Pith. sign in

REVIEW 4 major objections 4 minor 51 references

This paper claims that a dark SU(2) gauge symmetry broken to a residual Z3 naturally yields two stable dark matter particles, and that benchmark points exist satisfying the observed relic density along with current laboratory and cosmologic

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 14:58 UTC pith:YNFT2R7Z

load-bearing objection A new SU(2)D two-component DM model with real benchmark work, but the residual Z3 claim is probably wrong and the massless dark photon effects are ignored. the 4 major comments →

arxiv 2512.18568 v3 pith:YNFT2R7Z submitted 2025-12-21 hep-ph

Two-Component Dark Matter with an SU(2) Dark Sector

Shao-Long Chen , Wen-wen Jiang This is my paper
classification hep-ph PACS 95.35.+d12.60.Cn
keywords dark mattertwo-component dark matterSU(2) dark gauge symmetryZ3 residual symmetrydark photonHiggs portalrelic densitydark radiation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that multi-component dark matter can emerge from gauge symmetry breaking rather than from an imposed discrete symmetry. It proposes an SU(2) dark sector with a scalar triplet, doublet, and singlet; once the triplet acquires a vacuum expectation value, the dark gauge group breaks to a residual Z3 that protects the lightest charged states from decay. The two survivors are the massive dark gauge bosons X± and the lighter doublet scalar ρ1. Solving coupled Boltzmann equations with semi-annihilation and conversion processes, the authors find parameter choices that reproduce the total observed relic abundance while satisfying theoretical constraints and current experimental limits, with two explicit benchmark points given. The attractive feature is that the two-component structure is an automatic consequence of the symmetry-breaking pattern, not an input.

Core claim

The paper's claim is that an SU(2) dark gauge symmetry together with a dark scalar triplet, doublet, and singlet automatically produces two stable dark matter components. Once the triplet acquires a vacuum expectation value, the dark gauge group breaks to a residual Z3 symmetry; the lightest states carrying Z3 charge cannot decay, so the massive dark gauge bosons X± and the lighter doublet scalar ρ1 both survive as thermal relics. The authors formulate coupled Boltzmann equations for the two components, including semi-annihilation and particle-conversion processes, and scan parameters subject to perturbativity, unitarity, vacuum stability, Higgs invisible decay, dark radiation, ellipticity,

What carries the argument

The residual Z3 symmetry left over from the spontaneous breaking of SU(2)D is the load-bearing device. After the triplet vacuum expectation value, the fields transform with charges 1, ω, or ω², and the particle-number combination that keeps the lightest charged states from decaying is conserved exactly; it plays the role usually assigned to an imposed discrete symmetry. The second piece of machinery is the coupled set of Boltzmann equations, which track ρ1 and X± simultaneously and include semi-annihilations (ρ1ρ1 → X+X3 and X+ρ1* → ρ1X3) and conversion processes (ρ1ρ1* ↔ X+X−) that tie the two abundances together. These equations determine how the two relics share the total dark matter dens

Load-bearing premise

The load-bearing premise is that the cosmological abundances are set by ordinary tree-level freeze-out; the massless dark photon X3 would in reality mediate long-range forces that can enhance annihilation and self-interaction rates, an effect not included in the paper's calculation.

What would settle it

Take the benchmark parameters in Table II and recompute the coupled relic densities including Sommerfeld enhancement and bound-state formation from the massless X3 mediator; if the total ΩDM h² moves outside the Planck window by more than a few percent, or if the enhanced self-interaction cross section exceeds the ellipticity bound, the paper's benchmark-point claim is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Two-component freeze-out broadens the viable dark matter mass window: for gD around 0.1, the relic density can be matched for dark matter masses from roughly 100 to 1000 GeV, depending on the mass splitting between X± and ρ1.
  • The lighter scalar component ρ1 dominates the direct-detection signal; for the benchmark couplings it becomes accessible to next-generation experiments only above the current mass exclusions, about 65 GeV for gD = 0.15 and 83 GeV for gD = 0.099.
  • Annihilation of ρ1 into bottom-quark pairs is below present gamma-ray limits, so indirect detection is not yet a strong probe for these benchmark parameters.
  • Higgs invisible decays to the massless dark photon are extremely small (about 10⁻²¹ MeV), leaving the standard h1→ρ1ρ1 channel as the only practical Higgs-portal signature.
  • The dark-radiation constraint requires the dark and visible sectors to decouple above roughly 375 MeV, which sets a lower bound of about 7.5 GeV on the lighter dark matter mass.

