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Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings

This paper presents a high-precision lattice QCD calculation of the hadronic contributions to the running of the electromagnetic coupling and the electroweak mixing angle, revealing significant tensions with data-driven estimates at intermediate energy scales while providing a more precise determination of the coupling at the Z-pole that slightly deviates from global electroweak fit predictions.

Original authors: Alessandro Conigli, Dalibor Djukanovic, Georg von Hippel, Simon Kuberski, Harvey B. Meyer, Kohtaroh Miura, Konstantin Ottnad, Andreas Risch, Hartmut Wittig

Published 2026-09-03

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Original authors: Alessandro Conigli, Dalibor Djukanovic, Georg von Hippel, Simon Kuberski, Harvey B. Meyer, Kohtaroh Miura, Konstantin Ottnad, Andreas Risch, Hartmut Wittig

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

The universe is held together by a handful of fundamental forces, and the strength of these forces is not fixed; it changes depending on how close you are to the particles involved. Imagine the electromagnetic force, which governs how light and electricity behave. At very low energies, far from any atomic nucleus, this force has a specific, well-known strength. But as you move closer to the action, or as you probe it with higher energies, the force appears to get stronger. This change in strength is called "running," and it is a critical prediction of the Standard Model, the master theory that describes all known particles and forces. To test this model with extreme precision, scientists need to know exactly how much the force changes at the highest energies achievable in particle colliders, specifically at the energy level of the Z boson, a heavy particle that acts as a messenger of the weak nuclear force.

The difficulty lies in the fact that this change in strength is not caused by the simple, predictable interactions of light particles, but by the messy, complex world of the strong nuclear force. This force binds quarks together to form protons and neutrons, and its behavior is so tangled that standard mathematical tools often fail to describe it. To understand how the strong force influences the running of the electromagnetic force and the weak mixing angle (a value that determines how the electromagnetic and weak forces are related), scientists must calculate the contribution of these complex quark interactions from first principles. For decades, the best estimates for this contribution have come from piecing together experimental data from particle collisions, but these data-driven methods have recently shown signs of inconsistency and tension.

A team of researchers has now taken a different path, bypassing the experimental data entirely to calculate this contribution directly using a method called lattice quantum chromodynamics. Instead of relying on collision data, they built a digital grid that mimics the fabric of space-time and simulated the behavior of quarks and gluons on this grid. By running these simulations on some of the world's most powerful supercomputers, they were able to trace how the strength of the electromagnetic force evolves as the energy increases, all the way up to the mass of the Z boson. Their work provides a new, highly precise value for this running, one that is more than twice as accurate as the best previous estimates derived from experimental data.

The researchers focused on a specific range of energies, simulating the behavior of particles at distances smaller than a proton but larger than the smallest possible scale. They used a technique that breaks the problem into three distinct zones: a high-energy zone where the interactions are short and sharp, a middle zone, and a low-energy zone where the effects are long and diffuse. This "telescopic" approach allowed them to isolate different types of errors that usually plague such calculations. By treating each zone with a method tailored to its specific behavior, they could separate the effects of the computer grid itself from the actual physics of the particles. They also employed advanced noise-reduction strategies, effectively filtering out the random static that often obscures the signal in these complex simulations, allowing them to see the underlying physics with unprecedented clarity.

The results of this calculation reveal a significant discrepancy when compared to the traditional estimates based on experimental collision data. At certain energy levels, the new lattice calculation differs from the data-driven estimates by as much as seven standard deviations, a level of difference that in science usually indicates a fundamental problem with one of the methods or a hint of new physics. While the tension lessens as the energy increases, it remains substantial even at higher levels. When the researchers combined their new lattice results with established theories for the very highest energies, they arrived at a final value for the running of the electromagnetic coupling at the Z boson mass. This value is 0.027821, with a very small margin of error.

This new number is slightly different from the values predicted by global analyses that combine all known experimental data to fit the Standard Model. The difference is small, only about one to two standard deviations, but it is consistent enough to be noticed. It suggests that the traditional way of estimating these effects from experimental data might be missing something, or that the global fits need to be adjusted. The researchers also calculated how the weak mixing angle changes with energy, a key parameter for testing the Standard Model at low energies. Their calculation of this value agrees well with existing models but is far more precise, offering a new benchmark for future experiments.

The significance of this work extends beyond just a single number. It demonstrates that it is possible to calculate the effects of the strong force on fundamental constants with a precision that rivals, and in this case exceeds, the best methods based on experimental data. By providing a clean, first-principles determination, the team has offered a crucial cross-check for the entire field of particle physics. If future experiments confirm the tension between the lattice results and the data-driven estimates, it could point toward the existence of new particles or forces that have not yet been discovered. For now, the calculation stands as a rigorous test of the Standard Model, showing that the theory holds up under the most demanding scrutiny, even as it hints that there may be more to the story than we currently understand.

Technical Summary: Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings

Problem Statement
The electromagnetic coupling $\alpha$ and the electroweak mixing angle $\sin^2 \theta_W$ are fundamental parameters of the Standard Model (SM). Their energy dependence (running) is critical for interpreting high-energy collider data and searching for Beyond-Standard-Model (BSM) physics. However, the precision of these quantities at the $Z$ -pole and low energies is currently limited by the hadronic vacuum polarization (HVP) contribution. Traditional determinations rely on a data-driven dispersive approach using $e^+e^- \to \text{hadrons}$ cross-sections, which suffers from unresolved tensions in the $\pi^+\pi^-$ channel and requires model-dependent flavor separation for the weak mixing angle. Consequently, there is a need for a first-principles determination using Lattice QCD (LQCD) that avoids reliance on experimental cross-section data and provides a controlled error budget.

