⚛️ phenomenology

Global analysis of a minimally extended scotogenic model

This paper presents a global analysis of a minimally extended scotogenic model that simultaneously addresses neutrino masses, dark matter, and Standard Model vacuum instability, revealing specific constraints on parameter space, dark matter masses, and scalar sectors while remaining consistent with current flavor, electroweak, and cosmological observations.

Original authors: Huchan Lee, Sin Kyu Kang

Published 2026-06-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Huchan Lee, Sin Kyu Kang

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 Standard Model of particle physics as a magnificent, high-precision clock that has kept perfect time for decades. It explains how the universe works with incredible accuracy. However, this clock has three glaring flaws that the scientists in this paper are trying to fix:

The "Ghost" Problem: We know dark matter exists (it's the invisible glue holding galaxies together), but the clock doesn't have a part for it.
The "Whisper" Problem: Neutrinos (tiny, ghostly particles) are supposed to be weightless according to the clock, but experiments show they have a tiny bit of mass.
The "Crack" Problem: If you wind the clock too tight (look at very high energies), the mechanism starts to crack. The mathematical "spring" inside the clock (the Higgs field) becomes unstable and could theoretically snap, causing the universe to collapse.

The authors of this paper propose a minimal upgrade to this clock. They call it a "minimally extended scotogenic model." Think of "scotogenic" as a fancy word for "darkness-generated." They are adding just a few new gears and springs to fix all three problems at once without breaking the rest of the machine.

Here is a breakdown of their work using simple analogies:

1. The New Parts (The Model)

The scientists added two new ingredients to the recipe:

A New Scalar Field: Imagine the Higgs boson (the particle that gives things mass) as a central gear. They added a second, smaller gear that spins alongside it. These two gears mix together. This mixing is the key to fixing the "Crack Problem." It stabilizes the spring so the clock won't break at high speeds.
New Neutrinos: They added "right-handed" neutrinos. In the original clock, neutrinos were like shy ghosts that only interacted from the left. These new ones are bold and interact with the new scalar gear. This interaction is what generates the tiny mass for the neutrinos (fixing the "Whisper Problem") and creates a candidate for dark matter (fixing the "Ghost Problem").

2. The Safety Checks (Constraints)

Before they could say their new clock works, they had to run it through a series of rigorous safety inspections to make sure it didn't explode or run backward.

The "No-Runaway" Test: They checked if the energy of the new parts could ever go to infinity (which would be bad). They proved mathematically that the energy stays bounded, like a car that can't exceed its speed limit.
The "Stability" Test: They simulated the clock running from the present day all the way to the end of the universe (the Planck scale). They found that with their new gears, the clock remains stable and doesn't develop cracks.
The "Smoothness" Test: They checked that the mathematical forces between the parts don't get so strong that they break the laws of physics (perturbativity). They found that as long as the new parts aren't too "heavy" or "strong," the math stays smooth.

3. The Real-World Tests (Observables)

The scientists then asked: "If we built this clock, what would we see in the real world?" They looked at specific experiments to see if their new model matched reality.

The Muon's Spin (g-2): There was a famous mystery about how much a muon (a heavy cousin of the electron) wobbles. For a while, the clock's prediction didn't match the experiment. However, the paper notes that with the latest data, the mystery has largely disappeared. Their model predicts a tiny wobble, but it's so small it's currently invisible to our instruments.
Lepton Flavor Violation: They checked if particles could magically change their identity (e.g., a muon turning into an electron and a photon). In their model, this is like trying to turn a cat into a dog instantly—it's possible in theory, but the probability is so incredibly low (like winning the lottery every second for a billion years) that we won't see it in our current experiments.
The Invisible Decay: They looked at the Z boson (a heavy particle) decaying into invisible things. In the Standard Model, it decays into invisible neutrinos. In their model, it might decay into their new dark matter particles.
- The Result: Their prediction for how often this happens is very close to what we currently measure. In fact, a recent measurement by the ATLAS experiment (a giant particle detector) actually favors their model slightly more than the old average.

4. The Dark Matter Hunt

Who is the dark matter in their clock?

