WIMP Dark Matter within the dark photon portal

This paper investigates the viability of Dirac fermion and complex scalar dark matter up to 1 TeV mass within the dark photon portal framework by deriving thermal relic constraints and comparing predicted spin-independent scattering cross-sections against direct detection limits to identify allowed parameter space.

The Gist

The Big Picture: The Invisible Room and the Secret Door

Imagine the universe is a giant house. We can see the furniture, the lights, and the people (this is Normal Matter). But we know there is a whole other room attached to the house that we can't see, touch, or hear. This is the Dark Sector, filled with Dark Matter.

The big question in physics is: How do we talk to the people in that invisible room?

This paper explores a specific "secret door" between the two rooms called the Dark Photon. It's a hypothetical particle that acts like a bridge, allowing normal matter and dark matter to interact. The authors are testing two types of "guests" in the dark room:

1. Dirac Fermions: Think of these as heavy, solid bowling balls (particles with mass and spin).
2. Complex Scalars: Think of these as spinning tops or clouds (particles without spin).

They are checking if these guests can exist without breaking the rules of the universe.

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The Rules of the Game

To see if this "secret door" theory works, the scientists have to satisfy three very strict rules:

1. The "Goldilocks" Rule (Relic Density)

In the early universe, dark matter was created in huge amounts. As the universe cooled, most of it annihilated (destroyed itself) with its partner.

• The Rule: We need just enough dark matter left over today to match what we see in the universe (about 27% of everything).
• The Analogy: Imagine a party where people are leaving in pairs. If the "exit door" (annihilation) is too wide, everyone leaves, and the room is empty (too little dark matter). If the door is too narrow, the room gets overcrowded (too much dark matter). The scientists are trying to find the perfect door size so the party ends with exactly the right number of people left.

2. The "Silent Neighbor" Rule (Direct Detection)

We have giant, super-sensitive microphones (detectors like XENON and LZ) buried underground listening for dark matter bumping into atoms.

• The Rule: If dark matter is too "loud" (interacts too strongly with normal matter), our microphones would have heard it by now. Since we haven't heard anything, the interaction must be very quiet.
• The Twist: In this specific model, the dark matter interacts with protons (positive charge) but barely interacts with neutrons (neutral charge). It's like a ghost that only bumps into the furniture but not the walls. This means the "loudness" limits are actually a bit more lenient than we thought, but still very strict.

3. The "Precision Clock" Rule (Electroweak Constraints)

The universe has a very precise clock (electroweak physics) that measures how heavy the Z-boson (a known particle) is and how it behaves.

• The Rule: If our "secret door" (Dark Photon) is too big or too strong, it messes up the clock. The door must be small enough that the clock doesn't notice it's there.

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The Two Types of Guests

The paper tests two scenarios to see which one can survive these rules.

Scenario A: The Bowling Balls (Dirac Fermions)

• The Problem: These heavy particles are very eager to interact. To get the "Goldilocks" amount of dark matter left over, they need to interact strongly. But if they interact strongly, they become too "loud" for the underground microphones.
• The Result: It's almost impossible for these bowling balls to exist in this model. They are either too heavy (overcrowding the universe) or too loud (getting caught by detectors).
• The Exception: There is a tiny "sweet spot" called a Resonance. Imagine a swing. If you push it at exactly the right rhythm, it goes very high with very little effort. If the dark matter mass is exactly half the mass of the Z-boson (the "swing"), the interaction becomes efficient enough to clear the room without being too loud. But even then, the allowed area is tiny.

Scenario B: The Spinning Tops (Complex Scalars)

• The Advantage: These particles are naturally quieter. They don't interact as aggressively as the bowling balls.
• The Result: They have a much better chance of surviving the rules.
  • If the "secret door" is far from the resonance (the swing), they still struggle to clear the room fast enough.
  • However, if they are near the Resonance (the sweet spot), they can clear the room perfectly without being too loud.
• The Good News: For these spinning tops, there is a much wider range of "door sizes" (masses) that work. Specifically, a dark photon mass between 2 and 4 GeV (a specific weight) seems very promising.

