⚛️ phenomenology

Disentangling Dark Gauge Symmetries with Deep Learning on the Lund Jet Plane

This paper introduces a novel Monte Carlo simulation framework and a Neural Sorter Mamba Network to analyze Lund Jet Plane representations, successfully demonstrating the ability to distinguish between different dark gauge symmetries based on their unique radiation patterns while remaining robust against non-perturbative hadronization uncertainties.

Original authors: Jinmian Li, Junle Pei, Rao Zhang

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jinmian Li, Junle Pei, Rao Zhang

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 has a hidden "dark sector," a secret neighborhood where invisible particles live and interact. Physicists suspect these particles are held together by a force similar to the one that binds atoms in our world (called the strong force), but we don't know the exact rules of the game. Is the "dark force" built like a standard Lego set (a specific mathematical structure called SU(N)), or does it use a different kind of connector, like a round peg in a square hole (structures like SO(N) or Sp(2N))?

This paper is like a detective story where the authors try to figure out which "Lego set" the dark universe uses, just by looking at the debris left behind when these particles crash into each other.

Here is how they did it, explained simply:

1. Building a Custom Simulator

To solve this mystery, the team couldn't just use existing tools because those tools were built assuming the dark world looks exactly like our visible world. So, they built a brand-new video game engine (a Monte Carlo simulation).

The Analogy: Imagine you are trying to predict how a crowd of people will scatter when a bell rings. If you assume everyone is identical, you get one pattern. But if some people are heavy, some are light, and they are wearing different colored shirts, the pattern changes.
The Innovation: Their new engine can handle any type of "dark person" (particle), whether they are heavy or light, and regardless of the specific mathematical rules (gauge group) governing their interactions. It tracks every single "split" a particle makes as it breaks apart, creating a detailed map of the chaos.

2. The "Lund Jet Plane" Map

When these dark particles crash, they don't just vanish; they spray out a shower of smaller particles, like a firework exploding. The authors needed a way to visualize this explosion.

The Analogy: Think of the explosion as a complex tree with many branches. The authors used a special map called the Lund Jet Plane. Instead of a 3D explosion, this map flattens the tree onto a 2D grid.
- One axis shows how "wide" the branches are (the angle).
- The other axis shows how "hard" the branches are (the energy).
The Result: Different types of dark forces create different "forest patterns" on this map. Some forces make dense, thick trees; others make sparse, wispy ones. The map reveals the "fingerprint" of the underlying force.

3. The AI Detective (Neural Sorter Mamba Network)

Looking at these maps is hard for humans because the differences are subtle. So, the authors trained a super-smart AI to be the detective.

The Analogy: Imagine you have a pile of mixed-up jigsaw puzzles from different boxes. You need to sort them. A normal AI might look at the pieces one by one. This new AI, called a Neural Sorter Mamba Network, is special because it understands that the pieces of a puzzle have a specific order and history. It can look at the whole "story" of how the particle shower grew and instantly say, "This pattern belongs to the SO(N) box, not the SU(N) box."
The Power: The AI learned to spot tiny, almost invisible differences in the way the particles branch out, which are caused by the different mathematical rules of the dark forces.

4. The Heavyweights and the "Dead Zones"

The team also tested what happens if the dark particles are heavy (massive).

The Analogy: Imagine a runner (a light particle) who can sprint in any direction. Now imagine a runner carrying a heavy backpack (a massive particle). The heavy runner can't turn sharply; they have to keep going straight. This creates a "dead zone" where no turns happen.
The Finding: When the dark particles are heavy, the "firework" looks different. It creates a distinct gap or a separate "island" of activity on the map. The AI could still tell the groups apart, but it needed to see more explosions (more data) to be sure, because the heavy backpacks hid some of the subtle clues.

5. Ignoring the "Noise"

Finally, they tested if the AI could still work if they ignored the "soft" parts of the explosion—the messy, low-energy bits that are hard to predict.

The Analogy: Imagine trying to identify a song by listening to the music. Sometimes there is static noise (non-perturbative effects) at the beginning or end. The authors asked: "If we mute the static and only listen to the clear, loud notes, can we still tell the songs apart?"
The Result: Yes! For most of the different dark force types, the AI could still identify them perfectly even when the "noise" was cut out. This is great news because it means the AI is looking at the fundamental laws of physics, not just the messy details of how the particles settle down.

The Bottom Line

The paper proves that even if we can't see the dark sector directly, we can figure out its secret rules by:

Simulating how it would crash and spray particles.
Mapping those sprays onto a special 2D grid.
Using a smart AI to read the "fingerprint" of the spray.

This gives physicists a roadmap for the future: if we ever find evidence of this dark sector, we won't just know that it exists; we will be able to decode exactly what kind of force holds it together.

Technical Summary: Disentangling Dark Gauge Symmetries with Deep Learning on the Lund Jet Plane

Problem Statement
While dark sectors with new confining gauge symmetries are compelling candidates for Dark Matter (DM), experimentally discerning the underlying dark gauge group structure (e.g., distinguishing between $SU(N)$, $SO(N)$, and $Sp(2N)$) remains a significant phenomenological challenge. Most existing studies assume the dark sector mirrors the Standard Model's $SU(3)$ QCD structure. However, theoretical frameworks such as Neutral Naturalness and Composite Higgs models allow for arbitrary Lie groups. Distinguishing between these non-Abelian groups is difficult because their differences manifest in subtle variations of parton radiation patterns, which are further obscured by unknown non-perturbative hadronization dynamics. Current event generators (e.g., Pythia8, Herwig7) often lack the flexibility to simulate arbitrary gauge groups or handle massive dark gauge bosons with exact kinematic precision.

