⚛️ phenomenology

RelExt: A New Dark Matter Tool for the Exploration of Dark Matter Models

The paper introduces RelExt, a C++ program designed to efficiently scan the parameter spaces of Standard Model extensions with a Dark Matter candidate by automatically generating amplitudes, solving the Boltzmann equation for relic density, and integrating with other tools to test theoretical and experimental constraints.

Original authors: Rodrigo Capucha, Karim Elyaouti, Milada Margarete Mühlleitner, Johann Plotnikov, Rui Santos

Published 2026-01-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rodrigo Capucha, Karim Elyaouti, Milada Margarete Mühlleitner, Johann Plotnikov, Rui Santos

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 Big Picture: The Cosmic Puzzle

Imagine the universe is a giant, complex jigsaw puzzle. Scientists know that most of the puzzle pieces are missing—they call this missing stuff Dark Matter. We can't see it, but we know it's there because of how it pulls on galaxies (like an invisible hand).

The Standard Model of physics is the instruction manual we have for the pieces we can see (like atoms and light). But that manual has no instructions for the missing Dark Matter pieces. Scientists have built many new "instruction manuals" (models) that try to add a Dark Matter piece to the puzzle.

The problem? There are so many ways to build these new manuals, and so many numbers (parameters) to tweak, that finding the one version that actually fits the real universe is like trying to find a specific grain of sand on a beach by looking at every grain one by one. It takes forever.

The New Tool: RelExt

This paper introduces a new computer program called RelExt. Think of RelExt as a high-speed, smart metal detector for the beach of physics models.

Instead of a human manually digging through the sand, RelExt is a robot that can:

Read the blueprints: You give it the mathematical "blueprint" (called FeynRules files) of a new Dark Matter model.
Do the heavy lifting: It automatically calculates the complex math needed to see how these invisible particles would behave.
Find the match: It scans through millions of possible settings to find the specific combination that results in the exact amount of Dark Matter we observe in the universe today.

How It Works: The "Thermal Freeze-Out" Analogy

To understand what the program calculates, imagine a crowded dance party (the early universe).

The Party: The universe is hot and full of particles dancing and bumping into each other.
The Dance: Dark Matter particles are dancing with each other. Sometimes they bump and disappear (annihilate) into lighter particles. Sometimes lighter particles bump and create Dark Matter.
The Freeze: As the universe expands, the party gets bigger and the music slows down. The dancers get too far apart to bump into each other. The "dance" stops, and the number of Dark Matter particles left over is "frozen" in place.

RelExt's Job: It simulates this party. It calculates exactly how many dancers would be left over when the music stops. If the calculation matches the amount of Dark Matter astronomers actually see, that model is a "winner." If the calculation leaves too much or too little, the model is a "loser."

The Secret Sauce: The "Smart Search"

Older tools were like a person walking through the beach looking for a needle, checking one spot at a time. RelExt is different. It uses efficient search algorithms (like a smart GPS).

The "Random Walk": Imagine you are blindfolded and trying to find the top of a hill. You take a step. If you go up, you keep going that way. If you go down, you turn around. RelExt does this with numbers, quickly climbing toward the "peak" where the math works perfectly.
The "Grid": It can also lay a grid over the beach, checking the most promising spots first, and then zooming in on those areas to find the perfect match.

What Makes It Special?

The authors highlight three main features that make RelExt stand out:

It's a Universal Adapter: You don't need to be a coding wizard to use it. If you have the blueprint for a new model, RelExt can read it, automatically generate the math, and run the simulation. It's like a universal translator for physics models.
It's Future-Proof: The code is built so that scientists can easily add "advanced features" later (like more precise calculations) without rebuilding the whole thing.
It Plays Well with Others: RelExt is designed to talk to other tools the authors have built. One tool checks if the model breaks the laws of physics (theoretical constraints), and another checks if the model fits with real-world experiments. RelExt acts as the bridge, ensuring the model not only looks good on paper but also matches the "frozen" amount of Dark Matter we see in the sky.

The Results

The team tested RelExt on several existing models (like the "Complex Singlet Extension" or the "Two-Real-Singlet Model"). They compared their results with other famous tools (like MicrOMEGAs) and found that RelExt agrees very well with them (usually within a few percent).

They also showed that RelExt is much faster at finding the "winning" numbers. In one test, a random search found almost no good matches, but RelExt's smart search found hundreds of valid solutions in the same amount of time.

Summary

RelExt is a new, powerful computer program that helps physicists stop guessing and start finding. It automates the tedious math of simulating the early universe, allowing scientists to quickly test if their new ideas about Dark Matter actually add up to the reality we observe. It's a tool that turns a needle-in-a-haystack problem into a quick, efficient search.