Where Pith is reading between the lines

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

  • Because the X3 gauge boson is massless, long-range interactions could produce Sommerfeld enhancement and bound-state formation during freeze-out; the paper's tree-level Boltzmann treatment omits these, and including them could shift the quoted relic densities and halo self-interaction rates.
  • The Z3-from-SU(2)-breaking pattern is independent of the scalar realization; a fermionic dark multiplet version would inherit the same stability argument and could serve as an alternative two-component dark matter construction.
  • If future cosmic-microwave-background measurements tighten the effective number of neutrino species, the decoupling-temperature bound rises and the low-mass part of the favored region would be cut away, leaving only the heavier benchmarks.
  • The semi-annihilation and conversion channels make the two relic abundances interdependent through the mass ratio r = mX+/mρ1; measuring both masses and the shared relic fraction would therefore test the model in a way single-component models cannot.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a two-component dark matter model in which the Standard Model is extended by an SU(2)_D dark sector containing a scalar triplet, a scalar doublet, and a scalar singlet. The authors claim that the triplet VEV leaves a residual Z3 symmetry that stabilizes the lightest dark charged particles, identified as the vector bosons X± and the scalar ρ1. They present two benchmark points (BP1, BP2) that, they argue, satisfy relic density, direct and indirect detection, Higgs invisible decay, dark radiation, and ellipticity constraints. The paper contains the Lagrangian, mass spectra, Boltzmann equations, and a numerical analysis using FeynRules and micrOMEGAs.

Significance. If the central symmetry mechanism were correct, the model would be a novel realization of stable multi-component dark matter from a non-Abelian gauge group, and the simultaneous treatment of many experimental constraints would be valuable. The paper's strengths are the broad constraint list, the two explicit benchmark points, and the use of public packages (FeynRules, micrOMEGAs). However, the residual Z3 is not a genuine remnant of the proposed symmetry-breaking pattern, and the Boltzmann and reaction-rate treatments contain load-bearing errors. The viability claim is therefore not established in the current form.

major comments (4)
  1. [Sec. II.B, Eq. (15), and Abstract] The central claim that SU(2)_D is spontaneously broken to a residual Z3 is asserted without derivation and is inconsistent with the given VEV. With <Φ> = (v_D/√2)σ3, the unbroken generator is T3, which generates a continuous U(1); X3 is exactly massless, as Eq. (15) implicitly assumes. The stated charge assignments χ1∼ω^2, χ2∼ω, φ+∼ω, φ−∼ω^2 do not follow from the gauge structure. Under the actual U(1), the doublet components carry charges ±1/2 and the triplet/Goldstone modes carry charges 0, ±1, so the independent Z3 assignments are incompatible with the gauge interactions in Eq. (14). The stability of X± and ρ1 might instead be due to exact U(1) charge conservation, but the advertised Z3 mechanism is not present. This affects the Abstract, Introduction, Table I, and Sec. III.A and must be corrected before the central claim can be evaluated.
  2. [Sec. III.B, Eqs. (20) and (23)] The reverse-reaction terms are written with the equilibrium densities inverted. The standard collision term for ab→cd is proportional to [n_a n_b − n_a^eq n_b^eq n_c n_d/(n_c^eq n_d^eq)]. Equations (20) and (23) instead use n_hi^eq n_hj^eq/(n_hi n_hj) for the final-state densities, which is the reciprocal. As written, the equations do not satisfy detailed balance; if the final-state particles are overabundant, the net annihilation rate would be enhanced rather than suppressed. If the numerical code uses the correct formula, the manuscript equations still need to be corrected; if the code implements the printed equations, the relic densities in Table II are unreliable.
  3. [Sec. III.H, Eq. (40)] The claimed cross-section <σv>_{X+X−→X3X3} = π g_D^2/(4 m_X±^2) has the wrong parametric scaling. At tree level the amplitude contains two gauge couplings, so the cross-section must scale as g_D^4 (e.g., order π α_D^2/m^2). For g_D ≈ 0.1, this changes the rate by orders of magnitude. Since this cross-section feeds the dark radiation abundance and the bound T_dec ≳ 375 MeV, the constraint in Sec. III.H and the associated lower limit mχ ≳ 7.5 GeV need to be recomputed.
  4. [Sec. III.B, III.G, III.I] Because X3 is massless and the DM components carry U(1)_D charges, long-range interactions are an unavoidable part of the model. The paper computes tree-level relic densities without Sommerfeld enhancement in annihilation (Eqs. (18)–(23)) and uses a Rutherford self-scattering formula in Sec. III.I, but it does not treat Sommerfeld corrections for relic density or indirect detection. For α_D = g_D^2/(4π) ≈ 0.001–0.002, the enhancement at galactic velocities can be O(1–10), so the Fermi-LAT constraint (Sec. III.G) and the ellipticity limit (Sec. III.I) should be evaluated with this effect included, or the paper should demonstrate quantitatively that it is negligible.
minor comments (4)
  1. [Eq. (24)] The equilibrium yield for X+ is written with gρ1; it should instead use the number of degrees of freedom of X± (two polarizations). Also clarify how antiparticles are counted in Y_eq for ρ1 and X+.
  2. [Fig. 4 and Table II] It would be helpful to indicate explicitly where BP1 and BP2 lie in the relic-density plots.
  3. [Throughout] The notation ρ(∗)_1/ρ∗_1 is used inconsistently; please state clearly which fields are particles, antiparticles, and summed components in the Boltzmann equations.
  4. [General] There are numerous language and typographical errors (e.g., 'significantly' appears as 'significally'). A careful proofreading pass is needed.

