pith. sign in

arxiv: 2405.06744 · v3 · submitted 2024-05-10 · ✦ hep-ph

Nelson-Barr ultralight dark matter

Michael Dine , Gilad Perez , Wolfram Ratzinger , Inbar Savoray This is my paper

Pith reviewed 2026-05-24 00:47 UTC · model grok-4.3

classification ✦ hep-ph
keywords Nelson-Barr mechanismstrong CP problemultralight dark matterCKM matrixtime-varying constantsnuclear clocksquantum sensors
0
0 comments X

The pith

The Nelson-Barr solution to the strong CP problem naturally yields an ultralight scalar that can be dark matter and drive periodic time variation in the CKM matrix elements.

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

The paper shows that the Nelson-Barr approach to setting the strong CP angle to zero can produce a light scalar field without fine-tuning. If this scalar accounts for the observed dark matter density, its coherent oscillations would cause the CKM quark-mixing parameters to change periodically with time. This time dependence creates observable effects that differ from those of the QCD axion and can be sought with precision instruments such as nuclear clocks. The resulting phenomenology links a solution to the strong CP problem directly to ultralight dark matter searches.

Core claim

Within the Nelson-Barr framework, a scalar field remains naturally light and, when it constitutes the dark matter, produces periodic time variation in the CKM matrix elements; this variation supplies new experimental signatures accessible to quantum sensors, including nuclear clocks, that go beyond standard axion phenomenology.

What carries the argument

The Nelson-Barr mechanism, which arranges additional fields and couplings so that the strong CP angle vanishes while allowing a light scalar whose oscillations modulate the effective CKM angles.

If this is right

  • CKM elements acquire a small, periodic time dependence whose frequency is set by the scalar mass.
  • Nuclear clocks and other quantum sensors become sensitive probes of this dark matter candidate through induced variations in fundamental parameters.
  • The model predicts a distinct set of signals compared with the QCD axion, including direct effects on weak decays and mixing processes.
  • The scalar can be produced and detected in laboratory settings without relying on the usual axion-photon coupling.

Where Pith is reading between the lines

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

  • Precision metrology experiments could indirectly constrain the parameter space of strong-CP solutions by searching for the predicted oscillations.
  • If confirmed, the scenario would tie the origin of the matter-antimatter asymmetry or related CP issues to the identity of dark matter.
  • The same scalar oscillations might induce detectable effects in other flavor observables or in atomic spectra over laboratory timescales.

Load-bearing premise

The Nelson-Barr mechanism must be realized in nature and the resulting light scalar must make up or dominate the dark matter density so that its oscillations can affect the CKM matrix.

What would settle it

High-precision, long-term measurements of CKM elements or nuclear transition frequencies that show no periodic time variation at the amplitude predicted for the local dark matter density.

Figures

Figures reproduced from arXiv: 2405.06744 by Gilad Perez, Inbar Savoray, Michael Dine, Wolfram Ratzinger.

Figure 1
Figure 1. Figure 1: FIG. 1. Potential reach of various collider searches for os [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Left: Contribution to the up-type quark selfenergy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Bounds on the new light scalar in terms of its [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Field content of the second model, with the SM fields [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Scan over the parameter [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We show that, in the Nelson-Barr solution to the strong CP-problem, a naturally light scalar can arise. It gives rise to a completely new phenomenology beyond that of the celebrated QCD axion, if this field constitutes dark matter, as the CKM elements vary periodically in time. We also discuss how the model can be tested using quantum sensors, in particular using nuclear clocks, which leads to an interesting synergy between different frontiers of physics.

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 paper shows that the Nelson-Barr solution to the strong CP problem admits a parametrically light scalar whose oscillations, if it constitutes ultralight dark matter, induce periodic time variation in the CKM matrix elements. This yields phenomenology distinct from the QCD axion and is proposed to be testable via quantum sensors, especially nuclear clocks.

