Search for the production of dark Higgs in the framework of Mono-Z$^{\prime}$ portal at the FCC-ee simulated electron-positron collisions at $\sqrt{s} = 240$ GeV

This paper investigates the production of dark Higgs bosons via a Mono-Z' portal in simulated electron-positron collisions at the FCC-ee operating at 240 GeV, utilizing dimuon plus missing transverse energy events to set 95% confidence level upper limits on the dark Higgs mass in the absence of new physics discoveries.

The Gist

Imagine the universe is a giant, bustling city. For decades, we've had a very detailed map of this city called the Standard Model. It tells us about all the known citizens (particles like electrons and quarks) and how they interact. In 2012, we finally found the "Mayor" of this city, the Higgs boson, which gives everything else its mass.

But here's the problem: our map is incomplete. We know there are "dark" neighborhoods we can't see—places where Dark Matter and other mysterious forces might live. We need a new, more powerful flashlight to explore them.

This paper is a proposal for how to use a future, super-powerful flashlight called the FCC-ee (Future Circular Collider) to hunt for a specific type of invisible resident: the Dark Higgs.

Here is the story of the hunt, broken down into simple concepts:

1. The Setup: A High-Speed Dance

Imagine two dancers, an electron and a positron, spinning toward each other at nearly the speed of light in a giant circular ballroom (the collider). When they crash, they create a burst of energy that can spawn new particles.

The scientists are looking for a specific dance move:

• The crash creates a Z' boson (a new, heavy messenger particle).
• This Z' immediately splits into two things:
  1. A pair of muons (heavy cousins of electrons) that fly out and can be seen by our detectors.
  2. A Dark Higgs (the invisible suspect).

2. The Mystery: The "Missing" Partner

The Dark Higgs is tricky. It doesn't play by the rules of our visible world. It decays instantly into Dark Matter particles (let's call them "ghosts"). These ghosts don't leave a trace; they just vanish through the walls of the detector.

So, what does the detector see?

• It sees two muons flying out.
• It sees a huge gap in the energy balance. It's like watching a magic trick where the magician throws a ball, and suddenly, half the energy of the room is gone. The detector screams, "Something invisible took that energy!"

3. The Detective Work: Filtering the Noise

The problem is that the ballroom is noisy. Every day, millions of "fake" magic tricks happen where particles just fly off naturally, mimicking the missing energy. This is the Background Noise.

To find the real Dark Higgs, the scientists had to write a very strict set of rules (cuts) to filter out the noise:

• The Angle Rule: The two muons must be flying in a very specific direction relative to the missing energy. If they aren't "back-to-back" like a perfect dance pair, it's probably just noise.
• The Energy Rule: The missing energy must match the energy of the muons in a precise way.
• The Isolation Rule: The muons must be alone, not surrounded by a crowd of other particles (like a VIP in a club).

By applying these rules, the scientists managed to silence the noisy crowd and isolate the few events that looked like a real Dark Higgs signal.

4. The Results: What Did They Find?

The team ran a massive computer simulation of this experiment, assuming the FCC-ee runs for a long time with a huge amount of data (10.8 "inverse attobarns"—a unit that basically means "a lot of collisions").

• The Good News: If the Dark Higgs exists and has a mass between 20 and 80 GeV (lighter than our known Higgs), the FCC-ee would be able to spot it with incredible confidence (a "5-sigma" discovery, which in science means "we are 99.9999% sure this isn't a fluke").
• The Bad News (or the Safety Net): If they don't find it, they can still say something important. They can draw a line on the map and say, "The Dark Higgs is definitely not hiding in this mass range." They set a new, stricter limit than ever before, pushing the "exclusion zone" down to 20 GeV.

5. Why This Matters

Think of the Large Hadron Collider (LHC) as a sledgehammer. It smashes protons together with brute force. It's great for finding heavy things, but it's messy.

