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Probing of violation of Lorentz invariance by ultracold neutrons in the Standard Model Extension

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Pith's one-line read At current qBounce sensitivity, neutron transition frequencies would set first bounds on a Lorentz-violating coefficient.

desk verdict Useful projected SME neutron bounds from qBounce, with one load-bearing caveat: the cbar_ZZ constraint requires E0 to be externally calibrated rather than fitted from the measured transitions. read the letter →

arxiv 1908.01498 v1 pith:XNF3LFMS submitted 2019-08-05 hep-ph gr-qcnucl-ex

A. N. Ivanov , M. Wellenzohn , H. Abele This is my paper · ORCID
classification hep-phgr-qcnucl-ex PACS 11.10.Ef11.30.Cp12.60.-i14.20.Dh
keywords LorentzinvarianceviolationStandardModelExtensionultracoldneutronsqBounceexperimentquantumgravitationalstatesgravityresonancespectroscopyCPTneutronsector
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 tries to establish that the qBounce ultracold-neutron experiment, which already resolves the discrete quantum levels of neutrons bouncing in Earth's gravitational field, is a viable probe of Lorentz-invariance violation in the neutron sector of the Standard Model Extension. Using the existing sensitivity $\Delta E < 2\times10^{-15}\ \mathrm{eV}$, the authors calculate the shifts that Lorentz-violating interactions would produce in the transition frequencies between gravitational bound states, and convert those shifts into upper bounds on SME coefficients. The proposed bounds, of order $10^{-4}$ for the $\bar{c}_{XX}$, $\bar{c}_{YY}$, and $\bar{c}_{ZZ}$ coefficients and $2\times10^{-3}$ for $\bar{c}_{TT}$, would be new in the neutron sector, with $\bar{c}_{ZZ}$ having no prior estimate. The point of the exercise is to give current and future qBounce measurements a concrete set of target predictions.

What carries the argument

The carrying mechanism is the effective non-relativistic potential for Lorentz-violating interactions in the neutron sector, obtained by reducing the SME Dirac Hamiltonian to order $|\vec{p}|^3/m^3$; it is linear in the SME coefficients and contains both spin-independent momentum terms and spin-dependent spin-momentum terms. Its expectation values in the quantum-bouncer eigenstates $E_k^{(0)} = E_0|\xi_k|$, where $\xi_k$ are the Airy-function zeros, produce the first-order corrections to the binding energies. Degenerate perturbation theory handles the two spin states of each unpolarized level, non-degenerate perturbation theory applies to polarized states, and a sidereal rotation matrix maps laboratory-frame coefficients into the Sun-centered frame. This chain is what turns a transition-frequency measurement into individual coefficient bounds.

What would settle it

Measure the $|1\rangle\to|4\rangle$ transition frequency with uncertainty below $2\times10^{-15}$ eV and compare it with the gravity-only prediction $E_0(|\xi_4|-|\xi_1|)$: a match within uncertainties would mean the scale-shift interpretation has no support, while a resolvable discrepancy would be the signal the paper predicts.

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Extended reading notes

Core claim

The central result is a set of predicted bounds rather than a measured signal: if the qBounce sensitivity $\Delta E < 2\times10^{-15}\ \mathrm{eV}$ is attained, the measured transition frequencies between gravitational bound states of ultracold neutrons would constrain the neutron-sector SME coefficients to $|\bar{c}_{XX}|, |\bar{c}_{YY}|, |\bar{c}_{ZZ}| < 6.8\times10^{-4}$ and $|\bar{c}_{TT}| < 2.0\times10^{-3}$. The derivation uses first-order perturbation theory with the effective non-relativistic Lorentz-violating potential: the spin-independent part shifts level $k$ by $-(2\bar{c}_{zz}+\bar{c}_{00})E_k/3$, while spin-dependent parts shift polarized-state transition frequencies by combinations of $\bar{g}_{x0z}$, $\bar{g}_{y0z}$, and $\tilde{d}_z$. Expressing the laboratory-frame coefficients in the Sun-centered frame produces the tabulated constraints, and the paper notes that $|\bar{c}_{ZZ}|$ had not previously been estimated.

