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FeynRules and UFO models for $\nu$ SMEFT: operators of dimensions five and six

This paper presents the public release of FeynRules and UFO model files for the $\nu$ SMEFT, enabling the simulation of dimension-five and dimension-six operators involving right-handed neutrinos, including a specialized implementation for Majorana four-fermion operators compatible with MadGraph5_aMC@NLO.

Original authors: Arsenii Titov

Published 2026-08-18

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CC BY 4.0

Original authors: Arsenii Titov

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 as we know it is built on a foundation of particles and forces, a framework known as the Standard Model. For decades, this model has successfully explained how matter behaves, from the smallest atoms to the largest stars. Yet, it leaves a significant mystery unsolved: why do neutrinos, those ghostly particles that pass through everything, have mass? The simplest explanation suggests that there are hidden partners to the known neutrinos, particles that do not interact with light or ordinary matter in the usual way. These are called right-handed neutrinos. If they exist, they could be the key to understanding why the neutrinos we see are so incredibly light. However, if these hidden particles are too heavy to be created directly in our current particle accelerators, they might still leave subtle fingerprints on the universe. Physicists use a tool called effective field theory to describe these faint signals, treating the heavy, unseen particles as if they were simply modifying the rules of interaction for the lighter, visible ones. This approach allows scientists to search for new physics without needing to see the new particles directly, looking instead for tiny deviations in how known particles behave.

A researcher has now provided the essential tools for scientists to hunt for these specific deviations. They have released a new set of digital models that describe how right-handed neutrinos would influence the behavior of other particles, specifically focusing on interactions that involve five or six fundamental building blocks. In the language of particle physics, these are known as dimension-five and dimension-six operators. While the mathematical descriptions of these interactions have existed in theory for some time, they were difficult to use in practical computer simulations that predict what happens in particle colliders like the Large Hadron Collider. The researcher has translated these complex mathematical descriptions into a format that computer programs can read and use to simulate collisions. This work fills a critical gap, turning abstract theory into a practical toolkit for experimentalists who are looking for signs of new physics in the data.

The core of this achievement lies in solving a specific technical hurdle that has long blocked progress. When the researcher tried to include interactions involving four particles at once, where one of them is a heavy, invisible neutrino, the standard computer programs used for simulation would crash. This happened because the computer could not determine the correct direction of flow for the particles' internal properties, a necessary step for calculating probabilities. To get around this, the researcher invented a clever workaround. They introduced temporary, invisible helper fields that do not travel or exist as real particles but serve only to break down the complex four-particle interaction into simpler, two-step processes. Think of it as a bridge that allows traffic to cross a river that would otherwise be impassable; the bridge itself is not the destination, but it makes the journey possible. By using these auxiliary fields, the researcher successfully converted the problematic interactions into a format that the simulation software can handle without error.

The result is a comprehensive collection of model files that are now available to the global scientific community. These files cover a wide range of scenarios, including interactions where the heavy neutrino appears alone and cases where two of them appear together. The models also account for the possibility that there is only one type of heavy neutrino or three different types, though the current release focuses on the simplest case of just one generation. The researcher has rigorously tested their models to ensure they produce the correct results. They simulated specific particle collisions, such as those that might occur at the Large Electron-Positron collider or the Large Hadron Collider, and compared their computer-generated predictions with known analytical calculations. The results matched perfectly, confirming that the new models are reliable and ready for use.

This release is more than just a technical update; it is an invitation to explore the unknown. By making these models publicly available, the author has empowered experimental teams to design new searches and analyze existing data with a fresh perspective. The models are designed to be used in conjunction with the most advanced simulation tools available, allowing researchers to predict exactly what a signal from a heavy right-handed neutrino would look like in a detector. While the models currently focus on the most basic level of interaction, they lay the groundwork for future, more complex studies that could include higher levels of precision. The ultimate goal is to determine whether these hidden neutrinos exist and, if they do, to understand their role in the grand architecture of the universe. With these new tools in hand, the search for the missing pieces of the neutrino puzzle has moved from the realm of pure theory into the active phase of experimental discovery.

Technical Summary: FeynRules and UFO models for $\nu$ SMEFT

Problem Statement
The existence of right-handed (RH) neutrinos ( $N_R$ ) is a primary theoretical motivation for explaining the small masses of light neutrinos, often via the seesaw mechanism. When new physics exists at a scale $\Lambda$ significantly larger than the electroweak scale $v$ , the most general description of low-energy effects is the Standard Model Effective Field Theory extended with RH neutrinos ( $\nu$ SMEFT). While the phenomenology of $\nu$ SMEFT operators up to dimension six has been extensively studied in the literature, a critical gap existed in the availability of public computational tools. Specifically, there were no publicly available FeynRules and UFO model files capable of handling the full set of dimension-five and dimension-six operators involving $N_R$ for collider simulations. A particular technical hurdle was the incompatibility of direct UFO exports containing four-fermion operators with Majorana neutrinos with the MadGraph5_aMC@NLO framework, which cannot unambiguously define fermion flow in such interactions.

