⚛️ phenomenology

Probing Fermion-Portal Scalar Dark Matter through Charged Vector-Like Fermions at Future Muon Colliders

This paper investigates a minimal fermion-portal scalar dark matter model constrained by current experimental limits, demonstrating that its viable parameter space can be effectively probed through the discovery of lightest charged vector-like fermions at future 3 TeV and 10 TeV muon colliders via clean $e^+e^- + \cancel{E}_T$ signatures.

Original authors: Songshaptak De, Tapoja Jha, Najimuddin Khan, Madhurima Pandey

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Songshaptak De, Tapoja Jha, Najimuddin Khan, Madhurima Pandey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling city. For a long time, we thought we knew the population: the visible buildings (stars and galaxies) and the people walking around (ordinary matter). But recently, we've realized that 27% of the city is actually made of invisible "ghosts" called Dark Matter. We know they are there because they have gravity—they pull on the visible stuff—but we can't see them, touch them, or hear them. They don't talk to light.

This paper is like a detective story trying to figure out what these ghosts are made of and how we might finally catch a glimpse of them.

The Suspect: A New Theory of Dark Matter

The authors propose a specific theory about these ghosts. Imagine the Dark Matter is a shy, invisible person (a scalar particle) living in a house. In the past, scientists thought this person only interacted with the rest of the world through a specific door: the Higgs Portal (a connection to the Higgs field, which gives particles mass).

However, recent "police reports" (experiments looking for Dark Matter directly) have shown that if this person only uses the Higgs door, they would have been caught by now. The door is too obvious. So, the authors suggest a new theory: this Dark Matter person has secret backdoors.

These backdoors are Vector-Like Fermions. Think of these as heavy, charged bodyguards that the Dark Matter ghost can talk to.

The Setup: The Dark Matter ghost (the scalar) hangs out with these heavy bodyguards.
The Interaction: Instead of just using the Higgs door, the Dark Matter can now interact with the universe through these bodyguards. This allows the Dark Matter to "annihilate" (destroy each other) in a way that explains why there is just the right amount of it left over from the Big Bang, without getting caught by current detectors.

The Investigation: Checking the Alibis

Before looking for these particles, the authors had to make sure their theory didn't break the laws of physics. They ran a series of "stress tests":

Vacuum Stability: Is the universe's foundation strong enough to hold this new theory? (Yes, it is).
Unitarity: Do the math calculations stay reasonable, or do they explode into nonsense? (They stay reasonable).
The "Lepton Flavor" Test: Does this theory cause weird, forbidden reactions (like a muon turning into an electron and a photon)? The authors show that by tuning the "strength" of the interactions, they can avoid these forbidden reactions.

They found that if the Dark Matter is very light (around the mass of a proton) or very heavy (up to 2,000 times heavier), and if the "bodyguards" (the fermions) have specific masses and interaction strengths, the theory works perfectly. It explains the amount of Dark Matter we see and passes all current safety checks.

The Hunt: The Muon Collider

Now, how do we catch these "bodyguards"? The authors suggest using a new type of particle accelerator called a Muon Collider.

The Old Way (LHC): The Large Hadron Collider (LHC) is like a massive, chaotic demolition derby. It smashes protons together. It's great, but it's so messy (full of debris) that it's hard to spot a specific new particle if it's heavy or moves quietly. The LHC can currently see these bodyguards up to about 1.5 TeV (a unit of mass), but only if they are lucky.
The New Way (Muon Collider): A Muon Collider is like a high-speed, precision snooker table. Muons are elementary particles (like heavy electrons). When you smash them together, the collision is incredibly clean. There is very little "debris" or background noise.

The authors simulated what would happen if we built a Muon Collider with two different energy levels:

3 TeV (The "Standard" Future Machine): This machine could spot these heavy bodyguards up to a mass of 1.5 TeV.
10 TeV (The "Super" Future Machine): This powerhouse could spot them up to 2 TeV.

The Verdict

The paper concludes that the Muon Collider is a game-changer for this specific theory.

The "Clean Room" Advantage: Because the Muon Collider is so clean, it can see these heavy particles even when they are very massive, where the messy LHC would miss them.
The Sweet Spot: The authors found that the "sweet spot" for this theory (where the math works and Dark Matter exists) is almost entirely within the range that a 10 TeV Muon Collider could explore.
The Signal: If these particles exist, the Muon Collider would see them as a pair of high-energy electrons flying out in opposite directions, accompanied by a "missing energy" signature (the invisible Dark Matter ghost running away).

Summary

In short, the paper argues:

The simple idea that Dark Matter only talks to the Higgs field is likely dead because we haven't found it yet.
A more complex idea, where Dark Matter hangs out with heavy, charged "bodyguard" particles, is still alive and fits all the data.
The best way to prove this theory isn't to smash more protons (like the LHC), but to build a clean, high-energy Muon Collider.
If we build a 10 TeV Muon Collider, we have a very high chance of finding these bodyguards and finally understanding what Dark Matter is made of.

Technical Summary: Probing Fermion-Portal Scalar Dark Matter through Charged Vector-Like Fermions at Future Muon Colliders

Problem Statement
The nature of dark matter (DM) remains one of the most significant open questions in cosmology and particle physics. While the Standard Model (SM) successfully describes visible matter, it lacks a viable DM candidate. This work revisits a minimal extension of the SM featuring a real singlet scalar DM candidate ( $s$ ) stabilized by a discrete $Z_2$ symmetry, augmented by vector-like fermion singlets ( $E_S$ ) and doublets ( $F_D$ ).

