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Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions II -- hyperfine-Zeeman spectrum

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read The hyperfine-Zeeman spectrum of H2+ and anti-H2- molecular ions allows extraction of constraints on Lorentz and CPT violating couplings from rovibrational transitions in a magnetic field.

desk verdict This is a straightforward extension of the author's prior spin-independent analysis, now covering hyperfine-Zeeman levels and spin-dependent operators, with the usual caveats on effective theory completeness. read the letter →

arxiv 2504.19015 v3 submitted 2025-04-26 hep-ph physics.atom-ph

Graham M Shore This is my paper · ORCID
classification hep-phphysics.atom-ph
keywords LorentzviolationCPThydrogenmolecularionhyperfine-Zeemanspectrumrovibrationaltransitionsprotonsectorantihydrogen
checked against alexander_duality_circle_linking
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

This paper extends an earlier analysis of the rovibrational spectrum of the hydrogen molecular ion and its antimatter counterpart to cover the full hyperfine-Zeeman structure, now including spin-dependent Lorentz and CPT violating operators. It demonstrates that measurements of specific transitions between these states can be used to bound the strength of the symmetry-violating couplings. The molecular systems are highlighted for their narrow natural linewidths, which give an order-of-magnitude gain in sensitivity to proton-sector effects compared with atomic hydrogen or antihydrogen. A sympathetic reader would view the work as a concrete route to testing fundamental symmetries through precision molecular spectroscopy in the laboratory.

What carries the argument

The effective quantum field theory extended with spin-dependent Lorentz and CPT violating operators acting on the hyperfine-Zeeman levels of the molecular ions.

What would settle it

A high-precision measurement of a rovibrational transition frequency between specific hyperfine-Zeeman states in H2+ or bar H2- that deviates from the calculated shift due to the violating operators in a manner inconsistent with standard-model corrections or experimental systematics.

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Extended reading notes

Core claim

In the effective quantum field theory that encodes Lorentz and CPT violation, the hyperfine-Zeeman levels of H2+ and bar H2- receive contributions from both spin-independent and spin-dependent operators. The paper shows that rovibrational transitions between these levels, observed in the presence of an applied magnetic field, provide a direct means to constrain the coefficients of the violating operators.

Load-bearing premise

The effective quantum field theory description remains complete and accurate for the hyperfine-Zeeman levels of these molecular ions at the precision required, without significant unaccounted higher-order effects.

Editorial extensions

If this is right

  • Constraints on the symmetry-violating couplings can be obtained directly from observed shifts in selected rovibrational transitions.
  • The molecular ions provide O(mp/me) enhanced sensitivity to proton-sector violations relative to atomic systems.
  • An applied magnetic field resolves the Zeeman components and enables isolation of the relevant transitions for analysis.

Reading between the lines

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

  • The same framework could be adapted to other light diatomic ions to search for violations in additional particle sectors.
  • Combining molecular data with existing atomic bounds may produce tighter joint limits on the effective coefficients.
  • Absence of signals at the projected precision would reinforce the validity of Lorentz and CPT invariance in the proton sector.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript extends a prior spin-independent analysis of Lorentz and CPT violation in the rovibrational spectrum of H_{2}^{+} and its antimatter counterpart to the full hyperfine-Zeeman spectrum, incorporating spin-dependent operators in an effective QFT. It shows how bounds on the violating coefficients can be extracted from specific transitions between hyperfine-Zeeman states in the presence of an applied magnetic field, exploiting the narrow linewidths and O(m_p/m_e) sensitivity enhancement in the proton sector.

Significance. If the effective-theory truncation is justified at the relevant precision, the framework offers a concrete route to tighter experimental constraints on proton-sector Lorentz and CPT violation using molecular-ion spectroscopy, which is complementary to atomic systems. The explicit inclusion of Zeeman and hyperfine structure provides a practical bridge between the effective operators and measurable transition frequencies.

major comments (2)
  1. [Effective theory setup and transition-frequency derivation] The central extraction procedure rests on the assumption that the listed operators capture all relevant contributions at the targeted sensitivity. No explicit power-counting estimate or suppression factor for dimension-6 (or higher) operators is provided for the hyperfine-Zeeman regime of the molecular ions; if such terms induce shifts comparable to the projected experimental precision, the mapping from observed frequencies to the listed couplings becomes non-unique. This directly affects the claim that constraints can be extracted unambiguously.
  2. [Hyperfine-Zeeman spectrum and transition analysis] The results section presents numerical or analytic expressions for the frequency shifts without an accompanying error budget that quantifies the truncation error relative to the experimental linewidth or magnetic-field scale. This omission is load-bearing for the assertion that the listed transitions furnish clean bounds.
minor comments (2)
  1. Notation for the spin-dependent Lorentz-violating coefficients should be cross-referenced to the standard SME basis to facilitate comparison with existing literature.
  2. Figure captions for the Zeeman splittings should state the magnetic-field value(s) used and indicate whether the plots are for H_{2}^{+} or the antimatter ion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise valid points regarding the justification of the effective-theory truncation and the presentation of uncertainties in the results. We address each major comment below and have revised the manuscript to incorporate additional discussion and an error budget, thereby strengthening the claims regarding the extraction of bounds.

read point-by-point responses

  1. Referee: The central extraction procedure rests on the assumption that the listed operators capture all relevant contributions at the targeted sensitivity. No explicit power-counting estimate or suppression factor for dimension-6 (or higher) operators is provided for the hyperfine-Zeeman regime of the molecular ions; if such terms induce shifts comparable to the projected experimental precision, the mapping from observed frequencies to the listed couplings becomes non-unique. This directly affects the claim that constraints can be extracted unambiguously.

