⚛️ phenomenology

Differentiating Dimension-6 and Dimension-8 Effects in $ν$ SMEFT at the HL-LHC

This paper establishes a complete basis of dimension-eight operators in the $\nu$ SMEFT using Hilbert series formalism and demonstrates that their distinct kinematic signatures, specifically in right-handed neutrino pair production at the HL-LHC, can be experimentally distinguished from dimension-six effects using Boosted Decision Tree analysis.

Original authors: Manimala Mitra, Shakeel Ur Rahaman, Subham Saha, Michael Spannowsky

Published 2026-02-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Manimala Mitra, Shakeel Ur Rahaman, Subham Saha, Michael Spannowsky

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 highly detailed, working map of a city. It tells us exactly how the streets (forces) and buildings (particles) interact. However, this map has a missing neighborhood: it can't explain why neutrinos (ghostly, tiny particles) have mass. To fix this, scientists propose adding "Right-Handed Neutrinos" (RHNs) to the city.

This paper is like a team of urban planners and detectives trying to figure out how to spot these new RHNs at the High-Luminosity Large Hadron Collider (HL-LHC), which is essentially a giant, high-speed particle racetrack.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Low-Resolution" Map vs. The "High-Definition" Lens

For a long time, scientists have used a "Dimension-6" rulebook to predict how new particles might behave. Think of this rulebook as a low-resolution photo. It's good enough to see the general shape of things, but it misses the fine details.

The authors of this paper say, "Wait a minute. If we look with a high-definition lens (Dimension-8), we might see details that the low-resolution photo completely hides."

The Analogy: Imagine trying to identify a car by its shadow. A low-resolution shadow might just look like a blob. A high-resolution shadow might reveal the specific shape of the wheels or the roof rack. The paper argues that while the "blob" (Dimension-6) and the "detailed car" (Dimension-8) might look similar from a distance, they cast different shadows if you look closely enough.

2. The Tool: The "Hilbert Series" (The Ultimate Inventory List)

Before they could look for the particles, the team had to make sure they had a complete list of all the possible ways these new particles could interact.

The Analogy: Imagine you are building a massive Lego castle. Before you start, you need to know exactly how many unique brick combinations are possible so you don't miss any. The authors used a mathematical tool called the Hilbert Series to generate a complete, error-free inventory list of every possible "Lego structure" (operator) involving these new neutrinos at the highest level of detail (Dimension-8). They confirmed their list matched existing knowledge, ensuring they weren't missing any pieces.

3. The Hunt: Finding the "Ghost" Neutrinos

The team focused on a specific type of interaction where two Right-Handed Neutrinos are created along with some jets (sprays of particles).

The Scenario: Imagine a magic trick where two invisible ghosts (the neutrinos) are created, fly around, and then turn into visible objects (electrons and jets) before disappearing again.
The Twist: Sometimes, these ghosts are "lazy" and take a little while to turn visible. In physics terms, they travel a short distance before decaying, creating a displaced vertex.
- The Analogy: It's like a firework that is lit, flies a few feet away from the launchpad, and then explodes. Most fireworks explode immediately (standard particles), but these travel a bit first. This delay is a huge clue that helps scientists ignore the background noise of the racetrack.

4. The Challenge: Distinguishing the "Fake" from the "Real"

The tricky part is that the "low-resolution" rulebook (Dimension-6) also predicts events that look very similar to the "high-definition" events.

The Analogy: Imagine you are trying to tell the difference between a real diamond and a very good glass imitation. To the naked eye, they both sparkle. If you only look at the sparkle (the final result), you might get confused.
The Solution: The authors didn't just look at the final explosion; they looked at the trajectory and speed of the particles leading up to it.
- They used a Boosted Decision Tree (BDT). Think of this as a super-smart AI detective. They fed the AI 16 different clues (like the energy of the jets, the angle between particles, and the total mass of the system).
- The AI learned that the "high-definition" (Dimension-8) events have a slightly different "personality" or "gait" than the "low-resolution" (Dimension-6) events.

