Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Determination of a set of fundamental constants from molecular hydrogen ion spectroscopy: a modeling study

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper models how ultra-accurate spectroscopy of molecular hydrogen ions could determine key fundamental constants, including the triton charge radius, with up to 250-fold smaller uncertainty than today.

desk verdict A transparent, well-posed projection study: the headline 100-fold gains in mass ratios and triton radius are real only if two future theory milestones are met, and the paper says so clearly. read the letter →

arxiv 2505.05615 v1 pith:3LWRDOSS submitted 2025-05-08 physics.atom-ph

J.-Ph Karr , S. Schiller , V. I. Korobov , S. Alighanbari This is my paper · ORCID
classification physics.atom-ph
keywords molecularhydrogenionsfundamentalconstantsleast-squaresadjustmentmassratioschargeradiitritonprecisionspectroscopyQEDtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a carefully chosen set of vibration-rotation transition frequencies in molecular hydrogen ions, measured with 1-Hz accuracy, could determine the proton, deuteron, and triton-to-electron mass ratios and the triton charge radius with one to two hundred times smaller uncertainty than today's recommended values, without requiring any improvement in the QED theory uncertainty. It simulates a least-squares adjustment in which strongly correlated theoretical uncertainties cancel when many transitions are combined, so the data themselves pin down the unknown QED contributions. The practical payoff would be an independent, purely electronic route to the Rydberg constant and the proton and deuteron charge radii at nearly today's accuracy, plus a triton radius precise enough to test nuclear-structure predictions.

What carries the argument

The load-bearing object is the linearized least-squares adjustment, with sensitivity matrix $A_{ij}=\partial f_i/\partial z_j$ and a covariance matrix built from experimental uncertainties plus correlated QED uncertainties and uncorrelated numerical uncertainties. The numerical part is modeled by Eq. (8), $u(\beta_{v,N})=u_0(v+1)^{3/2}(N+1)^{1/2}$, where the Bethe logarithm is the hard-to-evaluate logarithmic QED correction, and the projected value is $u_0=1.25\times10^{-11}$. What makes the scheme work is that rows of $A$ point in different directions in constant space — in particular, high-lying vibrational transitions have mass sensitivities of the opposite sign to low-lying ones — so the adjustment can resolve combinations that a single transition or a few transitions cannot.

What would settle it

Rerun the least-squares adjustment with today's Bethe-logarithm uncertainties, $u_0=10^{-9}$, for exactly the transition sets in the paper's Tables IIc and III; the paper's own supplemental tables show the mass-ratio gains then fall to 6–40-fold, so any claim of a sustained 100-fold improvement would be refuted unless a path to $u_0\approx1.25\times10^{-11}$ is demonstrated.

Watch

Extended reading notes

Core claim

The central claim is that the correlation structure of the theoretical uncertainties is the resource: because the uncalculated QED terms shift all transition frequencies of a given type in nearly the same way, measuring a deliberately diversified set of transitions separates those shifts from the constants of interest. With 1-Hz measurements on five to nine optically accessible transitions in HD+, H2+, and D2+, the mass ratios $m_p/m_e$ and $m_d/m_e$ can be improved by factors of order 100–240; adding three transitions in two tritium-bearing isotopologues yields $m_t/m_e$ and the triton charge radius $r_t$ at roughly 150- and 200-fold improved uncertainty. All of this is projected at the current QED uncertainty of $8\times 10^{-12}$, with the main residual limitation shifted to the numerical uncertainty of the Bethe logarithm, assumed to be reducible by about two orders of magnitude.

Load-bearing premise

The projections rest on the assumption that the numerical uncertainty of the hardest QED correction (the Bethe logarithm) can be reduced roughly a hundredfold from today's value, and without that improvement the claimed gains shrink from more than a hundredfold to between sixfold and fortyfold.

