Inelastic Scalar Dark Matter Model

• Purely inelastic scalar dark matter is defined by two real scalar states where only off-diagonal (inelastic) Higgs portal interactions are present, eliminating tree-level elastic scattering.
• The model achieves the correct relic abundance via thermal coannihilation and supports a strongly first-order electroweak phase transition, potentially generating observable gravitational waves.
• Distinctive collider phenomenology arises from the long-lived excited state, leading to displaced decay signatures that complement astrophysical and cosmological probes.

A purely inelastic scalar dark matter (iDM) model describes a dark sector featuring two real scalar states—most commonly denoted as φ₁ (dark matter) and φ₂ (an excited partner)—where only inelastic transitions mediated by Standard Model (SM) portals are allowed at leading order. This design eliminates tree-level elastic scattering with SM targets, thereby evading stringent bounds from direct detection while maintaining efficient thermal coannihilation and rich phenomenological signatures in both astrophysics and collider experiments. The formulation, phenomenology, and experimental prospects of such models have evolved substantially, with theoretical and phenomenological studies highlighting connections to Higgs portal physics, cosmological phase transitions, and long-lived particle advents at colliders (Guo et al., 18 Aug 2025).

1. Model Architecture and Symmetry Structure

The central structure of a purely inelastic scalar dark matter model begins with a complex scalar field, φ̂, decomposed as

$$\hat{\phi} = \frac{1}{\sqrt{2}} (\hat{\phi}_1 + i \hat{\phi}_2)$$

where φ̂₁ and φ̂₂ are real fields. The interaction Lagrangian after imposing a global U(1) or appropriate discrete symmetry (often softly broken) is

$$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$

where λ_I is complex, and the last term introduces the Higgs portal.

A key requirement is that after symmetry breaking and mass diagonalization, the physical (mass) eigenstates φ₁, φ₂ are rotated mixtures of φ̂₁, φ̂₂ set by an angle θ: $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$ This basis enables tuning of all diagonal scalar-Higgs couplings to zero, ensuring that only off-diagonal, inelastic couplings (φ₁φ₂h and φ₁φ₂hh) survive after electroweak symmetry breaking.

The explicit Higgs portal interaction in the mass basis becomes: $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$ where $$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$.

This configuration ensures that elastic φ₁φ₁h (tree-level) couplings vanish, and consequently φ₁ cannot elastically scatter via Higgs exchange with nucleons at leading order.

2. Thermal History and (Co-)Annihilation

The cosmological viability of the model is determined by the freeze-out dynamics governed by coannihilation. The model assumes both φ₁ and φ₂ are populated in the early Universe and can interconvert via the inelastic interaction.

The effective annihilation cross section entering the Boltzmann equation is

$$\sigma_{\text{eff}} = \frac{g_1^2 \sigma_{11} + 2g_1g_2 (1+\Delta)^{3/2} e^{-x\Delta} \sigma_{12} + g_2^2 (1+\Delta)^3 e^{-2x\Delta} \sigma_{22}} {[g_1 + g_2(1+\Delta)^{3/2} e^{-x\Delta}]^2}$$

where Δ = (m₂ – m₁)/m₁ is the mass splitting, x = m₁/T, and $g_i$ the degrees of freedom (typically both 1). By construction, $\sigma_{11}$ (φ₁φ₁ annihilation) vanishes at tree level, and only inelastic ($\sigma_{12}$) and excited-state ($\sigma_{22}$) processes occur via the Higgs portal. Efficient relic annihilation is thus enabled by φ₁φ₂ and especially φ₂φ₂ annihilations, typically into SM fermion pairs and gauge bosons.

Relic abundance is set by solving

$$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$0

and

$$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$1

with $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$2 determined by $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$3 (fixed by relic abundance for given mass $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$4, mass splitting Δ, and total $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$5).

3. Direct Detection, Elastic Suppression, and Parameter Tuning

The defining feature of the model is the nullification of tree-level φ₁ elastic scattering via the Higgs portal, providing robust evasion of direct detection constraints.

This is achieved by tuning the basis such that the φ₁φ₁h coupling vanishes. The inelastic upscattering φ₁ N → φ₂ N requires the dark matter kinetic energy to exceed the mass splitting Δm—a condition not met for Δm ≳ O(100 MeV), thus suppressing even subdominant inelastic direct detection signals in underground detectors.

Since the elastic channel is forbidden at leading order, and inelastic transitions are kinematically inaccessible at typical nuclear recoil energies, even stringent future experiments such as DARWIN are projected to be insensitive within this parameter regime (Guo et al., 18 Aug 2025). Loop-induced elastic contributions remain subdominant but could become relevant at extreme precision.

4. Strong First Order Phase Transition and Gravitational Wave Production

The Higgs portal interactions not only determine DM freeze-out but also substantially reshape the scalar potential at high temperature. Thermal corrections can induce a multistep structure, enabling a strongly first-order electroweak phase transition.