Circularity Check

0 steps flagged

No significant circularity — relic-density benchmarks are standard fits and direct/indirect-detection rates are genuine outputs; the unproved 'residual Z₃' claim is a correctness gap, not a self-referential reduction.

full rationale

Derivation-chain audit: the paper contains no self-citation chain (refs [10]–[24] are independent prior SU(2)_D papers by other groups), no imported uniqueness theorem, and no ansatz smuggled in via citation; FeynRules/micrOMEGAs are external, non-overlapping machinery. The benchmark points in Table II are ordinary fits: g_D and m_ρ1 (and ancillary couplings) are chosen so that the two-component Boltzmann solution hits the Planck value Ω_DM h² = 0.1200 ± 0.0012 (Sec. III.B); the quantities then compared with data — σ_SI (Eqs. 36–39, Fig. 7), the Fermi-LAT dSph rate (Fig. 8), Higgs invisible width, ΔN_eff (Eqs. 40–48), and the ellipticity bound (Eqs. 49–56) — are genuine outputs of the fitted point, not refits of the same observables, so no fitted input is re-labeled as a prediction. The one load-bearing statement that is asserted rather than derived is the 'residual Z₃' of Sec. II.B: with ⟨φ0⟩ = v_D the triplet VEV leaves U(1)_T3, and the paper's own Eq. (15) gives only m²_X± = 2g_D²v_D² with X3 massless — the standard signature of an unbroken continuous U(1), not a discrete remnant. That is a correctness/derivation gap (with phenomenological corollaries: negligible Sommerfeld/bound-state effects are assumed in Eqs. (18)–(23) despite the massless X3), not a circular reduction: the stability conclusion for X± and ρ1 still follows from the actual U(1) charges, so the central existence claim is independent of the fitted parameters and is not equivalent to its input by construction.

Axiom & Free-Parameter Ledger

18 free parameters · 6 axioms · 4 invented entities

The model is constructed with a large parameter set; the benchmark points are hand-picked, so the ledger is dominated by chosen couplings and masses. No parameter is derived from first principles, and the central validity of the model rests on assumptions about the symmetry-breaking pattern and freeze-out dynamics.

