Constraint on ultralight Nelson-Barr dark matter from time-dependent nuclear decay

Chang-Jie Dai; Tong Li

REVIEW 2 major objections 4 minor 65 references

Constraint on ultralight Nelson-Barr dark matter from time-dependent nuclear decay

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The null observation of periodic variations in tritium beta decay excludes, at 95% confidence, a Nelson-Barr ultralight scalar dark matter with decay constant below 7.0e9 to 1.4e7 GeV for masses between 3.4e-23 and 1.7e-20 eV.

desk verdict The paper's central constraint is off by three orders of magnitude because it omits the direct CKM modulation of |V_ud|^2 that the model itself predicts. read the letter →

arxiv 2506.22081 v2 pith:AO3SV436 submitted 2025-06-27 hep-ph

Chang-Jie Dai , Tong Li This is my paper

classification hep-ph

keywords ultralightscalardarkmatterNelson-BarrmodelstrongCPproblemtritiumbetadecayperiodicnuclearratesCKMmatrixmodulationfrequentisthypothesistestingbindingenergy

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 tries to establish that the null observation of periodic variations in the tritium $\beta$-decay rate can be turned into a new exclusion on an ultralight scalar dark-matter candidate in the Nelson-Barr model, a proposal that solves the strong CP problem by spontaneous CP breaking. In that model the dark-matter scalar modulates the CKM matrix and the quark masses; the paper shows how those oscillations propagate through the neutron-proton mass difference and nuclear binding energies into a small oscillating residual in the tritium decay rate. Using twelve years of tritium data in which no statistically significant periodicity appears, it derives the quantitative claim: at 95% confidence the decay constant $f$ is excluded below $7.0\times 10^9$ to $1.4\times 10^7$ GeV for masses $m_\phi$ in the range $3.4\times 10^{-23}$ to $1.7\times 10^{-20}$ eV. If correct, this gives a new way to probe a strong-CP solution through ordinary radioactivity, with the bound set by the steadiness of a nuclear clock rather than by a dedicated dark-matter detector.

What carries the argument

The load-bearing object is the oscillating residual $I(\phi)=(\Gamma(\phi)-\langle\Gamma\rangle)/\langle\Gamma\rangle = 2.44\times 10^{-5}\phi/f$, which converts a dark-matter-induced frequency shift into a measurable fractional change in the decay rate. Two nuclear inputs carry the calculation: the $\phi$-dependent neutron-proton mass difference, fixed by the chiral relation $4c_5B_0(m_u-m_d)$, and the $\phi$-dependent tritium binding energy, built from a one-pion-exchange potential, a three-dimensional square-well deuteron wavefunction, and empirical SU(4) scaling from the deuteron to tritium. These enter the $\beta$-decay phase-space integral, and the resulting residual is tested with a Rice-distributed likelihood whose Type-I and Type-II error rates, with the ultralight-field amplitude treated as Rayleigh-distributed at low mass, produce the exclusion. A key structural identity is that the residual scales linearly with $1/f$, whereas the axion analogue scales as $1/f_a^2$, which is why the Nelson-Barr bound is weaker for a given decay constant.

What would settle it

A concrete check is to recompute the amplitude from first principles: the neutron-proton mass-difference coefficient in Eq. (23) and the final decay-rate coefficient in Eq. (41) have opposite signs and different magnitudes, so a corrected calculation would settle the absolute scale of the bound. Experimentally, a tritium dataset with greater statistical power that still shows no sinusoidal residual at the predicted amplitude and frequency would push the excluded $f$ higher, while a residual matching the predicted linear-in-$1/f$ scaling would confirm the Nelson-Barr interpretation.

Watch

Extended reading notes

Core claim

In the Nelson-Barr model, spontaneous CP breaking produces a pseudo-Nambu-Goldstone boson $\phi$; when this field is ultralight and forms the local dark matter, it oscillates at frequency $m_\phi$ and, through the field-dependent CKM matrix, induces periodic modulations of weak-interaction parameters and quark masses. The paper's central calculation follows these modulations into nuclear physics: the neutron-proton mass difference shifts through the QCD term $4c_5B_0(m_u-m_d)$, and the tritium binding energy shifts through a one-pion-exchange potential evaluated with a square-well deuteron wavefunction and SU(4) scaling. The combined effect is an oscillating residual in the tritium $\beta$-decay rate, $I(\phi)=2.44\times 10^{-5}\,\phi/f$, with $\phi/f = \sqrt{2\rho_{DM}}/(f m_\phi)\cos(m_\phi t+\delta)$. Applying a frequentist hypothesis test to the null tritium data yields the paper's main result: at 95% CL, decay constants below $7.0\times 10^9$ to $1.4\times 10^7$ GeV are excluded for $m_\phi$ between $3.4\times 10^{-23}$ and $1.7\times 10^{-20}$ eV.

