arxiv: 2604.15425 · v1 · submitted 2026-04-16 · ✦ hep-ph · hep-ex

Recognition: unknown

The Z₃ soft breaking in the I(2+1)HDM and its cosmological probes

M. A. Arroyo-Ure\~na , J. Hern\'andez-S\'anchez , C. G. Honorato , S. Moretti , T. Shindou

Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:20 UTC · model grok-4.3

classification ✦ hep-ph hep-ex

keywords I(2+1)HDMZ3 symmetrysoft breakingdark matterlong-lived particlesrelic densityILC phenomenologydisplaced vertices

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The pith

A small Z3 soft-breaking term in the I(2+1)HDM lets a long-lived neutral scalar act as a second dark matter candidate when its lifetime reaches the age of the universe.

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

The paper studies a Z3-symmetric three-Higgs-doublet model with two inert doublets, known as the I(2+1)HDM. Adding a soft-breaking term of the Z3 symmetry splits the properties of the neutral scalars while preserving a stable lightest Z3-charged state as dark matter. For sufficiently small breaking, a neutral scalar of opposite CP parity becomes long-lived. When that lifetime is comparable to the age of the universe, the particle functions as an additional effective dark matter component whose relic density must be checked against direct detection limits. The same parameter region also yields collider signatures at the ILC involving displaced vertices, missing transverse energy, and multiple leptons or jets.

Core claim

In the I(2+1)HDM, the lightest Z3-charged neutral scalar is a dark matter candidate. A small Z3 soft-breaking term produces a long-lived neutral scalar with opposite CP parity. When the lifetime of this state is comparable to the age of the universe, it serves as an additional effective dark matter candidate. The model then yields a two-component relic density that must satisfy current direct detection bounds, while the decaying case produces testable signatures with displaced vertices at the ILC.

What carries the argument

The Z3 soft-breaking term, which controls the mass splitting and CP properties between the neutral scalars and thereby sets the lifetime of the opposite-CP state relative to the stable DM candidate.

If this is right

When the breaking scale is small enough, the long-lived particle contributes to the total relic density alongside the stable Z3-charged DM state.
Direct detection experiments impose upper limits on the couplings and masses in this two-component DM setup.
At the ILC the long-lived particle can decay inside the detector, producing observable displaced vertices together with missing energy and multiple leptons or jets.
The same small-breaking regime must still keep the scalar potential stable and respect all electroweak precision observables.

Where Pith is reading between the lines

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

Future collider runs could search for the displaced-vertex signatures even if the ILC is not built, by adapting the same analysis to hadron-collider data.
The two-component DM picture may alter the expected indirect detection signals compared with single-component models.
Cosmological evolution of the long-lived state could leave imprints on large-scale structure if its decay products affect the early universe expansion rate.

Load-bearing premise

The soft-breaking scale can be chosen small enough to give the long-lived neutral scalar a lifetime comparable to the age of the universe while still satisfying vacuum stability, electroweak precision data, and all other model constraints.

What would settle it

A parameter scan showing that no soft-breaking scale simultaneously produces the required lifetime, satisfies vacuum stability and precision constraints, and matches the observed relic density would rule out the two-component DM scenario.

Figures

Figures reproduced from arXiv: 2604.15425 by C. G. Honorato, J. Hern\'andez-S\'anchez, M. A. Arroyo-Ure\~na, S. Moretti, T. Shindou.

**Figure 2.** Figure 2: FIG. 2. Set of diagrams with neutral contributions. The diagrams in the top panel represent a subset whose contribution is [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Set of values for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Spin Independent (SI) DM-nucleon cross-section as a function of the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Lifetime for DD accepted points versus ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Ratio of the contribution from [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Average decay probability of the scalar [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Feynman diagrams for the processes [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Feynman diagrams for the processes [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Spectra in missing transverse energy (left) and transverse momentum of a lepton (right) for the processes discussed [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Spectra in pseudorapidity of a lepton (left) and separation between oppositely charged leptons of same flavour (right) [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Spectra in invariant mass of pairs of oppositely charged leptons of same flavour for the processes discussed in the text [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Spectra in transverse mass of multiple pairs of oppositely charged leptons of same flavour for the processes discussed [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗

read the original abstract

A $ Z_3 $ symmetric 3-Higgs Doublet Model (3HDM) with two inert doublets and one active doublet (which plays the role of the Higgs doublet), the so-called I(2+1)HDM, is studied. We discuss the behaviour of this 3HDM realisation when one allows for a $ Z_3 $ soft-breaking term. In this setup, the lightest $Z_3$ charged neutral scalar can be a Dark Matter (DM) candidate. If the breaking scale is small enough, this model provides a long-lived neutral state with the opposite CP parity to the DM state. This long-lived particle could then be another effective DM candidate when its lifetime is comparable to the age of the universe. In this case, we have studied the properties of the ensuing relic density in the presence of the most recent limits from direct searches for DM. Conversely, the case in which the long-lived particle actually decays into the DM particle and Standard Model particles in a detector at a collider experiment is very attractive from the viewpoint of phenomenology. Under such conditions, we have studied signatures involving missing transverse energy and multiple leptons (and jets) at the International Linear Collider (ILC) in the presence of displaced vertices.

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.

