Inert Doublet Model: Dark Matter Extension

• Inert Doublet Model (IDM) is a minimal two-Higgs-doublet extension of the SM that incorporates an inert scalar doublet to yield a viable WIMP dark matter candidate.
• The framework enforces rigorous theoretical constraints—vacuum stability, bounded-from-below conditions, and unitarity—with electroweak precision tests restricting inert scalar mass splittings.
• Collider and astrophysical searches, along with relic density and direct detection limits, tightly bound IDM parameters and motivate extensions via dark portal scenarios.

The Inert Doublet Model (IDM) is a minimal, weakly-coupled extension of the Standard Model (SM), in which the scalar sector is enlarged by a second SU(2)$_L$ doublet that is neutral under an exact $\mathbb Z_2$ (or “inert”) symmetry. The preserved $\mathbb Z_2$ symmetry forbids couplings between the inert doublet and SM fermions, ensures the absence of flavor-changing neutral currents, and stabilizes the lightest inert scalar—rendering it a viable Weakly Interacting Massive Particle (WIMP) dark matter candidate. This framework realizes distinctive cosmological, phenomenological, and theoretical features and serves as a testbed for WIMP dark sectors, electroweak symmetry-breaking studies, and collider searches.

1. Model Structure and Scalar Potential

The IDM is defined as a restricted two-Higgs-doublet model (2HDM), with field content:

• $\Phi_1$: SU(2)$_L$ doublet ($Y=1/2$), $\mathbb Z_2$-even; acquires a vacuum expectation value (vev) $v \simeq 246$ GeV and plays the role of the SM Higgs doublet.
• $\Phi_2$: SU(2)$_L$ doublet ($Y=1/2$), $\mathbb Z_2$-odd; does not acquire a vev or Yukawa couplings to SM fermions—“inert.”

The most general, renormalizable, CP-conserving $\mathbb Z_2$-symmetric scalar potential is

$$V = \mu_1^2 |\Phi_1|^2 + \mu_2^2 |\Phi_2|^2 + \frac{\lambda_1}{2} |\Phi_1|^4 + \frac{\lambda_2}{2} |\Phi_2|^4 + \lambda_3 |\Phi_1|^2 |\Phi_2|^2 + \lambda_4|\Phi_1^\dagger\Phi_2|^2 + \frac{\lambda_5}{2}[(\Phi_1^\dagger\Phi_2)^2 + \text{h.c.}]$$

with all parameters chosen real.

After electroweak symmetry breaking, the physical states are:

• $h$: CP-even SM-like Higgs, mass $M_h^2 = \lambda_1 v^2$,
• $H^0$: inert CP-even neutral scalar, mass $M_{H^0}^2 = \mu_2^2 + \frac{1}{2} \lambda_L v^2$ with $\lambda_L = \lambda_3 + \lambda_4 + \lambda_5$,
• $A^0$: inert CP-odd neutral scalar, mass $$M_{A^0}^2 = \mu_2^2 + \frac{1}{2} (\lambda_3+\lambda_4-\lambda_5) v^2$$,
• $H^\pm$: charged inert scalar, mass $M_{H^\pm}^2 = \mu_2^2 + \frac{1}{2} \lambda_3 v^2$.

The IDM parameter basis is typically taken as $$\{M_{H^0}, M_{A^0}, M_{H^\pm}, \lambda_2, \lambda_L\}$$ and the model is defined up to an overall mass scale and physical couplings.

2. Theoretical Constraints and Vacuum Structure

Vacuum Structure

The inert vacuum,

$\langle\Phi_1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}0\v\end{pmatrix},\qquad \langle\Phi_2\rangle = 0,$

is the global minimum if

$$\frac{\mu_1^2}{\sqrt{\lambda_1}} > \frac{\mu_2^2}{\sqrt{\lambda_2}}$$

and all lower minima (with $v_2 \neq 0$) are disfavored.

