⚛️ phenomenology

Bounds on SMEFT affecting multi gauge and Higgs-gauge couplings using two and three body spin correlations in $e^-e^+\to 3l2j\slashed{E}$ process

This paper utilizes beam polarization, spin correlations, and machine learning techniques to constrain dimension-6 SMEFT Wilson coefficients affecting multi-gauge and Higgs-gauge couplings in $e^-e^+ \to 3l2j\slashed{E}$ events at future electron-positron colliders, demonstrating that Vector Boson Scattering-like processes provide tighter constraints on $c_W$ and $c_B$ while WWZ production dominates limits on the remaining couplings.

Original authors: Amir Subba, Ritesh K. Singh

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amir Subba, Ritesh K. Singh

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 as a giant, complex machine built by nature. For decades, physicists have been trying to understand the blueprints of this machine, known as the Standard Model. It's like a master recipe book that explains how particles (the ingredients) interact to create everything we see.

However, scientists suspect this recipe book is incomplete. There are missing pages, unexplained flavors, and maybe even secret ingredients we haven't found yet. This is where the concept of New Physics comes in.

This paper is a proposal for how to find those missing pages using a future, super-precise machine called an electron-positron collider. Here is a simple breakdown of what the authors are doing:

1. The Detective Work: Looking for "Glitches"

The authors are looking for tiny "glitches" in the machine's behavior. In physics, these glitches are called anomalous couplings.

The Analogy: Imagine you are watching a perfectly choreographed dance between three dancers (particles). In the Standard Model, they know exactly how to hold hands and spin. If you see them holding hands in a weird way, or spinning too fast, that's a "glitch."
The Goal: The paper focuses on three types of dances:
1. Triple Gauge Couplings: Three particles dancing together (like $W$ , $W$ , and $Z$ ).
2. Quartic Gauge Couplings: Four particles dancing together.
3. Higgs-Gauge Couplings: The "Higgs" particle (the one that gives mass to others) dancing with the force-carriers.

2. The Laboratory: A Clean Room vs. A Mud Pit

The authors are proposing to do this experiment at a future electron-positron collider (like the ILC or CLIC).

The Analogy: Think of the current Large Hadron Collider (LHC) as a mud pit. You smash two trucks together, and debris flies everywhere. It's chaotic, and it's hard to see the specific details of the crash because there's so much "noise."
The New Approach: The electron-positron collider is like a clean, sterile laboratory. You smash two tiny, perfectly controlled balls together. The environment is "clean," meaning you can see the exact details of the interaction without the mud obscuring your view. Plus, they can control the "spin" of the balls (polarization), like spinning a top clockwise or counter-clockwise, to test how the dance changes.

3. The Strategy: Sorting the Chaos with AI

When these particles collide, they don't just disappear; they turn into other particles: leptons (like electrons and muons) and jets (sprays of particles from quarks).

The Challenge: The collision produces a mix of different "dance styles." Some are the "Triple Gauge" dance, and some are the "Vector Boson Scattering" dance. They look very similar in the data.
The Solution: The authors used Boosted Decision Trees (BDT).
- Analogy: Imagine a bouncer at a club who has to sort people into two different VIP rooms based on how they walk and talk. The BDT is a smart algorithm that looks at the energy and angle of the particles and says, "This one belongs in the Triple Gauge room," and "That one belongs in the Scattering room." It sorts the data with about 96% accuracy.

4. The Secret Sauce: Reading the Spin

This is the most creative part of the paper. The authors aren't just counting how many times a dance happens; they are looking at how the dancers spin.

The Problem: Some of the "glitches" (New Physics) only show up if you look at the direction the particles are spinning. But if you mix up the "left-handed" and "right-handed" dancers, the signal cancels out and disappears.
The Solution: They need to know exactly which particle came from which "parent."
- The Jet Flavor Tagging: When a particle decays into a "jet" (a spray of particles), it's hard to tell if it came from an "up-type" or "down-type" quark. The authors trained an Artificial Neural Network (ANN)—a type of AI—to look at the spray of particles and guess the origin. It's like a sommelier tasting a wine and guessing the exact grape and region just by the smell. This AI got about 78% accuracy.
Why it matters: By knowing the "flavor" and the "spin," they can measure asymmetries.
- Analogy: If you spin a coin, it might land on heads 50% of the time. But if the coin is slightly weighted (New Physics), it might land on heads 55% of the time. By measuring these tiny imbalances in the spin directions, they can detect the "weight" of the new physics.

5. The Results: Tighter Constraints

The authors ran simulations to see how well this method works.

