Measurement and interpretation of inclusive $W\gamma$ production in proton-proton collisions at $\sqrt{s}=13$ TeV using the ATLAS detector

Using 140 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV, the ATLAS experiment presents precise differential cross-section measurements for inclusive $W\gamma$ production to test Standard Model predictions, constrain parton distribution functions, and set improved limits on anomalous weak-boson self-interactions within an effective field theory framework.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. Inside its 27-kilometer ring, tiny protons are accelerated to near the speed of light and smashed together, creating a chaotic explosion of new particles.

This paper is a detailed report card from the ATLAS experiment, one of the giant detectors watching these collisions. The scientists are studying a very specific, somewhat rare event: when a W boson (a heavy particle that carries the weak nuclear force) is created right next to a photon (a particle of light).

Here is the breakdown of what they did and why it matters, using simple analogies.

1. The Main Event: A "W" and a "Photon" Dating

Think of the W boson as a heavy, grumpy bouncer and the photon as a flash of light. Usually, they don't hang out together. But sometimes, when protons collide, they spawn a pair: a W boson and a photon.

The scientists collected data from 140 "inverse femtobarns" of collisions. To put that in perspective, if a femtobarn were a grain of sand, they analyzed a pile of sand the size of a small mountain. They looked at 264,000 specific events where this pair was created.

2. The "Zero" Trick: The Perfect Cancellation

One of the most fascinating things they looked for is called the Radiation Amplitude Zero (RAZ).

• The Analogy: Imagine two people shouting at a crowd. One shouts a note, and the other shouts the exact opposite note at the exact same time. To the crowd, it sounds like silence. The noise cancels itself out perfectly.
• In Physics: In the Standard Model (our current best theory of physics), the math predicts that at a specific angle and energy, the probability of creating this W-photon pair drops to zero. It's a "silence" in the data.
• The Test: The scientists checked if this "silence" actually happens. They found that if you stop the particles from shooting out extra "debris" (jets), the silence appears exactly where the theory predicts. This is a stress test for the laws of physics. If the silence wasn't there, it would mean our understanding of how forces work is broken.

3. The "Spin" and the "Pose"

The W boson isn't just a static ball; it spins. The scientists wanted to know how it was spinning when it was born.

• The Analogy: Imagine a figure skater. They can spin upright, lean to the side, or spin on their head. The way the W boson decays (breaks apart) tells us its "pose."
• The Innovation: The team didn't just look at the skater; they used a Neural Network (a type of AI) to act like a super-sleuth. They trained the AI to look at the debris from the collision and guess: "Was this a constructive interference (two waves adding up) or destructive interference (waves canceling out)?"
• Why it matters: This helps them hunt for "CP-violation." In simple terms, this is looking for a difference between how matter behaves versus how antimatter behaves. If the universe treats them differently, it helps explain why we exist at all (since the Big Bang should have created equal amounts of both, which would have annihilated each other).

4. The "Fake" Problem

In a collision, it's easy to get fooled. A jet of particles might look like a photon, or a heavy particle might look like a light one.

• The Analogy: Imagine trying to count real gold coins in a pile of plastic coins. You can't just pick them up; you have to use a magnet or a scale to tell them apart.
• The Solution: The team used "data-driven" methods. Instead of trusting the computer simulation 100%, they looked at control groups in the real data where they knew the coins were plastic. They used these to build a "fake rate" to subtract the plastic coins from their final count. This ensured their results were clean.

5. Hunting for "New Physics" (The EFT)

The Standard Model is great, but we know it's incomplete (it doesn't explain gravity or dark matter). Scientists use a framework called Effective Field Theory (EFT) to look for tiny cracks in the model.

• The Analogy: Imagine the Standard Model is a perfect, smooth table. EFT is like looking for microscopic scratches on the table that might reveal a hidden drawer underneath.
• The Result: They looked for "anomalous couplings"—weird interactions that shouldn't happen if the Standard Model is the whole story. They found no new cracks. The table is still smooth.
• The Win: However, by not finding cracks, they set tighter limits than ever before. They proved that if there is new physics hiding here, it's even more subtle than we thought. Specifically, they improved the sensitivity to one type of hidden interaction by a factor of 2.5.

6. The "Boost" Asymmetry

Finally, they looked at the "boost" of the particles.

• The Analogy: Imagine throwing a ball from a moving train. If you throw it forward, it goes faster relative to the ground. If you throw it backward, it goes slower.
• The Physics: The W boson is made from quarks inside the proton. The proton is mostly made of "up" quarks and "down" quarks. By seeing which way the W boson flies (forward or backward), the scientists can figure out the exact recipe of the proton's insides. Their measurements matched the current "recipe books" (Parton Distribution Functions) perfectly.

