W-boson helicity fractions in top decay as probes of dimension-6 and dimension-8 SMEFT operators

This paper presents a combined analysis of dimension-6 and dimension-8 SMEFT operators using W-boson helicity fractions in top-quark decays, demonstrating that including dimension-8 contributions significantly alters the constraints on dimension-6 coefficients due to non-trivial correlations and the necessity of a consistent EFT expansion treatment.

The Gist

Imagine the universe is built according to a very specific, incredibly detailed instruction manual called the Standard Model. For decades, this manual has explained almost everything we see, from the smallest atoms to the biggest stars. However, scientists know the manual is incomplete. It doesn't explain things like dark matter or why the universe has more matter than antimatter.

To find the missing pages, scientists look for tiny "glitches" in the instructions. They do this by smashing particles together at high speeds (like at the Large Hadron Collider) and watching how they behave.

The Detective Work: The Top Quark

In this paper, the authors are acting like detectives focusing on the top quark. Think of the top quark as the "heavyweight champion" of the particle world. It's the heaviest known particle and it decays (breaks apart) almost instantly into a W boson (a force carrier) and a bottom quark.

Because the top quark is so heavy and decays so quickly, it's a perfect laboratory to test if the "Standard Model" manual has any hidden errors. The authors are specifically looking at the spin (or "helicity") of the W boson produced in this decay. Imagine the W boson as a spinning top; it can spin in three different ways:

1. Longitudinal: Spinning along its path.
2. Left-handed: Spinning counter-clockwise.
3. Right-handed: Spinning clockwise.

In the current Standard Model, the "right-handed" spin is almost non-existent. If the scientists see more right-handed spins than expected, it's a huge clue that new physics is at play.

The "EFT" Toolkit: Dimension-6 vs. Dimension-8

To interpret these clues, scientists use a mathematical framework called SMEFT (Standard Model Effective Field Theory). You can think of this as a set of "correction lenses" they put over the Standard Model to see if there are subtle distortions.

• Dimension-6 Operators: These are the "standard" correction lenses. They have been studied for a long time. If you look at a photo through these lenses, you might see a slight blur or color shift that hints at something new.
• Dimension-8 Operators: These are "super-fine" correction lenses. They are much more subtle and were largely ignored in the past because they are harder to detect.

The Paper's Big Idea:
The authors argue that relying only on the standard lenses (Dimension-6) is like trying to solve a mystery with only half the evidence. They say that as our measurements get more precise, we must also look through the "super-fine" lenses (Dimension-8).

Why? Because the effect of the super-fine lenses (Dimension-8) is actually about the same size as the squared effect of the standard lenses. If you ignore the super-fine lenses but keep the squared standard ones, you might misinterpret the data. It's like trying to balance a scale: if you weigh the heavy items but forget to account for the tiny dust particles that add up to the same weight, your scale will be wrong.

What They Did

The team performed a massive statistical analysis (a "chi-squared fit") using real data from the ATLAS and CMS experiments at the Large Hadron Collider. They asked:

• "If we include both the standard lenses (Dimension-6) and the super-fine lenses (Dimension-8), how does our view of the top quark change?"

The Findings: A Shifting Landscape

Their results were surprising and important:

1. The Map Changes: When they added the Dimension-8 operators, the "allowed territory" for the standard operators shifted. Some areas that looked safe before now looked suspicious, and vice versa.
2. The "Flat" Spots: For some types of particles, the data was so ambiguous that the scientists couldn't pin down a specific value. It was like trying to find a specific spot on a perfectly flat, featureless plain; no matter where you look, the view is the same. They found that the new Dimension-8 operators created these "flat spots" or "degeneracies," making it harder to tell which specific correction was causing the effect.
3. The Dipole Operator: They found that one specific type of correction (called the dipole operator, $C_{uW}$) was tightly constrained. This is because it strongly affects the "right-handed" spin, which is the most sensitive part of the experiment.
4. The Others: The other corrections, especially the new Dimension-8 ones, were very loosely constrained. The data allowed for a huge range of values, meaning we need much better data to narrow them down.

The Bottom Line

The paper concludes that to truly understand the top quark and find new physics, we cannot just look at the "big" corrections (Dimension-6) and ignore the "small" ones (Dimension-8). They are intertwined.

If we want to solve the mystery of what lies beyond the Standard Model, we need to treat the "big" and "small" corrections as a team. Ignoring the small ones while trying to measure the big ones leads to a distorted picture. The authors suggest that future, more precise experiments (like the High-Luminosity LHC) will be needed to clear up the "flat spots" and finally pin down exactly what these new physics rules are.

Technical Summary

1. Problem Statement

While the Standard Model Effective Field Theory (SMEFT) has been extensively used to search for Beyond the Standard Model (BSM) physics via dimension-6 operators, current experimental precision at the LHC is reaching a level where neglecting higher-order terms may lead to biased interpretations.

• The Core Issue: In a consistent SMEFT expansion, contributions from dimension-8 operators (interfering with the Standard Model amplitude) enter at order $\mathcal{O}(\Lambda^{-4})$. This is the same order of magnitude as the squared dimension-6 terms ($\mathcal{O}(\Lambda^{-4})$) typically retained in analyses.
• The Gap: Previous studies often truncate the expansion at dimension-6 or treat dimension-8 effects as negligible. However, dimension-8 contributions can mimic, distort, or screen dimension-6 effects, creating degeneracies that alter the extracted constraints on Wilson coefficients.
• Specific Context: Top-quark decays ($t \to Wb$) are a prime laboratory for these studies due to the top's large coupling to the Higgs sector and the precision of W-boson helicity fraction measurements ($F_0, F_L, F_R$).

