The Scotogenic Model with Two Inert Doublets: Parameters Space and Electroweak Precision Tests

This paper investigates a scotogenic extension of the Standard Model with two inert doublets and three singlet Majorana fermions, identifying viable parameter spaces for radiative neutrino mass generation and dark matter while revealing that approximately 60% of these regions are excluded by recent CMS measurements of the $W$ boson mass due to constraints on the oblique parameter $\Delta T$.

The Gist

The Big Picture: Fixing the "Broken" Universe

Imagine the Standard Model of physics as a perfectly engineered Swiss Army Knife. It has a blade, a screwdriver, and a can opener, and it works flawlessly for almost everything we see in the universe.

But, there are two big problems:

1. The "Ghost" Problem: We know neutrinos (tiny, invisible particles) have mass, but this knife has no tool for that.
2. The "Dark Matter" Problem: We know most of the universe is made of invisible "Dark Matter," but this knife has no tool for that either.

This paper proposes adding two new tools to the Swiss Army Knife to fix these holes. The authors call this new setup the "Scotogenic Model with Two Inert Doublets."

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The New Tools: The "Invisible Twins"

To fix the knife, the authors add two new types of particles that don't interact with light (they are "inert" or invisible to us directly):

1. Three "Majorana Fermions": Think of these as three invisible, heavy ghosts.
2. Two "Inert Scalar Doublets": Think of these as two invisible, heavy twins (one male, one female version) that float around but never show up at parties (colliders) unless you hit them very hard.

The Magic Trick (Neutrino Mass):
In the original model, neutrinos were weightless. In this new model, the authors show that neutrinos get their mass through a loop of interactions.

• Analogy: Imagine you want to weigh a feather, but you don't have a scale. Instead, you bounce the feather off a trampoline (the new particles), and the way the trampoline bounces tells you the weight. The neutrinos get their mass by "bouncing" off these new invisible particles in a quantum loop.

The Dark Matter Candidate:
Because these new particles are "inert," the lightest one among them is stable. It can't decay into anything else.

• Analogy: It's like the last person standing in a game of musical chairs. Since everyone else has left the room, this one particle stays forever, floating through the universe as Dark Matter.

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The Challenge: The "Oblique Parameters" (The Tightrope Walk)

Now that we've added these new tools, we have to make sure they don't break the rest of the Swiss Army Knife. The universe is very sensitive; if we add too much weight to one side, the whole thing tips over.

Physicists use something called Oblique Parameters (S, T, and U) to measure how well the new tools fit.

• The Analogy: Imagine the universe is a tightrope walker.
  • Parameter T is the most sensitive. It measures if the new particles are "unbalanced" (like having one heavy foot and one light foot). If the new particles have different masses, the walker wobbles.
  • Parameter S is like checking if the walker is leaning too far forward or back.
  • Parameter U is a tiny wobble that usually doesn't matter.

The authors ran a massive computer simulation (a "parameter scan") to see if they could find a spot on the tightrope where the new particles exist without making the walker fall.

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The Results: What They Found

1. The "Resonance" Sweet Spot

The authors found that to get the right neutrino mass without using impossibly strong forces, the new particles need to be arranged just right.

• Analogy: Think of pushing a child on a swing. If you push at the exact right moment (resonance), a tiny push creates a huge swing.
• They found that if the masses of the new particles are very close to each other (like a perfect rhythm), the "swing" works perfectly. This allows the particles to have strong connections (Yukawa couplings) without breaking the laws of physics.

2. The "W Boson" Problem (The Plot Twist)

For a while, there was a measurement from the CDF experiment that said the W boson (a carrier of the weak force) was heavier than expected. This suggested the universe was wobbling, and our new "invisible twins" might be the reason.

However, a newer, more precise measurement from the CMS experiment said: "Actually, the W boson is exactly as heavy as the original Swiss Army Knife predicted. There is no wobble."

