Constraining the four-light quark operators in the SMEFT with multijet and VBF processes at linear level

This paper investigates constraints on ten four-light quark operators in the Standard Model Effective Field Theory by analyzing the interference between Standard Model and new physics contributions in multijet and vector boson fusion processes, while evaluating the sensitivity of differential distributions and the validity of the EFT approach by comparing linear and quadratic contributions.

The Gist

Imagine the Standard Model of particle physics as a perfectly tuned orchestra playing a familiar symphony. For decades, this music has matched what we hear in our experiments. But physicists suspect there might be a "ghost" in the machine—new, heavy particles that are too massive for our current accelerators to catch directly. These ghosts might be whispering subtle changes into the music, making the notes slightly sharper or the rhythm a bit off.

This paper is like a team of audio engineers trying to find those whispers. They are using a tool called the Standard Model Effective Field Theory (SMEFT). Think of SMEFT as a set of "knobs" on a mixing board. Each knob represents a possible interaction between four light quarks (the tiny building blocks of protons and neutrons). The scientists want to know: How far can we turn these knobs before the music sounds wrong?

Here is how they went about it, broken down into simple steps:

1. The Setup: A Digital Soundboard

The researchers built a massive digital simulation of the Large Hadron Collider (LHC), the world's biggest particle accelerator. They didn't just look at simple collisions; they simulated complex scenarios where particles smash together and spray out multiple jets (streams of particles) or collide alongside Z, W, or photon bosons (force-carrying particles).

They focused on ten specific "knobs" (operators) that control how four light quarks interact. In the real world, these interactions don't happen in the Standard Model, so if they see them, it's a sign of new physics.

2. The Method: Listening for the "Interference"

When a new particle interacts, it doesn't just add a new note; it interferes with the existing music.

• The Linear Effect: Imagine a new singer joining the orchestra. If they sing slightly out of tune with the existing melody, the sound waves cancel each other out or amplify them in specific places. This is the "interference" the paper focuses on. It's the most sensitive way to hear the new physics.
• The Quadratic Effect: If the new singer is very loud, their own voice might drown out the orchestra entirely. This is the "squared" contribution. The paper checks if this loud voice is so strong that it breaks the rules of their "mixing board" (the EFT approximation).

3. The Investigation: Scanning the Frequencies

The team ran their simulation for different types of "concerts":

• Multijet Production: Just a chaotic spray of particle jets.
• Z/W/Photon + Jets: A jet spray accompanied by a specific force carrier (like a Z boson).
• Flavor Tagging: They even simulated a "flavor filter" to see if they could spot jets made specifically of "bottom" or "charm" quarks, hoping this would help isolate specific knobs.

They looked at the shape of the data. If the knobs were turned, the distribution of particle energies and angles would change shape—like a hill becoming a peak or a valley.

4. The Findings: What They Heard

• The "Master Knob": Out of the ten knobs, one specific interaction (called $O^{(3)}_{qq}$) was the loudest. It affected almost every type of collision they simulated. The data suggests this knob is the most constrained (meaning we know the most about it).
• The "Silent Knobs": Some knobs (like those involving specific combinations of up and down quarks) didn't seem to interfere with the Standard Model music at all in these specific collisions. It's like trying to hear a whisper in a hurricane; the background noise was too loud, or the new sound didn't mix with the old one.
• The Sweet Spot: They found that looking at collisions with medium energy was the best strategy.
  • Too low energy: The new physics signal is too weak to hear.
  • Too high energy: The "loud voice" (quadratic effects) becomes so dominant that the simple "mixing board" model breaks down, and the math becomes unreliable.
  • Just right: The "interference" is clear, but the model is still valid.

5. The Conclusion: A Work in Progress

The paper concludes that while they can set limits on these knobs, the current precision isn't quite enough to rule out new physics completely.

• The Problem: The "noise" in their simulations (theoretical uncertainties) is sometimes as big as the signal they are looking for. It's like trying to hear a whisper when the orchestra is playing loudly and the microphones aren't perfectly calibrated.
• The Future: To find the ghosts, they need two things:
  1. Better Microphones: More precise calculations of how the Standard Model behaves (reducing theoretical errors).
  2. New Instruments: Different types of observables (measurements) that might be more sensitive to these specific interactions.

In short: The paper is a sophisticated "listening test" for the universe. They checked ten specific ways new physics could hide in particle collisions. They found that one specific interaction is the most likely to be hiding in plain sight, but to confirm it, we need to tune our instruments much more precisely before we can say for sure if the orchestra is playing a secret song.

Technical Summary

Technical Summary: Constraining Four-Light Quark Operators in the SMEFT with Multijet and VBF Processes

Problem Statement
The Standard Model Effective Field Theory (SMEFT) provides a framework to parametrize deviations from the Standard Model (SM) induced by heavy new physics (NP) at energy scales below a cutoff $\Lambda$. While global fits often focus on operators coupling to top quarks, Higgs bosons, or electroweak (EW) sectors, four-light quark (4LQ) operators have received less attention. These operators introduce four-fermion contact interactions involving $u, d, c, s, b$ quarks. Although they do not contribute at Leading Order (LO) to dominant top or Higgs processes, they can induce corrections at Next-to-Leading Order (NLO) via loops or contribute directly at LO to processes involving jets and EW bosons. The challenge lies in determining which currently measured observables can effectively constrain the ten independent 4LQ Wilson coefficients in the Warsaw basis and assessing the validity of the EFT expansion when probing high-energy tails where quadratic contributions ($O(1/\Lambda^4)$) may dominate over linear interference terms ($O(1/\Lambda^2)$).