Editorial extensions

If this is right

  • With five to nine 1-Hz measurements in HD+, H2+, and D2+, the proton and deuteron-to-electron mass ratios can be determined with uncertainties 100 to 240 times smaller than today's recommended values.
  • Adding just three transitions in two tritium-bearing molecules gives $m_t/m_e$ and $r_t$ at roughly 150- and 200-fold improved uncertainty, reaching the 0.9-attometer target for testing chiral effective field theory.
  • The Rydberg constant would be improved by about a factor of 1.8, while the proton and deuteron charge radii would be determined at nearly today's accuracy using only electronic H/D and MHI data, making possible a lepton-universality comparison.
  • A tritium-bearing MHI measurement combined with H2+ and HD+ data would yield $(m_p+m_d)/m_t$ with fractional uncertainty near $10^{-13}$, a stringent cross-check of independent mass-spectrometry results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the hundredfold numerical improvement is realized, the same correlated-uncertainty strategy could be applied to other molecular species whose QED uncertainties are dominated by state-independent terms, potentially sharpening constants beyond the systems studied here.
  • Because the adjustment is sensitive to the assumed correlation between theoretical uncertainties, calculating the two next-order relativistic and recoil QED corrections in a full three-body treatment, rather than improving the dominant higher-order terms, is the decisive theoretical investment for making the projections robust.
  • A successful triton-mass extraction from T-MHI spectroscopy would give an independent anchor for the endpoint of the tritium beta-decay spectrum, complementing the charge-radius test that is the paper's stated nuclear-physics motivation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper reports a modeling study of a future least-squares adjustment (LSA) of fundamental constants from ultra-high-precision spectroscopy of molecular hydrogen ions (MHI). Using the linearized Mohr-Taylor covariance framework, the authors compute sensitivity coefficients for a set of rovibrational transitions across five isotopologues, construct covariance matrices that include both estimated uncalculated-QED uncertainties and numerical Bethe-logarithm uncertainties, and optimize the choice of transitions under several experimental scenarios. The central result is that, assuming 1 Hz measurement uncertainties and no improvement in the estimated uncalculated-QED terms, the mass ratios mp/me, md/me, mt/me and the triton charge radius rt can be determined with 100- to 250-fold smaller uncertainties than CODATA 2022 values, while R∞, rp, and rd are recovered at roughly today's accuracy using only electronic H/D and MHI data. The paper also discusses implications for the triton radius test of chiral effective field theory, the triton mass and neutrino physics, QED tests, and beyond-standard-model searches.

Significance. If the projected improvements are realized, MHI spectroscopy would provide an independent, high-accuracy route to several fundamental constants, complementing Penning-trap mass spectrometry and muonic-atom measurements, and enabling a stringent test of chiral effective field theory via the triton charge radius. The paper's strengths are its use of the standard Mohr-Taylor LSA formalism, its fully tabulated inputs, and its explicit sensitivity analyses: the numerical-uncertainty scaling is stated, the perfect-correlation hypothesis is tested in Supplemental Material Sec. H, and the consequences of larger Bethe-logarithm uncertainties are quantified in SM Tables XII and XIV. The result is best read as a conditional projection rather than a current capability, because two of its key assumptions - an approximately 80-fold improvement in Bethe-logarithm numerical accuracy and near-perfect correlation of uncalculated QED uncertainties - are future theoretical milestones that are asserted rather than demonstrated. The paper is otherwise internally consistent and clearly discloses the dependence of the headline claims on these assumptions.

