Probing ALP-portal fermionic dark matter at the $e^+e^-$ colliders

This paper investigates the viability of axion-like particle (ALP) mediated fermionic dark matter by analyzing its relic density constraints and demonstrating the potential for detection at future electron-positron colliders through mono-photon plus missing energy signatures, which offer a distinct separation from Standard Model backgrounds.

The Gist

The Big Picture: A Ghostly Messenger

Imagine the universe is filled with a "Dark Sector"—a hidden world of particles that make up Dark Matter. We can't see them, and they don't talk to us (the "Standard Model" of normal matter) very easily.

This paper proposes a specific way these two worlds might talk: through a messenger particle called an Axion-Like Particle (ALP). Think of the ALP as a secret courier. It can carry messages between the hidden Dark Matter and the visible world.

The authors are asking: If we build a giant particle collider (like a super-powered racetrack for electrons and positrons), can we catch this courier in action?

The Setup: The "Ghost" and the "Flashlight"

In this scenario:

1. The Dark Matter: It's a heavy, invisible fermion (let's call it a "Ghost").
2. The Messenger (ALP): It's a light particle that loves to turn into two things: either two photons (light) or two Ghosts (Dark Matter).
3. The Trick: The authors focus on a specific "sweet spot" where the Messenger is just the right weight to turn into two Ghosts very efficiently. This is called the resonance. It's like pushing a child on a swing; if you push at exactly the right rhythm, the swing goes incredibly high with very little effort. Here, the "effort" is the energy needed to create Dark Matter, and the "swing" is the ALP.

The Experiment: The "Mono-Photon" Signal

The team suggests looking for a specific event at an electron-positron collider (like the proposed ILC).

The Scene:
Imagine two electrons and positrons crash into each other.

• The Signal: They produce a single, bright flash of light (a photon) and then... poof... nothing else is seen. The Messenger (ALP) was created, but instead of turning into light, it immediately turned into two invisible Ghosts (Dark Matter) and flew away.
• The Clue: Since the Ghosts are invisible, the only thing the detector sees is that one single photon. But because the Ghosts took energy away with them, the photon doesn't have as much energy as it should. This missing energy is the "smoking gun."

Why is this special?
Usually, when physicists look for missing energy, the background noise (other things happening in the collider that look similar) is huge. It's like trying to hear a whisper in a rock concert.
However, the authors found a unique feature:

• In most scenarios, the "flash" (photon) is just a random spark flying off the side (Initial State Radiation).
• In this specific model, the flash is created at the exact moment the Messenger is born.
• The Analogy: Imagine a magician. In a normal trick, a rabbit appears from a hat (random). In this trick, the rabbit appears because a specific card was pulled, and the card is right next to the rabbit. The relationship between the card and the rabbit is so specific that you can tell them apart from a random rabbit appearing.
• Because of this specific relationship, the "missing energy" pattern of the signal looks completely different from the background noise. It's like the signal is singing in a different key than the noise.

The Results: Can We See It?

The authors ran simulations to see if future colliders could spot this.

1. The "Sweet Spot" Works: They found that if the Messenger is about twice as heavy as the Dark Matter (the resonance), the Dark Matter can exist in the universe with the correct amount (relic density) without being ruled out by other experiments.
2. The Collider Test: At a future collider (specifically the ILC at 1 TeV energy), they showed that:
   • If you use polarized beams (like aligning the spins of the particles to filter out noise), the background noise drops dramatically.
   • The signal stands out clearly. They calculated that with enough data, they could see this signal with high confidence (5-sigma, which is the gold standard for discovery in physics).
   • They could even measure how strongly the Messenger talks to light (the ALP-photon coupling) with very high precision (about 1% accuracy).

What About Other Colliders?

The paper also checked the Large Hadron Collider (LHC), the biggest collider we have right now.

• The Verdict: The LHC is like a noisy construction site. The background noise is so loud and messy that the specific "whisper" of this signal gets drowned out. The authors conclude that while the LHC is great for many things, it is very difficult to find this specific type of Dark Matter there. The clean environment of an electron-positron collider is essential for this job.

Summary

The paper claims that:

1. There is a plausible model where Dark Matter talks to us via a "Messenger" (ALP).
2. This model works best if the Messenger is tuned to a specific "resonance" frequency.
3. Future electron-positron colliders can spot this by looking for a single flash of light accompanied by missing energy.
4. Because of the unique way the light is produced in this model, it is easy to distinguish from background noise, unlike at the current LHC.
5. If we build these colliders, we could not only find this Dark Matter but also measure exactly how it interacts with light.

Technical Summary

Technical Summary: Probing ALP-portal Fermionic Dark Matter at the $e^+e^-$ Colliders

Problem Statement
Axion-like particles (ALPs) serve as promising candidates for mediating interactions between the Standard Model (SM) and the dark sector. While GeV-scale ALPs cannot act as dark matter (DM) themselves due to instability over cosmological timescales, they can function as portals connecting a Dirac fermion DM candidate ($\Psi$) to the SM. A significant challenge in this framework is achieving the correct DM relic abundance via thermal freeze-out while evading stringent constraints from indirect detection experiments (e.g., Fermi-LAT, MAGIC) which typically rule out the annihilation channel $\Psi\bar{\Psi} \to \gamma\gamma$. Furthermore, probing such models at hadron colliders like the LHC is complicated by large QCD backgrounds and parton distribution function (PDF) uncertainties. This work investigates whether future high-energy electron-positron ($e^+e^-$) colliders can effectively probe this specific ALP-portal scenario, particularly in the resonant regime where the ALP mass ($m_a$) is approximately twice the DM mass ($m_\Psi$).

