Review of lattice results concerning low-energy particle physics

  • FLAG working group of FLAVIANET
  • G. Colangelo
  • S. Dürr
  • A. Jüttner
  • L. Lellouch
  • H. Leutwyler
  • V. Lubicz
  • S. Necco
  • C. T. Sachrajda
  • S. Simula
  • A. Vladikas
  • U. Wenger
  • H. Wittig
Review

Abstract

We review lattice results relevant for pion and kaon physics with the aim of making them easily accessible to the particle physics community. Specifically, we review the determination of the light-quark masses, the form factor f+(0), relevant for the semileptonic Kπ transition at zero momentum transfer as well as the ratio fK/fπ of decay constants and discuss the consequences for the elements Vus and Vud of the CKM matrix. 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. We introduce quality criteria and use these when forming averages. Although subjective and imperfect, these criteria may help the reader to judge different aspects of current lattice computations. Our main results are summarized in Sect. 1.2, but we stress the importance of the detailed discussion that underlies these results and constitutes the bulk of the present review.

References

  1. 1.
    Flavianet Lattice Averaging Group (FLAG), Review of lattice results concerning low energy particle physics. http://itpwiki.unibe.ch/flag
  2. 2.
    J. Laiho, E. Lunghi, R. Van de Water, 2+1 flavor lattice QCD averages. http://krone.physik.unizh.ch/~lunghi/webpage/LatAves
  3. 3.
    J. Laiho, E. Lunghi, R.S. Van de Water, Lattice QCD inputs to the CKM unitarity triangle analysis. Phys. Rev. D 81, 034503 (2010). arXiv:0910.2928 [hep-ph] ADSGoogle Scholar
  4. 4.
    K. Jansen, Lattice QCD: a critical status report. PoS LAT2008, 010 (2008). arXiv:0810.5634 [hep-lat] MathSciNetGoogle Scholar
  5. 5.
    C. Jung, Status of dynamical ensemble generation. PoS LAT2009, 002 (2009). arXiv:1001.0941 [hep-lat] Google Scholar
  6. 6.
    A. Bazavov et al. (MILC 09), Full nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks. Rev. Mod. Phys. 82, 1349–1417 (2010). arXiv:0903.3598 [math.PR] ADSCrossRefGoogle Scholar
  7. 7.
    M. Hasenbusch, Speeding up the hybrid Monte Carlo algorithm for dynamical fermions. Phys. Lett. B 519, 177–182 (2001). hep-lat/0107019 ADSMATHCrossRefGoogle Scholar
  8. 8.
    M. Lüscher, Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD. Comput. Phys. Commun. 165, 199–220 (2005). hep-lat/0409106 ADSCrossRefGoogle Scholar
  9. 9.
    C. Urbach, K. Jansen, A. Shindler, U. Wenger, HMC algorithm with multiple time scale integration and mass preconditioning. Comput. Phys. Commun. 174, 87–98 (2006). hep-lat/0506011 ADSCrossRefGoogle Scholar
  10. 10.
    M.A. Clark, A.D. Kennedy, Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudofermion fields. Phys. Rev. Lett. 98, 051601 (2007). hep-lat/0608015 ADSCrossRefGoogle Scholar
  11. 11.
    K.-I. Ishikawa, Recent algorithm and machine developments for lattice QCD. PoS LAT2008, 013 (2008). arXiv:0811.1661 [hep-lat] Google Scholar
  12. 12.
    D.J. Antonio et al. (RBC 07A), Localization and chiral symmetry in 3 flavor domain wall QCD. Phys. Rev. D 77, 014509 (2008). arXiv:0705.2340 [hep-lat] ADSGoogle Scholar
  13. 13.
    A. Bazavov et al. (MILC 10), Topological susceptibility with the asqtad action. Phys. Rev. D 81, 114501 (2010). arXiv:1003.5695 [hep-lat] ADSGoogle Scholar
  14. 14.
    S. Schaefer, R. Sommer, F. Virotta, Critical slowing down and error analysis in lattice QCD simulations. arXiv:1009.5228 [hep-lat]
  15. 15.
    M. Lüscher, Topology, the Wilson flow and the HMC algorithm. arXiv:1009.5877 [hep-lat]
  16. 16.
    S. Schaefer, Algorithms for lattice QCD: progress and challenges. arXiv:1011.5641 [hep-ph]
  17. 17.
    S. Dürr et al. (BMW 08), Ab-initio determination of light hadron masses. Science 322, 1224–1227 (2008). arXiv:0906.3599 [hep-lat] CrossRefGoogle Scholar
  18. 18.
    K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and φ 4 theory. Nucl. Phys. B 226, 187 (1983) MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    K. Symanzik, Continuum limit and improved action in lattice theories. 2. O(N) nonlinear sigma model in perturbation theory. Nucl. Phys. B 226, 205 (1983) MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    C.W. Bernard, M.F.L. Golterman, Partially quenched gauge theories and an application to staggered fermions. Phys. Rev. D 49, 486–494 (1994). hep-lat/9306005 ADSGoogle Scholar
  21. 21.
    S.R. Sharpe, Enhanced chiral logarithms in partially quenched QCD. Phys. Rev. D 56, 7052–7058 (1997). hep-lat/9707018. Erratum: Phys. Rev. D 62, 099901 (2000) ADSGoogle Scholar
  22. 22.
    M.F.L. Golterman, K.-C. Leung, Applications of partially quenched chiral perturbation theory. Phys. Rev. D 57, 5703–5710 (1998). hep-lat/9711033 ADSGoogle Scholar
  23. 23.
    S.R. Sharpe, R.L. Singleton Jr., Spontaneous flavor and parity breaking with Wilson fermions. Phys. Rev. D 58, 074501 (1998). hep-lat/9804028 ADSGoogle Scholar
  24. 24.
    W.-J. Lee, S.R. Sharpe, Partial flavor symmetry restoration for chiral staggered fermions. Phys. Rev. D 60, 114503 (1999). hep-lat/9905023 ADSGoogle Scholar
  25. 25.
    S.R. Sharpe, N. Shoresh, Physical results from unphysical simulations. Phys. Rev. D 62, 094503 (2000). hep-lat/0006017 ADSGoogle Scholar
  26. 26.
    G. Rupak, N. Shoresh, Chiral perturbation theory for the Wilson lattice action. Phys. Rev. D 66, 054503 (2002). hep-lat/0201019 ADSGoogle Scholar
  27. 27.
    O. Bär, G. Rupak, N. Shoresh, Simulations with different lattice Dirac operators for valence and sea quarks. Phys. Rev. D 67, 114505 (2003). hep-lat/0210050 ADSGoogle Scholar
  28. 28.
    C. Aubin, C. Bernard, Pion and kaon masses in staggered chiral perturbation theory. Phys. Rev. D 68, 034014 (2003). hep-lat/0304014 ADSCrossRefGoogle Scholar
  29. 29.
    O. Bär, G. Rupak, N. Shoresh, Chiral perturbation theory at O(a ∗∗2) for lattice QCD. Phys. Rev. D 70, 034508 (2004). hep-lat/0306021 ADSGoogle Scholar
  30. 30.
    C. Aubin, C. Bernard, Pseudoscalar decay constants in staggered chiral perturbation theory. Phys. Rev. D 68, 074011 (2003). hep-lat/0306026 ADSCrossRefGoogle Scholar
  31. 31.
    S. Aoki, Chiral perturbation theory with Wilson-type fermions including a ∗∗2 effects: N(f)=2 degenerate case. Phys. Rev. D 68, 054508 (2003). hep-lat/0306027 ADSGoogle Scholar
  32. 32.
