Dijet bounds on third-generation four-quark operators

✨

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 is a giant, complex machine, and the Standard Model is the instruction manual we've written for how it works. But we suspect there are hidden gears and levers (new physics) that the manual doesn't mention yet.

This paper is like a team of detectives trying to find those hidden gears by looking at the machine's "exhaust fumes." Specifically, they are looking at dijets—pairs of high-speed particle jets created when protons smash together at the Large Hadron Collider (LHC).

Here is the story of their investigation, broken down into simple concepts:

1. The Suspects: The "Third-Generation" Gang

In the particle world, there are three "families" of quarks (the building blocks of matter).

First and Second Generation: These are the "common citizens" (like up, down, strange, and charm quarks). They are everywhere and easy to spot.
Third Generation: These are the "elite" or "heavy hitters" (top and bottom quarks). They are rare, heavy, and hard to catch.

The scientists are looking for 10 specific "rules" (called operators) that might govern how these heavy third-generation quarks interact. The problem? Most of these rules only affect the heavy quarks directly. Since the LHC creates mostly light quarks, these rules seem invisible at first glance. It's like trying to find a rare, heavy gold coin in a pile of sand by only looking at the sand grains.

2. The Trick: The "Renormalization Group" (The Invisible Messenger)

The paper's big discovery is about a sneaky mechanism called Renormalization Group (RG) flow.

Think of the universe as a busy post office.

Tree Level (Direct): If you send a letter directly from the "Heavy Quark" department to the "Jet" department, it arrives instantly. But in our case, the "Heavy Quark" department is closed for business at the LHC (because top quarks aren't in the initial collision), so no direct letters get sent.
Loop Level (Indirect): However, the post office has a system where letters can be forwarded. A "Heavy Quark" rule can whisper a secret to a "Light Quark" rule through a series of intermediate steps (loops).

The authors realized that even if a rule only involves heavy quarks, the universe's "whispering network" (quantum loops) can pass that rule's influence down to the light quarks. By the time the light quarks smash together to make jets, they are carrying a faint echo of the heavy quark's rules.

3. The Magnifying Glass: PDFs (The Crowd Factor)

Here is the catch: The "whispering" process is very weak. It's like trying to hear a whisper in a hurricane. Usually, this would make the signal too faint to detect.

But, the LHC has a unique advantage: Parton Distribution Functions (PDFs).
Think of the proton as a crowded stadium. Inside, there are millions of light quarks (the crowd) but very few heavy quarks (the VIPs).

When the heavy quark rules try to influence the light quarks, they are trying to influence a massive crowd.
Even though the "whisper" is weak, the sheer number of light quarks (the crowd) amplifies the signal. It's like a single person shouting in a stadium; the sound might be weak, but if 10,000 people repeat it, it becomes a roar.

The paper calculates whether this "crowd amplification" is strong enough to overcome the weakness of the whisper.

4. The Results: Who Got Caught?

The team ran the numbers using data from the CMS experiment at the LHC.

The "Bottom" Quarks (The Easy Wins): Five of the rules involve four bottom quarks. Since bottom quarks are present in the proton (though rare), these rules show up directly in the jet data. The scientists set very tight limits on these. They effectively said, "If these rules exist, they are very weak."
The "Top" Quarks (The Elusive Ones): The other five rules involve top quarks. These only show up through the "whispering" (loop) mechanism.
- The Verdict: Even with the "crowd amplification" (PDFs), the signal was still too faint to catch these rules firmly. The limits on these top-quark rules are still very loose.
- The Silver Lining: However, the paper proves that the "whispering" does happen. It's not zero. It just means we need even more sensitive detectors or different types of experiments (like looking at the Higgs boson or precision electroweak measurements) to catch these specific rules.

5. The "Flat Direction" Mystery

One of the most interesting findings is a "flat direction." Imagine you have two knobs on a machine. If you turn one up and the other down by the exact same amount, the machine looks exactly the same.

The scientists found that for two specific rules involving bottom quarks, the data couldn't tell them apart if they were balanced against each other.
The Fix: By including the "whispering" effects (the loop corrections), they broke this symmetry. The extra data from the loops allowed them to distinguish between the two rules, resolving a mystery that direct measurements couldn't solve.

