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Open Access

FLAG review 2024

Y. Aoki ¹, T. Blum^2,3, S. Collins ⁴, L. Del Debbio⁵, M. Della Morte ⁶, P. Dimopoulos^7,8, X. Feng^9,10,11,12, M. Golterman¹³, Steven Gottlieb ¹⁴ et al. (Flavour Lattice Averaging Group (FLAG) )

Steven Gottlieb ¹⁴, R. Gupta¹⁵, G. Herdoíza¹⁶, P. Hernandez¹⁷, A. Jüttner ^18,19,20, T. Kaneko^21,22, E. Lunghi¹⁴, S. Meinel ²³, C. Monahan ^24,25, A. Nicholson²⁶, T. Onogi²⁷, P. Petreczky²⁸, A. Portelli ^1,5,20, A. Ramos¹⁷, S. R. Sharpe²⁹, J. N. Simone^30,31, S. Sint³¹, R. Sommer^32,33, N. Tantalo³⁴, R. Van de Water³⁰, A. Vaquero ^35,36, U. Wenger ³⁷, and H. Wittig^38,39 (Flavour Lattice Averaging Group (FLAG) )

¹RIKEN Center for Computational Science, Kobe 650-0047, Japan
²Physics Department, University of Connecticut, Storrs, Connecticut 06269-3046, USA
³RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA
⁴Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany
⁵Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
⁶IMADA, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
⁷Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, 43124 Parma, Italy
⁸INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/a (Campus), 43124 Parma, Italy
⁹School of Physics, Peking University, Beijing 100871, China
¹⁰Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
¹¹Center for High Energy Physics, Peking University, Beijing 100871, China
¹²State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
¹³Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA
¹⁴Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
¹⁵Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, New Mexico 87545, USA
¹⁶Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco 28049 Madrid, Spain
¹⁷IFIC (CSIC-UVEG), Parc Científic de la Universitat de València, E-46980 Paterna, Spain
¹⁸School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
¹⁹STAG Research Center, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
²⁰CERN, Theoretical Physics Department, Geneva, Switzerland
²¹High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan
²²Graduate Institute for Advanced Studies, SOKENDAI (The Graduate University for Advanced Studies), Ibaraki 305-0801, Japan
²³Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
²⁴Department of Physics, The College of William & Mary, Williamsburg, Virginia 23187, USA
²⁵Department of Physics, Colorado College, Colorado Springs, Colorado 80903, USA
²⁶Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27516-3255, USA
²⁷Department of Physics, The University of Osaka, Toyonaka 560-0043, Japan
²⁸Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
²⁹Physics Department, University of Washington, Seattle, Washington 98195-1560, USA
³⁰Fermi National Accelerator Laboratory, Batavia, Illinois 60510 USA
³¹School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
³²Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
³³Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
³⁴INFN, Sezione di Tor Vergata, c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy
³⁵Departamento de Física Teórica, Universidad de Zaragoza, Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain
³⁶Center for Astroparticles and High Energy Physics (CAPA), Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain
³⁷Albert Einstein Center for Fundamental Physics, Institut für Theoretische Physik, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
³⁸PRISMA Cluster of Excellence, Institut für Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099 Mainz, Germany
³⁹Helmholtz Institute Mainz and GSI Helmholtz Center for Heavy Ion Research, 64291 Darmstadt, Germany

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Phys. Rev. D 113, 014508 – Published 30 January, 2026

DOI: https://doi.org/10.1103/nfzp-p5dn

Abstract

We review lattice results related to pion, kaon, $D$ -meson, $B$ -meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_{+} (0)$ arising in the semileptonic $K \to π$ transition at zero momentum transfer, as well as the decay-constant ratio $f_{K} / f_{π}$ and its consequences for the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements $V_{u s}$ and $V_{u d}$ . We review the determination of the $B_{K}$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_{c}$ and $m_{b}$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant $α_{s}$ . We review the determinations of nucleon charges from the matrix elements of both isovector and flavor-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model.

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Article Text

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Phys. Rev. D 113, 014508– Published 30 January, 2026

Vol. 113, Iss. 1 — 1 January 2026

Received 11 February 2025
Accepted 27 June 2025

DOI: https://doi.org/10.1103/nfzp-p5dn

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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