A search for heavy axion-like particles in light-by-light scattering at the FCC-hh

This paper investigates the potential of the future 100 TeV FCC-hh collider to discover or exclude heavy axion-like particles (ALPs) via light-by-light scattering in pp, pPb, and PbPb collisions, demonstrating that the facility offers significant sensitivity to ALP masses up to the TeV scale compared to current LHC bounds.

The Gist

Imagine the universe is a giant, bustling marketplace. For decades, physicists have been looking for a specific, elusive type of "ghost" particle called an Axion-Like Particle (ALP). These particles are like the "dark matter" of the particle world: they might be everywhere, holding the universe together, but they are incredibly hard to catch because they barely interact with anything else.

This paper is a proposal for a new, super-powered "trap" to catch these ghosts, specifically the heavy ones, using a future machine called the FCC-hh.

Here is the breakdown of the research using simple analogies:

1. The Setting: The Ultimate Particle Collider

Think of the current Large Hadron Collider (LHC) as a very fast, very loud race track. It smashes protons together to see what breaks apart.
The FCC-hh (Future Circular Collider) is like building a race track that is seven times longer and can accelerate particles to speeds where they have seven times more energy. It's the difference between a bicycle sprint and a rocket launch.

2. The Method: The "Light-by-Light" Mirror Trick

Usually, to find a new particle, you smash two heavy things (like protons) together and look at the debris. But ALPs are tricky; they don't like to be made by smashing heavy things directly. They prefer to be born from light.

The researchers are looking at a rare phenomenon called Light-by-Light (LbL) scattering.

• The Analogy: Imagine two people throwing flashlights at each other in a dark room. Normally, the beams of light just pass right through each other like ghosts. But in the quantum world, under extreme conditions, those beams of light can actually bounce off each other.
• The Trap: If an ALP exists, it acts like a hidden trampoline in the middle of the room. When the light beams hit this invisible trampoline, they bounce differently than they would normally. By measuring exactly how the light bounces, the scientists can tell if a "ghost" (the ALP) was there.

3. The Three Different Hunting Grounds

The paper studies three different ways to set up this "flashlight trap" at the FCC-hh. Think of them as three different hunting strategies:

• Strategy A: The Proton vs. Proton (pp) Collision

  • The Analogy: This is like two snipers firing high-powered lasers at each other.
  • The Advantage: The lasers are incredibly powerful and focused. They can reach the highest energy levels.
  • The Result: This method is the best at finding very heavy ALPs (around 1,000 times heavier than a proton). It's the only way to see the "giants" in the dark.

• Strategy B: The Lead vs. Lead (PbPb) Collision

  • The Analogy: Instead of snipers, imagine two giant, glowing suns (Lead nuclei) passing each other. Because they are so huge and charged, they emit a massive, blinding flood of light (photons) around them.
  • The Advantage: Even though the "sun" collision happens less often, the sheer volume of light is so intense that it creates a "photon-photon collider." It's like having a floodlight instead of a laser.
  • The Result: This method is incredibly sensitive to medium-weight ALPs (around 250 GeV). The flood of light makes it easy to spot the "medium-sized" ghosts that the snipers might miss.

• Strategy C: The Proton vs. Lead (pPb) Collision

  • The Analogy: This is a mix. One sniper and one glowing sun.
  • The Result: It acts as a bridge, filling in the gaps between the other two methods.

4. The Findings: Why This Matters

The authors did the math (using complex equations that we can skip) and found some exciting results:

• The "Sweet Spot": They discovered that the FCC-hh will be able to find ALPs in mass ranges that the current LHC simply cannot reach.
• The Verdict: If these heavy ALPs exist, the FCC-hh is the perfect machine to find them.
  • If the ALP is a "medium" ghost, the Lead-Lead collisions will find it first.
  • If the ALP is a "heavy" ghost, the Proton-Proton collisions will find it.

The Bottom Line

This paper is a "blueprint" for a treasure hunt. It tells us that if we build this massive new collider (FCC-hh), we can use the unique properties of light and heavy atoms to scan the universe for these mysterious particles.

It's like saying: "We know there are hidden treasures in this cave. The LHC was a flashlight that could only see the entrance. The FCC-hh is a floodlight that can see the entire cave, and we have three different ways to sweep the floor to make sure we don't miss a single coin."

If successful, this could solve one of the biggest mysteries in physics: What is the dark matter holding our universe together?

Technical Summary

1. Problem Statement and Motivation

The paper addresses the search for heavy Axion-Like Particles (ALPs), which are hypothetical pseudoscalar bosons arising in theories beyond the Standard Model (BSM), such as string theory and Grand Unified Theories (GUTs). Unlike the QCD axion, ALPs have independent mass ($m_a$) and coupling ($f_a$) parameters, making them a flexible framework for new physics.

The specific challenge is to probe the heavy mass region (hundreds of GeV to several TeV) where current Large Hadron Collider (LHC) searches have limited sensitivity. The authors focus on Light-by-Light (LbL) scattering ($\gamma\gamma \to \gamma\gamma$), a rare Standard Model process occurring at the loop level. This process is highly sensitive to new physics contributions, particularly the resonant production of ALPs via the process $\gamma\gamma \to a \to \gamma\gamma$.

The study aims to determine the discovery and exclusion potential of the proposed Future Circular Collider in hadron-hadron mode (FCC-hh), operating at a center-of-mass energy of $\sqrt{s} = 100$ TeV, compared to current LHC limits.