Methodology
The authors investigate the interference between the SM and ten specific CP-conserving 4LQ operators (listed in Table I) using processes at the Large Hadron Collider (LHC) at $\sqrt{s} = 13$ TeV. The study focuses on:

1. Multijet production: Analyzing differential distributions in two dijet invariant mass ($M_{jj}$) regions: $[2.4, 3]$ TeV and $[6, 13]$ TeV.
2. Vector Boson Fusion (VBF) processes: $Z$+jets, $W$+jets, and $\gamma$+jets, where jets are produced in association with an EW boson.
3. Flavor-tagging: Simulating $b$- and $c$-tagging algorithms (emulating the ATLAS DL1r algorithm) applied to multijet events to enhance sensitivity to specific quark flavors.

Computational Framework:

• Generator: Simulations were performed using MadGraph5_aMC@NLO v3.5.4.
• Model: A Universal FeynRules Output (UFO) containing the SM and the ten 4LQ operators. The CKM matrix was assumed diagonal; light quarks were treated as massless.
• Matching and Showering: Events were generated at LO parton level with up to three jets, matched and merged using the MLM scheme, and subsequently showered with Pythia8.
• Observables: Differential cross-sections were analyzed for variables such as $\chi_{jj} = e^{|y_1 - y_2|}$ (related to the scattering angle), azimuthal distance between leading jets ($\Delta\phi_{jj}$), transverse momenta ($p_T$), and invariant masses.
• Statistical Analysis: Constraints were derived using a $\chi^2$ function comparing experimental data (or assumed SM distributions for cases lacking public data) against SM and interference predictions. The analysis considered both individual operator bounds (setting others to zero) and marginalised bounds.
• EFT Validity Check: The study explicitly compares the magnitude of the linear interference term ($O(1/\Lambda^2)$) against the squared dimension-6 contribution ($O(1/\Lambda^4)$) for the operator $O^{(3)}_{qq}$ to assess the validity of the truncated EFT expansion.

Key Contributions

• Systematic Operator Analysis: The paper provides a comprehensive LO analysis of all ten 4LQ operators across multiple high-multiplicity jet and VBF processes, identifying which operators interfere with specific SM channels (e.g., noting that $O^{(1)}_{ud}$, $O^{(1)}_{qu}$, and $O^{(1)}_{qd}$ do not interfere with SM QCD in dijet production).
• Shape Sensitivity: The authors demonstrate that despite the ten-dimensional coefficient space, the differential distributions for these processes typically collapse into only two distinct shapes (isotropic vs. forward-peaked), limiting the number of independent directions that can be constrained by current data.
• Flavor-Tagging Impact: The study quantifies the limited but non-negligible impact of $b$- and $c$-tagging on constraining specific operators, noting that while sensitivity to down-like or up-like quarks increases, the overall constraining power remains weaker than inclusive multijet measurements due to lower $p_T$ ranges and large scale uncertainties.
• EFT Validity Assessment: By computing the ratio of squared to linear contributions, the paper identifies the kinematic regimes where the EFT expansion remains valid (interference dominates) versus where it breaks down (squared terms dominate).

Results

• Constrained Directions: The analysis reveals that current measurements primarily constrain a single direction in the coefficient space, dominated by the operator $O^{(3)}_{qq}$. The most competitive bounds come from multijet production in the lower $M_{jj}$ region ($2.4 < M_{jj} < 3$ TeV), where experimental uncertainties are lowest.
• Operator Limits:
  • $O^{(3)}_{qq}$ is the most tightly constrained operator across all processes.
  • Operators that do not interfere with the dominant SM QCD channels (e.g., $O^{(1)}_{ud}$, $O^{(1)}_{qu}$, $O^{(1)}_{qd}$ in dijets) remain largely unconstrained at the individual level.
  • The combined constraints from all processes are similar to those derived from multijets alone.
• EFT Validity: For most processes, the squared contributions ($O(1/\Lambda^4)$) become comparable to or larger than the interference terms at the coefficient values corresponding to the derived limits, particularly in high-energy tails ($M_{jj} > 6$ TeV). The only exception is the low-$M_{jj}$ multijet region, where the interference term dominates, validating the linear EFT approach for those specific constraints.
• Uncertainties: The study highlights that theoretical uncertainties (scale variations) often exceed experimental uncertainties, particularly in the tails of distributions and for VBF processes. This limits the ability to probe subleading directions in the coefficient space or to distinguish between operators with similar shapes.

Significance and Claims
The paper concludes that while multijet and VBF processes offer a pathway to constrain 4LQ operators, current data is only sufficient to constrain a single linear combination of coefficients, heavily weighted toward $O^{(3)}_{qq}$. The authors modestly claim that:

1. Precision is Paramount: To improve constraints and probe additional directions in the coefficient space, significant improvements in both SM theoretical predictions (e.g., moving to NLO) and experimental precision are required.
2. Validity Limits: The validity of the SMEFT expansion is not guaranteed for the high-energy tails of current datasets, as quadratic contributions often dominate. Future analyses must carefully account for $O(1/\Lambda^4)$ terms or restrict analyses to kinematic regions where linear terms dominate.
3. Need for New Observables: Since existing differential variables show minimal shape differences between operators relative to uncertainties, the authors suggest that "out-of-the-box" observables are necessary to disentangle the contributions of different 4LQ operators.
4. Resource Availability: The authors provide a notebook containing all distributions and $\chi^2$ functions used, facilitating reproducibility and further study.

The work serves as a baseline for understanding the current reach of LHC data on four-light quark operators and underscores the challenges posed by theoretical uncertainties and the breakdown of the EFT expansion at high energies.