Measurement of the $W \to \mu \nu_\mu$ cross-sections as a function of the muon transverse momentum in $pp$ collisions at 5.02 TeV

Using LHCb data from proton-proton collisions at 5.02 TeV, this paper reports the differential and integrated cross-sections of $W \to \mu \nu_\mu$ production and demonstrates a proof-of-principle method for determining the $W$-boson mass from these measurements.

The Gist

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful particle smasher. Inside its giant ring, beams of protons (tiny building blocks of matter) zoom around at nearly the speed of light and crash into each other. When they collide, they create a shower of new particles, some of which are very heavy and unstable, like the W boson.

This paper is a report from the LHCb experiment, a giant camera and detector sitting at one of these collision points. Their job is to take a "snapshot" of these crashes and figure out exactly what happened.

Here is the story of what they did, explained simply:

1. The Goal: Catching a Ghost

The W boson is like a ghost. It lives for a tiny fraction of a second before it decays (falls apart) into other particles. In this experiment, they were looking for a specific type of decay: a W boson turning into a muon (a heavy cousin of the electron) and a neutrino (a ghostly particle that almost never interacts with anything).

Because the neutrino disappears without a trace, the scientists can't see the whole picture. They only see the muon. It's like trying to figure out the speed of a car that just drove through a wall, but you only see the tire tracks left behind. You have to use math and physics to guess what the car was doing.

2. The New Trick: A Detailed Map

In the past, scientists usually took a "blurry photo" of the whole event. They would count how many muons they saw and calculate an average.

In this paper, the LHCb team did something new. Instead of just counting, they created a detailed map. They looked at the muons and sorted them into 12 different "buckets" based on how fast they were moving sideways (transverse momentum).

• The Analogy: Imagine a river. In the past, they just measured the total amount of water flowing. Now, they measured the speed of the water at 12 different spots along the riverbank. This gives them a much more detailed picture of the flow.

3. Cleaning Up the Mess

The collision area is messy. Most of the time, the muons they see aren't from W bosons; they are "imposters" created by other common particles (like pions) that trick the detector.

• The Analogy: Imagine trying to find a specific celebrity in a crowded stadium. Most people look like the celebrity (wearing similar clothes). The scientists had to build a sophisticated filter to ignore the "lookalikes" and only count the real celebrity. They used a "cleanliness score" (called isolation) to check if the muon was hanging out alone (good) or surrounded by a crowd of other particles (bad).

4. The Big Reveal: Weighing the Ghost

Once they had their clean, detailed map of the muon speeds, they did something brilliant. They used this map to weigh the W boson.

Usually, weighing a particle is like trying to guess the weight of a balloon by looking at how fast it floats away. It's hard to get right. But because they had such a detailed map of the muon speeds, they could reverse-engineer the math to find the exact mass of the W boson.

The Result:
They calculated the mass of the W boson to be 80,369 MeV (a unit of mass in particle physics).

• They are very confident in this number. The "error bars" (the margin of uncertainty) are tiny, like measuring the width of a human hair from a mile away.
• This result matches what other experiments and global theories have predicted, which is a huge win for physics. It means our current understanding of the universe is working correctly.

5. Why This Matters

This paper is a "proof of principle." It's like a pilot test.

• The Dataset: They only used a small amount of data (100 "inverse picobarns," which is a tiny fraction of what the LHC produces in a year).
• The Future: They proved that this new, detailed method works. Now, they can apply this same technique to the massive amount of data they will collect in the future (Run 3 and beyond).
• The Payoff: With more data, they expect to measure the W boson's mass with even greater precision. If they find even a tiny difference between their measurement and the theory, it could mean there is new physics waiting to be discovered—something we don't know about yet!

Summary

The LHCb team took a small sample of particle collisions, cleaned up the noise, mapped out the speeds of the resulting particles in great detail, and used that map to weigh a fundamental particle of the universe. They found that the W boson weighs exactly what we thought it should, confirming our current laws of physics while paving the way for even more precise discoveries in the future.