major comments (3)
  1. [End Matter, Eq. (8) and SM Tables XII/XIV] The projected more-than-100-fold gains use u0_proj = 1.25e-11, about 80 times smaller than today's utoday_0 = 1e-9, with the only in-text justification being that Bethe-logarithm calculations 'can be improved in the future with dedicated efforts' and are 'more easily tractable' than QED terms. SM Table XIV shows that with utoday_0 the best mp/me uncertainty is 2.9e-12 (a factor of about 6 relative to CODATA) and rt is 2.6 am (a factor of about 33), so the headline improvement depends critically on this single numerical milestone. Please either provide a concrete computational roadmap or error-cost estimate for the required Bethe-logarithm improvement, or reformulate the abstract and conclusion to state that the >100-fold gains are conditional on approximately an 80-fold reduction in numerical Bethe-logarithm uncertainty.
  2. [End Matter, Eq. (6) and SM Sec. H, Table XVII] The model assumes r_k,ij = 1 for the correlations of uncalculated QED uncertainties, and the End Matter acknowledges that this assumption is 'not exact.' SM Table XVII shows that lowering r to 0.99 for the two me-alpha^6-type terms raises u(mp/me) from 0.071e-12 to 0.48e-12 and u(mt/me) from 0.15e-12 to 0.57e-12, which correspond to improvements of about 35-fold and 67-fold relative to CODATA, respectively - below the 'more than one-hundred-fold' claim. Because the central conclusion depends on near-perfect correlation, the paper should either quantify how close to 1 the correlation coefficients must be for the headline claim to hold, or soften the abstract to reflect the demonstrated sensitivity.
  3. [Introduction, p. 3, and End Matter, Eq. (5)] The statement that the prospective improvement requires 'no reduction of u(delta f_theor_i)' refers only to uncalculated QED terms, but the total theoretical uncertainty in Eq. (5) includes numerical Bethe-logarithm uncertainties, which the projections reduce by a factor of about 80. This distinction should be made explicit wherever the 'no QED theory improvement' claim is restated, especially in the abstract, to avoid the misleading impression that no theoretical progress at all is needed for the projected gains.
minor comments (3)
  1. [End Matter, Eq. (8)] The scaling law u(beta_v,N) = u0 (v+1)^{3/2}(N+1)^{1/2} is stated to be 'reasonably described' by results in Ref. [51], but no fit plot or extracted u0 values are given; providing the actual fit and residuals would strengthen confidence in this input.
  2. [Table I and transition numbering] The transition numbering in Table I skips from 6 to 8 and 9, then jumps to 13, 14, 18, 23, 24, while the text says transitions 1-12 are the same as in a previous paper; a reader cannot reconstruct the full set from Table I alone, so a complete mapping or a reference to SM Table IV at the point of first use would help.
  3. [Table II caption] The caption's phrase 'the rightmost three entries in the last line' is ambiguous because the last line contains five numerical entries; it should refer to the three constants R∞, rp, and rd explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are conditional LSA projections whose inputs are stated, prior, and not fitted to the projected constants.

full rationale

The paper is a forward modeling study: it computes the covariance of a linearized least-squares adjustment from assumed experimental and theoretical uncertainties and from sensitivity coefficients obtained from nonrelativistic wavefunctions and operator expectation values. The target constants (mass ratios, R∞, charge radii) are outputs of the LSA, not inputs. The paper explicitly states: "we do not include any CODATA 2022 information on the FCs of interest." CODATA values enter only as reference points for the linearization and for normalizing the reported ratios, which is not circular. The theoretical uncertainty model is inherited from prior published QED work (ref. [2]) and Bethe-logarithm calculations (ref. [51]); these are independent prior results and are not fitted to the projected constants. The assumed numerical Bethe-logarithm improvement (u0_proj = 1.25e-11) is a clearly labeled scenario parameter: "we assume a significantly lower value of u0 with respect to what can be estimated by matching with the results of [51], reflecting the fact that Bethe logarithm calculations can be improved in the future with dedicated efforts." The dependence of the headline gains on this assumption is disclosed in SM Tables XII and XIV, and the perfect-correlation assumption is explicitly called "not exact" and tested in SM Table XVII. Thus the abstract's "more than one-hundred-fold" claim is a conditional projection, not a result forced by definition or by self-citation. The use of author-owned prior work is real, independent support rather than load-bearing circularity. No step in the derivation reduces to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The projection depends on a handful of chosen parameters: the assumed 1 Hz experimental accuracy, the 100x-improved Bethe-log uncertainty scale, and the perfect-correlation model for QED errors. All three are stated and sensitivity-analyzed; they are not fitted to the target constants, but the headline factors are directly controlled by them.

free parameters (3)
  • u0_proj (Bethe-log numerical uncertainty scale) = 1.25e-11
    Eq. (8) sets the numerical uncertainty of each rovibrational level; the projected value is ~100x smaller than today's utoday_0=1e-9. This parameter determines whether the headline >100x constant improvements are reached (SM Tables XII/XIV show factors 6-40 at utoday_0).
  • u_proj(f_exp) experimental uncertainty = 1 Hz
    All transitions are assumed measurable to 1 Hz (Introduction, Experimental uncertainties). Chosen by hand as a realistic target; if experiments only reach 10 Hz the projected gains would shrink accordingly.
  • QED correlation coefficient r_k,ij = 1.0 (baseline); 0.99 and 0.95 in SM H
    Eq. (6) assumes perfect correlation of uncalculated QED uncertainties; SM Table XVII shows mass-ratio gains degrade by factors up to 14 when r=0.95.
assumptions (5)
  • standard math Linearized least-squares adjustment with normal errors (Eq. 2-3 of ref. [34]) is adequate for the projected uncertainties.
    Standard CODATA methodology; the linearization is valid because sensitivities do not change significantly over the final uncertainties.
  • domain assumption Uncalculated QED terms can be estimated by a delta-function effective potential with 100% uncertainty of the estimated contribution, following ref. [2].
    Adopted from the authors' previous QED error budget [2]; the perfect-correlation extension of this model is the enabler of the cancellation.
  • ad hoc to paper Bethe-logarithm numerical uncertainties follow the scaling u0 (v+1)^{3/2}(N+1)^{1/2} and are uncorrelated across states (Eqs. 7-8).
    The scaling law is matched to the uncertainties of ref. [51] and then extrapolated 100x downward; this is the load-bearing numerical-error model.
  • domain assumption The experimental data are uncorrelated and all at the same 1 Hz uncertainty (Eq. 4).
    A scenario assumption; correlations between measurements (e.g., common laser frequency reference) are neglected.
  • domain assumption Sensitivity coefficients A_ij from the nonrelativistic operator expectation values are sufficiently accurate for the LSA.
    Sensitivities are obtained in nonrelativistic approximation (End Matter), following ref. [35]; relativistic corrections to sensitivities are neglected.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Determination of a set of fundamental constants from molecular hydrogen ion spectroscopy: a modeling study." pith.science (2026). https://pith.science/paper/3LWRDOSS

