CP3-Origins-2013-040 DNRF90, DIAS-2013-40 FERMILAB-PUB-13-484-T, FTUAM-13-28 IFIC/13-76, IFT-UAM/CSIC-13-106 MITP/13-067, YITP-13-114 Review of lattice results concerning low energy particle physics April 18, 2014 FLAG Working Group S. Aoki1, Y. Aoki2,3, C. Bernard4, T. Blum5,3, G. Colangelo6, M. Della Morte7, S. D¨urr8, A. X. El-Khadra9, H. Fukaya10, R. Horsley11, A. J¨uttner12, T. Kaneko13, J. Laiho⋆14, L. Lellouch15,16, H. Leutwyler6, V. Lubicz17,18, E. Lunghi19, S. Necco6, T. Onogi10, C. Pena20, C. T. Sachrajda13, S. R. Sharpe21, S. Simula17, R. Sommer22, R. S. Van de Water23, A. Vladikas24, U. Wenger6, H. Wittig25 Abstract We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determi- nation of the light-quark masses, the form factor f+(0), arising in semileptonic K π transition at → zero momentum transfer, as well as the decay constant ratio fK/fπ of decay constants and its conse- quences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)L SU(2)R and SU(3)L SU(3)R Chiral × × Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on D- and B-meson decay constants, form factors, and mixing pa- rameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant αs. arXiv:1310.8555v2 [hep-lat] 17 Apr 2014 1 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku Kyoto 606-8502, Japan 2 Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University, Nagoya 464-8602, Japan 3 RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA 4 Department of Physics, Washington University, Saint Louis, MO 63130, USA 5 Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA and 6 Albert Einstein Center for Fundamental Physics, Institut f¨ur theoretische Physik, Universit¨at Bern, Sidlerstr. 5, 3012 Switzerland 7 CP3-Origins & Danish IAS, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark and IFIC (CSIC), c/ Catedr´atico Jos´eBeltr´an, 2. E-46980, Paterna, Spain 8 Bergische Universit¨at Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany and J¨ulich Supercomputing Center, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany 9 Department of Physics, University of Illinois, Urbana, IL 61801, USA 10 Department of Physics, Osaka University, Toyonaka, Osaka 560-0043 Japan 11 School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK 12 School of Physics & Astronomy, University of Southampton, SO17 1BJ, UK 13 High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan 14 SUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, United Kingdom 15 Aix-Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France 16 Universit´ede Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France 17 Dipartimento di Matematica e Fisica, Universit`aRoma Tre, Via della Vasca Navale 84, 00146 Rome, Italy 18 INFN, Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy 19 Physics Department, Indiana University, Bloomington, IN 47405, USA 20 Instituto de F´ısica Te´orica UAM/CSIC and Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, Cantoblanco 28049 Madrid, Spain 21 Physics Department, University of Washington, Seattle, WA 98195-1560, USA 22 NIC @ DESY, Platanenallee 6, 15738 Zeuthen, Germany 23 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA 24 INFN, Sezione di Tor Vergata, c/o Dipartimento di Fisica, Universit`adi Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 25 PRISMA Cluster of Excellence, Institut f¨ur Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099 Mainz, Germany ⋆ Present address: Department of Physics, Syracuse University, Syracuse, New York, USA 2 Contents 1 Introduction 6 1.1 FLAGenlargement................................. 7 1.2 General issues and summary of the main results . ........ 9 2 Quality criteria 12 2.1 Systematic errors and colour-coding . ....... 12 2.1.1 Light-quarkphysics .. .. .. .. .. .. .. .. 13 2.1.2 Heavy-quarkphysics . 15 2.2 Averagesandestimates . .. .. .. .. .. .. .. .. 17 2.3 Averaging procedure and error analysis . ....... 18 3 Masses of the light quarks 19 3.1 Contributions from the electromagnetic interaction . ............. 20 3.2 Pion and kaon masses in the isospin limit . ...... 24 3.3 Lattice determination of ms and mud ....................... 24 3.3.1 Nf = 2 lattice calculations . 25 3.3.2 Nf = 2 + 1 lattice calculations . 27 3.4 Lattice determination of mu and md ....................... 35 3.5 Estimates for R and Q ............................... 39 4 Leptonic and semileptonic kaon and pion decay and V and V 42 | ud| | us| 4.1 Experimental information concerning V , V , f (0) and f ±/f ± .... 42 | ud| | us| + K π 4.2 Lattice results for f+(0) and fK/fπ ........................ 43 4.3 Direct determination of f+(0) and fK± /fπ± ................... 45 4.4 TestingtheStandardModel . 51 4.5 AnalysiswithintheStandardModel . ..... 