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The "Z-Boson" Surprise

One of the key findings in this paper is that the Z-boson (a known particle from the Standard Model) isn't just sitting on the sidelines. Because of the mixing between the Dark Photon and the Z-boson, the Z-boson actually helps the dark matter annihilate.

• Analogy: Imagine you are trying to empty a room. Usually, you use one door (the Dark Photon). But in this model, a second door (the Z-boson) opens up right next to it. This makes it much easier to get the right number of people out, especially when the "swing" (resonance) is involved.

The Conclusion: What Does This Mean?

1. The "Heavy" Theory is Dead: If dark matter is made of heavy "bowling balls" (Dirac fermions) interacting via this portal, it's likely ruled out by current experiments. They are too loud.
2. The "Light" Theory is Alive: If dark matter is made of "spinning tops" (Complex Scalars), there is still a valid path.
3. The Sweet Spot: The most promising place to look is where the dark matter mass is roughly half the mass of the Z-boson (the resonance region). In this specific zone, the math works out perfectly: the universe has the right amount of dark matter, and it's quiet enough to have evaded our detectors so far.

In short: The universe is a very picky host. It won't let just any dark matter in. But if the dark matter is a "spinning top" and the secret door is tuned to a specific frequency (resonance), the universe is happy, and we might still find it in the future.

Technical Summary

1. Problem Statement

The paper addresses the search for Weakly Interacting Massive Particles (WIMPs) as dark matter (DM) candidates, specifically focusing on the hypercharge-mixing dark photon portal. While the "photon-mixing" dark photon model is well-studied, the "hypercharge-mixing" model (where the dark photon $A'$ kinetically mixes with the Standard Model hypercharge boson $B$) offers richer phenomenology but is more constrained by Electroweak Precision Observables (EWPO).

Key challenges addressed include:

• Mass Range: Previous studies often focused on sub-GeV masses or specific resonance regions. This work extends the analysis up to 1 TeV.
• Particle Types: It investigates both Dirac fermion and complex scalar DM candidates.
• Interference Effects: Unlike the photon-mixing model, the hypercharge-mixing model involves mixing between the dark photon ($A'$) and the $Z$ boson. This leads to $Z$-boson contributions to DM annihilation and scattering, as well as $A'-Z$ interference, which significantly alters constraints.
• Direct Detection Scaling: The paper re-evaluates direct detection limits, noting that in this specific model, the DM-neutron coupling vanishes at leading order, requiring a rescaling of experimental limits from $A^2$ (mass number squared) to $Z^2$ (atomic number squared).

2. Methodology

The authors employ a theoretical framework combining effective field theory, thermal freeze-out calculations, and experimental data comparison.

• Model Formalism:

  • The Lagrangian includes kinetic mixing between the dark photon field strength $F'_{\mu\nu}$ and the SM hypercharge field $B_{\mu\nu}$ with parameter $\epsilon$.
  • Diagonalization of the mass matrix yields physical states $Z$ and $A_D$ (dark photon) with a mixing angle $\alpha$.
  • DM candidates ($\chi$ for Dirac fermion, $\phi$ for complex scalar) couple to the dark sector via a gauge coupling $g_{\chi(\phi)}$.
  • Crucially, the physical $Z$ boson acquires a coupling to DM, and the dark photon acquires couplings to SM fermions (including neutrinos and axial-vector couplings).

• Constraints & Calculations:

  • Relic Density: The thermal relic abundance is calculated using the Boltzmann equation. The annihilation cross-section $\langle \sigma v \rangle$ is computed including $s$-channel exchanges of both $A'$ and $Z$, and their interference. The authors fix the observed relic density ($\Omega_{DM}h^2 = 0.1200 \pm 0.0012$) to derive lower limits on the mixing parameter.
  • Direct Detection: Spin-independent (SI) DM-nucleon scattering cross-sections are calculated. The authors demonstrate that the coupling to neutrons ($c_1^n$) vanishes at leading order due to the cancellation of electromagnetic components, meaning the scattering rate scales with $Z^2$ rather than $A^2$. Experimental limits from CRESST, DarkSide, XENON, PandaX, and LZ are rescaled accordingly.
  • EWPO & Collider Limits: Constraints on the mixing parameter $\epsilon$ are derived from LEP, SLC, Tevatron, and LHC data, extended to dark photon masses up to 3 TeV.
  • Numerical Tools: The model was implemented in FeynRules and micrOMEGAs for numerical simulation.

