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Precise QCD Predictions for the Scotogenic Model at Colliders

This paper presents the first next-to-leading order QCD predictions with parton-shower matching for the production and decay of scalars and heavy neutrinos in the Scotogenic model at 14 TeV and 100 TeV colliders, supported by the public release of the SM_Scoto FeynRules library and an analysis of future lepton collider prospects.

Original authors: Mohammed Boukidi, Camila Ramos, Richard Ruiz

Published 2026-09-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammed Boukidi, Camila Ramos, Richard Ruiz

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

The universe is built on a foundation of tiny, invisible particles that interact through fundamental forces. For decades, physicists have relied on a standard model to describe this microscopic world, a framework that successfully explains how matter behaves and how forces like electromagnetism work. However, this model has a glaring hole: it predicts that neutrinos, ghostly particles that stream through everything in the universe, should be massless. Yet, experiments have proven that these particles do have a tiny amount of mass, a fact that the standard model cannot explain on its own. To fix this, scientists have proposed various extensions to the theory, introducing new particles and hidden symmetries. One of the most compelling ideas is the Scotogenic model, which suggests that neutrinos gain their mass not through a direct interaction, but through a subtle, loop-like process involving new, invisible particles that also happen to be candidates for dark matter, the mysterious substance holding galaxies together.

A team of researchers has now taken a major step forward in testing this idea by creating the most precise predictions yet for how these new particles would behave if they were created in a particle collider. Using powerful computer simulations, they calculated exactly what would happen if the Large Hadron Collider, the world's most powerful particle accelerator, were to smash protons together at energies of 14 trillion electron volts, and also at a hypothetical future machine capable of reaching 100 trillion electron volts. Their work focuses on a specific set of new particles predicted by the Scotogenic model: heavy, invisible neutrinos and new types of scalar particles. The researchers did not just guess at the outcomes; they performed complex calculations that account for the messy, chaotic nature of particle collisions, including the way particles radiate energy and interact with the strong nuclear force. They found that these new particles would be produced in pairs, a requirement of the hidden symmetry that protects them, and they mapped out exactly how often this would happen and what energy signatures the resulting debris would leave behind.

The study reveals that if these particles exist within a certain mass range, they could be produced in significant numbers at current and future colliders. The researchers calculated that at the 14 trillion electron volt energy level, the production rates for these particles would be high enough to be detected, provided the particles are not too heavy. They also looked ahead to a potential 100 trillion electron volt collider, where the production rates would increase dramatically, making it possible to explore much heavier versions of these particles. Crucially, the team provided a detailed guide for experimentalists on how to distinguish these new particles from background noise. They showed that the particles would decay in specific ways, often turning into charged leptons like electrons or muons, or into invisible neutrinos, leaving behind distinct patterns of energy in the detector. The calculations included the effects of quantum corrections, which are tiny adjustments to the basic predictions that can significantly change the expected number of events, ensuring that the predictions are as accurate as possible.

Beyond the immediate predictions for the Large Hadron Collider, the researchers also explored how these particles might be discovered at future facilities, including electron-positron colliders and muon colliders. They found that at a future electron-positron machine, which would act as a "Z factory" producing billions of Z bosons, there is a chance to observe extremely rare decays where a Z boson transforms into a pair of these new heavy neutrinos along with other particles. While these events would be incredibly rare, the sheer number of Z bosons produced at such a facility could make them visible. Similarly, they looked at a proposed muon collider, where muons collide at energies of 10 trillion electron volts. Their simulations showed that even at these high energies, the production of these new scalar particles would be frequent enough to be studied in detail, offering a clean environment to measure their properties with high precision.

To make these findings accessible to the broader scientific community, the researchers also released a set of digital tools that allow other scientists to simulate these processes on their own computers. These tools, known as libraries, contain the mathematical rules and particle definitions needed to model the Scotogenic model within standard simulation software. This release is significant because it removes a major barrier to entry, allowing experimentalists to quickly test their own ideas against the new predictions without having to rebuild the complex theoretical framework from scratch. By providing these tools alongside their precise calculations, the team has laid the groundwork for a coordinated search for these particles, ensuring that when data from the next generation of colliders arrives, physicists will know exactly what to look for.

