Dark Matter and CP Violation in Some Symmetry-Constrained 3HDMs

Anton Kun\v{c}inas

REVIEW 3 major objections 6 minor 1 cited by

Dark Matter and CP Violation in Some Symmetry-Constrained 3HDMs

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The $S_3$-symmetric three-Higgs-doublet model, after all constraints, leaves a viable dark matter candidate in the 28.9–41.9 GeV window and none above about 500 GeV.

desk verdict A careful thesis compiling four peer-reviewed 3HDM papers, but the advertised DM window is halo-profile dependent and the author says so. read the letter →

arxiv 2506.08197 v1 pith:ZFJXZT6R submitted 2025-06-09 hep-ph

Anton Kunčinas This is my paper

classification hep-ph

keywords three-Higgs-doubletmodeldarkmatterS3symmetryCPviolationinertdoubletrelicdensitycontinuousmass-degeneratescalars

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 thesis tries to establish that three-Higgs-doublet models whose free parameters are aggressively constrained by symmetry make sharp, testable predictions about dark matter and CP violation. In one $S_3$-symmetric implementation with a complex vacuum (C-III-a), the surviving inert dark matter candidate is confined to about 28.9–41.9 GeV after theoretical and experimental cuts, a much lower window than the Inert Doublet Model's usual range, and no heavy candidate survives above roughly 500 GeV. In the $U(1)\times U(1)$-symmetric version, dark matter is stabilised by a continuous symmetry instead of a discrete remnant, producing two mass-degenerate neutral sectors and a broad viable range from about 45 GeV to 2000 GeV. The same framework classifies which vacuum configurations yield explicit or spontaneous CP violation, including a case generating the CKM phase without massless scalars and allowing O(MeV) neutral scalars. If right, the reduction in free parameters turns the scalar sector into a predictive place to look for dark matter and CP violation.

What carries the argument

The load-bearing object is the $S_3$-symmetric three-Higgs-doublet scalar potential, written in irreducible representations (one singlet, one pseudosinglet, one doublet) and organised into vacuum implementations labelled by their real or complex character, such as R-II-1a, C-III-a and C-V. The symmetry reduces the number of free couplings, and where a remnant $Z_2$ survives spontaneous symmetry breaking it protects one inert doublet as the dark matter candidate; the paper's mass predictions follow from scanning the constrained parameter space. For continuous symmetries, the analogous stabiliser is an unbroken $U(1)\times U(1)$, which the paper shows forces mass-degenerate neutral states and gives two independent inert sectors, the origin of the multi-component dark matter.

What would settle it

For the C-III-a model, compute the annihilation cross section of the 28.9–41.9 GeV candidate into Standard Model states for the surviving parameter points and compare with gamma-ray limits on dwarf spheroidal galaxies assuming a standard cuspy halo profile: if the predicted flux exceeds the observed upper limit, the claimed viable window is excluded. For the $U(1)\times U(1)$ model, a falsifying observation would be a dark sector with only one stable component and no second mass scale, or a scalar spectrum without the predicted mass-degenerate neutral states.

Watch

Extended reading notes

Core claim

The central claim is that the symmetry-constrained scalar sector does most of the model-building work, so that dark matter masses are not scanned over freely but are predicted by the implementation. After applying perturbativity, stability, unitarity, electroweak oblique-parameter and experimental constraints in the C-III-a implementation of the $S_3$-symmetric 3HDM, the viable dark matter region lies between about 28.9 and 41.9 GeV; the thesis attributes the absence of heavy candidates above about 500 GeV to the symmetry making some scalar interactions proportional to masses rather than free parameters. In the $U(1)\times U(1)$-symmetric 3HDM, the unbroken continuous symmetry stabilises two inert scalar sectors, giving multi-component dark matter with two independent mass scales and numerically allowed masses from about 45 GeV to 2000 GeV, with the decays of heavier inert states into lighter ones adjustable and able to shift the relic density. On the CP side, the paper claims that different $S_3$ assignments for quarks and different vacuum structures produce distinct CP phenomenology, that spontaneous CP violation arises in real-coupling vacua, and that the most general complex vacuum (C-V) can generate the CKM matrix from vacuum phases without unwanted massless scalars while allowing very light neutral scalars.