    S. Aoki, O. Bär, Twisted-mass QCD, O(a) improvement and Wilson chiral perturbation theory. Phys. Rev. D 70, 116011 (2004). hep-lat/0409006 ADSGoogle Scholar
  33. 33.
    S.R. Sharpe, R.S. Van de Water, Staggered chiral perturbation theory at next-to-leading order. Phys. Rev. D 71, 114505 (2005). hep-lat/0409018 ADSGoogle Scholar
  34. 34.
    S.R. Sharpe, J.M.S. Wu, Twisted mass chiral perturbation theory at next-to-leading order. Phys. Rev. D 71, 074501 (2005). hep-lat/0411021 ADSGoogle Scholar
  35. 35.
    O. Bär, C. Bernard, G. Rupak, N. Shoresh, Chiral perturbation theory for staggered sea quarks and Ginsparg–Wilson valence quarks. Phys. Rev. D 72, 054502 (2005). hep-lat/0503009 ADSGoogle Scholar
  36. 36.
    M. Golterman, T. Izubuchi, Y. Shamir, The role of the double pole in lattice QCD with mixed actions. Phys. Rev. D 71, 114508 (2005). hep-lat/0504013 ADSGoogle Scholar
  37. 37.
    J.-W. Chen, D. O’Connell, A. Walker-Loud, Two meson systems with Ginsparg–Wilson valence quarks. Phys. Rev. D 75, 054501 (2007). hep-lat/0611003 ADSCrossRefGoogle Scholar
  38. 38.
    J.-W. Chen, D. O’Connell, A. Walker-Loud, Universality of mixed action extrapolation formulae. J. High Energy Phys. 04, 090 (2009). arXiv:0706.0035 [hep-lat] ADSCrossRefGoogle Scholar
  39. 39.
    J.-W. Chen, M. Golterman, D. O’Connell, A. Walker-Loud, Mixed action effective field theory: an addendum. Phys. Rev. D 79, 117502 (2009). arXiv:0905.2566 [hep-lat] ADSGoogle Scholar
  40. 40.
    O. Bär, Chiral logs in twisted mass lattice QCD with large isospin breaking. arXiv:1008.0784 [hep-lat]
  41. 41.
    S. Dürr et al. (BMW 10), The ratio F K/F π in QCD. Phys. Rev. D 81, 054507 (2010). arXiv:1001.4692 [hep-lat] Google Scholar
  42. 42.
    S. Aoki et al. (PACS-CS 09), Physical point simulation in 2+1 flavor lattice QCD. Phys. Rev. D 81, 074503 (2010). arXiv:0911.2561 [hep-lat] ADSGoogle Scholar
  43. 43.
    J. Bijnens, G. Colangelo, G. Ecker, Double chiral logs. Phys. Lett. B 441, 437–446 (1998). hep-ph/9808421 ADSGoogle Scholar
  44. 44.
    G. Ecker, P. Masjuan, H. Neufeld, Chiral extrapolation of lattice data. Phys. Lett. B 692, 184–188 (2010). arXiv:1004.3422 [hep-ph] ADSGoogle Scholar
  45. 45.
    G. Ecker, Chiral extrapolation of SU(3) amplitudes. arXiv:1012.1522 [hep-ph]
  46. 46.
    W. Bietenholz et al. (QCDSF/UKQCD 10), Tuning the strange quark mass in lattice simulations. Phys. Lett. B 690, 436–441 (2010). arXiv:1003.1114 [hep-lat] ADSGoogle Scholar
  47. 47.
    W. Bietenholz et al. (QCDSF/UKQCD 10A), Flavour symmetry breaking and tuning the strange quark mass for 2+1 quark flavours. PoS LAT2010, 122 (2010). arXiv:1012.4371 [hep-lat] Google Scholar
  48. 48.
    A. Manohar, C.T. Sachrajda, Quark masses. J. Phys. G 37, 075021 (2010). Review of Particle Physics, p. 583 Google Scholar
  49. 49.
    B. Blossier et al. (ETM 07), Light quark masses and pseudoscalar decay constants from N f=2 Lattice QCD with twisted mass fermions. J. High Energy Phys. 04, 020 (2008). arXiv:0709.4574 [hep-lat] Google Scholar
  50. 50.
    J.C. Hardy, I.S. Towner, Superallowed 0+→0+ nuclear β decays: A new survey with precision tests of the conserved vector current hypothesis and the Standard Model. Phys. Rev. C 79, 055502 (2009). arXiv:0812.1202 [nucl-ex] ADSCrossRefGoogle Scholar
  51. 51.
    C. Pena, Twisted mass QCD for weak matrix elements. PoS LAT2006, 019 (2006). hep-lat/0610109. This is, to the best of our knowledge, the first time colour coding was used. It does not appear in the proceedings but in the slides, see http://www.physics.utah.edu/lat06/abstracts/sessions/plenary.html Google Scholar
  52. 52.
    R. Frezzotti, P.A. Grassi, S. Sint, P. Weisz (ALPHA 01), Lattice QCD with a chirally twisted mass term. J. High Energy Phys. 08, 058 (2001). hep-lat/0101001 Google Scholar
  53. 53.
    R. Frezzotti, G.C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement. J. High Energy Phys. 08, 007 (2004). hep-lat/0306014 ADSCrossRefGoogle Scholar
  54. 54.
    P. Boucaud et al. (ETM 07A), Dynamical twisted mass fermions with light quarks. Phys. Lett. B 650, 304–311 (2007). hep-lat/0701012 ADSGoogle Scholar
  55. 55.
    S. Dürr, Theoretical issues with staggered fermion simulations. PoS LAT2005, 021 (2006). hep-lat/0509026 Google Scholar
  56. 56.
    S.R. Sharpe, Rooted staggered fermions: good, bad or ugly? PoS LAT2006, 022 (2006). hep-lat/0610094 Google Scholar
  57. 57.
    A.S. Kronfeld, Lattice gauge theory with staggered fermions: how, where, and why (not). PoS LAT2007, 016 (2007). arXiv:0711.0699 [hep-lat] Google Scholar
  58. 58.
    M. Golterman, QCD with rooted staggered fermions. PoS CONFINEMENT8, 014 (2008). arXiv:0812.3110 [hep-ph] Google Scholar
  59. 59.
    A. Bazavov et al. (MILC 09A), MILC results for light pseudoscalars. PoS CD09, 007 (2009). arXiv:0910.2966 [hep-ph] Google Scholar
  60. 60.
    R. Baron, P. Boucaud, J. Carbonell, A. Deuzeman, V. Drach et al. (ETM 10), Light hadrons from lattice QCD with light (u,d), strange and charm dynamical quarks. J. High Energy Phys. 1006, 111 (2010). arXiv:1004.5284 [hep-lat] ADSCrossRefGoogle Scholar
  61. 61.
    L. Lellouch, Kaon physics: a lattice perspective. PoS LAT2008, 015 (2009). arXiv:0902.4545 [hep-lat] Google Scholar
  62. 62.
    M. Gell-Mann, R.J. Oakes, B. Renner, Behavior of current divergences under SU(3)×SU(3). Phys. Rev. 175, 2195–2199 (1968) ADSCrossRefGoogle Scholar
  63. 63.
    S. Aoki et al. (PACS-CS 08), 2+1 Flavor lattice QCD toward the physical point. Phys. Rev. D 79, 034503 (2009). arXiv:0807.1661 [hep-lat] ADSGoogle Scholar
  64. 64.
    S. Aoki et al. (PACS-CS 10), Non-perturbative renormalization of quark mass in N f=2+1 QCD with the Schroedinger functional scheme. J. High Energy Phys. 08, 101 (2010). arXiv:1006.1164 [hep-lat] ADSCrossRefGoogle Scholar
  65. 65.
    S. Dürr et al. (BMW 10A), Lattice QCD at the physical point: light quark masses. arXiv:1011.2403 [hep-lat]
  66. 66.