Methodology
This work presents an updated LQCD calculation of the hadronic contribution to the running of $\alpha$ and $\sin^2 \theta_W$ in the space-like momentum region up to $Q^2 = 12 \text{ GeV}^2$ . The calculation utilizes Coordinated Lattice Simulations (CLS) ensembles with $N_f = 2+1$ flavors of non-perturbatively $O(a)$ -improved Wilson fermions. Key methodological features include:

Ensemble Coverage: The analysis covers five lattice spacings ( $0.039 \text{ fm} \leq a \leq 0.087 \text{ fm}$ ) and a range of pion masses ( $130 \text{ MeV} \leq m_\pi \leq 420 \text{ MeV}$ ), including three ensembles at the physical pion mass.
Telescopic Window Strategy: A novel decomposition of the subtracted HVP function $\bar{\Pi}(Q^2)$ into High-Virtuality (HV), Mid-Virtuality (MV), and Low-Virtuality (LV) regions is employed. This strategy separates short-distance discretization effects (dominant in HV) from long-distance chiral dependence (dominant in LV), allowing for tailored fit models and a clean separation of systematic effects.
Flavor Decomposition: The calculation explicitly separates isovector ( $I=1$ ), isoscalar ( $I=0$ ), charm-connected, and $Z\gamma$ -mixing contributions. The isovector and charm-connected parts utilize subtracted kernels to cancel lattice artifacts, while the isoscalar and mixing terms exploit the cancellation of leading cutoff effects near the $SU(3)$ -symmetric point.
Noise Reduction: Statistical precision is enhanced via low-mode averaging (LMA), the bounding method for long-distance correlators, and explicit spectral reconstruction of the correlator tail for the most chiral ensembles.
Renormalization and Improvement: Two sets of non-perturbatively determined improvement coefficients and renormalization constants are used. Tree-level improvement is applied to short-distance correlators to further reduce cutoff effects.
Finite-Volume Corrections: Corrections are applied using the Hansen-Patella formalism for short distances and the Meyer-Lellouch-Lüscher formalism for long distances, with a specific focus on kaon propagation effects.
Euclidean Split Technique: To obtain the value at the $Z$ -pole, the lattice results at a threshold $Q_0^2$ are combined with perturbative QCD (pQCD) calculations of the Adler function. The high-energy running is computed using the AdlerPy package with inputs from FLAG24, and a small analytic continuation term connects the space-like and time-like regions.

Key Contributions and Results

$\Delta \alpha_{\text{had}}^{(5)}(M_Z^2)$ : The authors determine the hadronic contribution to the running of the electromagnetic coupling at the $Z$ -pole as:
$\Delta \alpha_{\text{had}}^{(5)}(M_Z^2) = 0.027821(34)_{\text{lat}}(35)_{\text{pQCD}}$
This result has a total relative uncertainty of $0.17\%$ , which is more than twice as precise as recent data-driven estimates.
Tension with Data-Driven Estimates: The lattice results for $\Delta \alpha_{\text{had}}^{(5)}(-Q^2)$ in the space-like region show significant tension with data-driven dispersive estimates based on $e^+e^-$ cross-sections. The discrepancy reaches up to $7\sigma$ around $Q^2 = 1 \text{ GeV}^2$ and remains above $4\sigma$ up to $Q^2 = 9 \text{ GeV}^2$ .
Electroweak Mixing Angle: The paper provides a precise determination of the hadronic contribution to the running of $\sin^2 \theta_W$ . The lattice framework allows for exact flavor separation of the $Z\gamma$ mixing function $\bar{\Pi}^{(Z,\gamma)}$ , avoiding the model-dependent reweighting required in phenomenological analyses. The results are in good agreement with phenomenological models but offer significantly higher precision.
Octet-Singlet Mixing: A precise determination of the octet-singlet mixing component $\bar{\Pi}^{(0,8)}$ is presented, showing a $\sim 2\sigma$ tension with the authors' previous 2022 results, attributed to the inclusion of finer lattice spacings and improved treatment of lattice artifacts.
Rational Approximation: The momentum dependence of the HVP functions is parametrized using Padé approximants, providing a continuous analytic representation for phenomenological applications.

Significance and Claims
The paper claims that this work represents a significant advancement in the precision of first-principles calculations for electroweak observables. By achieving a precision on $\Delta \alpha_{\text{had}}^{(5)}(M_Z^2)$ that surpasses data-driven methods by a factor of two, the authors provide a robust input for global electroweak fits. The persistent tension with data-driven estimates in the space-like region reinforces the discrepancy observed in the muon $g-2$ sector, suggesting potential issues with the experimental $R$ -ratio or the dispersive approach.

While the tension with dispersive results is reduced to a $1-2\sigma$ level at the $Z$ -pole due to the inclusion of the perturbative running, the authors note a mild $1-2\sigma$ deviation from values extracted from global electroweak fits, which typically favor smaller values. The study demonstrates that LQCD can now provide competitive, well-controlled determinations of the hadronic running of electroweak couplings, serving as a critical cross-check for the Standard Model and a probe for new physics. The authors conclude that future improvements will focus on extending the momentum range, incorporating full strong isospin-breaking and QED corrections, and further reducing uncertainties at large Euclidean times.

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