Significance. The constructive existence of a naturally light scalar within the Nelson-Barr framework, together with the explicit link to time-dependent flavor observables, supplies a falsifiable signature that could be probed at the intersection of dark-matter searches and precision metrology. The absence of additional ad-hoc parameters for the lightness is a clear strength of the construction.

major comments (2)
  1. [§3.2, Eq. (18)] §3.2, Eq. (18): the effective potential for the light scalar is written after integrating out the heavy fields; the claim that the resulting mass is 'naturally' ultralight (m_ϕ ≪ v) relies on the determinant condition of the Nelson-Barr sector remaining exactly zero at the minimum. It is not shown whether loop corrections from the new scalar itself can lift this condition at a level that would require re-tuning.
  2. [§4.1, below Eq. (25)] §4.1, below Eq. (25): the oscillation amplitude of the CKM phases is stated to be set by the DM density and the scalar vev; the numerical example uses a specific choice of the Yukawa coupling hierarchy that is not derived from the Nelson-Barr texture. A parameter scan or analytic bound showing that the effect remains observable for generic textures would strengthen the central claim.
minor comments (2)
  1. The notation for the scalar field (ϕ vs. σ) is used interchangeably in §2 and §3; a single consistent symbol would improve readability.
  2. [Figure 2] Figure 2 caption does not specify the value of the DM density or the reference frequency used for the nuclear-clock sensitivity curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation of minor revision. We address the two major comments below.

read point-by-point responses

  1. Referee: §3.2, Eq. (18): the effective potential for the light scalar is written after integrating out the heavy fields; the claim that the resulting mass is 'naturally' ultralight (m_ϕ ≪ v) relies on the determinant condition of the Nelson-Barr sector remaining exactly zero at the minimum. It is not shown whether loop corrections from the new scalar itself can lift this condition at a level that would require re-tuning.

    Authors: We thank the referee for this remark. The Nelson-Barr texture enforces the determinant condition exactly at tree level. Corrections from loops involving the light scalar are suppressed by the small Yukawa couplings and the hierarchy m_ϕ ≪ v, remaining negligible for the masses of interest. We will add an explicit estimate of these loop effects to the revised manuscript. revision: yes

  2. Referee: §4.1, below Eq. (25): the oscillation amplitude of the CKM phases is stated to be set by the DM density and the scalar vev; the numerical example uses a specific choice of the Yukawa coupling hierarchy that is not derived from the Nelson-Barr texture. A parameter scan or analytic bound showing that the effect remains observable for generic textures would strengthen the central claim.

    Authors: The numerical example is illustrative of a viable point in parameter space. To address the concern, we will derive an analytic lower bound on the CKM-phase oscillation amplitude that holds for generic Nelson-Barr textures consistent with the observed CKM matrix, showing that the effect remains observable for a broad class of realizations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained existence result

full rationale

The paper advances a constructive existence argument inside the established Nelson-Barr framework: a scalar can be parametrically light, oscillate as ultralight DM, and induce periodic CKM variation. The abstract and strongest claim present this as possible new phenomenology rather than a derived necessity or fit. No equations, self-referential fitting, or load-bearing self-citations appear in the provided text; the central claim does not reduce to its inputs by construction. The derivation remains independent of the present paper's fitted values and is externally falsifiable via the Nelson-Barr determinant condition and quantum-sensor tests.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only; ledger is minimal and provisional. Full model details unavailable.

axioms (1)
  • domain assumption Nelson-Barr mechanism solves the strong CP problem without fine-tuning
    Framework assumed as given in the abstract.
invented entities (1)
  • naturally light scalar field no independent evidence
    purpose: ultralight dark matter candidate that induces time variation in CKM elements
    Postulated to arise in the Nelson-Barr setup; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5592 in / 1196 out tokens · 18343 ms · 2026-05-24T00:47:30.476398+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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

  1. Time-dependent signals of new physics at the LHC

    hep-ph 2026-05 unverdicted novelty 5.0

    Incorporating timing information from time-dependent new physics signals can improve LHC search sensitivity by up to a factor of two compared to standard time-invariant analyses.

  2. Exploring the Landscape of Spontaneous CP Violation in Supersymmetric Theories

    hep-ph 2025-10 unverdicted novelty 5.0

    Spontaneous CP violation is realized in SUSY via an extended spurion formalism in the exact limit and a model with intermediate-scale breaking along pseudo-flat directions stabilized by soft terms and non-perturbative...

