High-precision measurement of the W boson mass with the CMS experiment

The CMS Collaboration reports a high-precision measurement of the W boson mass at 80,360.2 ± 9.9 MeV using 13 TeV proton-proton collision data, a result that agrees with Standard Model predictions and provides a stringent test against recent anomalies.

The Gist

The Great Cosmic Scale: How CMS Weighed the W Boson

Imagine the universe as a giant, intricate clockwork machine. For decades, physicists have been trying to understand the gears and springs that make it tick. One of the most important "gears" in this machine is a tiny particle called the W boson. It's the messenger that carries the "weak force," the invisible hand responsible for things like the sun shining (nuclear fusion) and radioactive decay.

In the Standard Model—the rulebook physicists use to describe the universe—there is a strict relationship between the weight of the W boson and its cousin, the Z boson. If you know the weight of the Z boson with incredible precision (which we do), the rulebook predicts exactly what the W boson should weigh.

The Puzzle:
Recently, a different team of scientists (the CDF collaboration) took a measurement of the W boson's weight and found it was significantly heavier than the rulebook predicted. It was like measuring a car's tire and finding it was the size of a tractor tire. This created a huge mystery: Is the rulebook wrong? Is there a hidden gear (a new, undiscovered particle) messing with the math?

The New Measurement:
The CMS collaboration at CERN's Large Hadron Collider (LHC) decided to settle this debate. They built a massive, high-tech scale to weigh the W boson again, but this time with unprecedented precision. Here is how they did it, using some everyday analogies:

1. The Invisible Ghost (The Neutrino)

The W boson is unstable; it breaks apart almost instantly. Usually, it splits into a muon (a heavy electron) and a neutrino. The problem? The neutrino is a "ghost." It passes through the detector without leaving a trace, like a ninja walking through a wall.

• The Analogy: Imagine you are trying to weigh a person (the W boson) who jumps off a diving board. You can see the splash (the muon), but you can't see the person (the neutrino) who jumped. However, you know the total energy of the jump. By measuring exactly how hard the splash hits the water and the direction it goes, you can calculate how heavy the person must have been, even though you never saw them.

2. The Perfect Scale (Muon Momentum)

To get the weight right, you need to measure the "splash" (the muon) with extreme precision. The CMS detector is like a giant, 3D camera that tracks the muon's path.

• The Calibration: Imagine you have a ruler, but it's slightly bent. If you use it to measure a table, your measurement will be wrong. The CMS team had to "straighten their ruler." They used a known, stable particle (the J/ψ particle, which decays into two muons) as a "standard weight" to calibrate their detector. They checked and re-checked their ruler until they were sure it was accurate to within a few parts per hundred-thousand.

3. The Crowd Control (Pileup)

The LHC smashes protons together billions of times. Sometimes, multiple collisions happen at the exact same time, creating a "crowd" of particles (called pileup). This is like trying to hear a single whisper in a stadium full of cheering fans.

• The Solution: The team used advanced computer algorithms to filter out the "noise" of the crowd and focus only on the specific "whisper" of the W boson decay they were interested in.

4. The Theory vs. Reality Check

The team didn't just trust their raw data; they compared it against the most sophisticated computer simulations (the "rulebook"). They used a technique called "in situ constraints."

• The Analogy: Imagine you are trying to guess the weight of a mystery box. Instead of just guessing, you put the box on a scale that also measures the air pressure, temperature, and humidity inside the room. You let the data itself tell you how to adjust your guess for those factors, rather than relying on a manual that might be outdated. This allowed them to reduce the uncertainty of their theoretical predictions significantly.

The Result: The Verdict is In

After analyzing over 117 million W boson events, the CMS team announced their result:

• The Measured Weight: 80,360.2 ± 9.9 MeV.
• The Prediction: The Standard Model predicted 80,353 ± 6 MeV.

What does this mean?
The CMS measurement is a perfect match for the Standard Model's prediction. It is like weighing the car tire again and confirming it is exactly the size the manual said it should be.