@misc{pith2026250505615,
  author       = {Pith},
  title        = {Pith review of: Determination of a set of fundamental constants from molecular hydrogen ion spectroscopy: a modeling study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LWRDOSS}},
  note         = {Machine review of arXiv:2505.05615}
}
read the original abstract

The rovibrational transition frequencies of molecular hydrogen ions (MHI) can be accurately computed using ab initio nonrelativistic quantum electrodynamics. A subset of the fundamental constants are required input. We analyze how, once upcoming ultra-high-accuracy spectroscopy data has been obtained, that subset of constants can be determined with greater accuracy. Our analysis shows that under realistic assumptions the uncertainties of the mass ratios of proton, deuteron and triton relative to the electron, and of the triton charge radius can be reduced more than onehundred-fold compared to today (CODATA 2022). Furthermore, the Rydberg constant, as well as the proton and deuteron charge radii can be determined with uncertainties similar to those of today, but solely using data from electronic systems. The implications are discussed.

Figures

Figures reproduced from arXiv: 2505.05615 by the authors.

Figure 1
Figure 1. FIG. 1. Upper plot: uncertainties of the [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Upper plot: the figure of merit FOM1, defined [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. LSA with T-MHI transitions [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. LSAs with T-MHI transitions [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Variational studies of ro-vibrational spectra of DT$^+$ and T$_2^+$ ions

    physics.atom-ph 2025-07 conditional novelty 6.0 of 10

    The last two hydrogen molecular ion isotopologues, DT+ and T2+, receive high-precision variational ro-vibrational energies, dipole transition amplitudes, relativistic corrections, and hyperfine coefficients.

Reference graph

Works this paper leans on

64 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    P. J. Mohr, D. B. Newell, B. N. Taylor, E. Tiesinga, Codata recommended values of the fundamental physical constants: 2022, Rev. Mod. Phys. 97 (2025) 025002. doi:10.1103/RevModPhys.97.025002. URL https://link.aps.org/doi/10.1103/ RevModPhys.97.025002

    doi:10.1103/revmodphys.97.025002 2025

  2. [2]

    V. I. Korobov, J.-P. Karr, Rovibrational spin-averaged transitions in the hydrogen molecular ions, Phys. Rev. A 104 (2021) 032806. doi:10.1103/PhysRevA.104.032806

    work page doi:10.1103/physreva.104.032806 2021

  3. [3]

    Alighanbari, G

    S. Alighanbari, G. S. Giri, F. L. Constantin, V. I. Ko- robov, S. Schiller, Precise test of quantum electrody- namics and determination of fundamental constants with HD+ ions, Nature 581 (2020) 152 – 158. doi:10.1038/ s41586-020-2261-5

    work page 2020

  4. [4]

    Patra, M

    S. Patra, M. Germann, J.-P. Karr, M. Haidar, L. Hilico, V. I. Korobov, F. M. J. Cozijn, K. S. E. Eikema, W. Ubachs, J. C. J. Koelemeij, Proton-electron mass ratio from laser spectroscopy of HD + at the part-per- trillion level, Science 369 (2020) 1238–1241. doi:10. 1126/science.aba0453

    work page 2020

  5. [5]