Load-bearing premise

The load-bearing premise is that the dark matter halo density profile used for indirect-detection bounds is the right one, since the thesis concedes that with a different halo profile the C-III-a model can be completely ruled out.

Editorial extensions

If this is right

If the C-III-a result holds, dark matter should be a light scalar of mass 28.9–41.9 GeV, below the usual inert-doublet window and potentially invisible to collider and indirect searches depending on the halo profile.
The $S_3$ implementations predict the absence of heavy dark matter above about 500 GeV, so a confirmed heavy thermal candidate would distinguish them from the Inert Doublet Model.
In the $U(1)\times U(1)$ 3HDM, dark matter must be multi-component with two mass scales between about 45 GeV and 2000 GeV, and the heavier component's decays into the lighter one can change the relic density.
The CP-violating C-V vacuum offers a way to generate the CKM phase from vacuum expectation values with no massless scalars, and permits O(MeV) neutral scalars that may evade detection while affecting cosmology or rare decays.

Reading between the lines

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

A direct extension: the 28.9–41.9 GeV window gives a concrete target for low-threshold direct-detection experiments and for Higgs invisible-width measurements; a null result in both would effectively shift the burden onto the halo-profile assumption.
The pattern that symmetry makes some scalar couplings proportional to masses may be generic: other tightly constrained multi-doublet scalar sectors could similarly exclude heavy thermal WIMPs, not by tuning but by symmetry alone.
If astronomical data ever fixes the Milky Way halo profile, the C-III-a window would become a sharp yes/no test of the model, converting what is now an astrophysical input into a discovery or exclusion channel.
For the $U(1)\times U(1)$ model, multi-component dark matter with two mass scales could be probed by structure-formation observables such as small-scale clustering differences between the two components, which the paper does not calculate.

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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

This is an honest compilation of four already-published papers (JHEP/PRD 2022–2024) on S3-symmetric and U(1)xU(1)-symmetric three-Higgs-doublet models. If you are looking for new results, there are none beyond the cited papers—the thesis itself says so. What you get is a systematic synthesis: vacuum classifications, symmetry analysis, CP-violation options, and the DM mass windows. That is a useful service to the subfield, and the presentation is clear.

The physics is mostly sound, but the headline numerical claim—the 28.9–41.9 GeV DM window in C-III-a—is exactly as fragile as the stress-test note says. The preface concedes that a different halo profile can rule the model out entirely. That is not a peripheral caveat; indirect-detection constraints are doing the work, and the J-factor uncertainty is orders of magnitude. The thesis does not marginalize over profiles or defend its choice. So I would not treat that window as a robust prediction. The lack of heavy DM above ~500 GeV and the two-scale 45–2000 GeV sector in the U(1)xU(1) case are less profile-dependent and rest on the same scalar-sector machinery, which passed peer review.

Minor issues: no code or parameter files are shipped, so the windows cannot be independently reproduced from this document; the abstract's claim about "different" mass ranges is not backed by a tabulated comparison; and there is a typo in Sec. 1.1 (scalar bosons called "spin-one"). None of this undermines the underlying work.

Who is this for? Someone entering multi-Higgs phenomenology, or a model builder wanting a one-stop map of what S3 and U(1)xU(1) 3HDMs allow. As a research posting it is a synthesis, not a new result paper. But the underlying papers were peer-reviewed and the synthesis is careful. I would send it to a referee if it were submitted as a review or as part of a thesis evaluation—the referee should focus on the halo-profile sensitivity, which the author already flags. If it were submitted as a new research preprint claiming the 28.9–41.9 GeV window as a prediction, I would want the profile dependence handled properly first.

