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

The axial coupling of the nucleon, g, is the strength of its coupling to the weak axial current of the standard model of particle physics, in much the same way as the electric charge is the strength of the coupling to the electromagnetic current. This axial coupling dictates the rate at which neutrons decay to protons, the strength of the attractive long-range force between nucleons and other features of nuclear physics. Precision tests of the standard model in nuclear environments require a quantitative understanding of nuclear physics that is rooted in quantum chromodynamics, a pillar of the standard model. The importance of g makes it a benchmark quantity to determine theoretically-a difficult task because quantum chromodynamics is non-perturbative, precluding known analytical methods. Lattice quantum chromodynamics provides a rigorous, non-perturbative definition of quantum chromodynamics that can be implemented numerically. It has been estimated that a precision of two per cent would be possible by 2020 if two challenges are overcome: contamination of g from excited states must be controlled in the calculations and statistical precision must be improved markedly. Here we use an unconventional method inspired by the Feynman-Hellmann theorem that overcomes these challenges. We calculate a g value of 1.271 ± 0.013, which has a precision of about one per cent.

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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.

We present physical results for a variety of light hadronic quantities obtained via a combined analysis of three 2+1 flavour domain wall fermion ensemble sets. For two of our ensemble sets we used the Iwasaki gauge action with β = 2.13 (a −1 = 1.75(4) GeV) and β = 2.25 (a −1 = 2.31(4) We also obtain values for the SU(2) chiral perturbation theory effective couplings,l 3 = 2.91(23) stat (7) sys andl 4 = 3.99(16) stat (9) sys .GeV3

We describe the computation of the amplitude A 2 for a kaon to decay into two pions with isospin I = 2. The results presented in the letter [1] from an analysis of 63 gluon configurations are updated to 146 configurations giving Re A 2 = 1.381(46) stat (258) syst 10 −8 GeV and Im A 2 = −6.54(46) stat (120) syst 10 −13 GeV . Re A 2 is in good agreement with the experimental result, whereas the value of Im A 2 was hitherto unknown. We are also working towards a direct computation of the K → (ππ) I=0 amplitude A 0 but, within the standard model, our result for Im A 2 can be combined with the experimental results for Re A 0 , Re A 2 and ε /ε to give Im A 0 /Re A 0 = −1.61(28) × 10 −4 . Our result for Im A 2 implies that the electroweak penguin (EWP) contribution to ε /ε is Re(ε /ε) EWP = −(6.25 ± 0.44 stat ± 1.19 syst ) × 10 −4 . 2

Abstract:We compute the hadronic matrix elements of the four-quark operators relevant for K 0 −K 0 mixing beyond the Standard Model. Our results are from lattice QCD simulations with n f = 2 + 1 flavours of domain-wall fermion, which exhibit continuum-like chiral-flavour symmetry. The simulations are performed at two different values of the lattice spacing (a ∼ 0.08 and a ∼ 0.11 fm) and with lightest unitary pion mass ∼ 300 MeV. For the first time, the full set of relevant four-quark operators is renormalised non-perturbatively through RI-SMOM schemes; a detailed description of the renormalisation procedure is presented in a companion paper. We argue that the intermediate renormalisation scheme is responsible for the discrepancies found by different collaborations. We also study different normalisations and determine the matrix elements of the relevant four-quark operators with a precision of ∼ 5% or better.