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arxiv: 2509.01682 · v2 · submitted 2025-09-01 · ✦ hep-ph

Towards a Unified Framework for Pseudo-Nambu-Goldstone Dark Matter and Electroweak Baryogenesis

Thomas Biek\"otter , Pedro Gabriel , Milada Margarete M\"uhlleitner , Rui Santos This is my paper

Pith reviewed 2026-05-18 19:12 UTC · model grok-4.3

classification ✦ hep-ph
keywords pseudo-Nambu-Goldstone dark matterelectroweak baryogenesiscomplex singlet 2HDMCP violationHiggs portalflavour alignmentdirect detectionLHC phenomenology
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The pith

The cS2HDM unifies a pseudo-Nambu-Goldstone dark matter candidate with electroweak baryogenesis by permitting CP violation under flavour alignment.

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

The paper introduces the complex singlet-extended two-Higgs-doublet model as a single framework that accommodates both a dark matter candidate and the mechanism for creating the observed matter-antimatter imbalance. It features a pseudo-Nambu-Goldstone boson whose direct interactions with ordinary matter are automatically small. The model keeps CP-violating phases active in the Higgs sector even when the 125 GeV Higgs matches Standard Model couplings, provided the Yukawa sector obeys flavour alignment. This combination lets the model pass current LHC, dark matter, and precision tests while supplying concrete predictions for future runs. A reader would care because it connects two separate puzzles with shared, experimentally accessible particles and couplings.

Core claim

The cS2HDM predicts a pseudo-Nambu-Goldstone DM candidate whose interactions with nuclei are naturally suppressed, while allowing for all sources of CP-violation under the assumption of flavour alignment in the Yukawa sector, which enables CP-violating interactions of the Higgs bosons even in the alignment limit. This feature makes the model attractive for studies of electroweak baryogenesis while accommodating a Higgs-portal DM candidate with standard thermal freeze-out.

What carries the argument

The complex singlet-extended 2-Higgs-Doublet Model (cS2HDM), which adds a complex singlet to two Higgs doublets to generate both the pseudo-Nambu-Goldstone DM and additional CP-violating phases that survive the alignment limit.

If this is right

  • The model yields testable LHC signals from additional Higgs bosons and dark matter production channels.
  • One-loop DM-nucleon scattering amplitudes are computed explicitly, including CP-violating effects.
  • Current Higgs signal strengths, electroweak precision data, and relic density are satisfied within viable parameter regions.
  • The electron EDM provides a direct constraint on the new CP-violating sources.

Where Pith is reading between the lines

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

  • Similar singlet extensions of the 2HDM could incorporate neutrino mass generation without spoiling the DM or baryogenesis features.
  • The public software package allows external groups to map the full viable space as new collider or EDM bounds appear.
  • If the alignment limit is only approximate, small deviations could produce measurable rates in flavour-violating Higgs decays.

Load-bearing premise

The assumption of flavour alignment in the Yukawa sector is required to permit CP-violating Higgs interactions even in the alignment limit.

What would settle it

A direct detection experiment reporting a DM-nucleon scattering rate significantly above the one-loop suppressed prediction, or an electron EDM measurement exceeding the model's CP-violating contributions, would rule out the central claim.

Figures

Figures reproduced from arXiv: 2509.01682 by Milada Margarete M\"uhlleitner, Pedro Gabriel, Rui Santos, Thomas Biek\"otter.