• It agrees with the Standard Model: The rulebook is holding up.
• It disagrees with the CDF result: The earlier measurement that suggested the W boson was "too heavy" was likely an outlier or had a hidden error.
• It's a huge step forward: This is the most precise measurement of the W boson ever made at a proton-proton collider.

Why Should You Care?

This isn't just about numbers on a chart. It's about the stability of our understanding of the universe.

• If the W boson had been heavier (as CDF suggested), it would have meant the Standard Model was broken, and we would need a whole new theory of physics to explain "new particles" we haven't found yet.
• Because the CMS result agrees with the Standard Model, it tells us that the universe is behaving exactly as we expect it to, at least at this level of precision. It closes the door on one theory of "new physics" while keeping the door open for others, urging scientists to keep looking for those hidden gears in the cosmic clockwork.

In short: The universe passed the test. The W boson is exactly where the rulebook says it should be, and the CMS team built the most precise scale in history to prove it.

Technical Summary

1. Problem and Motivation

The mass of the W boson ($m_W$) is a fundamental parameter of the Standard Model (SM) of particle physics. In the SM, $m_W$ and the Z boson mass ($m_Z$) are uniquely related through electroweak (EW) coupling strengths. Precise measurements of these masses serve as stringent tests of the SM; deviations could indicate "New Physics" arising from heavy, undiscovered particles interacting via quantum loops.

• The Discrepancy: While $m_Z$ is known with extreme precision (22 ppm), $m_W$ has historically been less precise. A recent measurement by the CDF Collaboration at Fermilab reported $m_W = 80,433.5 \pm 9.4$ MeV, which is in significant tension (approx. 7$\sigma$) with the SM global fit prediction ($m_W = 80,353 \pm 6$ MeV) and previous experimental averages.
• The Goal: The CMS Collaboration aims to provide an independent, high-precision measurement of $m_W$ at the Large Hadron Collider (LHC) to resolve this "W boson mass puzzle." Previous LHC measurements lacked the precision to definitively confirm or refute the CDF result or the SM prediction.

2. Methodology

The analysis utilizes proton-proton collision data collected by the CMS detector in 2016 at a center-of-mass energy of $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of $16.8 \text{ fb}^{-1}$.

Data Sample and Selection

• Event Selection: The analysis focuses on $W \to \mu\nu$ decays. Events are selected requiring exactly one isolated muon with transverse momentum $26 < p_T^\mu < 56$ GeV and pseudorapidity $|\eta^\mu| < 2.4$.
• Signal Purity: A transverse mass cut ($m_T^W > 40$ GeV) is applied to suppress backgrounds. The final sample contains approximately 117 million reconstructed $W \to \mu\nu$ events.
• Backgrounds: The dominant backgrounds are non-prompt muons (from heavy-flavor hadron decays) and prompt muons from $Z \to \mu\mu$ events where one muon is missed. These are estimated using data-driven methods (extended ABCD method) and Monte Carlo (MC) simulations.

Analysis Strategy

Unlike previous hadron collider measurements that relied heavily on the missing transverse momentum ($p_T^{miss}$) or transverse mass ($m_T$), this analysis focuses on the kinematic distributions of the charged muon ($p_T^\mu, \eta^\mu, q^\mu$).

• 3D Binned Likelihood Fit: The core of the measurement is a high-granularity binned maximum likelihood fit to a three-dimensional distribution of muon transverse momentum ($p_T^\mu$), pseudorapidity ($\eta^\mu$), and electric charge ($q^\mu$).
  • The distribution is divided into 48 $\eta^\mu$ bins, 30 $p_T^\mu$ bins, and 2 charge bins (totaling ~2,880 bins).
• Template Fitting: Signal and background templates are generated using state-of-the-art MC simulations (MINNLOPS interfaced with PYTHIA) and corrected with higher-order theoretical calculations (SCETLIB + DYTURBO) to achieve N$^3$LL+NNLO accuracy.