free parameters (18)
  • gD = 0.15 (BP1), 0.099 (BP2)
    Chosen to reproduce the observed relic density and pass constraints.
  • vD = 1000 GeV
    Hand-picked dark sector VEV scale.
  • sinα = 0.1
    Hand-picked Higgs-singlet mixing.
  • sinθ = 0.2
    Hand-picked doublet mixing angle.
  • mρ1 = 200 GeV (BP1), 300 GeV (BP2)
    DM mass input, chosen for phenomenology.
  • mρ2 = 3500 GeV
    Heavy partner mass, chosen large enough to decay.
  • mh2 = 200 GeV
    Heavy Higgs-like scalar mass, chosen for illustration.
  • mρ0 = 1000 GeV
    Triplet scalar mass, chosen by hand.
  • λs1 = 0.01 (BP1), 0.02 (BP2)
    Semi-annihilation coupling, hand-picked.
  • λs2 = 0.01
    Semi-annihilation coupling, hand-picked.
  • λχ = 0.1
    Scalar quartic, hand-picked.
  • λΦχ = 0.1
    Scalar quartic, hand-picked.
  • κϕ = 0.1
    Scalar cubic coupling, hand-picked.
  • λϕ = 0.1
    Scalar quartic, hand-picked.
  • λϕΦ = 0.01 (BP1), 0.08 (BP2)
    Scalar quartic, hand-picked.
  • κϕχ = 0.1
    Scalar cubic coupling, hand-picked.
  • λϕχ = 0.18 (BP1), 0.15 (BP2)
    Scalar quartic, hand-picked.
  • λϕH = 0.1
    Higgs portal quartic, hand-picked.
axioms (6)
  • ad hoc to paper Residual Z3 symmetry after SU(2)D triplet VEV
    Sec. II states the residual Z3 and lists Z3 charges but does not derive it; a triplet VEV standardly leaves a continuous U(1). This assumption is load-bearing for DM stability.
  • domain assumption Dark sector was in thermal equilibrium with the SM in the early universe
    Sec. III.H assumes both sectors thermalized, which sets the dark radiation constraint and initial abundances.
  • domain assumption Kinetic equilibrium is maintained during freeze-out
    Appendix A assumes kinetic equilibrium; standard in WIMP calculations but unverified for this multi-component model.
  • standard math Copositivity conditions are sufficient for vacuum stability
    Sec. III.C relies on the copositive matrix criterion of Refs. [29,30].
  • domain assumption Perturbativity and unitarity bounds (λ<4π, |Λ|<8π)
    Sec. III.C imposes these standard constraints without re-deriving them for this specific Lagrangian.
  • domain assumption Mass hierarchy mX < mρ2−mρ1 and absence of X+→ρ1ρ1 decays in the shown benchmarks
    Sec. III.A uses this hierarchy to keep X± stable; it is not enforced over the full scan, so some plotted points may have unstable X±.
invented entities (4)
  • SU(2)D gauge bosons X± and X3 no independent evidence
    purpose: X± is a DM component; X3 is a massless dark photon contributing to dark radiation and self-interactions.
    No tree-level coupling to SM; masses and couplings are input parameters, not uniquely predicted.
  • Scalar triplet Φ (ρ0, φ±) no independent evidence
    purpose: Breaks SU(2)D and generates masses for X±.
    The ρ0 mass is a free parameter and no unique collider signature is identified.
  • Scalar doublet χ (ρ1, ρ2) no independent evidence
    purpose: ρ1 is the second DM component; ρ2 mediates conversion processes.
    Masses are chosen by hand; the predicted direct-detection cross-section depends on the chosen parameters.
  • Real scalar singlet ϕ (h2) no independent evidence
    purpose: Mediates the Higgs portal connecting the dark sector to the SM.
    h2 at 200 GeV is a possible collider handle, but the paper does not develop a distinctive search signature.

pith-pipeline@v1.3.0-alltime-deepseek · 17925 in / 25439 out tokens · 261705 ms · 2026-08-03T14:58:53.401565+00:00 · methodology

0 comments
read the original abstract

We propose an extension to the standard model incorporating a dark sector with a non-Abelian SU(2) gauge symmetry. The model yields stable dark matter candidates, protected by a residual $Z_3$ symmetry arising after the spontaneous symmetry breaking. The dark sector interacts with the SM via a Higgs portal, facilitated from mixing between the SM Higgs doublet and a dark scalar singlet. The model features two distinct DM components. We analyze theoretical and experimental constraints, including perturbativity, unitarity, vacuum stability, dark matter relic density, direct detection, indirect detection, Higgs invisible decays, dark radiation, and ellipticity. Our findings identify viable parameter spaces that satisfy these constraints, as exemplified by two benchmark points.

Figures

Figures reproduced from arXiv: 2512.18568 by Shao-Long Chen, Wen-wen Jiang.