The Findings:
- Combining the "Triple Gauge" dance and the "Scattering" dance gives the best results. They complement each other, like using two different flashlights to see a dark room.
- The "spin" measurements (asymmetries) are actually more powerful than just counting the total number of events. Even though the LHC has more energy, the clean environment and the spin measurements at the future collider allow for much more precise measurements of these specific "glitches."
- They found that even with some "noise" (systematic errors), the results are dominated by the sheer amount of data (statistics), meaning the more they run the experiment, the clearer the picture becomes.

Summary

In short, this paper is a blueprint for a high-precision detective story.
Instead of smashing things apart in a chaotic mud pit, the authors propose using a clean, controlled laboratory where they can:

Sort the debris using smart algorithms.
Identify the specific ingredients using AI.
Measure the tiny spins and angles of the particles.

By doing this, they hope to catch the Standard Model "lying" about how particles interact, which would be the first step toward discovering the new, hidden laws of the universe.

1. Problem Statement

The Standard Model (SM) of particle physics has been highly successful, yet it leaves fundamental questions unanswered (e.g., neutrino masses, dark matter, the hierarchy problem). The Standard Model Effective Field Theory (SMEFT) provides a model-independent framework to parameterize potential new physics (NP) effects at energy scales ( $\Lambda$ ) much higher than the electroweak scale ( $v$ ) using higher-dimensional operators.

While previous studies have probed anomalous Triple Gauge Couplings (TGC) and Quartic Gauge Couplings (QGC) individually, a comprehensive analysis often treats these couplings in isolation. Furthermore, in certain diboson processes, the interference between the SM and dimension-6 ( $d=6$ ) operators can be suppressed due to helicity selection rules, making constraints on Wilson Coefficients (WCs) weak.

The specific challenges addressed in this work are:

How to simultaneously constrain a set of nine independent $d=6$ operators affecting TGC, Higgs-gauge (HVV), and Quartic Gauge (QGC) couplings.
How to overcome the suppression of SM- $d=6$ interference in high-energy processes.
How to utilize the clean environment of future $e^-e^+$ colliders (specifically at $\sqrt{s} = 1$ TeV) to extract these constraints using spin correlations and polarization, rather than relying solely on cross-section measurements.

2. Methodology

The authors propose a comprehensive analysis of the process $e^-e^+ \to 2j + 3l + \not{E}$ (two jets, three charged leptons, and missing energy) at a center-of-mass energy of 1 TeV with polarized beams.

A. Theoretical Framework

Operators: The study focuses on 12 bosonic $d=6$ operators in the HISZ basis. After accounting for linear dependencies and redefinitions to remove kinetic mixing, nine independent Wilson Coefficients ( $c_i$ ) are analyzed. These include CP-even ( $c_W, c_B, c_{WWW}, c_{WW}, c_{BB}, c_{BW}$ ) and CP-odd ( $c^f_W, c^f_{WWW}, c^f_{WW}, c^{\tilde{B}B}$ ) operators.
Processes: The final state arises from two dominant amplitudes:
1. Triple Gauge Boson Production (WWZ): $e^-e^+ \to W^+W^-Z$ .
2. Vector Boson Scattering (VBS): $e^-e^+ \to W^+W^-jj$ (via $e^-e^+ \to e^-e^+W^+W^-$ ).
Interference: The analysis considers terms up to $O(\Lambda^{-4})$ (interference of SM with $d=6$ and quadratic $d=6$ terms), neglecting higher-order terms which are found to be negligible compared to statistical noise.

B. Event Selection and Classification

Boosted Decision Trees (BDT): To separate the phase space of WWZ production from VBS-like events, the authors trained a BDT classifier.
- Features: Invariant masses ( $m_{e^+e^-}, m_{3l}$ ), transverse momenta ( $p_T$ ), pseudorapidity differences ( $\Delta\eta$ ), and transverse mass ( $m_T$ ).
- Performance: The classifier achieves $\approx 96\%$ accuracy in distinguishing the two subprocesses.