Referee Report

3 major / 6 minor

Summary. This is a PhD-thesis-style arXiv posting (hep-ph) that compiles four of the author's peer-reviewed papers on three-Higgs-doublet models (JHEP 2022, PRD 2022, JHEP 2023, JHEP 2024). In a unified notation it develops the S3-symmetric 3HDM scalar potential, its vacuum taxonomy (real/complex vacua with real/complex couplings; explicit versus spontaneous CP violation; accidental continuous symmetries; fermion assignments under S3), and a numerical dark-matter analysis of two implementations, R-II-1a (real vacuum) and C-III-a (complex vacuum). The headline results are a 'viable' dark-matter mass window of 28.9–41.9 GeV in C-III-a after theoretical and experimental cuts, the absence of dark-matter candidates heavier than about 500 GeV in the S3 implementations (attributed to mass–coupling proportionality), and, in the continuous-symmetry part, a classification of 3HDM stabilizations with a numerical study of the U(1)xU(1) model proposing a two-component dark sector with masses from about 45 GeV to 2000 GeV. A third theme is CP violation through the C-V vacuum, which can generate the CKM phase without unwanted massless scalars and can accommodate O(MeV) neutral scalars.

Significance. If the claims hold, the thesis provides a comprehensive, symmetry-constrained classification of 3HDM vacua with several falsifiable predictions: a light dark-matter window (28.9–41.9 GeV) distinct from typical inert-doublet ranges, a structurally motivated absence of dark matter above roughly 500 GeV in the S3 implementations, and a two-scale multi-component dark-matter spectrum in the U(1)xU(1) model, with mass-degenerate states as a model-discriminating signature. The strengths are the systematic and transparent taxonomy of vacua, the careful identification of which configurations yield explicit versus spontaneous CP violation (including a C-V solution with no massless scalars), and the honest disclosure in the Preface of the halo-profile sensitivity of the indirect-detection bounds. The numerical pipeline is standard (potential minimization, mass-squared matrices, freeze-out relic density, perturbativity/stability/unitarity/oblique cuts) and the constituent results have passed peer review in JHEP and PRD; the thesis-level added value is the synthesis and the comparative claims. The principal weakness is the conditional character of the headline mass window, discussed below.

major comments (3)

[Preface, p. xii; Abstract] The headline claim of the Preface—'a viable dark matter mass region was identified between approximately 28.9–41.9 GeV' for the C-III-a implementation—is not robust to the assumed dark-matter halo density profile, and the manuscript itself immediately concedes that 'depending on the dark matter halo distribution profile applied to the indirect dark matter detection constraints, this model may be completely ruled out.' The indirect-detection constraints enter as Cut 3 of the numerical pipeline (§6.5.3) and are computed from dwarf-spheroidal J-factors, which differ by roughly an order of magnitude among the standard profiles (NFW, Einasto, cored). In the material available to this reader, no fiducial profile is identified for the stated window, and no scan or marginalization over this uncertainty is presented. Because the window is also the basis of the abstract's comparative claim of 'different dark matter mass ranges,' it must be either explicitly conditioned on the assumed profile (naming the profile and showing the resulting constraint curve) or supplemented by a sensitivity scan over standard profiles. Pending that, the wording 'viable dark matter mass region' overstates the result; a formulation such as 'viable for the adopted benchmark halo profile' would correctly convey its status. Please also clarify whether direct-detection bounds, which are typically decisive for a 30–40 GeV candidate with Higgs-mediated scattering, are included among the applied constraints and, if so, where they are described.
[Preface, p. xii; §6.2–§6.5] The second headline claim—the 'lack of heavy dark matter candidates—typically heavier than 500 GeV in the Inert DoubleT Model' in both S3 implementations—is attributed to the structural property that 'some of the scalar interactions [are] proportional to masses rather than being free parameters.' This proportionality is a strong, falsifiable statement and is the thesis's justification for why the S3 models differ from the Inert Doublet Model, but the Preface never points to the specific relation. The relevant couplings should be exhibited explicitly (identifying which quartic couplings in §§6.2–6.4 are fixed in terms of physical masses by the S3-symmetric potential and its vacuum) and, ideally, their effect on the annihilation cross section should be shown. Alternatively, the claim should be softened to a numerical observation of the scanned parameter space. It would also strengthen the analysis to demonstrate that m_DM > 500 GeV is excluded by the physics (e.g., over-abundance for the mass-proportional couplings) rather than by the boundary of the scan performed in §6.5; a plot of the relic density versus m_DM through the heavy region would settle this.
[Preface, p. xiii; §7.14–§7.15] The continuous-symmetry claim—a multi-component dark sector in the U(1)xU(1) 3HDM with 'two independent mass scales' and allowed masses 'from about 45 GeV up to 2000 GeV'—needs a more precise statement of the relic-density computation. If the stabilizing U(1)xU(1) symmetry is exact, the lightest states of both sectors are absolutely stable and the final abundances are set by a coupled two-component Boltzmann system. The Preface's statement that 'the decays of the heavier inert states into the lighter ones ... can be adjusted, potentially altering the predicted relic density' is ambiguous as written, since a stable heavier state cannot decay into the lighter sector; the relevant processes must be number-changing or semi-annihilation reactions whose rates are controlled by specific quartic couplings. The thesis should name those processes, state which assumption fixes the relative abundances of the two stable species, and present the 45–2000 GeV range together with the assumption under which it is obtained, so the reader can distinguish a parameter-space statement from a robust prediction.