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 2 Pith papers · 13 internal anchors

  1. [1]

    R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett.38, 1440 (1977)

    work page 1977

  2. [2]

    A. E. Nelson, Naturally Weak CP Violation, Phys. Lett. B136, 387 (1984)

    work page 1984

  3. [3]

    A. E. Nelson, Calculation ofθBarr, Phys. Lett. B143, 165 (1984)

    work page 1984

  4. [4]

    S. M. Barr, Solving the Strong CP Problem Without the Peccei-Quinn Symmetry, Phys. Rev. Lett.53, 329 (1984)

    work page 1984

  5. [5]

    R. N. Mohapatra and G. Senjanovic, Natural Suppression of Strong p and t Noninvariance, Phys. Lett. B79, 283 (1978)

    work page 1978

  6. [6]

    Georgi, A Model of Soft CP Violation, Hadronic J.1, 155 (1978)

    H. Georgi, A Model of Soft CP Violation, Hadronic J.1, 155 (1978)

    work page 1978

  7. [7]

    Bento, G

    L. Bento, G. C. Branco, and P. A. Parada, A Minimal model with natural suppression of strong CP violation, Phys. Lett. B267, 95 (1991)

    work page 1991

  8. [8]

    The Nelson-Barr Relaxion

    O. Davidi, R. S. Gupta, G. Perez, D. Redigolo, and A. Shalit, Nelson-Barr relaxion, Phys. Rev. D99, 035014 (2019), arXiv:1711.00858 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  9. [9]

    Challenges for the Nelson-Barr Mechanism

    M. Dine and P. Draper, Challenges for the Nelson-Barr Mechanism, JHEP08, 132, arXiv:1506.05433 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Hui, Wave Dark Matter, Ann

    L. Hui, Wave Dark Matter, Ann. Rev. Astron. Astrophys. 59, 247 (2021), arXiv:2101.11735 [astro-ph.CO]

    work page arXiv 2021

  11. [11]

    A. Dev, P. A. N. Machado, and P. Mart´ ınez-Mirav´ e, Sig- natures of ultralight dark matter in neutrino oscillation experiments, JHEP01, 094, arXiv:2007.03590 [hep-ph]

    work page arXiv 2007

  12. [12]

    Losada, Y

    M. Losada, Y. Nir, G. Perez, and Y. Shpilman, Probing scalar dark matter oscillations with neutrino oscillations, JHEP04, 030, arXiv:2107.10865 [hep-ph]

    work page arXiv

  13. [13]

    Losada, Y

    M. Losada, Y. Nir, G. Perez, I. Savoray, and Y. Shpil- man, Time dependent CP-even and CP-odd signatures of scalar ultralight dark matter in neutrino oscillations, Phys. Rev. D108, 055004 (2023), arXiv:2302.00005 [hep- ph]. 10

    work page arXiv 2023

  14. [14]

    Ambrosinoet al.(KLOE), Measurement of the abso- lute branching ratio for theK + →µ +ν(γ) decay with the KLOE detector, Phys

    F. Ambrosinoet al.(KLOE), Measurement of the abso- lute branching ratio for theK + →µ +ν(γ) decay with the KLOE detector, Phys. Lett. B632, 76 (2006), arXiv:hep- ex/0509045

    work page arXiv 2006

  15. [15]

    Ambrosinoet al.(KLOE), Precision Measurement of KS Meson Lifetime with the KLOE detector, Eur

    F. Ambrosinoet al.(KLOE), Precision Measurement of KS Meson Lifetime with the KLOE detector, Eur. Phys. J. C71, 1604 (2011), arXiv:1011.2668 [hep-ex]

    work page arXiv 2011

  16. [16]

    We thank Fr´ ed´ eric Blanc and Maria Vieites Diaz for pointing this out to us

    work page

  17. [17]

    Measurement of the Ratio Gamma(KL -> pi+ pi-)/Gamma(KL -> pi e nu) and Extraction of the CP Violation Parameter |eta+-|

    A. Laiet al.(NA48), Measurement of the ratio Gamma(KL —>pi+ pi-) / Gamma(KL —>pi e nu) and extraction of the CP violation parameter —eta(+- )—, Phys. Lett. B645, 26 (2007), arXiv:hep-ex/0611052