• Parameter Space:

  • DM Mass ($m_{DM}$): Up to 1 TeV.
  • Mass Ratio ($R = M_{A'}/m_{DM}$): Tested at $R=3$ (off-resonance), $R=2.3$, and $R=2.05$ (near resonance).
  • Dark Coupling ($\alpha_D = g^2/4\pi$): Tested at $0.5, 0.05, 0.005$.
  • Variable $y$: A dimensionless combination $y = \epsilon^2 \alpha_D (m_{DM}/M_{A'})^4$ used to parameterize the constraints.

3. Key Contributions

1. Extension to TeV Scale: The analysis systematically covers the DM mass range from GeV to 1 TeV, a regime less explored in hypercharge-mixing dark photon models compared to sub-GeV scenarios.
2. Explicit $Z$-Boson Resonance: The paper explicitly quantifies the impact of the $Z$-boson resonance ($2m_{DM} \approx M_Z$) on annihilation cross-sections, showing it can reduce required couplings by orders of magnitude.
3. Rescaling of Direct Detection Limits: The authors rigorously derive that the DM-neutron coupling vanishes in the limit $\epsilon \ll 1$, necessitating a relaxation of direct detection bounds by a factor of $A^2/Z^2$. This is a critical correction for interpreting experimental null results in this specific model.
4. Scalar vs. Fermion Comparison: The study highlights a stark difference between Dirac fermion and complex scalar DM. Scalar DM annihilation is $p$-wave suppressed ($O(v^2)$), requiring significantly larger couplings to achieve the correct relic density, making it more difficult to satisfy constraints unless near resonance.

4. Results

• Dirac Fermion DM:

  • For $R=3$ (off-resonance), the entire GeV–TeV mass range is ruled out by direct detection constraints, even after rescaling.
  • Near the resonance ($R \approx 2$), small "islands" of allowed parameter space emerge, particularly for small $\alpha_D$ ($0.005$). However, these regions are very narrow and sensitive to the dark photon decay width.
  • The $Z$-boson resonance ($2m_\chi \approx M_Z$) provides a significant loophole, allowing lower cross-sections, but direct detection still excludes most of the parameter space.

• Complex Scalar DM:

  • For $R=3$, the model is completely ruled out because the required couplings to satisfy relic density exceed EWPO limits and direct detection bounds.
  • For $R \approx 2$ (near resonance), a viable region exists. Specifically, for $\alpha_D = 0.005$, allowed regions are found for:
    • $m_\phi < 6$ GeV
    • $m_\phi > 180$ GeV
    • A narrow region around $2m_\phi \approx M_Z$.
  • The allowed parameter space for scalar DM is broader than for fermions, particularly in the mass range suggested by recent deep inelastic scattering data ($M_{A'} \sim 2-4$ GeV).

• General Constraints:

  • The "eigenmass repulsion" region (where physical masses cannot be defined for certain $\epsilon$) excludes specific mass ranges.
  • Recent LZ5T limits effectively rule out scenarios with very heavy dark photons ($M_{A'} < 1$ TeV) unless the system is finely tuned to the resonance.

5. Significance

This work provides a comprehensive update on the viability of the hypercharge-mixing dark photon portal for WIMP dark matter.

• Phenomenological Insight: It clarifies that while the hypercharge-mixing model is heavily constrained, it is not entirely dead; specific resonance conditions ($R \approx 2$) and low dark couplings ($\alpha_D \sim 0.005$) allow for consistency with all current data.
• Experimental Guidance: The finding that DM-neutron coupling vanishes suggests that future direct detection experiments should focus on targets with high $Z/A$ ratios or re-analyze data with the $Z^2$ scaling in mind.
• Future Prospects: The paper identifies that the mass region of $2-4$ GeV for the dark photon remains a promising target for complex scalar DM, motivating future searches in deep inelastic scattering and direct detection experiments.
• Theoretical Robustness: By including $Z$-boson contributions and interference effects, the paper offers a more realistic assessment of the model's parameter space than previous studies that neglected the $Z$-portal effects.