The primary challenge addressed is the severe restriction placed on the "Higgs-portal" interaction (coupling $\kappa$ ) by the latest null results from direct DM detection experiments (e.g., LUX-ZEPLIN). In a pure Higgs-portal scenario, the parameter space allowing for the correct DM relic abundance is largely excluded, leaving only a narrow window near the Higgs resonance ( $m_{DM} \approx 62$ GeV). The paper investigates whether introducing new Yukawa interactions mediated by the vector-like fermions can open up a viable parameter space that satisfies both relic density requirements and direct detection limits, and whether this new physics can be probed at future muon colliders.

Methodology
The authors employ a multi-stage approach combining theoretical constraints, cosmological calculations, and collider phenomenology:

Model Construction and Theoretical Constraints:
- The model includes a scalar singlet $s$ , a vector-like doublet $F_D = (X^0_1, E^-_D)$ , and a singlet $E^-_S$ .
- The Lagrangian includes Higgs portal terms and new Yukawa couplings ( $Y_f, Y'_f$ ) connecting SM leptons to the new fermions.
- Theoretical viability is ensured by imposing vacuum stability, perturbative unitarity, and constraints on the scattering matrix eigenvalues.
- Experimental constraints from lepton flavor violation (LFV, specifically $\mu \to e\gamma$ ) and the anomalous magnetic moment of the muon ( $g-2$ ) are applied. The authors note that satisfying $g-2$ while respecting LFV limits requires specific tuning of Yukawa couplings (e.g., suppressing second-generation couplings).
Dark Matter Phenomenology:
- The relic density ( $\Omega_{DM}h^2$ ) is calculated using micrOMEGAs, interfaced with FeynRules and SPheno.
- The analysis focuses on the "Freeze-out" regime.
- The authors scan the parameter space ( $m_{DM}$ , $\kappa$ , $Y_f$ , and fermion masses) to identify regions where the observed relic abundance is reproduced.
- Key annihilation channels considered include Higgs-mediated $s$ -channel processes and new Yukawa-driven $t$ - and $u$ -channel processes involving the vector-like fermions. Co-annihilation processes are also evaluated.
Collider Simulation at Muon Colliders:
- The study focuses on the pair production of the lightest charged fermion: $\mu^+\mu^- \to E^+_1 E^-_1$ .
- The signal process is defined as $E^\pm_1 \to e^\pm s$ , resulting in a final state of $e^+e^- + \text{missing transverse energy} (\not{E}_T)$ .
- Simulations are performed for center-of-mass energies of $\sqrt{s} = 3$ TeV ( $L = 1 \text{ ab}^{-1}$ ) and $\sqrt{s} = 10$ TeV ( $L = 10 \text{ ab}^{-1}$ ).
- Tools: Events are generated using MadGraph5_aMC@NLO, showered/hadronized with PYTHIA 8, and detector effects simulated using Delphes 3.5 with a muon collider card.
- Backgrounds: Major SM backgrounds include $\mu^+\mu^- \to e^+e^-\nu\bar{\nu}$ , $e^+e^-jj$ , $e^+e^-\gamma\gamma$ , and $W^+W^-$ production.
- Analysis: A cut-based analysis is performed with "Loose," "Medium," and "Hard" kinematic selections on lepton $p_T$ and $\not{E}_T$ to maximize signal significance ( $S_{sig}$ ).

Key Contributions and Results

Expansion of Viable Parameter Space: The study demonstrates that while the pure Higgs-portal scenario is nearly ruled out by direct detection limits, the inclusion of vector-like fermions and their Yukawa couplings ( $Y_f \sim 0.35-0.45$ ) allows for a successful reproduction of the DM relic abundance over a wide mass range ( $m_{DM} \approx 10$ GeV to multi-TeV). In this regime, the relic density is dominated by $t$ - and $u$ -channel annihilation ( $ss \to \nu\bar{\nu}$ ) and co-annihilation, rendering the Higgs portal coupling $\kappa$ small enough to evade direct detection bounds.
Muon Collider Sensitivity:
- 3 TeV Collider: The authors project a $5\sigma$ discovery reach for the charged fermion mass ( $M_{E^\pm_1}$ ) up to approximately 1.5 TeV, provided there is a significant mass splitting between the fermion and the DM scalar.
- 10 TeV Collider: The reach extends to approximately 2 TeV.
- The analysis shows that the muon collider's clean environment and well-defined initial state allow for the probing of masses and parameter regions that are inaccessible to the High-Luminosity LHC (HL-LHC), which is limited to roughly 1.0–1.5 TeV due to large QCD backgrounds.
Benchmark Points: The paper provides specific benchmark points (BPs) illustrating the interplay between the DM mass, charged fermion mass, and Yukawa couplings that satisfy all constraints. It highlights that "compressed" spectra (small mass splitting) are more challenging to detect due to softer decay products, while large mass splittings yield high signal significance.

Significance and Claims
The paper claims that the proposed scenario—a real singlet scalar DM coupled to vector-like fermions—remains a viable and well-motivated framework despite stringent direct detection limits. The primary significance lies in the complementarity between direct detection experiments and future muon colliders.

The authors assert that:

The new Yukawa interactions are essential for maintaining the correct relic density in the absence of a strong Higgs portal.
Future muon colliders operating at 3 TeV and 10 TeV offer a unique opportunity to explore the charged fermion sector of this model, covering a substantial fraction of the phenomenologically viable parameter space that the HL-LHC cannot access.
Specifically, a 10 TeV muon collider could probe nearly the entire parameter space consistent with the observed DM relic abundance and current direct detection bounds, effectively testing the "fermion-portal" mechanism for dark matter.

The study concludes that the interplay between cosmological constraints and high-energy collider searches provides a robust strategy for testing this class of dark matter models in the coming decades.