Bounded-From-Below and Unitarity

Scalar couplings must satisfy positivity and perturbative unitarity constraints: $$\lambda_1 > 0,\; \lambda_2 > 0,\; \lambda_3 + 2\sqrt{\lambda_1\lambda_2}>0,\; \lambda_3 + \lambda_4 - |\lambda_5| + 2\sqrt{\lambda_1\lambda_2}>0$$ and all eigenvalues of the $2\to2$ scalar scattering matrix below $8\pi$. Updated unitarity studies restrict $|\lambda_3|,|\lambda_4|,|\lambda_5|$ significantly ($$|\lambda_3| < 16.33,\; |\lambda_4| < 5.93,\; |\lambda_5| < 8.21$$), with $M_{H^0} \lesssim 600$ GeV as an absolute perturbative bound on the dark scalar masses (Gorczyca et al., 2011).

Electroweak Precision and Collider Bounds

Oblique parameters ($S$, $T$, $U$) restrict mass splittings: typically $$|M_{H^\pm} - M_{A^0}|, |M_{H^\pm} - M_{H^0}| \lesssim 50-100$$ GeV for agreement with experimental $\Delta T$, enforcing near-degeneracy in the inert spectrum for high masses. LEP II searches exclude $M_{A^0} < 100$ GeV, $M_{H^0} < 80$ GeV for $M_{A^0} - M_{H^0} > 8$ GeV (0810.3924).

3. Dark Matter Phenomenology

Thermal Relic Abundance

The lightest inert scalar (typically $H^0$ by convention) is stable and a WIMP candidate. The relic abundance $\Omega_{H^0} h^2$ is governed by

$$\frac{dn_{H^0}}{dt} + 3 H n_{H^0} = -\langle\sigma v\rangle_{H^0 H^0 \rightarrow \text{SM}}\left[n_{H^0}^2 - n_{H^0,\,\text{eq}}^2\right]$$

Key regimes:

• Low mass ($M_{H^0} \lesssim M_W$): Dominated by $H^0 H^0 \to b\bar b$, $\tau^+\tau^-$ via Higgs s-channel exchange. Correct relic density is achieved for $|\lambda_L| \sim 10^{-3} - 10^{-2}$ outside resonance, and $O(10^{-4})$ near the Higgs-funnel ($M_{H^0} \simeq M_h/2$) (Goudelis, 2015, Honorez et al., 2010).
• Intermediate ($M_W < M_{H^0} \lesssim 115$ GeV): Annihilation into $VV$ ($V=W,\,Z$) via gauge interactions dominates. Subleading Higgs-mediated terms are important at large $|\lambda_L|$.
• High mass ($M_{H^0} \gtrsim 500$ GeV): Coannihilation with $A^0$ and $H^\pm$ becomes important; correct relic density exists only for small inert scalar mass splittings, $\lesssim 10$ GeV (Goudelis, 2015).

Inclusion of three-body annihilation channels such as $H^0 H^0 \to WW^*$ is essential, since they dominate in the intermediate regime and significantly suppress the allowed $|\lambda_L|$ and $M_{H^0}$ values (Honorez et al., 2010).

Direct and Indirect Detection

Spin-independent WIMP-nucleon scattering, mediated by Higgs exchange, gives: $$\sigma_{\text{SI}} = \frac{\lambda_L^2 f_N^2 \mu_r^2}{\pi m_h^4}$$ with $f_N \simeq 0.3$ ($\lambda_L$ is the $h H^0 H^0$ coupling), and current limits from XENON1T and LUX require $|\lambda_L| \lesssim 10^{-2}$ for $M_{H^0} \sim 100$ GeV (Goudelis, 2015, Kalinowski et al., 2019). Higgs invisible width bounds imply $|\lambda_L| \lesssim 0.01$ for $M_{H^0} < M_h/2$. Direct detection and relic density constraints together restrict viable parameter space to narrow bands, especially for $M_{H^0} \lesssim 100$ GeV.

Indirect detection, chiefly via $H^0 H^0\to WW, ZZ, b\bar b, t\bar t$, is subdominant for heavy $H^0$; bounds on $\langle\sigma v\rangle$ and gamma rays are consistent with parameter-space scans under current limits.

