Naive $T$-odd Drell-Yan angular coefficients as a probe of the dimension-8 SMEFT

This paper proposes using naive $T$-odd Collins-Soper moments ($A_6$ and $A_7$) in high-energy Drell-Yan processes at the high-luminosity LHC to probe dimension-8 $CP$-odd semi-leptonic four-fermion operators, potentially constraining new physics scales in the few-TeV range.

The Gist

Imagine you are a detective trying to solve a mystery at a massive, high-speed particle collision factory (the Large Hadron Collider, or LHC). You know the "Standard Model" is the rulebook for how particles behave, but you suspect there are hidden rules written in invisible ink that we haven't found yet.

This paper proposes a new, clever way to look for those hidden rules by watching how particles "dance" when they collide.

The Setup: The Particle Dance Floor

When protons smash together, they sometimes create a pair of electrons (or muons) that fly off in opposite directions. Physicists have been studying these pairs for decades. They usually look at how fast they are going or how heavy the pair is.

But this paper suggests looking at the angle at which they fly out. Imagine the particles are dancers spinning on a floor.

• The Standard Dance: In the known rules of physics, these dancers have a very specific, predictable spin pattern.
• The "Naive" T-Odd Twist: The authors are interested in a very specific, rare kind of spin called "naive T-odd." Think of this as a dance move that looks like it's happening in a mirror image of reality. In our normal world, this move is incredibly rare and almost invisible because it requires a very specific, complex sequence of events (like a dancer needing to trip over a specific thread in the carpet to do a backflip).

The Problem: The "Invisible" Clues

In the current rulebook (the Standard Model), these "mirror-image" dance moves are so faint that they are practically impossible to see. They are like a whisper in a hurricane. If you just listen to the crowd, you won't hear them.

However, the authors suspect that New Physics (heavy, unknown particles from the future) might be lurking in the background. If these new particles exist, they might act like a giant spotlight on the dance floor, making those rare "mirror-image" moves suddenly very loud and obvious.

The Solution: The Dimension-8 Operators

The authors use a mathematical toolkit called SMEFT (Standard Model Effective Field Theory). Think of this toolkit as a set of "what-if" lenses.

• Dimension-6 Lenses: These are the standard lenses everyone uses. They look for new physics that changes the speed of the dancers.
• Dimension-8 Lenses: These are the special, high-powered lenses the authors are using. They are looking for new physics that changes the spin (the angle) of the dancers.

Specifically, they are looking for "CP-odd" operators. In plain English, these are rules that treat matter and antimatter differently in a way that creates a "handedness" or a specific twist in the dance. The paper argues that these specific twists (called coefficients A6 and A7) are the perfect fingerprint for a specific type of heavy new physics involving gluons (the glue holding quarks together).

The Experiment: The High-Luminosity LHC

The authors ran simulations for the future High-Luminosity LHC (HL-LHC). Imagine the current collider is a camera taking 1 photo per second. The HL-LHC will be a camera taking 10,000 photos per second.

With this massive amount of data, they asked: If we watch the dance angles of millions of particle pairs, can we spot the "mirror-image" twist?

Their Findings:

1. The Signal is There: If new physics exists at a scale of about 1 to 2 TeV (which is like finding a new particle 10 times heavier than the Higgs boson), it would make the A6 and A7 dance moves much bigger than the Standard Model predicts.
2. The Blind Spot: They found that looking at just one angle (A6) or just the other (A7) isn't enough. It's like trying to solve a puzzle with only half the pieces. The "bad guys" (the new physics) can hide by making the A6 move look normal while messing up A7, or vice versa.
3. The "Flat Direction" Trap: When they tried to fit all the data at once, they found a mathematical trick where different combinations of new physics rules could cancel each other out, making it look like nothing happened. This is called a "flat direction." It's like two people pushing a car in opposite directions with equal force; the car doesn't move, and you can't tell who is pushing.
   • The Fix: You need to measure both angles (A6 and A7) simultaneously to break this tie and see the true signal.

The Conclusion: Why This Matters

This paper is a roadmap for experimentalists. It says:

"Don't just count how many particles you find. Look at how they are spinning. If you measure these specific angles (A6 and A7) with the future super-bright LHC, you might finally catch a glimpse of the heavy, invisible particles that are currently hiding in the shadows of our universe."

In a nutshell:
The authors are suggesting a new, highly sensitive "angle detector" to find heavy new physics. While the current rules say the signal should be tiny, new physics could turn that tiny signal into a shout. By measuring the specific "twist" of particle collisions at the future LHC, we might finally crack the code of what lies beyond our current understanding of the universe.

Technical Summary

1. Problem Statement

The study of angular distributions in the Drell-Yan (DY) process ($pp \to \ell^+\ell^-$) is a standard tool for testing the Standard Model (SM) and searching for new physics. While lower-order angular coefficients ($A_0$ through $A_4$) are well-studied, the higher-order "naive" T-odd coefficients ($A_5, A_6, A_7$) have received less attention.

• SM Suppression: In the SM, these coefficients are extremely small because they require both non-zero transverse momentum ($p_T$) of the lepton pair and complex phases in the amplitude, which only appear at the two-loop level ($O(\alpha_s^2)$).
• SMEFT Blind Spots: Previous studies of the High-Luminosity LHC (HL-LHC) potential in the DY process focused on inclusive cross-sections and dimension-6 operators. However, dimension-6 corrections to the DY $p_T$ spectrum are often dominated by soft gluon emissions (similar to the SM), masking new physics signals. Furthermore, many dimension-6 operators do not generate the specific angular structures associated with T-odd observables.
• The Gap: There is a need to identify observables that are sensitive to dimension-8 operators, specifically those that are CP-odd, which could reveal new physics in previously unexplored directions of the SMEFT parameter space.