Methodology
The author developed a suite of modular FeynRules model files and corresponding UFO implementations to address these gaps. The work is structured as follows:

Operator Implementation: The author extended the default SM model and the HeavyN model to include:
- Dimension-five and dimension-six two-fermion operators: These include interactions between $N_R$ and the Higgs/gauge sectors (e.g., dipole operators, Higgs-current couplings).
- Dimension-six four-fermion operators: These are categorized into "single- $N_R$ " (one $N_R$ field) and "pair- $N_R$ " (two $N_R$ fields) operators. The implementation focuses on the minimal case of $n_s=1$ (coupling to the first generation of RH neutrinos), though the framework allows for generalization.
Auxiliary-Field Strategy for Four-Fermion Operators: To resolve the MadGraph5_aMC@NLO incompatibility with Majorana fermions in four-fermion contact terms, the author implemented a specific workaround. Instead of exporting the contact interaction directly, they introduced auxiliary, non-propagating scalar and vector fields ( $U, V$ and $U_\mu, V_\mu$ ).
- These fields replace the four-fermion contact interaction with a sum of diagrams containing only three-point vertices.
- The auxiliary fields are defined with opposite-sign quadratic terms in the Lagrangian to cancel unwanted terms while preserving the desired effective interaction upon solving the equations of motion.
- A custom Python patch (patch_vSMEFT_aux_ufo.py) is applied to the generated UFO files to assign momentum-independent propagators ( $\pm i/\Lambda^2$ ) and ensure the auxiliary fields are treated as non-propagating internal states.
Model Construction:
- Modular Files: Separate .fr files were created for two-fermion interactions (vSMEFT_2-fermions_Higgs-N.fr), single- $N_R$ four-fermion interactions (vSMEFT_4-fermions_single-N.fr), and pair- $N_R$ four-fermion interactions (vSMEFT_4-fermions_pair-N.fr).
- Auxiliary Versions: Corresponding files using the auxiliary field method were created (..._aux.fr).
- Unified Model: A master file (vSMEFT.fr) combines all operators, which is then exported to a unified UFO model (vSMEFT_UFO).

Key Contributions

Public Release: The author released the first public set of FeynRules and UFO model files for $\nu$ SMEFT including dimension-five and dimension-six operators with RH neutrinos. These are available on GitHub.
MadGraph Compatibility: The paper provides a robust, generic implementation of four-fermion operators with Majorana neutrinos that is compatible with MadGraph5_aMC@NLO. This is achieved through the auxiliary-field formalism and the accompanying UFO patch, solving a known limitation in the toolchain.
Comprehensive Coverage: The models cover the full spectrum of relevant operators:
- Two-fermion operators (Higgs/gauge couplings).
- Single- $N_R$ four-fermion operators (involving quarks and leptons).
- Pair- $N_R$ four-fermion operators (including lepton-number-violating terms).
Validation: The models were validated against existing literature and analytical calculations.
- Two-fermion processes (e.g., $e^+e^- \to N_1N_1$ and $e^+e^- \to \nu N_1$ ) were compared with results from Ref. [43], showing perfect agreement.
- Single- $N_R$ production at the LHC ( $pp \to \mu N_1$ ) was compared with Ref. [26], reproducing cross-section trends.
- Four-fermion processes ( $d\bar{d} \to N_1N_1$ ) were simulated and compared against analytical cross-section formulas for both Majorana and Dirac cases, demonstrating perfect agreement between the numerical simulation and analytical results.

Results
The generated UFO models successfully reproduce known phenomenological results for processes involving heavy neutral leptons (HNLs) at both lepton colliders (LEP-like) and hadron colliders (LHC). The auxiliary-field implementation correctly reproduces the cross-sections for four-fermion contact interactions, distinguishing correctly between Majorana and Dirac fermion behaviors in the simulation. The models are currently suitable for tree-level computations, with a loop-induced Higgs-gluon-gluon vertex included as an effective coupling.

Significance
The paper claims that these model files fill a critical gap in the computational toolkit for $\nu$ SMEFT phenomenology. By providing publicly available, validated, and MadGraph-compatible files, the author enables further studies of collider phenomenology for RH neutrinos. The work is intended to assist both phenomenologists in performing simulations and experimental collaborations (particularly at the LHC) in implementing specific analyses within the $\nu$ SMEFT framework. The author notes that while the current version is tree-level, the modular structure allows for future extensions, such as NLO QCD computations or combination with the SMEFTsim package to study interplays between operators with and without RH neutrinos.

Technical Summary: FeynRules and UFO models for ν\nuνSMEFT

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Technical Summary: FeynRules and UFO models for $\nu$ SMEFT