This section summarizes state-of-the-art lattice calculations of the leptonic kaon and pion decay constants and the kaon semileptonic-decay form factor and provides an analysis in view of the Standard Model. With respect to the previous edition of the FLAG review [1] the data in this section has been updated. As in Ref. [1], when combining lattice data with experimental results, we take into account the strong SU(2) isospin correction, either obtained in lattice calculations or estimated by using chiral perturbation theory (χPT), both for the kaon leptonic decay constant fK± and for the ratio fK±/fπ± .

Jon A. Bailey ∗a†, A. Bazavov b, C. Bernard c, C. Bouchard e, C. DeTard, A.X. El-Khadra e, E.D. Freeland c, W. Freeman b, E. Gamiza,e, Steven Gottlieb e, f ,g, U.M. Heller h, J.E. Hetrick i, A.S. Kronfeld a, J. Laiho c, L. Levkova d, P.B. Mackenzie a, M.B. Oktay d, M. Di Pierro j , J.N. Simone a, R. Sugar k, D. Toussaint b, and R.S. Van de Water l aTheoretical Physics Department, Fermilab, Batavia, IL 605 1 , USA bDepartment of Physics, University of Arizona, Tucson, AZ 85 721, USA cDepartment of Physics, Washington University, St. Louis, M O 63130, USA dPhysics Department, University of Utah, Salt Lake City, UT 8 4112, USA ePhysics Department, University of Illinois, Urbana, IL 618 01, USA f Department of Physics, Indiana University, Bloomington, I N 47405, USA gNational Center for Supercomputing Applications, Univers ity of Illinois, Urbana, IL 61801, USA hAmerican Physical Society, One Research Road, Ridge, NY 119 61, USA iPhysics Department, University of the Pacific, Stockton, CA 95211, USA jSchool of Computing, DePaul University, Chicago, IL 60604, USA kDepartment of Physics, University of California, Santa Bar bara, CA 93106, USA l Department of Physics, Brookhaven National Laboratory, Up ton, NY 11973, USA

We calculate---for the first time in three-flavor lattice QCD---the hadronic matrix elements of all five local operators that contribute to neutral ${B}^{0}$- and ${B}_{s}$-meson mixing in and beyond the Standard Model. We present a complete error budget for each matrix element and also provide the full set of correlations among the matrix elements. We also present the corresponding bag parameters and their correlations, as well as specific combinations of the mixing matrix elements that enter the expression for the neutral $B$-meson width difference. We obtain the most precise determination to date of the SU(3)-breaking ratio $\ensuremath{\xi}=1.206(18)(6)$, where the second error stems from the omission of charm-sea quarks, while the first encompasses all other uncertainties. The threefold reduction in total uncertainty, relative to the 2013 Flavor Lattice Averaging Group results, tightens the constraint from $B$ mixing on the Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle. Our calculation employs gauge-field ensembles generated by the MILC Collaboration with four lattice spacings and pion masses close to the physical value. We use the asqtad-improved staggered action for the light-valence quarks and the Fermilab method for the bottom quark. We use heavy-light meson chiral perturbation theory modified to include lattice-spacing effects to extrapolate the five matrix elements to the physical point. We combine our results with experimental measurements of the neutral $B$-meson oscillation frequencies to determine the CKM matrix elements $|{V}_{td}|=8.00(34)(8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, $|{V}_{ts}|=39.0(1.2)(0.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and $|{V}_{td}/{V}_{ts}|=0.2052(31)(10)$, which differ from CKM-unitarity expectations by about $2\ensuremath{\sigma}$. These results and others from flavor-changing-neutral currents point towards an emerging tension between weak processes that are mediated at the loop and tree levels.

