Measurement of the $W$-boson production cross-sections in $pp$ collisions at $\sqrt{s}$ = 13 TeV in the forward region

Using 5.1 fb⁻¹ of proton-proton collision data at 13 TeV, the LHCb experiment performed a precision measurement of the forward $W$-boson production cross-sections via the $W \to \mu\nu$ decay channel, yielding results that significantly improve upon previous measurements and align well with next-to-next-to-leading order theoretical predictions.

The Gist

Imagine you are a detective trying to figure out the recipe for a giant, invisible cake that is baked every time two high-speed trains crash into each other. This cake is the proton, the tiny particle inside the atom that makes up everything around us.

The problem? You can't see the ingredients (called partons) directly. They are like flour, sugar, and eggs mixed so thoroughly in a batter that you can't tell them apart just by looking. To figure out the recipe, you have to smash the cakes together at incredible speeds and see what flies out the other side.

This paper is a report from the LHCb team at CERN (the world's largest particle physics lab). They are the detectives who specialize in looking at the "forward" direction—the side of the crash where the debris flies off at a sharp angle, rather than straight back.

Here is the story of their latest investigation, explained simply:

1. The Big Crash (The Experiment)

The team used the Large Hadron Collider to smash protons together at a speed that is 99.999999% the speed of light. They collected data from 5.1 billion billion of these collisions (a number so big it's hard to comprehend, represented as an "integrated luminosity" of 5.1 fb⁻¹).

They were specifically hunting for a very specific type of debris: a W-boson. Think of the W-boson as a "messenger particle." It's a heavy, unstable particle that appears for a split second after the crash and then immediately decays (breaks apart) into a muon (a heavy cousin of the electron) and a neutrino (a ghost-like particle that almost never interacts with anything).

2. The Detective Work (The Measurement)

Since the neutrino is a ghost that escapes undetected, the team couldn't see the whole picture. They had to find the muon and use math to figure out where the ghost went.

• The Net: They built a giant, high-tech net (the LHCb detector) that only catches particles flying in a specific direction (the "forward" region).
• The Filter: They filtered out millions of "fake" muons that look like the real thing but are actually just debris from other messy crashes (like pions or kaons pretending to be muons).
• The Count: After all the cleaning and filtering, they counted how many real W-bosons they found.

3. The Result: The Recipe is Refined

The main goal of this paper was to measure the cross-section. In everyday language, think of the cross-section as the "probability of catching a W-boson."

• The Old Map: Before this, scientists had a rough map of where these particles should appear. It was like a sketch drawn by a child—good enough to know where to look, but not precise enough for a master chef.
• The New Map: This paper provides a high-definition, 4K version of that map. They measured the probability with incredible precision.
  • They found that for every 1,754 W-bosons created, they caught about 1,178 of the "minus" charged ones.
  • The margin of error is tiny—like measuring the distance from London to New York and being off by only a few millimeters.

4. Why Does This Matter? (The "Parton Distribution Function")

This is the most important part. The W-boson doesn't just appear randomly; it is made from the "ingredients" inside the proton.

• The Analogy: Imagine the proton is a bag of marbles. Some marbles are heavy (quarks), some are light (gluons). The W-boson is formed when two specific marbles collide.
• The Discovery: By measuring exactly how often W-bosons appear in this specific "forward" direction, the LHCb team is essentially weighing the marbles inside the bag.
• The Impact: This helps scientists update the "Parton Distribution Functions" (PDFs). Think of PDFs as the instruction manual for the universe's building blocks. If the manual is wrong, our predictions for future particle physics experiments will be wrong. This paper says, "Hey, the manual needs a correction on pages 10 to 100 regarding how heavy the marbles are at these specific speeds."

5. The "Ghost" in the Machine (Systematic Uncertainties)

The scientists knew they couldn't just trust their eyes. Their detector might be slightly bent, or the computer simulation might be slightly off.

