Fermionic Dark Matter and New Scalar Production in $e^+e^- \to H^+H^-$ at Colliders

This paper investigates the pair production of charged scalars ($e^+e^- \to H^+H^-$) within the scotogenic model, demonstrating that the process is dominated by $t$-channel exchange of singlet right-handed fermions (including the dark matter candidate) and providing predictions for future high-energy $e^+e^-$ colliders to test the model's parameter space.

The Gist

Imagine the universe as a giant, complex machine. For a long time, scientists have had a "User Manual" for this machine called the Standard Model. It explains how most of the known particles (like electrons and protons) behave. But, like any old manual, it has missing pages. It can't explain two huge mysteries:

1. Dark Matter: The invisible "glue" that holds galaxies together, which we can't see but know is there.
2. Neutrino Masses: Tiny ghost-like particles that the manual says should have no weight, but experiments show they actually do.

This paper investigates a proposed "Supplement" to the User Manual called the Scotogenic Model. Think of this model as a new, secret workshop added to the machine. In this workshop, new particles are built to fix the missing pages.

The New Workshop: What's Inside?

The Scotogenic model introduces two main types of new workers:

• New Scalars (H+ and H-): Imagine these as charged, heavy twins. They are like new, heavy tools that can be created in particle collisions.
• New Fermions (N1, N2, N3): These are heavy, invisible particles. One of them, N1, is the star of the show because it is stable and invisible—it is the Dark Matter candidate.

The model has a special rule (called a Z2 symmetry) that acts like a security guard. It says, "All the old particles are allowed to leave the workshop, but the new ones must stay inside unless they pair up." This rule ensures that the Dark Matter particle (N1) never decays and stays around to hold galaxies together.

The Experiment: A High-Speed Collision

The authors of this paper asked a specific question: What happens if we smash an electron and a positron (their anti-particle) together at high speeds?

Specifically, they looked at the process where this collision creates a pair of those heavy new tools: H+ and H-.

To understand how this happens, they looked at three different "paths" or "routes" the particles could take to create this pair:

1. The Photon Route: Like two cars exchanging a glowing light beam to push each other apart.
2. The Z-Boson Route: Like exchanging a heavy, invisible baton.
3. The New Fermion Route (The Secret Path): This is the most interesting part. The collision creates the H+ and H- pair by exchanging the new, heavy Dark Matter particles (N1, N2, N3) in a "t-channel" (a sideways exchange).

The Detective Work: Checking the Rules

Before calculating the results, the authors had to make sure their new workshop didn't break any known laws of physics. They ran a series of strict tests:

• The "Ghost" Test (Neutrinos): The model must explain why neutrinos have mass. They checked if the math matches real-world measurements of how neutrinos change flavors.
• The "Rare Decay" Test: They checked if the new particles cause rare events (like a muon turning into an electron and a photon) that experiments have already said don't happen often. If the model predicted these happened too often, the model would be wrong.
• The "Cosmic Inventory" Test (Dark Matter): They calculated how much Dark Matter would be left over from the Big Bang. The amount must match what astronomers see in the universe today.

The Big Discovery

After running these strict tests, the authors found a very specific "safe zone" where the model works. In this zone:

• The new particles must be quite heavy (around 1,000 times heavier than a proton, or 1 TeV).
• The "Dark Matter" particle (N1) must be almost the same weight as the next heaviest particle (N2).

The Main Result:
When they calculated the probability (cross-section) of creating the H+ and H- pair, they found something surprising.

• The "Photon" and "Z-boson" routes (the standard paths) contribute very little.
• The "New Fermion Route" (the secret path involving the Dark Matter particles) is the dominant force. It is the main reason the H+ and H- pair gets created.

The Future: Looking for the Signal

The paper concludes by predicting what we would see if we built a super-powerful particle collider in the future.

• They calculated how the number of H+ and H- pairs would change as we increase the energy of the collision.
• They found that the signal would get stronger, reach a peak, and then drop off.

In simple terms: The paper says, "If you build a machine powerful enough to smash particles at these specific high energies, and you look for these specific heavy twins (H+ and H-), you will likely see them. And if you do, the reason you see them is mostly because of the invisible Dark Matter particles acting as the middlemen."

This doesn't just prove the model exists; it gives future scientists a specific "treasure map" (the energy levels and particle masses) to find this new physics.

Technical Summary

Technical Summary: Fermionic Dark Matter and New Scalar Production in $e^+e^- \to H^+H^-$ at Colliders

Problem Statement
The Standard Model (SM) fails to account for neutrino masses, dark matter (DM), and the baryon asymmetry of the universe. The scotogenic model, proposed by Ma, addresses these deficiencies by introducing a $Z_2$-odd scalar doublet ($\eta$) and three singlet right-handed Majorana fermions ($N_{1,2,3}$). In this framework, neutrino masses are generated radiatively at the one-loop level, and the lightest $Z_2$-odd particle serves as a DM candidate. While previous studies (e.g., Ref. [2]) investigated the process $e^+e^- \to H^+H^-$, they did so without incorporating the latest, stringent constraints on model parameters derived from updated neutrino oscillation data, cosmological DM relic density measurements, and direct detection limits. Consequently, accurate predictions for the production cross-section of charged Higgs pairs ($H^+H^-$) at future $e^+e^-$ colliders remained incomplete.