    B. Bloch-Devaux, Results from NA48/2 on ππ scattering lengths measurements in K ±π + π e ± ν and K ±π 0 π 0 π ± decays. PoS CONFINEMENT8, 029 (2008) Google Scholar
  67. 67.
    J. Gasser, A. Rusetsky, I. Scimemi, Electromagnetic corrections in hadronic processes. Eur. Phys. J. C 32, 97–114 (2003). hep-ph/0305260 ADSCrossRefGoogle Scholar
  68. 68.
    A. Rusetsky, Isospin symmetry breaking. PoS CD09, 071 (2009). arXiv:0910.5151 [hep-ph] Google Scholar
  69. 69.
    J. Gasser, Theoretical progress on cusp effect and K 4 decays. PoS KAON, 033 (2008). arXiv:0710.3048 [hep-ph] Google Scholar
  70. 70.
    H. Leutwyler, Light quark masses. PoS CD09, 005 (2009). arXiv:0911.1416 [hep-ph] Google Scholar
  71. 71.
    R.F. Dashen, Chiral SU(3)×SU(3) as a symmetry of the strong interactions. Phys. Rev. 183, 1245–1260 (1969) ADSCrossRefGoogle Scholar
  72. 72.
    A. Duncan, E. Eichten, H. Thacker, Electromagnetic splittings and light quark masses in lattice QCD. Phys. Rev. Lett. 76, 3894–3897 (1996). hep-lat/9602005 ADSCrossRefGoogle Scholar
  73. 73.
    T. Blum, T. Doi, M. Hayakawa, T. Izubuchi, N. Yamada (RBC 07), Determination of light quark masses from the electromagnetic splitting of pseudoscalar meson masses computed with two flavors of domain wall fermions. Phys. Rev. D 76, 114508 (2007). arXiv:0708.0484 [hep-lat] ADSGoogle Scholar
  74. 74.
    T. Blum et al. (Blum 10), Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED. Phys. Rev. D 82, 094508 (2010). arXiv:1006.1311 [hep-lat] ADSGoogle Scholar
  75. 75.
    A. Portelli et al. (BMW 10C), Electromagnetic corrections to light hadron masses. PoS LAT2010, 121 (2010). arXiv:1011.4189 [hep-lat] Google Scholar
  76. 76.
    C. Aubin et al. (MILC 04A), Results for light pseudoscalars from three-flavor simulations. Nucl. Phys. Proc. Suppl. 140, 231–233 (2005). hep-lat/0409041 ADSCrossRefGoogle Scholar
  77. 77.
    C. Aubin et al. (MILC 04), Light pseudoscalar decay constants, quark masses, and low energy constants from three-flavor lattice QCD. Phys. Rev. D 70, 114501 (2004). hep-lat/0407028 ADSGoogle Scholar
  78. 78.
    J. Bijnens, J. Prades, Electromagnetic corrections for pions and kaons: masses and polarizabilities. Nucl. Phys. B 490, 239–271 (1997). hep-ph/9610360 ADSCrossRefGoogle Scholar
  79. 79.
    J.F. Donoghue, A.F. Perez, The electromagnetic mass differences of pions and kaons. Phys. Rev. D 55, 7075–7092 (1997). hep-ph/9611331 ADSGoogle Scholar
  80. 80.
    S. Basak et al. (MILC 08), Electromagnetic splittings of hadrons from improved staggered quarks in full QCD. PoS LAT2008, 127 (2008). arXiv:0812.4486 [hep-lat] Google Scholar
  81. 81.
    C. Bernard, E.D. Freeland, Electromagnetic corrections in staggered chiral perturbation theory. PoS LAT2010, 084 (2010). arXiv:1011.3994 [hep-lat] Google Scholar
  82. 82.
    R. Urech, Virtual photons in chiral perturbation theory. Nucl. Phys. B 433, 234–254 (1995). hep-ph/9405341 ADSCrossRefGoogle Scholar
  83. 83.
    R. Baur, R. Urech, On the corrections to Dashen’s theorem. Phys. Rev. D 53, 6552–6557 (1996). hep-ph/9508393 ADSGoogle Scholar
  84. 84.
    R. Baur, R. Urech, Resonance contributions to the electromagnetic low energy constants of chiral perturbation theory. Nucl. Phys. B 499, 319–348 (1997). hep-ph/9612328 ADSCrossRefGoogle Scholar
  85. 85.
    B. Moussallam, A sum rule approach to the violation of Dashen’s theorem. Nucl. Phys. B 504, 381–414 (1997). hep-ph/9701400 ADSCrossRefGoogle Scholar
  86. 86.
    W. Cottingham, The neutron proton mass difference and electron scattering experiments. Ann. Phys. 25, 424 (1963) ADSCrossRefGoogle Scholar
  87. 87.
    R.H. Socolow, Departures from the Eightfold Way. 3. Pseudoscalar-meson electromagnetic masses. Phys. Rev. B 137, 1221–1228 (1965) MathSciNetADSGoogle Scholar
  88. 88.
    D.J. Gross, S.B. Treiman, F. Wilczek, Light quark masses and isospin violation. Phys. Rev. D 19, 2188 (1979) ADSGoogle Scholar
  89. 89.
    J. Gasser, H. Leutwyler, Quark masses. Phys. Rep. 87, 77–169 (1982) ADSCrossRefGoogle Scholar
  90. 90.
    T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low, J.E. Young, Electromagnetic mass difference of pions. Phys. Rev. Lett. 18, 759–761 (1967) ADSCrossRefGoogle Scholar
  91. 91.
    J. Gasser, H. Leutwyler (GL 85), Chiral perturbation theory: expansions in the mass of the strange quark. Nucl. Phys. B 250, 465 (1985) ADSCrossRefGoogle Scholar
  92. 92.
    G. Amoros, J. Bijnens, P. Talavera, QCD isospin breaking in meson masses, decay constants and quark mass ratios. Nucl. Phys. B 602, 87–108 (2001). hep-ph/0101127 ADSCrossRefGoogle Scholar
  93. 93.
    J. Gasser, H. Leutwyler (GL 84), Chiral perturbation theory to one loop. Ann. Phys. 158, 142 (1984) ADSCrossRefGoogle Scholar
  94. 94.
    B. Blossier et al. (ETM 10B), Average up/down, strange and charm quark masses with N f=2 twisted mass lattice QCD. Phys. Rev. D 82, 114513 (2010). arXiv:1010.3659 [hep-lat] ADSGoogle Scholar
  95. 95.
    J. Noaki et al. (JLQCD/TWQCD 08A), Convergence of the chiral expansion in two-flavor lattice QCD. Phys. Rev. Lett. 101, 202004 (2008). arXiv:0806.0894 [hep-lat] ADSCrossRefGoogle Scholar
  96. 96.
    M. Göckeler et al. (QCDSF/UKQCD 06), Estimating the unquenched strange quark mass from the lattice axial Ward identity. Phys. Rev. D 73, 054508 (2006). hep-lat/0601004 Google Scholar
  97. 97.
    D. Becirevic et al. (SPQcdR 05), Non-perturbatively renormalised light quark masses from a lattice simulation with N f=2. Nucl. Phys. B 734, 138–155 (2006). hep-lat/0510014 ADSCrossRefGoogle Scholar
  98. 98.
    M. Della Morte et al. (ALPHA 05), Non-perturbative quark mass renormalization in two-flavor QCD. Nucl. Phys. B 729, 117–134 (2005). hep-lat/0507035 ADSCrossRefGoogle Scholar
  99. 99.
    M. Göckeler et al. (QCDSF/UKQCD 04), Determination of light and strange quark masses from full lattice QCD. Phys. Lett. B 639, 307–311 (2006). hep-ph/0409312 Google Scholar
  100. 100.