Department of Physics and Astronomy, Seoul National University, Seoul 151-742, South Korea Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA Department of Physics, Washington University, St. Louis, Missouri 63130, USA Physics Department, College of William and Mary, Williamsburg, Virginia 23187, USA Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA Department of Physics, University of Illinois, Urbana, Illinois 61801, USA Department of Physics, Syracuse University, Syracuse, New York 13244, USA Liberal Arts Department, School of the Art Institute of Chicago, Chicago, Illinois 60603, USA CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, Granada E-18002, Spain Department of Physics, Indiana University, Bloomington, Indiana 47405, USA American Physical Society, Ridge, New York 11961, USA Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany Department of Physics, University of Colorado, Boulder, Colorado 80309, USA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA Department of Physics, University of California, Santa Barbara, California 93106, USA Department of Physics, University of Arizona, Tucson, Arizona 85721, USA (Received 3 April 2015; published 10 August 2015)

(Fermilab Lattice and MILC Collaborations) 1Michigan State University, East Lansing, Michigan, USA 2University of Utah, Salt Lake City, Utah, USA 3University of Illinois, Urbana, Illinois, USA 4Fermi National Accelerator Laboratory, Batavia, Illinois, USA 5Universidad de Granada, Granada, Spain 6Indiana University, Bloomington, Indiana, USA 7American Physical Society, Ridge, New York, USA 8University of Kentucky, Lexington, Kentucky, USA 9Syracuse University, Syracuse, New York, USA 10University of Chicago, Chicago, Illinois, USA 11Brookhaven National Laboratory, Upton, New York, USA 12University of Colorado, Boulder, Colorado, USA 13RIKEN-BNL Research Center, Upton, New York, USA 14Argonne National Laboratory, Argonne, Illinois, USA 15University of California, Santa Barbara, California, USA 16University of Arizona, Tucson, Arizona, USA

Department of Physics and Astronomy, Seoul National University, Seoul, South Korea Physics Department, Brookhaven National Laboratory, Upton, New York, USA Department of Physics, Washington University, St. Louis, Missouri, USA Department of Physics, The Ohio State University, Columbus, Ohio, USA Department Physics and Astronomy, University of Utah, Salt Lake City, Utah, USA Physics Department, University of Illinois, Urbana, Illinois, USA Department of Physics, Syracuse University, Syracuse, New York, USA Fermi National Accelerator Laboratory, Batavia, Illinois, USA Department of Physics, Benedictine University, Lisle, Illinois, USA Liberal Arts Department, School of the Art Institute of Chicago, Chicago, Illinois, USA CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, Granada, Spain Department of Physics, Indiana University, Bloomington, Indiana, USA American Physical Society, Ridge, New York, USA SUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom Department of Physics, University of Colorado, Boulder, Colorado, USA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York, USA Department of Physics, University of California, Santa Barbara, California, USA Department of Physics, University of Arizona, Tucson, Arizona, USA (Received 11 March 2014; published 19 June 2014)

We compute the $B\to\pi\ell\nu$ semileptonic form factors and update the determination of the CKM matrix element $|V_{ub}|$. We use the MILC asqtad ensembles with $N_f=2+1$ sea quarks at four different lattice spacings in the range $a \approx 0.045$~fm to $0.12$~fm. The lattice form factors are extrapolated to the continuum limit using SU(2) staggered chiral perturbation theory in the hard pion limit, followed by an extrapolation in $q^2$ to the full kinematic range using a functional $z$-parameterization. The extrapolation is combined with the experimental measurements of the partial branching fraction to extract $|V_{ub}|$. Our preliminary result is $|V_{ub}|=(3.72\pm 0.14)\times 10^{-3}$, where the error reflects both the lattice and experimental uncertainties, which are now on par with each other.

We report on the status of the Fermilab-MILC calculation of the form factor f_+^{K pi}(0), needed to extract the CKM matrix element |V_{us}| from experimental data on K semileptonic decays. The HISQ formulation is used in the simulations for the valence quarks, while the sea quarks are simulated with the asqtad action (MILC N_f=2+1 configurations). We discuss the general methodology of the calculation, including the use of twisted boundary conditions to get values of the momentum transfer close to zero and the different techniques applied for the correlators fits. We present initial results for lattice spacings a=0.12fm and a=0.09fm, and several choices of the light quark masses.

We present results from an ongoing lattice study of the lowest lying charmonium and bottomonium level splittings using the Fermilab heavy quark formalism. Our objective is to test the performance of this action on MILC-collaboration ensembles of (2+1) flavors of light improved staggered (asqtad) quarks. Measurements are done on 16 ensembles with degenerate up and down quarks of various masses, thus permitting a chiral extrapolation, and over lattice spacings ranging from 0.09 fm to 0.18 fm, thus permitting study of lattice-spacing dependence. We examine combinations of the mass splittings that are sensitive to components of the effective quarkonium potential.