2. Methodology

The authors employ the Standard Model Effective Field Theory (SMEFT) framework to parametrize heavy new physics effects.

• Theoretical Framework:
  • They focus on dimension-8 operators involving four fermions and a gluon field-strength tensor ($G_{\mu\nu}$).
  • They specifically isolate CP-odd operators (denoted with a tilde, e.g., $\tilde{O}$) which contribute to the "naive" T-odd moments $A_6$ and $A_7$.
  • The analysis distinguishes between operators that generate $p_T$ via external leg emission (dominated by SM-like soft gluons) and those generating $p_T$ directly from a hard contact vertex (dimension-8 level).
• Angular Moments:
  • The differential cross-section is expanded in spherical harmonics (Collins-Soper frame).
  • The authors derive that $A_6$ and $A_7$ are odd under the combined Parity and "naive" Time-reversal ($AP$) transformation.
  • In the SM, these are generated at $O(\alpha_s^2)$. In SMEFT, CP-odd dimension-8 operators generate these terms at tree-level, providing a significant enhancement over the SM background at high $p_T$.
• Simulation and Fitting:
  • Collider Setup: Simulations assume $\sqrt{s} = 14$ TeV and an integrated luminosity of $3 \text{ ab}^{-1}$ (HL-LHC).
  • Kinematic Cuts: Selection criteria mimic realistic experimental acceptance (e.g., lepton $p_T > 20/25$ GeV, dilepton $p_T > 100$ GeV).
  • Binning: Data is binned in invariant mass ($m_{\ell\ell}$) and transverse momentum ($p_T$), focusing on regions well above the Z-peak ($m_{\ell\ell} > 300$ GeV) to maximize sensitivity to contact interactions.
  • Statistical Analysis: The authors perform both non-marginalized (1D) and marginalized (multi-dimensional) fits to simulated data to extract bounds on Wilson coefficients.

3. Key Contributions

1. Identification of a New Probe: The paper establishes that the naive T-odd moments $A_6$ and $A_7$ are uniquely sensitive to CP-odd dimension-8 semi-leptonic operators containing a gluon field strength.
2. Tree-Level Enhancement: It demonstrates that while these moments are loop-suppressed in the SM, they receive tree-level contributions from specific dimension-8 SMEFT operators, making them ideal probes for high-$p_T$ physics.
3. Degeneracy Analysis: The authors provide a detailed analytical and numerical study of the correlations (flat directions) between the Wilson coefficients of the relevant dimension-8 operators. They show that while individual coefficients can be constrained, simultaneous fits reveal strong degeneracies.
4. Complementarity: The study highlights that measuring both $A_6$ and $A_7$ is crucial to breaking these degeneracies, as their correlation structures differ.

4. Results

• Sensitivity to High Scales:
  • In non-marginalized (1D) fits (activating one Wilson coefficient at a time), the HL-LHC can probe effective UV scales ($\Lambda$) up to 9 TeV for certain operators.
  • In fully marginalized fits (activating all 7 relevant Wilson coefficients simultaneously), the constraints weaken significantly due to parameter degeneracies. However, the study shows that effective scales in the 1–2 TeV range can still be probed.
• Operator Hierarchy:
  • The left-handed current operators (e.g., $C_{\ell^2 q^2 g}^{(1)}$) generally provide the largest contributions.
  • The CP-odd operators are invisible in inclusive cross-section measurements (due to cancellation in the absence of absorptive parts) but become dominant in the $A_6$ and $A_7$ angular moments.
• Flat Directions:
  • The analysis reveals "flat directions" in the parameter space where different combinations of Wilson coefficients cancel each other's effects on the observables.
  • Analytical relations (Eqs. 34–37) are derived showing how dependent coefficients relate to independent ones to nullify the SMEFT contribution.
  • The correlation matrices (Figs. 13–14) confirm that the $A_6$ and $A_7$ fits are complementary; combining them reduces the size of the allowed parameter space compared to using either alone, though strong correlations remain.

5. Significance

• Probing CP-Violation: This work opens a new window for testing CP-violation in the quark-lepton sector at the energy frontier, specifically targeting operators that are invisible to traditional inclusive measurements.
• Dimension-8 Physics: It reinforces the necessity of studying dimension-8 operators in the SMEFT, as dimension-6 operators often fail to provide distinct signatures in high-$p_T$ DY processes due to soft-gluon dominance.
• Experimental Guidance: The paper provides concrete targets for the ATLAS and CMS experiments at the HL-LHC. It argues that extending angular coefficient measurements beyond the Z-peak region is essential to fully map the SMEFT parameter space.
• Future Prospects: The results suggest that while degeneracies exist, a broad-based experimental program measuring multiple angular moments ($A_6$ and $A_7$) is required to disentangle the underlying UV physics, potentially distinguishing between different new physics models that yield identical dimension-6 results.

In summary, Petriello and Şimşek propose that the "naive" T-odd angular coefficients in Drell-Yan scattering are a powerful, previously underutilized tool for discovering heavy CP-odd new physics at the HL-LHC, capable of probing energy scales up to several TeV.