Scale setting is of central importance in lattice QCD. It is required to predict dimensional quantities in physical units. Moreover, it determines the relative lattice spacings of computations performed at different values of the bare coupling, and this is needed for extrapolating results into the continuum. Thus, we calculate a new quantity, w ₀, for setting the scale in lattice QCD, which is based on the Wilson flow like the scale t ₀ (M. Luscher, JHEP 08 (2010) 071). It is cheap and straightforward to implement and compute. In particular, it does not involve the delicate fitting of correlation functions at asymptotic times. It typically can be determined on the few per-mil level. We compute its continuum extrapolated value in 2 + 1-flavor QCD for physical and non-physical pion and kaon masses, to allow for mass-independent scale setting even away from the physical mass point. We demonstrate its robustness by computing it with two very different actions (one of them with staggered, the other with Wilson fermions) and by showing that the results agree for physical quark masses in the continuum limit.

We extend our earlier lattice-QCD analysis of heavy-quark correlators to smaller lattice spacings and larger masses to obtain new values for the c mass and QCD coupling, and, for the first time, values for the b mass: mc(3\,\,GeV,nf=4)=0.986(6)\,\,GeV, $\alpha${}MS\ifmmode\bar\else\textasciimacron\fi{}(MZ,nf=5)=0.1183(7), and mb(10\,\,GeV,nf=5)=3.617(25)\,\,GeV. These are among the most accurate determinations by any method. We check our results using a nonperturbative determination of the mass ratio mb($\mu${},nf)/mc($\mu${},nf); the two methods agree to within our 1% errors and taken together imply mb/mc=4.51(4). We also update our previous analysis of $\alpha${}MS\ifmmode\bar\else\textasciimacron\fi{} from Wilson loops to account for revised values for r1 and r1/a, finding a new value $\alpha${}MS\ifmmode\bar\else\textasciimacron\fi{}(MZ,nf=5)=0.1184(6); and we update our recent values for light-quark masses from the ratio mc/ms. Finally, in the Appendix, we derive a procedure for simplifying and accelerating complicated least-squares fits.

We present results of lattice QCD simulations with mass-degenerate up and down and mass-split strange and charm (N _f = 2 + 1 + 1) dynamical quarks using Wilson twisted mass fermions at maximal twist. The tuning of the strange and charm quark masses is performed at two values of the lattice spacing a ≈ 0:078 fm and a ≈ 0:086 fm with lattice sizes ranging from L ≈ 1:9 fm to L ≈ 2:8 fm. We measure with high statistical precision the light pseudoscalar mass m _PS and decay constant f _PS in a range 270 ≲ m _PS ≲ 510 MeV and determine the low energy parameters f ₀ and \( {\bar{l}_{3,4}} \) of SU(2) chiral perturbation theory. We use the two values of the lattice spacing, several lattice sizes as well as different values of the light, strange and charm quark masses to explore the systematic effects. A first study of discretisation effects in light-quark observables and a comparison to N _f = 2 results are performed.

Lattice QCD with $N_f=2$ flavours of sea quark is used to explore the spectrum and decay of a $J^{PC}=1^{-+}$ spin-exotic hybrid meson. We test lattice determination of $S$-wave decay amplitudes at threshold using $b_1\to \pi \omega$ where agreement with data is found. We find a hybrid meson state at 2.2(2) GeV with a partial width to $\pi b_1$ of 400(120) MeV and to $\pi f_1$ of 90(60) MeV.

We update our previous determination of both the decay constant and the mass of the Ds meson using the highly improved staggered quark formalism. We include additional results at two finer values of the lattice spacing along with improved determinations of the lattice spacing and improved tuning of the charm and strange quark masses. We obtain mDs=1.9691(32)\,\,GeV, in good agreement with experiment, and fDs=0.2480(25)\,\,GeV. Our result for fDs is 1.6$\sigma${} lower than the most recent experimental average determined from the Ds leptonic decay rate and using Vcs from Cabibbo-Kobayashi-Maskawa unitarity. Combining our fDs with the experimental rate we obtain a direct determination of Vcs=1.010(22), or alternatively 0.990+0.013-0.016 using a probability distribution for statistical errors for this quantity which vanishes above 1. We also include an accurate prediction of the decay constant of the $\eta${}c, f$\eta${}c=0.3947(24)\,\,GeV, as a calibration point for other lattice calculations.