Key Technical Innovations

1. In Situ Constraints (Theory Nuisance Parameters - TNPs): Instead of relying on external measurements of the W boson transverse momentum ($p_T^W$) spectrum (which are limited by $p_T^{miss}$ resolution), the analysis uses a novel "Theory Nuisance Parameter" approach. Theoretical uncertainties regarding perturbative QCD corrections and non-perturbative effects are parameterized and constrained in situ by the data itself during the likelihood fit. This significantly reduces the dependence on theoretical modeling.
2. Helicity Fit (Cross-Check): An alternative "helicity fit" was performed where the W boson polarization and $p_T^W$ spectrum are extracted simultaneously with $m_W$. This relaxes assumptions about W boson production models, providing a robust cross-check.
3. Muon Momentum Calibration: The muon momentum scale is calibrated to a precision of a few parts per hundred-thousand using $J/\psi \to \mu\mu$ decays. Independent validations are performed using $\Upsilon(1S) \to \mu\mu$ and $Z \to \mu\mu$ decays. The analysis uses a continuous variable helix (CVH) fit for track reconstruction to minimize biases from magnetic field and material modeling.
4. W-like $m_Z$ Validation: To validate the analysis chain, a "W-like" measurement of $m_Z$ was performed using $Z \to \mu\mu$ events where one muon is treated as a neutrino. The consistency of this result with the world-average $m_Z$ confirms the robustness of the momentum scale and theoretical modeling.

3. Key Contributions

• Unprecedented Precision: This is the first $m_W$ measurement by the CMS Collaboration, achieving a precision of 9.9 MeV, which is comparable to the CDF result and significantly better than previous LHC measurements (e.g., ATLAS, LHCb).
• Resolution of the $p_T^W$ Modeling Issue: By using in situ profiling of theoretical uncertainties rather than external $p_T^W$ constraints, the analysis mitigates one of the largest systematic uncertainties in hadron collider mass measurements.
• Comprehensive Uncertainty Budget: The analysis provides a detailed breakdown of uncertainties, demonstrating that the dominant sources are now the muon momentum calibration (4.8 MeV) and Parton Distribution Functions (PDFs) (4.4 MeV), while theoretical modeling of $p_T^W$ has been reduced to a subleading contribution.

4. Results

The measured W boson mass is:
$$m_W = 80,360.2 \pm 9.9 \text{ MeV}$$
(Breakdown: $\pm 2.4$ MeV statistical, $\pm 9.6$ MeV systematic)

• Comparison with SM: The result is in excellent agreement with the Standard Model global electroweak fit prediction ($80,353 \pm 6$ MeV).
• Comparison with CDF: The result is in significant tension (approx. 4.5$\sigma$) with the CDF Collaboration measurement ($80,433.5 \pm 9.4$ MeV).
• Comparison with Other Experiments: The result is consistent with previous measurements from LEP, D0, ATLAS, and LHCb, but with superior precision.

5. Significance

• Resolving the Anomaly: This measurement strongly supports the Standard Model prediction and challenges the CDF result, suggesting that the CDF measurement may be affected by unaccounted systematic effects or that the tension requires further investigation.
• Benchmark for New Physics: Achieving a precision of ~10 MeV brings the direct experimental measurement of $m_W$ to a level comparable to the indirect SM prediction. This allows for a sensitive test of the SM; any future deviation at this precision level would be a clear sign of New Physics.
• Methodological Advancement: The techniques developed for in situ constraint of theoretical uncertainties and the high-granularity likelihood fitting set a new standard for precision electroweak measurements at hadron colliders, paving the way for even more precise measurements in future LHC runs.

In conclusion, this paper represents a major milestone in particle physics, providing the most precise single measurement of the W boson mass to date (excluding the CDF result) and reinforcing the validity of the Standard Model's electroweak sector.