⚛️ high-energy theory

Precision Measurements of the Electroweak Mixing Angle in the Region of the Z pole

This paper presents the most precise single-experiment measurement of the effective leptonic weak mixing angle to date, $\sin^2\theta^\ell_{\rm eff} = 0.23154 \pm 0.00024$ , extracted from CMS Drell-Yan data at 13 TeV using enhanced PDF constraints, which shows excellent agreement with the Standard Model prediction.

Original authors: Arie Bodek, Hyon-San Seo, Un-Ki Yang

Published 2026-09-23

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CC BY 4.0

Original authors: Arie Bodek, Hyon-San Seo, Un-Ki Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

At the heart of modern physics lies a quest to understand the fundamental rules that govern the universe, specifically how particles interact to create the matter we see around us. One of the most important rules in this rulebook is a number known as the weak mixing angle. Think of this number as a precise setting on a cosmic dial that determines how the force responsible for radioactive decay, called the weak force, mixes with the force of electromagnetism. If this dial is set even slightly differently than our current best guess, it would mean our entire understanding of the universe is missing a piece, perhaps pointing to the existence of new, undiscovered particles or forces. For decades, scientists have tried to measure this dial with extreme precision, primarily using machines that smash electrons and positrons together. However, a new study suggests that we can now measure this same dial with equal precision using the world's largest particle collider, which smashes protons together.

The researchers behind this study, working with data from the Large Hadron Collider, focused on a specific type of collision where protons produce a heavy particle called a Z boson, which then immediately decays into a pair of leptons, such as electrons or muons. By carefully analyzing the direction in which these leptons fly after the collision, the team could calculate the value of the weak mixing angle. The challenge in using a proton collider, rather than an electron collider, is that protons are complex bundles of smaller particles called quarks. When two protons collide, it is often unclear which specific quark was moving forward and which was moving backward, making it difficult to determine the direction of the collision with certainty. This uncertainty creates a "fog" in the data, known as parton distribution function uncertainty, which can blur the final measurement.

To clear this fog, the team did not just look at the Z boson collisions in isolation. They combined that data with two other types of measurements taken from the same machine: the asymmetry of particles produced when a W boson decays, and the ratio of how often W bosons are produced compared to Z bosons. By feeding all three sets of measurements into a sophisticated computer model, the researchers were able to tighten the constraints on the internal structure of the proton. This process effectively sharpened the focus of their experiment, allowing them to extract a much clearer value for the weak mixing angle. The result was a measurement of 0.23154 with a very small margin of error, making it the most precise single measurement of its kind to date.

This new value aligns almost perfectly with the theoretical prediction made by the Standard Model of particle physics, which is the current best theory we have for how the universe works. The agreement is so close that the difference between the measurement and the prediction is smaller than the width of a human hair compared to the size of a football field. This finding is significant because it rules out a specific alternative theory that had gained attention due to a conflicting measurement of a related particle's mass. That alternative theory, known as the Two Higgs Doublet Model, predicted a different value for the weak mixing angle, but the new data does not support that prediction. Instead, the results reinforce the Standard Model, suggesting that the current theory remains robust even under the most rigorous testing.

The study also demonstrates a powerful new method for future research. By using the extra data to refine the understanding of the proton's internal structure, the team showed that other experiments at the Large Hadron Collider, including those by different research groups, can now achieve higher precision with their own data. As the collider continues to operate and collect more data in the coming years, this technique promises to push the boundaries of measurement even further. For now, the work confirms that the fundamental dial of the weak force is set exactly where our best theories say it should be, closing the door on some speculative ideas while keeping the path open for even deeper exploration of the subatomic world.

Problem and Motivation
The effective leptonic weak mixing angle, $\sin^2 \theta^\ell_{\text{eff}}$ , is a fundamental parameter of the Standard Model (SM) of particle physics. While the SM predicts this value with high accuracy using inputs such as the $Z$ boson mass ( $M_Z$ ), the Fermi constant, the fine structure constant, the Higgs mass, quark masses, and the QCD strong coupling ( $\alpha_s$ ) at $M_Z$ , deviations between measurement and prediction can signal new physics. Recent tensions exist in the electroweak sector; specifically, the CDF collaboration's measurement of the $W$ boson mass ( $M_W$ ) is approximately 9 standard deviations higher than the SM prediction, whereas the CMS measurement agrees with the SM. The CDF $M_W$ value, if correct, would imply a shift in $\sin^2 \theta^\ell_{\text{eff}}$ to approximately 0.23110 in a Two Higgs Doublet Model (2HDM), contrasting with the 2026 SM indirect prediction of $0.23152 \pm 0.00006$ .