• They used Z-bosons (a stable, well-known cousin of the W-boson) as a "calibration weight." If they knew exactly how heavy the Z-boson should be, they could adjust their scales to make sure the W-boson measurements were accurate.
• They checked their work against different computer models (like comparing a recipe written by a French chef vs. a Japanese chef) to ensure the result was solid.

The Bottom Line

This paper is a precision measurement that tells us exactly how often nature produces a specific heavy particle when protons smash together at high speeds.

It's like taking a blurry, low-resolution photo of a rare bird and replacing it with a crystal-clear, high-definition image. This new image helps physicists understand the fundamental "DNA" of matter (the proton) much better than before, ensuring that our theories about how the universe works are as accurate as possible.

In short: They smashed atoms, caught the rare debris, cleaned up the mess, and used the count to update the universe's instruction manual with extreme precision.

Technical Summary

1. Problem and Motivation

The primary motivation for this study is the need for precise constraints on Proton Parton Distribution Functions (PDFs). PDFs describe the momentum distribution of quarks and gluons within the proton and are fundamental inputs for calculating cross-sections at hadron colliders. They currently represent the dominant source of systematic uncertainty in many precision Standard Model (SM) measurements.

• Kinematic Gap: While experiments like ATLAS, CMS, and ALICE cover central rapidity regions, the LHCb experiment offers unique forward acceptance ($2 < \eta < 5$). This region probes parton momentum fractions ($x$) ranging from approximately $10^{-4}$ to $10^{-1}$, covering both small-$x$ and large-$x$ regimes that are theoretically challenging and poorly constrained by other experiments.
• Goal: To perform a high-precision measurement of the inclusive $W^+$ and $W^-$ boson production cross-sections in the decay channel $W \to \mu\nu$ using the full Run 2 dataset ($\sqrt{s}=13$ TeV), thereby providing stringent constraints on PDFs and testing perturbative Quantum Chromodynamics (pQCD) at Next-to-Next-to-Leading Order (NNLO).

2. Methodology

Data Sample

• Dataset: Proton-proton collisions collected by the LHCb detector during Run 2 (2016–2018).
• Integrated Luminosity: $5.1 \text{ fb}^{-1}$.
• Kinematic Fiducial Region:
  • Muon transverse momentum: $25 < p_T^\mu < 55$ GeV.
  • Muon pseudorapidity: $2.0 < \eta^\mu < 4.5$.

Event Selection and Reconstruction

• Signal: Muons identified via the muon system, satisfying strict quality criteria (track fit quality, momentum resolution $\sigma(p)/p < 0.06$).
• Background Suppression:
  • Heavy Flavor: Rejected by requiring low energy deposition in the hadronic calorimeter around the muon track and strict impact parameter significance.
  • QCD Multijets: Suppressed using track isolation ($P_{T}^{track} < 3.0$ GeV in a cone $\Delta R < 0.5$).
  • Z Boson Veto: Events with a second opposite-charge muon ($p_T > 25$ GeV) are rejected to remove $Z \to \mu^+\mu^-$ contamination.
• Yield Extraction: A binned maximum-likelihood fit is performed on the muon $p_T$ spectrum across 36 bins (18 $\eta$ intervals $\times$ 2 charges). The fit model includes signal ($W \to \mu\nu$) and various backgrounds (Heavy Flavor, Electroweak, QCD).

Corrections and Calibration

• Momentum Scale & Resolution:
  • Scale: Calibrated using $\Upsilon(1S) \to \mu^+\mu^-$ decays to correct for detector misalignment.
  • Curvature Bias: Addressed using the pseudomass method on $Z \to \mu^+\mu^-$ decays to determine charge-dependent curvature biases ($\delta$).
  • Smearing: A multi-parameter smearing model is applied to simulation to match the data's momentum resolution, fitted using $J/\psi$, $\Upsilon(1S)$, and $Z$ invariant mass spectra.
• Efficiency: Determined via a data-driven tag-and-probe method using $Z \to \mu^+\mu^-$ events. Efficiency is factorized into tracking ($\epsilon_{trk}$), identification ($\epsilon_{ID}$), and trigger ($\epsilon_{trig}$) components.
• Theoretical Corrections:
  • FSR: Final-state radiation corrections are applied using a ratio of ResBos 2 predictions with and without the Photos implementation to map bare-level kinematics to Born-level definitions.
  • Generator Weighting: Simulation samples (generated with Pythia) are re-weighted using DYTurbo (NNLO accuracy) to correct for higher-order QCD effects, including angular coefficients and unpolarized cross-sections.