Summary

This paper is a triumph of indirect detection.

Old Way: Look for the heavy particles directly. (Hard to do).
New Way in this paper: Look for the echoes of heavy particles in the light particles, amplified by the sheer number of light particles in the proton.

The Bottom Line:
The scientists successfully used the "echoes" to set new, tighter rules for how bottom quarks behave. However, for the top quarks, the echoes are still too faint to pin them down. They proved that the "whispering network" of the universe is real and important, but for the heaviest particles, we might need a bigger microphone (more data or different experiments) to hear them clearly.

1. Problem Statement

The Standard Model Effective Field Theory (SMEFT) provides a model-independent framework to search for Beyond the Standard Model (BSM) physics. However, a significant challenge arises from the large number of dimension-six operators and their mixing under Renormalization Group (RG) flow.

The Specific Issue: Third-generation four-quark operators (involving top and bottom quarks) are often weakly constrained by direct measurements because they do not contribute to standard light-jet production at tree level.
The Gap: While operators involving four bottom quarks ( $b$ ) can contribute directly to dijet production (due to the bottom quark PDF), operators involving top quarks ( $t$ ) or mixed $t/b$ states generally do not. Previous analyses often neglected the indirect constraints generated by RG evolution, which can mix third-generation operators into first- and second-generation operators that do contribute to dijet production.
Goal: To determine if LHC dijet measurements can constrain the full set of ten third-generation four-quark operators by including leading-logarithmic (LL) RG effects up to the two-loop order.

2. Methodology

The authors employ a combination of theoretical calculations and phenomenological analysis using LHC data.

Theoretical Framework:

Operators: The analysis focuses on ten dimension-six operators in the Warsaw basis involving third-generation quarks ( $i=j=k=l=3$ ): $Q_{QQ}^{(1,3)}$ , $Q_{tt}$ , $Q_{bb}$ , $Q_{tb}^{(1,8)}$ , $Q_{Qt}^{(1,8)}$ , and $Q_{Qb}^{(1,8)}$ .
RG Evolution: The core theoretical innovation is the inclusion of RG mixing effects:
- One-Loop (LL): Operators involving $b$ quarks (e.g., $Q_{Qb}^{(8)}$ ) mix into light-quark operators via QCD penguin diagrams.
- Two-Loop (LL): Purely right-handed top operators ( $Q_{tt}$ ) do not mix at one-loop to light quarks. Instead, they require a "double-penguin" insertion (two-loop level) to generate light-quark currents. This induces double-logarithmic ( $\alpha_s^2 L^2$ ) corrections.
Matrix Elements: The authors calculate the squared matrix elements for $2 \to 2$ partonic scattering processes (e.g., $bb \to bb$ , $bd \to bd$ , $uu \to uu$ ) including the SM contribution, linear BSM terms, and quadratic BSM terms. They explicitly derive the logarithmically enhanced terms induced by RG running from a high scale $\Lambda$ down to the dijet invariant mass scale $M_{jj}$ .

Phenomenological Analysis:

Observable: The study utilizes the normalized dijet angular distribution ( $1/\sigma d\sigma/d\chi$ ), where $\chi = (1+|\cos\hat{\theta}|)/(1-|\cos\hat{\theta}|)$ . This observable is sensitive to contact interactions and broad resonances.
Data: The analysis uses CMS Run 2 data (36 fb $^{-1}$ ) covering seven $M_{jj}$ intervals, specifically focusing on the three lowest bins ( $2.4 < M_{jj} < 4.2$ TeV) where sensitivity is highest.
Simulation:
- PDFs: CT14nnlo.
- Scales: Renormalization ( $\mu_R$ ) and factorization ( $\mu_F$ ) scales set to $M_{jj}$ .
- Corrections: NLO QCD and EW corrections are incorporated via rescaling factors derived from CMS SM predictions. NNLO QCD effects are neglected as they are subleading ( $\sim 5\%$ ) in the relevant $\chi$ region.
Statistical Approach: A $\chi^2$ fit is performed. Wilson coefficients are profiled (treated as nuisance parameters) to account for correlations between operators. Limits are extracted at 95% Confidence Level (CL) using $\Delta\chi^2 = 3.84$ (1D) and $5.99$ (2D).