    S. Aoki et al. (JLQCD 02), Light hadron spectroscopy with two flavors of O(a)-improved dynamical quarks. Phys. Rev. D 68, 054502 (2003). hep-lat/0212039 ADSGoogle Scholar
  101. 101.
    A. Ali Khan et al. (CP-PACS 01), Light hadron spectroscopy with two flavors of dynamical quarks on the lattice. Phys. Rev. D 65, 054505 (2002). hep-lat/0105015. Erratum: Phys. Rev. D 66, 059901 (2003) ADSGoogle Scholar
  102. 102.
    M. Constantinou et al. (ETM 10C), Non-perturbative renormalization of quark bilinear operators with N f=2 (tmQCD) Wilson fermions and the tree-level improved gauge action. J. High Energy Phys. 08, 068 (2010). arXiv:1004.1115 [hep-lat] ADSCrossRefGoogle Scholar
  103. 103.
    A. Bazavov et al. (MILC 10A), Staggered chiral perturbation theory in the two-flavor case and SU(2) analysis of the MILC data. PoS LAT2010, 083 (2010). arXiv:1011.1792 [hep-lat] Google Scholar
  104. 104.
    C. McNeile, C.T.H. Davies, E. Follana, K. Hornbostel, G.P. Lepage (HPQCD 10), High-precision c and b masses, and QCD coupling from current-current correlators in lattice and continuum QCD. Phys. Rev. D 82, 034512 (2010). arXiv:1004.4285 [hep-lat] ADSGoogle Scholar
  105. 105.
    S. Dürr et al. (BMW 10B), Lattice QCD at the physical point: Simulation and analysis details. arXiv:1011.2711 [hep-lat]
  106. 106.
    Y. Aoki et al. (RBC/UKQCD 10A), Continuum limit physics from 2+1 flavor domain wall QCD. arXiv:1011.0892 [hep-lat]
  107. 107.
    C.T.H. Davies et al. (HPQCD 09), Precise charm to strange mass ratio and light quark masses from full lattice QCD. Phys. Rev. Lett. 104, 132003 (2010). arXiv:0910.3102 [hep-ph] ADSCrossRefGoogle Scholar
  108. 108.
    C. Allton et al. (RBC/UKQCD 08), Physical results from 2+1 flavor domain wall QCD and SU(2) chiral perturbation theory. Phys. Rev. D 78, 114509 (2008). arXiv:0804.0473 [hep-lat] ADSGoogle Scholar
  109. 109.
    T. Ishikawa et al. (CP-PACS/JLQCD 07), Light quark masses from unquenched lattice QCD. Phys. Rev. D 78, 011502 (2008). arXiv:0704.1937 [hep-lat] ADSGoogle Scholar
  110. 110.
    Q. Mason, H.D. Trottier, R. Horgan, C.T.H. Davies, G.P. Lepage (HPQCD 05), High-precision determination of the light-quark masses from realistic lattice QCD. Phys. Rev. D 73, 114501 (2006). hep-ph/0511160 ADSGoogle Scholar
  111. 111.
    C. Aubin et al. (HPQCD/MILC/UKQCD 04), First determination of the strange and light quark masses from full lattice QCD. Phys. Rev. D 70, 031504 (2004). hep-lat/0405022 ADSGoogle Scholar
  112. 112.
    J. Garden, J. Heitger, R. Sommer, H. Wittig (ALPHA 99), Precision computation of the strange quark’s mass in quenched QCD. Nucl. Phys. B 571, 237–256 (2000). hep-lat/9906013 ADSCrossRefGoogle Scholar
  113. 113.
    A.T. Lytle, Non-perturbative calculation of Z m using Asqtad fermions. PoS LAT2009, 202 (2009). arXiv:0910.3721 [hep-lat] Google Scholar
  114. 114.
    M. Lüscher, R. Narayanan, P. Weisz, U. Wolff, The Schrödinger functional: a renormalizable probe for non-Abelian gauge theories. Nucl. Phys. B 384, 168–228 (1992). hep-lat/9207009 ADSCrossRefGoogle Scholar
  115. 115.
    G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa, A. Vladikas, A general method for nonperturbative renormalization of lattice operators. Nucl. Phys. B 445, 81–108 (1995). hep-lat/9411010 ADSCrossRefGoogle Scholar
  116. 116.
    I. Allison et al. (HPQCD 08), High-precision charm-quark mass from current-current correlators in lattice and continuum QCD. Phys. Rev. D 78, 054513 (2008). arXiv:0805.2999 [hep-lat] ADSGoogle Scholar
  117. 117.
    A.I. Vainshtein et al., Sum rules for light quarks in quantum chromodynamics. Sov. J. Nucl. Phys. 27, 274 (1978) Google Scholar
  118. 118.
    S. Narison, Strange quark mass from e + e revisited and present status of light quark masses. Phys. Rev. D 74, 034013 (2006). hep-ph/0510108 ADSGoogle Scholar
  119. 119.
    M. Jamin, J.A. Oller, A. Pich, Scalar form factor and light quark masses. Phys. Rev. D 74, 074009 (2006). hep-ph/0605095 ADSGoogle Scholar
  120. 120.
    K.G. Chetyrkin, A. Khodjamirian, Strange quark mass from pseudoscalar sum rule with \(O(\alpha _{s}^{4})\) accuracy. Eur. Phys. J. C 46, 721–728 (2006). hep-ph/0512295 ADSCrossRefGoogle Scholar
  121. 121.
    C.A. Dominguez, N.F. Nasrallah, R. Röntsch, K. Schilcher, Light quark masses from QCD sum rules with minimal hadronic bias. Nucl. Phys. Proc. Suppl. 186, 133–136 (2009). arXiv:0808.3909 [hep-ph] ADSCrossRefGoogle Scholar
  122. 122.
    K. Nakamura et al. (PDG 10), Review of particle physics. J. Phys. G 37, 075021 (2010) ADSGoogle Scholar
  123. 123.
    K. Maltman, J. Kambor, m u+m d from isovector pseudoscalar sum rules. Phys. Lett. B 517, 332–338 (2001). hep-ph/0107060 ADSGoogle Scholar
  124. 124.
    T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, The four-loop β-function in quantum chromodynamics. Phys. Lett. B 400, 379–384 (1997). hep-ph/9701390 ADSGoogle Scholar
  125. 125.
    K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Strong coupling constant with flavour thresholds at four loops in the \(\overline{\mathrm{MS}}\) scheme. Phys. Rev. Lett. 79, 2184–2187 (1997). hep-ph/9706430 ADSCrossRefGoogle Scholar
  126. 126.
    K.G. Chetyrkin, A. Retey, Renormalization and running of quark mass and field in the regularization invariant and \(\overline{\mathrm{MS}}\) schemes at three and four loops. Nucl. Phys. B 583, 3–34 (2000). hep-ph/9910332 ADSCrossRefGoogle Scholar
  127. 127.
    S. Bethke, The 2009 World Average of α s(M Z). Eur. Phys. J. C 64, 689–703 (2009). arXiv:0908.1135 [hep-ph] ADSCrossRefGoogle Scholar
  128. 128.
    S. Weinberg, The problem of mass. Trans. New York Acad. Sci. 38, 185–201 (1977) Google Scholar
  129. 129.
    H. Leutwyler, The ratios of the light quark masses. Phys. Lett. B 378, 313–318 (1996). hep-ph/9602366 ADSGoogle Scholar
  130. 130.
    R. Kaiser, The η and the η′ at large N c. Diploma work, University of Bern, 1997 Google Scholar
  131. 131.