minor comments (6)

[§1.1] The sentence 'Apart from the spin-one gauge bosons, there are spin-one bosons. These are referred to as scalar bosons' should read 'spin-zero bosons'; as printed, the sentence is self-contradictory.
[Introduction, first paragraph] The sentence 'Among the simplest extensions capable of addressing many of its shortcomings the multi-Higgs-doublet models' is a fragment and is missing the verb 'are.'
[§1.7.6 and §1.7.7] Equations (1.7.58) and (1.7.73), and the surrounding discussion of the Lee–Quigg–Thacker bound, contain corrupted square-root symbols ('/radicaltp/radicalvertex/radicalvertex√') in the posted version, so the decay width and the bounds (about 1008 GeV and about 713 GeV) are unreadable; the document should be regenerated from the LaTeX source so that the radicals render properly.
[§2.2] The sentence introducing eq. (2.2.5)—'Let us consider a particle with mass m_χ^{-2} ~ <sigma_ann v>'—is dimensionally inconsistent with the relation that follows (rho_chi/rho ~ x/(M_Planck m_chi <sigma v>)); the intended scaling should be corrected so that the text and the equation agree.
[§3.6] The remark after eq. (3.6.19)—'Notice that in the first line we want a new doublet to be invariant under the action of D2(a), i.e., "it" does not hold information about being constructed out of the pseudosinglet representation'—is unclear and should be rephrased.
[Figure 2.1 caption] The caption of Figure 2.1 contains a typo: 'temeprature' should be 'temperature.'

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the dark matter mass windows are outputs of an externally benchmarked parameter scan, not fits to the claimed ranges; the halo-profile caveat is a robustness issue, not a circular reduction.

