Search for dark photons at future e$^+$e$^-$ colliders

This paper presents a study using fully simulated events in the ILD detector at the ILC Higgs factory to investigate the sensitivity of future high-energy $e^+e^-$ colliders to dark photons ($A_D$) decaying into muon pairs, highlighting their potential to probe mixing parameters and masses beyond the reach of current experiments like Belle II.

The Gist

Imagine the universe is a giant, bustling city. We know the "visible" part of the city—the people, buildings, and cars—is only about 15% of the total population. The other 85% is a mysterious "Dark Sector" that we can't see, touch, or hear, except for the fact that it has gravity (it pulls on things).

For decades, scientists have been trying to find the residents of this Dark Sector. They've been looking for "WIMPs" (Weakly Interacting Massive Particles), but so far, the city has been empty-handed.

This paper proposes a new way to look for them. Instead of looking for the heavy, invisible residents directly, the authors suggest looking for a "Dark Messenger" called a Dark Photon.

The Story of the Dark Messenger

Think of the Dark Sector as a secret, walled-off neighborhood. The only way to get a message in or out is through a tiny, almost invisible door. In physics, this door is called a "Portal."

The "Dark Photon" is the messenger that walks through this door. It's a particle that lives in the dark world but has a tiny, magical ability to "mix" with our normal light (photons). Because of this mixing, we might be able to create a Dark Photon in our particle accelerators and watch it decay back into normal particles (like muons, which are heavy cousins of electrons).

The Hunt at the "Higgs Factory"

The paper focuses on a specific type of particle collider called the ILC (International Linear Collider), which is being designed to be a "Higgs Factory." Think of this as a high-speed racetrack where we smash electrons and positrons together at incredible speeds.

The scientists asked: If we smash these particles together, can we spot the Dark Photon?

They ran a massive computer simulation of the ILD detector (the giant camera that would take pictures of the collisions) to see what would happen.

The Challenge: Finding a Needle in a Haystack

The problem is that the Dark Photon is very shy.

1. It's rare: It doesn't show up often.
2. It's quiet: When it does show up, it decays instantly into two muons.
3. The Haystack: The "haystack" is the background noise of millions of other collisions that look exactly the same but aren't Dark Photons.

The team simulated millions of collisions to see if they could distinguish the "signal" (the Dark Photon) from the "noise" (standard background).

The Big Surprise: The "Theory" Was Too Optimistic

Here is the most important part of the paper, explained simply:

The Old Map vs. The Real Terrain
Previous studies (the "Old Map") predicted that the ILC would be able to find Dark Photons with very high precision. They assumed that the detector was perfect and that the math was simple. They thought, "If we look hard enough, we'll see a sharp, clear peak in the data."

The Reality Check
When the authors used a full, realistic simulation (the "Real Terrain"), they found that the detector isn't perfect.

• The Blur: Just like a camera lens that isn't perfectly sharp, the detector smears out the energy measurements slightly.
• The Angle: The particles fly off at weird angles, and the detector misses some of them, especially if they are light or fly too close to the edge of the machine.

The Result:
Because of these real-world imperfections, the "Old Map" was too optimistic.

• At lower masses, the new realistic limits are actually worse than the old predictions.
• At higher masses, the limit is about twice as high (meaning it's harder to find the particle) than the simple theory suggested.

It's like thinking you can hear a whisper from a mile away because the air is clear, but then realizing there's actually a strong wind and some traffic noise that makes the whisper much harder to hear.

Why This Matters

Even though the limits are "worse" than the simple theory suggested, the paper concludes that the ILC is still a fantastic place to look for Dark Photons.

• Precision: Unlike the massive LHC (which smashes protons and creates a lot of messy debris), the ILC is like a scalpel. It creates very clean collisions with very little background noise.
• The Future: The paper also projects what would happen if we built even bigger versions of this collider (LCF 550 and 1000). They show that with higher energy, we could hunt for Dark Photons that are much heavier, pushing the boundaries of what we know about the universe.

The Takeaway

This paper is a "reality check" for the physics community. It says: "We are going to build the best camera in the world to look for Dark Photons. Our simple math said we'd find them easily, but when we simulated the real camera, we realized it's a bit harder than we thought. However, with our high-tech camera and clean environment, we are still the best hunters in the game."

It's a story of humility in science: realizing that the real world is messier than our equations, but that with enough precision and patience, we can still find the hidden secrets of the universe.

Technical Summary

1. Problem Statement

The paper addresses the search for Dark Photons ($A_D$), hypothetical particles arising from a "dark sector" with a $U(1)$ symmetry that kinetically mixes with the Standard Model (SM) photon via a mixing parameter $\epsilon$.

• Context: While Weakly Interacting Massive Particles (WIMPs) have not been detected, Feebly Interacting Particles (FIPs) in a dark sector offer an alternative explanation for Dark Matter. The Dark Photon acts as a "vector portal" between the visible and dark sectors.
• Challenge: Previous theoretical estimates for exclusion limits at future colliders (specifically Higgs factories like the ILC) were overly optimistic. They relied on simplified assumptions regarding detector resolution and background levels, often ignoring the complex interplay of multiple scattering, angular acceptance, and event-by-event kinematic variations.
• Goal: To provide a realistic assessment of the sensitivity to Dark Photons at the International Linear Collider (ILC) operating at 250 GeV (Higgs factory stage) using the ILD (International Large Detector) concept, and to extrapolate these findings to future Linear Collider Facility (LCF) upgrades at 550 GeV and 1000 GeV.