They found two main suspects: a new heavy particle (a fermion) and a new heavy scalar particle (a boson).
The Fermion: If this is the dark matter, it must weigh between 120 and 350 GeV (about 130 to 370 times the mass of a proton).
The Scalar: If the scalar is the dark matter, it must weigh between 350 and 600 GeV.
The Good News: Their model predicts that these particles interact with normal matter so weakly that current detectors (like XENON1T) haven't seen them yet, but future, more sensitive detectors might.

5. The "DESI" Twist

One of the most interesting findings involves the inverted hierarchy of neutrino masses.

Imagine neutrino masses as a stack of blocks. There are two ways to stack them: "Normal" (lightest on top) or "Inverted" (heaviest on top).
The paper ran a simulation and found that if a recent experiment called DESI (which measures the expansion of the universe) is correct, the "Inverted" stack is impossible. The math simply doesn't add up. If DESI is right, the universe must have a "Normal" stack, and the "Inverted" option is ruled out.

Summary

The authors built a slightly upgraded version of the universe's rulebook. They added just enough new parts to explain dark matter and neutrino masses while fixing a theoretical instability that threatened to break the Standard Model at high energies.

The verdict?

The model is mathematically stable and safe.
It fits current data perfectly.
It predicts that dark matter particles are likely in the "medium-heavy" range (120–600 GeV).
It suggests that if the DESI experiment is confirmed, the "Inverted" neutrino mass pattern is dead.
Most importantly, it predicts that the "weird" effects (like particles changing identity) are so tiny that we won't see them soon, but the "invisible" decay of the Z boson might give us a clue in the near future.

They didn't invent a new engine for a car; they just tightened a few bolts and added a stabilizer to an engine that was already running, making sure it won't break down at top speed.

Technical Summary: Global Analysis of a Minimally Extended Scotogenic Model

Problem Statement
The Standard Model (SM) successfully describes a wide range of phenomena but fails to account for non-zero neutrino masses, the existence of dark matter (DM), and the metastability of the electroweak (EW) vacuum at high energies. The original scotogenic model addresses neutrino masses and DM but inherits the SM's vacuum instability issue, where the Higgs quartic coupling runs to negative values at scales around $10^9$ GeV. Furthermore, recent updates to the muon anomalous magnetic moment ( $\Delta a_\mu$ ) using lattice QCD inputs suggest a reduced tension (0.6 $\sigma$ ) between SM predictions and experimental data, challenging previous New Physics (NP) interpretations that relied on large Yukawa couplings. This work addresses the need for a model that simultaneously stabilizes the EW vacuum, explains neutrino masses and DM, and remains consistent with precision observables without invoking artificially enhanced parameters.

Methodology
The authors propose a minimally extended scotogenic model incorporating an additional gauge-singlet scalar field $S$ and a global $U(1)_X$ symmetry. The scalar $S$ acquires a non-zero vacuum expectation value (VEV) and mixes with the SM Higgs. The model includes right-handed (RH) neutrinos and an inert scalar doublet $\eta$ .

Key methodological components include:

Theoretical Constraints: The study enforces bounded-from-below (BFB) conditions, vacuum stability (verified using the tool Vevacious), and Renormalization Group (RG)-driven perturbativity bounds. The RG analysis ensures couplings remain perturbative up to the Planck scale, avoiding Landau poles.
Renormalization Scheme: To handle the extended scalar sector and mixing, the authors adopt an alternative tadpole scheme where one-particle-irreducible (1PI) contributions are combined with explicit tadpole insertions. This ensures gauge invariance and UV finiteness.
Observable Calculation: Precision and flavor observables are computed by calculating the full contributions in both the SM and the BSM framework separately, then taking the difference ( $\Delta = \text{BSM} - \text{SM}$ $Δ = BSM - SM$ ) to isolate NP effects. This approach is applied to:
- Muon anomalous magnetic moment ( $\Delta a_\mu$ ) and radiative decays ( $\ell_\alpha \to \ell_\beta \gamma$ ).
- Three-body decays ( $\ell_\alpha \to 3\ell_\beta$ ) and $\mu \to e$ conversion.
- Oblique parameters ( $S, T, U$ ).
- Leptonic and invisible decays of $Z$ and $H$ bosons ( $Z/H \to \ell_\alpha \ell_\beta$ , $Z/H \to \text{invisible}$ ).
Numerical Scan: A Markov Chain Monte Carlo (MCMC) algorithm is employed to scan the high-dimensional parameter space. The scan is constrained by the one-loop SM Higgs mass ( $125.20 \pm 0.11$ GeV) and the relic density ( $\Omega_{\text{CDM}}h^2 = 0.120 \pm 0.012$ ). The analysis avoids large Yukawa couplings ( $g_X \lesssim 0.2$ ) to maintain perturbativity.

