Probing EFT breakdown in the tails of $W^+ W^-$ observables

This paper demonstrates that clipping Effective Field Theory (EFT) simulations at the new physics scale $\Lambda$ via an invariant mass cut ($M_{WW} < \Lambda$) is insufficient to guarantee EFT validity across different observables and operator orders, while also highlighting the risks of introducing model-dependent form factors and proposing alternative transverse mass cuts for more robust sensitivity studies.

The Gist

Imagine you are a detective trying to solve a mystery about a hidden world of "new physics" that exists beyond our current understanding. You have a powerful magnifying glass called Effective Field Theory (EFT). This tool allows you to look at particle collisions (like those at the Large Hadron Collider) and spot tiny clues that suggest new, heavier particles might exist, even if you can't see those heavy particles directly.

However, there's a catch: your magnifying glass only works if the mystery isn't too complex. If the energy of the collision gets too high (too close to the scale of the new physics), the magnifying glass cracks, and your clues become meaningless. This is the "breakdown" of the theory.

The paper by Gillies, Banfi, and Martin is about making sure you don't accidentally use your cracked magnifying glass. They are studying a specific type of particle crash: two "W bosons" (heavy force-carrying particles) smashing into each other.

Here is the breakdown of their investigation using simple analogies:

1. The Problem: The Invisible Scale

To know if your magnifying glass is working, you need to know the total energy of the crash. In this specific experiment, the total energy is determined by the mass of the two W bosons combined ($M_{WW}$).

The Catch: One of the W bosons decays into a particle that is invisible (a neutrino), like a ghost slipping out of the room. Because you can't see the ghost, you can't measure the total energy of the crash directly. You are flying blind.

2. The Old Trick: "Clipping" the Simulation

Since you can't measure the total energy, physicists have been using a shortcut. They run computer simulations of the crash and tell the computer: "If the total energy looks like it's getting too high, just pretend it didn't happen. Cut it off."

In the paper, they call this "Clipping on Simulation" (CoS). It's like telling a video game engine: "If a car goes faster than 100 mph, delete it from the screen."

The Flaw: The authors found that this trick is too loose. Even if you tell the computer to delete high-energy crashes, the "ghost" particles (the neutrinos) mess up the math. You might delete a high-energy crash, but the remains of that crash (the visible particles) still look like they belong in the high-energy zone. So, you end up analyzing data that is actually broken, thinking it's safe.

3. The Better Trick: Finding a Better Proxy

Since you can't see the total energy ($M_{WW}$), you need a "proxy"—a visible clue that acts like a stand-in for the total energy.

• The Old Proxy ($M_{e\mu}$): Previously, physicists used the combined mass of the two visible electrons/muons left behind. The authors show this is a bad stand-in. It's like trying to guess the weight of a truck by weighing only the driver's shoes. The driver's shoes (the visible particles) don't change much even if the truck (the total energy) gets huge.
• The New Proxy ($M_{T3}$): The authors tested three different "transverse mass" variables (ways of calculating momentum sideways). They found that one, called $M_{T3}$, is a much better stand-in. It tracks the total energy of the crash much more closely, like weighing the driver and the cargo in the back.

4. The Solution: Cut the Data, Not the Simulation

The authors propose a new rule for the experiment:
Instead of telling the computer to "clip" the simulation (which is messy and mathematically questionable), we should put a hard cut on the actual data we collect.

We say: "We will only look at crashes where our new proxy ($M_{T3}$) is below a certain safe limit."

This is safer because:

1. It applies to the real data, not just the simulation.
2. It ensures that the data we are analyzing is actually within the range where our "magnifying glass" (EFT) works.
3. It avoids the mathematical weirdness of "clipping" the simulation, which the authors argue is like trying to fix a broken theory by pasting a band-aid on it (a "form factor") rather than fixing the theory itself.

5. The Trade-off: Sensitivity vs. Safety

The paper also notes a funny trade-off.

• The "Safe" Proxy ($M_{T3}$): It keeps the data safe and valid, but because it's so accurate, it filters out a lot of data. It's like being a very strict bouncer who only lets in people who are definitely under the age limit.
• The "Loose" Proxy ($M_{e\mu}$): It lets in more data, but some of it might be "fake" (invalid).

Surprisingly, the authors found that even though $M_{T3}$ is "safer," using the old, looser proxy ($M_{e\mu}$) actually gave them better sensitivity to find new physics in this specific setup. Why? Because the "safe" proxy was so strict it threw away the very high-energy events where the new physics clues are strongest.

Summary

The paper is a warning and a guide for particle physicists:

1. Don't trust the "clipping" method (cutting the simulation) alone; it leaves you with broken data.
2. Don't trust the old proxy (lepton mass) to tell you if the data is safe.
3. Use the new proxy ($M_{T3}$) to define a safe zone for your data.
4. Be careful: Being too safe might hide the very clues you are looking for.

The ultimate goal is to ensure that when physicists claim they have found evidence of "new physics," they haven't accidentally been looking at a broken version of their own theory.

Technical Summary

Technical Summary: Probing EFT Breakdown in the Tails of $W^+W^-$ Observables

Problem Statement
Model-independent Effective Field Theory (EFT) fits are increasingly used to interpret collider data, reducing the space of possible Beyond the Standard Model (BSM) theories to a finite set of operators suppressed by a new physics scale $\Lambda$. The validity of these fits relies on the condition that the interaction energy scale $E$ remains well below $\Lambda$ ($E \ll \Lambda$). In processes involving $W^+W^-$ production at the LHC, the relevant energy scale is the invariant mass of the diboson pair, $M_{WW}$. However, $M_{WW}$ is not directly observable in fully leptonic decay channels due to missing neutrino energy.

