We study the leading discretization errors for staggered fermions by first constructing the continuum effective Lagrangian including terms of O(a 2 ), and then constructing the corresponding effective chiral Lagrangian. The terms of O(a 2 ) in the continuum effective Lagrangian completely break the SU (4) flavor symmetry down to the discrete subgroup respected by the lattice theory. We find, however, that the O(a 2 ) terms in the potential of the chiral Lagrangian maintain an SO(4) subgroup of SU (4). It follows that the leading discretization errors in the pion masses are SO(4) symmetric, implying three degeneracies within the seven lattice irreducible representations. These predictions hold also for perturbatively improved versions of the action. These degeneracies are observed, to a surprising degree of accuracy, in existing data. We argue that the SO(4) symmetry does not extend to the masses and interactions of other hadrons (vector mesons, baryons, etc), nor to higher order in a 2 . We show how it is possible that, for physical quark masses of O(a 2 ), the new SO(4) symmetry can be spontaneously broken, leading to a staggered analogue of the Aoki-phase of Wilson fermions. This does not, however, appear to happen for presently studied versions of the staggered action.

We report the Standard Model evaluation of the indirect CP violation parameter εK using inputs determined from lattice QCD: the kaon bag parameterBK , ξ0, |Vus| from the K 3 and Kµ2 decays, and |V cb | from the axial current form factor for the exclusive decayB → D * ν at zero-recoil. The theoretical expression for εK is thoroughly reviewed to give an estimate of the size of the neglected corrections, including long distance effects. The Wolfenstein parametrization (|V cb |, λ,ρ,η) is adopted for CKM matrix elements which enter through the short distance contribution of the box diagrams. For the central value, we take the Unitarity Triangle apex (ρ,η) from the angle-only fit of the UTfit collaboration and use Vus as an independent input to fix λ. We find that the Standard Model prediction of εK with exclusive V cb (lattice QCD results) is lower than the experimental value by 3.4σ. However, with inclusive V cb (results of the heavy quark expansion), there is no gap between the Standard Model prediction of εK and its experimental value. For the calculation of εK , we perform the renormalization group running to obtain ηcc at next-to-next-to-leading-order; we find η NNLO cc = 1.72(27).

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

We present nonperturbative results for the constants needed for on-shell O(a) improvement of bilinear operators composed of Wilson fermions. We work at ␤ϭ6.0 and 6.2 in the quenched approximation. The calculation is done by imposing axial and vector Ward identities on correlators similar to those used in standard hadron mass calculations. A crucial feature of the calculation is the use of nondegenerate quarks. We also obtain results for the constants needed for off-shell O(a) improvement of bilinears, and for the scale-and scheme-independent renormalization constants, Z A , Z V , and Z S /Z P . Several of the constants are determined using a variety of different Ward identities, and we compare their relative efficacies. In this way, we find a method for calculating c V that gives smaller errors than that used previously. Wherever possible, we compare our results with those of the ALPHA Collaboration ͑who use the Schrödinger functional͒ and with one-loop tadpole-improved perturbation theory.

We evaluate the continuum limit of the valence ͑quenched͒ approximation to the mass of the lightest scalar quarkonium state, for a range of different quark masses, and to the mixing energy between these states and the lightest scalar glueball. Our results support the interpretation of f 0 (1710) as composed mainly of the lightest scalar glueball.