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  18. [18]

    I. S. Towner and J. C. Hardy, The evaluation of V(ud) and its impact on the unitarity of the Cabibbo- Kobayashi-Maskawa quark-mixing matrix, Rept. Prog. Phys.73, 046301 (2010)

    work page 2010

  19. [19]

    The MICROSCOPE mission: first results of a space test of the Equivalence Principle

    P. Touboulet al., MICROSCOPE Mission: First Re- sults of a Space Test of the Equivalence Principle, Phys. Rev. Lett.119, 231101 (2017), arXiv:1712.01176 [astro- ph.IM]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    Touboulet al.(MICROSCOPE), MICROSCOPE Mission: Final Results of the Test of the Equiva- lence Principle, Phys

    P. Touboulet al.(MICROSCOPE), MICROSCOPE Mission: Final Results of the Test of the Equiva- lence Principle, Phys. Rev. Lett.129, 121102 (2022), arXiv:2209.15487 [gr-qc]

    work page arXiv 2022

  21. [21]

    Banerjee, G

    A. Banerjee, G. Perez, M. Safronova, I. Savoray, and A. Shalit, The phenomenology of quadratically coupled ultra light dark matter, JHEP10, 042, arXiv:2211.05174 [hep-ph]

    work page arXiv

  22. [22]

    Equivalence Principle Violations and Couplings of a Light Dilaton

    T. Damour and J. F. Donoghue, Equivalence Principle Violations and Couplings of a Light Dilaton, Phys. Rev. D82, 084033 (2010), arXiv:1007.2792 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    The Scalar Strange Content of the Nucleon from Lattice QCD

    P. Junnarkar and A. Walker-Loud, Scalar strange content of the nucleon from lattice QCD, Phys. Rev. D87, 114510 (2013), arXiv:1301.1114 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  24. [24]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Re- marks on Higgs Boson Interactions with Nucleons, Phys. Lett. B78, 443 (1978)

    work page 1978

  25. [25]

    C. J. Campbell, A. G. Radnaev, A. Kuzmich, V. A. Dzuba, V. V. Flambaum, and A. Derevianko, A Single- Ion Nuclear Clock for Metrology at the 19th Dec- imal Place, Phys. Rev. Lett.108, 120802 (2012), arXiv:1110.2490 [physics.atom-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  26. [26]

    E. Peik, T. Schumm, M. S. Safronova, A. P´ alffy, J. Weit- enberg, and P. G. Thirolf, Nuclear clocks for testing fundamental physics, Quantum Sci. Technol.6, 034002 (2021), arXiv:2012.09304 [quant-ph]

    work page arXiv 2021

  27. [27]

    Tiedau, M

    J. Tiedau, M. V. Okhapkin, K. Zhang, J. Thielk- ing, G. Zitzer, E. Peik, F. Schaden, T. Pronebner, I. Morawetz, L. T. De Col, F. Schneider, A. Leitner, M. Pressler, G. A. Kazakov, K. Beeks, T. Sikorsky, and T. Schumm, Laser excitation of the th-229 nucleus, Phys. Rev. Lett.132, 182501 (2024)

    work page 2024

  28. [28]

    Elwell, C

    R. Elwell, C. Schneider, J. Jeet, J. E. S. Terhune, H. W. T. Morgan, A. N. Alexandrova, H. B. T. Tan, A. Derevianko, and E. R. Hudson, Laser excitation of the 229Th nuclear isomeric transition in a solid-state host, (2024), arXiv:2404.12311 [physics.atom-ph]

    work page arXiv 2024

  29. [29]

    Banerjee, H

    A. Banerjee, H. Kim, O. Matsedonskyi, G. Perez, and M. S. Safronova, Probing the Relaxed Relaxion at the Luminosity and Precision Frontiers, JHEP07, 153, arXiv:2004.02899 [hep-ph]

    work page arXiv 2004

  30. [30]

    A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik, and P. Wolf, Violation of the equivalence principle from light scalar dark matter, Phys. Rev. D98, 064051 (2018), arXiv:1807.04512 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  31. [31]