Key Contributions and Results

Vacuum Stability: The introduction of the singlet scalar $S$ modifies the RG running of the quartic couplings, allowing the SM-like quartic coupling to remain positive across the energy range, thereby restoring EW vacuum stability.
Neutrino Mass Hierarchies: The study scans both Normal (NH) and Inverted (IH) hierarchies.
- The IH scenario yields a minimum sum of neutrino masses $\sum m_{\nu_i} \approx 9.94 \times 10^{-11}$ GeV.
- The NH scenario yields a range of $5.87 \times 10^{-11}$ to $9.99 \times 10^{-11}$ GeV.
- The authors note that if the recent DESI BAO bound ( $\sum m_{\nu_i} < 6.42 \times 10^{-11}$ GeV) is confirmed, the IH scenario would be ruled out, and a significant portion of the NH parameter space would be excluded.
Dark Matter Candidates: Two viable DM candidates emerge from the scan:
- Fermionic: The lightest RH neutrino $\chi_1$ , with a mass range of 120–350 GeV.
- Scalar: The CP-odd scalar $\eta_I$ , with a mass range of 350–600 GeV.
- The CP-even scalar $\eta_R$ is consistently slightly heavier than $\eta_I$ and is not a viable DM candidate.
- The spin-independent proton cross-sections for these candidates lie well below current experimental bounds (XENON1T) and even the neutrino floor, indicating that direct detection constraints do not significantly restrict the model's parameter space.
Flavor and Precision Observables:
- Suppression of Flavor Violation: Due to the constraints on Yukawa couplings from neutrino oscillation data and the perturbativity requirement, all lepton flavor-violating (LFV) observables (including $\Delta a_\mu$ , $\mu \to e\gamma$ , and $\mu \to e$ conversion) are suppressed by many orders of magnitude (e.g., $\Delta a_\mu \sim 10^{-14}$ ), far below current experimental sensitivities.
- Oblique Parameters: The $S, T, U$ parameters are less suppressed than LFV observables. The dominant contribution arises from the mass splitting between the SM-like Higgs and the SM Higgs. These parameters are projected to be within the reach of future precision measurements (e.g., FCC-ee).
- Lepton Flavor Universality: The ratios $R_{\tau\mu}$ and $R_{\tau e}$ for $Z$ and $H$ decays are consistent with SM predictions within $2\sigma$ .
- Infrared (IR) Contributions: For $H \to \ell\ell$ decays, IR contributions from bremsstrahlung are found to be negligible for heavy leptons ( $\tau, \mu$ ) but comparable in magnitude to vertex corrections for light leptons ( $e$ ).
$Z \to \text{Invisible}$ Decay: The NP contribution to the invisible $Z$ width is driven primarily by diagonal contributions, resulting in a numerical enhancement of $\sim 0.0004$ . The results are compatible with the world average at the $3\sigma$ level. Notably, the authors find their results are favored by the recent ATLAS measurement (which has a broader uncertainty) at the $3\sigma$ level.

Significance and Claims
The paper claims to provide a comprehensive global analysis of a minimally extended scotogenic model that resolves the vacuum instability of the original model while maintaining consistency with neutrino data and DM constraints. A distinguishing feature of this work is the rejection of "large-Yukawa-driven" analyses; instead, it operates within a regime where couplings are small enough to satisfy RG perturbativity bounds up to the Planck scale. Consequently, the model predicts that flavor-violating observables are unobservable in near-future experiments, shifting the focus of potential discovery to precision electroweak measurements (oblique parameters) and the invisible decay width of the $Z$ boson. The authors emphasize that their findings on the $Z \to \text{Invisible}$ decay align with recent ATLAS data, offering a viable direction for NP searches under the updated muon $g-2$ landscape.