Load-bearing premise

The derivation assumes that the relative experimental uncertainty of the qBounce transition frequencies can be attributed entirely to the overall energy scale $E_0$, even though the dominant Lorentz-violating shift is exactly proportional to $E_k$ and therefore rescales the spectrum without changing the ratios of transition frequencies.

Editorial extensions

If this is right

  • A qBounce run at the assumed sensitivity would produce the first bound on $|\bar{c}_{ZZ}|$, a neutron-sector SME coefficient with no previous estimate.
  • Polarized-UCN measurements separate the spin-independent combination $(2\bar{c}_{zz}+\bar{c}_{00})$ from the spin-dependent coefficients $\bar{g}_{x0z}$, $\bar{g}_{y0z}$, and $\tilde{d}_z$ by comparing non-spin-flip and spin-flip transition frequencies.
  • Time-averaged measurements constrain the Sun-centered-frame combinations given in the paper's Table I, while resolving the sidereal period $T_\oplus = 23\ \mathrm{hr}\ 56\ \mathrm{min}$ would test additional coefficients that average away over a full day.
  • Each improvement in sensitivity, from $\Delta E<2\times10^{-15}\ \mathrm{eV}$ toward $10^{-17}\ \mathrm{eV}$ and ultimately $10^{-21}\ \mathrm{eV}$, tightens all tabulated bounds by the corresponding factor.

Reading between the lines

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

  • Because the dominant $\bar{c}$ shift is proportional to $E_k$, the current qBounce frequency-ratio data carry no information about $(2\bar{c}_{zz}+\bar{c}_{00})$ unless the absolute scale $E_0$ is pinned down independently; a dedicated measurement of one absolute transition energy is the cleanest way to close that gap.
  • The sidereal-time dependence in the rotation matrix suggests a test the paper does not spell out: searching for a 23-hour-56-minute modulation of any transition frequency would isolate off-diagonal coefficients such as $\bar{c}_{XZ}$ and $\bar{g}_{XTZ}$ that cancel in time-averaged data.
  • The same effective-potential reduction could be applied to the gravitational-sector coefficients of the SME; the paper announces this as future work, so it is a natural next step rather than a result claimed here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper applies the Standard-Model Extension (SME) formalism for Lorentz violation to the quantum gravitational states of ultracold neutrons (UCNs) bouncing in the Earth's gravitational field. Using the non-relativistic effective potential of Kostelecky and Lane, the authors compute first-order corrections to transition frequencies between gravitational bound states for unpolarized and polarized UCNs, treating the twofold spin degeneracy of unpolarized states through a secular equation. They rotate the relevant SME coefficients from the laboratory frame to the canonical Sun-centered frame and, using the qBounce sensitivity delta_E < 2 x 10^-15 eV, they project upper bounds on neutron-sector SME coefficients. The headline results are |cbar_XX|, |cbar_YY|, |cbar_ZZ| < 6.8 x 10^-4 and |cbar_TT| < 2.0 x 10^-3, with the cbar_ZZ limit claimed as the first estimate of this coefficient. The paper also derives the Heisenberg spin-evolution equation for UCNs under Lorentz violation.

Significance. If the projected constraints are realized, this work would provide a new probe of neutron-sector Lorentz violation and the first estimate of cbar_ZZ. The calculation is careful and mostly correct, with strengths including the explicit treatment of the degenerate spin subspace for unpolarized UCNs, the absence of fitted parameters in the model, and the transparent rotation to the Sun-centered frame. The claim that cbar_ZZ has not been previously estimated is a concrete potential contribution. The authors are appropriately cautious in the discussion, describing the results as a theoretical basis for future experiments. The main caveat is that the central cbar constraint depends on an external calibration of the absolute energy scale, a point addressed in the major comments.