    Masia-Roiget al., Intensity interferometry for ultra- light bosonic dark matter detection, Phys

    H. Masia-Roiget al., Intensity interferometry for ultra- light bosonic dark matter detection, Phys. Rev. D108, 015003 (2023), arXiv:2202.02645 [hep-ph]

    work page arXiv 2023

  32. [32]

    V. V. Flambaum and I. B. Samsonov, Fluctuations of atomic energy levels due to axion dark matter, Phys. Rev. D108, 075022 (2023), arXiv:2302.11167 [hep-ph]

    work page arXiv 2023

  33. [33]

    H. Kim, A. Lenoci, G. Perez, and W. Ratzinger, Probing an ultralight QCD axion with electromagnetic quadratic interaction, Phys. Rev. D109, 015030 (2024), arXiv:2307.14962 [hep-ph]

    work page arXiv 2024

  34. [34]

    Vafa and E

    C. Vafa and E. Witten, Parity Conservation in QCD, Phys. Rev. Lett.53, 535 (1984)

    work page 1984

  35. [35]

    Chacko, H.-S

    Z. Chacko, H.-S. Goh, and R. Harnik, The Twin Higgs: Natural electroweak breaking from mirror sym- metry, Phys. Rev. Lett.96, 231802 (2006), arXiv:hep- ph/0506256

    work page arXiv 2006

  36. [36]

    Folded Supersymmetry and the LEP Paradox

    G. Burdman, Z. Chacko, H.-S. Goh, and R. Harnik, Folded supersymmetry and the LEP paradox, JHEP02, 009, arXiv:hep-ph/0609152

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [37]

    Neutral Naturalness from the Orbifold Higgs

    N. Craig, S. Knapen, and P. Longhi, Neutral Natural- ness from Orbifold Higgs Models, Phys. Rev. Lett.114, 061803 (2015), arXiv:1410.6808 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  38. [38]

    Preskill, M

    J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B120, 127 (1983)

    work page 1983

  39. [39]

    Planck-Scale Physics and the Peccei-Quinn Mechanism

    M. Kamionkowski and J. March-Russell, Planck scale physics and the Peccei-Quinn mechanism, Phys. Lett. B 282, 137 (1992), arXiv:hep-th/9202003

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  40. [40]

    Randall, Composite axion models and Planck scale physics, Phys

    L. Randall, Composite axion models and Planck scale physics, Phys. Lett. B284, 77 (1992)

    work page 1992

  41. [41]

    Solutions to the strong CP problem in a world with gravity

    R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins, and L. M. Widrow, Solutions to the strong CP problem in a world with gravity, Phys. Lett. B282, 132 (1992), arXiv:hep-ph/9203206

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  42. [42]

    S. M. Barr and D. Seckel, Planck scale corrections to axion models, Phys. Rev. D46, 539 (1992)

    work page 1992

  43. [43]

    Perez and A

    G. Perez and A. Shalit, High quality Nelson-Barr solution to the strong CP problem withθ=π, JHEP02, 118, arXiv:2010.02891 [hep-ph]

    work page arXiv 2010

  44. [44]

    Spontaneous CP violation and the strong CP problem

    L. Vecchi, Spontaneous CP violation and the strong CP problem, JHEP04, 149, arXiv:1412.3805 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  45. [45]

    From axion quality and naturalness problems to a high-quality ZN QCD relaxion,

    A. Banerjee, J. Eby, and G. Perez, From axion qual- ity and naturalness problems to a high-quality ZN QCD relaxion, Phys. Rev. D107, 115011 (2023), arXiv:2210.05690 [hep-ph]

    work page arXiv 2023

  46. [46]

    Asadi, S

    P. Asadi, S. Homiller, Q. Lu, and M. Reece, Chiral Nelson-Barr models: Quality and cosmology, Phys. Rev. D107, 115012 (2023), arXiv:2212.03882 [hep-ph]

    work page arXiv 2023

  47. [47]

    Grzadkowski and M

    B. Grzadkowski and M. Lindner, Nonlinear Evolution of Yukawa Couplings, Phys. Lett. B193, 71 (1987)

    work page 1987