53 4.6 Direct determination of fK and fπ ........................ 54 5 Low-energy constants 59 5.1 SU(2)Low-EnergyConstants . 60 5.1.1 Quark-mass dependence of pseudoscalar masses and decay constants . 60 5.1.2 Two-point correlation functions in the epsilon-regime.......... 61 5.1.3 Energy levels of the QCD Hamiltonian in a box and δ-regime . 63 5.1.4 Other methods for the extraction of the Low-Energy Constants . 64 5.1.5 Pionformfactors.............................. 65 5.1.6 Lattice determinations . 66 5.2 SU(3)Low-EnergyConstants . 77 5.2.1 Quark-mass dependence of pseudoscalar masses and decay constants . 77 5.2.2 Chargeradius................................ 78 5.2.3 Partially quenched formulae . 78 5.2.4 Lattice determinations . 78 6 Kaon B-parameter BK 83 6.1 Indirect CP-violation and ǫK ........................... 83 6.2 Lattice computation of BK ............................ 86 3 7 D-meson decay constants and form factors 93 7.1 Leptonic decay constants fD and fDs ....................... 93 7.2 Semileptonic form factors for D πℓν and D Kℓν ............. 98 → → 7.2.1 Results for f+(0).............................. 101 7.3 Determinations of V and V and test of second-row CKM unitarity . 103 | cd| | cs| 8 B-meson decay constants, mixing parameters and form factors 108 8.1 Leptonic decay constants fB and fBs ....................... 110 8.2 Neutral B-mesonmixingmatrixelements . 117 8.3 Semileptonic form factors for B decays to light flavours . 124 8.3.1 Parameterizations of heavy-to-light semileptonic form factors . 125 8.3.2 Form factors for B πℓν and B Kℓν ................ 128 → s → 8.3.3 Form factors for rare and radiative B semileptonic decays to light flavours132 8.4 Semileptonic form factors for B Dℓν, B D ℓν, and B Dτν ...... 133 → → ∗ → 8.4.1 B D decays ............................ 133 (s) → (s) 8.4.2 B D decays............................... 135 → ∗ 8.4.3 B D( )τν decays............................. 136 → ∗ 8.4.4 Ratios of B and Bs semileptonic decay form factors . 137 8.4.5 Summary.................................. 138 8.5 Determination of V ............................... 138 | ub| 8.6 Determination of V ............................... 141 | cb| 9 The strong coupling αs 144 9.1 Introduction.................................... 144 9.1.1 Scheme and scale dependence of αs and ΛQCD ............. 145 9.1.2 Overview of the review of αs ....................... 146 9.2 Discussion of criteria for computations entering the averages. 146 9.3 αs fromtheSchr¨odingerFunctional. 150 9.3.1 Generalconsiderations . 150 9.3.2 Discussion of computations . 151 9.4 αs fromthepotentialatshortdistances . 153 9.4.1 Generalconsiderations . 153 9.4.2 Discussion of computations . 154 9.5 αs from the vacuum polarisation at short distances . ..... 155 9.5.1 Generalconsiderations . 155 9.5.2 Discussion of computations . 156 9.6 αs from observables at the lattice spacing scale . ..... 156 9.6.1 Generalconsiderations . 156 9.6.2 Continuumlimit .............................. 157 9.6.3 Discussion of computations . 158 9.7 αs fromcurrenttwo-pointfunctions . 161 9.7.1 Generalconsiderations . 161 9.7.2 Discussion of computations . 163 9.8 αs fromQCDvertices ............................... 164 9.8.1 Generalconsiderations . 164 9.8.2 Discussion of computations . 165 9.9 Summary ...................................... 167 4 9.9.1 Thepresentsituation . .. .. .. .. .. .. .. 167 9.9.2 Our range for α(5) ............................. 169 MS (Nf ) 9.9.3 Ranges for [r0Λ] and ΛMS ....................... 172 9.9.4 Conclusions................................. 174 A Glossary 176 A.1 Latticeactions .................................. 176 A.1.1 Gaugeactions ............................... 176 A.1.2 Light-quarkactions . 178 A.1.3 Heavy-quarkactions . 185 A.2 Settingthescale ................................. 195 A.3 Matchingandrunning .............................. 195 A.4 Chiralextrapolation .. .. .. .. .. .. .. .. 197 A.5 Summary of simulated lattice actions . ....... 199 B Notes 205 B.1 Notes to section 3 onquarkmasses........................ 205 B.2 Notes to section 4 on V and V ....................... 214 | ud| | us| B.3 Notes to section 5 onLow-EnergyConstants . 222 B.4 Notes to section 6 on Kaon B-parameter BK .................. 229 B.5 Notes to section 7 on D-meson decay constants and form factors . 237 B.5.1 D(s)-mesondecayconstants . 237 B.5.2 D πℓν and D Kℓν formfactors. .. .. .. .. .. 244 → → B.6 Notes to section 8 on B-meson decay constants, mixing parameters, and form factors........................................ 248 B.6.1 B(s)-mesondecayconstants . 248 B.6.2 B(s)-mesonmixingmatrixelements. 256 B.6.3 B πℓν formfactor............................ 262 → B.6.4 B Dℓν and B D ℓν form factors and R(D)...........