    H. Leutwyler, On the 1/N-expansion in chiral perturbation theory. Nucl. Phys. Proc. Suppl. 64, 223–231 (1998). hep-ph/9709408 ADSCrossRefGoogle Scholar
  132. 132.
    J.A. Oller, L. Roca, Non-perturbative study of the light pseudoscalar masses in chiral dynamics. Eur. Phys. J. A 34, 371–386 (2007). hep-ph/0608290 ADSGoogle Scholar
  133. 133.
    J. Gasser, H. Leutwyler, η→3π to one loop. Nucl. Phys. B 250, 539 (1985) ADSCrossRefGoogle Scholar
  134. 134.
    J. Kambor, C. Wiesendanger, D. Wyler, Final state interactions and Khuri-Treiman equations in η→3π decays. Nucl. Phys. B 465, 215–266 (1996). hep-ph/9509374 ADSCrossRefGoogle Scholar
  135. 135.
    A.V. Anisovich, H. Leutwyler, Dispersive analysis of the decay η→3π. Phys. Lett. B 375, 335–342 (1996). hep-ph/9601237 ADSGoogle Scholar
  136. 136.
    C. Ditsche, B. Kubis, U.-G. Meissner, Electromagnetic corrections in η→3π decays. Eur. Phys. J. C 60, 83–105 (2009). arXiv:0812.0344 [hep-ph] ADSCrossRefGoogle Scholar
  137. 137.
    G. Colangelo, S. Lanz, E. Passemar, A new dispersive analysis of η→3π. PoS CD09, 047 (2009). arXiv:0910.0765 [hep-ph] Google Scholar
  138. 138.
    J. Bijnens, K. Ghorbani, η→3π at two loops in chiral perturbation theory. J. High Energy Phys. 11, 030 (2007). arXiv:0709.0230 [hep-ph] MathSciNetADSCrossRefGoogle Scholar
  139. 139.
    M. Antonelli et al., An evaluation of |V us| and precise tests of the Standard Model from world data on leptonic and semileptonic kaon decays. Eur. Phys. J. C 69, 399–424 (2010). arXiv:1005.2323 [hep-ph] ADSCrossRefGoogle Scholar
  140. 140.
    J. Gasser, G.R.S. Zarnauskas, On the pion decay constant. Phys. Lett. B 693, 122–128 (2010). arXiv:1008.3479 [hep-ph] ADSGoogle Scholar
  141. 141.
    J.L. Rosner, S. Stone, Decay constants of charged pseudoscalar mesons. J. Phys. G 37, 075021 (2010). Review of Particle Physics, p. 861 Google Scholar
  142. 142.
    V. Cirigliano, H. Neufeld, A note on isospin violation in P l2(γ) decays. arXiv:1102.0563 [hep-ph]
  143. 143.
    I.S. Towner, J.C. Hardy, An improved calculation of the isospin-symmetry-breaking corrections to superallowed Fermi beta decay. Phys. Rev. C 77, 025501 (2008). arXiv:0710.3181 [nucl-th] ADSCrossRefGoogle Scholar
  144. 144.
    G.A. Miller, A. Schwenk, Isospin-symmetry-breaking corrections to superallowed Fermi beta decay: formalism and schematic models. Phys. Rev. C 78, 035501 (2008). arXiv:0805.0603 [nucl-th] ADSCrossRefGoogle Scholar
  145. 145.
    N. Auerbach, Coulomb corrections to superallowed beta decay in nuclei. Phys. Rev. C 79, 035502 (2009). arXiv:0811.4742 [nucl-th] ADSCrossRefGoogle Scholar
  146. 146.
    H. Liang, N. Van Giai, J. Meng, Isospin corrections for superallowed Fermi beta decay in self-consistent relativistic random-phase approximation approaches. Phys. Rev. C 79, 064316 (2009). arXiv:0904.3673 [nucl-th] ADSCrossRefGoogle Scholar
  147. 147.
    G.A. Miller, A. Schwenk, Isospin-symmetry-breaking corrections to superallowed Fermi beta decay: radial excitations. Phys. Rev. C 80, 064319 (2009). arXiv:0910.2790 [nucl-th] ADSCrossRefGoogle Scholar
  148. 148.
    I.S. Towner, J.C. Hardy, Comparative tests of isospin-symmetry-breaking corrections to superallowed 0+→0+ nuclear beta decay. arXiv:1007.5343 [nucl-th]
  149. 149.
    E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, Determination of m s and |V us| from hadronic tau decays. J. High Energy Phys. 01, 060 (2003). hep-ph/0212230 ADSCrossRefGoogle Scholar
  150. 150.
    E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, V us and m s from hadronic τ decays. Phys. Rev. Lett. 94, 011803 (2005). hep-ph/0408044 ADSCrossRefGoogle Scholar
  151. 151.
    K. Maltman, A mixed τ-electroproduction sum rule for V us. Phys. Lett. B 672, 257–263 (2009). arXiv:0811.1590 [hep-ph] ADSGoogle Scholar
  152. 152.
    A. Pich, R. Kass, talks given at CKM 08, Rome, Italy, 2008. http://ckm2008.roma1.infn.it
  153. 153.
    E. Gamiz, M. Jamin, A. Pich, J. Prades, F. Schwab, Theoretical progress on the V us determination from τ decays. PoS KAON, 008 (2008). arXiv:0709.0282 [hep-ph] Google Scholar
  154. 154.
    K. Maltman, C.E. Wolfe, S. Banerjee, J.M. Roney, I. Nugent, Status of the hadronic τ determination of V us. Int. J. Mod. Phys. A 23, 3191–3195 (2008). arXiv:0807.3195 [hep-ph] ADSCrossRefGoogle Scholar
  155. 155.
    K. Maltman, C.E. Wolfe, S. Banerjee, I.M. Nugent, J.M. Roney, Status of the hadronic τ decay determination of |V us|. Nucl. Phys. Proc. Suppl. 189, 175–180 (2009). arXiv:0906.1386 [hep-ph] ADSCrossRefGoogle Scholar
  156. 156.
    M. Beneke, M. Jamin, α s and the τ hadronic width: fixed-order, contour-improved and higher-order perturbation theory. J. High Energy Phys. 09, 044 (2008). arXiv:0806.3156 [hep-ph] ADSCrossRefGoogle Scholar
  157. 157.
    I. Caprini, J. Fischer, α s from τ decays: contour-improved versus fixed-order summation in a new QCD perturbation expansion. Eur. Phys. J. C 64, 35–45 (2009). arXiv:0906.5211 [hep-ph] ADSCrossRefGoogle Scholar
  158. 158.
    S. Menke, On the determination of α s from hadronic τ decays with contour-improved, fixed order and renormalon-chain perturbation theory. arXiv:0904.1796 [hep-ph]
  159. 159.
    P.A. Boyle et al. (RBC/UKQCD 10), Kπ form factors with reduced model dependence. Eur. Phys. J. C 69, 159–167 (2010). arXiv:1004.0886 [hep-lat] ADSCrossRefGoogle Scholar
  160. 160.
    P.A. Boyle et al. (RBC/UKQCD 07), K l3 semileptonic form factor from 2+1 flavour lattice QCD. Phys. Rev. Lett. 100, 141601 (2008). arXiv:0710.5136 [hep-lat] ADSCrossRefGoogle Scholar
  161. 161.
    V. Lubicz, F. Mescia, L. Orifici, S. Simula, C. Tarantino (ETM 10D), Improved analysis of the scalar and vector form factors of kaon semileptonic decays with N f=2 twisted-mass fermions. PoS LAT2010, 316 (2010). arXiv:1012.3573 [hep-lat] Google Scholar
  162. 162.