full rationale

The thesis's central claims are parameter-space statements about S3-symmetric and U(1)xU(1)-symmetric 3HDMs. The scalar potentials, minimization conditions, mass-squared matrices, and interaction vertices are derived within the thesis (Chapters 5-7), and the relic-density target is the external Planck measurement. The constraints applied (perturbativity, stability, unitarity, oblique parameters, direct and indirect detection) are standard external benchmarks. The 28.9-41.9 GeV window for C-III-a is presented as an output of a scan with cuts, not as a parameter fitted to that window, so no fitted input is renamed as a prediction. The Preface's explicit concession that 'depending on the dark matter halo distribution profile applied to the indirect dark matter detection constraints, this model may be completely ruled out' identifies an astrophysical modeling sensitivity that sits between the scalar-sector calculation and the viability statement; it is a robustness caveat, not a circular reduction, because the scalar-sector calculation itself does not assume the window. Self-citation is present (the thesis is based on four of the author's papers), but the numerical analyses and mass matrices are substantially reproduced in the thesis itself, and no load-bearing step reduces to an unverified self-citation. No equation is exhibited that equals its input by construction, and no parameter is fitted to the claimed mass ranges. Therefore no significant circularity is found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis is a constrained scan over multi-Higgs-doublet potentials. The free parameters are the couplings of the S3-symmetric potential and the scanned parameter ranges; the claimed mass windows are outputs of these scans under constraints, not closed-form predictions, so the window endpoints inherit the scan choices. The external inputs are the standard freeze-out cosmology, the Planck relic density benchmark, and the halo profile used for indirect detection limits. The thesis introduces no new entities: the dark matter candidates are the model's own neutral scalars, and the C-V light scalars are likewise part of the scalar spectrum.

free parameters (3)

Scalar quartic couplings lambda_i of the S3-symmetric potential = scanned ranges (values in Chs. 5-6)
Model parameters constrained by perturbativity, stability, unitarity, and oblique parameters; the resulting dark matter mass windows are functions of these scan choices.
Bilinear mass parameters mu^2_i of the scalar potential = scanned ranges
Set the scalar spectrum; the claimed 28.9-41.9 GeV window for C-III-a arises from a scan over these parameters rather than a closed-form prediction.
Yukawa representation assignments and vacuum phases = S3 singlet/doublet choices and complex phases
Different quark representations under S3 combined with various vacuum configurations produce the different CP phenomenologies described in the abstract; these are scanned choices, not derived.

assumptions (4)

domain assumption Standard freeze-out thermal relic formalism (Boltzmann equation) determines the dark matter abundance
Invoked in the Preface and Ch. 2 to convert annihilation cross sections into relic densities; assumes no non-thermal production mechanism.
domain assumption The classified vacuum implementations are the relevant global minima of the S3-symmetric potential
Ch. 5 identifies implementations by vacuum structure; if an implementation is only a local minimum, the mass spectrum and dark matter analysis for it would be invalid.
standard math S3 representation theory (character table, tensor products, invariant construction as developed in Ch. 3)
Used to construct the invariant scalar potential and Yukawa structures; standard group theory with no special assumptions.
domain assumption Renormalizable dimension-4 scalar potential and standard CP classification (explicit versus spontaneous)
The entire analysis stays within the renormalizable 3HDM framework; higher-dimension operators and exotic CP definitions are neglected.

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Cite this review

Pith. "Pith review of Dark Matter and CP Violation in Some Symmetry-Constrained 3HDMs." pith.science (2026). https://pith.science/paper/ZFJXZT6R

@misc{pith2026250608197,
  author       = {Pith},
  title        = {Pith review of: Dark Matter and CP Violation in Some Symmetry-Constrained 3HDMs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFJXZT6R}},
  note         = {Machine review of arXiv:2506.08197}
}

abstract

Multi-Higgs-doublet models can accommodate a dark matter candidate. An underlying symmetry not only ensures the candidate's stability but also helps control the number of free parameters. We consider two scenarios. First, we explore dark matter candidates in the $S_3$-symmetric three-Higgs-doublet model. Notably, our findings reveal that the cases we discuss allow for different dark matter mass ranges compared to other candidates within three-Higgs-doublet models. Additionally, we investigate an alternative to conventional stabilisation via discrete symmetries by examining stabilisation through continuous symmetries. This alternative approach exhibits a unique characteristic-the emergence of mass-degenerate states. We also explore the impact of CP violation in the $S_3$-symmetric three-Higgs-doublet model. Our analysis investigates how different representations of quarks under the $S_3$ group, combined with various vacuum configurations, lead to distinct phenomenological consequences, including the possibility of very light neutral scalars.

