We review lattice results related to pion, kaon, - and -meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor , arising in semileptonic transition at zero momentum transfer, as well as the decay-constant ratio of decay constants and its consequences for the CKM matrix elements and . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of and Chiral Perturbation Theory and review the determination of the parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on - and -meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant .

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor , arising in the semileptonic transition at zero momentum transfer, as well as the decay constant ratio and its consequences for the CKM matrix elements and . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of and Chiral Perturbation Theory. We review the determination of the parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for and (also new compared to the previous review), 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. Finally, we review the status of lattice determinations of the strong coupling constant .

We review lattice results related to pion, kaon, D-and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the lightquark masses, the form factor f + (0), arising in semileptonic K → π transition at zero momentum transfer, as well as the decay-constant ratio f K / f π of decay constants 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 and review the determination of the B K parameter of neutral kaon mixa e-mail: gilberto@itp.unibe.ch ing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on D-and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant α s .

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 $$f_+(0)$$f+(0) arising in the semileptonic $$K \rightarrow \pi $$K→π transition at zero momentum transfer, as well as the decay constant ratio $$f_K/f_\pi $$fK/fπ and its consequences for the CKM matrix elements $$V_{us}$$Vus and $$V_{ud}$$Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of $$SU(2)_L\times SU(2)_R$$SU(2)L×SU(2)R and $$SU(3)_L\times SU(3)_R$$SU(3)L×SU(3)R Chiral Perturbation Theory. We review the determination of the $$B_K$$BK 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 $$m_c$$mc and $$m_b$$mb 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 $$\alpha _s$$αs. 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.

We present a quenched lattice calculation of the lowest order (O(α 2 )) hadronic contribution to the anomalous magnetic moment of the muon which arises from the hadronic vacuum polarization.A general method is presented for computing entirely in Euclidean space, obviating the need for the usual dispersive treatment which relies on experimental data for e + e − annihilation to hadrons.While the result is not yet of comparable precision to those state-of-the-art calculations, systematic improvement of the quenched lattice computation to this level is straightforward and well within the reach of present computers. Including the effects of dynamical quarks is conceptually trivial, the computer resources required are not.

We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing a, (a −1 = 1.73 (3) GeV and a −1 = 2.28 (3) GeV). On the coarser lattice, with 24 3 × 64 × 16 points (where the 16 corresponds to L s , the extent of the 5 th dimension inherent in the domain wall fermion (DWF) formulation

We present results of our numerical calculation of the mass spectrum for isospin one-half and spin one-half nonstrange baryons, i.e., the ground and excited states of the nucleon, in quenched lattice QCD. We use a new lattice discretization scheme for fermions, domain wall fermions, which possess almost exact chiral symmetry at nonzero lattice spacing. We make a systematic investigation of the negative-parity N* spectrum by using two distinct interpolating operators at ␤ϭ6/g 2 ϭ6.0 on a 16 3 ϫ32ϫ16 lattice. The mass estimates extracted from the two operators are consistent with each other. The observed large mass splitting between this state, N*(1535), and the positive-parity ground state, the nucleon N(939), is well reproduced by our calculations. We have also calculated the mass of the first positive-parity excited state and find that it is heavier than the negative-parity excited state for the quark masses studied.

2 Results computed in lattice QCD+QED are presented for the electromagnetic mass splittings of the low lying hadrons. These are used to determine the renormalized, non-degenerate, light quark masses. It is found that m M S u = 2.24 (10) (34), m M S d = 4.65 (15) (32), and m M S s = 97.6 (2.9) (5.5) MeV at the renormalization scale 2 GeV, where the first error is statistical and the second systematic.We find the lowest order electromagnetic splitting (m π + − m π 0 ) QED = 3.38 (23) MeV, the splittings including next-to-leading order, (m π + − m π 0 ) QED = 4.50 (23) MeV, (m K + − m K 0 ) QED = 1.87(10) MeV, and the m u = m d contribution to the kaon mass difference, (m K + − m K 0 ) (mu−m d ) = −5.840(96) MeV. All errors are statistical only, and the next-to-leading order pion splitting is only approximate in that it does not contain all next-to-leading order contributions. We also computed the proton-neutron mass difference, including for the first time, QED interactions in a realistic 2+1 flavor calculation. We find (m p − m n ) QED = 0.383 (68) MeV, (m p − m n ) (mu−m d ) = −2.51(14) MeV (statistical errors only), and the total m p − m n = −2.13(16)(70) MeV, where the first error is statistical, and the second, part of the systematic error. The calculations are carried out on QCD ensembles generated by the RBC and UKQCD collaborations, using domain wall fermions and the Iwasaki gauge action (gauge coupling β = 2.13 and lattice cutoff a −1 ≈ 1.78 GeV). We use two lattice sizes, 16 3 and 24 3 ( (1.8 fm) 3 and (2.7 fm) 3 ), to address finite volume effects. Non-compact QED is treated in the quenched approximation.The valence pseudo-scalar meson masses in our study cover a range of about 250 to 700 MeV, though we use only those up to about 400 MeV to quote final results.We present new results for the electromagnetic low energy constants in SU(3) and SU(2) partially-quenched chiral perturbation theory to the next-to-leading order, obtained from fits to our data. Detailed analysis of systematic errors in our results and methods for improving them are discussed. Finally, new analytic results for SU(2) L × SU(2) R -plus-kaon chiral perturbation theory, including the one-loop logs proportional to α em m, are given. 3