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Lattice calculation of the $π^{0}$ , $η$ and $η^{'}$ transition form factors and the hadronic light-by-light contribution to the muon $g - 2$

Antoine Gérardin ^1,*, Willem E. A. Verplanke ^1,†, Gen Wang ¹, Zoltan Fodor^2,3,4,5, Jana N. Guenther², Laurent Lellouch ¹, Kalman K. Szabo^2,3, and Lukas Varnhorst²

¹Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille 13009, France
²Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
³Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52428 Jülich, Germany
⁴Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary
⁵Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA

^*Contact author: antoine.gerardin@cpt.univ-mrs.fr
^†Contact author: willem.verplanke@cpt.univ-mrs.fr

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Phys. Rev. D 111, 054511 – Published 21 March, 2025

DOI: https://doi.org/10.1103/PhysRevD.111.054511

Abstract

In this paper we present a first ab initio calculation of the $π^{0}$ , $η$ and $η^{'}$ transition form factors performed with physical light-quark masses. We provide a complete parametrization of the form factors that includes both single and double-virtual kinematics. Our results are compared with experimental measurements of the form factors in the spacelike region and with the measured two-photon decay widths. In a second step, our parametrizations of the transition form factors are used to compute the dominant pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon $g - 2$ . Our final result reads $a_{μ}^{hlbl, ps-pole} = (85.1 \pm 5.2) \times 10^{- 11}$ . Although the pion-pole is dominant, we confirm that, together, the $η$ and $η^{'}$ provide roughly half of its contribution.

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Phys. Rev. D 111, 054511– Published 21 March, 2025

Vol. 111, Iss. 5 — 1 March 2025

Received 20 June 2023
Accepted 28 February 2025

DOI: https://doi.org/10.1103/PhysRevD.111.054511

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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Lattice calculation of the $π^{0}$ , $η$ and $η^{'}$ transition form factors and the hadronic light-by-light contribution to the muon $g - 2$

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Lattice calculation of the π0, η and η′ transition form factors and the hadronic light-by-light contribution to the muon g−2

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Lattice calculation of the $π^{0}$ , $η$ and $η^{'}$ transition form factors and the hadronic light-by-light contribution to the muon $g - 2$