Figure 1
Figure 1. Figure 1: Left: one-loop self-energy diagram that gives rise to a CP-violating mixing between A0 and Hi under the presence of a bosonic CP-violating coupling A0H+H−. Right: one-loop diagram that gives rise to a coupling between a seemingly CP-odd state A0 and the W boson under the presence of the same CP-violating coupling considered in the left diagram. See text for details. Then, only a mixing between the three CP… view at source ↗
Figure 2
Figure 2. Figure 2: Set of points without (with) vacuum stability constraints in blue (orange) in three different plane 1 projections. The points are obtained in a limit where analytical conditions are available [56]. The plot shows that in this limit our strategy to obtain the conditions numerically works. randomly generated 10,000 parameter points that feature a bounded scalar potential according to our numerical procedure.… view at source ↗
Figure 3
Figure 3. Figure 3: Allowed parameter space after imposing perturbative unitarity constraints in a projection plane with mH2 = M on the horizontal axis and mH4 = mH± on the vertical axis, keeping all other parameters fixed as shown on top of the plot. Different tones of green correspond to three values of the CP-violating mixing angleα4. the value of α4, i.e. the more CP-violating mixing is present in the scalar sector (see a… view at source ↗
Figure 4
Figure 4. Figure 4: Left: predictions for the T-parameter in a CP-conserving scenario in the {mH, mA} plane, where mH is the mass of the second CP-even doublet Higgs state, and mA is the mass of the CP-odd Higgs state. The mixing angles α are set according to the alignment limit given in Eq. (26), such that the singlet field corresponding to the Higgs boson with a mass of mH3 = 400 GeV is decoupled, and there is no CP-violati… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Predicted relic abundance in the CP-conserving limit with Yukawa type I with a SM-like 1 Higs boson at mh1 = 125.1 GeV, a singlet-like Higgs boson at mh3 = 95 GeV, and the other doublet-like Higgs-bosons at M = mh2 = mh4 = mH± = 800 GeV. The remaining free parameters are set as follows: tan β = 3, α1 = β, α4,5,6 = 0, vS = 1000 GeV, λ Im 5 = λ Re 6 = λ Re 7 = λ Re 11 = 0, and the mixing angles α2 = α3… view at source ↗
Figure 6
Figure 6. Figure 6: Feynman diagrams for the scattering process χf → χf at the classical level (left) and at the quantum level, with vertex corrections (center) and propagator corrections (right). The red vertex is in general CP violating. The hashed blobs represent loop corrections that give rise to non-vanishing amplitudes in the limit of zero momentum transfer ⃗p = 0. 3.4.2 Direct Detection Due to the pNG-nature of the DM … view at source ↗
Figure 7
Figure 7. Figure 7: Prediction for the spin-independent DM-proton scattering cross section in the absence of CP￾violation in a benchmark scenario taken from Ref. [22]. In the right plot, the predicted values of the scattering cross section are rescaled with the ratio ξPlanck = (Ωh 2 )FO/(Ωh 2 )Planck, where (Ωh 2 )FO is the predicted DM relic abundance assuming standard freeze-out mechanism and (Ωh 2 )Planck is the experiment… view at source ↗
Figure 8
Figure 8. Figure 8: Predicted values of the squared Wilson coefficients 1 c s 1 , c v 1 , c s 10 and c v 10 as defined in Eq. (70) as a function of the DM mass mχ in the top left, top right, bottom left and bottom right plot, respectively, normalized to 1/v2 , and for different values of the mixing angles α4 and α5. The other free parameters are fixed as shown in [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Example diagram with a charged Higgs boson loop contributing to the electron EDM at the two-loop level. one-loop contributions to the eEDM are subleading barring sizable destructive interferences at the two-loop level. Taking this into account, in our analysis we consider the two-loop Barr-Zee type contributions [116]. An example diagram is shown in [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Predicted values of the eEDM |de| (see text for details) as a function of α6 (left) and α4 = α5 (right) for ξ Im ℓ = −1.5, ξ Im ℓ = 0 and ξ Im ℓ = 1.5 (left) and ξ Im ℓ = −0.1, ξ Im ℓ = 0 and ξ Im ℓ = 0.1 (right) in blue, orange and green, respectively, with the other free parameters set as shown on top of the plot. The horizontal gray dashed lines indicate the experimental 90% confidence-level upper limi… view at source ↗
Figure 11
Figure 11. Figure 11: Predicted values of the electron EDM as a function of the DM candidate mass for a parameter space scan. The horizontal black lines represent the experimental upper limits from the ACME [121, 122] and JILA [123] collaborations. upper bounds on the eEDM as presented by the ACME Collaboration in 2014 [121] (solid) and 2018 [122] (dash-dotted), and more recently by the JILA in 2023 [123] (dashed). One can see… view at source ↗
Figure 12
Figure 12. Figure 12: Predicted values of the squared non-relativistic Wilson coefficients c s 1 (left) and c v 1 (right) plotted against the DM mass mχ for a scan of the parameter space. Also shown with the solid and dashed black lines are the observed and expected upper limits set by the LZ experiment, respectively [103]. The green and orange bands show the 1σ and 2σ uncertainty bands, respectively. The color of each point i… view at source ↗
Figure 13
Figure 13. Figure 13: Predicted values of the squared non-relativistic Wilson coefficients c s 1 (left) and c v 1 (right) re￾scaled with the ratio of predicted over observed relic abundance ξPlanck plotted against the DM mass mχ for a scan of the parameter space. The red points are allowed by the JILA 2023 electron EDM constraint while the gray points are excluded. presence of CP-mixed interactions.18 This is in contrast to th… view at source ↗
Figure 14
Figure 14. Figure 14: Parameter space excluded by LHC Higgs data in planes with sin(β − α1) on the horizontal axes 1 and the flavour alignment parameters on the vertical axes: ξ Im u (left), ξ Im d (centre), and ξ Im ℓ (right). The dashed and solid black lines indicate the exclusion region at 2σ and 3σ confidence level, respectively, obtained with HiggsTools. The region above the orange dot-dashed lines are excluded by cross s… view at source ↗
read the original abstract