    I. V. Kortunov, S. Alighanbari, M. G. Hansen, G. S. Giri, V. I. Korobov, S. Schiller, Proton-electron mass ratio by high-resolution optical spectroscopy of ion ensembles in the resolved-carrier regime, Nat. Phys. 17 (2021) 569–

    work page 2021

  6. [6]

    B. Roth, J. C. J. Koelemeij, H. Daerr, S. Schiller, Rovi- brational spectroscopy of trapped molecular hydrogen ions at millikelvin temperatures, Phys. Rev. A 74 (2006) 040501. doi:10.1103/PhysRevA.74.040501

    work page doi:10.1103/physreva.74.040501 2006

  7. [7]

    Bressel, A

    U. Bressel, A. Borodin, J. Shen, M. Hansen, I. Ernsting, S. Schiller, Manipulation of individual hyperfine states in cold trapped molecular ions and application to HD + frequency metrology, Phys. Rev. Lett. 108 (2012) 183003. doi:10.1103/PhysRevLett.108.183003

    work page doi:10.1103/physrevlett.108.183003 2012

  8. [8]

    Alighanbari, I

    S. Alighanbari, I. V. Kortunov, G. S. Giri, S. Schiller, Test of charged baryon interaction with high-resolution vibrational spectroscopy of molecular hydrogen ions, Nature Physics 19 (2023) 1263–1269. doi:10.1038/ s41567-023-02088-2

    work page 2023

Show all 64 references
  1. [9]

    Alighanbari, M

    S. Alighanbari, M. R. Schenkel, V. I. Korobov, S. Schiller, to appear, Nature

  2. [10]

    Schiller, Precision spectroscopy of molecular hydrogen ions: an introduction, Contemporary Physics 63 (2022) 247–279

    S. Schiller, Precision spectroscopy of molecular hydrogen ions: an introduction, Contemporary Physics 63 (2022) 247–279. doi:10.1080/00107514.2023.2180180

    2022

  3. [11]

    K¨ ohler, S

    F. K¨ ohler, S. Sturm, A. Kracke, G. Werth, W. Quint, K. Blaum, The electron mass from g-factor measure- ments on hydrogen-like carbon 12C5+, At. Mol. Opt. Phys. 48 (2015) 144032. doi:10.1088/0953-4075/48/ 14/144032

    2015 doi

  4. [12]

    Heiße, S

    F. Heiße, S. Rau, F. K¨ ohler-Langes, W. Quint, G. Werth, S. Sturm, K. Blaum, High-precision mass spectrometer for light ions, Phys. Rev. A 100 (2019) 022518. doi: 10.1103/PhysRevA.100.022518

    2019 doi

  5. [13]

    A first study [20] by us unfortunately contained a numer- ical error, corrected in [21]

  6. [14]

    Holzapfel, F

    D. Holzapfel, F. Schmid, N. Schwegler, O. Stadler, M. Stadler, A. Ferk, J. P. Home, D. Kienzler, Quan- tum control of a single H + 2 molecular ion (2024). arXiv: 2409.06495. URL https://arxiv.org/abs/2409.06495

    2024 arXiv

  7. [15]

    Schiller, D

    S. Schiller, D. Bakalov, V. I. Korobov, Simplest molecules as candidates for precise optical clocks, Phys. Rev. Lett. 113 (2014) 023004. doi:10.1103/PhysRevLett.113. 023004

    2014 doi

  8. [16]

    Karr, H + 2 and HD +: candidates for a molecular clock, J

    J.-P. Karr, H + 2 and HD +: candidates for a molecular clock, J. Mol. Spectrosc. 300 (2014) 37 – 43. doi:10. 1016/j.jms.2014.03.016

    2014

  9. [17]

    Kullie, S

    O. Kullie, S. Schiller, Solution of the two-center Dirac equation with 20-digit precision using the finite-element technique, Phys. Rev. A 105 (2022) 052801. doi:10. 1103/PhysRevA.105.052801

    2022

  10. [18]

    H. D. Nogueira, J.-P. Karr, High-precision solution of the Dirac equation for the hydrogen molecular ion using a basis-set expansion, Phys. Rev. A 107 (2023) 042817. doi:10.1103/PhysRevA.107.042817

    2023 doi

  11. [19]

    Given the higher theory accuracy for H/D/T (and the availability of the muonic variantsµH,µD), one may ask whether the present idea could in principle be applied to these systems as well. The answer is affirmative, but in practice it would be very hard to obtain usefully high ...