C. Spin Correlations and Flavor Tagging

Spin Observables: The core innovation is the use of angular distributions of decay products to reconstruct spin density matrices.
- For VBS (2 spin-1 bosons): 80 parameters are extracted via 2-body spin correlations.
- For WWZ (3 spin-1 bosons): 728 parameters are extracted via 3-body spin correlations.
- These are converted into asymmetries ( $A_{ij}$ ) based on angular correlators ( $\cos\theta, \sin\phi$ , etc.).
Flavor Tagging (Jet Identification): Parity-odd asymmetries require distinguishing between up-type and down-type quarks in the hadronic decays of $W$ $W$ bosons.
- Artificial Neural Network (ANN): An ANN was developed to tag the two hardest jets as up/down-type.
- Inputs: Jet constituents (charged/neutral particle counts, kaon/pion counts, displaced tracks).
- Performance: $\approx 78\%$ accuracy for WWZ events and $\approx 70\%$ for VBS events.
Beam Polarization: The analysis utilizes initial beam polarization $(\eta_3, \xi_3) = (\mp 0.8, \pm 0.6)$ to enhance sensitivity to specific couplings and suppress backgrounds.

D. Statistical Analysis

Chi-Squared Fit: A binned $\chi^2$ analysis is performed over 7 kinematic bins of the hardest lepton's transverse momentum ( $p_T$ ).
MCMC: Markov Chain Monte Carlo (Metropolis-Hastings algorithm) is used to derive marginalized 95% Confidence Level (CL) limits on all nine WCs simultaneously, accounting for correlations.
Systematics: The study evaluates the impact of systematic errors (ranging from 0% to 10%) and integrated luminosity ($100$ to $3000 \text{ fb}^{-1}$ ).

3. Key Contributions

Simultaneous Multi-Process Analysis: Unlike previous works that often isolate TGC or QGC, this paper simultaneously probes TGC, HVV, and QGC operators using a single final state ( $3l2j/\not{E}$ ) that captures both WWZ and VBS topologies.
Utilization of Spin Correlations: The paper demonstrates that spin correlation asymmetries (derived from angular distributions) provide significantly tighter constraints than cross-section measurements alone, particularly for CP-odd operators where cross-section sensitivity is low.
Jet Flavor Tagging for Asymmetries: The implementation of an ANN for jet flavor tagging allows the recovery of parity-odd observables that would otherwise average to zero, a crucial step for constraining CP-violating operators.
Complementarity of Channels: The study quantifies the complementary nature of VBS and WWZ channels:
- VBS provides superior constraints on $c_B$ and $c_W$ .
- WWZ dominates the constraints for the remaining 7 coefficients (including $c_{WWW}, c_{WW}, c^f_W$ , etc.) due to the larger number of available spin observables (728 vs. 80).

4. Results

Wilson Coefficient Limits:
- At $\sqrt{s}=1$ TeV with $L=1000 \text{ fb}^{-1}$ and zero systematics, the combined analysis yields limits in the range of $[-0.3, +0.4] \text{ TeV}^{-2}$ for most coefficients.
- $c_B$ and $c_W$ : Best constrained by VBS.
- $c_{WWW}, c_{WW}, c^f_W, c^f_{WWW}, c^f_{WW}, c^{\tilde{B}B}$ : Best constrained by WWZ.
- $c_{BB}$ : Receives comparable contributions from both channels.
Impact of Observables:
- Cross-section measurements alone provide weak constraints (except for $c_{WW}$ ).
- Asymmetries (spin correlations) dominate the $\chi^2$ contribution, reducing the allowed parameter space by a factor of 2 to 4 compared to cross-section-only analyses.
Robustness:
- The limits are statistics-dominated. Even with conservative systematic errors of 10%, the constraints degrade by less than 20%, indicating the method is robust against experimental uncertainties.
- Increasing luminosity from $100 \text{ fb}^{-1}$ to $3000 \text{ fb}^{-1}$ significantly tightens the bounds (roughly scaling with $1/\sqrt{L}$ ).

5. Significance

This work establishes that future $e^-e^-$ colliders (like ILC, CLIC, or FCC-ee) are ideal facilities for precision tests of the electroweak sector, even at energies lower than the LHC.

Overcoming Helicity Suppression: By exploiting spin correlations and polarization, the study bypasses the helicity selection rules that suppress $d=6$ interference in high-energy hadron colliders.
Model-Independent Constraints: The derived limits on the nine Wilson Coefficients provide a rigorous, model-independent benchmark for new physics searches.
Methodological Advancement: The successful integration of machine learning (BDT for phase space selection and ANN for jet flavor tagging) with spin density matrix formalism offers a blueprint for analyzing complex multi-boson final states in future collider experiments.

In conclusion, the paper argues that the combination of polarized beams, high luminosity, and sophisticated spin-correlation observables makes the $e^-e^+ \to 3l2j/\not{E}$ channel a "golden mode" for probing the structure of the electroweak symmetry breaking and potential new physics at the TeV scale.