We propose the complex singlet-extended 2-Higgs-Doublet Model (cS2HDM), a spin-0 Dark Matter (DM) model with a Higgs sector consisting of two Higgs doublets and a complex singlet, as a benchmark for LHC DM searches. The model predicts a pseudo-Nambu-Goldstone DM candidate whose interactions with nuclei are naturally suppressed, while allowing for all sources of CP-violation under the assumption of flavour alignment in the Yukawa sector, which enables CP-violating interactions of the Higgs bosons even in the alignment limit. This feature makes the model attractive for studies of electroweak baryogenesis while accommodating a Higgs-portal DM candidate with standard thermal freeze-out. We confront the model with a comprehensive set of theoretical and experimental constraints, including Higgs-boson signal strength measurements, searches for additional Higgs bosons, DM relic abundance and direct detection, as well as electroweak precision observables and the electron EDM, with emphasis on the impact of the new CP-violating sources. For DM direct detection, we perform a one-loop computation of DM-nucleon scattering including CP-violating effects. We provide a public software package to facilitate future phenomenological studies of the cS2HDM.

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 / 1 minor

Summary. The manuscript proposes the complex singlet-extended 2-Higgs-Doublet Model (cS2HDM) as a benchmark for LHC DM searches. It features a pseudo-Nambu-Goldstone DM candidate with naturally suppressed nuclear interactions and incorporates all sources of CP violation for electroweak baryogenesis under the assumption of flavour alignment in the Yukawa sector, which permits CP-violating Higgs couplings even in the alignment limit. The work confronts the model with theoretical and experimental constraints including Higgs signal strengths, additional Higgs searches, DM relic abundance, direct detection (via one-loop DM-nucleon scattering including CP effects), electroweak precision observables, and the electron EDM, while providing a public software package.

Significance. If the flavour alignment remains consistent with the new CP-violating phases introduced by the complex singlet, the model offers a unified framework linking pseudo-Nambu-Goldstone DM with electroweak baryogenesis in a Higgs-portal setup. The one-loop direct detection calculation and public code are positive features that support reproducibility and further phenomenological studies.