  12. [20]

    Schiller, J.-P

    S. Schiller, J.-P. Karr, Prospects for the determina- 7 tion of fundamental constants with beyond-state-of- the-art uncertainty using molecular hydrogen ion spec- troscopy, Phys. Rev. A 109 (2024) 042825. doi:10.1103/ PhysRevA.109.042825

    2024

  13. [21]

    Schiller, J.-P

    S. Schiller, J.-P. Karr, Prospects for the determination of fundamental constants with beyond-state-of-the-art un- certainty using molecular hydrogen ion spectroscopy: Er- ratum. Subm. to, Phys. Rev. A (2025)

    2025

  14. [22]

    J.-P. Karr, L. Hilico, J. C. J. Koelemeij, V. I. Korobov, Hydrogen molecular ions for improved determination of fundamental constants, Phys. Rev. A 94 (2016) 050501. doi:10.1103/PhysRevA.94.050501

    2016 doi

  15. [23]

    Carrington, I

    A. Carrington, I. R. McNab, C. A. Montgomerie, Obser- vation of the 2 pσu− 1sσg Electronic Spectrum of D + 2 , Phys. Rev. Lett. 61 (1988) 1573–1575. doi:10.1103/ PhysRevLett.61.1573

    1988

  16. [24]

    Carrington, I

    A. Carrington, I. R. McNab, C. A. Montgomerie, J. M. Brown, Microwave spectra of the HD + and D + 2 ions at their dissociation limits, Molecular Physics 66 (1989) 1279–1289. doi:10.1080/00268978900100881

    1989 doi

  17. [25]

    Carrington, I

    A. Carrington, I. R. McNab, C. A. Montgomerie, R. A. Kennedy, Electronic spectrum (2p σu - 1sσg) of the D + 2 ion, Molecular Physics 67 (1989) 711–738. doi:10.1080/ 00268978900101401

    1989

  18. [26]

    Carrington, C

    A. Carrington, C. A. Leach, A. J. Marr, R. E. Moss, C. H. Pyne, T. C. Steimle, Microwave spectra of the D + 2 and HD+ ions near their dissociation limits, J. Chem. Phys. 98 (1993) 5290–5301. doi:10.1063/1.464928

    1993 doi

  19. [27]

    H. A. Cruse, C. Jungen, F. Merkt, Hyperfine struc- ture of the ground state of para-D + 2 by high-resolution Rydberg-state spectroscopy and multichannel quantum defect theory, Phys. Rev. A 77 (2008) 042502. doi: 10.1103/PhysRevA.77.042502

    2008 doi

  20. [28]

    Beyer, F

    M. Beyer, F. Merkt, High-resolution photoelectron spec- troscopy and calculations of the highest bound levels of D + 2 below the first dissociation threshold, Journal of Physics B: Atomic, Molecular and Optical Physics 50 (2017) 154005. doi:10.1088/1361-6455/aa7ac9

    2017 doi

  21. [30]

    C. G. Parthey, A. Matveev, J. Alnis, R. Pohl, T. Udem, U. D. Jentschura, N. Kolachevsky, T. W. H¨ ansch, Preci- sion Measurement of the Hydrogen-Deuterium 1 S− 2S Isotope Shift, Phys. Rev. Lett. 104 (2010) 233001. doi: 10.1103/PhysRevLett.104.233001

    2010 doi

  22. [31]

    C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Pre- dehl, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, T. W. H¨ ansch, Im- proved Measurement of the Hydrogen 1 S− 2S Tran- sition Frequen...

    2011 doi

  23. [32]

    Matveev, C

    A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Ko- lachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Lau- rent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, T. W. H¨ ansch, Precision mea- surement of the h...

    2013 doi

  24. [33]

    The correlations existing between the FC values may be disregarded for the following consideration

  25. [34]

    P. J. Mohr, B. N. Taylor, CODATA recommended values of the fundamental physical constants: 1998, Rev. Mod. Phys. 72 (2000) 351–495. doi:10.1103/RevModPhys.72. 351

    2000 doi

  26. [35]

    J.-P. Karr, J. C. J. Koelemeij, Extraction of spin- averaged rovibrational transition frequencies in HD + for the determination of fundamental constants, Molecular Physics 121 (2023) e2216081. doi:10.1080/00268976. 2023.2216081

    2023

  27. [36]

    Here, we make the assumption that the solutions are not too far from ini- tial reference values, such that the first iteration already gives a very good approximation

    In the CODATA procedure, the LSA has to be iterated until convergence, due to the fact that the sensitivity co- efficients themselves depend on FC values. Here, we make the assumption that the solutions are not too far from ini- tial reference values, such that the first itera...