major comments (3)
  1. [Section III.C, Eq. (20)] The spin-independent Lorentz-violating shift, delta_E_k = -(2 cbar_zz + cbar_00) E_k/3, is exactly proportional to the unperturbed energy E_k. It therefore rescales every transition frequency by the same factor, leaving the ratios nu_31/nu_41 invariant. The bound in Eq. (20) is meaningful only if the absolute energy scale E0 is fixed by external inputs (the neutron mass and a gravimeter value of g) and the measured absolute transition frequencies are compared with the predicted spectrum. If the qBounce analysis instead calibrates E0 or g from the same measured transition frequencies, this common-mode shift is absorbed into the fitted E0 and Eq. (20) yields no constraint. The manuscript never states this external-calibration requirement; the sentence 'relative experimental uncertainties should be attributed to E0' invites the opposite reading. This is the load-bearing assumption for the claimed first bound on cbar_ZZ and must be stated explicitly and justified.
  2. [Section III.C, Eq. (20), derivation] The algebraic route to the numerical bound |2 cbar_zz + cbar_00| < 6 delta_E E4/(E4^2 - E1^2) = 3.4 x 10^-3 is not shown. The result appears to follow from adding the non-spin-flip bound 3 delta_E/(E4 - E1) and the spin-flip bound 3 delta_E/(E4 + E1) of Eq. (19). The manuscript should present this derivation explicitly, because Eq. (20) is the quantitative basis for all cbar constraints in Table I and the final conclusions.
  3. [Section IV, text after Eq. (25) and Table I] The derivation of the individual bounds on cbar_XX and cbar_YY is logically unclear. From Eq. (25) and the negligible cbar_Q one obtains |cbar_ZZ| < 6.8 x 10^-4. The sentences 'the experimental data ... giving cbar_ZZ = (cbar_XX + cbar_YY)/2' and '... giving cbar_XX = cbar_YY = cbar_ZZ' do not constitute a derivation; the correct route is |cbar_XX|, |cbar_YY| < |cbar_ZZ| + |cbar_XX - cbar_YY|/2, which with the quoted existing limits gives the table entries. This passage should be rewritten for clarity.
minor comments (6)
  1. [Section III.B, Eq. (16)] The left-hand side of Eq. (16) uses index k (E^(1)_{k sigma}), while the integrand uses p sigma'; the indices should be made consistent.
  2. [Section IV] The notation alternates between cbar_Q and ctilde_Q, and between bbar and btilde, for what appear to be the same quantities; a consistent notation should be adopted.
  3. [Section III.A] The sentence 'For the numerical analysis we shall use only the corrections where the second term is proportional to (E_p + E_q)' requires a justification, since the discarded branch with (E_p - E_q) also contributes to the unpolarized transition frequencies.
  4. [General] There are numerous typographical and grammatical errors, including 'we analyze a dynamics', 'metrices', and 'the ILL laboratory ... with the values theta = 45.166670 N and phi = 5.716670 E'. A careful proofreading is needed.
  5. [Section IV] The step from the sidereal-time-dependent expressions in Eq. (23) to the time-averaged bounds in Eqs. (24) and (25) is not written out; the averaging procedure should be stated.
  6. [References] Reference [12] ends with 'YYY', indicating an incomplete entry; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SME constraints are projected from an independent experimental sensitivity and an external effective Hamiltonian, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain is self-contained and non-circular. The SME non-relativistic potential is taken from Kostelecky and Lane (J. Math. Phys. 40, 6245 (1999)), an external derivation from the SME Lagrangian, and the qBounce sensitivity ΔE < 2×10^-15 eV is an experimental input from Cronenberg et al. [37], not a parameter fitted to the SME coefficients being bounded. The bound in Eq. (20), |2 cbar_zz + cbar_00| < 6 ΔE E4/(E4^2 - E1^2), follows algebraically from requiring |δν_pq| < ΔE/2π for the level shifts computed in Eqs. (15) and (17); the coefficients cbar_munu are independent SME parameters, not defined in terms of ΔE. The subsequent conversion to the Sun-centered frame and the bound |cbar_ZZ| < 6.8×10^-4 uses external Kostelecky-Russell constraints on cbar_Q, which are physically independent inputs. The fact that the spin-independent SME shift δE_k = -(2 cbar_zz + cbar_00) E_k/3 is proportional to E_k means the constraint is, in effect, a bound on a common-mode rescaling of the energy scale E0; this is a physics caveat about external calibration of E0 via m and g, not a definitional circularity. The paper states that relative experimental uncertainties are attributed to E0, but it does not fit E0 from the same data and then call that fit a prediction. Self-citations to qBounce are experimental references and are not load-bearing theoretical inputs. No circular step can be exhibited from the paper's own equations.