    V. Lubicz, F. Mescia, S. Simula, C. Tarantino (ETM 09A), Kπℓν semileptonic form factors from two-flavor lattice QCD. Phys. Rev. D 80, 111502 (2009). arXiv:0906.4728 [hep-lat] ADSGoogle Scholar
  163. 163.
    D. Brömmel et al. (QCDSF 07), Kaon semileptonic decay form factors from N f=2 non-perturbatively O(a)-improved Wilson fermions. PoS LAT2007, 364 (2007). arXiv:0710.2100 [hep-lat] Google Scholar
  164. 164.
    C. Dawson, T. Izubuchi, T. Kaneko, S. Sasaki, A. Soni (RBC 06), Vector form factor in K l3 semileptonic decay with two flavors of dynamical domain-wall quarks. Phys. Rev. D 74, 114502 (2006). hep-ph/0607162 ADSGoogle Scholar
  165. 165.
    N. Tsutsui et al. (JLQCD 05), Kaon semileptonic decay form factors in two-flavor QCD. PoS LAT2005, 357 (2006). hep-lat/0510068 Google Scholar
  166. 166.
    M. Ademollo, R. Gatto, Nonrenormalization theorem for the strangeness violating vector currents. Phys. Rev. Lett. 13, 264–265 (1964) ADSCrossRefGoogle Scholar
  167. 167.
    G. Furlan, F. Lannoy, C. Rossetti, G. Segré, Symmetry-breaking corrections to weak vector currents. Nuovo Cimento 38, 1747 (1965) CrossRefGoogle Scholar
  168. 168.
    J. Gasser, H. Leutwyler, Low-energy expansion of meson form factors. Nucl. Phys. B 250, 517–538 (1985) ADSCrossRefGoogle Scholar
  169. 169.
    D. Becirevic, G. Martinelli, G. Villadoro, The Ademollo-Gatto theorem for lattice semileptonic decays. Phys. Lett. B 633, 84–88 (2006). hep-lat/0508013 ADSGoogle Scholar
  170. 170.
    J.M. Flynn, C.T. Sachrajda (RBC 08), SU(2) chiral perturbation theory for Kl3 decay amplitudes. Nucl. Phys. B 812, 64–80 (2009). arXiv:0809.1229 [hep-ph] ADSMATHCrossRefGoogle Scholar
  171. 171.
    F. Farchioni, G. Herdoiza, K. Jansen, M. Petschlies, C. Urbach et al. (ETM 10E), Pseudoscalar decay constants from N f=2+1+1 twisted mass lattice QCD. PoS LAT2010, 128 (2010). arXiv:1012.0200 [hep-lat] Google Scholar
  172. 172.
    A. Bazavov et al. (MILC 10), Results for light pseudoscalar mesons. PoS LAT2010, 074 (2010). arXiv:1012.0868 [hep-lat] Google Scholar
  173. 173.
    J. Noaki et al. (JLQCD/TWQCD 09A), Chiral properties of light mesons with N f=2+1 overlap fermions. PoS LAT2009, 096 (2009). arXiv:0910.5532 [hep-lat] Google Scholar
  174. 174.
    C. Aubin, J. Laiho, R.S. Van de Water (Aubin 08), Light pseudoscalar meson masses and decay constants from mixed action lattice QCD. PoS LAT2008, 105 (2008). arXiv:0810.4328 [hep-lat] Google Scholar
  175. 175.
    Y. Kuramashi (PACS-CS 08A), PACS-CS results for 2+1 flavor lattice QCD simulation on and off the physical point. PoS LAT2008, 018 (2008). arXiv:0811.2630 [hep-lat] Google Scholar
  176. 176.
    E. Follana, C.T.H. Davies, G.P. Lepage, J. Shigemitsu (HPQCD/UKQCD 07), High precision determination of the π, K, D and D s decay constants from lattice QCD. Phys. Rev. Lett. 100, 062002 (2008). arXiv:0706.1726 [hep-lat] ADSCrossRefGoogle Scholar
  177. 177.
    S.R. Beane, P.F. Bedaque, K. Orginos, M.J. Savage (NPLQCD 06), f K/f π in full QCD with domain wall valence quarks. Phys. Rev. D 75, 094501 (2007). hep-lat/0606023 ADSGoogle Scholar
  178. 178.
    B. Blossier et al. (ETM 09), Pseudoscalar decay constants of kaon and D-mesons from N f=2 twisted mass Lattice QCD. J. High Energy Phys. 07, 043 (2009). arXiv:0904.0954 [hep-lat] Google Scholar
  179. 179.
    G. Schierholz et al. (QCDSF/UKQCD 07), Probing the chiral limit with clover fermions I: The meson sector, talk given at Lattice 2007, Regensburg, Germany, PoS LAT2007, 133. http://www.physik.uni-regensburg.de/lat07/hevea/schierholz.pdf
  180. 180.
    H. Leutwyler, M. Roos (LR 84), Determination of the elements V us and V ud of the Kobayashi-Maskawa matrix. Z. Phys. C 25, 91 (1984) ADSGoogle Scholar
  181. 181.
    P. Post, K. Schilcher, K l3 form factors at order p 6 in chiral perturbation theory. Eur. Phys. J. C 25, 427–443 (2002). hep-ph/0112352 ADSCrossRefGoogle Scholar
  182. 182.
    J. Bijnens, P. Talavera, K l3 decays in chiral perturbation theory. Nucl. Phys. B 669, 341–362 (2003). hep-ph/0303103 ADSCrossRefGoogle Scholar
  183. 183.
    M. Jamin, J.A. Oller, A. Pich, Order p 6 chiral couplings from the scalar form factor. J. High Energy Phys. 02, 047 (2004). hep-ph/0401080 ADSCrossRefGoogle Scholar
  184. 184.
    V. Cirigliano et al., The Green function and SU(3) breaking in K l3 decays. J. High Energy Phys. 04, 006 (2005). hep-ph/0503108 ADSCrossRefGoogle Scholar
  185. 185.
    A. Kastner, H. Neufeld, The K l3 scalar form factors in the Standard Model. Eur. Phys. J. C 57, 541–556 (2008). arXiv:0805.2222 [hep-ph] ADSCrossRefGoogle Scholar
  186. 186.
    V. Bernard, M. Oertel, E. Passemar, J. Stern, Dispersive representation and shape of the K 3 form factors: robustness. Phys. Rev. D 80, 034034 (2009). arXiv:0903.1654 [hep-ph] ADSGoogle Scholar
  187. 187.
    V. Bernard, E. Passemar, Chiral extrapolation of the strangeness changing form factor. J. High Energy Phys. 04, 001 (2010). arXiv:0912.3792 [hep-ph] ADSCrossRefGoogle Scholar
  188. 188.
    E. Passemar, Dispersive approach to K 3 form factors, in NA62 Physics Handbook Workshop (CERN 2009) (2009) Google Scholar
  189. 189.
    E. Passemar, Precision SM calculations and theoretical interests beyond the SM in K 2 and K 3 decays. PoS KAON09, 024 (2009). arXiv:1003.4696 [hep-ph] Google Scholar
  190. 190.
    S. Di Vita et al., Vector and scalar form factors for K- and D-meson semileptonic decays from twisted mass fermions with N f=2. PoS LAT2009, 257 (2009). arXiv:0910.4845 [hep-ph] Google Scholar
  191. 191.
    D. Becirevic et al. (SPQcdR 04), The Kπ vector form factor at zero momentum transfer on the lattice. Nucl. Phys. B 705, 339–362 (2005). hep-ph/0403217 ADSCrossRefGoogle Scholar
  192. 192.
    R. Kowalewski, T. Mannel, Determination of V cb and V ub. J. Phys. G 37, 075021 (2010). Review of Particle Physics, p. 1014 Google Scholar
  193. 193.