While the most precise extractions of $\sin^2 \theta^\ell_{\text{eff}}$ historically come from $e^+e^-$ colliders (LEP and SLD), recent advances in experimental and analysis techniques at the Large Hadron Collider (LHC) have made measurements derived from the forward-backward asymmetry ( $A_{FB}$ ) in Drell-Yan dilepton events competitive. However, LHC extractions face distinct challenges, primarily dominated by uncertainties in Parton Distribution Functions (PDFs), which differ from the systematic uncertainties in $e^+e^-$ measurements.

Methodology
This paper presents an updated analysis of the CMS Drell-Yan $A_{FB}$ measurements at a center-of-mass energy of 13 TeV to extract a more precise value of $\sin^2 \theta^\ell_{\text{eff}}$ . The analysis utilizes the polar-angle distribution of the negative lepton in the Collins-Soper frame. The extraction is based on the angular coefficient $A_4$ (where $A_4 = 8A_{FB}/3$ ), which is sensitive to $\sin^2 \theta^\ell_{\text{eff}}$ near the $Z$ pole and to PDFs at other mass regions.

The core of the methodology involves a "profiling" technique to constrain PDF uncertainties. The authors employ the open-source code xFitter to minimize a $\chi^2$ function that includes correlated experimental and theoretical uncertainties via nuisance parameters. The analysis proceeds in three stages of constraint:

Baseline: Reproducing the original CMS analysis using the 13 TeV $A_4$ data (63 data points) across 19 modern PDF sets (including CTEQ, NNPDF, MSHT, and PDF4LHC21).
Extended Constraint 1: Incorporating the CMS measurement of the $W$ boson decay lepton charge asymmetry ( $W_{\text{asym}}$ ) at 13 TeV (18 additional data points). This provides constraints on the $d/u$ quark ratio.
Extended Constraint 2: Incorporating the CMS measurements of the $W/Z$ fiducial cross-section ratios at 5.02 TeV and 13 TeV (2 additional data points). This provides further constraints on the strange-quark distribution, addressing discrepancies found in certain PDF sets (e.g., CT18) regarding high-energy strange-quark densities.

The theoretical predictions utilize the POWHEG 2 event generator at MiNNLOPS accuracy in QCD, matched to PYTHIA 8.2 for parton showering, with PHOTOS for photon radiation and POWHEG-Z ew for next-to-leading order virtual weak corrections.

Key Contributions and Results
The primary contribution of this work is the reduction of the total uncertainty in the extraction of $\sin^2 \theta^\ell_{\text{eff}}$ by leveraging additional CMS measurements to constrain PDFs.

Uncertainty Reduction: For the nominal CT18Znnlo PDF set, the uncertainty in $\sin^2 \theta^\ell_{\text{eff}}$ was reduced from $0.00056$ (in the original CMS analysis) to $0.00032$ after profiling with $A_4$ alone. By further including $W_{\text{asym}}$ and $W/Z$ cross-section ratios, the total uncertainty was reduced to $0.00024$, with the PDF contribution specifically reduced to $0.00017$.
Final Value: The analysis yields a value of $\sin^2 \theta^\ell_{\text{eff}} = 0.23154 \pm 0.00024$ .
PDF Consistency: The profiling significantly reduces the spread of results across different PDF sets. After the full profiling, 18 of the 19 tested PDF sets yield values within half a standard deviation of the nominal CT18Znnlo result. The inclusion of $W/Z$ ratios notably reduces the discrepancy between PDF sets with symmetric versus asymmetric strange-quark distributions.
Comparison: The result is in excellent agreement with the 2026 SM indirect prediction ( $0.23152 \pm 0.00006$ ) and is competitive with the most precise $e^+e^-$ collider measurements. It is also consistent with the 2HDM prediction derived from the CDF $M_W$ value, though the central value aligns more closely with the SM.

Significance
The paper claims that this measurement represents the most precise single-experiment determination of $\sin^2 \theta^\ell_{\text{eff}}$ to date. Its significance lies in demonstrating that LHC measurements, when augmented with complementary asymmetry and cross-section ratio data to constrain PDFs, can achieve precision rivaling that of dedicated $e^+e^-$ colliders. Furthermore, the profiled PDF parameters derived from this analysis are presented as a reusable input for future extractions, including potential improvements for LHCb and ATLAS measurements. The authors note that further improvements are anticipated with the integration of Run III data (326 fb $^{-1}$ at 13.6 TeV), which could potentially reduce the total uncertainty to approximately $0.00020$.

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