Search for Dark Matter in 2HDMS at LHC and future Lepton Colliders

This paper investigates the phenomenological prospects of the Two Higgs Doublet and Complex Singlet Scalar Extension (2HDMS) model as a dark matter candidate, demonstrating through benchmark scans and collider analyses that while the High Luminosity LHC may only offer hints of new physics, future lepton colliders serve as highly efficient discovery probes for this framework.

The Gist

Imagine the universe is a giant, bustling city. For decades, we've known the rules of this city (the Standard Model of particle physics), but we've noticed something strange: there's a massive, invisible crowd moving through the streets that we can't see, but we can feel their gravity pulling on buildings and cars. This invisible crowd is Dark Matter. We know it's there, but we have no idea what it's made of.

This paper is like a detective's roadmap for a team of scientists trying to catch a glimpse of these invisible citizens. They are testing a specific theory called the 2HDMS (Two Higgs Doublet and Complex Singlet Model). Think of this theory as a "renovation plan" for the city's central power plant (the Higgs field).

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

1. The New Neighborhood (The Model)

In our current understanding, the Higgs field is like a single family living in a house. This new theory suggests the Higgs family has expanded. They now have:

• Two Doublets: Like two extra wings added to the house.
• A Singlet: A secret, hidden basement apartment.

The "Dark Matter" in this story is a ghostly particle living in that secret basement (the singlet). It's invisible to our eyes but interacts with the rest of the house through a special "portal" (the Higgs bosons).

2. The Detective Work (The Constraints)

Before the detectives can go hunting, they have to make sure their theory doesn't break the laws of physics. They ran millions of simulations to check:

• Does the house stand up? (Theoretical stability).
• Does it match the police reports? (Experimental data from current colliders like the LHC).
• Does it explain the invisible crowd? (Dark Matter density in the universe).

They found several "suspects" (called Benchmarks) that fit all the rules. Some suspects are light (like a feather), some are medium-sized (like a dog), and some are heavy (like a boulder).

3. The Hunt: Three Different Hunting Grounds

The scientists realized that catching these invisible ghosts requires different tools depending on how heavy the ghost is. They compared three different "hunting grounds" (colliders):

A. The Giant Net (The High-Luminosity LHC)

• The Tool: The Large Hadron Collider (LHC) is like a massive, chaotic demolition derby. It smashes protons together at incredible speeds.
• The Strategy: They look for "missing energy." If a collision happens and energy suddenly disappears, it might have been carried away by a Dark Matter ghost.
• The Result: The LHC is great at finding heavy things, but it's like trying to find a needle in a haystack made of other needles. The background noise is so loud that for many of their "suspects," the LHC can only give a weak hint (a "maybe"), not a definitive "gotcha."

B. The Precision Sniper (Electron-Positron Colliders)

• The Tool: Machines like the ILC or FCC-ee. These are like clean, quiet shooting galleries. They smash electrons and positrons together.
• The Strategy: Because the environment is so clean, they can spot tiny, subtle signals. They look for a "Mono-Z" event: a Z-particle (a messenger) flying off alone, leaving the Dark Matter ghost behind.
• The Result: This is the best place to catch the "Light" and "Medium" ghosts. It's like using a high-powered telescope to spot a small bird in a clear sky. The LHC would miss these, but these machines would see them clearly.

C. The Heavy Hammer (The Muon Collider)

• The Tool: A proposed machine that smashes muons (heavy cousins of electrons).
• The Strategy: Muons are heavy, so they can punch through to much higher energies than electrons.
• The Result: This is the only tool capable of catching the "Heavy" ghosts. If the Dark Matter is as heavy as a boulder, the LHC and electron colliders are too weak to produce it. The Muon collider is the only one with enough "muscle" to create these heavy particles and catch them in the act.

4. The "95 GeV" Mystery

The paper also mentions a strange clue: a faint signal seen years ago at an old collider (LEP) and recently at the LHC, suggesting a particle exists at 95 GeV.

• The Analogy: Imagine finding a footprint that doesn't quite match any known animal.
• The Solution: The scientists checked if their "renovated house" (2HDMS) could explain this footprint. They found that yes, it can! They adjusted their "suspects" to include this 95 GeV particle, making their theory even more attractive.

The Big Takeaway

The main conclusion of this paper is a lesson in teamwork:

• The LHC is a powerful brute force, but it might miss the subtle or very heavy clues.
• Electron Colliders are the precision experts, perfect for the lighter, elusive suspects.
• Muon Colliders are the heavy hitters, needed for the massive, hard-to-reach suspects.