2. Methodology

The study employs a rigorous, full-simulation approach rather than relying on analytical approximations.

• Signal Process: The primary search channel is $e^+e^- \to \gamma_{ISR} A_D \to \gamma_{ISR} \mu^+ \mu^-$. The Dark Photon is produced via Initial State Radiation (ISR) and decays promptly into a muon pair.
• Simulation Chain:
  • Event Generation: Used Whizard (v3.0) with a Unified Feynrules output (UFO model) to generate signal events.
  • Detector Simulation: Events were processed through ILD full simulation (ddsim) based on Geant4, modeling the detector geometry, magnetic fields, and material interactions.
  • Reconstruction: The Marlin framework was used for full event reconstruction.
• Background Handling: The analysis included the entire fully simulated Standard Model background, not just the dominant $e^+e^- \to \mu^+\mu^- + \gamma_{ISR}$ process. It accounted for t-channel processes involving undetected beam-remnant electrons and misidentified ISR photons.
• Selection Criteria: Events were selected based on the presence of exactly two muons and an isolated photon, with no other significant activity.
• Analysis Strategy:
  • The search focused on a narrow resonance peak in the dimuon invariant mass ($m_{\mu\mu}$) distribution.
  • Crucially, the study utilized event-by-event mass resolution ($\sigma_m$) rather than a fixed global resolution. This accounts for variations in momentum resolution due to multiple scattering (dominant at $p \lesssim 100$ GeV) and the polar angle of the muons.
  • The search window was optimized for each event based on its specific $\sigma_m$ to maximize sensitivity.

3. Key Contributions

• Realistic Detector Performance Modeling: The paper demonstrates that previous theoretical limits (e.g., from the 2019 EPPSU briefing book) were too optimistic by a factor of 2 to 4. This discrepancy arises because simplified models assumed constant momentum resolution ($\sigma(1/p_T)$) and ignored the significant loss of acceptance for low-mass Dark Photons where muons are emitted at small angles (outside the barrel tracker).
• Full Simulation Validation: By using full Geant4 simulation, the authors established that the observed peak width is determined by detector resolution, not the intrinsic width of the $A_D$ (which is extremely narrow, $c\tau < 1$ nm).
• Recasting Framework: The authors developed a method to extrapolate results from the ILC250 baseline to higher energy configurations (LCF550 and LCF1000). They derived a scaling factor ($\sqrt{k} \approx 20$) relating signal and background levels, allowing for accurate limit predictions at higher masses and energies without re-running full simulations for every scenario.
• Polarization Analysis: The study evaluated the impact of beam polarization, noting that while signal and background share similar polarization dependencies, likelihood weighting can improve sensitivity at lower masses where other backgrounds dominate.

4. Results

• Exclusion Limits at ILC250:
  • The full simulation reveals that the exclusion limit on the mixing parameter $\epsilon$ is significantly weaker than naive theory estimates.
  • At $m_{A_D} = 100$ GeV, the limit is 4 times higher (less sensitive) than the theoretical estimate.
  • At the kinematic limit ($\sim 250$ GeV), the limit is 2 times higher than the estimate.
  • Below the Z-mass ($M_Z$), the limits are weaker than those expected from the HL-LHC due to larger non-Z backgrounds and correct error estimation.
• Efficiency and Acceptance:
  • Detection efficiency for the dimuon pair is only $\sim 25\%$ for $m_{A_D} = 10$ GeV due to angular acceptance cuts (muons falling outside the tracker).
  • Efficiency approaches 100% for $m_{A_D} \ge 100$ GeV.
• Future Projections (LCF):
  • LCF250: Increasing luminosity from 2 to 3 ab$^{-1}$ only improves the $\epsilon$ limit by a factor of $\sim 0.9$ (due to the $\epsilon \propto L^{-1/4}$ scaling).
  • LCF550 & LCF1000: These facilities extend the mass reach significantly. For $m_{A_D} > 250$ GeV, the background is dominated purely by $e^+e^- \to \mu^+\mu^-$, and the signal-to-background ratio scales predictably with mass resolution.
  • The study provides exclusion curves for LCF250, 550, and 1000, showing that while HL-LHC may compete at lower masses, high-energy $e^+e^-$ colliders offer superior sensitivity for $m_{A_D} > 100$ GeV due to lower backgrounds and superior momentum resolution.

5. Significance

This paper is significant because it shifts the paradigm for Dark Photon searches at future colliders from theoretical idealization to experimental realism.

1. Correction of Expectations: It corrects the community's understanding of the ILC's potential, showing that while the ILC is a powerful tool, its sensitivity is not infinite and is heavily dependent on accurate modeling of detector effects (multiple scattering, angular acceptance).
2. Detector-Driven Physics: It highlights that the performance of the ILD detector (specifically its tracking resolution and acceptance) is the limiting factor for low-mass Dark Photon searches, not just the luminosity.
3. Strategic Roadmap: By providing a validated method to recast results for higher energy upgrades (LCF), the paper offers a concrete roadmap for the Linear Collider community to optimize search strategies for the Vector Portal across a wide mass range (from GeV to TeV scales).
4. Complementarity: It reinforces the role of $e^+e^-$ colliders as the premier facility for probing the Dark Photon parameter space in the mass range above Belle II's reach but below the optimal sensitivity of hadron colliders, provided the detector resolution is properly accounted for.