational transition. Through a comparison of experimental and ab initio frequencies of this and previously measured transitions, we implement a test of the low-energy quantum physics of baryon interaction and motion. The results may also be interpreted as a test of Weinberg’s quantum mechanics extension. Furthermore, we compare the value of the fundamental constant combination *μ*/*m*_{e} = *m*_{p}*m*_{d}/(*m*_{p} + *m*_{d})*m*_{e} determined from our measurement with the value obtained from mass spectrometry experiments. This may also be regarded as a test of the quantum behaviour of baryons, revealing a moderate tension of 1.7 times the combined uncertainty. Combining our measurement result with some previous ones on HD^{+}, we obtain a least-squares-adjusted value for *μ*/*m*_{e} as well as a bound for the force between the proton and deuteron in a scenario that is beyond the Standard Model of particle physics. No evidence of a deviation from conventional quantum physics is found.

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## Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Code availability

No custom code or software was used for analysis and presentation of the data associated with this paper.

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## Acknowledgements

We are indebted to C.-J. Kwong and V. A. Vogt for contributing to the reference laser characterization, to E. Wiens for assistance with the frequency comb measurements and to M. G. Hansen for the maser characterization. The studies of additional rovibrational transitions were performed with crucial contributions by M. R. Schenkel. We thank V. I. Korobov for providing the BSM force sensitivities and the spin-averaged frequency value, as well as for helpful comments on the paper. We thank him and J.-Ph. Karr for the communication of results on the hyperfine structure before publication. O. Kullie provided some theoretical data. We are indebted to J. Leute and P. Uhrich for their gracious help with the maser stability characterization. We thank G. Morigi for a helpful discussion. We appreciate the important suggestions of the reviewers. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 786306, ‘PREMOL’ (S.S.)) and from both Deutsche Forschungsgemeinschaft (DFG) and the state of Nordrhein-Westfalen via grants INST-208/737-1 FUGG (S.S.), INST-208/774-1 FUGG (S.S.) and INST-208/796-1 FUGG (S.S.).

## Author information

### Authors and Affiliations

### Contributions

S.A. and I.V.K. contributed equally to this work. S.A., I.V.K. and G.S.G. performed the experiments and analysed the data. I.V.K. developed and characterized the laser system. S.A. maintained the apparatus. S.S. performed the analyses and wrote the paper. All authors contributed to review of the data and paper contents.

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*Nature Physics* thanks Piotr Wcislo and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Spectral characteristics of the 1.15 *μ*m spectroscopy laser.

**Left**: (red) spectrum analyzer trace of the heterodyne beat of the reference laser 1 (1.06 *μ* m) and an independent sub-Hz-linewidth, low-drift frequency-stable laser of nearly same optical frequency (reference laser 2, 1.06 *μ* m); (inset, blue) the heterodyne beat between the spectroscopy laser and the spectrally closest mode of the frequency comb, while the spectroscopy laser is phase-locked to the comb and the comb is phase-locked to the reference laser 1. The solid lines are the lorentzian fits to the data points. The frequency offsets are arbitrary. The linewidths of the two beats and additional characterizations indicate that the spectroscopy laser has a linewidth of less than 10 Hz. Resolution bandwidth of the spectrum analyzer: 1 Hz, red trace - single sweep, sweep time 6 s, blue trace - average of 10 sweeps, time span of 1 min. The error bars are smaller than data markers. **Right**: (blue) Allan deviation of the frequency of the spectroscopy laser, measured relative to a hydrogen maser (maser 1) via the frequency comb; (green) Allan deviation of maser 1 relative to a second independent maser (maser 2). The shown values are the Allan deviation of the beat frequency divided by \(\sqrt{2}\); (red) Allan deviation of the beat of the reference laser 1 with reference laser 2. The error bars are estimated as 68% confidence intervals.

### Extended Data Fig. 2 Sensitivity of transitions and their ratios to BSM forces.

The sensitivity is defined as *X*^{−1}(*d**X*/*d**β*), where \(X={f}_{{v}^{{\prime} }}\) or \({{{{\mathcal{R}}}}}_{a,b}\), and is computed as described in Methods, Sec. 1.5. *β* has units of energy. **(a)**: case of a BSM force between proton and deuteron, proportional to *β*_{pd}. **(b)**: case of a BSM force between electron and proton, proportional to *β*_{ep}. Full lines denote positive values, dashed lines negative values. The brown, cyan, and gray lines in panel (b) are very close.

