Heavy singlet fermionic dark matter with $Z_4$ symmetry

Imagine the universe is a giant, bustling party. We know most of the guests (the "Standard Model" particles like electrons and protons), but there's a massive, invisible crowd we can't see: Dark Matter. For decades, physicists thought these invisible guests were "WIMPs" (Weakly Interacting Massive Particles)—like shy people who occasionally bump into the visible crowd. But recent experiments have looked hard for these bumps and found nothing. The invisible guests seem even shyer than we thought.

This paper proposes a new way to understand these shy guests. Instead of bumping into us directly, they hang out in a private, VIP lounge (the "secluded sector") that is only loosely connected to the main party.

Here is the breakdown of their theory, using simple analogies:

1. The New Rules of the Party (The Model)

The authors introduce a new rule for the universe called $Z_4$ symmetry. Think of this as a strict bouncer at the VIP lounge door.

The Guest ( $\chi$ ): The Dark Matter candidate is a "Majorana fermion." In our analogy, it's a guest who is their own twin. The bouncer ( $Z_4$ ) says, "You cannot exist alone; you need a partner to enter."
The Key ( $S_0$ ): To get a mass (to become a real "thing"), this guest needs a key. The paper introduces a new invisible scalar particle ( $S_0$ ) that acts as this key. When the universe cooled down, this key was turned, giving the Dark Matter guest its mass.

2. The Two Higgs Bosons (The Doormen)

In this model, there isn't just one "Higgs" (the famous particle that gives things mass). There are two:

$h_1$ : The original Higgs we already know and have seen at the Large Hadron Collider (LHC).
$h_2$ : A new, heavier Higgs boson that lives in the VIP lounge.

These two Higgs bosons are like two doormen who can switch places or blend together. The amount they blend is called the mixing angle ( $\theta$ ).

Small Mixing: The doormen stay mostly separate. The VIP lounge stays very private.
Large Mixing: The doormen blend more, letting the VIPs peek out into the main party.

3. How Dark Matter "Dances" (Annihilation)

In the early universe, Dark Matter particles were dancing around and bumping into each other, disappearing (annihilating) into other particles.

The Old Problem: If they danced with the visible crowd (Standard Model particles), we would have seen them by now in direct detection experiments.
The New Solution: In this "secluded" scenario, the Dark Matter guests mostly dance with each other inside the VIP lounge, turning into pairs of the new heavy Higgs ( $h_2$ ). They rarely dance with the visible crowd.
The Result: Because they stay in their own lane, they don't trigger the alarms (direct detection experiments), but they still manage to leave just the right amount of Dark Matter behind today (the "relic density").

4. The Heavyweights (The Main Discovery)

The authors focused on a specific region: Heavy Dark Matter.

They found that even if the Dark Matter is very heavy (much heavier than a proton), the "mixing angle" (how much the VIPs peek out) doesn't have to be microscopic.
The Analogy: Imagine the VIPs are so heavy that they can't easily jump over the fence to the main party. Because they are so heavy, the fence doesn't need to be locked tight (a tiny mixing angle) to keep them in. A "moderately small" gap is enough.
Why this matters: If the gap is "moderately small" (not zero), we might actually be able to see the new Higgs doorman ( $h_2$ ) at the LHC collider!

5. Hunting for the New Higgs

The paper suggests two ways to catch a glimpse of this secluded Dark Matter at the Large Hadron Collider:

The Invisible Exit: If the new Higgs ( $h_2$ ) decays into Dark Matter, it disappears. We would see a "missing energy" signal—like a magician making a ball vanish, leaving only a gap in the data.
The Visible Flash: If the mixing is slightly larger, the new Higgs might decay into visible particles (like Z bosons) and Dark Matter. This would look like a clean, sharp signal (a resonance peak) in the data, surrounded by a little bit of missing energy.

The Bottom Line

This paper argues that we shouldn't give up on finding Dark Matter just because it's shy. By assuming Dark Matter lives in a secluded "VIP lounge" and is very heavy, we can explain why we haven't found it yet. Furthermore, this setup predicts that the new heavy Higgs boson ( $h_2$ ) might be within reach of our current or future particle colliders, even if the Dark Matter itself remains hidden. The "mixing angle" doesn't need to be zero; it just needs to be small enough to hide the Dark Matter, but big enough to let us see the new Higgs.

Technical Summary: Heavy Singlet Fermionic Dark Matter with $Z_4$ Symmetry

Problem Statement
The nature of dark matter (DM) remains a central mystery in modern physics. While the Weakly Interacting Massive Particle (WIMP) paradigm offers a natural explanation for the observed relic abundance via thermal freeze-out, current null results from direct detection experiments have imposed stringent constraints on the interaction cross-section between DM and Standard Model (SM) particles. This creates a tension: WIMP models typically require large annihilation cross-sections to reproduce the observed relic density, yet direct detection limits force the corresponding couplings to be extremely small. The "secluded DM" scenario is proposed as a solution, where DM annihilates primarily into a hidden sector mediator rather than directly into SM particles, thereby suppressing direct detection signals while maintaining the correct relic abundance.