Current experimental analyses often rely on proxies, such as the dilepton invariant mass $M_{e\mu}$, or apply "clipping" cuts on the EFT simulation (enforcing $M_{WW} < \Lambda$ at the generator level) while leaving the Standard Model (SM) simulation uncut. The authors identify a critical flaw in these approaches: the correlation between $M_{e\mu}$ and $M_{WW}$ changes significantly as one moves from the SM to dimension-6 and dimension-8 EFT contributions. Consequently, a cut on $M_{WW}$ applied only to the simulation does not guarantee that the corresponding $M_{e\mu}$ bins are free from dominant dimension-8 contributions, potentially invalidating the EFT expansion even within the selected phase space.

Methodology
The authors investigate the breakdown of EFT validity in $W^+W^-$ production, specifically focusing on the gluon-fusion ($gg$) channel where bosonic dimension-6 and dimension-8 operators dominate. They compare three distinct methods for ensuring EFT validity:

1. Clipping on Simulation (CoS): Enforcing $M_{WW} < \Lambda$ only during the generation of the dimension-6 EFT contribution, effectively modifying the Wilson coefficient with a step function $\Theta(1 - p^2/\Lambda^2)$.
2. Comparison Bin-by-Bin (CBB): Directly comparing the squared contributions of dimension-6 and dimension-8 operators ($\sigma^{(6)}$ vs. $\sigma^{(8)}$) to determine which bins satisfy the convergence criterion $\sigma^{(6)} > 2\sigma^{(8)}$.
3. Cut on Data (CoD): Applying a kinematic cut directly to the data (or fiducial selection) using a proxy observable that correlates strongly with $M_{WW}$.

The study utilizes the operators $O_{GH}^{(6)}$ and $O_{3}^{(8)}$, which generate amplitudes growing with energy. Calculations are performed at Next-to-Leading Logarithmic (NLL) accuracy using MCFM-RE, with fiducial cuts based on ATLAS analyses. The authors compute the conditional expectation value $E[M_{WW} | X]$ for various observables $X$ (including $M_{e\mu}$, $M_{T1}$, $M_{T2}$, and $M_{T3}$) across SM, dimension-6 squared, and dimension-8 squared contributions to assess their correlation with the true energy scale.

Key Contributions and Results

• Failure of the CoS Method: The authors demonstrate that imposing $M_{WW} < \Lambda$ on the EFT simulation alone is insufficient. Even with a strict cut (e.g., $M_{WW} < 1.65$ TeV for $\Lambda = 1$ TeV), the resulting $M_{e\mu}$ distribution still contains bins where the dimension-8 squared contribution dominates the dimension-6 contribution. This occurs because the correlation between $M_{e\mu}$ and $M_{WW}$ shifts order-by-order in the EFT expansion; high-$M_{WW}$ events contribute significantly to lower-$M_{e\mu}$ bins in the dimension-8 regime.
• Poor Correlation of $M_{e\mu}$: The analysis of $E[M_{WW} | M_{e\mu}]$ reveals that while $M_{e\mu}$ correlates with $M_{WW}$ in the SM, this relationship degrades significantly for dimension-8 operators. For dimension-8, the relation shifts such that $M_{e\mu} \approx M_{WW}/4$ at lower values, meaning $M_{e\mu}$ is a poor proxy for determining EFT validity in high-energy tails.
• Superiority of Transverse Mass Observables: The authors evaluate three transverse mass observables ($M_{T1}, M_{T2}, M_{T3}$). They find that $M_{T3}$ correlates most closely with $M_{WW}$ across all EFT orders. The conditional expectation $E[M_{WW} | M_{T3}]$ remains stable, and the ratio of dimension-8 to dimension-6 contributions follows the theoretical $M^4/\Lambda^4$ scaling much more accurately for $M_{T3}$ than for $M_{e\mu}$.
• Sensitivity Studies: Using High-Luminosity LHC (HL-LHC) projections, the authors compare constraints on Wilson coefficients derived from different methods.
  • The CBB method (bin-by-bin comparison) provides the most robust validity check but is computationally cumbersome and results in looser constraints because it discards many bins.
  • The CoD method (cutting on $M_{T3} < \Lambda$) yields constraints comparable to the CoS method but with the advantage of being applicable directly to data.
  • Interestingly, while $M_{T3}$ is a better proxy for $M_{WW}$, the $M_{e\mu}$ distribution provides tighter constraints when using the CBB method. This is because the EFT-valid region for $M_{e\mu}$ extends to higher energies where sensitivity to new physics is greater, whereas the stricter correlation of $M_{T3}$ with $M_{WW}$ restricts the valid bins to lower energies where sensitivity is reduced.

Significance and Claims
The paper claims that the standard practice of clipping EFT simulations at the generator level ($M_{WW} < \Lambda$) is not a rigorous method for ensuring EFT validity in differential distributions, as it fails to account for the shifting correlations between observables and the true energy scale at higher operator dimensions.

The authors propose that implementing a cut on the observable $M_{T3}$ directly on the data (CoD) is a viable alternative that ensures EFT validity without requiring complex bin-by-bin re-evaluations for every new physics scale. Furthermore, they highlight a subtle theoretical implication: applying a step-function cut only to the EFT simulation (CoS) effectively modifies the SMEFT expansion by introducing a form factor. This modification breaks the strict truncation of the EFT at dimension-6, potentially impacting the model independence of the resulting fits. The study concludes that while $M_{e\mu}$ offers better sensitivity, $M_{T3}$ offers a more reliable proxy for defining the region of EFT validity, suggesting a trade-off between sensitivity and theoretical rigor in future experimental analyses.