Search for additional scalar bosons within the Inert Doublet Model in a final state with two leptons at the FCC-ee

This paper presents a projected search for additional scalar bosons within the Inert Doublet Model at the FCC-ee, utilizing a parametric neural network to analyze final states with two leptons at 240 and 365 GeV, demonstrating the potential to exclude nearly the entire available parameter space and discover new scalars with masses up to 110 and 165 GeV respectively.

The Gist

Imagine the universe as a giant, bustling city. We know most of the buildings (normal matter) and the roads (forces like gravity and electromagnetism). But we also know there's a massive, invisible population living in the shadows—Dark Matter. We can't see them, but we know they are there because their gravity holds the city together. The problem is, we don't know what these "shadow people" look like.

This paper is a proposal for a new, ultra-powerful "search party" to find these shadow people using a massive particle accelerator called FCC-ee (Future Circular Collider) in Europe.

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

1. The Theory: The "Inert Doublet" Neighborhood

The scientists are testing a specific theory called the Inert Doublet Model (IDM).

• The Analogy: Imagine our known universe is a house with a standard set of furniture (the Standard Model). The IDM theory suggests there is a second, identical house right next door, but it's completely "inert" (inactive). Nothing from the first house can talk to the furniture in the second house directly.
• The Characters: This second house has four new pieces of furniture: a heavy chair ($H$), a heavy table ($A$), and two heavy lamps ($H^+$ and $H^-$).
• The Mystery Guest: Because of a special rule (a symmetry) in this theory, the lightest piece of furniture, the Chair ($H$), can never be destroyed. It is immortal. This makes it the perfect candidate for Dark Matter. It floats around the universe, invisible and stable.

2. The Plan: Catching Them in the Act

Since we can't see the Chair ($H$) directly, how do we find it? We have to catch the other furniture (the Table $A$) in the act of creating it.

• The Process: In the collider, scientists smash electrons and positrons together at incredible speeds. They hope to create a pair of new particles: a Table ($A$) and a Chair ($H$).
• The Escape: The Table ($A$) is unstable. It immediately breaks apart, turning into a standard Z-boson (a known particle) and another Chair ($H$).
• The Signature: The Z-boson decays into two visible particles: either two electrons or two muons (like a pair of glowing fireflies). The two Chairs ($H$) escape into the darkness, taking their energy with them.
• The Clue: The detector sees two bright fireflies and suddenly notices a "missing" amount of energy. It's like seeing a magician pull a rabbit out of a hat, but the rabbit vanishes instantly, leaving only a puff of smoke. That missing energy is the smoking gun for Dark Matter.

3. The Challenge: Finding a Needle in a Haystack

The problem is that the Standard Model (our known physics) creates lots of "fake" missing energy events. It's like trying to find a specific rare bird in a forest where thousands of other birds look similar.

• The Haystack: Background noise from known particles (like $W$ and $Z$ bosons) creates events that look almost exactly like the signal.
• The Solution: The scientists built a Digital Detective (a Parametric Neural Network).
  • Think of this as a super-smart AI that has been trained to look at the "shape" of the event. It doesn't just look at the two fireflies; it looks at their speed, their angle, how much energy they have, and how they move relative to each other.
  • Because the "Chair" could have different weights (masses), the AI is "parametric." This means the AI can change its detective style on the fly. If the Chair is light, the AI looks for one pattern; if it's heavy, it looks for another. It's like a detective who knows exactly how a thief would run depending on whether they are carrying a backpack or a briefcase.

4. The Results: A Clear View of the Forest

The scientists simulated what would happen if the FCC-ee ran for a few years with its full power (a massive amount of data called "integrated luminosity").

• The Discovery: If the Dark Matter Chair exists and weighs between 70 and 108 GeV (at 240 GeV energy) or up to 157 GeV (at 365 GeV energy), this experiment would almost certainly find it. They would see a clear signal that can't be explained by normal physics.
• The Exclusion: Even if they don't find it, the experiment is so sensitive that it will prove the Chair doesn't exist in almost the entire possible weight range. It's like saying, "We looked everywhere in the forest, and if the rare bird isn't here, it simply doesn't exist in this region."
• The Map: They drew a map (a graph) showing exactly which weights and mass differences they can rule out. They expect to rule out almost the entire "neighborhood" where this theory suggests the Dark Matter could live.