5. The Result: The AI Wins

The team ran simulations for two different scenarios: one with light neutrinos and one with heavy neutrinos.

The Outcome: The AI detective was able to separate the "real" Dimension-8 signals from the "fake" Dimension-6 signals with incredible accuracy.
The Score: In statistical terms, they achieved a "5-sigma" confidence level (which is the gold standard in physics, meaning the result is almost certainly real, not a fluke). In fact, for some scenarios, the confidence was over 17 sigma.
The Takeaway: Even without using the "displaced vertex" clue (the fact that the particle traveled a bit before exploding), just looking at the energy and angles of the crash was enough to tell the difference between the two types of physics.

Summary

In short, this paper says:

We have a complete, high-definition list of how new neutrinos might behave (Dimension-8).
We know that older, lower-resolution theories (Dimension-6) might look similar.
But, by using smart computer analysis to look at the specific details of particle crashes, we can reliably tell the difference between the two.
This means that when the next big collider runs, we won't just see "something new"; we might be able to tell exactly what kind of new physics is happening, even if it's a subtle, high-level effect.

The paper concludes that we need to start looking for these "high-definition" effects now, because they are distinct and detectable, offering a clearer picture of the universe beyond our current maps.

Technical Summary: Differentiating Dimension-6 and Dimension-8 Effects in $\nu$ SMEFT at the HL-LHC

Problem Statement
The Standard Model (SM) fails to explain observed neutrino masses and mixings, necessitating physics beyond the SM (BSM). The Type I seesaw mechanism, extended with heavy right-handed neutrinos (RHNs), is a compelling framework. When RHNs are light enough to be accessible at collider energies, they cannot be integrated out at the electroweak scale, requiring an Effective Field Theory (EFT) that includes the RHN sector, known as $\nu$ SMEFT.

While collider phenomenology for dimension-5 (D5) and dimension-6 (D6) operators in $\nu$ SMEFT has been extensively studied, dimension-8 (D8) effects remain comparatively unexplored. A critical challenge arises because a double insertion of a D6 operator carries the same suppression scale ( $\sim 1/\Lambda^4$ ) as a single D8 operator insertion. Consequently, distinguishing between leading D6 effects and genuine D8 contributions is essential for probing the underlying UV completion and differentiating among competing models. This paper addresses the construction of the D8 operator basis in $\nu$ SMEFT and investigates whether D8 effects can be experimentally distinguished from D6 effects at the High-Luminosity LHC (HL-LHC).

Methodology
The authors employ a systematic approach combining theoretical construction with collider phenomenology:

Theoretical Framework (Hilbert Series):
- The complete basis of dimension-8 operators in $\nu$ SMEFT is derived using the Hilbert Series (HS) formalism. This method utilizes group-theoretic techniques to systematically classify independent operators, accounting for gauge symmetries ( $SU(3)_C \times SU(2)_w \times U(1)_Y$ ) and redundancies from equations of motion and integration by parts.
- The authors focus on the $\Psi^4 X$ class of operators (four fermions and one field strength tensor) that conserve Baryon ( $B$ ) and Lepton ( $L$ ) numbers. Specifically, they identify the operator $O^{(1,2)}_{N2q2B}$ as the representative for their study, which mediates the partonic process $pp \to NNjj$ via an intermediate $Z$ -boson.
- The study also considers relevant D6 operators ( $O_{qN}$ and $O_{duNe}$ ) that generate similar final states to serve as competing hypotheses.
Phenomenological Analysis:
- Process: The analysis focuses on the pair production of RHNs in association with jets ( $pp \to jjNN$ ), followed by the decay of RHNs into electron-jet final states ( $N \to ejj$ ).
- Decay Dynamics: The RHN decay is mediated by active-sterile mixing ( $\chi$ ) for D6 operators ( $O_{qN}$ ) and D8 operators ( $O^{(1,2)}_{N2q2B}$ ), leading to displaced vertices. In contrast, the D6 operator $O_{duNe}$ introduces a contact interaction allowing prompt $N \to ejj$ decays.
- Simulation: Signal events are simulated at Leading Order (LO) using MadGraph5_aMC@NLO with Pythia8 for parton showering and hadronization. Standard Model backgrounds are neglected because the benchmark scenarios involve long-lived RHNs (displaced decays), rendering SM backgrounds negligible.
- Discrimination Strategy: A Boosted Decision Tree (BDT) multivariate analysis is employed to distinguish the D8 signal from two D6 scenarios:
  - Case (i): D8 ( $O^{(1,2)}_{N2q2B}$ ) vs. D6 ( $O_{qN}$ ).
  - Case (ii): D8 ( $O^{(1,2)}_{N2q2B}$ ) vs. D6 ( $O_{duNe}$ ).
- Kinematic Variables: The BDT utilizes 16 kinematic observables, including invariant masses ( $M_{4j2e}$ , $M_{j1j2}$ ), transverse momenta of jets and electrons, and angular separations ( $\Delta R$ ).

