Probing the Higgs Portal to a Strongly-Interacting Dark Sector at the FCC-ee

This paper proposes a machine learning strategy utilizing graph neural networks to detect Higgs-induced semi-visible jets from a confining dark sector at the Future Circular Collider, demonstrating the ability to constrain exotic Higgs branching ratios to the permille level across a wide parameter space.

The Gist

Imagine the universe as a giant, bustling city. We know a lot about the "visible" part of this city—the buildings, the people, the cars, and the traffic. This is what physicists call the Standard Model. But we also know there's a massive, invisible "dark sector" hiding in the shadows, making up most of the city's mass (Dark Matter), yet we've never seen a single person from there.

This paper is a proposal for how to catch a glimpse of this invisible neighborhood using a super-powered microscope called the FCC-ee (Future Circular Collider), which is planned to be built in the future.

Here is the story of their search, explained simply:

1. The Secret Tunnel (The Higgs Portal)

The scientists propose that the Higgs boson (a famous particle discovered a few years ago) acts like a secret tunnel or a "portal" connecting our visible city to the invisible dark sector.

If this tunnel exists, the Higgs boson could occasionally decay (break apart) not into the usual particles we know, but into "dark quarks." These are the building blocks of the dark world.

2. The "Semi-Visible" Ghosts (Semi-Visible Jets)

Once these dark quarks are created, they don't stay alone. They immediately start a chaotic party, similar to how a drop of ink spreads in water. This process is called "hadronization."

• The Problem: Some of the resulting dark particles are stable and invisible (they just fly away like ghosts). Others are unstable and decay back into normal particles we can see (like light or electrons).
• The Result: Instead of a clean, invisible signal, the scientists expect to see "Semi-Visible Jets." Imagine a firework that explodes. Usually, you see the whole burst. But in this scenario, the firework explodes, and half the sparks are visible light, while the other half are invisible smoke that vanishes instantly. You see a messy, partial explosion.

3. The Two Scenarios: The "Heavy" vs. The "Light"

The team realized there are two main ways this could happen, and they need different strategies to find them:

• Scenario A: The "Heavy" Invisible (High Invisible Fraction)
  Here, most of the dark particles are invisible ghosts. The explosion leaves a huge amount of missing energy.

  • The Strategy: It's like looking for a thief who ran away with a heavy safe. You can easily spot them because the safe is missing from the room. The scientists use simple math (kinematics) to look for events where a lot of energy is unaccounted for. This works well.

• Scenario B: The "Light" Invisible (Low Invisible Fraction)
  Here, most of the dark particles decay back into visible stuff. The explosion looks almost exactly like a normal firework, with only a tiny bit of invisible smoke.

  • The Problem: This is like looking for a thief who stole a single coin. The room looks almost the same as before, so it's very hard to tell if a theft happened. The "missing energy" is too small to be a useful clue.

4. The Super-Smart Detective (The Graph Neural Network)

To catch the "Light" invisible thieves (Scenario B), the scientists couldn't just look at the energy. They needed to look at the shape of the explosion.

They used a type of Artificial Intelligence called a Graph Neural Network (GNN). Think of this AI as a master detective who doesn't just look at what exploded, but how it exploded.

• The Analogy: Imagine you have two piles of confetti. One pile was thrown by a human (a normal particle), and the other by a machine (a dark semi-visible jet). To the naked eye, they look like random colorful bits. But the AI looks at the "family tree" of every single piece of confetti—how they split, how they moved, and how they relate to each other.
• The AI learns that the "dark" confetti has a unique, messy pattern that normal confetti doesn't have. This allows the scientists to spot the signal even when the missing energy is tiny.

5. The Results: A Powerful New Lens

The paper concludes that this combined strategy is incredibly powerful:

• For the "Heavy" invisible cases: Simple energy checks work great.
• For the "Light" invisible cases: The AI "super-detective" is essential. Without it, the signal would be lost in the background noise. With it, the scientists can detect these exotic events even when they are extremely rare.

The Bottom Line:
The authors show that the future FCC-ee collider, using this mix of simple physics checks and advanced AI, could probe the Higgs boson's connection to the dark sector with extreme precision. They could potentially rule out (or discover) these dark interactions at a level of one part in a thousand (the permille level). This would be a massive step forward in understanding what the "dark sector" of our universe actually looks like.

