Measurement of the $W$-boson angular coefficients and transverse momentum in $pp$ collisions at $\sqrt{s}=$ 13 TeV with the ATLAS detector

Using 338 pb$^{-1}$ of low-pile-up proton-proton collision data at $\sqrt{s}=13$ TeV collected by the ATLAS detector in 2017 and 2018, this paper presents the first measurement of the full set of $W$-boson angular coefficients and differential cross-sections as a function of transverse momentum in the full lepton phase space, showing agreement with theoretical predictions incorporating QCD corrections up to order $\alpha_S^2$.

The Gist

Imagine the Large Hadron Collider (LHC) as a massive, high-speed racetrack where tiny particles called protons are smashed together at nearly the speed of light. When they collide, they sometimes create a short-lived particle called a W-boson. Think of the W-boson as a "messenger" that instantly decays (falls apart) into two other particles: a charged lepton (like an electron or a muon) and a ghostly, invisible particle called a neutrino.

This paper is a report from the ATLAS experiment, one of the giant detectors at the LHC, describing how they managed to take a very precise "snapshot" of how these W-bosons behave.

Here is the breakdown of what they did and found, using simple analogies:

1. The Challenge: The Invisible Ghost

The main problem with studying W-bosons is that they produce a neutrino. Neutrinos are like ghosts; they pass right through the detector without leaving a trace. You can't see them, so you can't know exactly where they went or how fast they were moving.

• The Paper's Solution: The scientists used a clever trick of "deduction." They knew the total energy and mass of the system before the crash. By measuring the visible particles (the electron or muon) and the "missing" energy (the recoil of the debris), they could mathematically guess the neutrino's path.
• The Analogy: Imagine you are in a dark room and you hear a glass shatter. You can't see the glass, but you can hear the sound and feel the vibration. By knowing the laws of physics, you can guess exactly where the glass was and how hard it was thrown, even though you never saw it. The ATLAS team did this for billions of collisions.

2. The "Low Pile-Up" Advantage

Usually, when the LHC runs, it smashes protons together so frequently that hundreds of collisions happen at the exact same time. This is called "pile-up." It's like trying to listen to a single conversation in a crowded, noisy stadium. The noise makes it hard to hear the details.

• The Paper's Solution: For this specific study, they used data from special "low-luminosity" runs where the collisions were much more spread out.
• The Analogy: They turned the stadium down to a whisper. Instead of a roaring crowd, they had a quiet library. This allowed them to hear every detail of the "conversation" between the particles with incredible clarity. This low-noise environment was crucial for measuring the invisible neutrino's momentum accurately.

3. Measuring the "Spin" (Angular Coefficients)

When a W-boson is created, it isn't just sitting still; it has a "spin" or orientation, like a spinning top. The way it falls apart (decays) depends on which way it was spinning. The scientists wanted to measure nine different numbers (called angular coefficients) that describe this spin and how the decay products fly out.

• The Analogy: Imagine throwing a spinning frisbee. If it spins one way, the wind might catch it differently than if it spins another way. By watching exactly where the frisbee lands and how it tumbles, you can figure out exactly how it was spinning when you threw it.
• The Achievement: This is the first time anyone has measured the full set of these nine numbers for the W-boson. Previously, they had only measured two of them, or had to guess the rest based on measurements of a different particle (the Z-boson). This paper fills in the whole picture.

4. The Results: A Perfect Match

The team measured these spin numbers across different ranges of speed (transverse momentum). They then compared their real-world data to the predictions made by Quantum Chromodynamics (QCD), which is the complex mathematical theory that describes how the strong force works inside atoms.

• The Finding: The measurements matched the theoretical predictions almost perfectly.
• The Analogy: It's like building a super-accurate weather model that predicts rain, wind, and temperature. When the actual storm hits, the real weather matches the model's prediction exactly. This confirms that our current understanding of how these particles interact is correct.

5. Why This Matters (According to the Paper)

The paper states that these measurements are important for two main reasons:

1. Testing the Theory: It proves that our current mathematical models of the "strong force" (QCD) are working correctly up to very high levels of precision.
2. Helping Other Measurements: Scientists are currently trying to measure the exact mass of the W-boson with extreme precision. To do that, they need to understand exactly how it spins and moves. This paper provides the "rulebook" for that spin, helping to reduce errors in those future mass measurements.

Summary

In short, the ATLAS collaboration used a quiet, low-noise period at the LHC to catch a clear glimpse of a W-boson falling apart. By using math to track the invisible "ghost" neutrino, they mapped out the particle's spin in full detail for the first time. The result? The universe behaved exactly as the complex equations predicted, giving scientists a high-confidence check on their understanding of the fundamental building blocks of matter.

Technical Summary

Technical Summary: Measurement of the W-boson Angular Coefficients and Transverse Momentum in pp Collisions at $\sqrt{s} = 13$ TeV

Problem and Motivation
The angular distributions of lepton pairs produced via the Drell–Yan process serve as sensitive probes of the underlying quantum chromodynamics (QCD) dynamics in vector-boson production. These distributions are governed by spin correlations between initial-state partons and final-state leptons, mediated by a spin-1 intermediate state. While the full set of eight angular coefficients for the $Z$ boson has been previously measured by the ATLAS Collaboration and others, direct measurements for the $W$ boson have been limited. Prior to this work, only two angular coefficients ($A_2$ and $A_3$) had been measured as a function of the $W$-boson transverse momentum ($p_T^W$) by the CDF Collaboration. Other $W$-boson polarization measurements exist but are related to specific coefficients ($A_0$ and $A_4$) and often rely on fiducial phase-space restrictions.