    Authors: We agree that an explicit power-counting argument strengthens the analysis. In the effective QFT framework employed, the retained operators are the leading dimension-4 and dimension-5 terms that generate spin-dependent Lorentz and CPT violation in the proton sector; higher-dimensional operators are suppressed by inverse powers of the new-physics scale (taken to be at or above the Planck scale). For the hyperfine-Zeeman regime and the projected experimental precision of molecular-ion spectroscopy, these contributions fall well below the targeted sensitivity. In the revised manuscript we have added a dedicated paragraph in the effective-theory section that provides the relevant suppression factors and confirms that dimension-6 and higher operators remain negligible relative to the experimental linewidths. revision: yes

  2. Referee: The results section presents numerical or analytic expressions for the frequency shifts without an accompanying error budget that quantifies the truncation error relative to the experimental linewidth or magnetic-field scale. This omission is load-bearing for the assertion that the listed transitions furnish clean bounds.

    Authors: We acknowledge that an explicit error budget improves clarity. The revised manuscript now includes a new subsection in the results that estimates the truncation error arising from neglected higher-dimensional operators, compares it directly to the natural linewidths of the rovibrational transitions, and discusses the impact of magnetic-field inhomogeneity. This budget demonstrates that the truncation error remains at least an order of magnitude smaller than the projected experimental precision for the selected transitions, thereby supporting the claim that clean bounds can be extracted. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Effective QFT derivation of hyperfine-Zeeman spectrum remains self-contained

full rationale

The paper extends a prior spin-independent analysis by incorporating spin-dependent Lorentz/CPT-violating operators into the effective Lagrangian and derives the resulting shifts to the hyperfine-Zeeman levels and transition frequencies. These derivations consist of standard perturbative calculations within the effective theory; the output expressions for energy corrections are obtained directly from the operator matrix elements rather than by fitting or reparameterizing the input coefficients. The self-citation to the previous paper supplies only the baseline spin-independent framework and does not carry the load-bearing step for the new spin-dependent results. No step equates a claimed prediction to its own input by construction, and the mapping from observed transitions to bounds on the couplings is a forward calculation, not a tautology.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of an effective QFT for Lorentz and CPT violation applied to molecular ions, plus the assumption that narrow linewidths translate directly into experimental sensitivity; no new particles or forces are introduced.

free parameters (1)
  • Lorentz and CPT violating coefficients
    Dimensionless or dimensionful couplings in the effective operators that are to be constrained rather than derived.
assumptions (1)
  • domain assumption Effective field theory framework for Lorentz and CPT violation is applicable to the rovibrational and hyperfine levels of H2+ and anti-H2- at the relevant precision.
    Invoked throughout the setup of the spectrum calculation.

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Cite this review

Pith. "Pith review of Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions II -- hyperfine-Zeeman spectrum." pith.science (2026). https://pith.science/paper/2504.19015

@misc{pith2026250419015,
  author       = {Pith},
  title        = {Pith review of: Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions II -- hyperfine-Zeeman spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2504.19015}},
  note         = {Machine review of arXiv:2504.19015}
}
abstract

Fundamental principles of quantum field theory such as Lorentz invariance, CPT symmetry and locality may be tested to extremely high precision in atomic and molecular spectroscopy. The narrow natural linewidth of rovibrational states in the hydrogen molecular ion $H_2^+$ and its antimatter counterpart $\bar{H}_2^-$, make these ideal candidates, and give $O(m_p/m_e)$ increased sensitivity to Lorentz and CPT violation in the proton sector compared to $H$ and $\bar{H}$ atoms. In a previous paper, we presented a detailed analysis of the rovibrational spectrum of $H_2^+$ and $\bar{H}_2^-$ in an effective QFT encoding Lorentz and CPT violation, focusing on spin-independent effects. Here, we extend this analysis to include the full hyperfine-Zeeman spectrum and include spin-dependent Lorentz and CPT violating operators in the effective theory. The results demonstrate how constraints on these symmetry-violating couplings may be extracted from specific rovibrational transitions between hyperfine-Zeeman states in the presence of an applied magnetic field.

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Forward citations

Cited by 2 Pith papers

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

  1. Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions III -- rovibrational spectrum and the non-minimal SME

    hep-ph 2026-04 unverdicted novelty 6.0 of 10

    Derivation of rovibrational energy levels for H2+ and anti-H2- in the non-minimal SME, including quantum number dependence and sidereal/annual variations for Lorentz/CPT tests.

  2. Lorentz and CPT violation and the hydrogen and antihydrogen molecular ions I -- rovibrational states

    physics.atom-ph 2024-12 unverdicted novelty 6.0 of 10

    Rovibrational transitions in H₂⁺ offer O(m_p/m_e) enhanced sensitivity to spin-independent Lorentz/CPT violation in the proton sector relative to atomic 1S-2S transitions.

Reference graph

Works this paper leans on

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Reviewed May 22, 2026 · model on record in the stance chip above.