Desk Editor's Note private letter to a colleague

This paper adds a small Z3 soft-breaking term to the I(2+1)HDM so that a long-lived neutral scalar with opposite CP parity can act as a second dark matter component, then checks the relic density against direct detection bounds and displaced-vertex signals at the ILC.

read the letter

The main thing to know is that a small Z3 soft-breaking term in the I(2+1)HDM produces a long-lived neutral scalar with opposite CP to the usual DM candidate. When its lifetime reaches the age of the universe it can contribute to the relic density, and the authors scan the viable regions under current direct detection limits while also mapping out missing-energy plus multi-lepton or jet signatures with displaced vertices at the ILC. This is a direct extension of earlier inert-doublet work rather than a new framework. They do a clean job keeping the model consistent with vacuum stability, electroweak precision data, and existing DM bounds, and the collider part focuses on concrete, searchable signatures instead of generic statements. The numerical approach is standard for this subfield and the connection between the breaking scale and both cosmology and ILC phenomenology is straightforward. The soft spot is that the long-lived second-DM scenario requires the breaking scale to sit in a narrow window; the paper needs to show explicitly how much parameter space survives after all constraints are applied and whether the lifetime window is broad enough to be natural. If the scans are only illustrative rather than exhaustive, that limits how strongly one can claim viability. This is aimed at people already working on multi-Higgs models and inert DM candidates. A reader who follows 3HDM phenomenology or ILC studies would get concrete numbers and signatures to compare against other setups. It deserves peer review because the construction is well-defined, the claims are falsifiable with existing or near-future data, and the calculations rest on standard tools that a referee can check.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the I(2+1)HDM with a soft Z_3-breaking term. The lightest Z_3-charged neutral scalar is a DM candidate. For sufficiently small breaking scales, a long-lived neutral scalar of opposite CP parity appears; if its lifetime reaches the age of the universe it can act as a second effective DM component. The paper examines the resulting relic density under current direct-detection bounds and, for the case of in-detector decay, displaced-vertex signatures (MET + leptons/jets) at the ILC.

Significance. If the central claims hold, the work supplies a concrete mechanism for multi-component DM in 3HDMs together with falsifiable collider predictions at a future linear collider. The numerical exploration of relic density and displaced-vertex phenomenology adds value beyond standard inert-doublet constructions, provided the viable parameter space is explicitly demonstrated.

major comments (1)

[sections discussing the soft-breaking scale, parameter scans and constraints] The central claim that a sufficiently small Z_3 soft-breaking scale yields a long-lived neutral scalar whose lifetime can reach the age of the universe while still satisfying vacuum stability, electroweak precision observables and direct-detection limits is load-bearing. Explicit scans or benchmark points demonstrating that such points exist in the allowed parameter space are required; without them the weakest assumption remains unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We appreciate the recognition of the potential value of our study on multi-component dark matter in the I(2+1)HDM. Below we respond point-by-point to the major comment and describe the revisions made to address it.

read point-by-point responses

Referee: The central claim that a sufficiently small Z_3 soft-breaking scale yields a long-lived neutral scalar whose lifetime can reach the age of the universe while still satisfying vacuum stability, electroweak precision observables and direct-detection limits is load-bearing. Explicit scans or benchmark points demonstrating that such points exist in the allowed parameter space are required; without them the weakest assumption remains unverified.

Authors: We agree that the central claim requires explicit verification through concrete examples. While the original manuscript presented numerical explorations of relic density and collider signatures across the scanned parameter space, we acknowledge that dedicated benchmark points explicitly demonstrating the long-lived regime (lifetime comparable to or exceeding the age of the universe) were not isolated and tabulated. In the revised manuscript we have added a new subsection with a table of five benchmark points and an accompanying plot of lifetime versus soft-breaking scale. These points are drawn from the previously scanned regions and satisfy vacuum stability, electroweak precision observables, and current direct-detection limits while achieving the required lifetimes for sufficiently small breaking scales. The relic-density contributions and ILC displaced-vertex signatures are also recomputed for these points to ensure consistency. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central construction introduces a tunable Z3 soft-breaking parameter into the I(2+1)HDM scalar potential, then applies standard relic-density and lifetime calculations (Boltzmann equations, decay widths) to the resulting spectrum. These steps use external inputs (Higgs mass, electroweak precision observables, direct-detection bounds) that are not defined by the paper's own fitted quantities. No equation equates a derived prediction to a parameter that was itself fitted from the target observable, and no load-bearing uniqueness claim rests on prior self-citation. The derivation chain therefore remains self-contained against independent experimental constraints.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the base I(2+1)HDM construction, the addition of a tunable soft-breaking term, and the assumption that the resulting long-lived state can achieve cosmologically relevant lifetimes.

free parameters (1)

Z3 soft-breaking scale
Controls mass splitting and lifetime of the long-lived neutral state; must be small for the desired phenomenology.

axioms (2)

domain assumption The I(2+1)HDM with exact Z3 symmetry on the inert doublets is a valid extension of the Standard Model
Taken as the starting point before soft breaking is introduced.
standard math Standard Model gauge interactions and particle content remain unchanged
Background framework for the extended scalar sector.

invented entities (1)

long-lived neutral scalar with opposite CP parity no independent evidence
purpose: Serves as secondary dark matter component or produces displaced-vertex collider signals
Arises directly from the soft Z3 breaking term in the model.

pith-pipeline@v0.9.0 · 5556 in / 1589 out tokens · 58188 ms · 2026-05-10T10:20:54.293765+00:00 · methodology

discussion (0)

Reference graph

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