The validation of QCD modeling for $W$-boson production, particularly for precision measurements such as the $W$-boson mass, currently relies on extrapolations from $Z$-boson data, introducing larger uncertainties. A model-independent approach that exploits the spin-1 nature of the gauge boson and the spin-1/2 nature of decay leptons is required to isolate production dynamics without detailed modeling of polarization and decay. This paper addresses the need for a simultaneous measurement of the full set of $W$-boson angular coefficients and the differential cross-section in the full phase space of the decay leptons.

Methodology
The analysis utilizes proton-proton collision data recorded by the ATLAS detector at $\sqrt{s} = 13$ TeV during 2017 and 2018. A key feature of this dataset is the low instantaneous luminosity, resulting in a mean pile-up of approximately two interactions per bunch crossing. This low pile-up environment significantly improves the resolution of the hadronic recoil, which is critical for reconstructing the neutrino kinematics. The dataset corresponds to an integrated luminosity of $338 \text{ pb}^{-1}$.

The methodology relies on the formalism used for previous $Z$-boson angular coefficient extractions. The five-dimensional differential cross-section for $W \to \ell\nu$ decay is decomposed into a sum of nine harmonic polynomials depending on the lepton polar ($\theta$) and azimuthal ($\phi$) angles in the Collins-Soper (CS) frame, multiplied by dimensionless angular coefficients $A_i$ ($i=0$ to $7$).

A primary challenge in $W$-boson analysis is the inability to fully reconstruct the longitudinal momentum of the neutrino ($p_z^\nu$). The analysis infers $p_z^\nu$ by imposing the $W$-boson mass constraint ($m_W$), resulting in a quadratic equation with two possible real solutions. To resolve the resulting twofold sign ambiguity in $\cos\theta_{CS}$, one solution is chosen randomly with equal probability. For events where the quadratic equation yields no real solution (approximately 15% at low $p_T^W$ rising to 60% at high $p_T^W$), the neutrino kinematics are adjusted via a momentum imbalance constraint to ensure a unique solution.

The extraction of the angular coefficients and the differential cross-section ($d\sigma/dp_T^W$) is performed using a template fit to the two-dimensional reconstructed angular distributions in the $(\cos\theta_{CS}, \phi_{CS})$ plane. The fit is conducted in bins of the reconstructed transverse momentum $p_T^{\ell\nu}$, with bin boundaries defined from 0 to 600 GeV. The analysis is performed separately for $W^-$ and $W^+$ channels and for electron and muon decay modes, which are subsequently combined. Backgrounds, particularly from QCD multijet production, are estimated using data-driven template fits in anti-isolation regions, while other backgrounds (top, diboson, $Z$) are modeled using Monte Carlo (MC) simulations (Powheg+Pythia8, Sherpa).

Key Contributions and Results
This paper presents the first simultaneous measurement of the full set of $W$-boson angular coefficients ($A_0$ through $A_7$) and the differential cross-section as a function of $p_T^W$ in the full phase space of the decay leptons.

• Angular Coefficients: The coefficients $A_0$, $A_2$, $A_3$, and $A_4$ are measured to be significantly different from zero. The precision achieved for $A_3$ exceeds that of any previous measurement. The coefficients $A_1$, $A_5$, $A_6$, and $A_7$ are also reported, though their extraction is limited by statistical sensitivity. The $A_7$ coefficient, expected to be non-zero due to the parity-violating nature of the weak interaction, shows a slight deviation from zero in the intermediate $p_T$ range.
• Lam-Tung Relation: The relation $A_0 - A_2 = 0$, a consequence of the spin-1 nature of the gluon at leading order, is studied. The data lack the precision to probe potential deviations from zero expected from higher-order diagrams.
• Differential Cross-Section: The differential cross-section $d\sigma/dp_T^W$ is measured with a precision of approximately 3% at low $p_T^W$.
• Comparison with Theory: All results are compared with theoretical predictions incorporating finite-order QCD corrections up to order $\alpha_S^2$ (NNLO) and resummation at NNLL accuracy, calculated using the DYTurbo program, as well as MC generators (Powheg+Pythia8 and Sherpa 2.2.1). The measurements show good agreement with the DYTurbo predictions across the full spectrum. The MC predictions describe the low-$p_T^W$ region reasonably well, while Sherpa 2.2.1 provides a good description at higher transverse momenta, whereas Powheg+Pythia8 shows worse agreement for $p_T^W > 40$ GeV.

Significance
The paper claims that this measurement provides the first direct experimental insight into the $W$-boson transverse momentum spectrum and its angular coefficients in the full phase space, bypassing the need for extrapolations from $Z$-boson data. The low pile-up dataset enabled high-precision measurements of neutrino-related observables, which are typically challenging due to reconstruction ambiguities. The results offer stringent comparisons with state-of-the-art QCD predictions and contribute to the validation of modeling used in precision measurements, such as the determination of the $W$-boson mass. The analysis demonstrates that the full set of angular coefficients can be extracted simultaneously, providing a comprehensive test of the spin correlations and QCD dynamics in $W$-boson production.