Methodology
The authors develop a comprehensive framework combining a novel Monte Carlo (MC) simulation with advanced deep learning to decode dark sector radiation patterns.

Generalized Monte Carlo Parton Shower:
- Group-Theoretic Tagging: The authors construct a simulation framework applicable to arbitrary gauge groups ($SU(N)$, $SO(N)$, $Sp(2N)$). They implement a "group-theoretic tagging" procedure where partons carry discrete gauge-group labels. Color dipoles are formed between emitters and spectators with matching conjugate tags, generalizing the standard QCD color-flow picture.
- Exact Kinematics: To handle massive dark quarks and gauge bosons, the framework employs an exact three-body phase-space parameterization. Instead of relying on massless approximations, the algorithm analytically solves the cubic kinematic equation for each branching. This captures full mass effects, including "dead-cone" thresholds and mass-induced gaps in the phase space.
- Local Recoil: A local recoil scheme is used where the spectator absorbs longitudinal recoil, ensuring exact four-momentum conservation without ad hoc prescriptions.
- Running Couplings: The simulation incorporates the Renormalization Group Equation (RGE) evolution of the dark coupling $\alpha(\mu)$ , which varies significantly depending on the specific gauge group and number of flavors ( $N_f$ ).
Data Representation and Network Architecture:
- Lund Jet Plane (LJP): The final-state jet constituents are mapped onto the Lund Jet Plane, a 2D representation ( $\ln(1/\Delta)$ vs. $\ln(k_T)$ ) that separates perturbative radiation from non-perturbative hadronization.
- Neural Sorter Mamba Network (NS-MambaNet): To classify jets, the authors utilize a specialized deep learning architecture. The process involves:
  - Set Encoding: Treating the jet tree as an unordered set of nodes (10-dimensional feature vectors per node) processed by Transformer-based self-attention.
  - Neural Sorting: A Multi-Head Neural Sorter predicts optimal sorting strategies to convert the unordered set into $K$ sequential representations.
  - Mamba Backbone: These sequences are processed by 6 Mamba layers (Selective State Space Models) to efficiently capture causal correlations in the emission history with linear time complexity.
  - Fusion and Pooling: A Gated Fusion module (Mixture-of-Experts) integrates the sorted sequences, followed by Query Attention Pooling to generate a global jet embedding for classification.

Key Contributions

Simulation Framework: The development of the first MC parton shower framework capable of simulating arbitrary gauge groups with exact massive kinematics and a unified dipole tagging scheme.
Massive Kinematics: The derivation and implementation of exact phase-space boundaries for massive splittings, explicitly modeling the impact of massive dark gauge bosons on the Lund Jet Plane topology.
Deep Learning Architecture: The introduction of NS-MambaNet, which effectively handles the permutation invariance of radiation patterns while capturing the sequential causal structure of parton showers.
Systematic Analysis: A systematic investigation of radiation patterns across $SU(2)$, $SU(4)$, $Sp(4)$, $Sp(8)$, $SO(5)$, and $SO(7)$ under various mass benchmarks and infrared cutoffs.

Results

Massless Limit: In the massless limit ( $m_V = 0$ ), the NS-MambaNet achieves high classification accuracy for distinct groups (e.g., 81% for $SU(2)$, 75% for $SO(5)$). While some groups (e.g., $SU(4)$, $Sp(8)$, $SO(7)$) show degeneracy in single-jet classification, statistical accumulation allows for 95% confidence level (C.L.) discrimination with very few jets (e.g., 14 jets for $SU(4)$ vs. $Sp(8)$).
Mass Effects: Introducing a finite dark gauge boson mass ( $m_V$ ) creates a distinct "soft radiation island" in the LJP at low $k_T$ . As $m_V$ increases, the kinematically allowed phase space shrinks, and the discrimination power degrades, requiring significantly more jets (e.g., rising from 20 to 123 jets for $SU(4)$ vs. $Sp(8)$ as $m_V$ goes from 0.1 GeV to 1.0 GeV).
Robustness to Non-Perturbative Effects: The model demonstrates robustness against non-perturbative uncertainties. Even under stringent infrared $k_T$ cuts (e.g., $k_T > 1.0$ GeV), the classifier maintains high efficiency for most pairs. However, pairs with similar Casimir ratios (e.g., $SU(4)$ vs. $Sp(8)$) suffer significant performance degradation when soft emissions are removed, as the soft tail is crucial for breaking leading-order Casimir degeneracies.

Significance
The paper claims that the perturbative footprints of dark gauge symmetries are encoded in the multi-dimensional phase space of the parton shower and can be disentangled using the proposed framework. The work provides a roadmap for the "characterization phase" post-discovery of a dark sector. By combining precision kinematics, first-principles representations (LJP), and state-of-the-art deep learning, the authors argue that future experiments can move beyond merely detecting dark jets to reconstructing the ultraviolet nature of the dark interaction and decoding the fundamental symmetries governing Dark Matter. The study explicitly notes that while it assumes dark jet constituents are measurable, the specific phenomenological signatures (e.g., semi-visible jets) are model-dependent and outside the scope of this discrimination-focused analysis.