Methodology
This work revisits a singlet fermionic dark matter model extended by a $Z_4$ symmetry. The model introduces:

A Majorana fermion $\chi$ carrying a $Z_4$ charge, serving as the DM candidate. The $Z_4$ symmetry forbids a bare Majorana mass term for $\chi$ .
A new singlet scalar $S_0$ with a non-zero vacuum expectation value (vev) $v_0$ . Upon spontaneous symmetry breaking (SSB), $S_0$ generates the mass for $\chi$ via a Yukawa interaction ( $m_\chi = y_{sf}v_0$ ).
A mixing between the SM Higgs doublet $H$ and the new scalar $S_0$ , resulting in two physical Higgs mass eigenstates: $h_1$ (identified as the observed SM Higgs) and $h_2$ (a new heavy Higgs).

The study focuses on the "secluded" region where interactions between DM and SM particles are negligible. The authors analyze the model using four free parameters: the new Higgs mass $m_2$ , the Yukawa coupling $y_{sf}$ , the DM mass $m_\chi$ , and the mixing angle $\sin\theta$ .

The methodology involves:

Theoretical Formulation: Deriving the mass matrix and mixing relations for the scalar sector and calculating the DM-nucleon scattering cross-section for direct detection constraints.
Numerical Simulation: Solving the Boltzmann equations for DM abundance using the micrOMEGAs 6.0 package, with the model implemented via FeynRules. The analysis considers the freeze-out mechanism, focusing on the annihilation channel $\chi\chi \to h_2h_2$ .
Parameter Scanning: Performing random scans over the parameter space ( $m_\chi \in [40, 10000]$ GeV, $m_2 \in [200, 2000]$ GeV, $y_{sf} \in [0.001, 3.14]$ , $\sin\theta \in [10^{-6}, 0.2]$ ) to identify regions satisfying the Planck relic density constraint ( $\Omega_{DM}h^2 = 0.1198 \pm 0.0012$ ) and direct detection limits.
Secluded Definition: Defining the secluded scenario by requiring the contribution of the $\chi\chi \to h_2h_2$ channel to the total relic density to be effectively 100% (negligible contribution from SM particle production).

Key Contributions and Results

Dominance of the $h_2$ Channel: The analysis confirms that in the secluded region, DM relic density is dominated by the $t$ -channel annihilation process $\chi\chi \to h_2h_2$ . The cross-section for this process is proportional to $\cos^4\theta$ , making it largely insensitive to small mixing angles.
Mass Hierarchy and Resonance: The study highlights the critical role of the mass hierarchy between $m_\chi$ $m_{χ}$ and $m_2$ $m_{2}$ .
- When $m_\chi < m_2/2$ , SM particle production (e.g., $b\bar{b}$ ) can dominate unless $\sin\theta$ is extremely small.
- When $m_\chi > m_2$ , the $\chi\chi \to h_2h_2$ channel dominates regardless of $\sin\theta$ , provided the process is kinematically allowed.
- Resonance effects occur near $m_\chi \approx m_2/2$ and $m_\chi \approx m_2$ , significantly enhancing the annihilation cross-section.
Viable Parameter Space: The authors identify viable parameter spaces where the model satisfies both relic density and direct detection constraints.
- For heavy DM ( $m_\chi > 2$ TeV), the allowed values for $y_{sf}$ are constrained to a narrow region $[0.2, 1.5]$ .
- The mixing angle $\sin\theta$ is not required to be vanishingly small. The study finds that for TeV-scale DM, moderately small mixing angles of $\sin\theta \sim \mathcal{O}(0.01 - 0.04)$ are permissible in the secluded scenario.
Upper Bounds on Mixing: The upper bound on $\sin\theta$ for a strictly secluded scenario increases with $m_\chi$ . For example, with $m_\chi = 400$ GeV, the bound is $\sin\theta \lesssim 0.003$ ; for $m_\chi = 2500$ GeV, the bound relaxes to $\sin\theta \lesssim 0.042$ .
Collider Prospects: The paper argues that the non-zero mixing angle allows for the production of the new Higgs $h_2$ $h_{2}$ at colliders (e.g., LHC) via gluon-gluon fusion or vector boson fusion, with cross-sections proportional to $\sin^2\theta$ $sin^{2} θ$ .
- If $\sin\theta$ is small and $y_{sf}$ is large, $h_2$ decays invisibly ( $h_2 \to \chi\chi$ ), leading to signatures with large missing transverse energy (MET), potentially probed via associated production like $pp \to h_2 + Z$ .
- If $\sin\theta$ is larger, $h_2$ may be observable via visible decay modes (e.g., $h_2 \to ZZ \to 4l$ ) accompanied by MET from the invisible decay branch.

Significance
The paper claims to demonstrate that the secluded dark matter scenario does not necessitate an extremely small mixing angle between the SM Higgs and the new scalar mediator. Instead, for heavy DM masses, a mixing angle of $\mathcal{O}(0.01 - 0.04)$ is sufficient to suppress direct detection signals while maintaining the correct relic abundance. This finding is significant because it expands the viable parameter space for such models and suggests that secluded DM is potentially accessible to future collider experiments, including the High-Luminosity LHC and proposed $e^+e^-$ colliders, through the search for the new scalar $h_2$ . The work provides a systematic mapping of the parameter space, clarifying the interplay between the DM mass, the new Higgs mass, and the mixing angle in determining the phenomenology of the model.

Heavy singlet fermionic dark matter with Z4Z_4Z4​ symmetry