Load-bearing premise

The whole result depends on the size of the tiny periodic wiggle the authors compute in the tritium decay rate; that size comes from approximate nuclear calculations, and an error there would move the excluded region.

Editorial extensions

If this is right

A Nelson-Barr scalar with $f$ below roughly $10^7$--$10^{10}$ GeV and mass in the quoted window would imprint a periodic modulation on the tritium decay rate that the twelve-year dataset does not contain, so that parameter region is ruled out.
At the same mass and decay constant, the excluded region is about three orders of magnitude weaker than the axion-model bound because the Nelson-Barr residual carries an extra $(m_q/v)^2$ one-loop suppression and scales as $1/f$.
For masses below $10^{-16}$ eV the scalar-field amplitude must be described as a Rayleigh-distributed stochastic variable; the paper's bound already includes this correction, so it remains valid when the coherence time is longer than the observation time.
The result adds radioactive-decay clocks to the set of probes of the Nelson-Barr parameter space, complementing equivalence-principle tests and searches for CKM-matrix variations.

Reading between the lines

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

Beyond the paper, the same residual formula could be applied to other beta-emitting isotopes; nuclei with larger $Q$-values would give a larger $\delta E$/keV lever arm and could push the excluded $f$ further at the same mass.
Beyond the paper, an independent recalculation of the amplitude from first principles would be a useful check: the printed coefficient changes sign between Eq. (23) and Eq. (41), and although the current bound depends on the square of the amplitude and may be insensitive to that sign, a corrected coefficient would shift the absolute scale of the exclusion.
Beyond the paper, the linear-$1/f$ scaling implies that combining several isotopes could help distinguish the Nelson-Barr scalar from axion-like dark matter in future positive detections, since the two models predict different amplitude-versus-frequency relations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, the punchline: the main numerical result is not trustworthy. The authors derive a time-dependent |V_ud|^2 in Eq. (17) with fractional amplitude around -0.09 φ/f, but when they compute the tritium decay rate in Sec. III.C, they only propagate the phase-space shift from the endpoint energy, which is about 2.44e-5 φ/f. Since Γ ∝ |V_ud|^2, the direct CKM term dominates by three orders of magnitude and should be included. I checked the algebra: T_ud ≈ 0.23 and κ ≈ 0.2 give δ|V_ud|^2/|V_ud|^2 ≈ -0.092 φ/f. This is an internal zero-parameter issue, not a question of nuclear modeling. Correcting it changes the 95% CL exclusion by a factor of ~4000 in f, making the bound much stronger and the comparison with the axion limit qualitatively different.

Credit where it's due: the idea of probing Nelson-Barr scalar DM through nuclear decays is novel, the paper is clearly structured, and the frequentist pipeline is verified by reproducing the axion limit from Ref. [14]. The nuclear physics chain (OPE potential, square-well wavefunction, SU(4) scaling) is a reasonable first pass, though the authors don't propagate uncertainties from these approximations.

The other soft spots are secondary. There is a sign mismatch between Eq. (23) and Eq. (41) that suggests a slip in the intermediate algebra; the bound uses the squared amplitude so the sign cancels, but the magnitude also looks off by about 60%. The parameters κ and log(Λ/v) have order-one freedom that isn't reflected in the error analysis. These are fixable. The missing CKM term is not.

Who is this for? Someone working on ultralight scalar DM in Nelson-Barr models or on nuclear decay as a DM probe would want to see this, but only in a corrected form. As it stands, the paper's central constraint is not a prediction of the model. I'd recommend sending it to review, but with the expectation of major revision: the authors need to include the direct CKM modulation and redo the limit. The idea is worth refereeing, but the current numbers should not be quoted.