Key Contributions

Operator Basis Construction: The paper provides a complete and independent basis of dimension-8 operators in $\nu$ SMEFT involving RHNs, derived via the Hilbert Series method. The counting of these operators is cross-checked and found to be in full agreement with existing literature.
Phenomenological Differentiation: The study explicitly demonstrates that D8 contributions can be experimentally distinguished from D6 effects, even when they produce similar final states.
Benchmark Scenarios: Two benchmark points (BP1: $M_N = 30$ GeV; BP2: $M_N = 500$ GeV) are analyzed with a cutoff scale $\Lambda = 1$ TeV. The analysis accounts for existing constraints from ATLAS and CMS searches for displaced leptons and jets, noting that current bounds on D6 operators are conservative for the specific signal topology considered.

Results

Production Cross-Sections: The D6 operator $O_{qN}$ provides the dominant production contribution due to weaker cutoff suppression ( $\sim 1/\Lambda^2$ ) compared to D8 operators ( $\sim 1/\Lambda^4$ ). However, the D8 operator $O^{(1,2)}_{N2q2B}$ yields distinct kinematic distributions.
Kinematic Distinctions: The D8 operator contributions exhibit harder spectra in invariant masses ( $M_{4j2e}$ , $M_{j1j2}$ ) and jet transverse momenta compared to D6 scenarios.
Classification Performance: The BDT analysis achieves high discriminatory power:
- For BP1 ( $M_N = 30$ GeV), the significance for distinguishing D8 from $O_{qN}$ is $17.28\sigma$ (AUC = 0.924), and from $O_{duNe}$ is $17\sigma$ (AUC = 0.799).
- For BP2 ( $M_N = 500$ GeV), the significance is $10.68\sigma$ (AUC = 0.901) against $O_{qN}$ and $13.20\sigma$ (AUC = 0.866) against $O_{duNe}$ .
Robustness: Crucially, this discrimination is achieved using only prompt kinematic information, without relying on observables related to decay lengths or displaced vertices, although the inclusion of such information would further enhance sensitivity.

Significance and Claims
The authors claim that their results demonstrate dimension-eight operators in $\nu$ SMEFT can give rise to experimentally resolvable signatures at the HL-LHC. The study establishes that multivariate analysis techniques are powerful tools for resolving the underlying EFT structure, specifically for differentiating between single D8 insertions and double D6 insertions.

The paper concludes that dimension-eight operators should be included in collider EFT interpretations, particularly in scenarios involving light right-handed neutrinos. The work provides a consistent foundation for studying higher-dimensional effects and suggests that future studies could explore additional D8 operator classes and incorporate broader sets of dimension-six backgrounds to further strengthen discovery potential. The authors remain modest regarding future implications, noting that while the current analysis is robust, extending the study to include more advanced machine-learning strategies and a wider range of operators is a direction for future work.