Methodology
The authors perform a comprehensive analysis of the pair production process $e^+e^- \to H^+H^-$ within the scotogenic model. The theoretical framework includes:

• Lagrangian Formulation: The study utilizes the scotogenic Lagrangian, which extends the SM scalar sector with the doublet $\eta$ and includes Yukawa couplings between the SM leptons, the new scalars, and the singlet fermions.
• Amplitude Calculation: The scattering amplitude is computed at tree level, accounting for contributions from $s$-channel photon ($\gamma$) and $Z$-boson exchange, as well as $t$-channel exchange of the singlet fermions $N_{1,2,3}$. The total cross-section formula is derived, explicitly separating the contributions from gauge bosons and the new fermionic sector.
• Constraint Implementation: The parameter space is rigorously scanned and constrained by:
  • Neutrino Data: Normal ordering (NO) of neutrino masses, global oscillation data (mixing angles, $\Delta m^2$ ratios), and the sum of neutrino masses ($\sum m_i < 0.12$ eV).
  • Lepton Flavor Violation (LFV): Experimental upper bounds on branching ratios, specifically $BR(\mu \to e\gamma) < 4.2 \times 10^{-13}$.
  • Dark Matter Relic Density: The observed value of $\Omega h^2$, calculated via co-annihilation processes involving $N_1$ and $N_2$.
  • Direct Detection: Constraints from DM-nucleon scattering mediated by Higgs exchange, addressed by setting specific coupling values ($\lambda_{3,4} = 0.01$) to evade current limits.
  • Collider Limits: Lower bounds on scalar masses from LHC searches and $W/Z$ width measurements.

The analysis assumes $N_1$ as the DM candidate with $N_2$ nearly degenerate in mass, while $N_3$ is heavier. A systematic scan is performed over the masses of the new particles ($M_{1,2,3}, m_{H^\pm}, m_0$) and Yukawa couplings ($Y_{1,2,3}$).

Key Contributions

1. Updated Cross-Section Analysis: The paper provides the first calculation of the $e^+e^- \to H^+H^-$ cross-section that fully integrates modern constraints from neutrino oscillation data, DM relic density, and direct detection limits, which were previously omitted in similar studies.
2. Decomposition of Contributions: The authors explicitly evaluate and separate the individual contributions of the photon, $Z$-boson, and the new singlet fermions ($N_{1,2,3}$) to the total cross-section.
3. Identification of Dominant Mechanism: The study demonstrates that, within the allowed parameter space, the $t$-channel exchange of the singlet fermions (particularly the DM component $N_1$) dominates the production cross-section, overshadowing the standard gauge boson contributions.
4. Benchmark Scenarios: The authors identify specific benchmark points in the high-mass regime (near or above 1 TeV) that simultaneously satisfy all theoretical and experimental constraints.

Results

• Parameter Space Constraints: The viable parameter space is highly restricted.
  • LFV constraints ($\mu \to e\gamma$) require heavy charged Higgs masses and large fermion masses.
  • Neutrino mass ratio constraints ($|\Delta m^2_{31}|/\Delta m^2_{21}$) and relic density requirements impose specific relationships between the Yukawa couplings and particle masses.
  • Simultaneous satisfaction of all constraints is only possible for new particle masses close to or exceeding the 1 TeV scale.
• Cross-Section Behavior: For the identified benchmark points (e.g., $M_1 \approx 1005$ GeV, $m_{H^\pm} \approx 1405$ GeV), the total cross-section $\sigma(e^+e^- \to H^+H^-)$ increases with center-of-mass energy ($\sqrt{s}$), reaches a maximum, and then decreases.
• Dominance of Fermionic Exchange: The analysis confirms that the contribution from the exchange of singlet fermions $N_{1,2,3}$ is the dominant term in the cross-section, while the $Z$-boson contribution is the smallest.
• Coupling Requirements: Satisfying the $\mu \to e\gamma$ constraint requires Yukawa couplings of order $O(10^{-1})$, while relic density constraints for $M_1 < 100$ GeV would require larger couplings, which are excluded by LFV bounds. This reinforces the necessity of the high-mass (TeV scale) scenario.

Significance
The paper asserts that future high-energy $e^+e^-$ colliders can probe the process $e^+e^- \to H^+H^-$ to test the scotogenic framework. By measuring the cross-section and its energy dependence, these experiments can either confirm the dominance of the fermionic exchange mechanism predicted by the model or impose even stricter constraints on the model parameters. The study highlights that while the model remains viable, the relevant new physics scale is pushed to the TeV range, making the detection of these processes a sensitive test for the interplay between fermionic dark matter and neutrino mass generation.