In short: You can't catch all the criminals with just one type of police car. To solve the mystery of Dark Matter, we need a fleet of different vehicles. This paper provides the map showing exactly which car to use for which suspect, proving that future colliders (especially the Muon collider) are essential for finally seeing the invisible.

Technical Summary

1. Problem Statement

The Standard Model (SM) lacks a viable Dark Matter (DM) candidate. While the discovery of the 125 GeV Higgs boson solidified the SM, it also opened the door for extended scalar sectors that could explain DM. Recent experimental anomalies, specifically a $\sim 2\sigma$ excess in the $b\bar{b}$ channel at LEP and a $\sim 3\sigma$ excess in the $\gamma\gamma$ channel at the LHC (CMS/ATLAS) around 95 GeV, suggest the existence of a light scalar.

The authors investigate the Two Higgs Doublet and Complex Singlet Scalar Extension (2HDMS) model. This model extends the SM with two Higgs doublets and a complex singlet scalar. The imaginary part of the singlet serves as a stable DM candidate ($A_S$) stabilized by a $Z_2'$ symmetry. The core challenge is to identify benchmark scenarios within this model that satisfy all theoretical and experimental constraints (including DM relic density, direct/indirect detection limits, and Higgs signal strengths) while determining the optimal collider strategy (HL-LHC vs. future lepton colliders) to discover such DM.

2. Methodology

The study employs a systematic, multi-stage approach:

• Model Definition: The 2HDMS potential includes Type-II 2HDM interactions and a complex singlet. The model has 15 free parameters in the mass basis, including scalar masses ($h_1, h_2, h_3, A, H^\pm, A_S$), mixing angles, and couplings.
• Constraint Scanning: An exhaustive scan of the parameter space is performed to find points consistent with:
  • Theoretical constraints: Boundedness-from-below (bfb), tree-level unitarity, and CP conservation.
  • Experimental constraints: 125 GeV Higgs properties (signal strengths), invisible Higgs decay limits (ATLAS/CMS), flavor physics ($b \to s\gamma$, $B_s \to \mu^+\mu^-$), electroweak precision tests (S, T, U parameters), and the 95 GeV excess (for specific benchmarks).
  • DM Constraints: Planck relic density ($\Omega h^2$), LUX-ZEPLIN (LZ) direct detection limits, and Fermi-LAT indirect detection limits.
• Benchmark Selection: Six representative benchmarks are selected covering three DM mass regimes:
  • Light DM: $m_{A_S} \approx 55$ GeV and $70$ GeV.
  • Intermediate DM: $m_{A_S} \approx 156$ GeV and $400$ GeV.
  • Heavy DM: $m_{A_S} \approx 1000$ GeV.
  • Some benchmarks include a 95 GeV scalar ($h_1$) to explain the LEP/LHC excess; others do not.
• Collider Simulation:
  • Tools: MadGraph5_aMC@NLO (LHC), WHIZARD (Lepton/Muon colliders), Pythia (showering/hadronization), Delphes (detector simulation), and MadAnalysis5 (analysis).
  • Processes: Analyses cover Gluon Fusion (GGF), Vector Boson Fusion (VBF), and associated production ($b\bar{b}H$, $t\bar{t}H$).
  • Final States: Mono-jet, di-jet, $b\bar{b}$, $t\bar{t}$, mono-photon, mono-Z, all accompanied by Missing Transverse Energy (MET).
  • Facilities: HL-LHC ($\sqrt{s}=14$ TeV, 3000 fb$^{-1}$), $e^+e^-$ colliders (ILC, FCC-ee, CLIC, CEPC), and Muon Colliders ($\sqrt{s}=1, 3, 10$ TeV).

3. Key Contributions

• Comprehensive Benchmarking: The paper provides a rigorous set of benchmark points that simultaneously satisfy the 95 GeV excess (where applicable), DM relic density, and collider constraints, filling a gap in previous studies that often treated these separately.
• Complementarity Analysis: It establishes a clear hierarchy of discovery potential across different collider types. It demonstrates that while the HL-LHC has limited reach for heavy or singlet-dominated scenarios, future lepton colliders offer superior sensitivity.
• Muon Collider Advantage: A significant contribution is the demonstration of the Muon Collider's unique capability to probe heavy DM and heavy mediators via $t\bar{t}$ and $b\bar{b}$ associated production channels, leveraging the enhanced muon-Yukawa coupling compared to electron-Yukawa coupling.
• Singlet-Dominated Challenges: The study highlights the "challenging scenarios" where the mediator is singlet-dominated. While these can satisfy DM constraints, their production cross-sections at lepton colliders are suppressed, necessitating future 100 TeV hadron colliders (FCC-hh/SPPC) for discovery.

