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The running of the electroweak gauge couplings from first principles

This paper presents a high-precision, first-principles lattice QCD calculation of the hadronic running of electroweak gauge couplings that achieves permille-level accuracy for low virtualities and yields a significantly more precise determination of the electromagnetic coupling at the Z-boson mass scale compared to recent phenomenological estimates.

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

Published 2026-07-07

📖 4 min read🧠 Deep dive

CC BY 4.0

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

Imagine the universe is built on a set of fundamental rules, like the laws of physics in a giant video game. Two of the most important "settings" in this game are the strength of the electromagnetic force (how light and electricity work) and the "weak mixing angle" (which governs how particles decay). Scientists call these the "couplings."

However, these settings aren't fixed numbers. They change depending on how much energy you are looking at, much like how a rubber band feels stiffer when you stretch it further. This change is called "running."

The problem is that calculating exactly how these settings change is incredibly difficult because of the "hadronic" part. Hadrons are particles made of quarks (like protons and neutrons), and they are messy, chaotic, and don't follow simple math rules. For decades, scientists tried to guess this messy part by looking at data from particle collisions (like watching the debris after a crash to guess what the car looked like). But recently, different teams got different answers from this debris, creating a confusing disagreement.

What This Paper Does
Instead of looking at the crash debris, the authors in this paper decided to build the car from scratch, piece by piece, using a method called Lattice QCD.

Think of the universe as a giant, 3D grid (like a massive chessboard made of space and time). The researchers used supercomputers to simulate the laws of physics on this grid. They didn't guess; they calculated the behavior of the quarks and gluons (the tiny building blocks) directly from the fundamental equations of nature.

The Key Findings

A New, Sharper Picture: By using this "from scratch" method, they calculated how the electromagnetic force changes with energy with extreme precision. Their result is about twice as precise as the best previous estimates that relied on experimental data.
The Disagreement: When they compared their "from scratch" calculation to the old "debris analysis" (data from electron-positron collisions), they found a massive clash. At certain energy levels, their numbers were off by 7 standard deviations. In the world of science, this is like flipping a coin 10 times and getting heads every single time, then being told it's a fair coin. It suggests the old data-based method might be missing something important.
The Final Score: They combined their precise low-energy grid calculation with high-energy math rules (perturbative QCD) to give a final, ultra-precise value for the electromagnetic coupling at the "Z-pole" (a specific high-energy point used in particle physics). Their new value is 0.027821, with a tiny margin of error.

Looking Ahead
The paper also acts as a roadmap for the future. It asks: "How do we get even more precise?"

They created a model to see what it would take to reach the precision needed for next-generation particle colliders (like the proposed FCC-ee). They found that simply doing more of the same isn't enough. To win, they need a two-pronged approach:

Better Grids: Improve the "from scratch" calculations by about 50%.
Higher Energy: Push the calculations to higher energy levels on the grid.

If they do both, they can reach a level of precision that will allow future experiments to spot new, hidden physics that we can't see yet.

In Summary
This paper is a triumph of "first principles" computing. Instead of relying on messy experimental data that seems to be giving conflicting answers, the authors built a digital simulation of the universe's fundamental forces. They found that the old way of measuring these forces might be wrong, and they provided a much more accurate map for future scientists to use as they search for new physics.

Technical Summary: The Running of Electroweak Gauge Couplings from First Principles

Problem Statement
The electromagnetic coupling $\alpha(M_Z^2)$ and the weak mixing angle $\sin^2\theta_W$ are fundamental parameters of the Standard Model (SM) critical for interpreting high-energy collider data and searching for Beyond-Standard Model (BSM) physics. The precision of these parameters is currently limited by the hadronic contribution to their running, denoted as $\Delta\alpha^{(5)}_{\text{had}}$ . Traditional determinations rely on dispersive relations using experimental $e^+e^- \to \text{hadrons}$ cross-section data. However, these data-driven approaches suffer from unresolved tensions, particularly in the $\pi^+\pi^-$ channel, and model-dependent flavor separations for the weak mixing angle. These uncertainties hinder the sensitivity of future precision experiments, such as those planned at the FCC-ee, which aim for absolute precisions on $\alpha(M_Z^2)$ in the range of $(3-5) \times 10^{-5}$ .