The Bottom Line

This paper is a massive success story for the Standard Model.

1. It confirmed that the "silence" (Radiation Amplitude Zero) happens exactly as predicted.
2. It confirmed that the W boson spins the way we expect.
3. It confirmed that the proton's internal recipe is correct.
4. It found no evidence of new, exotic physics yet.

Why is finding "nothing" good?
In science, finding "nothing" is actually finding a very strong boundary. It tells us that if there is a "New Physics" monster hiding in the dark, it is very good at hiding. The ATLAS team has built a tighter cage around where that monster could be, forcing theorists to come up with smarter ideas for where to look next.

They used 140 fb⁻¹ of data (the most ever for this specific process) and clever AI tools to prove that, for now, the universe is behaving exactly as the 1970s theory predicted.

Technical Summary

1. Problem and Motivation

The production of a $W$ boson in association with a photon ($W\gamma$) in proton-proton collisions is a critical process for testing the Standard Model (SM) and searching for Beyond the Standard Model (BSM) physics.

• Radiation Amplitude Zero (RAZ): The $s$-channel scattering amplitude contains the $WW\gamma$ triple gauge coupling. Interference between $s$-, $t$-, and $u$-channel amplitudes leads to a "radiation amplitude zero" (a complete cancellation) at specific kinematic points. This effect is sensitive to the gauge structure of the electroweak interaction and is removed by higher-order QCD corrections or deviations in the $WW\gamma$ coupling.
• Effective Field Theory (EFT): Inclusive $W\gamma$ production is sensitive to dimension-six operators in the Standard Model Effective Field Theory (SMEFT) that induce anomalous weak-boson self-interactions.
• CP-Violation and Interference: A "no-interference theorem" suggests that for diboson production, interference between SM and BSM amplitudes is suppressed due to different $W$ boson helicities. However, this interference can be restored by measuring angular correlations or decay angles. Specifically, CP-odd operators require CP-odd observables to be detected, as their interference terms integrate to zero in CP-even observables.
• Parton Distribution Functions (PDFs): The process probes the PDFs of the proton, particularly the distinction between valence and sea quark contributions via boost asymmetries.

Previous measurements at the LHC (ATLAS and CMS) have been limited in their sensitivity to the $W$ boson spin density matrix and CP-violating operators. This paper aims to provide a comprehensive set of differential cross-section measurements to address these gaps.

2. Methodology

Data and Detector:

• Dataset: Proton-proton collision data collected by the ATLAS experiment at $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of 140 fb$^{-1}$ (Run 2).
• Decay Channel: The analysis focuses on the $W\gamma \to \ell\nu\gamma$ channel, where $\ell = e$ or $\mu$.
• Selection: Events are selected requiring exactly one lepton, at least one photon, and significant missing transverse momentum ($p_T^{miss} > 40$ GeV). Tight identification and isolation criteria are applied to leptons and photons. A jet veto is applied for specific observables (RAZ and CP-sensitive) to suppress higher-order QCD radiation.

Monte Carlo (MC) Simulation:

• Signal Modeling: Two generators are used: Sherpa 2.2.11 (NLO for 0/1 jet, LO for up to 4 jets) and MadGraph5_aMC@NLO (NLO for 0/1 jet, matched to Pythia8).
• Backgrounds:
  • Prompt backgrounds ($Z\gamma$, $t\bar{t}\gamma$, $W\gamma$ with $\tau$ decays) are modeled using MC.
  • Non-prompt/Fake backgrounds (jets faking photons/leptons, electrons faking photons) are estimated using data-driven techniques (template fits, ABCD methods, matrix method, and fake factors).
  • Pile-up backgrounds are estimated using the longitudinal separation ($\Delta z$) between the photon and lepton vertices.

Key Observables (16 total):

1. Kinematic: $p_T^\gamma$, $p_T^\ell$, $\eta^\gamma$, $m_{\ell\gamma}$, $m_{T}^{\ell\nu\gamma}$.
2. Radiation Amplitude Zero: $\Delta\eta_{\ell\gamma}$ (measured in a jet-veto region).
3. Polarization: Double-differential cross sections in terms of $W$ decay angles $(\theta_f, \phi_f)$ defined in a specific reference frame (Figure 2).
4. CP-Sensitive:
   • $\Delta\phi_{\ell\gamma}$ (ordered azimuthal angle).
   • $O_{NN}$: An optimized observable constructed using a Multi-Layer Perceptron (MLP) neural network. The NN is trained to distinguish between constructive interference, destructive interference, and SM contributions for CP-odd operators ($O_{\tilde{W}}$ and $O_{H\tilde{W}B}$).
5. PDF-Sensitive: Boost asymmetry $A_{W\gamma}^{boost}$, defined by the forward/central photon production relative to the lepton.