3. Key Contributions

Two-Loop RG Mixing: The paper explicitly demonstrates that two-loop double-logarithmic corrections are necessary to probe purely right-handed top operators ( $Q_{tt}$ ) via dijet production. This bridges the gap between operators that are "invisible" at tree level in dijet channels and those that are observable.
Quantitative Assessment of PDF Enhancement: The authors investigate whether the large parton distribution function (PDF) enhancement for light quarks (relative to bottom quarks) can overcome the loop suppression ( $\alpha_s/4\pi$ ) of the RG-induced contributions. They find that while PDF enhancement is significant ( $10^3$ – $10^5$ ), it is insufficient to fully compensate for the loop suppression, resulting in weak constraints for top-dominated operators.
Resolution of Flat Directions: The analysis shows that RG effects lift the "flat direction" in parameter space (specifically the degeneracy between $C_{QQ}^{(1)}$ and $C_{QQ}^{(3)}$ ) that exists at tree level, allowing for closed 95% CL regions in the 2D parameter space.
Comprehensive Constraints: The study provides the first set of constraints specifically targeting the full suite of third-generation four-quark operators using dijet data, including the indirect limits derived from RG mixing.

4. Results

The derived 95% CL limits (in units of $1/\text{TeV}^2$ at the EW scale $v=250$ GeV) are summarized as follows:

Strong Constraints (Bottom-dominated):
- Operators involving four bottom quarks ( $Q_{QQ}^{(1,3)}$ , $Q_{bb}$ , $Q_{Qb}^{(1,8)}$ ) are directly constrained.
- Limits are competitive with or stronger than existing literature (e.g., $C_{bb} \in [-4.7, 4.7]$ ).
- These bounds are primarily driven by tree-level contributions.
Weak Constraints (Top-dominated/Mixed):
- Operators involving top quarks ( $Q_{tt}$ , $Q_{Qt}^{(1,8)}$ , $Q_{tb}^{(1,8)}$ ) yield significantly weaker bounds (e.g., $C_{tt} \in [-850, 330]$ ).
- Despite the inclusion of two-loop double-logarithmic effects, the constraints remain an order of magnitude weaker than those from Electroweak Precision Observables (EWPO) or direct top-quark production measurements ( $t\bar{t}$ , $t\bar{t}t\bar{t}$ ).
- The PDF enhancement of the induced light-quark operators is not enough to overcome the loop suppression.
Asymmetry: The limits for $C_{tt}$ and $C_{tb}^{(8)}$ are noticeably asymmetric, indicating that linear interference terms play a significant role for these specific operators, unlike the quadratic-dominated constraints for the bottom operators.

5. Significance and Implications

Necessity of RG Running: The work underscores that interpreting LHC high-energy data in the SMEFT framework requires including RG-induced mixing. Ignoring these effects would lead to incorrect conclusions about the "blind spots" of dijet searches.
Complementarity: The results highlight the complementarity of different LHC observables. While dijet data provides the strongest constraints for bottom-quark operators, Electroweak Precision Observables and top-quark production remain the primary probes for top-quark operators.
EFT Validity: The authors note that the derived limits for top-dominated operators are quite weak, raising questions about the validity of the EFT expansion in these specific channels (as the limits approach the cutoff scale). They emphasize that robust constraints on the full operator set require a global fit combining dijet, top-production, and EW precision data.
Future Directions: The paper sets a precedent for including higher-order (two-loop) RG effects in SMEFT phenomenology, suggesting that similar "double-logarithmic" mixing effects should be considered in other BSM searches where tree-level contributions are absent.

In conclusion, while dijet measurements cannot currently provide stringent constraints on top-quark four-quark operators even with two-loop RG improvements, they significantly tighten the bounds on bottom-quark operators and demonstrate the critical role of RG evolution in connecting high-energy collider data to low-energy effective parameters.