Combined effective field theory interpretation of measurements sensitive to quartic gauge boson couplings in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector

The ATLAS collaboration presents a combined effective field theory interpretation of vector-boson scattering and tri-boson measurements from 13 TeV proton-proton collisions to constrain anomalous quartic gauge boson couplings arising from dimension-8 operators, providing confidence intervals with and without unitarity constraints.

The Gist

Imagine the universe is a giant, cosmic kitchen. For decades, physicists have been cooking up recipes based on the Standard Model, which is like the ultimate cookbook of how particles (the ingredients) interact. This cookbook has worked perfectly for everything we've tested so far.

However, scientists suspect there might be "secret ingredients" or "hidden spices" in the universe that we haven't found yet. These could be new particles or forces that are too heavy or too rare to see directly with our current tools (like the Large Hadron Collider, or LHC).

This paper is about a team of chefs (the ATLAS Collaboration) trying to taste-test the universe's soup to see if there are any subtle, strange flavors that don't match the standard recipe.

The Big Idea: The "Quartic" Mystery

In the Standard Model, particles usually bump into each other in pairs or triplets. But sometimes, four particles can interact all at once. This is called a Quartic Gauge Coupling.

Think of it like a dance:

• Standard Model: Usually, two people dance together, or maybe a trio forms a circle.
• The Mystery: Sometimes, four people try to dance in a square formation. The Standard Model says, "This dance is very specific and follows strict rules."
• The Goal: The scientists want to see if the four dancers are breaking the rules or doing something weird. If they are, it's a sign of "New Physics."

The Detective Work: The "Effective Field Theory" (EFT)

Since the scientists can't see the heavy "secret ingredients" directly, they use a clever trick called Effective Field Theory.

Imagine you are trying to figure out what a giant, invisible elephant is doing in a room just by looking at the footprints in the dust. You can't see the elephant, but you can see how the dust is disturbed.

• The Footprints: These are the tiny deviations in the data.
• The Elephant: This is the new, heavy physics we can't see yet.
• The Theory: The scientists use a mathematical framework (EFT) to predict what kind of footprints the elephant would leave if it were there. They look for "Wilson coefficients," which are basically numbers that tell us how big the elephant's footprints are.

The Experiment: A Massive Combination

The paper describes a massive effort to combine data from eight different experiments (analyses) that the ATLAS detector has been running.

• The Analogy: Imagine eight different detectives investigating the same crime scene, but each one is looking at a different clue (one looks at the mud, one at the broken window, one at the footprints).
• The Problem: If they work alone, their clues might be too weak to solve the case.
• The Solution: They decided to put all their notes together into one giant "Super-Detective Report." By combining the data from 140 trillion collisions (140 fb⁻¹ of data), they created a much sharper picture.

They looked at two main types of "crime scenes":

1. Vector Boson Scattering (VBS): Like two billiard balls hitting each other and sending two other balls flying off.
2. Tri-boson Production: Like three balls colliding at once.

The Results: No "Elephant" Found (Yet)

After crunching the numbers, the team found that the footprints match the Standard Model perfectly.

• The Verdict: There is no evidence of the "elephant" (new physics) breaking the rules of the four-particle dance.
• The Constraint: While they didn't find new physics, they did something very important: they tightened the net. They now know exactly how big the elephant's footprints could be before we would have seen them.
  • Analogy: Before, we knew the elephant's footprints couldn't be bigger than a swimming pool. Now, thanks to this combined study, we know they can't be bigger than a bathtub. If the elephant is there, it's very small, very quiet, or very far away.

The "Safety Valve": Unitarity

There was a tricky part of the math. If the "secret ingredients" were too strong, the math would break down (like a recipe that says "add infinite sugar"). This is called violating unitarity.

To keep things realistic, the scientists applied a "safety valve" (a clipping threshold). They said, "Okay, if the new physics gets too strong at high energies, we'll assume it stops there." Even with this safety valve, they found no evidence of new physics, but they set even stricter limits on where that new physics could be hiding.

Why This Matters

Even though they didn't find a new particle, this paper is a huge success because:

1. It's the most comprehensive check yet: It's the first time so many different angles of the same problem have been looked at together.
2. It sets the rules: It tells future physicists, "Don't look for new physics in this specific range; it's not there." This saves time and money.
3. It proves the Standard Model is tough: The universe's "recipe book" is holding up against the most rigorous taste tests we can perform.