Analysis Steps:

• Unfolding: Detector-level distributions are unfolded to the particle level using D'Agostini's iterative method to correct for inefficiency and resolution.
• Systematics: Comprehensive treatment of experimental uncertainties (lepton/photon/jet energy scales, luminosity, pile-up) and theoretical uncertainties (scale variations, PDFs, generator choice).
• EFT Interpretation: Constraints on Wilson coefficients are derived using a profile-likelihood fit, comparing data to SMEFT predictions (including interference and pure dimension-six terms).

3. Key Contributions

1. First Double-Differential Measurement of $W$ Polarization: The paper presents the first measurement of the double-differential cross section as a function of the $W$ decay angles $(\theta_f, \phi_f)$ in $W\gamma$ production, providing direct sensitivity to the $W$ boson spin-density matrix.
2. Neural Network for CP-Violation: A novel application of a neural network to construct the $O_{NN}$ observable. This method significantly enhances sensitivity to CP-odd operators by optimally separating constructive and destructive interference contributions, overcoming the limitations of simple angular observables.
3. Comprehensive RAZ Study: Detailed measurement of the radiation amplitude zero effect in the $\Delta\eta_{\ell\gamma}$ distribution under jet-veto conditions, testing the interplay between LO predictions and higher-order QCD corrections.
4. Improved PDF Constraints: Measurement of the boost asymmetry for $W^+\gamma$ and $W^-\gamma$ separately, offering new constraints on the valence vs. sea quark contributions in the proton.
5. State-of-the-Art Theory Comparison: The data is compared against four theoretical predictions: Sherpa, MG5+Py8, Geneva (NNLO), and Geneva+Py8+EWvirt. The inclusion of virtual EW corrections is shown to be crucial for agreement at high momentum scales.

4. Results

• Kinematic Distributions: The measured differential cross sections ($p_T^\gamma$, $\eta^\gamma$, $m_{\ell\gamma}$) generally agree with theoretical predictions. However, Sherpa shows large theoretical uncertainties, while MG5+Py8 and Geneva tend to overestimate the cross-section at high $p_T^\gamma$. The Geneva+Py8+EWvirt prediction (including NLO EW virtual corrections) provides the best agreement across all kinematic regions.
• Radiation Amplitude Zero: A clear dip at $|\Delta\eta_{\ell\gamma}| = 0$ is observed, consistent with the RAZ prediction. Theoretical predictions are slightly below data in the jet-veto region, possibly due to underestimated uncertainties from the jet veto.
• Polarization: The measured angular distributions $(\theta_f, \phi_f)$ are in good agreement with SM predictions, confirming the expected $W$ boson polarization states.
• CP-Sensitive Observables: The distributions of $\Delta\phi_{\ell\gamma}$ and $O_{NN}$ agree well with SM expectations. The $O_{NN}$ observable demonstrates superior separation of interference terms compared to $\Delta\phi_{\ell\gamma}$.
• Boost Asymmetry: The measured boost asymmetries are $A_{W^-\gamma}^{boost} = -0.091 \pm 0.012(\text{stat}) \pm 0.017(\text{syst})$ and $A_{W^+\gamma}^{boost} = 0.033 \pm 0.012(\text{stat}) \pm 0.009(\text{syst})$, consistent with Sherpa predictions using NNPDF3.0nnlo PDFs.
• EFT Constraints:
  • Constraints were placed on Wilson coefficients for operators $O_W$, $O_{HWB}$, $O_{\tilde{W}}$, and $O_{H\tilde{W}B}$.
  • The sensitivity to the CP-odd operator $O_{H\tilde{W}B}$ is improved by a factor of 2.5 compared to previous measurements in other final states (e.g., $H \to 4\ell$). With $k$-factors applied, the improvement is a factor of 5.
  • Constraints on $O_W$ and $O_{HWB}$ are competitive with previous CMS and ATLAS results.

5. Significance

This paper represents a milestone in the precision study of diboson production at the LHC.

• Validation of SM: It confirms the SM prediction for the $WW\gamma$ triple gauge coupling and the radiation amplitude zero effect with high precision.
• New Physics Sensitivity: By utilizing neural networks to construct CP-odd observables, the analysis significantly lowers the threshold for detecting CP-violating new physics, setting the most stringent limits to date on the $O_{H\tilde{W}B}$ operator.
• Theoretical Benchmark: The results provide a rigorous test for state-of-the-art theoretical calculations, highlighting the necessity of including both NNLO QCD and NLO EW corrections for accurate predictions at high energies.
• Global Impact: The data serves as a vital input for global PDF fits and global EFT fits, helping to disentangle the contributions of different parton flavors and constraining the parameter space of BSM physics models.