The $Z_3$ soft breaking in the I(2+1)HDM and its cosmological probes

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.

• The paper shows that introducing a soft Z3 breaking term in the I(2+1)HDM splits the inert scalar masses, resolving dark matter candidate degeneracy.
• Detailed parameter scans and relic density calculations demonstrate that both single- and two-component dark matter scenarios can satisfy stringent detection constraints.
• Loop-induced A1 decays produce distinctive collider signatures, including displaced vertices and multi-lepton events, which offer promising experimental tests.

Theoretical Framework and Motivation

The paper "The $Z_3$ soft breaking in the I(2+1)HDM and its cosmological probes" (2604.15425) investigates a specific extension of the Standard Model (SM) Higgs sector: a three-Higgs-doublet scenario known as the I(2+1)HDM, containing two inert doublets and one active Higgs doublet. Imposing a $Z_3$ discrete symmetry ensures stability of neutral scalars from the inert doublets, providing viable dark matter (DM) candidates. The study focuses on the consequences of introducing a soft breaking term for the $Z_3$ symmetry, which generates distinctive phenomenology in both cosmological and collider contexts.

The motivation is grounded in the well-recognized limitations of the SM to explain dark matter, baryogenesis, and neutrino masses, with extended scalar sectors offering solutions without violating experimental constraints such as $\rho = 1$ at tree level. The $Z_3$ symmetry classifies the three doublets by their charges and, in its exact form, leads to a "Hermaphrodite DM" scenario: two degenerate DM candidates of opposite CP parities. However, degeneracy complicates experimental identification, and direct detection limits are stringent due to unsuppressed $ZH_1A_1$ couplings. The soft breaking of $Z_3$ removes this degeneracy and allows for only one stable DM candidate (or two, if the heavier CP-odd scalar is sufficiently long-lived), relaxing constraints and enabling unique collider signatures.

Model Structure and Parameter Constraints

The scalar potential consists of $Z_3$-symmetric terms and a soft-breaking term of the form $-\mu_{12}^2 (\phi_1^\dagger \phi_2) + {\rm h.c.}$ This term, localized in the inert sector, introduces explicit mass splittings between the CP-even ($H_1$) and CP-odd ($Z_3$0) scalars. The physical eigenstates' mass matrices and couplings, including mixing angles and vertices relevant for DM annihilation and scattering, are explicitly parameterized. The paper details the constraints imposed by perturbativity, unitarity, boundedness from below, LEP and LHC searches, as well as direct and indirect DM detection data.

Notably, the coupling $Z_3$1 vanishes at tree-level by enforcing $Z_3$2, a condition leading to $Z_3$3. This nullifies dangerous direct detection rates, with loop-induced effects parametrized and evaluated. The scan of parameter space yields viable DM scenarios only in regions with small $Z_3$4 and narrow mass splittings, tightly respecting current DD and ID bounds.

Dark Matter Phenomenology and Relic Density

The analysis distinguishes between two regimes:

1. Two-component DM scenario: For $Z_3$5 GeV$Z_3$6, both $Z_3$7 and $Z_3$8 are cosmologically stable (or $Z_3$9 is effectively stable due to its lifetime exceeding the universe's age), contributing to the relic density.
2. Single-component DM with unstable $Z_3$0: For $Z_3$1 of a few GeV$Z_3$2, $Z_3$3 undergoes slow decay to $Z_3$4 and SM particles via loop-induced vertices.

Relic density calculations utilize micrOMEGAs, incorporating all relevant annihilation and co-annihilation channels, including Higgs-portal and gauge interactions. The paper demonstrates that viable parameter points exist satisfying Planck satellite constraints, with the relic density saturating at $Z_3$5 or shared by $Z_3$6 and $Z_3$7 depending on $Z_3$8 and the mass splitting $Z_3$9.

Figure 1: Ratio of the contribution from $\rho = 1$0 and $\rho = 1$1 to the total relic density, illustrating two-component and single-component DM regimes as controlled by the lifetime of $\rho = 1$2.