Figure 1
Figure 1. Figure 1: FIG. 1. The Feynman diagrams of DM annihilation processes. The relevant processes are shown [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. DM semi-annihilation processes in the model. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The Feynman diagrams for the DM conversion processes [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Left panel: The DM relic density versus the value of the dark sector coupling [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Feynman diagrams for the Higgs invisible decay to the dark photons. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scattering of the DM with the SM fermions in the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The spin-independent DM-nucleon scattering cross-section as a function of the DM mass [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Indirect detection limit on the DM self-annihilation process [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

51 extracted references · 43 linked inside Pith

  1. [1]

    Coulomb logarithm

    The total relic abundance is obtained by multiplying the solution by a factor of two to account for the contributions from the antiparticles. The tree-level interactions changing the number density ofρ 1 andX + are ρ1ρ∗ 1 →h 1h1 , h1h2 , h2h2 , W+W − , ZZ , f¯f , X+X − , X3X3 ; ρ1ρ1 →X +X3 ; X +ρ∗ 1 →ρ 1X3 , ρ1h1 , ρ1h2 ; X +X − →h 1h1 , h1h2 , h2h2 , X3X...

  2. [2]

    + i 2 gDX3(∂µρ1 ·ρ ∗ 1 +∂ µρ2 ·ρ ∗

    + i√ 2 gDX − µ (∂µρ1 ·ρ 2 −∂ µρ2 ·ρ 1) +h.c. + i 2 gDX3(∂µρ1 ·ρ ∗ 1 +∂ µρ2 ·ρ ∗

  3. [3]

    + 1 4 g2 DX 2 3 (ρ1ρ∗ 1 +ρ 2ρ∗ 2) + 1 2 g2 DX + µ X − µ (ρ1ρ∗ 1 +ρ 2ρ∗ 2)

    +h.c. + 1 4 g2 DX 2 3 (ρ1ρ∗ 1 +ρ 2ρ∗ 2) + 1 2 g2 DX + µ X − µ (ρ1ρ∗ 1 +ρ 2ρ∗ 2). The gauge bosonX 3 remains massless, while the masses of the charged gauge bosons are m2 X ± = 2g2 Dv2 D .(15) 6 III. PHENOMENOLOGY AND CONSTRAINTS A. Dark matter After the spontaneous dark gauge symmetry breaking, the particles carrying non-zeroZ 3 charge can serve as dark m...

  4. [4]

    Pospelov, A

    M. Pospelov, A. Ritz, and M. B. Voloshin, Phys. Lett. B662, 53 (2008), arXiv:0711.4866 [hep-ph]

    Pith/arXiv arXiv 2008

  5. [5]

    X. Chu, T. Hambye, and M. H. G. Tytgat, JCAP05, 034 (2012), arXiv:1112.0493 [hep-ph]

    Pith/arXiv arXiv 2012

  6. [6]

    S.-M. Choi, J. Kim, P. Ko, and J. Li, JHEP09, 028 (2021), arXiv:2103.05956 [hep-ph]

    Pith/arXiv arXiv 2021

  7. [7]

    A. Das, S. Gola, S. Mandal, and N. Sinha, Phys. Lett. B829, 137117 (2022), arXiv:2202.01443 [hep-ph]

    Pith/arXiv arXiv 2022

  8. [8]

    Ko and T

    P. Ko and T. Nomura, Phys. Rev. D94, 115015 (2016), arXiv:1607.06218 [hep-ph]

    Pith/arXiv arXiv 2016

  9. [9]

    Choi and H

    S.-M. Choi and H. M. Lee, JHEP09, 063 (2015), arXiv:1505.00960 [hep-ph]

    Pith/arXiv arXiv 2015

  10. [10]

    B´ elanger, A

    G. B´ elanger, A. Pukhov, C. E. Yaguna, and ´O. Zapata, JHEP09, 030 (2020), arXiv:2006.14922 [hep-ph]

    Pith/arXiv arXiv 2020

  11. [11]

    Bhattacharya, P

    S. Bhattacharya, P. Ghosh, and N. Sahu, JHEP02, 059 (2019), arXiv:1809.07474 [hep-ph]

    Pith/arXiv arXiv 2019

  12. [12]

    Arcadi, C

    G. Arcadi, C. Gross, O. Lebedev, Y. Mambrini, S. Pokorski, and T. Toma, JHEP12, 081 (2016), arXiv:1611.00365 [hep-ph]

    Pith/arXiv arXiv 2016

  13. [13]

    S. Baek, P. Ko, and W.-I. Park, JCAP10, 067 (2014), arXiv:1311.1035 [hep-ph]

    Pith/arXiv arXiv 2014

  14. [14]