The work also addresses the broader context of how these particles fit into the universe. The researchers confirmed that the model remains consistent with current experimental limits, such as the non-observation of certain rare decays and the constraints on dark matter interactions. They showed that the model can naturally explain the tiny masses of neutrinos while simultaneously providing a stable candidate for dark matter, the lightest of the new particles that would never decay. This dual role makes the Scotogenic model particularly attractive, as it solves two of the biggest mysteries in physics at once. The simulations indicate that if the new particles are within the reach of current or near-future machines, they could be discovered within the next decade, potentially revolutionizing our understanding of the fundamental building blocks of reality.

In the end, this paper serves as a comprehensive roadmap for the next phase of particle physics. It moves the Scotogenic model from a purely theoretical concept to a testable hypothesis with concrete, quantitative predictions. By combining advanced computational techniques with a clear understanding of the underlying physics, the researchers have provided the community with the tools and the targets needed to hunt for these elusive particles. Whether the new particles are found at the Large Hadron Collider, a future 100 trillion electron volt machine, or a specialized muon collider, the path to discovery is now clearly marked, offering a genuine hope of uncovering the hidden sector of the universe that governs the mass of neutrinos and the nature of dark matter.

Technical Summary: Precise QCD Predictions for the Scotogenic Model at Colliders

Problem and Motivation
The Scotogenic model, a radiative neutrino mass framework extending the Standard Model (SM) with a second Higgs doublet ( $\Phi_2$ ) and three generations of right-handed Majorana neutrinos ( $\nu_R$ ), offers a solution to the smallness of neutrino masses and provides a dark matter candidate via an exact $Z_2$ parity. While previous studies have explored the model's phenomenology at the Large Hadron Collider (LHC), they largely relied on Leading Order (LO) calculations. With the anticipated high-luminosity (HL) phase of the LHC and the prospect of future 100 TeV colliders, there is a need for more precise theoretical predictions that include Next-to-Leading Order (NLO) Quantum Chromodynamics (QCD) corrections matched to parton showers (PS). This paper addresses the lack of such precision predictions for the production and decay of Scotogenic scalars ( $\eta$ ) and heavy neutrinos ( $N$ ).

Methodology
The authors implement the Scotogenic model Lagrangian into the FeynRules framework (v2.3.49), adapting the SM sector and omitting the explicit one-loop neutrino mass term ( $L_{1-loop}^\nu$ ) to treat Yukawa couplings and scalar masses as independent phenomenological parameters.

Library Generation: Using NLOCT and FeynArts, the authors extract UV and $R_2$ counterterms up to $\mathcal{O}(\alpha_s)$ . These are packaged into a public set of Universal FeynRules Output (UFO) libraries, termed the SM Scoto UFO libraries, compatible with contemporary high-energy physics software.
Simulation Pipeline: Numerical simulations are performed using MadGraph5_aMC@NLO (v3.7.2) for matrix element generation (including virtual and real radiative corrections via MadLoop and MadFKS) and Pythia8 (v8.306) for parton showering and hadronization.
Parameter Space: The study focuses on a simplified benchmark where the heavy neutrinos $N_2$ and $N_3$ are decoupled ( $m_{N_{2,3}} = 10^{10}$ GeV), and the scalar masses are degenerate ( $m_{\eta^\pm} = m_{\eta_R} = m_{\eta_I} \equiv m_\eta$ ). Couplings are restricted to the $\tau$ -flavor sector ( $Y_{\ell N_1} = \delta_{\ell\tau}$ ).
Scales: Calculations are performed at $\sqrt{s} = 14$ TeV (LHC) and a hypothetical $\sqrt{s} = 100$ TeV collider. Factorization and renormalization scales are set dynamically to half the sum of final-state transverse energies.

Key Contributions

Public Release: The first public release of the SM Scoto UFO libraries, enabling NLO+PS simulations of the Scotogenic model.
NLO QCD Predictions: The first NLO-accurate predictions for total and differential cross-sections for various production channels of Scotogenic scalars in hadronic collisions.
Decay Validation: A comprehensive calculation of decay widths and lifetimes for scalars and heavy neutrinos, validating the numerical implementation against known analytical limits and including non-zero lepton masses.

