FLAG Review 2021

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.