In short: The ATLAS team combined all their data to check if four particles were dancing to a new, secret rhythm. They listened very carefully, and while they didn't hear the new music, they proved that if the music is playing, it's so quiet and subtle that we need even better ears to hear it next time.

Technical Summary

1. Problem and Motivation

The Standard Model (SM) of particle physics has been rigorously tested at the Large Hadron Collider (LHC), yet no direct evidence of new heavy particles (TeV-scale resonances) has been found. Consequently, the search for new physics has shifted toward detecting subtle deviations in SM predictions caused by high-energy phenomena that are too heavy to be produced directly.

• The Gap: While constraints on dimension-6 operators in the SM Effective Field Theory (SMEFT) are well-established, many Beyond the Standard Model (BSM) scenarios (e.g., composite Higgs, warped extra dimensions) predict Anomalous Quartic Gauge Couplings (aQGCs). These effects first appear at dimension-8 in the EFT expansion.
• The Challenge: Dimension-8 operators are numerous (tens of thousands), and probing them individually is experimentally impractical. Furthermore, dimension-8 effects are often suppressed or degenerate with dimension-6 effects in certain processes.
• The Goal: To provide the most comprehensive constraints to date on aQGCs by combining multiple independent ATLAS measurements sensitive to vector-boson scattering (VBS) and tri-boson production, utilizing the full Run 2 dataset ($140 \text{ fb}^{-1}$).

2. Methodology

Theoretical Framework

The analysis utilizes the Éboli model, a specific subset of dimension-8 operators that respect SM $SU(2)_L \times U(1)_Y$ gauge symmetry and generate aQGCs.

• Operators: The model considers 21 operators categorized as scalar ($O_{S0}, O_{S1}, O_{S2}$), tensor ($O_{T0} \dots O_{T9}$), and mixed ($O_{M0} \dots O_{M7}$).
• Reductions: $O_{M6}$ is redundant; $O_{T3}$ and $O_{T4}$ are excluded due to simulation limitations in previous analyses. $O_{S0}$ and $O_{S2}$ are varied simultaneously with equal coefficients ($f_{S02}$). This leaves 17 independent Wilson coefficients ($f_i$) to be constrained.
• Cross-Sections: The cross-section is expanded as:
  $$\sigma = |\mathcal{M}_{SM}|^2 + \sum_i \frac{f_i}{\Lambda^4} 2\text{Re}(\mathcal{M}^{(8)}_i \mathcal{M}^*_{SM}) + \sum_i \left(\frac{f_i}{\Lambda^4}\right)^2 |\mathcal{M}^{(8)}_i|^2 + \sum_{i<j} \dots$$
  The analysis includes linear (interference), quadratic, and cross-term contributions.

Input Measurements

The combination integrates 8 distinct analyses performed by ATLAS using $140 \text{ fb}^{-1}$ of data at $\sqrt{s}=13$ TeV:

1. VBS Channels:
   • $VVjj \to (\ell\ell jj / \ell\nu jj / \nu\nu jj) jj$ (Semileptonic)
   • $W^\pm W^\pm jj \to \ell^\pm \nu \ell^\pm \nu jj$ (Same-sign)
   • $WZjj \to \ell^+\ell^-\ell^\pm \nu jj$
   • $ZZjj \to \ell^+\ell^-\ell^+\ell^- jj$
   • $W\gamma jj \to \ell^\pm \nu \gamma jj$
   • $Z\gamma jj \to \nu\bar{\nu}\gamma jj$
   • $Z\gamma jj \to \ell^+\ell^-\gamma jj$
2. Tri-boson Channel:
   • $W\gamma\gamma \to \ell^\pm \nu \gamma\gamma$

Data Harmonization:

• Binning: Inputs were re-binned or re-interpreted to ensure full coverage of the operator space. For example, the $Z(\nu\nu)\gamma jj$ analysis was expanded from a single overflow bin to an 8-bin distribution.
• Template Replacement: For analyses that did not originally provide templates for all 17 operators, a "template replacement" procedure was developed, approximating missing kinematic shapes using existing operators with the lowest $\chi^2$ difference at the particle level.
• Overlap Handling: A non-negligible overlap between the control region of the $W^\pm W^\pm jj$ analysis and the signal region of $WZjj$ was resolved by dropping the overlapping control region.