Figures

Figures reproduced from arXiv: 2506.08197 by the authors.

**Figure 1.1.** In the physical basis of the SM of Particle Physics there are twelve (not in [PITH_FULL_IMAGE:figures/full_fig_p019_1_1.png] view at source ↗

**Figure 1.2.** Interactions between leptons and vector bosons are presented in terms of [PITH_FULL_IMAGE:figures/full_fig_p028_1_2.png] view at source ↗

**Figure 1.3.** The scalar potential for the case of µ 2 > 0 (left) and for the case of µ 2 < 0 (right) with minima specified. Let us consider perturbations around the minimum, which will be given by: ϕ = v + η. (1.7.14) 18 [PITH_FULL_IMAGE:figures/full_fig_p033_1_3.png] view at source ↗

Figures from the paper (33 more)

**Figure 1.4.** Figure 1.4: Cartoon of the complex scalar potential with [PITH_FULL_IMAGE:figures/full_fig_p035_1_4.png]

**Figure 1.5.** Figure 1.5: The SM Higgs boson branching ratios, and their uncertainties, and the total [PITH_FULL_IMAGE:figures/full_fig_p042_1_5.png]

**Figure 1.6.** Figure 1.6: Main production and decay channels of the measured Higgs boson state at [PITH_FULL_IMAGE:figures/full_fig_p044_1_6.png]

**Figure 2.1.** Figure 2.1: The 2018 Planck map of the temeprature anisotropies of the CMB, extracted [PITH_FULL_IMAGE:figures/full_fig_p050_2_1.png]

**Figure 2.2.** Figure 2.2: The primordial abundances of 4He, D, 3He, and 7Li as predicted by the standard model of the BBN—the bands show the 95% CL range [8]. Boxes indicate the observed light element abundances. The narrow blue vertical band indicates the CMB measure of the cosmic baryon den…

**Figure 3.1.** Figure 3.1: A bijective function, f : X → Y , as a permutation of a particular set X into another set Y . For example, f(a) = b. The set of elements in Sn is the set of all possible permutations of n distinct objects. 43 [PITH_FULL_IMAGE:figures/full_fig_p058_3_1.png]

**Figure 3.2.** Figure 3.2: An equilateral triangle with its vertices provided, [PITH_FULL_IMAGE:figures/full_fig_p064_3_2.png]

**Figure 3.3.** Figure 3.3: Depiction of symmetries of an equilateral triangle. [PITH_FULL_IMAGE:figures/full_fig_p065_3_3.png]

**Figure 4.1.** Figure 4.1: A cartoon of the relic density as a function of the DM mass in the IDM. [PITH_FULL_IMAGE:figures/full_fig_p109_4_1.png]

**Figure 5.1.** Figure 5.1: Then, one has to take into account different orderings of the subindices of [PITH_FULL_IMAGE:figures/full_fig_p126_5_1.png]

**Figure 5.2.** Figure 5.2: Scatter plots of vevs in the C-V implementation with [PITH_FULL_IMAGE:figures/full_fig_p140_5_2.png]

**Figure 5.3.** Figure 5.3: Scatter plots of masses that satisfy constraints in the C-V implementation [PITH_FULL_IMAGE:figures/full_fig_p140_5_3.png]

**Figure 6.1.** Figure 6.1: Sketch of allowed DM mass ranges in the IDM and various 3HDMs up to 1 [PITH_FULL_IMAGE:figures/full_fig_p163_6_1.png]

**Figure 6.2.** Figure 6.2: Determining the parameter space of the C-III-a case based on the input of [PITH_FULL_IMAGE:figures/full_fig_p181_6_2.png]

**Figure 6.3.** Figure 6.3: A cartoon showing a general approach to how different implementations are [PITH_FULL_IMAGE:figures/full_fig_p194_6_3.png]

**Figure 6.4.** Figure 6.4: Scatter plots of masses of the R-II-1a (first two columns) and C-III-a (last [PITH_FULL_IMAGE:figures/full_fig_p195_6_4.png]