• The Impact: This is bad news for the authors' model.
• The Result: Because the new particles do cause a tiny wobble (especially in Parameter T), and the real world says "no wobble allowed," 60% of the possible versions of this model are now ruled out.
• Analogy: It's like you designed a new car engine that you thought would make the car go faster. But then the speed limit police (CMS) said, "The speed limit is exactly what we thought, and your engine makes it go too fast." So, you have to throw away 60% of your engine designs.

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The Conclusion: Is the Model Dead?

No, but it's in the hospital.

• The Good News: The model still works! There are still 40% of the scenarios where the new particles are heavy enough or balanced enough that they don't upset the W boson measurement. In these scenarios, the model successfully explains neutrino mass and Dark Matter.
• The Bad News: The "easy" versions of the model (where the new particles are light and easy to find) are mostly gone. The surviving versions require the new particles to be heavier and more carefully balanced.

Summary for the General Public:
This paper is a detective story. The authors built a new theory to explain two mysteries of the universe (neutrino mass and dark matter). They tested it against the strict rules of physics and found that while the theory is clever and mathematically sound, a recent measurement of a specific particle (the W boson) has eliminated most of the "easy" ways the theory could be true. The theory survives, but only in its most "tuned-up" and delicate forms.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics fails to explain the origin of non-zero neutrino masses and the nature of Dark Matter (DM). While the original Scotogenic Model (proposed by E. Ma) addresses both by introducing a single inert scalar doublet and singlet Majorana fermions stabilized by a $Z_2$ symmetry, this work investigates a more complex extension: the Scotogenic Model with Two Inert Doublets ($\Phi_1, \Phi_2$) and three singlet Majorana fermions ($N_i$).

The primary challenges addressed in this study are:

• Parameter Space Viability: Determining if the extended scalar sector (two doublets instead of one) allows for radiative neutrino mass generation without requiring non-perturbative Yukawa couplings.
• Electroweak Precision Tests (EWPT): Assessing how the additional scalar degrees of freedom (mass splittings and mixing) affect the oblique parameters ($S, T, U$) and the $W$ boson mass ($M_W$).
• Recent Experimental Constraints: Re-evaluating the model's viability in light of the conflicting $W$ boson mass measurements (CDF-II vs. CMS), specifically utilizing the recent CMS result which aligns with the SM.

2. Methodology

The authors performed a comprehensive numerical scan of the model's free parameters, subjecting 5,000 benchmark points (BPs) to a rigorous set of constraints:

• Theoretical Constraints:
  • Perturbativity: Scalar quartic couplings and Yukawa interactions are bounded ($|\lambda_i|, |h_{\alpha,k}|^2 \leq 4\pi$).
  • Vacuum Stability: The scalar potential must be bounded from below.
  • Unitarity: Scattering amplitudes for scalar processes must satisfy $|\Lambda_i| < 8\pi$.
• Experimental Constraints:
  • Lepton Flavor Violation (LFV): Strict limits on decays $\mu \to e\gamma$, $\tau \to e\gamma$, and $\tau \to \mu\gamma$.
  • Higgs Physics: Constraints on Higgs decay channels ($h \to \gamma\gamma, \gamma Z$) matching SM predictions within $\sim 10\%$.
  • Collider Limits: Charged scalar masses $m_{H^\pm} > 78$ GeV (from LEP-II).
  • Electroweak Precision Observables (EWPO): Global fits for oblique parameters $S, T, U$ and the $W$ boson mass shift $\Delta M_W$.
• Analytical Framework:
  • Neutrino Mass Generation: Calculated via 12 one-loop diagrams involving the two doublets and three fermions. The mass matrix depends on Yukawa couplings, Majorana masses ($M_k$), and scalar mixing angles ($\theta_H, \theta_A, \theta_C$).
  • Oblique Parameters: Derived analytical formulas for $\Delta S, \Delta T, \Delta U$ incorporating the full mixing matrices of the charged and neutral scalar sectors.
  • Statistical Analysis: A $\chi^2$ function was constructed using experimental central values and correlations for $S, T, U$ to identify the $1\sigma$ viable region.

