Sensitivity of the FCC-ee to axion-like particles at different center-of-mass energies

This paper investigates the sensitivity of the proposed FCC-ee collider to axion-like particles (ALPs) coupling to electroweak gauge bosons across all planned center-of-mass energies, demonstrating that the facility can detect ALPs with couplings as low as a few $10^{-6} \mathrm{GeV}^{-1}$ via the three-photon final state and potentially probe their electroweak structure for masses below the Z-boson mass.

The Gist

Imagine the universe as a giant, complex puzzle. For decades, scientists have been using a box of pieces called the "Standard Model" to solve it. It's a great box, but it's missing a few pieces. It can't explain things like why the universe is made of matter instead of anti-matter, or what "dark matter" is (the invisible stuff holding galaxies together).

To find the missing pieces, scientists are planning to build a massive new machine called the FCC-ee. Think of this machine as a super-powered, ultra-precise camera that smashes electrons and positrons (tiny particles of light and anti-light) together at incredible speeds.

This paper is a "blueprint" for how this new camera could spot a very specific, elusive ghost-like particle called an Axion-Like Particle (ALP).

The Ghost in the Machine

ALPs are theoretical particles. They are like cosmic ghosts: they are very light, very hard to catch, and they barely interact with normal matter. If they exist, they might be the missing pieces of our puzzle, or even the dark matter itself.

The scientists in this paper asked a simple question: "If we smash particles together at the FCC-ee, can we spot these ALPs, and how small can they be?"

The "Three-Light" Trick

To find these ghosts, the scientists looked for a specific magic trick.

1. The Setup: They imagine an electron and a positron crashing into each other.
2. The Magic: In this crash, a photon (a particle of light) is kicked out, and an ALP is created.
3. The Reveal: The ALP is unstable. It immediately splits apart into two more photons.

So, the final result of the crash is three flashes of light (three photons) flying out in a specific pattern. The background noise of the universe usually produces random flashes, but the ALP would produce a very specific, organized trio.

The Different "Speeds" of the Machine

The FCC-ee isn't just one speed; it's like a car that can drive at four different, very specific speeds to catch different types of targets:

• The Z-Pole (Slow & Steady): This is the most crowded, high-luminosity run. It's like scanning a crowded room with a magnifying glass. It's best at finding very weak, subtle interactions (tiny couplings) but can only see lighter ALPs.
• The High-Speed Runs (WW, ZH, tt): These are faster, more energetic collisions. They are like using a high-powered telescope. They can't see the faintest whispers, but they can spot heavier, more energetic ALPs that the slow run would miss.

The paper maps out how well the machine works at each of these speeds.

The Detective Work: Filtering the Noise

The real challenge is that the universe is noisy. When you smash particles, you get billions of random flashes of light. Finding the "three-photon" signal is like trying to find three specific fireflies in a stadium full of fireworks.

The authors designed a set of rules (filters) to clean up the data:

• The "Recoil" Check: They calculate exactly how much energy the "kicked out" photon should have based on the ALP's mass. If the numbers don't match, it's not the ghost.
• The "Angle" Check: They look at the angles between the three flashes. The ALP's ghosts leave a specific geometric signature that random fireworks don't.

What They Found

After running millions of simulations on a computer (using a virtual version of the FCC-ee detector called "IDEA"), they found:

1. Sensitivity: The FCC-ee will be incredibly sensitive. At the "Z-Pole" speed, it could detect ALPs with couplings as weak as one part in a hundred thousand. That's like hearing a whisper from across a football field.
2. Mass Range: By combining all the different speeds of the machine, they can search for ALPs ranging from 5 GeV to 320 GeV. This covers a huge territory that current machines (like the LHC) haven't fully explored yet.
3. The "Sweet Spot": For ALPs between 90 and 300 GeV, this new method is much better than what we can do today. It could potentially rule out (or find) these particles where other experiments have failed.
4. Cracking the Code: If they find an ALP, this method doesn't just say "it's there." It can also tell them how the ALP interacts with the forces of nature (specifically, whether it talks more to the "photon" force or the "Z boson" force). This helps scientists understand the underlying structure of the universe.

The Bottom Line

This paper is a feasibility study. It says: "If we build the FCC-ee and run it at these specific speeds, we have a very strong chance of finding these elusive Axion-Like particles, or at least proving they don't exist in this mass range."

It's a roadmap for the next generation of particle physics, showing us exactly where to look for the missing pieces of the universe's puzzle.

Technical Summary

Technical Summary: Sensitivity of the FCC-ee to Axion-Like Particles at Different Center-of-Mass Energies

Problem Statement
The Standard Model (SM) of particle physics fails to explain several experimental observations, including dark matter, neutrino masses, and the strong CP problem. Axion-like particles (ALPs), a class of pseudoscalar states arising from spontaneously broken global symmetries, serve as a generic probe for new physics across a wide range of energy scales. While ALPs have been constrained by collider searches (ATLAS, CMS, LHCb), beam-dump experiments, and astrophysical bounds, significant parameter space remains unexplored, particularly regarding their couplings to electroweak gauge bosons. The Future Circular Collider in electron-positron mode (FCC-ee) offers a high-luminosity, clean environment capable of precision measurements sensitive to tiny deviations from the SM. This study addresses the need to quantify the FCC-ee's sensitivity to ALPs across its full range of proposed center-of-mass energies ($E_{CM}$), extending previous analyses limited to the Z pole.