The FCC-ee is like a scalpel. It smashes electrons and positrons together in a clean, controlled environment. Because the starting energy is so precise, if something is missing, we know exactly how much is missing.

The Bottom Line:
This paper is a blueprint for a future treasure hunt. It tells us that if we build this specific type of collider and look for "missing energy" paired with two muons, we have a very high chance of either:

1. Discovering a new particle that explains what Dark Matter is (the "Dark Higgs").
2. Proving that this specific type of Dark Higgs doesn't exist, which forces scientists to rewrite their theories and look elsewhere.

It's about turning the "dark" parts of our universe's map into something we can finally see.

Technical Summary

1. Problem Statement

The discovery of the Standard Model (SM) Higgs boson in 2012 opened the door to searching for "exotic" decays into invisible particles, potentially linking the Higgs sector to Dark Matter (DM). While the Large Hadron Collider (LHC) has placed stringent limits on invisible Higgs decays and heavy $Z'$ bosons, a specific mass range remains unexplored: light neutral gauge bosons ($Z'$) below 90 GeV and associated light dark Higgs bosons ($h_D$).

Previous experiments (LEP, LHC Run 1 & 2) have excluded heavy $Z'$ masses (0.6–5.15 TeV) and set limits on invisible decays, but they lack sensitivity to light $Z'$ bosons that do not couple strongly to quarks. The Future Circular Collider in Electron-Positron mode (FCC-ee) offers a unique environment with high luminosity, clean collision conditions, and precise energy control to probe this "light dark sector" via the Mono-$Z'$ portal mechanism.

2. Methodology

Theoretical Framework

The study utilizes a Simplified Dark Higgs (DH) Model within the Mono-$Z'$ portal framework:

• Process: $e^+e^- \to Z' h_D$, where the $Z'$ decays into a dimuon pair ($Z' \to \mu^+\mu^-$) and the dark Higgs decays invisibly into dark matter particles ($h_D \to \chi\bar{\chi}$).
• Mass Assumptions:
  • $M_{Z'} \leq 90$ GeV.
  • $M_{h_D} = M_{Z'}$ if $M_{Z'} < 125$ GeV; otherwise $M_{h_D} = 125$ GeV.
• Couplings:
  • Lepton coupling ($g_l$): Varied between 0.00051 and 0.003 (constrained by LEP/LHC limits).
  • Dark sector coupling ($g_D$): Fixed at 1.0 (recommended by the LHC Dark Matter Working Group).

Simulation and Detector Model

• Collider: FCC-ee operating at a center-of-mass energy $\sqrt{s} = 240$ GeV with an integrated luminosity of 10.8 ab$^{-1}$ (Run 1 scenario).
• Event Generation:
  • Signal: Generated using WHIZARD 3.1.1 with Initial State Radiation (ISR) effects.
  • Backgrounds: Standard Model processes including $Z/\gamma \to \mu^+\mu^-$, $Z/\gamma \to \tau^+\tau^-$, $WW$, $ZZ$, and $t\bar{t}$.
  • Showering/Hadronization: Handled by Pythia 6.24.
  • Detector Simulation: Fast simulation of the IDEA detector concept using DELPHES 3.

Event Selection Strategy

The analysis targets the final state: Two muons + Large Missing Transverse Energy ($E_{miss}$).

1. Pre-selection:
   • Two opposite-charge muons with $p_T > 5$ GeV and $|\eta| < 2.5$.
   • Isolation cut: $\Sigma p_T / p_T^\mu < 0.1$ within $\Delta R = 0.5$.
   • Dimuon invariant mass ($M_{\mu\mu}$) < 120 GeV.
2. Recoil Mass Calculation:
   • The mass of the invisible system is reconstructed using energy-momentum conservation:
     $$M_{rec} = \sqrt{s + M_{\mu\mu}^2 - 2\sqrt{s}E_{\mu\mu}}$$
   • A cut of $M_{rec} < 90$ GeV is applied to suppress SM $ZH$ and $ZZ$ backgrounds.
3. Final Selection Cuts:
   • Energy Balance: $|E_{miss} - E_{\mu\mu}| / E_{\mu\mu} < 0.4$.
   • Azimuthal Angle: $\Delta\phi(\mu\mu, \vec{E}_{miss}) > 3.0$ rad.
   • 3D Angle: $\cos(\text{Angle}_{3D}) < -0.8$ (ensuring back-to-back topology).
   • Angular Separation: $\Delta R(\mu^+\mu^-) < 1.7$.