  28. [37]

    spin- averaged

    The considered transitions have a hyperfine structure. In experimental works on HD + so far [3–5, 8], a few hy- perfine components were measured, from which a “spin- averaged” transition frequency was extracted using the theoretical hyperfine structure. In order to reach the h...

  29. [38]

    Amroun, V

    A. Amroun, V. Breton, J.-M. Cavedon, B. Frois, D. Goutte, F. Juster, P. Leconte, J. Martino, Y. Mizuno, X.-H. Phan, S. Platchkov, I. Sick, S. Williamson, 3H and 3He electromagnetic form factors, Nuclear Physics A 579 (1994) 596–626. doi:10.1016/0375-9474(94)90925-3

    1994 doi

  30. [39]

    V. I. Korobov, L. Hilico, J.-P. Karr, Fundamental Tran- sitions and Ionization Energies of the Hydrogen Molecu- lar Ions with Few ppt Uncertainty, Phys. Rev. Lett. 118 (2017) 233001. doi:10.1103/PhysRevLett.118.233001

    2017 doi

  31. [40]

    Medina Restrepo, E

    M. Medina Restrepo, E. G. Myers, Mass Difference of Tri- tium and Helium-3, Phys. Rev. Lett. 131 (2023) 243002. doi:10.1103/PhysRevLett.131.243002

    2023 doi

  32. [41]

    Delaunay, J.-P

    C. Delaunay, J.-P. Karr, T. Kitahara, J. C. J. Koele- meij, Y. Soreq, J. Zupan, Self-Consistent Extraction of Spectroscopic Bounds on Light New Physics, Phys. Rev. Lett. 130 (2023) 121801. doi:10.1103/PhysRevLett. 130.121801

    2023 doi

  33. [42]

    Schiller, V

    S. Schiller, V. I. Korobov, Canceling spin-dependent con- tributions and systematic shifts in precision spectroscopy of molecular hydrogen ions, Phys. Rev. A 98 (2018) 022511. doi:10.1103/PhysRevA.98.022511

    2018 doi

  34. [43]

    M. R. Schenkel, S. Alighanbari, S. Schiller, Laser spec- troscopy of a rovibrational transition in the molecular hydrogen ion H + 2 , Nature Physics 20 (2024) 383–388. doi:10.1038/s41567-023-02320-z

    2024 doi

  35. [46]

    J.-P. Karr, L. Hilico, High accuracy results for the energy levels of the molecular ions H+ 2 , D+ 2 and HD+, up to J = 2, Journal of Physics B: Atomic, Molecular and Optical Physics 39 (2006) 2095–2105. doi:10.1088/0953-4075/ 39/8/024

    2006 doi

  36. [47]

    V. I. Korobov, Ro-vibrational states of H + 2 . Variational calculations, Molecular Physics 116 (1) (2017) 93–98. doi:10.1080/00268976.2017.1367427

    2017

  37. [48]

    D. T. Aznabayev, A. K. Bekbaev, V. I. Korobov, Leading-order relativistic corrections to the rovibrational spectrum of H + 2 and HD + molecular ions, Phys. Rev. A 99 (2019) 012501. doi:10.1103/PhysRevA.99.012501

    2019 doi

  38. [50]

    Tian, L.-Y

    Q.-L. Tian, L.-Y. Tang, Z.-X. Zhong, Z.-C. Yan, T.-Y. Shi, Oscillator strengths between low-lying ro-vibrational states of hydrogen molecular ions, The Journal of Chemi- cal Physics 137 (2012) 024311. doi:10.1063/1.4733988

    2012 doi

  39. [51]

    V. I. Korobov, Z.-X. Zhong, Bethe logarithm for the H + 2 and HD+ molecular ions, Phys. Rev. A 86 (2012) 044501. doi:10.1103/PhysRevA.86.044501

    2012 doi

  40. [52]

    V. I. Korobov, L. Hilico, J.-P. Karr, Calculation of the relativistic Bethe logarithm in the two-center problem, Phys. Rev. A 87 (2013) 062506. doi:10.1103/PhysRevA. 87.062506

