Eur. Phys. J. C (2022) 82: 869
https://doi.org/10.1140/epjc/s10052-022-10536-1

Review

FLAG Review 2021

¹ RIKEN Center for Computational Science, 650-0047, Kobe, Japan
² Physics Department, University of Connecticut, 06269-3046, Storrs, CT, USA
³ RIKEN BNL Research Center, Brookhaven National Laboratory, 11973, Upton, NY, USA
⁴ Albert Einstein Center for Fundamental Physics, Institut für Theoretische Physik, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland
⁵ Institut für Theoretische Physik, Universität Regensburg, 93040, Regensburg, Germany
⁶ CP3-Origins and IMADA, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark
⁷ Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, 43124, Parma, Italy
⁸ INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/a (Campus), 43124, Parma, Italy
⁹ University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany
¹⁰ Jülich Supercomputing Center, Forschungszentrum Jülich, 52425, Jülich, Germany
¹¹ School of Physics, Peking University, 100871, Beijing, China
¹² Collaborative Innovation Center of Quantum Matter, 100871, Beijing, China
¹³ Center for High Energy Physics, Peking University, 100871, Beijing, China
¹⁴ State Key Laboratory of Nuclear Physics and Technology, Peking University, 100871, Beijing, China
¹⁵ Department of Physics, Osaka University, 560-0043, Toyonaka, Osaka, Japan
¹⁶ Department of Physics and Astronomy, San Francisco State University, 94132, San Francisco, CA, USA
¹⁷ Department of Physics, Indiana University, 47405, Bloomington, IN, USA
¹⁸ Los Alamos National Laboratory, Theoretical Division T-2, 87545, Los Alamos, NM, USA
¹⁹ High Energy Accelerator Research Organization (KEK), 305-0801, Tsukuba, Japan
²⁰ School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), 305-0801, Tsukuba, Japan
²¹ American Physical Society (APS), One Research Road, 11961, Ridge, NY, USA
²² Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain
²³ IFIC (CSIC-UVEG), Parc Científic de la Universitat de València, 46980, Paterna, Spain
²⁴ Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD, Edinburgh, UK
²⁵ School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, UK
²⁶ STAG Research Center, University of Southampton, Highfield, SO17 1BJ, Southampton, UK
²⁷ CERN, Theoretical Physics Department, Geneva, Switzerland
²⁸ Department of Physics, University of Arizona, 85721, Tucson, AZ, USA
²⁹ Department of Physics, The College of William and Mary, 23187, Williamsburg, VA, USA
³⁰ Theory Center, Thomas Jefferson National Accelerator Facility, 23606, Newport News, VA, USA
³¹ Department of Physics and Astronomy, University of North Carolina, 27516-3255, Chapel Hill, NC, USA
³² Physics Department, Brookhaven National Laboratory, 11973, Upton, NY, USA
³³ Physics Department, University of Washington, 98195-1560, Seattle, WA, USA
³⁴ Fermi National Accelerator Laboratory, 60510, Batavia, IL, USA
³⁵ INFN, Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Rome, Italy
³⁶ School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
³⁷ John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, 15738, Zeuthen, Germany
³⁸ Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489, Berlin, Germany
³⁹ INFN, Sezione di Tor Vergata, c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Rome, Italy
⁴⁰ PRISMA Cluster of Excellence, Institut für Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099, Mainz, Germany

^u a.juttner@soton.ac.uk

Received: 3 February 2022
Accepted: 31 May 2022
Published online: 4 October 2022

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 $K\to \pi$ transition at zero momentum transfer, as well as the decay constant ratio $f_K/f_{\pi }$ and its consequences for the CKM matrix elements $V_{\mathit{us}}$ and $V_{\mathit{ud}}$. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $SU{(2)}_L\times SU{(2)}_R$ and $SU{(3)}_L\times SU{(3)}_R$ Chiral Perturbation Theory. 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 ${\alpha }_s$. We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities.