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arxiv: 2604.15518 · v1 · submitted 2026-04-16 · ✦ hep-ph · physics.atom-ph

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

Graham M Shore
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:11 UTC · model grok-4.3

classification ✦ hep-ph physics.atom-ph
keywords Lorentz violationCPT violationStandard Model Extensionrovibrational spectrummolecular ionshydrogenantihydrogenSME couplings
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The pith

Rovibrational transitions in H2+ and anti-H2- test Lorentz and CPT violation by isolating SME couplings through quantum number dependence.

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

The paper derives the rovibrational spectrum of the hydrogen molecular ion H2+ and its antimatter counterpart anti-H2- within the Standard Model Extension, an effective theory that incorporates possible Lorentz and CPT violation. It performs a complete first-principles analysis of the molecular dynamics using the spherical tensor representation of the SME couplings and systematically extends the treatment to the non-minimal SME. The resulting energy levels show explicit dependence on rotational and vibrational quantum numbers, which allows separation of the contributions from different violation parameters and predicts sidereal and annual variations in transition frequencies due to Earth's motion. This framework indicates that spectroscopy of these ions could reach sensitivities around 10^{-17} and thereby open new channels for detecting symmetry breaking.

Core claim

The rovibrational spectrum of H2+ and anti-H2- is derived comprehensively from first principles in the spherical tensor representation of the SME couplings, with a systematic extension to the non-minimal SME, a full accounting of quantum-number dependence in both high and low background magnetic fields, and an extended treatment of sidereal and annual variations arising from rotations and Lorentz boosts.

What carries the argument

The spherical tensor representation of the SME couplings, applied to a first-principles analysis of the molecular dynamics of H2+ and anti-H2-.

If this is right

  • The spectrum becomes sensitive to an extended range of SME couplings beyond those accessible in the minimal SME.
  • Individual effects of different SME couplings can be isolated by examining the dependence of transition frequencies on rotational and vibrational quantum numbers.
  • Sidereal and annual variations in the transition frequencies arise from both Earth's rotation and Lorentz boosts, providing additional observable signals.
  • These features together enhance the opportunities to detect Lorentz and CPT symmetry breaking through rovibrational spectroscopy of the molecular ions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The quantum-number isolation method could be adapted to other precision molecular systems to cross-check results from different physical regimes.
  • Laboratory measurements in controlled magnetic fields could be combined with sidereal monitoring to separate time-dependent signals from constant SME coefficients.
  • If confirmed, the approach would supply independent constraints on non-minimal SME operators that are difficult to access in atomic or particle systems.

Load-bearing premise

The molecular dynamics of H2+ and anti-H2- can be analyzed completely from first principles in the spherical tensor representation of the SME couplings without missing higher-order effects or requiring significant approximations.

What would settle it

A high-precision measurement of rovibrational transition frequencies in H2+ or anti-H2- that fails to exhibit the predicted quantum-number-dependent shifts or sidereal variations matching the SME spherical-tensor formulas would falsify the claimed sensitivity and isolation of individual couplings.

read the original abstract

Rovibrational transitions in the hydrogen and antihydrogen molecular ions $H_2^+$ and $\overline{H}_2^-$ offer the possibility of testing Lorentz and CPT symmetry to extremely high precision, in principle attaining $O(10^{-17})$. In this paper, the third in a series, we give a comprehensive derivation of the rovibrational spectrum of $H_2^+$ and $\overline{H}_2^-$ in the SME, an effective quantum field theory incorporating Lorentz and CPT violation. New developments described here include a complete analysis of the molecular dynamics from first principles in terms of the spherical tensor representation of the SME couplings, the systematic extension of our previous results to the non-minimal SME, a full description of the quantum number dependence of the rovibrational energy levels in the spherical tensor formalism with both high and low background magnetic fields, and an extended discussion of sidereal and annual variations of transition frequencies arising from both rotations and Lorentz boosts. The resulting sensitivity of the rovibrational spectrum to an extended range of SME couplings, together with the ability to isolate their individual effects using the quantum number dependence of the transition frequencies, enhances the opportunities to detect Lorentz and CPT symmetry breaking through rovibrational spectroscopy of $H_2^+$ and $\overline{H}_2^-$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript derives the rovibrational spectrum of H₂⁺ and H̄₂⁻ within the Standard Model Extension (SME), extending prior work to the non-minimal sector. It performs a first-principles analysis of the molecular dynamics using the spherical-tensor decomposition of SME coefficients, obtains the explicit quantum-number dependence of the energy levels for both high- and low-field regimes, and incorporates sidereal and annual modulations arising from rotations and boosts. The central result is an enlarged set of accessible SME couplings whose individual contributions can be isolated via transition-frequency quantum-number scaling, potentially reaching O(10^{-17}) sensitivity.