Referee Report

2 major / 4 minor

Summary. This paper considers ultralight scalar dark matter in a Nelson-Barr solution to the strong CP problem, where the pseudo-Goldstone field φ oscillates and induces time-dependent modulations of CKM matrix elements and quark masses. The authors calculate the resulting periodic modulation of the tritium β-decay rate through the neutron-proton mass difference and nuclear binding energies, evaluate the phase-space integral, and apply a frequentist hypothesis test with stochastic-amplitude marginalization to the null JRC tritium data. They obtain a 95% CL exclusion on the decay constant f below 7.0×10^9 to 1.4×10^7 GeV for masses m_φ in the range 3.4×10^-23 to 1.7×10^-20 eV, and they compare this with existing CKM-variation and equivalence-principle constraints.

Significance. If the calculation were correct, the claimed exclusion would be a new and interesting probe of ultralight scalar dark matter in the Nelson-Barr framework, using a genuine null dataset and a statistical pipeline that partly validates itself by reproducing the axion limit of Ref. [14]. The paper is clearly organized and the frequentist/stochastic-amplitude treatment follows standard methods. However, the central decay-rate amplitude in Eq. (46) omits the direct |V_ud|^2 modulation that the same model predicts in Eq. (17); until that term is included, the numerical exclusion region cannot be considered a prediction of the model. This is the main load-bearing issue.

major comments (2)

[§III.C, Eqs. (17) and (46)] The decay-rate residual in Eq. (46) includes only the phase-space shift δI^β/I^β, but the model itself predicts an oscillating CKM element: Eq. (17) gives |V_ud|^2 ≈ |V^0_ud|^2 (1 − 2 T_ud κ φ/f). With the inputs quoted in §II, κ∼0.2 and T_ud∼0.2, this implies δ|V_ud|^2/|V_ud|^2 ≈ −8×10^-2 φ/f, which is about three orders of magnitude larger than the +2.44×10^-5 φ/f phase-space term. Since the tritium decay rate is proportional to |V_ud|^2 times the phase-space integral, Eq. (46) should be replaced by δΓ/Γ ≈ δ|V_ud|^2/|V_ud|^2 + δI^β/I^β. As written, the amplitude, the coefficient b in Eq. (61), and the exclusion region in Fig. 1 are not the model's prediction. Including the direct CKM term would strengthen the constraint on f by roughly three orders of magnitude, and the resulting region should be checked against the CKM-variation limits shown in Fig. 1.
[§III.A–C, Eqs. (23), (40), (41)] There is an internal sign and normalization inconsistency in the derivation of δM_i − δM_f. Eq. (23) predicts a negative coefficient, (m_n − m_p) ≈ (1.55 − 1.75×10^-7 φ/f) MeV. Combining this with Eq. (38) and B_f − B_i ≈ −0.76 MeV in Eq. (39) gives a φ/f coefficient of approximately −1.75×10^-7 − 0.76×(4.55×10^-7/8.1) ≈ −2.18×10^-7 MeV, not +1.33×10^-7 MeV as stated in Eq. (41). The sign flip and the mismatch in magnitude indicate at least one intermediate step is inconsistent. Because the bound is derived from the squared amplitude, the sign error alone would not change the limit, but the magnitude discrepancy and the unexplained sign change should be corrected before the amplitude can be considered reliable.

minor comments (4)

[§II and Eq. (5)] The display of the nonzero couplings lacks commas between g1 ≠ 0 and ˜g2 ≠ 0, and the footnote 'vector-like up-type 1 quark' appears to contain a typographical artifact.
[§IV.A and Eq. (60)] In the sentence before Eq. (53), 'TakingAs = 0' should be 'Taking As = 0', and in Eq. (60) the notational chain 'α = ... ≥ β = ...' is confusing because the two error rates are written as a single continued equality.
[Fig. 1] The label 'MACROSCOPE' in Fig. 1 should read 'MICROSCOPE'.
[§IV.B] In the discussion of stochastic amplitudes, the meaning of the ensemble average and the observation-time averaging in Eq. (54) could be clarified; in particular, the relation between the fixed amplitude Φ_DM and the Rayleigh-distributed Φ0 is stated but not formally derived.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the tritium constraint is derived from an external null measurement and an independently computed nuclear amplitude, with only a minor non-load-bearing self-citation to Ref. [14].