Comprehensive parameter scans highlight that only extremely small mass splittings avoid DD bounds, and larger values rapidly suppress $\rho = 1$3 relic density. The dependence of lifetime on $\rho = 1$4 is directly calculated, with BPs provided for each scenario.

Loop-Induced Decay and Collider Signatures

A distinctive element is the loop-induced decay of $\rho = 1$5, mediated by $\rho = 1$6 and other neutral scalar loops. The decay width is computed using FeynCalc and LoopTools, producing explicit Passarino-Veltman expressions for the effective coupling. This decay can yield observable displaced vertices at colliders, introducing phenomenological opportunities for scenarios where $\rho = 1$7 is unstable but long-lived.

Figure 2: Average decay probability of the scalar $\rho = 1$8 for the three scenarios (A, B, and C); continuous and dashed lines correspond to different detector regions relevant for displaced vertex searches.

Numerical simulations at the International Linear Collider (ILC) are conducted for representative BPs, incorporating realistic beam polarization and luminosity. Processes such as $\rho = 1$9 and $Z_3$0 are calculated at parton level, including both direct production and secondary decays of $Z_3$1.

Figure 3: Feynman diagrams for the processes $Z_3$2, illustrating dominant production and decay channels leading to multi-lepton final states.

Figure 4: Spectra in missing transverse energy (left) and lepton transverse momentum (right), demonstrating characteristic kinematic features for signal separation.

Figure 5: Spectra in lepton pseudorapidity (left) and separation between oppositely charged leptons (right), providing handles for event selection at the ILC.

Figure 6: Invariant mass of pairs of oppositely charged leptons, directly probing the mass spectrum of the inert scalar sector.

Figure 7: Transverse mass spectra for multiple lepton pairs, further characterizing signal topology and underlying mass structure.

The predicted cross-sections for exotic signatures (multi-leptons plus $Z_3$3 with displaced vertices) are quantified. The paper asserts that these are distinctive and potentially accessible at future lepton colliders, contingent on $Z_3$4 and the underlying mass spectrum.

Implications and Outlook

The study delineates a framework wherein the I(2+1)HDM with soft $Z_3$5 breaking can yield both single- and multi-component DM scenarios, subject to experimentally viable parameter choices. The phenomenology is exceptionally rich, with collider signals ranging from invisible production to displaced vertices in multi-lepton final states. Constraints from direct and indirect detection, cosmological relic abundance, and collider searches are synergistically respected.

Bold claims include the potential explanation for DM as multi-component, the compatibility with current DD constraints via loop-induced suppression of dangerous couplings, and the possibility to probe the model at future colliders through unique signatures not accessible in traditional single-doublet or $Z_3$6-protected scalar DM models.

The theoretical implication is that discrete symmetry breaking in extended scalar sectors naturally accommodates both cosmological stability and collider accessibility, offering a robust pathway for DM model-building. Practically, the interplay between mass splittings, loop-induced couplings, and lifetime calculations provides guidance for designing next-generation DM search strategies.

Looking ahead, further detailed studies on cosmological impact (e.g., Big Bang Nucleosynthesis, structure formation) and comprehensive signal-to-background analyses at colliders will be necessary. The methodological advances in parameter scanning and loop calculation are broadly applicable to other BSM scenarios with extended scalar sectors and non-trivial symmetry properties.

Conclusion

This paper systematically explores the consequences of $Z_3$7 soft breaking in a three-Higgs-doublet scenario with two inert doublets. It demonstrates that the resulting scalar spectrum allows for both cosmologically viable multi-component and single-component DM, with direct and indirect detection constraints satisfied via loop-level effects. The model predicts distinctive collider signatures characterized by displaced vertices and multi-lepton final states, potentially accessible at the ILC or similar facilities. The theoretical and practical framework established in this study enhances the landscape of scalar DM models and provides clear guidance for future experimental and theoretical directions.