Dark axion portal at $Z$ boson factories

This paper investigates the phenomenological prospects of the Dark Axion Portal at $Z$ boson factories, demonstrating that gauge invariance necessitates a significant $Z$-dark photon-ALP coupling that enables efficient detection of dark sector particles through displaced decays and missing energy signatures across LEP, FCC-ee, and forward physics detectors.

The Gist

Imagine the universe is a giant, bustling city. We know most of the residents: the people we can see, touch, and measure (like electrons and photons). But we suspect there's a "Dark Sector" living in the shadows—a neighborhood of invisible creatures that make up most of the city's mass (Dark Matter) but refuse to interact with us directly.

This paper is about a specific "secret tunnel" that connects our visible city to this dark neighborhood. The author, Krzysztof Jodłowski, is proposing a new way to find these hidden residents by looking at a very specific type of particle collision: the Z boson.

Here is the story of the paper, broken down into simple concepts:

1. The "Dark Axion Portal" (The Secret Tunnel)

Think of the Dark Axion Portal as a special door that connects our world to the dark one.

• The Characters:
  • The Axion (ALP): A ghostly, light particle (like a whisper).
  • The Dark Photon: A heavy, invisible cousin of the light photon (like a silent shadow).
  • The Z Boson: A heavy, energetic particle produced in our particle accelerators (like a massive delivery truck).

Usually, scientists thought this door only opened when the Axion met a regular photon (light). But the author points out a crucial rule of physics: Gauge Invariance.

• The Analogy: Imagine you have a key that opens a door to a "Light Room." Because of the laws of the building's architecture, that same key must also open a door to a "Z Room" next door. You can't have one without the other.
• The Discovery: The paper argues that because the Axion and Dark Photon are connected to light, they are automatically connected to the Z boson. This creates a new, powerful way to produce these dark particles: The Z boson can decay into an Axion and a Dark Photon.

2. The Hunt: Catching Ghosts at the "Z Factories"

The author suggests we hunt for these particles at places that produce massive amounts of Z bosons, which he calls "Z Factories."

• The Factories:
  • LEP: The old, retired factory (like a vintage car museum).
  • FCC-ee: The brand new, super-powerful factory of the future.
  • LHC & FPF: The massive industrial complexes where we smash protons together.

How do we catch them?
Since these dark particles don't interact with our detectors, they are invisible. However, they don't last forever. They eventually decay (break apart) into things we can see.

• The "Displaced Decay" (The Delayed Explosion):
  Imagine the Z boson creates a ghost particle. This ghost flies a few meters away from the crash site before it suddenly explodes into a flash of light (a photon) or a pair of electrons/positrons.
  • Why is this cool? In a normal crash, everything happens instantly at the center. If you see an explosion happening a few meters away from the center, you know something invisible flew there first. This is the "smoking gun."

3. The Detective Work: What the Paper Found

The author ran simulations (computer models) to see how well different detectors could catch these ghosts.

• The Old Factory (LEP): Even though it's old, the data from LEP is so clean that it already sets very strict rules on how heavy these particles can be. It's like finding a fingerprint on a 20-year-old window that proves a burglar was there.
• The New Factory (FCC-ee): This is the big winner. Because it produces billions more Z bosons than LEP, it can spot these particles even if they are very rare or very heavy. The author predicts FCC-ee will improve our search sensitivity by a factor of 10 or more.
• The Long-Distance Detectors (FASER & MATHUSLA):
  • Imagine the ghost particle is very long-lived. It flies way past the main detector, into a separate, smaller building miles away.
  • MATHUSLA is like a giant, empty warehouse designed specifically to catch these long-distance ghosts. The paper shows that MATHUSLA is perfect for catching the "heavy" dark particles that travel far before decaying.

4. The "Two-Door" Strategy

The paper highlights a clever strategy using two types of signals:

1. The "Missing Energy" Signal: The Z boson disappears, and we see nothing but a photon. (Like a magician making a rabbit vanish, leaving only a hat).
2. The "Displaced Decay" Signal: The Z boson creates a ghost that flies away and explodes later. (Like a time-delayed firecracker).

By using both methods, we cover all bases. If the particles are short-lived, we catch them near the crash. If they are long-lived, we catch them far away.