Technical Summary

1. Problem and Motivation

The production of massive electroweak vector bosons ($W$ and $Z$) is a fundamental process for testing the Standard Model and constraining Parton Distribution Functions (PDFs). While the LHCb collaboration has previously measured $W \to \mu\nu$ cross-sections, these were integrated over muon transverse momentum ($p_T$) because the shape of the $p_T$ distribution was historically required to subtract hadronic backgrounds.

The primary challenges addressed in this work are:

1. Differential Measurement: Developing a method to measure the differential cross-section ($d\sigma/dp_T$) without relying on the $p_T$ shape for background subtraction, thereby allowing for a more robust extraction of physics parameters.
2. $W$-Boson Mass ($m_W$) Determination: Utilizing the newly measured differential cross-sections to determine the $W$-boson mass. Traditionally, $m_W$ is measured via direct fits to the charged lepton $p_T$ or transverse mass distributions. This paper proposes a novel "proof-of-principle" approach: using the corrected differential cross-sections as input for a theoretical fit to extract $m_W$.
3. Forward Region Constraints: Providing constraints on PDFs at high and low momentum fractions ($x$) in the forward pseudorapidity region ($2 < \eta < 5$), which complements measurements from central detectors (ATLAS/CMS).

2. Methodology

Dataset and Detector

• Data: Proton-proton collisions at a center-of-mass energy $\sqrt{s} = 5.02$ TeV, recorded by the LHCb experiment in 2017.
• Luminosity: $100 \pm 2 \text{ pb}^{-1}$.
• Kinematic Region: Muons with $2.2 < \eta < 4.4$ and $p_T > 28$ GeV.
• Selection: Events triggered by a hardware muon trigger ($p_T > 5$ GeV). Signal candidates are isolated muons ($p_T > 28$ GeV, isolation $< 8$ GeV). Backgrounds from $Z \to \mu\mu$ are suppressed by vetoing events with a second muon.

Background Modeling and Calibration

• Hadronic Background: Light charged hadrons (pions, kaons, protons) misidentified as muons constitute the largest background.
  • Misidentification probabilities are parametrized as a function of momentum using simulation.
  • A data-driven calibration is performed using a high-$p_T$ hadron-enriched sample from 13 TeV data (since no dedicated hadron trigger existed at 5.02 TeV).
  • An isolation calibration factor ($0.93 \pm 0.02$) is derived from $Z \to \mu\mu$ data to correct the simulation.
• Muon Momentum Corrections:
  • Curvature Bias: A "pseudomass" method is applied to correct charge-dependent momentum biases ($q/p \to q/p + \delta$) caused by magnetic field misalignment, using $Z \to \mu\mu$ candidates.
  • Smearing: Simulation is smeared to match the data's momentum resolution, with parameters ($\alpha, \sigma_1, \sigma_2$) fitted to the $Z \to \mu\mu$ mass distribution.

Differential Cross-Section Measurement

• Fit Strategy: A simultaneous fit is performed on a 2D distribution of muon $p_T$ (12 bins, 28–52 GeV) and isolation (8 bins).
• Likelihood Function: A binned likelihood minimizes the difference between observed candidates and model predictions ($\lambda_i$), which include signal, hadronic background, and other electroweak backgrounds.
• Unfolding: The signal component is modeled using a response matrix ($R_{ij}$) that maps true $p_T$ to reconstructed $p_T$. The fit extracts the differential cross-sections directly without assuming a specific shape for $d\sigma/dp_T$.