4. Collider Signatures and Experimental Limits

Hadron Colliders

Main LHC production modes are electroweak Drell–Yan processes: $$pp \to Z^* \to A^0 H^0,\quad pp \to W^* \to H^\pm H^0,\quad pp \to \gamma^*/Z^* \to H^+ H^-$$ Decay chains are $A^0 \to Z^{(*)} H^0$ and $H^\pm \to W^\pm H^0$. The cleanest signature is $$pp \to A^0 H^0 \to \ell^+ \ell^- + E_T^{\text{miss}}$$ (Goudelis, 2015). Cross sections range down from $\sim 1$ pb for light inert scalars, rapidly falling with mass.

LEP II and LHC searches place definitive lower mass bounds. At LHC, recasts of dilepton+$E_T^{\text{miss}}$ supersymmetry searches and invisible Higgs or vector-boson-fusion searches exclude $M_{H^0} \lesssim 55$ GeV for wide parameter regions, leaving only the Higgs-funnel and compressed-spectrum scenarios (Sengupta, 2015, Lahiri et al., 28 Nov 2025). Compressed spectra with $M_{A^0} - M_{H^0} \lesssim 5$ GeV and $M_{H^0} \sim 62.5 - 64$ GeV are viable but challenging to probe.

Lepton and Muon Colliders

At $e^+e^-$ and future muon colliders, pair production via $e^+e^- \to H^+H^-,\,AH$ allows robust tests up to multi-TeV scales; significance is enhanced by clean leptonic final states and permissive cross sections. High-energy muon colliders ($\sqrt{s}=10$ TeV) via vector-boson fusion (VBF) grant access to nearly degenerate spectra well beyond LHC reach (Ghosh et al., 8 Aug 2025, Braathen et al., 20 Nov 2024).

5. Extensions: Axion Sector, Vector-Like Quarks, and High-Scale Completions

PQ-assisted IDM and Two-Component Dark Matter

A compelling next step is to supplement the IDM by a global $U(1)_{PQ}$ Peccei–Quinn symmetry, which is spontaneously broken to yield an axion and a residual $\mathbb Z_2$ that stabilizes $H^0$ (Ghosh et al., 1 Jul 2024). The scalar sector then comprises the standard IDM states plus a PQ scalar $\eta$ and a vector-like quark $\Psi$. The combined $H^0$ (WIMP) + axion scenario allows the dark matter relic to be split between WIMP and axion contributions, thereby populating the previously under-abundant “desert” $M_{H^0} \in [100,550]$ GeV. The PQ sector introduces couplings,

$$\mathcal{L} \supset f_\Psi \eta^* \bar \Psi_L \Psi_R + f\,\bar q_L \Phi_2 \Psi_R + {\rm h.c.}$$

with

$M_\Psi = f_\Psi f_a/\sqrt{2}.$

Vector-like quarks act as a dark portal with distinct LHC signatures: $\Psi \to t H^-$, $b H^0$, $b A^0$ (Ghosh et al., 1 Jul 2024).

Vector-Like Quark Extensions and Relic Density

A simpler extension introduces $Z_2$-odd singlet vector-like quarks $\xi$, opening new coannihilation and $t$-channel diagrams, which re-populate the relic density for heavy $H^0$ and allow much smaller $|\lambda_L|$ to satisfy direct detection limits. Benchmark scenarios illustrate viable IDM+VLQ regions at $M_{H^0} \gtrsim 550$ GeV and $m_\xi \gtrsim 600$ GeV (Das et al., 23 Dec 2024).

Classical Scale-Invariance and Coleman-Weinberg Mechanism

The Coleman–Weinberg mechanism has been embedded into the IDM via introduction of a new hidden sector scalar $\Phi$, dynamically generating all mass scales. Scalar mixing modifies Higgs-portal couplings, both for the relic density and for SI cross sections. The allowed DM strip is pushed to higher $M_{H^0}$ ($\gtrsim 900$ GeV) for fixed quartics (Plascencia, 2015). Stable models up to the Planck scale with vacuum stability and perturbativity can be constructed