We present an update of our calculations of the decay constants of the D, D_s, B, and B_s mesons in unquenched 2+1 flavor QCD. We use the MILC library of improved staggered gauge ensembles at lattice spacings 0.09, 0.12, and 0.15 fm, clover heavy quarks with the Fermilab normalizations, and improved staggered light valence quarks.

We present the first lattice QCD calculation of the form factor for B-> D* l nu with three flavors of sea quarks. We use an improved staggered action for the light valence and sea quarks (the MILC configurations), and the Fermilab action for the heavy quarks. The form factor is computed at zero recoil using a new double ratio method that yields the form factor more directly than the previous Fermilab method. Other improvements over the previous calculation include the use of much lighter light quark masses, and the use of lattice (staggered) chiral perturbation theory in order to control the light quark discretization errors and chiral extrapolation. We obtain for the form factor, F_{B-> D*}(1)=0.921(13)(20), where the first error is statistical and the second is the sum of all systematic errors in quadrature. Applying a 0.7% electromagnetic correction and taking the latest PDG average for F_{B-> D*}(1)|V_cb| leads to |V_cb|=(38.7 +/- 0.9_exp +/- 1.0_theo) x 10^-3.

We present new results for the leptonic decay constants fB and fD+ determined in 2 + 1 flavor lattice QCD at lattice spacings a = 0.09, 0.12 and 0.15 fm. Results are obtained using the MILC collaboration gauge configuration ensembles, clover heavy quarks in the Fermilab interpretation and improved staggered light quarks. Decay constants, computed at partially quenched combinations of the valence and sea light quark masses, are used to determine the low-energy parameters of staggered chiral perturbation theory. The physical decay constants are found in an extrapolation using the parameterized chiral formula.

We have greatly extended an earlier calculation of the charmonium spectrum on three flavor dynamical quark ensembles by using more recent ensembles generated by the MILC collaboration.The heavy quarks are treated using the Fermilab formulation.The charmonium state masses are in reasonable agreement with the observed spectrum; however, some of the spin splittings may still be too small.

The heavy-quark expansion for inclusive semi-leptonic B decays introduces̄ Λ andλ1, which are matrix elements in heavy-quark effective field theory. We re view how they can be obtained from an analysis of the heavy quark mass dependence of heavy-ligh t meson masses in lattice QCD. We present preliminary results for the bottom quark mass, mb, usingΛ̄ andλ1 for theBs meson from a 2+1 sea-flavor unquenched calculation.

We present the first three-flavor lattice QCD calculations for D-->pilnu and D-->Klnu semileptonic decays. Simulations are carried out using ensembles of unquenched gauge fields generated by the MILC Collaboration. With an improved staggered action for light quarks, we are able to simulate at light quark masses down to 1/8 of the strange mass. Consequently, the systematic error from the chiral extrapolation is much smaller than in previous calculations with Wilson-type light quarks. Our results for the form factors at q(2)=0 are f(D-->pi)(+)(0)=0.64(3)(6) and f(D-->K)(+)(0)=0.73(3)(7), where the first error is statistical and the second is systematic, added in quadrature. Combining our results with experimental branching ratios, we obtain the Cabibbo-Kobayashi-Maskawa matrix elements |V(cd)|=0.239(10)(24)(20) and |V(cs)|=0.969(39)(94)(24), where the last errors are from experimental uncertainties.

We present the first lattice QCD calculation with realistic sea quark content of the D+-meson decay constant f(D+). We use the MILC Collaboration's publicly available ensembles of lattice gauge fields, which have a quark sea with two flavors (up and down) much lighter than a third (strange). We obtain f(D+)=201+/-3+/-17 MeV, where the errors are statistical and a combination of systematic errors. We also obtain f(Ds)=249+/-3+/-16 MeV for the Ds meson.

The recently developed Symanzik-improved staggered-quark discretization allows unquenched lattice-QCD simulations with much smaller (and more realistic) quark masses than previously possible. To test this formalism, we compare experiment with a variety of nonperturbative calculations in QCD drawn from a restricted set of "gold-plated" quantities. We find agreement to within statistical and systematic errors of 3% or less. We discuss the implications for phenomenology and, in particular, for heavy-quark physics.