major comments (2)
  1. [Abstract and Yukawa sector definition] Abstract and model definition section: The central claim that flavour alignment in the Yukawa sector enables CP-violating Higgs interactions even in the alignment limit while suppressing FCNCs is load-bearing for both the baryogenesis mechanism and the overall viability. The complex singlet vev and CP-odd phases in the scalar potential can induce effective off-diagonal operators at one loop or via mixing; the manuscript does not explicitly verify that the alignment conditions are preserved under RG evolution or protected by an additional symmetry.
  2. [DM direct detection calculation] DM direct detection and constraint scan section: The one-loop computation of DM-nucleon scattering is presented as including CP-violating effects and yielding naturally suppressed rates, but the manuscript provides no detailed breakdown of the loop contributions, renormalization scheme, or quantitative comparison to tree-level results that would substantiate the suppression claim across the scanned parameter space.
minor comments (1)
  1. [Model setup] The notation for the scalar potential parameters and mixing angles could be introduced with a dedicated table or explicit definitions at the start of the model section to improve clarity for readers unfamiliar with extended 2HDMs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and Yukawa sector definition] Abstract and model definition section: The central claim that flavour alignment in the Yukawa sector enables CP-violating Higgs interactions even in the alignment limit while suppressing FCNCs is load-bearing for both the baryogenesis mechanism and the overall viability. The complex singlet vev and CP-odd phases in the scalar potential can induce effective off-diagonal operators at one loop or via mixing; the manuscript does not explicitly verify that the alignment conditions are preserved under RG evolution or protected by an additional symmetry.

    Authors: We appreciate the referee drawing attention to this foundational aspect. The flavour alignment is imposed directly on the Yukawa matrices at the electroweak scale, ensuring that the CP-violating phases introduced by the complex singlet (which enter only through the scalar potential) do not generate tree-level FCNCs. The effective Higgs-fermion couplings remain flavour-diagonal by construction in the alignment limit. While the manuscript does not perform an explicit RG analysis, the alignment conditions are protected at the relevant low-energy scale in the same manner as in other aligned 2HDM realisations; any higher-scale mixing effects are suppressed by the small portal couplings. We will add a clarifying paragraph in Section 2 discussing this protection and the scale at which the alignment is applied. revision: partial

  2. Referee: [DM direct detection calculation] DM direct detection and constraint scan section: The one-loop computation of DM-nucleon scattering is presented as including CP-violating effects and yielding naturally suppressed rates, but the manuscript provides no detailed breakdown of the loop contributions, renormalization scheme, or quantitative comparison to tree-level results that would substantiate the suppression claim across the scanned parameter space.

    Authors: We agree that additional technical details would improve clarity. The one-loop amplitude receives contributions from CP-odd and CP-even Higgs exchanges, with the dominant suppression arising from the derivative couplings characteristic of the pseudo-Nambu-Goldstone DM candidate. The calculation is performed in the on-shell scheme. In the revised manuscript we will expand the relevant subsection to list the individual diagram classes, state the renormalization scheme explicitly, and include a supplementary figure showing the ratio of one-loop to tree-level cross sections for benchmark points across the scanned parameter space. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The cS2HDM is introduced as an explicit model extension with new fields, a scalar potential, and the flavour-alignment assumption stated outright in the abstract and model definition. All subsequent claims (suppressed DM-nucleon scattering at one loop, CP-violating sources compatible with alignment limit, relic density via thermal freeze-out) are derived from these inputs via standard calculations rather than by re-expressing fitted quantities or prior self-citations as new predictions. No load-bearing step reduces by the paper's own equations to a tautological redefinition of its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on introducing a new complex singlet scalar and assuming flavour alignment to achieve both the DM suppression and the CP violation needed for baryogenesis; these are not derived from more fundamental principles within the paper.

free parameters (1)
  • Higgs-sector parameters (masses, mixing angles, couplings, vevs)
    Multiple continuous parameters of the extended Higgs potential that are constrained by data rather than fixed by the model construction.
axioms (1)
  • domain assumption Flavour alignment in the Yukawa sector
    Invoked to allow CP-violating Higgs interactions while remaining in the alignment limit; stated explicitly in the abstract.
invented entities (1)
  • Complex singlet scalar field no independent evidence
    purpose: To generate the pseudo-Nambu-Goldstone boson that serves as the dark matter candidate
    New field postulated to extend the Higgs sector; no independent evidence outside the model is provided.

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