    M.E. Fisher, V. Privman, First-order transitions breaking O(n) symmetry: Finite-size scaling. Phys. Rev. B 32, 447–464 (1985) MathSciNetADSCrossRefGoogle Scholar
  194. 194.
    E. Brezin, J. Zinn-Justin, Finite size effects in phase transitions. Nucl. Phys. B 257, 867 (1985) ADSCrossRefGoogle Scholar
  195. 195.
    J. Gasser, H. Leutwyler, Light quarks at low temperatures. Phys. Lett. B 184, 83 (1987) ADSGoogle Scholar
  196. 196.
    J. Gasser, H. Leutwyler, Thermodynamics of chiral symmetry. Phys. Lett. B 188, 477 (1987) ADSGoogle Scholar
  197. 197.
    J. Gasser, H. Leutwyler, Spontaneously broken symmetries: effective Lagrangians at finite volume. Nucl. Phys. B 307, 763 (1988) ADSCrossRefGoogle Scholar
  198. 198.
    P. Hasenfratz, H. Leutwyler, Goldstone boson related finite size effects in field theory and critical phenomena with O(N) symmetry. Nucl. Phys. B 343, 241–284 (1990) ADSCrossRefGoogle Scholar
  199. 199.
    G. Colangelo, J. Gasser, H. Leutwyler (CGL 01), ππ scattering. Nucl. Phys. B 603, 125–179 (2001). hep-ph/0103088 ADSCrossRefGoogle Scholar
  200. 200.
    F.C. Hansen, Finite size effects in spontaneously broken SU(N)×SU(N) theories. Nucl. Phys. B 345, 685–708 (1990) ADSCrossRefGoogle Scholar
  201. 201.
    F.C. Hansen, H. Leutwyler, Charge correlations and topological susceptibility in QCD. Nucl. Phys. B 350, 201–227 (1991) ADSCrossRefGoogle Scholar
  202. 202.
    H. Leutwyler, A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD. Phys. Rev. D 46, 5607–5632 (1992) MathSciNetADSGoogle Scholar
  203. 203.
    P.H. Damgaard, M.C. Diamantini, P. Hernandez, K. Jansen, Finite-size scaling of meson propagators. Nucl. Phys. B 629, 445–478 (2002). hep-lat/0112016 ADSMATHCrossRefGoogle Scholar
  204. 204.
    P.H. Damgaard, P. Hernandez, K. Jansen, M. Laine, L. Lellouch, Finite-size scaling of vector and axial current correlators. Nucl. Phys. B 656, 226–238 (2003). hep-lat/0211020 ADSMATHCrossRefGoogle Scholar
  205. 205.
    S. Aoki, H. Fukaya, Chiral perturbation theory in a theta vacuum. Phys. Rev. D 81, 034022 (2010). arXiv:0906.4852 [hep-lat] ADSGoogle Scholar
  206. 206.
    F. Bernardoni, P.H. Damgaard, H. Fukaya, P. Hernandez, Finite volume scaling of Pseudo Nambu–Goldstone Bosons in QCD. J. High Energy Phys. 10, 008 (2008). arXiv:0808.1986 [hep-lat] ADSCrossRefGoogle Scholar
  207. 207.
    P.H. Damgaard, H. Fukaya, The chiral condensate in a finite volume. J. High Energy Phys. 01, 052 (2009). arXiv:0812.2797 [pdf] ADSCrossRefGoogle Scholar
  208. 208.
    H. Leutwyler, Energy levels of light quarks confined to a box. Phys. Lett. B 189, 197 (1987) MathSciNetADSGoogle Scholar
  209. 209.
    P. Hasenfratz, The QCD rotator in the chiral limit. Nucl. Phys. B 828, 201–214 (2010). arXiv:0909.3419 [hep-th] ADSMATHCrossRefGoogle Scholar
  210. 210.
    F. Niedermayer, C. Weiermann, The rotator spectrum in the δ-regime of the O(n) effective field theory in 3 and 4 dimensions. Nucl. Phys. B 842, 248–263 (2011). arXiv:1006.5855 [hep-lat] ADSMATHCrossRefGoogle Scholar
  211. 211.
    M. Weingart, The QCD rotator with a light quark mass. arXiv:1006.5076 [hep-lat]
  212. 212.
    A. Hasenfratz, P. Hasenfratz, F. Niedermayer, D. Hierl, A. Schafer, First results in QCD with 2+1 light flavors using the fixed-point action. PoS LAT2006, 178 (2006). hep-lat/0610096 Google Scholar
  213. 213.
    W. Bietenholz et al. (QCDSF 10), Pion in a box. Phys. Lett. B 687, 410–414 (2010). arXiv:1002.1696 [hep-lat] ADSGoogle Scholar
  214. 214.
    P. Di Vecchia, G. Veneziano, Chiral dynamics in the large N limit. Nucl. Phys. B 171, 253 (1980) ADSCrossRefGoogle Scholar
  215. 215.
    Y.-Y. Mao, T.-W. Chiu (TWQCD 09), Topological susceptibility to the one-loop order in chiral perturbation theory. Phys. Rev. D 80, 034502 (2009). arXiv:0903.2146 [hep-lat] ADSGoogle Scholar
  216. 216.
    L. Giusti, M. Lüscher (CERN 08), Chiral symmetry breaking and the Banks–Casher relation in lattice QCD with Wilson quarks. J. High Energy Phys. 03, 013 (2009). arXiv:0812.3638 [hep-lat] ADSCrossRefGoogle Scholar
  217. 217.
    T. Banks, A. Casher, Chiral symmetry breaking in confining theories. Nucl. Phys. B 169, 103 (1980) MathSciNetADSCrossRefGoogle Scholar
  218. 218.
    E.V. Shuryak, J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD. Nucl. Phys. A 560, 306–320 (1993). hep-th/9212088 ADSCrossRefGoogle Scholar
  219. 219.
    J.J.M. Verbaarschot, I. Zahed, Spectral density of the QCD Dirac operator near zero virtuality. Phys. Rev. Lett. 70, 3852–3855 (1993). hep-th/9303012 ADSCrossRefGoogle Scholar
  220. 220.
    J.J.M. Verbaarschot, The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way. Phys. Rev. Lett. 72, 2531–2533 (1994). hep-th/9401059 MathSciNetADSCrossRefGoogle Scholar
  221. 221.
    J.J.M. Verbaarschot, T. Wettig, Random matrix theory and chiral symmetry in QCD. Annu. Rev. Nucl. Part. Sci. 50, 343–410 (2000). hep-ph/0003017 ADSCrossRefGoogle Scholar
  222. 222.
    S.M. Nishigaki, P.H. Damgaard, T. Wettig, Smallest Dirac eigenvalue distribution from random matrix theory. Phys. Rev. D 58, 087704 (1998). hep-th/9803007 ADSGoogle Scholar
  223. 223.
    P.H. Damgaard, S.M. Nishigaki, Distribution of the k-th smallest Dirac operator eigenvalue. Phys. Rev. D 63, 045012 (2001). hep-th/0006111 MathSciNetADSGoogle Scholar
  224. 224.
    F. Basile, G. Akemann, Equivalence of QCD in the epsilon-regime and chiral random matrix theory with or without chemical potential. J. High Energy Phys. 12, 043 (2007). arXiv:0710.0376 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  225. 225.
    G. Akemann, P.H. Damgaard, J.C. Osborn, K. Splittorff, A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential. Nucl. Phys. B 766, 34–67 (2007). hep-th/0609059 MathSciNetADSMATHCrossRefGoogle Scholar
  226. 226.
    C. Lehner, S. Hashimoto, T. Wettig, The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential. J. High Energy Phys. 06, 028 (2010). arXiv:1004.5584 [hep-lat] ADSCrossRefGoogle Scholar
  227. 227.