The critical parameters are:

• $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$6 (vacuum energy released relative to radiation): Evaluated from

$$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$7

• $$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$8 (inverse timescale of nucleation): Defined by

$$\mathcal{L} \supset (\partial_\mu \hat{\phi})^\dagger (\partial^\mu \hat{\phi}) - \frac{1}{2}\mu_1^2 \hat{\phi}_1^2 - \frac{1}{2}\mu_2^2 \hat{\phi}_2^2 - \lambda_\phi (\hat{\phi}^\dagger \hat{\phi})^2 - 2\lambda_I \hat{\phi}^2 |H|^2 + h.c. + V_{SM}(H)$$9

• $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$0 (nucleation temperature), with $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$1.

These parameters determine the gravitational wave energy density spectrum, for example the contribution from sound waves

$$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$2

where $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$3 is the fraction of vacuum energy converted into bulk motion, $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$4 the bubble wall velocity, $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$5 the spectral shape.

Predicted signals for benchmark points (e.g., $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$6 between 142–155 GeV, $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$7–0.2, $$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$8–0.07) peak in the deci- to milli-Hz range, within the reach of future GW interferometers such as U-DECIGO, BBO, and LISA (Guo et al., 18 Aug 2025).

5. Long-Lived Excited State and Collider Phenomenology

The excited state φ₂, separated from φ₁ by a small Δm, is long-lived due to phase space suppression in its decay via an off-shell SM Higgs, primarily to

$$\begin{pmatrix} \phi_1 \ \phi_2 \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \hat{\phi}_1 \ \hat{\phi}_2 \end{pmatrix}$$9

where $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$0 denotes SM fermions (in practice, muons dominate when heavier decay modes are kinematically forbidden).

The proper lifetime of φ₂ can be of order centimeters, making displaced decays a prominent feature. At the HL-LHC, production proceeds via

$$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$1

with a high-p$$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$2 ISR jet for triggering, followed by φ₂ decay inside the detector. Displaced muon-jets (DMJ) and time-delayed methods (TDM) are optimal search strategies, employing requirements on transverse impact parameter ($$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$3 1 mm), radial vertex displacement ($$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$4 30 cm), and timing ($$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$5 0.3 ns).

A favorable parameter window is identified where the production cross-section, displaced muon-jet efficiency, and lifetime result in observable signals without conflicting with cosmological or low-energy constraints.

6. Parameter Space and Complementarity

The key physical parameter set after diagonalization is: $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$6 with $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$7 the ground state mass, $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$8, $$\mathcal{L}_{\text{eff}} \supset - \lambda_{12} \phi_1 \phi_2 (2 v_h h + h^2)$$9 the off-diagonal inelastic Higgs portal coupling, and $$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$0 the dark quartic coupling.

The combined requirements of the correct relic abundance, efficient strongly first-order phase transition (for detectable GWs), and collider signatures delineate an optimal region: for $$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$1 in the 142–155 GeV range, $$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$2 between 0.1–0.2, and modest fractional splitting $$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$3 0.05–0.07, all signatures are maximally accessible (Guo et al., 18 Aug 2025).

This overlap is depicted as:

Parameter	Phenomenological Role	Optimal Range
$$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$4	DM, FOEWPT, collider	142–155 GeV
$$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$5	coannihilation, direct detection	0.05–0.07
$$\lambda_{12} = \mp \frac{1}{2v_h} \sqrt{(\mu_2^2 - \mu_1^2)\lambda_{I,r}}$$6	relic density, phase transition, signals	0.1–0.2

Simultaneous fulfillments of all cosmological, astrophysical, and collider criteria can be achieved within this compact region.

7. Theoretical and Methodological Implications

This model provides a blueprint for constructing scalar dark matter sectors that are robust to the constraints from current and next-generation direct detection experiments by forbidding all elastic leading order couplings to SM fields. The use of complex portal couplings and mass matrix rotation offers a direct realization of purely inelastic interactions.

From a model-building perspective, the internal mass splitting and inelastic interaction structure are essential for enabling thermal freeze-out/coannihilation, rendering the correct abundance, and facilitating strong phase transitions. The lack of tree-level elastic couplings is not an artifact of fine-tuning but an inevitable outcome of symmetry and mixing engineering. Collider, GW, and cosmological probes are therefore inherently complementary for these models.

The signatures—collider LLPs, gravitational waves, and suppressed elastic nuclear scattering—together define a pathway toward either the confirmation or exclusion of the purely inelastic scalar dark matter framework.

Summary

Purely inelastic scalar dark matter models constitute a theoretically economical and phenomenologically rich framework where tree-level elastic interactions with SM states are evaded, leaving only inelastic transitions at leading order. The correct relic abundance is obtained via coannihilation driven by off-diagonal Higgs portal couplings. Strongly first-order electroweak phase transitions are generic, yielding promising prospects for gravitational wave detection, while the long-lived excited state φ₂ precipitates distinctive displaced muon-jet signals at colliders. A well-motivated parameter region exists where all these phenomena are concurrently visible, making purely inelastic scalar dark matter a prime target for a multidomain search strategy encompassing astrophysical, cosmological, and collider observatories (Guo et al., 18 Aug 2025).