Dark Higgs-strahlung in Hidden Sector Physics

• Dark Higgs-strahlung is defined as the process in which a dark Higgs boson is radiated from a dark photon, initiating spontaneous dark symmetry breaking.
• This process yields unique experimental signatures such as displaced vertices, missing energy, and altered resonance features observable at colliders.
• Its investigation combines fixed-order and resummed calculations to refine constraints on dark gauge coupling, mixing parameters, and the dark sector mass spectrum.

Dark Higgs-strahlung refers to processes in which a Higgs boson or a dark photon radiates a dark Higgs boson, with the dark Higgs being responsible for spontaneous breaking of a dark gauge symmetry and mass generation in the hidden sector. The phenomenology of dark Higgs-strahlung arises both in direct production at colliders and in dark matter annihilation, and it generates distinctive experimental signatures—often involving displaced vertices, missing energy, and non-standard resonance structures—providing unique access to the mass generation mechanism and the couplings of the dark sector.

1. Theoretical Frameworks and Mechanisms

Dark Higgs-strahlung is grounded in models where a hidden sector with its own gauge group (often U(1)′) is broken via a dark Higgs mechanism. The resulting spectrum contains a dark photon (A′) with mass $m_{A′}=g′v$, and a dark Higgs (ϕ) with mass $m_ϕ = \sqrt{λ/2} \, v$, where $v$ is the vacuum expectation value (vev) of the dark Higgs field and $λ$ the quartic coupling (Acanfora et al., 12 Aug 2025, Li et al., 25 Jun 2025, Weihs et al., 2011).

Schematically, dark Higgs-strahlung can proceed via:

• e⁺e⁻ → γ A′(→χχ) [+ ϕ radiation]: At e⁺e⁻ colliders, the photon-associated production of a dark photon (with invisible decay modes) is altered by the emission of a dark Higgs from A′, leading to e⁺e⁻ → γ A′s (Li et al., 25 Jun 2025).
• e⁺e⁻ → A′* → ϕ χχ: Off-shell dark photon production with subsequent ϕ emission yields final states containing a (long-lived) dark Higgs and missing energy (Acanfora et al., 12 Aug 2025).
• e⁺e⁻ → γ A′* → γϕχχ: The addition of initial-state radiation (ISR), producing a photon and dark Higgs-strahlung via an off-shell dark photon.
• Dark matter annihilation: Processes like χχ → ϕ ϕ, or χχ → f̄ f ϕ (via s- or t-channel mediators), relevant for indirect detection and cosmic ray signals (Luo et al., 2013, Kumar et al., 2016).

The underlying Lagrangian after symmetry breaking generically includes a dark sector Yukawa piece and A′–ϕ couplings:

$$\mathcal{L}_{\rm dark} \supset g′^2 v\, A′_μ A′^μ \, ϕ + \text{(dark gauge and kinetic terms)}$$

This vertex enables dark Higgs emission from a dark photon leg in production or decay processes, directly linking observable rates and kinematics to the mechanism of mass generation in the hidden sector.

2. Collider Phenomenology: Experimental Signatures

At electron–positron colliders such as Belle and Belle II, dark Higgs-strahlung can be exploited via distinctive search strategies:

• Mono-photon + missing energy: The standard signature for invisible dark photon production, e⁺e⁻ → γ A′, is modified by the emission of a dark Higgs, leading to e⁺e⁻ → γ A′s. Both the dark photon and dark Higgs are assumed to decay invisibly (or outside the detector), resulting in a continuum in the missing mass distribution and an increase in event yield (Li et al., 25 Jun 2025). The dark Higgs emission causes the missing mass squared Mₓ² spectrum to develop a tail for Mₓ² > m_{A′}².
• Visible displaced vertices with missing energy: If the dark Higgs is the lightest dark sector state, it may have a long lifetime and decay to SM particles via mixing, producing a displaced vertex within the detector. The processes e⁺e⁻ → A′* → ϕχχ or e⁺e⁻ → γA′* → γϕχχ generate a visible two-track displaced vertex (from ϕ → μ⁺μ⁻ or ππ) with associated missing energy, a signature with exceptionally low Standard Model backgrounds (Acanfora et al., 12 Aug 2025).

Specific kinematic features are highly discriminating:

Observable	Channel	Feature in Presence of Dark Higgs-strahlung
Missing mass	γ + invisible	Peak at m_{A′}, then tail for Mₓ > m_{A′}
Photon energy	γϕ + missing	Peaks at “hard” Eγ; possible bimodality
Invariant mass	Displaced tracks	Reconstructs m_{ϕ}
Displaced vertex	ϕ → SM	At mm–cm scale, depending on mixing angle

These signatures can be efficiently targeted using vertexing, kinematic cuts, and photon measurements. In the ϕ and γϕ channels at Belle II, resonance structures in the missing mass and photon energy distributions further isolate signal from background (Acanfora et al., 12 Aug 2025). The analyses exploit dedicated variable combinations, for example

$v = \frac{E_γ - E̅}{E₀ - E̅} \, \frac{m_{χχ}^2 - m_{A′}^2}{m_{ϕ}^2 - m_{A′}^2}$

with v ≈ 0 for signal, where $E̅$ ($E₀$) is the energy corresponding to the hard (soft) photon solution.

