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We present results for several light hadronic quantities ($f_\pi$, $f_K$, $B_K$, $m_{ud}$, $m_s$, $t_0^{1/2}$, $w_0$) obtained from simulations of 2+1 flavor domain wall lattice QCD with large physical volumes and nearly-physical pion masses at two lattice spacings. We perform a short, O(3)%, extrapolation in pion mass to the physical values by combining our new data in a simultaneous chiral/continuum `global fit' with a number of other ensembles with heavier pion masses. We use the physical values of $m_\pi$, $m_K$ and $m_\Omega$ to determine the two quark masses and the scale - all other quantities are outputs from our simulations. We obtain results with sub-percent statistical errors and negligible chiral and finite-volume systematics for these light hadronic quantities, including: $f_\pi$ = 130.2(9) MeV; $f_K$ = 155.5(8) MeV; the average up/down quark mass and strange quark mass in the $\bar {\rm MS}$ scheme at 3 GeV, 2.997(49) and 81.64(1.17) MeV respectively; and the neutral kaon mixing parameter, $B_K$, in the RGI scheme, 0.750(15) and the $\bar{\rm MS}$ scheme at 3 GeV, 0.530(11).Comment: 131 pages, 30 figures. Updated to match published versio

We present a first-principles lattice QCD þ QED calculation at physical pion mass of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. The total contribution of up, down, strange, and charm quarks including QED and strong isospin breaking effects is a

We have simulated QCD using 2 þ 1 flavors of domain wall quarks and the Iwasaki gauge action on a ð2:74 fmÞ 3 volume with an inverse lattice scale of a À1 ¼ 1:729ð28Þ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617, and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter B K , and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulas from both approaches fit our data for light quarks, we find the higher-order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the baryon, and the and K mesons to set the lattice scale and determine the quark masses. We then find f ¼ 124:1ð3:6Þ stat  ð6:9Þ syst MeV, f K ¼ 149:6ð3:6Þ stat ð6:3Þ syst MeV, and f K =f ¼ 1:205ð0:018Þ stat ð0:062Þ syst . Using nonperturbative renormalization to relate lattice regularized quark masses to regularization independent momentum scheme masses, and perturbation theory to relate these to MS, we find m MS ud ð2 GeVÞ ¼ 3:72ð0:16Þ stat ð0:33Þ ren ð0:18Þ syst MeV, m MS s ð2 GeVÞ ¼ 107:3ð4:4Þ stat ð9:7Þ ren ð4:9Þ syst MeV, and mud : ms ¼ 1:28:8ð0:4Þ stat ð1:6Þ syst . For the kaon bag parameter, we find B MS K ð2 GeVÞ ¼ 0:524ð0:010Þ stat ð0:013Þ ren  ð0:025Þ syst . Finally, for the ratios of the couplings of the vector mesons to the vector and tensor currents (f V and f T V , respectively) in the MS scheme at 2 GeV we obtain f T =f ¼ 0:687ð27Þ; f T K à =f K à ¼ 0:712ð12Þ, and f T =f ¼ 0:750ð8Þ.

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We present a general class of unbiased improved estimators for physical observables in lattice gauge theory computations which significantly reduces statistical errors at modest computational cost. The idea can be easily adapted to other branches of physics and computational science that employ Monte Carlo methods. The error reduction techniques, referred to as covariant approximation averaging, utilize approximations which are covariant under lattice symmetry transformations. We observe cost reductions from the new method compared to the traditional one, for fixed statistical error, of 16 times for the nucleon mass at M $ 330 MeV (domain-wall quark) and 2.6-20 times for the hadronic vacuum polarization at M $ 315 MeV (Asqtad quark). These cost reductions should improve with decreasing quark mass and increasing lattice sizes.As nonperturbative computations using lattice gauge theory are applied to a wider range of physically interesting observables, it is increasingly important to find numerical strategies that provide precise results. In Monte Carlo simulations our reach to important physics is still often limited by statistical uncertainties. Examples include hadronic contributions to the muon's anomalous magnetic moment [1], nucleon form factors and structure functions [2], including nucleon electric dipole moments [3-6], hadron matrix elements relevant to flavor physics (e.g., K ! amplitudes) [7], and multihadron state physics [8], to name only a few. In addition, there are many examples of Monte Carlo simulation applied to condensed matter physics [9,10], many-body systems [11,12] and cold gases [13]. As a generalization of low-mode averaging (LMA) [14,15], we present a class of unbiased statistical error reduction techniques, utilizing approximations that are covariant under lattice symmetry transformations. LMA has worked well in cases where low eigenmodes of the Dirac operator dominate [16]: low energy constants in the " regime [14,17-20], pseudoscalar meson masses, decay constants [21-23], and so on. With a modest increase in computational cost, the generalized method can reduce statistical errors by an order of magnitude, or more, even in cases where LMA fails. Unlike LMA, we account for all modes of the Dirac operator, averaging over (most of) the lattice volume, with modest additional computational cost. The all-to-all methods [24,25] implement this stochastically for the higher modes, while treating the low modes exactly. For expectation values invariant under translations, statistics effectively increase by averaging over the whole lattice.The all-to-all method is advantageous when the stochastic noise introduced in the target observable is comparable to, or smaller than, the gauge field fluctuations of the ensemble [26], which typically holds only for many random source vectors per measurement. The error reduction techniques presented here, which do not rely on stochastic noise, are potentially more effective, provided an inexpensive approximation can be found for the desired observable.In lattice ...

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We report on the first realistic ab initio calculation of a hadronic weak decay, that of the amplitude A2 for a kaon to decay into two π-mesons with isospin 2. We find Re A2 = (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 Im A2 = −(6.83±0.51 stat ±1.30 syst ) 10 −13 GeV. Moreover combining our result for Im A2 with experimental values of Re A2, Re A0 and ǫ ′ /ǫ, we obtain the following value for the unknown ratio Im A0/Re A0 within the Standard Model: Im A0/Re A0 = −1.63(19)stat(20)syst × 10 −4 . One consequence of these results is that the contribution from Im A2 to the direct CP violation parameter ǫ ′ (the so-called Electroweak Penguin, EWP, contribution) is Re(ǫ ′ /ǫ)EWP = −(6.52 ± 0.49 stat ± 1.24 syst ) × 10 −4 . We explain why this calculation of A2 represents a major milestone for lattice QCD and discuss the exciting prospects for a full quantitative understanding of CP-violation in kaon decays.