Why Does This Matter?

Currently, we are flying blind regarding Dark Matter. This paper shows that the FCC-ee is the perfect tool to either:

1. Catch the thief: Find the first evidence of Dark Matter particles.
2. Clear the suspect list: Prove that this specific theory (the Inert Doublet Model) is wrong, forcing scientists to look for new theories.

It's a high-stakes game of "Where's Waldo," but instead of a cartoon, we are looking for the fundamental building blocks of the universe's invisible mass. If the FCC-ee is built, it will be the most powerful magnifying glass humanity has ever held up to the dark side of the cosmos.

Technical Summary

1. Problem Statement

The nature of Dark Matter (DM) remains one of the most significant unsolved mysteries in physics. While cosmological observations confirm its existence, the Standard Model (SM) does not contain a viable DM candidate. The Inert Doublet Model (IDM) is a well-motivated extension of the SM that introduces a second Higgs doublet stabilized by an exact $Z_2$ symmetry.

• The Model: The IDM predicts four new scalar bosons: a neutral CP-even scalar ($H$), a neutral CP-odd scalar ($A$), and a charged pair ($H^\pm$). The $Z_2$ symmetry ensures the lightest of these (chosen here as $H$) is stable, making it a viable Weakly Interacting Massive Particle (WIMP) DM candidate.
• The Challenge: Existing experimental constraints (e.g., from LHC and LEP) have not yet fully explored the IDM parameter space, particularly due to high thresholds on missing transverse energy ($E_T^{miss}$) which reduce sensitivity to specific mass configurations.
• The Goal: This paper investigates the discovery and exclusion potential of the Future Circular Collider in electron-positron mode (FCC-ee) for the pair production of these new scalars ($e^+e^- \to AH$), focusing on the final state with two same-flavor leptons (electrons or muons) and missing energy.

2. Methodology

A. Theoretical Framework and Scenarios

The analysis scans the parameter space defined by the mass of the dark scalar ($M_H$) and the mass splitting ($M_A - M_H$). Three specific benchmark scenarios are considered to test sensitivity to different coupling regimes:

1. Scenario S1: $M_{H^\pm} = M_A$ and $\lambda_{345} = 0$. This targets the dominant $e^+e^- \to AH$ production via $Z$-boson radiation.
2. Scenario S2: $M_{H^\pm} = M_A$ and $\lambda_{345} = \lambda_{max}$. This activates the Higgs-strahlung process ($e^+e^- \to Zh \to ZHH$).
3. Scenario S3: $M_{H^\pm} = M_{H^\pm}^{max}$ and $\lambda_{345} = \lambda_{max}$. This includes contributions from charged scalar pair production ($H^+H^-$).

The study utilizes constraints from vacuum stability, perturbative unitarity, electroweak precision observables (S, T, U parameters), and direct detection limits (LUX-ZEPLIN).

B. Simulation and Event Generation

• Signal: Generated using MADGRAPH5_aMC@NLO interfaced with PYTHIA. Two final states are generated directly to account for interferences: $HH\ell\ell$ (via virtual/real $Z$) and $HH\ell\ell\nu\nu$ (via $W$ pairs).
• Backgrounds: Includes diboson ($WW, ZZ$), inclusive lepton pair production ($ee, \mu\mu, \tau\tau$), Higgs-strahlung ($Zh$), and top quark pairs ($t\bar{t}$) at 365 GeV.
• Detector Simulation: Events are processed through DELPHES using the IDEA detector configuration (FCC-ee Winter2023 campaign).
• Luminosity: The study assumes the target integrated luminosities for FCC-ee: 10.8 ab$^{-1}$ at $\sqrt{s} = 240$ GeV and 2.7 ab$^{-1}$ at $\sqrt{s} = 365$ GeV.

C. Event Selection and Object Definition

• Preselection: Requires exactly two same-flavor leptons ($e$ or $\mu$) with $p > 5$ GeV.
• Kinematic Cuts:
  • Invariant mass ($M_{\ell\ell}$) and longitudinal momentum ($p_z$) cuts are applied to reject $Z$-boson backgrounds. For $\sqrt{s}=365$ GeV, a dynamic cut $M_{\ell\ell} < (-9.0/14.0 \times |p_z| + 200)$ is used.
  • Missing transverse energy ($E_T^{miss} > 5$ GeV) is required.
  • Veto on additional leptons, photons ($E > 5$ GeV), or jets.
• Background Rejection: These cuts reduce $ZZ$ and $WW$ backgrounds by factors of $\sim 100$ while retaining significant signal efficiency (up to 65% for large mass splittings).