Significance. If the derivations are complete, the work materially strengthens the case for using molecular-ion spectroscopy as a probe of Lorentz and CPT violation. The spherical-tensor formalism, the systematic inclusion of non-minimal operators, and the explicit mapping of quantum numbers to coefficient combinations constitute genuine technical advances over the earlier papers in the series. These features supply concrete, falsifiable predictions that experimental groups can use to design targeted measurements.

major comments (2)
  1. [non-minimal SME extension] Section on non-minimal SME extension (around the discussion following Eq. (the Hamiltonian in spherical-tensor form)): the claim of a 'systematic extension' without missing higher-order effects is load-bearing for the completeness assertion in the abstract. It is not shown explicitly that all dimension-6 operators that can contribute to the rovibrational Hamiltonian at the target precision have been retained; an omitted term linear in a particular tensor component could alter the quoted quantum-number dependence.
  2. [low magnetic field regime] The treatment of the low-background magnetic-field regime: the perturbative expansion used to obtain the energy shifts must be checked against the size of the SME coefficients themselves. If any SME term is comparable to the Zeeman splitting, the ordering assumed in the derivation breaks down and the isolation of individual couplings via quantum numbers is compromised.
minor comments (3)
  1. [formalism] Notation for the spherical-tensor components of the SME coefficients is introduced without a compact summary table; a single table collecting the mapping between Cartesian and spherical-tensor indices would improve readability.
  2. [sidereal and annual variations] The discussion of annual variations would benefit from an explicit statement of the reference frame (e.g., Sun-centered celestial equatorial frame) and the precise boost velocity used in the transformation of the coefficients.
  3. [figures] A few typographical inconsistencies appear in the labeling of quantum numbers (v, J, M) between the text and the figure captions; these should be harmonized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for raising these important points regarding the completeness of our analysis. We provide point-by-point responses below.

read point-by-point responses

  1. Referee: Section on non-minimal SME extension (around the discussion following Eq. (the Hamiltonian in spherical-tensor form)): the claim of a 'systematic extension' without missing higher-order effects is load-bearing for the completeness assertion in the abstract. It is not shown explicitly that all dimension-6 operators that can contribute to the rovibrational Hamiltonian at the target precision have been retained; an omitted term linear in a particular tensor component could alter the quoted quantum-number dependence.

    Authors: We agree that an explicit demonstration of completeness strengthens the claim. The spherical-tensor decomposition used in the paper is designed to include all possible contributions from the non-minimal SME coefficients at dimension 6. However, to address the referee's concern directly, we will revise the manuscript by adding a short discussion or appendix that classifies the relevant operators and shows that no omitted linear terms affect the energy level expressions at the stated precision. This revision will not change the results but will make the systematic nature more transparent. revision: yes

  2. Referee: The treatment of the low-background magnetic-field regime: the perturbative expansion used to obtain the energy shifts must be checked against the size of the SME coefficients themselves. If any SME term is comparable to the Zeeman splitting, the ordering assumed in the derivation breaks down and the isolation of individual couplings via quantum numbers is compromised.

    Authors: The perturbative expansion in the low-field regime is valid because the SME coefficients are constrained to be very small by existing experiments, ensuring that SME-induced shifts remain much smaller than the Zeeman splitting for any realistic laboratory magnetic field. For instance, typical Zeeman energies are on the scale of 10^{-6} eV or larger, while the target precision corresponds to relative effects at 10^{-17}, implying SME coefficients far below the threshold where the ordering would break down. We will add a clarifying statement in the revised manuscript to explicitly note this validity condition based on current bounds, thereby addressing the potential caveat without altering the derived expressions. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior series papers; central derivation independent from first principles

full rationale

The paper presents a comprehensive derivation of the rovibrational spectrum from first principles using the spherical tensor representation of SME couplings, with systematic extension to the non-minimal SME and analysis of quantum number dependence. It references prior results in the series (self-citation), but this is not load-bearing for the central claims: the molecular dynamics analysis, energy level expressions, and sensitivity to SME couplings are derived anew without reducing to fitted parameters, self-definitions, or unverified self-citations. No predictions are shown to be equivalent to inputs by construction, and the work is self-contained against external SME benchmarks. This yields a normal low score for minor series self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the SME as the correct effective theory for Lorentz/CPT violation and the validity of the spherical tensor decomposition for molecular dynamics. No new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The Standard Model Extension provides a complete effective field theory description of Lorentz and CPT violation at the relevant energy scales.
    All spectrum calculations are performed within the SME framework.
  • standard math Spherical tensor methods fully capture the angular momentum algebra and coupling structure in the molecular Hamiltonian.
    Invoked for the complete analysis of quantum number dependence.

pith-pipeline@v0.9.0 · 5535 in / 1333 out tokens · 58102 ms · 2026-05-10T10:11:52.421556+00:00 · methodology

discussion (0)

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Reference graph

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