Methodology
The study employs an effective field theory (EFT) approach where the ALP ($a$) interacts with SM electroweak gauge bosons and the fermionic DM via dimension-5 operators. The Lagrangian includes derivative couplings between the ALP and DM, and couplings to the $U(1)_Y$ and $SU(2)_L$ field strength tensors.

• Phenomenological Framework: The authors assume a mass hierarchy where $m_a \gtrsim 2m_\Psi$ and $m_a < M_W, M_Z$. The DM relic density is calculated by solving the Boltzmann equation, considering both standard thermal freeze-out and the effects of early kinetic decoupling (EKD) in the resonant regime.
• Resonant Enhancement: To satisfy the observed relic density ($\Omega h^2 \approx 0.12$) without violating indirect detection limits, the analysis focuses on the resonant region where $m_a/m_\Psi \approx 2$. In this regime, the annihilation cross-section is enhanced, allowing for a larger ALP decay constant ($f_a$) and weaker couplings.
• Collider Analysis: The study simulates the process $e^+e^- \to \gamma a$, where the ALP decays invisibly into a DM pair ($a \to \Psi\bar{\Psi}$), resulting in a mono-photon plus missing energy ($E_{\text{miss}}$) signature. Simulations were performed using MadGraph5_aMC@NLO for event generation, Pythia for showering, and Delphes with ILC-specific detector cards for simulation.
• Statistical Analysis: A cut-based analysis was performed at $\sqrt{s} = 1$ TeV with an integrated luminosity of $5 \text{ ab}^{-1}$. The sensitivity was evaluated using signal significance ($Z$) and a binned $\chi^2$ analysis to estimate the precision of the ALP-photon coupling ($g_{a\gamma\gamma}$) measurement. Beam polarization effects were also investigated.

Key Contributions and Results

1. Relic Density and Resonance: The authors demonstrate that for a mass ratio $r = m_a/m_\Psi \approx 2.02$ and $f_a \sim \mathcal{O}(200)$ GeV, the model can reproduce the correct DM relic abundance. The analysis shows that while early kinetic decoupling can lead to an overabundance of DM, adjusting the mass ratio slightly closer to the resonance (e.g., $r \approx 2.012$) compensates for this effect, maintaining consistency with observations.
2. Distinctive Signal Kinematics: A primary finding is the unique kinematic signature of the signal compared to the SM background. In the proposed model, the photon is produced at the interaction vertex (final state radiation-like topology due to the ALP-photon coupling), whereas the dominant SM backgrounds ($e^+e^- \to \nu\bar{\nu}\gamma$) arise from initial state radiation (ISR).
   • Missing Energy ($E_{\text{miss}}$): The signal distribution peaks sharply at $E_{\text{miss}} \approx \sqrt{s}/2$ (500 GeV for $\sqrt{s}=1$ TeV), whereas the SM background exhibits a double-peak structure with a dominant peak near $\sqrt{s}$ and a sub-dominant peak near 500 GeV.
   • Pseudorapidity ($\eta_\gamma$): The signal and background also differ in pseudorapidity distributions.
3. Sensitivity at ILC:
   • Applying cuts on $E_{\text{miss}}$ and $\eta_\gamma$ significantly suppresses the SM background.
   • With unpolarized beams, a significance of $Z \approx 2.85\sigma$ is achieved for the benchmark point.
   • Utilizing beam polarization ({Pe$^+$: Pe$^-$} = {-20%: +80%}), the significance increases to $Z \approx 7.64\sigma$, allowing for a $5\sigma$ discovery with an integrated luminosity of $2.2 \text{ ab}^{-1}$.
   • The study estimates that the ALP-photon coupling ($g_{a\gamma\gamma}$) can be measured with a precision of approximately 0.87% to 1.87% using the $\chi^2$ method, depending on polarization.
4. Comparison with Other Facilities: The paper notes that the High-Luminosity LHC (HL-LHC) struggles to probe this specific channel due to the lack of a clear distinction between signal and background in transverse momentum and missing transverse energy distributions. Similarly, while muon colliders offer a complementary opposite-sign muon channel, the $e^+e^-$ mono-photon channel provides a unique advantage due to the clean environment and specific kinematic features of the ALP-photon interaction.

Significance and Claims
The paper claims that the proposed ALP-portal fermionic DM model offers a "spectacular distinction" between the signal and SM background at $e^+e^-$ colliders, driven by the specific nature of the ALP-photon interaction. This distinction allows for the probing of ALP-photon couplings as small as $\mathcal{O}(10^{-5}) \text{ GeV}^{-1}$, a precision difficult to achieve at the HL-LHC. The work highlights that the mono-photon signal at future linear colliders (like the ILC) is not only capable of discovering such DM scenarios but also of precisely measuring the coupling parameters that govern the DM relic density. The authors conclude that this approach opens a new direction for ALP searches, leveraging the clean environment of lepton colliders to overcome the limitations of hadron collider searches for invisible decays.