full rationale

The paper's central claim, the 95% CL exclusion on the Nelson-Barr decay constant f, is not fed back into the derivation at any step. The oscillating residual in Eq. (47) is computed from model parameters taken from Ref. [30] (κ ∼ 0.2, θ12 ∼ 0.1, log(ΛUV/(v/√2)) ∼ 1) plus standard nuclear inputs, and the excluded region is obtained by comparing that amplitude with the JRC tritium null result via the frequentist framework of Ref. [52]. No parameter is fitted to the tritium data, and no 'prediction' is equivalent by construction to an input. The only overlapping citation is Ref. [14] (co-authored by T. Li), which is used as a data source, a statistical template, and a benchmark for reproducing the axion limit; it does not assume the Nelson-Barr signal, so it is not load-bearing in a circular sense. Two internal consistency issues do exist but are correctness concerns rather than circularity: the sign of the neutron-proton mass-difference coefficient in Eq. (23) is inconsistent with the positive phase-space shift used in Eq. (41), and Eq. (46) propagates only the phase-space modulation while the direct |V_ud|^2 modulation from Eq. (17) is not included in the decay-rate residual. These affect the magnitude of the predicted signal but do not make the derivation circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the Nelson-Barr model borrowed from Ref. [30], the nuclear approximations from Refs. [38] and [40], and the statistical framework from Ref. [52]. The paper introduces no new entities. The main free choices are kappa and the log(Lambda) factor, both order-one and both affecting the final bound linearly.

free parameters (2)

kappa = sin(2 theta12) sin(2 theta0) = 0.2
Chosen following Ref. [30] to match observed CKM mixing. It sets the amplitude of the CKM oscillation that drives the decay-rate signal. Not fitted to tritium data.
log(Lambda/(v/sqrt(2))) = 1
One-loop quark mass correction factor. Set to 1 as a 'conservative estimate' in Ref. [30]. This order-one choice directly scales the neutron-proton mass difference modulation.

assumptions (5)

domain assumption Nelson-Barr model with vector-like up-type quark and complex scalar as given in Ref. [30]
Sec. II; the entire time-varying CKM and quark mass structure is imported from that model.
domain assumption The scalar field phi constitutes the local dark matter with rho_DM = 0.45 GeV/cm^3 and oscillates as phi/f = sqrt(2 rho_DM)/(f m_phi) cos(m_phi t)
Eq. (16); standard virialized ultralight DM ansatz.
standard math Chiral perturbation theory expression for the QCD contribution to the neutron-proton mass difference (Eq. 22) and one-pion exchange potential for the deuteron (Eq. 24)
Sec. III A and III B; adopted from Refs. [38] and [40].
domain assumption SU(4) scaling law (Eq. 36) relating tritium binding energy to deuteron binding energy
Sec. III B, Eq. (36); empirical universal behavior from Ref. [49], used to extrapolate from deuteron to tritium.
domain assumption Gaussian noise and the likelihood from Centers et al. [52]
Sec. IV A, Eq. (50); statistical model for the null data.

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Cite this review

Pith. "Pith review of Constraint on ultralight Nelson-Barr dark matter from time-dependent nuclear decay." pith.science (2026). https://pith.science/paper/AO3SV436

@misc{pith2026250622081,
  author       = {Pith},
  title        = {Pith review of: Constraint on ultralight Nelson-Barr dark matter from time-dependent nuclear decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AO3SV436}},
  note         = {Machine review of arXiv:2506.22081}
}

read the original abstract

Many experiments have notably reported anomalies in radioactive decay rates with periodic variations and challenged the traditional belief of time-independent decays. This periodicity could potentially be explained by the presence of a periodic dark matter (DM) candidate. In this work, we investigate the impact of time-dependent nuclear decay rates on the ultralight scalar DM in Nelson-Barr solution to the strong CP problem. The light scalar DM field in this framework induces periodic modulations of both CKM matrix elements and quark mass parameters. These modulations generate corresponding periodic perturbations in the neutron-proton mass difference and nuclear binding energies. Consequently, nuclear decay rates receive oscillating residuals. By analyzing the tritium decay rate, we derive new exclusion bounds on the Nelson-Barr DM parameters.

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Figures reproduced from arXiv: 2506.22081 by the authors.

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