The Bottom Line

This paper is a roadmap for the future of particle physics. It tells us:

1. Don't ignore the Z boson: It's a direct highway to the dark sector that we haven't fully exploited yet.
2. The future is bright: The upcoming FCC-ee collider and the new forward detectors (MATHUSLA/FASER) are perfectly designed to find these particles.
3. We are close: If these "Dark Axion Portal" particles exist with masses above a certain threshold (0.1 GeV), we have a very high chance of finding them in the next decade.

In short, the author is saying: "We found a secret door in the Z boson. If we look at the right places with the right tools, we might finally catch a glimpse of the dark universe."

Technical Summary

1. Problem Statement and Motivation

The Dark Axion Portal (DAP) is a non-minimal dark sector (DS) model connecting an Axion-Like Particle (ALP, denoted as $a$) and a Dark Photon (DP, denoted as $\gamma'$) to the Standard Model (SM). The defining interaction is the coupling of the ALP to both the SM photon ($\gamma$) and the dark photon ($\gamma'$), described by the Lagrangian term:
$$\mathcal{L} \supset \frac{g_{a\gamma\gamma'}}{2} a F_{\mu\nu} \tilde{F}_D^{\mu\nu}$$
where $g_{a\gamma\gamma'}$ is the coupling constant.

The Core Issue: Previous phenomenological studies of the DAP have largely focused on the $a\gamma\gamma'$ coupling. However, the author argues that SM gauge invariance ($SU(2)_L \times U(1)_Y$) necessitates the generation of a related coupling between the ALP, the Z boson, and the dark photon ($aZ\gamma'$).

• Unlike standard ALP models where couplings to $\gamma$ and $Z$ are independent parameters, in the DAP, the ratio $|g_{aZ\gamma'}/g_{a\gamma\gamma'}|$ is fixed by the Weinberg angle ($\theta_W$) to be $\tan \theta_W$.
• This coupling, while negligible for astrophysical constraints (due to $Z$-mediated suppression), becomes the dominant production mechanism for DAP particles at high-energy colliders operating at the $Z$-pole, specifically $e^+e^-$ factories.

The paper investigates the experimental prospects of detecting DAP particles via this $g_{aZ\gamma'}$ coupling at Z boson factories (LEP, FCC-ee) and forward physics detectors (FASER, MATHUSLA) at the LHC and future 100 TeV colliders.

2. Methodology

The study employs a combination of analytical calculations and Monte Carlo simulations to evaluate production rates and decay signatures.

A. Theoretical Framework

• Gauge Invariance: The author derives the $g_{aZ\gamma'}$ coupling from the pre-symmetry breaking Lagrangian involving $B_{\mu\nu}$ (hypercharge) and $W_{\mu\nu}$ (weak isospin) tensors. After Electroweak Symmetry Breaking (EWSB), this yields the relation $g_{aZ\gamma'} = -\tan \theta_W g_{a\gamma\gamma'}$.
• Production Mechanisms:
  • Z Boson Factories: Dominant production via the decay $Z \to a \gamma'$.
  • High Energy ($e^+e^-$): $e^+e^- \to a \gamma'$ via $\gamma^*/Z^*$ annihilation.
  • Hadron Colliders (LHC/FPF): Production via heavy vector meson decays (e.g., $J/\psi \to a \gamma'$) and $Z \to a \gamma'$.
• Decay Signatures:
  • Displaced Decays: Since DAP particles are feebly coupled, they act as Long-Lived Particles (LLPs). The paper analyzes:
    • Two-body decays: $\gamma' \to \gamma a$ and $a \to \gamma \gamma'$.
    • Three-body decays: $\gamma' \to f\bar{f}a$ and $a \to f\bar{f}\gamma'$ (where $f$ is a charged SM fermion). These are crucial for detectors without electromagnetic calorimeters (like MATHUSLA) or for reconstructing vertices.
  • Missing Energy Signatures: $Z \to \gamma + \text{inv.}$ (where "inv" is the undetected dark sector pair) and $Z \to \text{inv.}$

