2HDM+a: Extended Higgs & Dark Matter

• 2HDM+a is an extended Higgs sector model that adds a pseudoscalar mediator to the standard 2HDM to address flavor anomalies, dark matter, and new collider signals.
• It features mixing between the doublet CP-odd state and a singlet pseudoscalar, resulting in unique Yukawa structures and novel decay channels.
• The model’s viability is constrained by collider searches, flavor observables, and cosmological data, guiding future experimental probes.

The Two Higgs Doublet Model plus a Pseudoscalar ("2HDM+a," Editor's term) is an extended Higgs sector framework that supplements the conventional two-Higgs-doublet model (2HDM) with an additional pseudoscalar degree of freedom, typically a real gauge-singlet. This model has been advanced to simultaneously address diverse open questions in particle physics and cosmology including flavor anomalies, dark matter relic abundance, and beyond-Standard Model collider signals. The presence of a new pseudoscalar mixing with the doublet CP-odd Higgs enables nontrivial flavor structure, new decay channels, and dark sector interactions.

1. Model Structure and Scalar Sector

The 2HDM+a extends the scalar content of the 2HDM, whose physical states after electroweak symmetry breaking include two CP-even neutral Higgses (h, H), one CP-odd pseudoscalar (A), and a charged Higgs pair (H±). The model adds a real pseudoscalar singlet ($a$), leading to mixing between the doublet CP-odd state and the new singlet, typically parametrized by a mixing angle θ. The physical scalar spectrum then contains:

• The SM-like Higgs boson h (with mass ≈125 GeV and couplings determined by the alignment limit),
• A heavier CP-even state H,
• Two CP-odd pseudoscalars (A, a), with one predominantly singlet-like,
• Charged scalars H±.

The mixing is controlled via quartic couplings and explicit mass-mixing terms in the scalar potential. In the Type-III 2HDM+a (Liu et al., 2015), the scalar potential and mixing lead to couplings of the form:

$$\mathcal{L}_{\text{dark}} = -y_x (\cos\theta\,a + \sin\theta\,A)\,\bar{X}i\gamma_5 X$$

where $X$ is a Dirac fermion dark matter candidate.

2. Yukawa Structure and Flavor Physics

Yukawa interactions in the 2HDM+a are governed by the underlying 2HDM type (I, II, X, Y) and Natural Flavor Conservation (NFC) implemented by discrete symmetries. The assignment of doublets to up-type quarks, down-type quarks, and leptons determines the scaling factors for the new scalar couplings. In the Type-III scenario (Liu et al., 2015), tree-level flavor violation is present due to unsuppressed non-diagonal Yukawa couplings, which can induce processes such as $h \to \mu\tau$ with branching ratios determined by the flavor-violating parameters (PUT, $p_{\mu\tau}$) and scalar mixing:

$$\operatorname{BR}(h \to \mu\tau) = \frac{m_h}{16\pi\Gamma_h}\left(|y_{h\mu\tau}|^2 + |y_{h\tau\mu}|^2\right)$$

This structure enables simultaneous explanation for LHC observations of lepton flavor violating decays and the muon $g-2$ anomaly via one-loop and Barr–Zee diagrams, subject to constraints from other flavor-changing processes.

3. Dark Matter and Pseudoscalar Mediator

A central feature of 2HDM+a is the ability to accommodate a stable dark matter candidate and mediate dark sector interactions through the new pseudoscalar $a$. After mixing, $a$ gains effective couplings to SM particles. The relic abundance and indirect signals (e.g., the Galactic Center gamma-ray excess) are governed by the annihilation cross section:

$$\sigma v \sim \frac{3\, y_x^2\, p_{bb}^2}{32\pi} \frac{m_\chi^2}{(s - m_a^2)^2 + m_a^2\Gamma_a^2}$$

where $p_{bb}$ is the effective coupling to $b$ quarks. The preferred parameter space for accommodating observed dark matter abundance ($\Omega h^2 \approx 0.12$) and indirect signals includes a light pseudoscalar mediator with $m_a$ in the range 30–95 GeV, a dark matter mass $m_X \approx 30$ GeV, and mixing/decay parameters tuned to evade collider constraints while maintaining sufficient annihilation cross section (Liu et al., 2015).

