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This paper presents a formulation of lattice fermions applicable to all quark masses, large and small. We incorporate interactions from previous light-fermion and heavy-fermion methods, and thus ensure a smooth connection to these limiting cases. The couplings in improved actions are obtained for arbitrary fermion mass m q , without expansions around small-or large-mass limits. We treat both the action and external currents. By interpreting on-shell improvement criteria through the lattice theory's Hamiltonian, one finds that cutoff artifacts factorize into the form b n (m q a)͓pa͔ s n where p is a momentum characteristic of the system under study, s n is related to the dimension of the nth interaction, and b n (m q a) is a bounded function, numerically always of order 1 or less. In heavy-quark systems p is typically rather smaller than the fermion mass m q . Therefore, artifacts of order (m q a) s do not arise, even when m q aտ1. An important by-product of our analysis is an interpretation of the Wilson and Sheikholeslami-Wohlert actions applied to nonrelativistic fermions. ͓S0556-2821͑97͒03607-2͔

Dramatic progress has been made over the last decade in the numerical study of quantum chromodynamics (QCD) through the use of improved formulations of QCD on the lattice (improved actions), the development of new algorithms and the rapid increase in computing power available to lattice gauge theorists. In this article we describe simulations of full QCD using the improved staggered quark formalism, "asqtad" fermions. These simulations were carried out with two degenerate flavors of light quarks (up and down) and with one heavier flavor, the strange quark. Several light quark masses, down to about 3 times the physical light quark mass, and six lattice spacings have been used. These enable controlled continuum and chiral extrapolations of many low energy QCD observables. We review the improved staggered formalism, emphasizing both advantages and drawbacks. In particular, we review the procedure for removing unwanted staggered species in the continuum limit. We then describe the asqtad lattice ensembles created by the MILC Collaboration.All MILC lattice ensembles are publicly available, and they have been used extensively by a number of lattice gauge theory groups. We review physics results obtained with them, and discuss the impact of these results on phenomenology. Topics include the heavy quark potential, spectrum of light hadrons, quark masses, decay constant of light and heavy-light pseudoscalar mesons, semileptonic form factors, nucleon structure, scattering lengths and more. We conclude with a brief look at highly promising future prospects. PACS numbers: 12.38.Gc, 11.15.Ha 3. Staggered fermions 16 4. Chirally invariant fermions 21 C. Numerical simulations 25 D. Asqtad improved staggered fermions 29 E. Highly improved staggered fermions 32 III. Staggered chiral perturbation theory and "rooting" 34 A. Chiral effective theory for staggered quarks 34 B. Extensions of staggered chiral perturbation theory 41 C. The issue of rooting 45 IV. Overview of the MILC lattice ensembles 56 A. Algorithms and algorithm tests 57 B. The static potential and determining the lattice spacing 62 C. Tuning the strange quark mass 68 D. The topological susceptibility 68 V. Spectroscopy of light hadrons 71 A. Hadron mass computations 72 B. Correlated fits 76 C. Results for some light hadrons 79 3 D. Flavor singlet spectroscopy 83 E. Scalar mesons f 0 and a 0 84 F. Summary 88 VI. Results for the light pseudoscalar mesons 88 A. Motivation 88 B. From correlators to lattice masses and decay constants 88 C. Other computations of f π and f K 95 VII. Heavy-light mesons: masses and decay constants 96 A. Heavy quarks on the lattice 97 1. Nonrelativistic QCD 98 2. Wilson fermions with the Fermilab interpretation 98 3. The HISQ action 99 B. Lattice calculations of masses and decay constants 100 C. Results for masses, decay constants, and CKM matrix elements 104 VIII. Semileptonic form factors 107 A. D → πℓν and D → Kℓν 107 B. B → πℓν and |V ub | 109 C. B → Dℓν and B → D * ℓν 113 IX. Other computations using MILC lattices 116 A. Determination of ...

We present the first unquenched lattice-QCD calculation of the hadronic form factors for the exclusive decay B → D ν at nonzero recoil. We carry out numerical simulations on fourteen ensembles of gauge-field configurations generated with 2+1 flavors of asqtad-improved staggered sea quarks. The ensembles encompass a wide range of lattice spacings (approximately 0.045 to 0.12 fm) and ratios of light (up and down) to strange sea-quark masses ranging from 0.05 to 0.4.For the b and c valence quarks we use improved Wilson fermions with the Fermilab interpretation, while for the light valence quarks we use asqtad-improved staggered fermions. We extrapolate our results to the physical point using rooted staggered heavy-light meson chiral perturbation theory.We then parameterize the form factors and extend them to the full kinematic range using modelindependent functions based on analyticity and unitarity. We present our final results for f + (q 2 ) and f 0 (q 2 ), including statistical and systematic errors, as coefficients of a series in the variable z and the covariance matrix between these coefficients. We then fit the lattice form-factor data jointly with the experimentally measured differential decay rate from BaBar to determine the CKM matrix element, |V cb | = (39.6 ± 1.7 QCD+exp ± 0.2 QED ) × 10 −3 . As a byproduct of the joint fit we obtain the form factors with improved precision at large recoil. Finally, we use them to update our calculation of the ratio R(D) in the Standard Model, which yields R(D) = 0.299(11).