First Lattice Calculation of the QED Corrections to Leptonic Decay Rates.

The leading-order electromagnetic and strong isospin-breaking corrections to the ratio of K_{µ2} and π_{µ2} decay rates are evaluated for the first time on the lattice, following a method recently proposed. The lattice results are obtained using the gauge ensembles produced by the European Twisted Mass Collaboration with N_{f}=2+1+1 dynamical quarks. Systematic effects are evaluated and the impact of the quenched QED approximation is estimated. Our result for the correction to the tree-level K_{µ2}/π_{µ2} decay ratio is -1.22(16)%, to be compared to the estimate of -1.12(21)% based on chiral perturbation theory and adopted by the Particle Data Group.

Review of lattice results concerning low-energy particle physics: Flavour Lattice Averaging Group (FLAG).

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 [Formula: see text], arising in the semileptonic [Formula: see text] transition at zero momentum transfer, as well as the decay constant ratio [Formula: see text] and its consequences for the CKM matrix elements [Formula: see text] and [Formula: see text]. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of [Formula: see text] and [Formula: see text] Chiral Perturbation Theory. We review the determination of the [Formula: see text] 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 [Formula: see text] and [Formula: see text] (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 [Formula: see text].

Standard Model Prediction for Direct CP Violation in K→ππ Decay.

We report the first lattice QCD calculation of the complex kaon decay amplitude A_{0} with physical kinematics, using a 32³×64 lattice volume and a single lattice spacing a, with 1/a=1.3784(68) GeV. We find Re(A_{0})=4.66(1.00)(1.26)×10(-7) GeV and Im(A_{0})=-1.90(1.23)(1.08)×10(-11) GeV, where the first error is statistical and the second systematic. The first value is in approximate agreement with the experimental result: Re(A_{0})=3.3201(18)×10(-7) GeV, while the second can be used to compute the direct CP-violating ratio Re(ϵ^{'}/ϵ)=1.38(5.15)(4.59)×10^{-4}, which is 2.1σ below the experimental value 16.6(2.3)×10(-4). The real part of A_{0} is CP conserving and serves as a test of our method while the result for Re(ϵ^{'}/ϵ) provides a new test of the standard model theory of CP violation, one which can be made more accurate with increasing computer capability.

K(L) - K(S) mass difference from lattice QCD.

We report on the first complete calculation of the K_{L}-K_{S} mass difference, ΔM_{K}, using lattice QCD. The calculation is performed on a 2+1 flavor, domain wall fermion ensemble with a 330 MeV pion mass and a 575 MeV kaon mass. We use a quenched charm quark with a 949 MeV mass to implement Glashow-Iliopoulos-Maiani cancellation. For these heavier-than-physical particle masses, we obtain ΔM_{K}=3.19(41)(96)×10^{-12} MeV, quite similar to the experimental value. Here the first error is statistical, and the second is an estimate of the systematic discretization error. An interesting aspect of this calculation is the importance of the disconnected diagrams, a dramatic failure of the Okubo-Zweig-Iizuka rule.

Review of lattice results concerning low-energy particle physics.

We review lattice results related to pion, kaon, [Formula: see text]- and [Formula: see text]-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 [Formula: see text], arising in semileptonic [Formula: see text] transition at zero momentum transfer, as well as the decay-constant ratio [Formula: see text] of decay constants and its consequences for the CKM matrix elements [Formula: see text] and [Formula: see text]. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of [Formula: see text] and [Formula: see text] Chiral Perturbation Theory and review the determination of the [Formula: see text] 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 [Formula: see text]- and [Formula: see text]-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 [Formula: see text].

Emerging understanding of the ΔI=1/2 rule from lattice QCD.

There has been much speculation as to the origin of the ΔI=1/2 rule (ReA0/ReA2≃22.5). We find that the two dominant contributions to the ΔI=3/2, K→ππ correlation functions have opposite signs, leading to a significant cancelation. This partial cancelation occurs in our computation of ReA2 with physical quark masses and kinematics (where we reproduce the experimental value of A2) and also for heavier pions at threshold. For ReA0, although we do not have results at physical kinematics, we do have results for pions at zero momentum with mπ≃420 MeV [ReA0/ReA2=9.1(2.1)] and mπ≃330 MeV [ReA0/ReA2=12.0(1.7)]. The contributions which partially cancel in ReA2 are also the largest ones in ReA0, but now they have the same sign and so enhance this amplitude. The emerging explanation of the ΔI=1/2 rule is a combination of the perturbative running to scales of O(2 GeV), a relative suppression of ReA2 through the cancelation of the two dominant contributions, and the corresponding enhancement of ReA0. QCD and electroweak penguin operators make only very small contributions at such scales.

K→(ππ)(I=2) decay amplitude from lattice QCD.

We report on the first realistic ab initio calculation of a hadronic weak decay, that of the amplitude A(2) for a kaon to decay into two π mesons with isospin 2. We find ReA(2)=(1.436±0.063(stat)±0.258(syst))10(-8) GeV in good agreement with the experimental result and for the hitherto unknown imaginary part we find ImA(2)=-(6.83±0.51(stat)±1.30(syst))10(-13) GeV. Moreover combining our result for ImA(2) with experimental values of ReA(2), ReA(0), and ε'/ε, we obtain the following value for the unknown ratio ImA(0)/ReA(0) within the standard model: ImA(0)/ReA(0)=-1.63(19)(stat)(20(syst)×10(-4). One consequence of these results is that the contribution from ImA(2) to the direct CP violation parameter ε' (the so-called Electroweak Penguin contribution) is Re(ε'/ε)(EWP)=-(6.52±0.49(stat)±1.24(syst))×10(-4). We explain why this calculation of A(2) represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of CP violation in kaon decays.

η and η' mesons from lattice QCD.

The large mass of the ninth pseudoscalar meson, the η', is believed to arise from the combined effects of the axial anomaly and the gauge field topology present in QCD. We report a realistic, 2+1-flavor, lattice QCD calculation of the η and η' masses and mixing which confirms this picture. The physical eigenstates show small octet-singlet mixing with a mixing angle of θ=-14.1(2.8)°. Extrapolation to the physical light quark mass gives, with statistical errors only, mη=573(6) MeV and mη'=947(142) MeV, consistent with the experimental values of 548 and 958 MeV.

Kl3 semileptonic form factor from (2+1)-flavor lattice QCD.

We present the first results for the K13 form factor from simulations with 2+1 flavors of dynamical domain wall quarks. Combining our result, namely, f+(0)=0.964(5) with the latest experimental results for Kl3 decays leads to |V us|=0.2249(14), reducing the uncertaintity in this important parameter. For the O(p6) term in the chiral expansion we obtain Delta f=-0.013(5).