    Borah, E

    D. Borah, E. Ma, and D. Nanda, Phys. Lett. B842, 137981 (2023), arXiv:2212.11847 [hep-ph]

    Pith/arXiv arXiv 2023

  15. [15]

    Frigerio, N

    M. Frigerio, N. Grimbaum-Yamamoto, and T. Hambye, SciPost Phys.15, 177 (2023), arXiv:2212.11918 [hep-ph]

    Pith/arXiv arXiv 2023

  16. [16]

    Otsuka, T

    H. Otsuka, T. Shimomura, K. Tsumura, Y. Uchida, and N. Yamatsu, Phys. Rev. D106, 115033 (2022), arXiv:2210.08696 [hep-ph]

    Pith/arXiv arXiv 2022

  17. [17]

    Borah, E

    D. Borah, E. Ma, and D. Nanda, Phys. Lett. B835, 137539 (2022), arXiv:2204.13205 [hep-ph]

    Pith/arXiv arXiv 2022

  18. [18]

    Baouche, A

    N. Baouche, A. Ahriche, G. Faisel, and S. Nasri, Phys. Rev. D104, 075022 (2021), arXiv:2105.14387 [hep-ph]

    Pith/arXiv arXiv 2021

  19. [19]

    Ko, New Phys

    P. Ko, New Phys. Sae Mulli66, 966 (2016)

    2016

  20. [20]

    Hambye, JHEP01, 028 (2009), arXiv:0811.0172 [hep-ph]

    T. Hambye, JHEP01, 028 (2009), arXiv:0811.0172 [hep-ph]

    Pith/arXiv arXiv 2009

  21. [21]

    F. Chen, J. M. Cline, and A. R. Frey, Phys. Rev. D80, 083516 (2009), arXiv:0907.4746 [hep-ph]

    Pith/arXiv arXiv 2009

  22. [22]

    J. L. Diaz-Cruz and E. Ma, Phys. Lett. B695, 264 (2011), arXiv:1007.2631 [hep-ph]

    Pith/arXiv arXiv 2011

  23. [23]

    T. A. Chowdhury and S. Saad, JCAP10, 014 (2021), arXiv:2107.11863 [hep-ph]

    Pith/arXiv arXiv 2021

  24. [24]

    P. Ko, T. Nomura, and H. Okada, Phys. Rev. D103, 095011 (2021), arXiv:2007.08153 [hep- ph]

    Pith/arXiv arXiv 2021

  25. [25]

    S.-M. Choi, Y. Hochberg, E. Kuflik, H. M. Lee, Y. Mambrini, H. Murayama, and M. Pierre, JHEP10, 162 (2017), arXiv:1707.01434 [hep-ph]. 23

    Pith/arXiv arXiv 2017

  26. [26]

    S.-M. Choi, H. M. Lee, Y. Mambrini, and M. Pierre, JHEP07, 049 (2019), arXiv:1904.04109 [hep-ph]

    Pith/arXiv arXiv 2019

  27. [27]

    Coleppa, K

    B. Coleppa, K. Loho, and A. Sarkar, Eur. Phys. J. C84, 144 (2024), arXiv:2307.14873 [hep-ph]

    Pith/arXiv arXiv 2024

  28. [28]

    Alloul, N

    A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014), arXiv:1310.1921 [hep-ph]

    Pith/arXiv arXiv 2014

  29. [29]

    Alguero, G

    G. Alguero, G. Belanger, F. Boudjema, S. Chakraborti, A. Goudelis, S. Kraml, A. Mjallal, and A. Pukhov, Comput. Phys. Commun.299, 109133 (2024), arXiv:2312.14894 [hep-ph]

    Pith/arXiv arXiv 2024

  30. [30]

    Aghanimet al.(Planck), Astron

    N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  31. [31]

    Arhrib, inWorkshop on Noncommutative Geometry, Superstrings and Particle Physics (2000) arXiv:hep-ph/0012353

    A. Arhrib, inWorkshop on Noncommutative Geometry, Superstrings and Particle Physics (2000) arXiv:hep-ph/0012353

    Pith/arXiv arXiv 2000

  32. [32]

    Ping and F

    L. Ping and F. Y. Yu, Linear Algebra and its Applications194, 109 (1993)

    1993

  33. [33]