Technical Summary

Technical Summary: Probing the Higgs Portal to a Strongly-Interacting Dark Sector at the FCC-ee

Problem Statement
The Standard Model (SM) fails to explain phenomena such as dark matter (DM) and neutrino masses, suggesting the existence of a "dark sector" (DS). The Hidden Valley (HV) scenario proposes a strongly-coupled DS that interacts with the SM via mediators, potentially including the Higgs boson. If the Higgs decays into dark quarks ($q_d$), the subsequent dark confinement and hadronization can produce "semi-visible jets" (SVJs). These final states contain a mixture of visible SM particles (from unstable dark hadrons) and invisible particles (stable dark hadrons). While high-missing-energy signatures are detectable, SVJs with a low fraction of invisible states ($r_{inv}$) mimic SM backgrounds, particularly at hadron colliders where QCD activity is high. This work investigates the potential to probe these signatures at the Future Circular Collider in electron-positron mode (FCC-ee), leveraging its clean environment and reduced QCD background.

Methodology
The study constructs benchmark models within the HV framework where the Higgs boson acts as a portal to a dark $SU(3)_d$ sector. Two distinct regimes are analyzed:

1. High-$r_{inv}$ Regime: Motivated by DM relic abundance, this scenario assumes a significant fraction of dark hadrons are stable. The model includes dark pions ($\pi_d$), vector mesons ($\rho_d$), and $\eta'_d$, with specific decay channels to SM leptons, quarks, and photons. The invisible fraction $r_{inv}$ is a derived quantity dependent on hadronization parameters ($p_v, p_\eta, \Lambda$).
2. Low-$r_{inv}$ Regime: A model-agnostic parametric approach where $r_{inv}$ is an explicit input parameter, and the dark shower consists primarily of dark pions decaying to SM particles or invisibly.

Simulation and Analysis Strategy

• Event Generation: Signal ($e^+e^- \to ZH \to \ell^+\ell^- q_d\bar{q}_d$) and background ($ZZ, WW, ZH$) events are generated using Pythia8 with the Hidden Valley module and simulated through the IDEA detector parameterization in Delphes3. The center-of-mass energy is set to 240 GeV with an integrated luminosity of 10.8 ab$^{-1}$.
• Kinematic Selections: Events are selected based on a clean $Z \to \ell^+\ell^-$ recoil topology. Cuts include lepton isolation, di-lepton mass ($85 < m_{\ell\ell} < 95$ GeV), and Higgs recoil mass ($120 < m_{rec} < 130$ GeV).
  • For high-$r_{inv}$, selections include missing energy ($30 < E_{miss} < 90$ GeV) and azimuthal separation ($\min \Delta\phi < 1.7$).
  • For low-$r_{inv}$, where $E_{miss}$ is insufficient for discrimination, a tighter $\Delta\phi$ cut ($< 0.7$) is applied to suppress backgrounds like $WW$.
• Machine Learning (GNN): To address the challenge of low-$r_{inv}$ signals where kinematic cuts fail, the authors employ a graph neural network (LundNet). This tagger represents jets as Lund graphs encoding full clustering history, node kinematics ($k_t, \Delta, z, \psi, m$), and particle-type energy fractions ($f_e, f_\mu, f_\gamma, f_h$). The network distinguishes SVJ substructure from SM QCD jets.

Key Contributions and Results

• Sensitivity in High-$r_{inv}$: Kinematic selections alone provide good signal-background separation, achieving sensitivity to the exotic Higgs branching ratio ($BR(H \to q_d\bar{q}_d)$) at the percent level. The inclusion of the GNN tagger improves this sensitivity by nearly three orders of magnitude, reaching the permille level ($O(10^{-2}\%)$).
• Sensitivity in Low-$r_{inv}$: In regimes with small invisible fractions, standard kinematic cuts are ineffective as they remove too much signal or fail to suppress backgrounds. The GNN tagger is crucial here, restoring sensitivity and enabling constraints on $BR(H \to q_d\bar{q}_d)$ down to the permille level across the full parameter space.
• Coupling Constraints: Translating these branching ratio limits into constraints on the Higgs-dark quark coupling ($y$), the study finds expected 95% CL upper limits ranging from $4.1 \times 10^{-4}$ to $1.3 \times 10^{-3}$ across the parameter space.
• Background Suppression: The study demonstrates that the FCC-ee environment, free from overwhelming QCD backgrounds typical of the LHC, allows for precise identification of SVJs. The GNN effectively exploits differences in jet substructure, particularly the energy fractions of leptons and photons and the number of pseudo-jets, to discriminate signal from SM $ZH$ and $ZZ$ backgrounds.

Significance
The paper claims that the proposed strategy effectively probes a wide parameter space for Higgs-portal dark sectors at the FCC-ee. It highlights that while optimized kinematic selections suffice for high-missing-energy scenarios, modern jet-substructure tagging (specifically GNNs) is essential for discovering or constraining models with predominantly visible final states. The results suggest the FCC-ee can constrain Higgs exotic branching ratios into dark quarks at the permille level, significantly surpassing current LHC limits (approx. 18%). The authors note that the simplified modeling strategy can be extended to hadron colliders (FCC-hh, LHC) for complementary searches, though this specific study focuses on the $e^+e^-$ environment.