3. Quantitative Impact on Searches and Interpretation

Dark Higgs-strahlung modifies both the total event yield and the shape of experimental distributions. For mono-photon searches (e.g., BaBar (Li et al., 25 Jun 2025)), inclusion of the e⁺e⁻ → γ A′s channel increases the integrated signal cross section (σ_signal), while the kinematic distribution of Mₓ² broadens and shifts to higher values. Careful treatment is necessary:

• For m_{A′} + m_{ϕ} ~ √s, a fixed-order computation suffices: the correction factor S_cxs = σ_no-FSR / σ_signal decreases the inferred limit on the kinetic mixing parameter ε by ~10% for low masses (m < 0.1 GeV).
• For very light dark sector masses (m ≪ √s), the collinear enhancement requires a dark sector shower algorithm, merging fixed-order and resummed contributions.
• The broadening of the Mₓ² spectrum (encoded by S_fit) slightly reduces sensitivity, but the overall effect is dominated by the cross section increase; net improvement in limits can be up to 3% tightening in ε for low masses, with near cancellation in the multi-GeV region.

For displaced vertex plus missing energy signatures, the expected background is negligible, allowing sensitivity scaling nearly linearly with integrated luminosity. At Belle II, the anticipated large dataset and advanced vertexing technology permit detailed mapping of the dark sector parameter space, including accessible range for mixing angle θ and dark Higgs mass m_{ϕ} (Acanfora et al., 12 Aug 2025, Jaegle, 2012). The possibility to simultaneously detect visible and invisible channels (via ϕ → visible and A′ → invisible decays) provides complementary coverage.

4. Connections to Mass Generation and Dark Sector Structure

Dark Higgs-strahlung provides a direct probe of the mass generation mechanism in the dark sector:

• The presence and couplings of the dark Higgs are required for dark photon mass via spontaneous symmetry breaking, as $m_{A′} = q_ϕ g′ w$ (Acanfora et al., 12 Aug 2025, Li et al., 25 Jun 2025).
• The observation of processes such as e⁺e⁻ → ϕχχ or γϕχχ gives empirical evidence for the underlying Higgs mechanism in the dark sector, analogous to the SM Higgs mechanism.
• Accessing both the dark photon and dark Higgs masses and decay widths enables simultaneous determination of the dark gauge coupling, Higgs vev, and the mixing parameters.

By studying both visible and invisible decay modes, as well as resonance structures in kinematic variables, one can reconstruct the hidden sector spectrum: $m_{A′}$, $m_{ϕ}$, $g′$, and their ratios. Constraints on the mixing parameters θ (for Higgs mixing) and ε (for kinetic mixing) can be improved by orders of magnitude over existing limits, especially for low mass and low coupling regimes.

5. Broader Implications for Dark Matter and Indirect Detection

Dark Higgs-strahlung processes are critical in various dark matter settings:

• In scenarios where the dark Higgs is the lightest dark sector particle, it may itself constitute dark matter if sufficiently long-lived (Mondino et al., 2020). Its relic abundance can be set via freeze-in or freeze-out, mediated through suppressed kinetic mixing and dark gauge coupling. The lifetime can naturally exceed the age of the Universe if $m_{ϕ} \leq m_{A′}$ and ε, $g′$ are very small.
• Dark Higgs-strahlung in dark matter annihilation, e.g., χχ → f̄ f ϕ (s-channel or t-channel) (Luo et al., 2013, Kumar et al., 2016), lifts chirality suppression for Majorana or Dirac dark matter, enabling s-wave annihilation and modifying the indirect detection signatures in gamma rays, cosmic rays, and neutrinos.
• Self-interactions of dark Higgs dark matter can resolve small-scale structure issues, with cross sections per mass potentially in the range of astrophysical interest (Mondino et al., 2020).

6. Comparative Perspectives and Experimental Strategies

A comparison with standard dark photon searches reveals the unique power of dark Higgs-strahlung channels:

Search Mode	Standard A′ Decay	Dark Higgs-strahlung	Advantage
Fully visible (e.g., μ⁺μ⁻)	Resonance in dilepton	Displaced vertex / resonance	Background suppression, new kinematics
Fully invisible	Mono-photon	Flatter missing mass shape	Sensitivity to mass generation, broadening
Semi-visible (prompt + missing)	Not present	Displaced + missing energy	Dual handle from displaced vertex, MET

The presence of highly unusual signals—displaced decays with or without ISR photons, sharp resonance features in reconstructed variables, and modified spectrum shapes—substantially enhances signal-background separation and provides robust coverage of new physics parameter space. Initial studies indicate that with anticipated Belle II luminosities, regions in coupling-mass parameter space previously unconstrained can now be benchmarked or excluded (Acanfora et al., 12 Aug 2025, Jaegle, 2012).

Future directions include dedicated triggers for displaced vertices, optimization of angular and energy cuts for ISR photons, and combined analysis of multiple channels to improve reconstruction of dark sector couplings and mass scales.

7. Summary

Dark Higgs-strahlung—defined as the emission of a dark Higgs boson from a dark photon or Higgs parent—serves as a uniquely incisive probe of hidden sector dynamics, the nature of dark gauge symmetry breaking, and the structure of dark mass spectra. It is characterized by cross section enhancement and modified kinematic distributions in missing mass and event topologies that escape traditional dark photon search methodologies. Its role is pronounced both in collider environments (notably at Belle II and future e⁺e⁻ machines) and in shaping the phenomenology of indirect dark matter detection, influencing observational strategies across a wide range of experimental platforms (Acanfora et al., 12 Aug 2025, Li et al., 25 Jun 2025, Mondino et al., 2020, Luo et al., 2013, Kumar et al., 2016, Jaegle, 2012). The systematic paper of these channels will remain central to resolving fundamental questions about the dark sector and its connections to Standard Model physics.