2. Methodology

The authors perform a combined analysis of dimension-6 and a representative subset of dimension-8 SMEFT operators using W-boson helicity fractions as observables.

Theoretical Framework

• Lagrangian Expansion: The analysis retains all terms up to $\mathcal{O}(\Lambda^{-4})$. This includes:
  1. SM–Dimension-6 interference ($\mathcal{O}(\Lambda^{-2})$).
  2. Squared Dimension-6 amplitudes ($\mathcal{O}(\Lambda^{-4})$).
  3. SM–Dimension-8 interference ($\mathcal{O}(\Lambda^{-4})$).
  • Terms of $\mathcal{O}(\Lambda^{-6})$ and higher are neglected.
• Operator Basis:
  • Dimension-6: Uses the Warsaw basis. Four CP-even operators directly affect the $t \to Wb$ vertex: $Q^{(3)}_{Hq}$, $Q^{(3)}_{Hud}$, $Q_{uW}$, and $Q_{dW}$.
  • Dimension-8: A representative subset of five operators is selected that contribute at tree level to the on-shell $Wtb$ vertex. This includes CP-even operators ($Q^{(2)}_{q2H4D}$, $Q_{udH4D}$, $Q^{(1)}_{quWH3}$, $Q^{(1)}_{qdWH3}$) and one CP-odd operator (which vanishes for CP-even observables at this order).
• Form Factors: The operators modify the effective $Wtb$ vertex form factors ($f_{V,L/R}$ and $f_{T,L/R}$). The authors derive the shifts in these form factors induced by both dimension-6 and dimension-8 Wilson coefficients.

Statistical Analysis

• Observables: The analysis uses the longitudinal ($F_0$) and left-handed ($F_L$) W-boson helicity fractions. The right-handed fraction ($F_R$) is derived via the normalization condition $F_0 + F_L + F_R = 1$.
• Data: Combined ATLAS and CMS measurements at $\sqrt{s} = 8$ TeV (Run 1) are used as the baseline for the $\chi^2$ fit.
• Fit Strategy:
  • One-Parameter Fits: Scanning individual Wilson coefficients while fixing others to zero.
  • Two-Parameter Fits: Scanning pairs of coefficients (one dimension-6 and one dimension-8) simultaneously to visualize correlations and degeneracies.
• Scale: The analysis is performed at a reference new physics scale $\Lambda = 1$ TeV.

3. Key Contributions

1. Consistent $\mathcal{O}(\Lambda^{-4})$ Treatment: The paper provides one of the first decay-level phenomenological analyses that consistently includes dimension-8 interference terms alongside squared dimension-6 terms, ensuring a theoretically consistent EFT expansion.
2. Identification of Degeneracies: The study explicitly demonstrates how dimension-8 operators introduce non-trivial correlations and "flat directions" in the parameter space that are invisible in dimension-6-only analyses.
3. Operator Classification: It categorizes the impact of specific operators:
   • Vector Operators: Modify the vector structure; some induce flat directions in helicity ratios.
   • Dipole (Tensor) Operators: Introduce tensor structures absent in the SM, lifting helicity suppression and strongly affecting $F_R$.
4. Validation: The dimension-6 constraints derived in the paper are validated against existing combined ATLAS/CMS results, showing broad compatibility despite different analysis strategies (helicity fractions vs. differential distributions).

4. Key Results

• Impact on Dimension-6 Constraints: Including dimension-8 operators significantly alters the allowed parameter space for dimension-6 coefficients.
  • Constraints generally become weaker and more asymmetric.
  • The geometry of confidence regions changes from simple ellipses to elongated bands or non-elliptic shapes (cusps), indicating strong non-linear correlations.
• Specific Coefficient Constraints (at $\Lambda = 1$ TeV):
  • Dipole Operator ($C_{uW}$): Exhibits the tightest constraints ($95\%$ CL: $[-0.125, 0.598]$) due to its direct impact on the suppressed right-handed helicity fraction.
  • Vector Operators ($C^{(3)}_{Hud}, C_{dW}$): Show intermediate sensitivity ($\mathcal{O}(1)$ bounds).
  • Dimension-8 Operators: Generally weakly constrained, with allowed ranges extending up to $\mathcal{O}(10^3)$.
  • Flat Directions: Coefficients $C^{(3)}_{Hq}$ and $C^{(2)}_{q2H4D}$ exhibit approximate flat directions in one-parameter scans, preventing the extraction of finite confidence intervals with current data.
• Correlations: Two-parameter fits reveal that dimension-6 and dimension-8 operators can compensate for each other. For example, the $(C_{uW}, C^{(1)}_{quWH3})$ plane shows a narrow open band, indicating a strong anti-correlation where the two operators' effects on helicity fractions cancel out over a large region of parameter space.

5. Significance and Conclusion

• Necessity of Dimension-8: The results prove that neglecting dimension-8 contributions while retaining squared dimension-6 terms leads to an incomplete and potentially biased interpretation of SMEFT constraints. The interference between SM and dimension-8 operators is phenomenologically relevant at current precision levels.
• Limitations of Current Data: W-boson helicity fractions alone are insufficient to fully constrain the SMEFT operator space due to existing degeneracies.
• Future Outlook: To lift these degeneracies and achieve a global determination of the $Wtb$ interaction, future analyses must:
  • Incorporate complementary observables (single-top production, top-pair differential distributions).
  • Utilize higher-precision data from the High-Luminosity LHC (HL-LHC).
  • Include Next-to-Leading Order (NLO) QCD corrections to SMEFT, which are expected to be relevant as experimental uncertainties shrink.

In summary, this work establishes a rigorous framework for interpreting top-quark decay data in the SMEFT context, highlighting that a consistent treatment of the EFT expansion (up to dimension-8) is critical for accurate BSM searches as experimental precision improves.