$W$-Boson Mass Determination

• Theoretical Model: The measured $d\sigma/dp_T$ is fitted against theoretical templates generated using DYTurbo (NNLO accuracy) and Pythia8.
• Parameters: The fit varies $m_W$, the strong coupling constant $\alpha_s$, and a non-perturbative parameter $g$ (affecting the $p_T$ resummation).
• PDF Sets: Results are averaged over three PDF sets: NNPDF3.1, MSHT20, and CT18.
• Validation: The model is validated against $Z$-boson $p_T$ data at 5.02 TeV to constrain $\alpha_s$ and $g$ before applying it to the $W$ mass fit.

3. Key Contributions

1. First Differential $W \to \mu\nu$ Measurement at LHCb: This is the first time LHCb has measured the $W \to \mu\nu$ cross-section differentially in muon $p_T$, overcoming previous limitations where the $p_T$ shape was tied to background subtraction.
2. Novel $m_W$ Extraction Method: It demonstrates a new pathway to determine the $W$-boson mass by fitting the differential cross-section rather than the kinematic distributions of the final state particles. This decouples the physics modeling from the detector resolution modeling in a unique way.
3. Comprehensive Systematic Treatment: The analysis introduces a full 24 $\times$ 24 covariance matrix accounting for statistical and systematic uncertainties (including charge-dependent momentum biases, hadronic background modeling, and unfolding), which is crucial for the precision $m_W$ fit.

4. Results

Integrated Cross-Sections

The measured cross-sections (integrated over $2.2 < \eta < 4.4$ and $28 < p_T < 52$ GeV) are:

• $\sigma(W^+ \to \mu^+\nu_\mu) = 300.9 \pm 2.4 \text{ (stat)} \pm 3.8 \text{ (syst)} \pm 6.0 \text{ (lumi)} \text{ pb}$
• $\sigma(W^- \to \mu^-\bar{\nu}_\mu) = 236.9 \pm 2.1 \text{ (stat)} \pm 2.7 \text{ (syst)} \pm 4.7 \text{ (lumi)} \text{ pb}$
  These results are consistent with theoretical predictions at $\mathcal{O}(\alpha_s^2)$ using various PDF sets.

Differential Cross-Sections

Differential cross-sections were measured in 12 bins of $p_T$ (2 GeV width). The results show the expected falling spectrum, with $W^+$ cross-sections generally higher than $W^-$ due to the proton's valence quark composition.

$W$-Boson Mass

Using the differential cross-sections as input, the $W$-boson mass was determined as:
$$m_W = 80369 \pm 130 \text{ (exp)} \pm 33 \text{ (theory)} \text{ MeV}$$

• Experimental Uncertainty (130 MeV): Dominated by the charge-independent momentum bias and statistical limitations of the small dataset.
• Theoretical Uncertainty (33 MeV): Includes PDF uncertainties, QCD scale variations, and QED radiation modeling.
• Significance: This result is compatible with the global electroweak fit and previous direct measurements (LEP, Tevatron, LHC), validating the new methodology despite the limited dataset.

5. Significance and Future Outlook

• Proof of Principle: The analysis successfully proves that differential cross-sections can be used to extract $m_W$ with competitive precision, offering a complementary perspective to traditional kinematic fits.
• Systematic Understanding: The detailed breakdown of systematic uncertainties (e.g., the strong anticorrelation between $W^+$ and $W^-$ charge-dependent momentum biases) provides a blueprint for future high-precision measurements.
• Future Potential: The authors note that applying this method to the full LHCb Run 2 dataset ($\sqrt{s}=13$ TeV) would increase the statistics by two orders of magnitude. This could reduce the statistical uncertainty on $m_W$ to approximately 12 MeV, potentially making LHCb a leading experiment in $W$-mass precision measurements.
• PDF Constraints: The differential measurements provide new constraints on PDFs in the forward region, particularly at high $x$ for $W^-$ and low $x$ for $W^+$, which are critical for reducing uncertainties in other LHC physics processes.

In conclusion, this paper establishes a robust framework for differential $W$-boson measurements at LHCb and demonstrates the viability of using these measurements for high-precision electroweak parameter determination.