Methodology
The study investigates the production of ALPs in association with a photon ($e^+e^- \to a\gamma$) followed by the decay of the ALP into two photons ($a \to \gamma\gamma$), resulting in a three-photon ($3\gamma$) final state.

• Theoretical Framework: The analysis utilizes an effective Lagrangian where the ALP couples to electroweak gauge bosons. The benchmark scenario sets the Wilson coefficient for the $SU(2)$ coupling ($C_{WW}$) to zero, implying the ALP couples primarily to the $U(1)$ field ($C_{BB}$), and consequently to photons. The study also recasts results for varying ratios of $C_{WW}/C_{BB}$ to explore the sensitivity to the underlying electroweak structure.
• Simulation: Signal events ($e^+e^- \to a\gamma \to 3\gamma$) and the irreducible background ($e^+e^- \to \gamma\gamma\gamma$) were generated using MG5aMC@NLO v.3.6.3 with the ALP UFO model. Showering and hadronization were simulated with PYTHIA8, followed by a fast detector simulation using Delphes configured with the IDEA detector card (FCC-PED "Winter2023").
• Kinematics and Selection:
  • Preselection: Events were required to have exactly three reconstructed photons ($E \ge 2$ GeV, $|\eta| \le 3$). The invariant mass of the three photons was constrained to the center-of-mass energy ($E_{CM} \pm \sigma$), and a minimum 3D opening angle ($\Delta\alpha > 0.02$) was enforced to ensure photon resolution.
  • Photon Assignment: To distinguish the recoil photon ($\gamma_r$) from the ALP decay photons ($\gamma_1, \gamma_2$), the analysis minimized a variable $M^2_{cut}$ based on the recoil energy formula and the ALP mass hypothesis.
  • Final Selection: A multivariate selection was optimized for each ALP mass point ($m_a$) and $E_{CM}$ using the CMS Combine tool. Discriminating variables included $M_{cut}$, the polar angle of the decay photon in the ALP rest frame ($\cos\theta_{\gamma_1}$), the azimuthal angle ($\phi_{\gamma_1}$), and the opening angle $\Delta\alpha_{\gamma_1\gamma_2}$.
• Statistical Analysis: Sensitivity was derived using a binned likelihood fit on the $|\cos\theta_{\gamma_r}|$ distribution, employing the modified frequentist $CL_s$ criterion with the profile likelihood ratio as the test statistic.

Key Contributions

1. Extended Mass Range: The study updates and extends previous FCC-ee ALP sensitivity projections from the Z pole (previously limited to $m_a < 85$ GeV) to cover the full FCC-ee energy spectrum, probing ALP masses up to 320 GeV.
2. Multi-Energy Analysis: It provides a comprehensive sensitivity study across all four planned FCC-ee running modes: the Z pole (91 GeV), WW threshold (160 GeV), ZH threshold (240 GeV), and $t\bar{t}$ threshold (340–365 GeV).
3. Coupling Structure Dependence: Unlike previous studies focusing solely on the $C_{\gamma\gamma}$ coupling, this work explicitly analyzes the dependence of sensitivity on the ratio $r \equiv C_{WW}/C_{BB}$. It demonstrates how the $3\gamma$ channel's sensitivity varies with $r$, particularly highlighting the "photophobic" case ($C_{WW} = -C_{BB}$) where sensitivity drops significantly for $m_a < m_Z$.
4. Combined Reach: The paper quantifies the improvement in sensitivity achieved by combining data from different collision energies, showing gains of up to a factor of 1.6 in the 80–150 GeV mass range compared to individual runs.

Results

• Sensitivity Limits: For the benchmark scenario ($C_{WW}=0$), the FCC-ee is projected to detect ALPs with couplings down to a few $\times 10^{-6}$ GeV$^{-1}$ during the Z pole run and $\sim 10^{-5}$ GeV$^{-1}$ during the higher energy runs (WW, ZH, $t\bar{t}$).
• Mass Coverage: The Z pole run offers the best sensitivity for low masses ($m_a < 100$ GeV). Higher energy runs are essential for probing masses up to 320 GeV, where the Z pole sensitivity vanishes due to kinematic limits.
• Comparison with LHC: For ALP masses between 90 and 300 GeV, the projected sensitivity of the $3\gamma$ final state at FCC-ee significantly extends the current exclusion limits set by the LHC.
• Model Dependence: The sensitivity is highly dependent on the Wilson coefficient ratio $r$ for $m_a < m_Z$. In the photophobic limit ($r \approx -1$), the production cross-section is suppressed, leading to a loss of sensitivity. Conversely, for $m_a > m_Z$, the sensitivity becomes largely independent of $r$ due to the opening of additional decay channels ($a \to \gamma Z, ZZ$) and production modes.

Significance
The paper concludes that the FCC-ee possesses strong potential to explore the ALP phase space, particularly by leveraging its high luminosity at the Z pole and its ability to reach higher masses at elevated energies. A key finding is that the $3\gamma$ channel is sensitive to both $C_{\gamma\gamma}$ and $C_{\gamma Z}$, unlike the $\gamma\gamma$-fusion channel which is primarily sensitive to $C_{\gamma\gamma}$. This dual sensitivity allows the FCC-ee to potentially constrain the relative values of the ALP couplings to the $SU(2)$ and $U(1)$ bosons, thereby probing the underlying electroweak structure of ALP interactions. The study emphasizes that while the Z pole run is optimal for small couplings, the combination of all runs is necessary to cover the full mass range up to 320 GeV and to distinguish between different theoretical models of ALP couplings.