Results

Decay Phenomenology:
- For scalar masses $m_\eta \sim 400$ GeV to 1 TeV, the total decay width $\Gamma_{\eta^\pm}$ ranges from $\sim 20$ to $60$ GeV, corresponding to lifetimes of $\sim 10^{-25}$ s.
- Heavy neutrino decays ( $N_1 \to \ell \eta, \nu \eta$ ) remain narrow ( $\Gamma_{N_1} \sim 20-40$ GeV) even at $m_{N_1} \sim 1$ TeV, with lifetimes well beyond the detector scale ( $\tau_N \gg 10^{-24}$ s).
Inclusive Production at $\sqrt{s} = 14$ TeV:
- Drell-Yan (DY): The dominant production mechanisms are Charged-Current (CCDY: $qq' \to W^* \to \eta^\pm \eta^0$ $q q^{'} \to W^{*} \to η^{\pm} η^{0}$ ) and Neutral-Current (NCDY: $qq \to \gamma^*/Z^* \to \eta^+\eta^-, \eta_R\eta_I$ $q q \to γ^{*} / Z^{*} \to η^{+} η^{-}, η_{R} η_{I}$ ).
  - NLO cross-sections for $m_\eta = 100-1400$ GeV range from $1.1 \times 10^3$ fb down to $5.8 \times 10^{-3}$ fb (CCDY) and $6.3 \times 10^2$ fb down to $2.7 \times 10^{-3}$ fb (NCDY).
  - The NLO K-factors ( $K_{NLO} = \sigma_{NLO}/\sigma_{LO}$ ) are consistently between 1.16 and 1.22.
  - The ratio $\sigma_{CCDY}/\sigma_{NCDY}$ spans 1.77 to 2.12, driven by PDF effects (sea vs. valence quarks) and gauge charges.
- Gluon Fusion (GF): Loop-induced $gg \to \eta\eta$ is suppressed by 1–2 orders of magnitude compared to NCDY. The authors approximate NLO effects using a fixed $K_{N3LL} \approx 3.0$ , resulting in cross-sections of $45$ fb to $3.1$ ab.
- Associated Production ( $\eta\eta V$ ): Cross-sections are roughly one order of magnitude lower than GF, with $K_{NLO} \approx 1.20-1.27$ .
- Photon Fusion (AF): The $γγ \to \eta^+\eta^-$ channel is negligible (1–2 orders of magnitude below CCDY) with large scale uncertainties ( $\sim \pm 15\%$ ).
Inclusive Production at $\sqrt{s} = 100$ TeV:
- Cross-sections increase by approximately a factor of 10 for quark-initiated channels and 20 for gluon-initiated channels compared to 14 TeV.
- Scale uncertainties for DY channels decrease slightly (1–2%), while PDF uncertainties reduce significantly due to probing lower momentum fractions ( $x$ ).
- The ratio $\sigma_{CCDY}/\sigma_{NCDY}$ remains consistent with the 14 TeV results (1.71–2.04).
Differential Distributions:
- Transverse momentum ( $p_T$ ) distributions plateau at $p_T \approx 0.88 m_\eta$ before falling off as a power law.
- Rapidity distributions show a migration to higher $|y|$ as $m_\eta/\sqrt{s}$ decreases, with events reaching $|y| \sim 4$ at 100 TeV.
- Differential K-factors are largely flat across the kinematic ranges studied.
Future Colliders:
- $e^+e^-$ (FCC-ee): The paper estimates branching ratios for rare $Z$ decays ( $Z \to \ell\ell N_1 N_1$ , $Z \to \nu\nu N_1 N_1$ ). For light $N_1$ ( $<40$ GeV) and $m_\eta \sim 250$ GeV, branching ratios can reach $10^{-10}$ , potentially within the sensitivity of the FCC-ee ( $L \sim 150$ ab $^{-1}$ ).
- $\mu^+\mu^-$ (10 TeV): Production rates for $\mu^+\mu^- \to \eta\eta$ , $\eta\eta\gamma$ , and $\eta\eta\nu\bar{\nu}$ are calculated. The $\eta\eta\nu\bar{\nu}$ channel dominates ( $\sigma \sim 0.1-10$ fb) due to EW logarithms, while pair production dominates near threshold. These rates are well within the reach of integrated luminosities of $1-10$ ab $^{-1}$ .

Significance and Claims
The paper claims to provide the first NLO-accurate predictions for the Scotogenic model at hadron colliders, moving beyond the LO approximations prevalent in previous literature. By releasing the SM Scoto UFO libraries, the authors aim to facilitate rigorous, high-precision tests of the model at the HL-LHC and future colliders. The work highlights that while inclusive production rates are dominated by Drell-Yan processes with modest K-factors ( $\sim 1.2$ ), the precise modeling of PDFs and gauge charges is crucial for distinguishing the Scotogenic model from other radiative neutrino mass scenarios. The authors conclude that the model's collider phenomenology remains under-studied compared to tree-level Seesaw models and that their tools and predictions are essential for guiding experimental searches for new rare $Z$ decays and multi-TeV scalar production.

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