### Extended Data Fig. 3 Comparison of results on the fundamental mass ratio *μ*/*m*_{e}.

*μ* = *m*_{p}*m*_{d}/(*m*_{p} + *m*_{d}) is the reduced proton-deuteron mass. “Penning traps” (dark cyan): computed using the CODATA 2018 value for *m*_{e}/u (that relies mostly on ref. ^{21}), *m*_{d}/u^{23}, *m*_{d}/*m*_{p}^{24}. CODATA 2018 (black): *μ*/*m*_{e} computed using only the CODATA values for *m*_{e}/u, *m*_{p}/u, *m*_{d}/u. The vertical line corresponds to zero of *x* axis. Note that the value and uncertainty of the “CODATA 2018” entry is significantly correlated with the “Penning traps” entry. LSA-1 is based on *f*_{1} (cyan) and *f*_{5} (brown); LSA-1b is based on *f*_{0} (green), *f*_{1} (cyan) and *f*_{5} (brown). LSA-1b and LSA-3 (ED Table 5) are nearly identical. As can be seen, the results of the least-squares adjustments deviate from the respective weighted means of the included frequencies. This is due to the assumption of near-perfect correlation *r*_{QED} of the theoretical uncertainties. The error bars represent standard uncertainty.

### Extended Data Fig. 4 Hyperfine structure energy levels of HD^{+} in the two relevant rovibrational levels.

Shown are the values \({E}_{{{{\rm{spin}}}},p}^{({{{\rm{theor}}}})}/h\) for the ground vibrational level (*v* = 0, *N* = 0) (left) and \({E}_{{{{\rm{spin}}}},q}^{({{{\rm{theor}}}})}/h\) for the excited vibrational level \(({v}^{{\prime} }=5,{N}^{{\prime} }=1)\) (right) of the \({}^{2}{\Sigma }_{g}^{+}\) ground electronic state. The vertical axis is broken for better visualization. The numbers next to the arrows indicate the line numbers *i*(*p*, *q*), where *p* (*q*) denotes the particular lower (upper) spin state. In this work, line 12 and line 21 were measured. *G*_{1} refers to the sum of electron and proton spin, *G*_{2} to the sum of *G*_{1} and deuteron spin, and *F* to the sum of *G*_{2} and rotational angular momentum *N*. The ground rovibrational level has zero rotational angular momentum, *N* = 0, giving rise to 4 spin states with total angular momentum *F* = 0, 1, 2. The upper rovibrational level \(({v}^{{\prime} }=5,\,{N}^{{\prime} }=1)\) has 10 spin states, having values \({F}^{{\prime} }=0,\,1,\,2,\,3\). For both studied lines 12 and 21, the lower and upper hyperfine states have the same particle spin coupling \({G}_{1}={G}_{1}^{{\prime} }\) and \({G}_{2}={G}_{2}^{{\prime} }\). The line 12 measured here is analogous to line 12 in ref. ^{3} and to line 12 in ref. ^{4}. The line 21 is analogous to line 16 in ref. ^{3}. Spin states with total angular momentum *F* are (2*F* + 1)-fold degenerate in zero magnetic field. We denote the lines and corresponding transition frequencies in zero external fields by *f*_{line i}. *i* indicates the position in the list {*f*_{spin,i}} of the spin frequency values of all spin components in ascending order.

## Supplementary information

### Supplementary Information

Supplementary Sections A and B, Fig. 1 and Table 1.

## Source data

### Source Data Fig. 2

Coordinates of each data point and their uncertainties.

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Alighanbari, S., Kortunov, I.V., Giri, G.S. *et al.* Test of charged baryon interaction with high-resolution vibrational spectroscopy of molecular hydrogen ions.
*Nat. Phys.* **19**, 1263–1269 (2023). https://doi.org/10.1038/s41567-023-02088-2

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DOI: https://doi.org/10.1038/s41567-023-02088-2