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

Alessandro Conigli ^1,2,*, Dalibor Djukanovic ^1,2, Georg von Hippel ³, Simon Kuberski ⁴, Harvey B. Meyer ^1,3,4, Kohtaroh Miura⁵, Konstantin Ottnad ³, Andreas Risch ⁶, and Hartmut Wittig ^1,2,3

¹Helmholtz Institute Mainz, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
²GSI Helmholtz Centre for Heavy Ion Research, 64291 Darmstadt, Germany
³PRISMA⁺⁺ Cluster of Excellence and Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
⁴Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
⁵Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
⁶Department of Physics, University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany

^*Contact author: aconigli@uni-mainz.de

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Phys. Rev. D 114, 034508 – Published 13 August, 2026

DOI: https://doi.org/10.1103/bk78-wtvd

Abstract

We present an update of our lattice QCD determination of the hadronic contribution to the running of the electromagnetic coupling, $Δ α_{had}^{(5)} (- Q^{2})$ , and of the electroweak mixing angle in the spacelike momentum region up to $Q^{2} = 12 {GeV}^{2}$ . The calculation is based on coordinated lattice simulations ensembles with $N_{f} = 2 + 1$ flavors of $O (a)$ -improved Wilson fermions, covering five lattice spacings between 0.039 and 0.085 fm and a range of pion masses, including the physical point. A refined analysis employing a telescopic window strategy allows for a clean separation of systematic effects across Euclidean distance scales. Statistical precision is further enhanced through low-mode averaging, combined with a spectral reconstruction of the vector-vector correlator at long distances on the most chiral ensembles. We confirm significant tensions of up to $7 σ$ at spacelike virtualities around $Q^{2} = 1 {GeV}^{2}$ between our lattice results for $Δ α_{had}^{(5)} (- Q^{2})$ and the corresponding data-driven estimates based on $e^{+} e^{-}$ cross section data. Combining our lattice data with perturbative QCD via the Euclidean split technique, we obtain at the $Z$ pole $Δ α_{had}^{(5)} (M_{Z}^{2}) = 0.027821 (34)_{lat} (35)_{pQCD}$ , which is more than two times more precise than recent data-driven estimates. Our result deviatesslightly, by $1 - 2 σ$ , from the value produced by global electroweak fits. For the electroweak mixing angle, we present the hadronic contribution to its running and provide a precise determination of the octet-singlet mixing component ${\bar{Π}}^{(0, 8)}$ , in good agreement with phenomenological models but with significantly higher precision.

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Phys. Rev. D 114, 034508– Published 13 August, 2026

Vol. 114, Iss. 3 — 1 August 2026

Received 14 January 2026
Accepted 6 July 2026

DOI: https://doi.org/10.1103/bk78-wtvd

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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

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Running of the Electroweak Gauge Couplings from First Principles

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