Systematic Uncertainties

Uncertainties are evaluated for:

• Reconstruction efficiency (statistical, parameterization, selection criteria).
• Generator modeling (scale variations, PDF sets: NNPDF3.1, MSHT20, CT18).
• QCD background estimation (misidentification rates, hadron abundances).
• Track smearing and curvature bias.
• Luminosity determination (2.0%).

3. Key Contributions

1. Unprecedented Precision: This measurement utilizes the largest LHCb dataset to date ($5.1 \text{ fb}^{-1}$), resulting in significantly reduced statistical and systematic uncertainties compared to previous Run 1 and early Run 2 results.
2. Differential Measurements: The paper provides differential cross-sections in 18 pseudorapidity bins, offering a granular view of the kinematic dependence crucial for PDF fits.
3. Advanced Background Modeling: The use of data-driven misidentification rates parameterized by momentum and the specific treatment of QCD backgrounds (decay-in-flight vs. punch-through) sets a new standard for forward region analyses.
4. Comprehensive Uncertainty Budget: A detailed breakdown of systematic uncertainties, including the specific impact of generator choices (DYTurbo vs. Powheg) and PDF sets, highlights the robustness of the result.

4. Results

The integrated production cross-sections for $W$ bosons decaying to muons are measured as:

• $W^+$ Cross-section:
  $$\sigma_{W^+ \to \mu^+\nu} = 1754.2 \pm 1.5 (\text{stat}) \pm 11.9 (\text{syst}) \pm 35.1 (\text{lumi}) \text{ pb}$$
• $W^-$ Cross-section:
  $$\sigma_{W^- \to \mu^-\nu} = 1178.1 \pm 1.3 (\text{stat}) \pm 9.7 (\text{syst}) \pm 23.6 (\text{lumi}) \text{ pb}$$

Ratios to Z Boson:
The ratios of $W$ to $Z$ cross-sections are also determined to minimize luminosity and some systematic uncertainties:

• $R_{W^+Z} = 8.932 \pm 0.012 (\text{stat}) \pm 0.081 (\text{syst})$
• $R_{W^-Z} = 5.999 \pm 0.009 (\text{stat}) \pm 0.061 (\text{syst})$

Comparison with Theory:

• The results show excellent agreement with theoretical predictions calculated at NNLO in pQCD using the ResBos 2 generator coupled with modern PDF sets (CT18, NNPDF4.0, MSHT20).
• The experimental uncertainties are now comparable to the theoretical uncertainties, marking a transition where these measurements can drive improvements in PDF fits rather than just being limited by theory.

5. Significance

• PDF Constraints: These measurements provide critical constraints on the up and down quark distributions in the proton, particularly in the small-$x$ region ($x \sim 10^{-4}$) and the large-$x$ region, where previous data was sparse. This is essential for reducing uncertainties in predictions for future high-energy physics processes, including Higgs boson production and searches for new physics.
• SM Validation: The agreement with NNLO QCD predictions validates the Standard Model in the forward kinematic regime, a region sensitive to non-perturbative QCD dynamics and resummation effects.
• Future Impact: The precision achieved (total uncertainty $\sim 2\%$) demonstrates the maturity of the LHCb experiment's forward physics program and establishes a benchmark for future high-luminosity runs (Run 3 and beyond). The results are expected to be incorporated into global PDF analyses (e.g., by NNPDF, CT, and MSHT collaborations) to refine the proton structure model.