2. Methodology

2.1 Collision Modes and Kinematics

The authors analyze three distinct collision modes at the FCC-hh to exploit different kinematic advantages:

1. **Proton-Proton ($pp$):**$\sqrt{s} = 100$ TeV.
2. Proton-Lead ($p$Pb): $\sqrt{s_{NN}} = 62.8$ TeV.
3. Lead-Lead (PbPb): $\sqrt{s_{NN}} = 39.4$ TeV.

The study utilizes the Equivalent Photon Approximation (EPA). In heavy-ion collisions (PbPb, pPb), the large nuclear charge ($Z=82$) generates an intense flux of quasi-real photons, scaling as $Z^2$ (for pPb) and $Z^4$ (for PbPb) relative to protons. This effectively turns these collisions into high-luminosity photon-photon colliders, compensating for the lower integrated luminosity of ion beams compared to proton beams.

2.2 Theoretical Framework

• Lagrangian: The interaction is described by the effective Lagrangian term $\mathcal{L} \supset -\frac{1}{4f_a} a F_{\mu\nu} \tilde{F}^{\mu\nu}$, where $f_a$ is the ALP-photon coupling.
• Cross-Section Calculation: The differential cross-section for $\gamma\gamma \to \gamma\gamma$ is calculated by summing the squared helicity amplitudes. The total amplitude is a coherent sum of the Standard Model (SM) contribution (fermion and $W$-boson loops) and the ALP contribution (resonant s-channel).
  • The ALP amplitude includes the Breit-Wigner propagator: $\frac{1}{s - m_a^2 + i m_a \Gamma_a}$.
  • The total width $\Gamma_a$ is dominated by the decay into two photons: $\Gamma(a \to \gamma\gamma) = m_a^3 / (4\pi f_a^2)$.
• Integration: The total cross-sections are obtained by integrating the photon-photon luminosity ($dL_{\gamma\gamma}/dW$) over the invariant mass $W$ of the photon pair, using specific photon distribution functions for protons (based on form factors) and nuclei (based on Bessel functions and nuclear radius).

2.3 Statistical Analysis

The authors calculate the expected number of signal ($S$) and background ($B$) events using integrated luminosities of:

• $L_{pp} = 30 \text{ ab}^{-1}$
• $L_{pPb} = 27 \text{ pb}^{-1}$
• $L_{PbPb} = 110 \text{ nb}^{-1}$

Significance is evaluated using asymptotic formulae for:

• Exclusion: 95% Confidence Level (C.L.) limits ($SS_{excl} \leq 1.645$).
• Discovery: $3\sigma$ and $5\sigma$ discovery potentials ($SS_{disc} > 3$ and $5$).
• Systematic uncertainties ($\delta$) are assumed to be negligible as the exclusion significance shows weak dependence on $\delta$ in their calculations.

3. Key Contributions

1. Systematic Comparison of Collision Modes: The paper provides the first unified analysis comparing $pp$, $p$Pb, and PbPb collisions at the 100 TeV scale for heavy ALP searches. It highlights the complementarity of these modes:
   • PbPb: Superior sensitivity in the intermediate mass range due to the $Z^4$ enhancement of the photon flux.
   • pp: Dominates in the high-mass (TeV) range due to the "harder" photon spectrum and larger kinematic reach ($W_{max}$).
2. Kinematic Threshold Analysis: The authors explain the behavior of exclusion limits as a function of ALP mass ($m_a$). They identify "plateaus" and "dips" in the sensitivity curves, attributing them to the interplay between the ALP mass, the imposed kinematic cuts (e.g., $m_{\gamma\gamma} > 1000$ GeV), and the photon flux spectral shape (soft for ions, hard for protons).
3. Updated Projections: The study updates the search prospects for heavy ALPs, moving beyond the LHC's 13 TeV reach to the 100 TeV regime.

4. Key Results

• Cross-Section Behavior: The ALP contribution to the LbL cross-section decreases rapidly as $m_a$ increases, particularly for ion-ion collisions where the photon flux drops off sharply at high energies.
• Optimal Mass Regions for Sensitivity:
  • PbPb Collisions: The strongest exclusion limits are achieved around $m_a \simeq 250$ GeV. The sensitivity drops significantly for masses approaching 1 TeV due to the soft photon spectrum of lead nuclei.
  • pp and pPb Collisions: The strongest limits are achieved around $m_a \simeq 1$ TeV. The hard photon spectrum of protons allows access to higher invariant masses where the ALP resonance can still be produced efficiently.
• Exclusion and Discovery Limits:
  • The 95% C.L. exclusion limits on the coupling $f_a^{-1}$ reach values significantly lower than current LHC bounds in the TeV mass region.
  • For $m_a \approx 1$ TeV, the FCC-hh ($pp$ mode) can probe couplings roughly one order of magnitude stronger (lower $f_a^{-1}$) than the current best LHC limits (CMS/TOTEM and ATLAS).
  • The $5\sigma$ discovery reach extends to ALP masses well into the multi-TeV range, depending on the coupling strength.

5. Significance and Conclusion

The paper concludes that the FCC-hh offers exceptional physics potential for discovering or constraining heavy Axion-Like Particles. The study demonstrates that:

1. Complementarity is Key: No single collision mode covers the entire mass spectrum. PbPb collisions are essential for the intermediate mass window (hundreds of GeV), while $pp$ collisions are indispensable for the TeV-scale frontier.
2. Beyond LHC: The FCC-hh will significantly extend the exclusion limits on the ALP-photon coupling, particularly in the high-mass region ($m_a > 500$ GeV) where LHC sensitivity is currently limited by kinematic reach and lower photon fluxes.
3. Robustness: The results are robust against higher-order QCD/QED corrections, which are numerically small for the relevant kinematic regions.

This work establishes a comprehensive benchmark for future ALP searches at the 100 TeV frontier, guiding experimental strategies for the FCC-hh program.