Technical Summary of RelExt: A New Dark Matter Tool for the Exploration of Dark Matter Models

Problem Statement
The nature of Dark Matter (DM) remains one of the most significant open questions in contemporary particle physics. While the Standard Model (SM) is structurally complete, it lacks a viable DM candidate. Theoretical extensions often introduce DM candidates stabilized by discrete symmetries (e.g., $Z_2$ ), but verifying these models requires calculating the thermal relic density and ensuring it matches observational constraints. Existing tools such as SuperIso Relic, DarkSUSY, MicrOMEGAs, and MadDM calculate relic densities based on the freeze-out mechanism. However, finding parameter configurations within complex Beyond-the-Standard-Model (BSM) spaces that reproduce the measured relic density within experimental uncertainties can be computationally intensive. Furthermore, many existing tools are either specialized for specific models (e.g., Supersymmetry) or require manual input of thermally averaged cross-sections, limiting their flexibility for arbitrary new models.

Methodology
The authors present RelExt, a C++ program designed to compute the thermal relic density for arbitrary SM extensions featuring a DM candidate stabilized by a discrete $Z_2$ symmetry. The methodology integrates several distinct components:

Model Implementation: Users provide FeynRules model files. RelExt utilizes the packages FeynArts and FeynCalc to automatically generate tree-level squared amplitudes for all DM $2 \to 2$ (co-)annihilation processes. These amplitudes are tokenized to optimize computation and converted into C++ code.
Amplitude and Width Calculation: The code automatically identifies $s$ -channel mediators and calculates their total decay widths. For scalar and pseudoscalar mediators, it incorporates higher-order QCD corrections and off-shell decay effects (e.g., to gauge bosons) following the implementation of HDECAY. For gauge bosons and top quarks, widths are taken from input files or calculated at leading order. To ensure numerical stability near resonances, the code employs the Breit-Wigner formula for propagators.
Thermally Averaged Cross Section (TAC): The code computes the TAC by integrating the cross-sections over the center-of-mass energy and temperature. It utilizes adaptive integration methods (Simpson's 3/8 and Gauss-Kronrod) and handles resonance peaks separately. Channels heavily suppressed by Boltzmann factors are neglected based on a user-defined threshold ( $B_\epsilon$ ).
Boltzmann Equation Solver: The evolution of the DM yield is determined by solving the Boltzmann equation. The code offers two modes: a fast freeze-out approximation and a full numerical solution using an adaptive Dormand-Prince method.
Parameter Search Algorithms: A distinct feature of RelExt is its implementation of three efficient search algorithms to find parameter sets satisfying the relic density constraint:
- Monte Carlo Grid: A grid-based approach tracking the "best cells" (parameter regions closest to the target relic density) to guide subsequent sampling.
- Single-Parameter Search: A gradient-based method (using bisection or gradient descent) to adjust a single parameter to reach the target.
- Random Walk: A multi-parameter random walk algorithm that iteratively adjusts parameters to minimize the difference between the computed and target relic density.

Key Contributions

Automation and Flexibility: RelExt allows users to implement arbitrary BSM models with a single DM candidate by solely providing FeynRules files. It automatically generates the necessary amplitudes, widths, and C++ code, removing the need for manual matrix element derivation.
Efficient Scan Procedures: Unlike many existing tools that focus on calculation, RelExt prioritizes the search for viable parameter spaces. Its integrated scan algorithms are designed to efficiently locate regions where the model reproduces the observed relic density within user-specified uncertainties.
Modular Architecture: The code is designed to be easily linked with other tools from the authors' collaboration, such as ScannerS (for theoretical and experimental constraints) and BSMPT (for phase history and gravitational wave signals), facilitating comprehensive model testing.
Pre-Implemented Models: The initial release includes pre-tested implementations of the Complex Singlet Extension (CxSM), the N2HDM (Dark Doublet Phase), the CP in the Dark model (CPVDM), the Two-Real-Singlet Model (TRSM), and the BDM5.

Results and Validation
The authors validated RelExt by comparing its relic density calculations against MicrOMEGAs for several random parameter points and dedicated scans across the pre-implemented models.

Agreement: For the real singlet model and the CPVDM, RelExt and MicrOMEGAs show very good agreement, with relative differences generally below 2%.
Discrepancies: Larger deviations (up to ~9%) were observed in specific scenarios, particularly near $s$ -channel resonances involving mediators with large total widths (e.g., in the N2HDM dark doublet phase). The authors attribute this to different implementations of total widths in the propagators: RelExt applies the width throughout the integration range, whereas MicrOMEGAs restricts it to a finite area around the resonance.
Performance: In the CPVDM example, the combination of the "best cells" tracking and the random walk algorithm (Method 3) demonstrated significantly higher efficiency than simple random scanning, producing roughly 100 times more viable points within the $2\sigma$ range of the observed relic density in the same computational time.

Significance
The paper claims that RelExt provides a powerful and flexible tool for constraining DM models. By automating the generation of amplitudes and implementing efficient search algorithms, it enables researchers to systematically explore the parameter spaces of arbitrary BSM models to identify regions consistent with the measured relic density. This capability is crucial for determining the phenomenological implications of such models for current and future collider and direct detection experiments. The authors note that the code is extensible to Next-to-Leading Order (NLO) corrections and future releases aim to include multi-component DM and freeze-in mechanisms. The tool is publicly available via GitHub.