Eur. Phys. J. C (2020) 80: 113
https://doi.org/10.1140/epjc/s10052-019-7354-7

Review

FLAG Review 2019

Flavour Lattice Averaging Group (FLAG)

¹ Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, 606-8502, Sakyo-ku, Kyoto, Japan
² High Energy Accelerator Research Organization (KEK), 305-0801, Tsukuba, Japan
³ RIKEN BNL Research Center, Brookhaven National Laboratory, 11973, Upton, NY, USA
⁴ Laboratoire de Physique Théorique (UMR8627), CNRS, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France
⁵ Physics Department, University of Connecticut, 06269-3046, Storrs, CT, USA
⁶ Albert Einstein Center for Fundamental Physics, Institut für Theoretische Physik, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland
⁷ Institut für Theoretische Physik, Universität Regensburg, 93040, Regensburg, Germany
⁸ CP3-Origins and IMADA, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark
⁹ Centro Fermi – Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Compendio del Viminale, Piazza del Viminiale 1, 00184, Rome, Italy
¹⁰ c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Rome, Italy
¹¹ University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany
¹² Jülich Supercomputing Center, Forschungszentrum Jülich, 52425, Jülich, Germany
¹³ Department of Physics, Osaka University, 560-0043, Toyonaka, Osaka, Japan
¹⁴ Dept. of Physics and Astronomy, San Francisco State University, 94132, San Francisco, CA, USA
¹⁵ Department of Physics, Indiana University, 47405, Bloomington, IN, USA
¹⁶ Los Alamos National Laboratory, Theoretical Division T-2, 87545, Los Alamos, NM, USA
¹⁷ School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), 305-0801, Tsukuba, Japan
¹⁸ American Physical Society (APS), One Research Road, 11961, Ridge, NY, USA
¹⁹ Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049, Cantoblanco, Madrid, Spain
²⁰ Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD, Edinburgh, UK
²¹ School of Physics & Astronomy, University of Southampton, SO17 1BJ, Southampton, UK
²² Institute of Physics, National Chiao-Tung University, 30010, Hsinchu, Taiwan
²³ Centre for High Energy Physics, Chung-Yuan Christian University, 32023, Chung-Li, Taiwan
²⁴ Physics Department, Columbia University, 10027, New York, NY, USA
²⁵ Dept. of Physics and Astronomy, University of North Carolina, 27516-3255, Chapel Hill, NC, USA
²⁶ School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
²⁷ Physics Department, University of Washington, 98195-1560, Seattle, WA, USA
²⁸ Fermi National Accelerator Laboratory, 60510, Batavia, IL, USA
²⁹ Sezione di Roma Tre, INFN, Via della Vasca Navale 84, 00146, Rome, Italy
³⁰ John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, 15738, Zeuthen, Germany
³¹ Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489, Berlin, Germany
³² Sezione di Tor Vergata, INFN, c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Rome, Italy
³³ PRISMA Cluster of Excellence, Institut für Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099, Mainz, Germany
³⁴ RIKEN Center for Computational Science, 650-0047, Kobe, Japan

^u a.juttner@soton.ac.uk

Received: 26 March 2019
Accepted: 5 September 2019
Published online: 11 February 2020

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 D- and B-meson decay constants, form factors, and mixing parameters. 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 ${\alpha }_s$. Finally, in this review we have added a new section reviewing results for nucleon matrix elements of the axial, scalar and tensor bilinears, both isovector and flavor diagonal.