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arxiv: 2412.09730 · v4 · submitted 2024-12-12 · ⚛️ physics.atom-ph · hep-ph

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

Graham M. Shore This is my paper

Pith reviewed 2026-05-23 06:58 UTC · model grok-4.3

classification ⚛️ physics.atom-ph hep-ph
keywords Lorentz violationCPT violationhydrogen molecular ionrovibrational transitionsprecision testsantihydrogenfundamental symmetries
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0 comments X

The pith

Rovibrational transitions in hydrogen molecular ions enhance sensitivity to Lorentz and CPT violation by the proton-to-electron mass ratio in the hadron sector.

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

This paper examines the rovibrational spectrum of the hydrogen molecular ion H2+ and its antimatter version in a low-energy effective theory that adds Lorentz and CPT violation. It concentrates on spin-independent couplings whose current limits come from the 1S-2S transition in atomic hydrogen and antihydrogen. The central result is that rovibrational transitions gain an extra sensitivity factor of order the proton-to-electron mass ratio in the proton sector compared with atomic transitions. This factor could push experimental bounds to one part in 10^17. A sympathetic reader would care because the narrow natural linewidths of these molecular levels open a route to tighter tests of fundamental symmetries using existing and future precision spectroscopy.

Core claim

In the low-energy effective theory incorporating Lorentz and CPT violation, the rovibrational energy levels of the hydrogen molecular ion exhibit an enhanced sensitivity of O(mp/me) to the spin-independent violating couplings in the proton sector, relative to the sensitivity in atomic transitions such as the 1S-2S line.

What carries the argument

The low-energy effective theory with spin-independent Lorentz and CPT violating couplings applied to the rovibrational spectrum of the molecular ion.

If this is right

  • Rovibrational frequencies can improve existing bounds on the violating parameters beyond the precision of atomic 1S-2S data.
  • The mass-ratio enhancement applies specifically in the proton (hadron) sector.
  • The same framework covers both H2+ and anti-H2- , enabling direct CPT comparisons.
  • Narrow linewidths of rovibrational levels support the higher precision needed to exploit the enhancement.

Where Pith is reading between the lines

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

  • Experimental programs to produce and trap antihydrogen molecular ions would directly test the predicted CPT symmetry between matter and antimatter spectra.
  • The enhancement mechanism could be checked by comparing atomic and molecular transitions within the same experimental apparatus.
  • Similar mass-ratio factors may appear in other light molecular ions, suggesting a broader class of systems for Lorentz-violation searches.

Load-bearing premise

The low-energy effective theory with spin-independent Lorentz and CPT violating couplings applies directly to the rovibrational spectrum without additional molecular-structure corrections that would alter the mass-ratio enhancement factor.

What would settle it

A calculation or measurement showing that molecular-structure effects cancel or remove the O(mp/me) enhancement factor in the sensitivity to the violating couplings.

read the original abstract

The extremely narrow natural linewidths of rovibrational energy levels in the molecular hydrogen ion $\textrm{H}_2^{\,+}$, and the prospect of synthesising its antimatter counterpart $\overline{\textrm{H}}_2^{\,-}$, make it a promising candidate for high-precision tests of fundamental symmetries such as Lorentz and CPT invariance. In this paper, we present a detailed analysis of the rovibrational spectrum of the (anti-)hydrogen molecular ion in a low-energy effective theory incorporating Lorentz and CPT violation. The focus is on the spin-independent couplings in this theory, for which the best current bounds come from measurements of the 1S-2S transition in atomic hydrogen and antihydrogen. We show that in addition to the improvement in these bounds from the increased precision of the transition frequencies, potentially reaching 1 part in $10^{17}$, rovibrational transitions have an enhanced sensitivity to Lorentz and CPT violation of $O(m_p/m_e)$ in the proton (hadron) sector compared to atomic transitions.

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 / 2 minor

Summary. The manuscript analyzes the rovibrational spectrum of the hydrogen molecular ion H₂⁺ (and its antimatter counterpart) within a low-energy effective field theory that includes Lorentz- and CPT-violating terms. Focusing on the spin-independent sector, it claims that rovibrational transitions exhibit an O(m_p/m_e) enhancement in sensitivity to proton-sector violations relative to the atomic 1S–2S transition, with projected experimental reach of order 10^{-17}.

Significance. If the claimed enhancement survives a full molecular-structure calculation, the work would open a new experimental channel for bounding Lorentz/CPT violation in the hadron sector that is complementary to atomic spectroscopy and leverages the narrow natural linewidths of molecular ions. The manuscript supplies the first systematic treatment of this system in the effective theory.

major comments (2)
  1. [abstract, §1] The central claim of an O(m_p/m_e) enhancement (abstract and §1) rests on the assertion that matrix elements of the spin-independent LV operators over Born-Oppenheimer rovibrational wave functions do not introduce compensating factors from reduced-mass dependence or electronic screening. No explicit evaluation of these matrix elements or demonstration that higher-order corrections remain sub-dominant is visible in the provided text; this must be supplied before the enhancement can be regarded as established.
  2. [abstract] The projection to 10^{-17} sensitivity (abstract) is stated without an accompanying error budget that folds in the theoretical uncertainty of the molecular LV matrix elements themselves. If those matrix elements carry relative uncertainties larger than 10^{-3}, the projected improvement over existing atomic bounds is lost.
minor comments (2)
  1. Notation for the effective LV coefficients should be aligned with the standard SME literature (e.g., explicit listing of which c_{μν} or a_μ components are retained).
  2. Figure captions should state the numerical values of the LV coefficients used for the plotted shifts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below. Both points identify areas where the manuscript can be strengthened by additional explicit material, and we will incorporate the requested content in the revised version.

read point-by-point responses

  1. Referee: [abstract, §1] The central claim of an O(m_p/m_e) enhancement (abstract and §1) rests on the assertion that matrix elements of the spin-independent LV operators over Born-Oppenheimer rovibrational wave functions do not introduce compensating factors from reduced-mass dependence or electronic screening. No explicit evaluation of these matrix elements or demonstration that higher-order corrections remain sub-dominant is visible in the provided text; this must be supplied before the enhancement can be regarded as established.

    Authors: We agree that an explicit evaluation strengthens the central claim. In the revised manuscript we will add a dedicated subsection (in §2 or a new §3) that evaluates the relevant spin-independent LV matrix elements explicitly over the Born-Oppenheimer rovibrational wave functions. This calculation will demonstrate that reduced-mass and electronic-screening corrections do not cancel the O(m_p/m_e) enhancement factor and that higher-order Born-Oppenheimer corrections remain sub-dominant at the level of the quoted sensitivity. The abstract and §1 will be updated with a brief reference to this new material. revision: yes

  2. Referee: [abstract] The projection to 10^{-17} sensitivity (abstract) is stated without an accompanying error budget that folds in the theoretical uncertainty of the molecular LV matrix elements themselves. If those matrix elements carry relative uncertainties larger than 10^{-3}, the projected improvement over existing atomic bounds is lost.

    Authors: We accept that a quantitative error budget is required to support the projected sensitivity. The revised manuscript will include a new paragraph (or short subsection) that propagates the estimated theoretical uncertainties of the LV matrix elements (arising from finite-basis truncation, non-adiabatic corrections, and numerical integration) into the final sensitivity figure. This budget will be presented alongside the experimental linewidth and statistical considerations, confirming that the 10^{-17} reach remains valid provided the matrix-element uncertainties stay below the 10^{-3} relative level. revision: yes

Circularity Check

0 steps flagged

No circularity: O(m_p/m_e) enhancement is a physical scaling from molecular reduced-mass dependence vs. atomic electronic transitions, not a fit or self-citation reduction.

full rationale

The paper applies a standard low-energy effective theory for spin-independent Lorentz/CPT violation to the rovibrational spectrum of H2+ and derives the relative sensitivity scaling directly from the Born-Oppenheimer separation and reduced-mass factors in the molecular Hamiltonian. This O(m_p/m_e) factor is a fixed physical constant arising from the proton-to-electron mass ratio in the rovibrational energy expression compared to the electron-mass-scaled atomic 1S-2S transition; it is not obtained by fitting parameters to data, renaming a known result, or invoking a self-citation chain. The abstract and claimed derivation contain no equations that reduce the enhancement to an input by construction, and the central claim remains independent of any fitted LV coefficients. This is the normal case of a self-contained effective-theory calculation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the paper invokes the standard low-energy effective theory for Lorentz/CPT violation but supplies no further parameter count or derivation details.

axioms (1)
  • domain assumption The low-energy effective theory incorporating Lorentz and CPT violation applies to the rovibrational states of the (anti)hydrogen molecular ion.
    Framework stated in the abstract as the basis for the spectrum analysis.

pith-pipeline@v0.9.0 · 5703 in / 1235 out tokens · 28620 ms · 2026-05-23T06:58:18.038360+00:00 · methodology

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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.

  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

    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 II -- hyperfine-Zeeman spectrum

    hep-ph 2025-04 unverdicted novelty 5.0

    Extends effective QFT treatment of Lorentz and CPT violation to the hyperfine-Zeeman spectrum of H2+ and anti-H2- ions, demonstrating extraction of constraints on spin-dependent couplings from rovibrational transition...

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