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

No free parameters are fitted; the projected bounds scale linearly with the external input ΔE taken from ref. [37]. No new particles or forces are introduced.

assumptions (5)
  • domain assumption The effective non-relativistic Lorentz-violating potential Φ_LV (Eq. 4) from Kostelecky-Lane is correct to order O(|p|^3/m^3) and linear in LV coefficients.
    This potential is the Hamiltonian whose matrix elements produce all reported bounds; it is imported from Ref. [47] and not rederived in the paper.
  • domain assumption Contributions from gravitational-sector SME coefficients and chameleon/symmetron fields are negligible.
    Section I and the discussion state these are neglected based on constraints from Refs. [16] and [37,42-45].
  • ad hoc to paper The qBounce sensitivity ΔE < 2×10^-15 eV applies to future polarized UCN transition-frequency measurements.
    All Table I bounds are projections built by inserting this sensitivity into Eqs. (19)-(20), but no polarized UCN transition data exist yet.
  • domain assumption The rotation matrix Eq. (22) and sidereal time-averaging correctly map laboratory-frame coefficients to the canonical Sun-centered frame.
    Section IV relies on this standard SME transformation to express final bounds in Sun-centered coordinates.
  • domain assumption The experimental uncertainty of measured transition frequencies is attributed to E0, and the LV correction does not alter the ratios of transition frequencies.
    Section III.C converts the relative uncertainties of ν_31 and ν_41 into ΔE0 and then uses this to bound coefficient combinations.

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Pith. "Pith review of Probing of violation of Lorentz invariance by ultracold neutrons in the Standard Model Extension." pith.science (2026). https://pith.science/paper/XNF3LFMS

@misc{pith2026190801498,
  author       = {Pith},
  title        = {Pith review of: Probing of violation of Lorentz invariance by ultracold neutrons in the Standard Model Extension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNF3LFMS}},
  note         = {Machine review of arXiv:1908.01498}
}
read the original abstract

We analyze a dynamics of ultracold neutrons (UCNs) caused by interactions violating Lorentz invariance within the Standard Model Extension (SME) (Colladay and Kostelecky, Phys. Rev. D55, 6760 (1997) and Kostelecky, Phys. Rev. D69, 105009 (2004)). We use the effective non-relativistic potential for interactions violating Lorentz invariance derived by Kostelecky and Lane (J. Math. Phys. 40, 6245 (1999)) and calculate contributions of these interactions to the transition frequencies of transitions between quantum gravitational states of UCNs bouncing in the gravitational field of the Earth. Using the experimental sensitivity of qBounce experiments we make some estimates of upper bounds of parameters of Lorentz invariance violation in the neutron sector of the SME which can serve as a theoretical basis for an experimental analysis. We show that an experimental analysis of transition frequencies of transitions between quantum gravitational states of unpolarized and polarized UCNs should allow to place some new constraints in comparison to the results adduced by Kostelecky and Russell in Rev. Mod. Phys. 83, 11 (2011); edition 2019, arXiv: 0801.0287v12 [hep-ph].

Figures

Figures reproduced from arXiv: 1908.01498 by the authors.

Figure 1
Figure 1. FIG. 1: The position of the ILL laboratory of the qBounce expe [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.