FLAG Review 2021

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 →π transition at zero momentum transfer, as well as the decay constant ratio f_K/f_π and its consequences for the CKM matrix elements V_us and V_ud . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)_L× SU(2)_R and SU(3)_L× 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 α _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.