    Chakrabortty, P

    J. Chakrabortty, P. Konar, and T. Mondal, Phys. Rev. D89, 095008 (2014), arXiv:1311.5666 [hep-ph]

    Pith/arXiv arXiv 2014

  34. [34]

    Aaboudet al.(ATLAS), Phys

    M. Aaboudet al.(ATLAS), Phys. Rev. Lett.122, 231801 (2019), arXiv:1904.05105 [hep-ex]

    Pith/arXiv arXiv 2019

  35. [35]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Phys. Lett. B78, 443 (1978)

    1978

  36. [36]

    Beringeret al.(Particle Data Group), Phys

    J. Beringeret al.(Particle Data Group), Phys. Rev. D86, 010001 (2012)

    2012

  37. [37]

    Belanger, F

    G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun.185, 960 (2014), arXiv:1305.0237 [hep-ph]

    Pith/arXiv arXiv 2014

  38. [38]

    Aprileet al.(XENON), Phys

    E. Aprileet al.(XENON), Phys. Rev. Lett.121, 111302 (2018), arXiv:1805.12562 [astro- ph.CO]

    Pith/arXiv arXiv 2018

  39. [39]

    Aprileet al.(XENON), Phys

    E. Aprileet al.(XENON), Phys. Rev. Lett.131, 041003 (2023), arXiv:2303.14729 [hep-ex]

    Pith/arXiv arXiv 2023

  40. [40]

    Aalberset al.(LZ), Phys

    J. Aalberset al.(LZ), Phys. Rev. Lett.131, 041002 (2023), arXiv:2207.03764 [hep-ex]

    Pith/arXiv arXiv 2023

  41. [41]

    Ruppin, J

    F. Ruppin, J. Billard, E. Figueroa-Feliciano, and L. Strigari, Phys. Rev. D90, 083510 (2014), arXiv:1408.3581 [hep-ph]

    Pith/arXiv arXiv 2014

  42. [42]

    Ackermannet al.(Fermi-LAT), Phys

    M. Ackermannet al.(Fermi-LAT), Phys. Rev. Lett.115, 231301 (2015), arXiv:1503.02641 [astro-ph.HE]

    Pith/arXiv arXiv 2015

  43. [43]

    Wallisch,Cosmological Probes of Light Relics, Ph.D

    B. Wallisch,Cosmological Probes of Light Relics, Ph.D. thesis, Cambridge U. (2018), arXiv:1810.02800 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  44. [44]

    B. D. Fields, K. A. Olive, T.-H. Yeh, and C. Young, JCAP03, 010 (2020), [Erratum: JCAP 11, E02 (2020)], arXiv:1912.01132 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  45. [45]

    Akita and M

    K. Akita and M. Yamaguchi, JCAP08, 012 (2020), arXiv:2005.07047 [hep-ph]

    Pith/arXiv arXiv 2020

  46. [46]

    J. J. Bennett, G. Buldgen, P. F. De Salas, M. Drewes, S. Gariazzo, S. Pastor, and Y. Y. Y. Wong, JCAP04, 073 (2021), arXiv:2012.02726 [hep-ph]

    Pith/arXiv arXiv 2021

  47. [47]

    Giunti and C

    C. Giunti and C. W. Kim,Fundamentals of Neutrino Physics and Astrophysics(2007)

    2007

  48. [48]

    K. N. Abazajianet al.(Topical Conveners: K.N. Abazajian, J.E. Carlstrom, A.T. Lee), As- tropart. Phys.63, 66 (2015), arXiv:1309.5383 [astro-ph.CO]. 24

    Pith/arXiv arXiv 2015

  49. [49]

    J. L. Feng, M. Kaplinghat, H. Tu, and H.-B. Yu, JCAP07, 004 (2009), arXiv:0905.3039 [hep-ph]

    Pith/arXiv arXiv 2009

  50. [50]

    Agrawal, F.-Y

    P. Agrawal, F.-Y. Cyr-Racine, L. Randall, and J. Scholtz, JCAP05, 022 (2017), arXiv:1610.04611 [hep-ph]

    Pith/arXiv arXiv 2017

  51. [51]

    D. A. Buote, T. E. Jeltema, C. R. Canizares, and G. P. Garmire, Astrophys. J.577, 183 (2002), arXiv:astro-ph/0205469

    Pith/arXiv arXiv 2002