**Figure 6.5.** Figure 6.5: Constraints on the angles of R-II-1a (top) and C-III-a (bottom) from the [PITH_FULL_IMAGE:figures/full_fig_p196_6_5.png]

**Figure 6.6.** Figure 6.6: Regions in the tan β−mH+ plane that satisfy the B¯ → X(s)γ constraint. Left: A logarithmic scale representation extending to larger values of tan β and mH+ . Right: A linear scale representation focusing on the small tan β region. The yellow region represents a 3-σ t…

**Figure 6.7.** Figure 6.7: Scatter plots of masses that satisfy Cut 1 and Cut 2 constraints. Left: the [PITH_FULL_IMAGE:figures/full_fig_p199_6_7.png]

**Figure 6.8.** Figure 6.8: Exploring the CP properties of the SM-like Higgs boson-fermion couplings in [PITH_FULL_IMAGE:figures/full_fig_p202_6_8.png]

**Figure 6.9.** Figure 6.9: Trilinear self-interactions of the SM-like Higgs boson normalised to the SM [PITH_FULL_IMAGE:figures/full_fig_p203_6_9.png]

**Figure 6.10.** Figure 6.10: Contribution to X annihilation channels at high DM masses. 189 [PITH_FULL_IMAGE:figures/full_fig_p204_6_10.png]

**Figure 6.11.** Figure 6.11: The case of R-II-1a. The absolute value of the trilinear portal coupling [PITH_FULL_IMAGE:figures/full_fig_p205_6_11.png]

**Figure 6.12.** Figure 6.12: Dark matter relic density for the R-II-1a model (first two panels) and the [PITH_FULL_IMAGE:figures/full_fig_p206_6_12.png]

**Figure 6.** Figure 6: adopted from Ref. [29] and re-evaluated in light of the new report from [PITH_FULL_IMAGE:figures/full_fig_p206_6.png]

**Figure 6.13.** Figure 6.13: The spin-independent DM-nucleon cross-section compatible with XENON1T [PITH_FULL_IMAGE:figures/full_fig_p206_6_13.png]

**Figure 6.14.** Figure 6.14: The DM self-annihilation cross-section is plotted as a function of the DM [PITH_FULL_IMAGE:figures/full_fig_p207_6_14.png]

**Figure 6.15.** Figure 6.15: Allowed mass regions of the R-II-1a DM candidate. Blue: relic density and [PITH_FULL_IMAGE:figures/full_fig_p208_6_15.png]

**Figure 6.16.** Figure 6.16: Scatter plots showing mass distributions that comply with constraints. Left [PITH_FULL_IMAGE:figures/full_fig_p209_6_16.png]

**Figure 7.1.** Figure 7.1: The considered 3HDMs follow a hierarchy of symmetry-breaking patterns, [PITH_FULL_IMAGE:figures/full_fig_p212_7_1.png]

**Figure 7.2.** Figure 7.2: Allowed couplings for some U(1)-based 3HDMs. The red terms indicate potential soft symmetry-breaking terms. The underlying symmetry for each case is listed beneath the corresponding block, and the total numbers of bilinear and quartic terms are provided to the right …

**Figure 7.3.** Figure 7.3: Panels (a), (b): relic density as a function of mass for the two DM candidates. [PITH_FULL_IMAGE:figures/full_fig_p252_7_3.png]

**Figure 7.4.** Figure 7.4: Mass scatter plots of parameters satisfying all constraints. Left: masses of [PITH_FULL_IMAGE:figures/full_fig_p253_7_4.png]

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Forward citations

Cited by 1 Pith paper

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

The Hilbert Series and the Flavor Invariants of the 3HDM
hep-th 2026-04 unverdicted novelty 6.0 of 10

The full multigraded Hilbert series of the 3HDM is computed in closed form, and a basis of SU(3)-invariant operators is constructed up to cubic order in the quartic couplings.

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