Statistical Model

• Likelihood: A joint simultaneous likelihood model was constructed across all channels.
• Parameters: Wilson coefficients ($f_i/\Lambda^4$) are parameters of interest; systematic uncertainties are nuisance parameters.
• Correlations: Systematic uncertainties (jet energy scale, lepton ID, luminosity, etc.) are treated as fully correlated across analyses if they originate from the same source. Signal modeling uncertainties (QCD scale, PDFs) are correlated for VBS processes but uncorrelated for $W\gamma\gamma$.
• Unitarity Constraints: To address the violation of perturbative unitarity at high energies (where EFT breaks down), a clipping procedure was applied. Contributions from dimension-8 operators are set to zero above a specific energy threshold (clipping scale, e.g., 1.5 TeV). The final "unitarized" limits are the intersection of the experimental confidence intervals and theoretical unitarity bounds.

3. Key Contributions

1. First Comprehensive Combination: This is the first global combination of ATLAS VBS and tri-boson measurements specifically targeting dimension-8 aQGCs.
2. Operator Coverage: It provides constraints on 17 independent Wilson coefficients simultaneously, a significant expansion over previous single-channel or single-operator studies.
3. Methodological Rigor: The paper introduces robust methods for handling operator degeneracies and missing templates via template replacement, validated by non-closure studies.
4. Unitarity Handling: It systematically applies unitarity constraints (clipping) to provide physically meaningful limits, comparing results with and without these constraints to demonstrate the impact of high-energy tails.

4. Results

Confidence Intervals (CI)

Results are presented as 68% and 95% confidence level intervals for the Wilson coefficients ($f_i/\Lambda^4$ in $\text{TeV}^{-4}$).

• Without Unitarity Constraints:

  • The combination improves individual channel constraints by up to 20% for scalar and mixed-type operators.
  • The $Z(\nu\nu)\gamma jj$ channel is the dominant driver for tensor-type operators ($f_{T0} \dots f_{T9}$).
  • The $VV$-semileptonic channel drives constraints on scalar ($f_{S02}, f_{S1}$) and mixed ($f_{M0}, f_{M1}, f_{M7}$) operators.
  • Example 95% CL observed intervals (no unitarization):
    • $f_{S02}/\Lambda^4$: $[-4.96, 4.90]$
    • $f_{T0}/\Lambda^4$: $[-0.32, 0.22]$

• With Unitarity Constraints (Clipping at 1.5 TeV):

  • Applying the clipping threshold widens the intervals significantly (by an order of magnitude) because the high-energy sensitivity is removed.
  • However, the relative contribution of different channels becomes more balanced. For instance, the $WZjj$ channel becomes as sensitive as the previously dominant $Z(\nu\nu)\gamma jj$ channel under unitarization.
  • The combined bounds improve upon previous published results by 17% to 96%, depending on the coefficient.
  • Positivity Bounds: Theoretical positivity bounds (derived from causality and locality) are overlaid, drastically reducing the allowed phase space for certain coefficients.

Simultaneous Fits

• 2D Fits: Confidence contours for pairs of operators (e.g., $f_{M2}$ vs $f_{M3}$) show that cross-terms are highly suppressed between different operator families (scalar vs. tensor) but can be significant within families.
• Profiled Fits: When all 17 coefficients are allowed to float simultaneously, the intervals change by only a few percent for coefficients with negligible cross-terms. However, for coefficients with significant cross-terms (e.g., $f_{M7}, f_{T0} \dots f_{T9}$), the profiled intervals differ more substantially from the 1D fits.

5. Significance

• Competitiveness: The constraints achieved are competitive with, and in many cases superior to, similar results from the CMS collaboration.
• BSM Sensitivity: By constraining dimension-8 operators, this analysis probes energy scales ($\Lambda$) up to several TeV (e.g., $\Lambda \approx 2\text{--}3$ TeV for $f_{T1}$), providing indirect sensitivity to new physics models like composite Higgs or extra dimensions that might not manifest as direct resonances.
• Foundation for Future Runs: This combination establishes a robust statistical framework for interpreting future high-luminosity LHC data (HL-LHC) and future colliders, demonstrating how to effectively combine diverse final states to constrain complex EFT parameter spaces.
• Validation of EFT: The successful application of unitarity clipping and the comparison of linear vs. quadratic contributions validate the use of EFT in the high-energy regime of the LHC, provided appropriate theoretical safeguards are applied.

In conclusion, this paper represents a milestone in the indirect search for new physics, offering the most stringent and comprehensive limits on anomalous quartic gauge couplings to date, while rigorously addressing the theoretical limitations of the EFT approach.