4. Key Results

A. Benchmark Performance at HL-LHC

• Light DM ($m_{A_S} < 100$ GeV): The HL-LHC shows very low sensitivity. For $m_{A_S} \approx 55$ GeV, significance is $\sim 0.3\sigma$; for $m_{A_S} \approx 70$ GeV (singlet-like), it reaches $\sim 0.55\sigma$ (GGF) and $\sim 1.94\sigma$ (VBF).
• Intermediate DM ($m_{A_S} \approx 156$ GeV): The $b\bar{b} + \text{MET}$ channel (associated production) yields a significance of $\sim 1.95\sigma$ at HL-LHC. This is competitive with VBF and suggests $b\bar{b}$ associated production is a viable channel for heavy Higgs searches in Type-II 2HDM.
• Heavy DM ($m_{A_S} \ge 1000$ GeV): Production cross-sections at 14 TeV are negligible ($< 10^{-6}$ fb). The HL-LHC cannot probe these scenarios.

B. Performance at Future $e^+e^-$ Colliders (ILC, FCC-ee, CEPC)

• Low Mass Dominance: These colliders excel at low DM masses via the Mono-Z + MET channel (Higgsstrahlung).
  • DM55w95: At $\sqrt{s}=250$ GeV, significance reaches 11$\sigma$ (1 ab$^{-1}$).
  • DM70: At $\sqrt{s}=250$ GeV, significance reaches 3$\sigma$ (3 ab$^{-1}$).
• Limitation: Sensitivity drops significantly for intermediate and heavy DM masses due to kinematic thresholds and the lack of resonant production for singlet-like mediators.

C. Performance at Future Muon Colliders

• Intermediate Mass ($m_{A_S} \approx 156$ GeV):
  • At $\sqrt{s}=3$ TeV, the $b\bar{b} + \text{MET}$ channel achieves 6.3$\sigma$ (3 ab$^{-1}$).
  • Remarkably, at $\sqrt{s}=1$ TeV, the Mono-photon + MET channel (mediated by s-channel $h_3$) achieves 3$\sigma$ (3 ab$^{-1}$), outperforming $e^+e^-$ colliders at the same energy due to enhanced muon couplings.
• Heavy Mass ($m_{A_S} \approx 1000$ GeV):
  • Only the $\sqrt{s}=10$ TeV Muon Collider can probe the heavy benchmark (DM1000w95).
  • The $t\bar{t} + \text{MET}$ channel yields a significance of $\sim 3\sigma$ (10 ab$^{-1}$). This is the only viable discovery channel for such heavy masses in the considered scenarios.

D. Challenging Scenarios (Singlet-Dominated Mediators)

• Benchmarks like DM400 and DM1000 (without the 95 GeV excess) feature singlet-dominated mediators.
• Production cross-sections at lepton colliders are extremely small ($< 10^{-3}$ fb), rendering them invisible even at high luminosity.
• Solution: Future 100 TeV hadron colliders (FCC-hh/SPPC) are required, where cross-sections increase by factors of 100–1000, making these scenarios accessible.

5. Significance and Conclusion

This paper provides a definitive roadmap for Dark Matter searches in the 2HDMS framework. The primary conclusions are:

1. Collider Complementarity: No single machine can cover the entire parameter space.
   • $e^+e^-$ Colliders are the undisputed leaders for Light DM ($< 100$ GeV) via Mono-Z signatures.
   • Muon Colliders are essential for Intermediate and Heavy DM ($> 100$ GeV), particularly via associated production ($b\bar{b}$, $t\bar{t}$) and Mono-photon channels, leveraging the unique muon-Yukawa coupling.
   • HL-LHC offers only marginal hints (up to $\sim 2\sigma$) for specific intermediate scenarios but lacks the reach for heavy DM.
2. Discovery Potential: The study proves that future lepton colliders are not just precision machines but powerful discovery probes for DM, often surpassing the HL-LHC in sensitivity for specific BSM scenarios.
3. Guidance for Future Experiments: The authors recommend that future experimental programs prioritize the Muon Collider for heavy DM searches and the $e^+e^-$ colliders for light DM, while acknowledging the need for 100 TeV hadron colliders to tackle singlet-dominated "dark sector" scenarios.

In summary, the 2HDMS model offers a rich phenomenology where the interplay between DM mass, mediator nature (doublet vs. singlet), and collider type dictates the discovery strategy. The work strongly motivates the construction of high-energy lepton colliders, particularly Muon Colliders, as the next frontier in the search for Dark Matter.