    2013 doi

  41. [53]

    Epelbaum, H.-W

    E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81 (2009) 1773–

    2009

  42. [54]

    Epelbaum, U.-G

    E. Epelbaum, U.-G. Meißner, Chiral Dynamics of Few- and Many-Nucleon Systems, Annual Review of Nuclear and Particle Science 62 (2012) 159–185. doi:10.1146/ annurev-nucl-102010-130056

    2012

  43. [55]

    Epelbaum, H

    E. Epelbaum, H. Krebs, P. Reinert, High-Precision Nu- clear Forces From Chiral EFT: State-of-the-Art, Chal- lenges, and Outlook, Frontiers in Physics 8 (2020) 98. doi:10.3389/fphy.2020.00098

    2020

  44. [56]

    Machleidt, D

    R. Machleidt, D. Entem, Chiral effective field theory and nuclear forces, Physics Reports 503 (2011) 1–75. doi: 10.1016/j.physrep.2011.02.001

    2011 doi

  45. [57]

    Filin, private comm

    A. Filin, private comm. (2023). URL https://indico.uis.no/event/2/ contributions/523/

    2023

  46. [58]

    A. A. Filin, D. M¨ oller, V. Baru, E. Epelbaum, H. Krebs, P. Reinert, High-accuracy calculation of the deuteron charge and quadrupole form factors in chiral effective field theory, Phys. Rev. C 103 (2021) 024313. doi: 10.1103/PhysRevC.103.024313

    2021 doi

  47. [59]

    Filin, private comm

    A. Filin, private comm. (2024)

    2024

  48. [60]

    Maisenbacher, V

    L. Maisenbacher, V. Wirthl, A. Grinin, A. Matveev, R. Pohl, T. W. H¨ ansch, T. Udem, Hydrogen spectroscopy as a test of the Standard Model to below 1 part per tril- lion, in: PSAS 2024: International Conference on Preci- sion Physics of Simple Atomic Systems, 2024. URL https:/...

    2024

  49. [61]

    Schuhmann, L

    The CREMA Collaboration, K. Schuhmann, L. M. P. Fernandes, F. Nez, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D. S. Covita, A. J. Dax, M. Diepold, B. Franke, S. Galtier, A. L. Gou- vea, J. G¨ otzfried, T. Graf, T. W. H¨ ansch, M. Hilde- brandt, P. Indelicato, ...

    2023 arXiv

  50. [62]

    Aker, etal., Direct neutrino- mass measurement based on 259 days of KATRIN data, Science 388 (2025) 180–185

    KATRIN Collaboration, M. Aker, etal., Direct neutrino- mass measurement based on 259 days of KATRIN data, Science 388 (2025) 180–185. doi:10.1126/science. adq9592

    2025 doi

  51. [63]

    A. K. Bekbaev, V. I. Korobov, M. Dineykhan, Hyperfine structure and relativistic corrections to ro-vibrational energies of HT + ions, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (2013) 175101. doi: 10.1088/0953-4075/46/17/175101

    2013 doi

  52. [64]

    Bethe logarithm

    S. Schmidt, M. Willig, J. Haack, R. Horn, A. Adam- czak, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, P. Carvalho, T.-L. Chen, L. M. P. Fernandes, T. Graf, M. Guerra, T. W. H¨ ansch, M. Hildebrandt, Y.-C. Huang, P. Indelicato, L. Julien, K. Kirch, A. Knecht, F. Kottmann, J....

    2018

  53. [67]

    The nuclear spin of the triton is the same as that of the proton, I = 1/2, and the magnetic moments are similar

    T-MHI spectroscopy The masses of T-MHI are similar to that of HD +, therefore sympathetic cooling by 9Be+ will be possible. The nuclear spin of the triton is the same as that of the proton, I = 1/2, and the magnetic moments are similar. Therefore, the spin structure of the lev...

  54. [68]

    A first-generation experiment shall be performed using a discharge in a cell contain- ing warm T 2 gas

    Atomic tritium spectroscopy The T-REX project [64] aims at developing two-photon 1s-2s spectroscopy in T and to extractrt in a similar man- ner as done for H and D. A first-generation experiment shall be performed using a discharge in a cell contain- ing warm T 2 gas. Later, w...

  55. [573]

    doi:10.1038/s41567-020-01150-7

    doi

  56. [1825]

    doi:10.1103/RevModPhys.81.1773

    doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.