Methodology
This work presents a high-precision, first-principles calculation of the hadronic vacuum polarization (HVP) using Lattice QCD. The authors employ the following methodological framework:

Time-Momentum Representation (TMR): The subtracted HVP, $\bar{\Pi}(q^2)$ , is computed via the Euclidean vector correlator $G(x_0)$ using a kernel integral. This approach directly encodes the HVP without relying on experimental $R$ -ratios.
Telescopic Window Decomposition: To optimize chiral-continuum extrapolations and isolate physics at different Euclidean distances, the HVP is decomposed into three kinematic regions: High-Virtuality (HV, short-distance), Mid-Virtuality (MV, intermediate), and Low-Virtuality (LV, long-distance). This allows for tailored treatment of discretization effects and noise reduction.
Lattice Ensembles: The analysis utilizes 27 CLS (Coordinated Lattice Simulations) ensembles with lattice spacings $a \in [0.039, 0.085]$ fm and pion masses ranging down to the physical point.
Noise Reduction and Systematics: Long-distance noise is mitigated using low-mode averaging and spectral reconstruction. Finite-volume effects are corrected using a hybrid Hansen-Patella and Meyer-Lellouch-Lüscher framework. Systematic uncertainties are estimated via model averaging (using the Akaike Information Criterion) over various fit ansätze and ensemble cuts.
Isospin Breaking: Corrections for isospin breaking are estimated by combining lattice results for quark-connected contributions with an independent phenomenological estimate based on specific models.
Euclidean Split Technique: To obtain the value at the Z-pole ( $M_Z^2$ ), the lattice results at space-like momenta ( $Q^2 \lesssim 12 \text{ GeV}^2$ ) are matched with perturbative QCD (pQCD) at a scale $Q_0^2 = 9 \text{ GeV}^2$ . The perturbative evolution is handled using the AdlerPy package (cross-checked with an updated pQCDAdler code).

Key Contributions and Results

Precision Determination: The authors calculate the running of $\alpha$ and $\sin^2\theta_W$ for virtualities $0.25 \text{ GeV}^2 \leq Q^2 \leq 12 \text{ GeV}^2$ . The final result for the hadronic contribution to the running of the electromagnetic coupling at the Z-pole is:
$\Delta\alpha^{(5)}_{\text{had}}(M_Z^2) = 0.027821(34)_{\text{lat}}(35)_{\text{pQCD}}$
This yields a total relative uncertainty of 0.17% (1.7‰), representing an improvement by a factor of approximately two over recent phenomenological determinations.
Discrepancy with Data-Driven Estimates: At $Q^2 \simeq 1 \text{ GeV}^2$ , the lattice determination deviates from estimates based on $e^+e^-$ measurements by up to 7 $\sigma$ . While this tension decreases at higher virtualities, it remains significant (approx. 4.5 $\sigma$ ) at $Q^2 = 9 \text{ GeV}^2$ . At the Z-pole, the discrepancy with dispersive results is reduced to the 1–2 $\sigma$ level due to the additional uncertainty introduced by the perturbative running.
Weak Mixing Angle: The paper provides a precise lattice determination of the octet-singlet mixing contribution, finding it approaches a constant at large virtualities consistent with phenomenological models but with significantly higher precision.
Future Improvement Scenarios: The authors construct a parametric model to assess the path toward the FCC-ee target precision of $3 \times 10^{-5}$ $3 \times 1 0^{- 5}$ . They find that:
- Increasing the matching scale $Q_0^2$ alone yields diminishing returns.
- Achieving the FCC-ee target requires a combination of moderate lattice improvements (reducing lattice uncertainty by $\geq 50\%$ ) and an increased matching scale ( $Q_0^2 \sim 20 \text{ GeV}^2$ ).
- Reaching the ambitious statistical sensitivity of $10^{-5}$ (proposed in recent literature) would require a five-fold improvement in both lattice and pQCD inputs, alongside reduced truncation uncertainties.

Significance
The paper claims to provide a robust, first-principles input for future electroweak precision tests, significantly reducing the dominant non-perturbative uncertainty in the determination of $\alpha(M_Z^2)$ . By achieving per-mille precision for virtualities up to $12 \text{ GeV}^2$ , the work offers a critical alternative to data-driven methods, particularly in regions where experimental data shows tensions. The results serve as a benchmark for interpreting future collider data and probing BSM physics. The authors emphasize that while their current precision already surpasses phenomenological estimates, further progress in both lattice QCD and perturbative QCD inputs is necessary to fully exploit the physics potential of next-generation facilities like the FCC-ee.

Technical Summary: The Running of Electroweak Gauge Couplings from First Principles

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