In symmetry-enforced inert setups (e.g., the IDM (0911.2457), symmetric 2HDM (Bossi et al., 2018)), the dark sector scalars ($H$, $A$, $H^\pm$) carry odd parity under a discrete $Z_2$ symmetry, making the lightest neutral state absolutely stable and an automatic dark matter candidate.

4. Collider Phenomenology and Electroweak Production

The presence of additional neutral and charged Higgs states in 2HDM+a modifies collider signals. Lepton-specific and "flipped" Yukawa structures can enhance leptonic decays and facilitate new search channels:

• Processes such as $pp\to h\to AZ(Z^*)\to b\bar{b} \ell^+\ell^-$ allow detection of light pseudoscalars via lepton tags (Ghosh et al., 21 May 2025).
• Electroweak production mechanisms can dominate for light scalar-pseudoscalar pairs ($hA$), with resonant $Z^*$ enhancement if $m_h + m_A < m_Z$ (Enberg et al., 2017, Enberg et al., 2018).
• Charged Higgs decays ($H^\pm \to W^\pm A$) require careful treatment in the off-shell regime; a full $1\to4$-body approach increases branching ratios in threshold regions, extending experimental reach (Moretti et al., 2023).
• Dedicated analyses using tau-ID algorithms and boosted decision trees demonstrate robust multi-lepton invariant mass peaks that distinguish signal from background in leptophilic scenarios (Hashemi, 2017).

5. Theoretical and Experimental Constraints

Parameter space exploration is bounded by perturbative unitarity, vacuum stability, and compatibility with LHC constraints on Higgs signal strengths, invisible decays, and heavy-scalar searches. Flavor physics observables (e.g., $b\to s\gamma$) set lower bounds on charged Higgs masses, particularly severe in Type-II models [$M_{H^\pm}>800$ GeV; (Arcadi et al., 2023)], while in lepton-specific configurations the bounds are milder.

In the symmetric 2HDM (Bossi et al., 2018), the interchange symmetry and residual $Z_2$ enforce universal Yukawa couplings and stabilize the dark sector, resulting in distinct collider signatures (absence of tree-level fermionic couplings for $H$, $A$, $H^\pm$) and invisible Higgs decays ($h\to HH$).

Electroweak precision tests (S,T,U parameters) and limits from LEP further constrain scalar mass splittings, while phase transition analyses require vacuum stability at finite temperature.

6. Cosmological Implications and Multi-Step Phase Transitions

The 2HDM+a model accommodates multi-step cosmological phase transitions, including scenarios of spontaneous CP violation at high temperature followed by restoration at present (Si et al., 21 Oct 2024). The transitions can be first-order or cross-over, involving nonzero VEVs of CP-even Higgs doublets and temporary VEVs for pseudoscalars. Gravitational wave spectra produced by bubble nucleation and collision have frequencies and amplitudes potentially detectable by planned GW observatories (e.g., U-DECIGO), with the transition strength and spectrum dependent on the sequence and nature of symmetry breaking.

The model also admits axion dark matter via a global PQ symmetry breaking in extensions with two singlet scalars, addressing the strong CP problem and generating the axion-gluon coupling relevant for cosmological axion production (Arcadi et al., 2023).

7. Prospects and Open Questions

The 2HDM+a provides a flexible framework unifying mechanisms for flavor anomalies, dark matter, collider signals, baryogenesis, gravitational wave production, and axion physics. Further systematic studies of parameter correlations—especially addressing fine-tuning resulting from perturbativity and boundedness-from-below—will constrain viable regions. Proposed experimental searches (multi-lepton final states, invisible Higgs decays, photon-photon collider signatures, rare electron and muon transitions, axion haloscopes, and light $Z'$ boson probes) offer routes for testing the rich phenomenology predicted. The interplay between scalar and dark sector parameters, flavor structures, and cosmological dynamics defines the frontier of research in this class of models.