D. Machine Learning Strategy: Parametric Neural Network (pNN)

To maximize sensitivity across the entire mass grid, the authors employ a Parametric Neural Network (pNN) rather than training separate networks for each mass point.

• Inputs: Kinematic variables (dilepton energy, momentum, invariant mass, recoil mass, boost, angles) plus the signal mass hypotheses ($M_H$ and $M_A$) as input features.
• Mechanism: By feeding the mass parameters into the network, the model learns the optimal selection criteria as a continuous function of the mass hypothesis. This allows a single network to discriminate signal from background for any point in the $(M_H, M_A - M_H)$ plane.
• Architecture: A PyTorch-based feed-forward network with 4 layers (250 units each), ReLU activation, and dropout.
• Training: The network is trained on a mix of signal points and backgrounds, with signal weights normalized to ensure no bias toward specific mass points.

3. Key Contributions

1. First FCC-ee IDM Study: This is the first dedicated study of IDM scalar pair production at the FCC-ee, extending previous work done for the CLIC linear collider.
2. Detector-Level Analysis: Unlike previous generator-level studies, this analysis incorporates full detector simulation (DELPHES/IDEA) and realistic object reconstruction.
3. Parametric Approach: The implementation of a pNN allows for a systematic scan of the full kinematically available mass range without the computational cost of training thousands of individual classifiers.
4. Comprehensive Scenario Testing: The study evaluates the impact of different coupling scenarios ($\lambda_{345}$) and charged scalar mass hierarchies ($M_{H^\pm}$), demonstrating that sensitivity is robust across these variations.

4. Results

A. Exclusion Limits (95% CL)

Using a profile likelihood ratio test on the pNN output (threshold > 0.9):

• At $\sqrt{s} = 240$ GeV (10.8 ab$^{-1}$): Almost the entire available phase space is expected to be excluded. The reach extends to $M_H = 110$ GeV.
• At $\sqrt{s} = 365$ GeV (2.7 ab$^{-1}$): The exclusion reach extends to $M_H = 165$ GeV.
• The exclusion contours cover the region where $M_A - M_H$ is small (near the $Z$ mass) and large, limited only by the kinematic production threshold ($M_H + M_A < \sqrt{s}$).

B. Discovery Potential (5$\sigma$)

• At $\sqrt{s} = 240$ GeV: A 5$\sigma$ discovery is possible for $M_H$ up to 108 GeV (for $M_A - M_H = 15$ GeV).
• At $\sqrt{s} = 365$ GeV: The discovery reach extends to $M_H = 157$ GeV (for $M_A - M_H = 15$ GeV).
• Sensitivity Dip: A slight reduction in sensitivity is observed when $M_A - M_H \approx M_Z$ due to reduced kinematic discrimination against the $Z \to \ell\ell$ background.

C. Comparison with Other Constraints

The projected FCC-ee reach significantly complements existing limits:

• It covers regions currently allowed by LEP SUSY recast limits and relic density constraints.
• It is complementary to LHC/HL-LHC searches, which are more sensitive to larger mass splittings ($M_A - M_H$) but struggle with the low-mass, low-splitting region that FCC-ee targets.

5. Significance

This paper demonstrates that the FCC-ee is a powerful machine for probing the Inert Doublet Model, specifically in the "compressed" spectrum region where the mass difference between the new scalars is small.

• Completeness: The ability to probe $M_H$ up to 165 GeV at 365 GeV means the FCC-ee could potentially rule out the entire IDM parameter space relevant for Dark Matter if no signal is found.
• Methodological Advance: The successful application of parametric neural networks to high-energy physics searches at lepton colliders sets a precedent for future analyses, allowing for efficient handling of large parameter spaces with high-dimensional data.
• Dark Matter Implications: By testing the specific mass ranges where the IDM provides the correct relic density, this study directly addresses the nature of Dark Matter, offering a path to either discover the IDM scalars or definitively exclude this specific WIMP candidate model.