B. Simulation and Validation

• Benchmark Scenarios: Two mass hierarchies are considered:
  1. $m_{\gamma'} \gg m_a$ (Heavy Dark Photon).
  2. $m_a \gg m_{\gamma'}$ (Heavy ALP).
• Simulation Tool: The author adapted the simulation framework used for Heavy Neutral Leptons (HNL) with magnetic dipole portals (referencing works [19, 20]) to the DAP model.
• Validation: The simulation was validated by reproducing existing LEP bounds and FCC projected sensitivities for the HNL dipole portal. The DAP simulation parameters (luminosities, detector lengths, cuts) were aligned with LEP and FCC-ee specifications.
• Detectors Analyzed:
  • LEP: $Z$-pole ($\sqrt{s} \approx 91$ GeV) and LEP2 ($\sqrt{s} \approx 205$ GeV).
  • FCC-ee: High luminosity $Z$-pole run ($145 \text{ ab}^{-1}$).
  • Forward Detectors: FASER2 and MATHUSLA at LHC and the proposed 100 TeV FPF@FCC.

3. Key Contributions

1. Identification of the $Z$-Boson Portal: The paper establishes that the $g_{aZ\gamma'}$ coupling is an unavoidable consequence of SM gauge invariance in the DAP model, with a fixed strength relative to the photon coupling.
2. Re-evaluation of LEP Limits: The study recalculates LEP constraints using the $Z \to a \gamma'$ production channel. It demonstrates that LEP limits are significantly stronger than previously thought for this specific coupling, often surpassing future Belle II projections.
3. Complementarity of Signatures: The work highlights the necessity of combining displaced decay searches (sensitive to semi-visible decays like $f\bar{f} + \text{inv}$) with missing energy searches ($Z \to \gamma + \text{inv}$ or $Z \to \text{inv}$).
   • Note: MATHUSLA, lacking a calorimeter, relies heavily on the three-body semi-visible decays, while FASER2 can utilize the single-photon signature.
4. FCC-ee Sensitivity Projection: The paper provides the first detailed sensitivity projections for FCC-ee, showing it can improve upon LEP limits by more than an order of magnitude, particularly in the mass range $m \gtrsim 0.1$ GeV.

4. Results

• LEP Constraints:
  • LEP data (specifically $Z \to \gamma + \text{inv.}$ and $Z \to \text{inv.}$) sets the current strongest terrestrial bounds on the DAP parameter space for masses $m \gtrsim 0.1$ GeV.
  • These limits are comparable to or stronger than projected limits from Belle II.
• FCC-ee Projections:
  • FCC-ee will significantly extend the exclusion reach.
  • The displaced decay signature at FCC-ee is projected to be the most stringent terrestrial probe for masses $m \gtrsim 1$ GeV.
  • The improvement over LEP is driven by the massive increase in integrated luminosity ($145 \text{ ab}^{-1}$ vs $0.2 \text{ fb}^{-1}$).
• Forward Physics (LHC & FPF@FCC):
  • At the LHC, production via heavy vector meson decays (e.g., $J/\psi$) dominates over $Z$ decays for forward detectors.
  • At the proposed 100 TeV FPF@FCC, the $Z \to a \gamma'$ decay becomes the dominant production mode due to the high $Z$ yield.
  • FASER2 and MATHUSLA at 100 TeV will probe the parameter space for both short and long-lived regimes, with MATHUSLA covering the large transverse momentum regime and FASER2 covering highly collimated particles.
• Mass Dependence:
  • For $m \gtrsim 0.1$ GeV, the $g_{aZ\gamma'}$ coupling allows efficient production.
  • The three-body decay channels (semi-visible) contribute at the percent level for masses $> 1$ GeV, making them critical for detectors like MATHUSLA.

5. Significance

• Theoretical Consistency: The paper corrects a gap in previous DAP phenomenology by enforcing SM gauge invariance, revealing a "hidden" production channel ($Z \to a \gamma'$) that was previously overlooked.
• Experimental Strategy: It provides a roadmap for current and future experiments. It argues that Z boson factories are the premier discovery machines for the DAP in the GeV mass range, outperforming beam-dump experiments and B-factories in specific coupling regimes.
• Complementarity: The study emphasizes that no single signature is sufficient. A robust search strategy must combine:
  • Missing energy ($Z \to \text{inv}$).
  • Single photon + missing energy ($Z \to \gamma + \text{inv}$).
  • Displaced vertices with charged tracks ($f\bar{f} + \text{inv}$).
• Future Outlook: The results strongly motivate the inclusion of DAP searches in the physics programs of FCC-ee and the Forward Physics Facility at the 100 TeV collider, suggesting that these facilities could either discover the Dark Axion Portal or rule out significant portions of its parameter space.