Statistical Analysis

• Method: Profile Likelihood Ratio test using the $M_{rec}$ distribution.
• Significance: Calculated as $S_L = \sqrt{-2\ln\lambda}$.
• Limits: Exclusion limits derived using the $CL_s$ modified frequentist method at 95% Confidence Level (CL). Systematic uncertainties (10% flat uncertainty) are treated as nuisance parameters.

3. Key Contributions

• First FCC-ee Study on Light Mono-$Z'$: This work specifically targets the unexplored region of light $Z'$ ($<90$ GeV) and light dark Higgs bosons, a domain inaccessible to the LHC due to QCD backgrounds and trigger thresholds.
• Optimized Selection Criteria: The paper demonstrates that specific kinematic cuts (particularly the back-to-back topology and energy balance) can effectively suppress the overwhelming $Z/\gamma \to \mu^+\mu^-$ background while retaining signal efficiency.
• Comprehensive Sensitivity Mapping: The study provides exclusion contours and discovery potential across a wide range of $M_{h_D}$ (20–80 GeV) and coupling constants ($g_l$).

4. Results

Discovery Potential

• 5$\sigma$ Discovery:
  • For $M_{h_D} = 30$ GeV and $g_l = 0.003$, a 5$\sigma$ discovery is achievable with an integrated luminosity of 1.21 ab$^{-1}$.
  • For $M_{h_D} = 60$ GeV, the required luminosity increases to 3.5 ab$^{-1}$.
  • With the full Run 1 luminosity (10.8 ab$^{-1}$), signals with $M_{h_D} > 20$ GeV are expected to exceed the 5$\sigma$ threshold.
• Efficiency: The selection cuts maintain a flat efficiency for signal events with high muon transverse momentum ($p_T > 40$ GeV) while suppressing diboson ($WW, ZZ$) and Drell-Yan backgrounds by several orders of magnitude.

Exclusion Limits

• Mass Exclusion: If no signal is observed, the FCC-ee can exclude dark Higgs masses in the range 20 GeV to 80 GeV for a coupling of $g_l = 0.003$.
• Coupling Sensitivity: The experiment loses sensitivity for couplings $g_l < 0.00051$.
• Comparison to Previous Limits: This analysis significantly improves upon LEP limits, which excluded masses only down to ~60 GeV (leptonic) and ~112 GeV (hadronic). The FCC-ee extends the lower bound of exclusion down to 20 GeV.

5. Significance

This study highlights the FCC-ee as a premier facility for "Dark Sector" physics, specifically for light mediators that are invisible to hadron colliders.

• Complementarity: It fills a critical gap in the global search for Dark Matter, covering the low-mass region ($M_{Z'} < 90$ GeV) where LHC sensitivity is limited by trigger thresholds and background noise.
• Precision: The clean environment of $e^+e^-$ collisions allows for the precise reconstruction of the recoil mass, enabling the detection of invisible decays with high resolution.
• Physics Impact: Establishing or excluding the existence of a light dark Higgs coupled to a light $Z'$ would provide crucial insights into the nature of Dark Matter and the mechanism of electroweak symmetry breaking in hidden sectors.

In conclusion, the paper demonstrates that with 10.8 ab$^{-1}$ of data at 240 GeV, the FCC-ee has the potential to either discover a light dark Higgs boson or set world-leading exclusion limits, effectively probing the Mono-$Z'$ portal parameter space that has remained unexplored by previous colliders.