3. Key Contributions

• Mechanism for Sizable Yukawa Couplings: The paper identifies specific regions in the scalar mixing plane ($s^2_H, s^2_A$) where constructive interference occurs. This allows for naturally large Yukawa couplings (necessary for observable neutrino masses) without violating perturbativity, particularly when scalar mass splittings are small or tuned to resonance structures.
• Differential Sensitivity of Oblique Parameters: The study clarifies that in this two-doublet framework:
  • $\Delta T$ is the dominant parameter, highly sensitive to mass splittings between charged and neutral scalars.
  • $\Delta S$ and $\Delta U$ remain small across the viable parameter space, as they depend primarily on logarithmic terms or are suppressed by the specific mixing structure.
• Impact of CMS $W$-Mass Measurement: The authors explicitly contrast the model's predictions against the recent CMS measurement ($M_W \approx 80,360$ MeV) versus the anomalous CDF-II result. They demonstrate that the CMS result, being consistent with the SM, imposes a much tighter constraint on the model than the CDF-II result.

4. Key Results

• LFV Constraints: The model comfortably satisfies current LFV bounds, particularly for charged scalar masses $m_{H^\pm} \gtrsim 200$ GeV. The decay $\mu \to e\gamma$ provides the strongest constraint, disfavoring scenarios with light charged scalars and large Yukawa couplings.
• Yukawa Coupling Enhancement: In scenarios with nearly degenerate neutral scalars (Set A), the parameter space exhibits "resonance ridges" where Yukawa couplings peak due to the suppression of the loop function eigenvalues ($\Lambda'_i$). As mass splittings increase (Sets B and C), these resonances broaden, allowing sizable couplings over a wider range of mixing angles.
• Electroweak Precision & $W$ Mass:
  • Approximately 60.6% of the viable parameter space is excluded by the recent CMS $W$ boson mass measurement.
  • The exclusion is driven by the shift $\Delta M_W$, which is directly linked to $\Delta T$. Large mass splittings between charged and neutral scalars generate large $\Delta T$, pushing the predicted $M_W$ outside the $1\sigma$ CMS range.
  • Mass Dependence: For light charged scalars ($m_{H^\pm} \lesssim 250$ GeV), large mass splittings are permitted. However, for heavier charged scalars ($m_{H^\pm} \gtrsim 300$ GeV), the mass splittings must be very small ($\delta_{charged} \lesssim 0.1$) to satisfy EWPT.
• Dark Matter: While not the primary focus of the scan, the model supports both fermionic ($N_1$) and scalar (lightest $H^0_1$ or $A^0_1$) DM candidates.

5. Significance

This work significantly refines the phenomenological landscape of the Scotogenic Model with two inert doublets.

1. Theoretical Viability: It proves that the extended scalar sector does not inherently require fine-tuned, non-perturbative couplings to generate neutrino masses; rather, specific mixing configurations naturally enhance the couplings.
2. Phenomenological Filtering: By applying the latest CMS $W$-mass data, the authors effectively halve the viable parameter space, providing a clear target for future collider searches. The result that $\Delta T$ is the primary driver of constraints offers a specific signature (charged-neutral mass splitting) for experimentalists to look for.
3. Model Discrimination: The analysis highlights how the two-doublet extension differs from the single-doublet version, particularly regarding the complexity of the mixing angles and the specific resonance structures that allow for large Yukawa couplings.

In conclusion, the paper establishes that while the two-inert-doublet scotogenic model is theoretically robust and consistent with most low-energy constraints, it is under significant pressure from high-precision electroweak measurements, specifically the $W$ boson mass, which rules out the majority of the parameter space unless the scalar spectrum is nearly degenerate or the charged scalars are relatively light.