    C. Lehner, J. Bloch, S. Hashimoto, T. Wettig, Geometry dependence of RMT-based methods to extract the low-energy constants Sigma and F. arXiv:1101.5576 [hep-lat]
  228. 228.
    L. Del Debbio, L. Giusti, M. Lüscher, R. Petronzio, N. Tantalo (CERN-TOV 05), Stability of lattice QCD simulations and the thermodynamic limit. J. High Energy Phys. 02, 011 (2006). hep-lat/0512021 ADSCrossRefGoogle Scholar
  229. 229.
    H. Fukaya et al., Two-flavor lattice QCD in the epsilon-regime and chiral random matrix theory. Phys. Rev. D 76, 054503 (2007). arXiv:0705.3322 [hep-lat] ADSGoogle Scholar
  230. 230.
    C.B. Lang, P. Majumdar, W. Ortner, The condensate for two dynamical chirally improved quarks in QCD. Phys. Lett. B 649, 225–229 (2007). hep-lat/0611010 ADSGoogle Scholar
  231. 231.
    T. DeGrand, Z. Liu, S. Schaefer, Quark condensate in two-flavor QCD. Phys. Rev. D 74, 094504 (2006). hep-lat/0608019 ADSGoogle Scholar
  232. 232.
    P. Hasenfratz et al., 2+1 Flavor QCD simulated in the epsilon-regime in different topological sectors. J. High Energy Phys. 11, 100 (2009). arXiv:0707.0071 [hep-lat] ADSCrossRefGoogle Scholar
  233. 233.
    T. DeGrand, S. Schaefer, Parameters of the lowest order chiral Lagrangian from fermion eigenvalues. Phys. Rev. D 76, 094509 (2007). arXiv:0708.1731 [hep-lat] ADSGoogle Scholar
  234. 234.
    J.F. Donoghue, J. Gasser, H. Leutwyler, The decay of a light Higgs boson. Nucl. Phys. B 343, 341–368 (1990) ADSCrossRefGoogle Scholar
  235. 235.
    J. Bijnens, G. Colangelo, P. Talavera (BCT 98), The vector and scalar form factors of the pion to two loops. J. High Energy Phys. 05, 014 (1998). hep-ph/9805389 ADSGoogle Scholar
  236. 236.
    R. Frezzotti, V. Lubicz, S. Simula (ETM 08), Electromagnetic form factor of the pion from twisted-mass lattice QCD at N f=2. Phys. Rev. D 79, 074506 (2009). arXiv:0812.4042 [hep-lat] ADSGoogle Scholar
  237. 237.
    T. Kaneko et al. (JLQCD/TWQCD 08), Pion vector and scalar form factors with dynamical overlap quarks. PoS LAT2008, 158 (2008). arXiv:0810.2590 [hep-lat] Google Scholar
  238. 238.
    R. Baron et al. (ETM 09C), Light meson physics from maximally twisted mass lattice QCD. J. High Energy Phys. 08, 097 (2010). arXiv:0911.5061 [hep-lat] ADSCrossRefGoogle Scholar
  239. 239.
    J. Gasser, C. Haefeli, M.A. Ivanov, M. Schmid, Integrating out strange quarks in ChPT. Phys. Lett. B 652, 21–26 (2007). arXiv:0706.0955 [hep-ph] ADSGoogle Scholar
  240. 240.
    J. Gasser, C. Haefeli, M.A. Ivanov, M. Schmid, Integrating out strange quarks in ChPT: terms at order p 6. Phys. Lett. B 675, 49–53 (2009). arXiv:0903.0801 [hep-ph] ADSGoogle Scholar
  241. 241.
    H. Fukaya et al. (JLQCD/TWQCD 10), Determination of the chiral condensate from QCD Dirac spectrum on the lattice. Phys. Rev. D 83, 074501 (2011). arXiv:1012.4052 [hep-lat] ADSGoogle Scholar
  242. 242.
    H. Fukaya et al. (JLQCD 09), Determination of the chiral condensate from 2+1-flavor lattice QCD. Phys. Rev. Lett. 104, 122002 (2010). arXiv:0911.5555 [hep-lat] ADSCrossRefGoogle Scholar
  243. 243.
    T.-W. Chiu, T.-H. Hsieh, P.-K. Tseng (TWQCD 08), Topological susceptibility in 2+1 flavors lattice QCD with domain-wall fermions. Phys. Lett. B 671, 135–138 (2009). arXiv:0810.3406 [hep-lat] ADSGoogle Scholar
  244. 244.
    T.W. Chiu et al. (JLQCD/TWQCD 08B), Topological susceptibility in (2+1)-flavor lattice QCD with overlap fermion. PoS LAT2008, 072 (2008). arXiv:0810.0085 [hep-lat] Google Scholar
  245. 245.
    F. Bernardoni, P. Hernandez, N. Garron, S. Necco, C. Pena (Bernardoni 10), Probing the chiral regime of N f=2 QCD with mixed actions. Phys. Rev. D 83, 054503 (2011). arXiv:1008.1870 [hep-lat] ADSGoogle Scholar
  246. 246.
    S. Aoki et al. (JLQCD/TWQCD 07A), Topological susceptibility in two-flavor lattice QCD with exact chiral symmetry. Phys. Lett. B 665, 294–297 (2008). arXiv:0710.1130 [hep-lat] ADSGoogle Scholar
  247. 247.
    K. Jansen, A. Shindler (ETM 09B), The epsilon regime of chiral perturbation theory with Wilson-type fermions. PoS LAT2009, 070 (2009). arXiv:0911.1931 [hep-lat] Google Scholar
  248. 248.
    A. Hasenfratz, R. Hoffmann, S. Schaefer (HHS 08), Low energy chiral constants from epsilon-regime simulations with improved Wilson fermions. Phys. Rev. D 78, 054511 (2008). arXiv:0806.4586 [hep-lat] ADSGoogle Scholar
  249. 249.
    H. Fukaya et al. (JLQCD/TWQCD 07), Lattice study of meson correlators in the epsilon-regime of two-flavor QCD. Phys. Rev. D 77, 074503 (2008). arXiv:0711.4965 [hep-lat] ADSGoogle Scholar
  250. 250.
    R. Baron et al. (ETM 11), Light hadrons from N f=2+1+1 dynamical twisted mass fermions. PoS LAT2010, 123 (2010). arXiv:1101.0518 [hep-lat] Google Scholar
  251. 251.
    P.A. Boyle et al. (RBC/UKQCD 08A), The pion’s electromagnetic form factor at small momentum transfer in full lattice QCD. J. High Energy Phys. 07, 112 (2008). arXiv:0804.3971 [hep-lat] ADSGoogle Scholar
  252. 252.
    G. Colangelo, S. Dürr (CD 03), The pion mass in finite volume. Eur. Phys. J. C 33, 543–553 (2004). hep-lat/0311023 ADSCrossRefGoogle Scholar
  253. 253.
    S. Aoki et al. (JLQCD/TWQCD 09), Pion form factors from two-flavor lattice QCD with exact chiral symmetry. Phys. Rev. D 80, 034508 (2009). arXiv:0905.2465 [hep-lat] ADSGoogle Scholar
  254. 254.
    S. Dürr, \(M_{\pi}^{2}\) versus m q: Comparing CP-PACS and UKQCD data to chiral perturbation theory. Eur. Phys. J. C 29, 383–395 (2003). hep-lat/0208051 ADSCrossRefGoogle Scholar
  255. 255.
    L. Del Debbio, L. Giusti, M. Lüscher, R. Petronzio, N. Tantalo (CERN-TOV 06), QCD with light Wilson quarks on fine lattices (I): first experiences and physics results. J. High Energy Phys. 02, 056 (2007). hep-lat/0610059 ADSCrossRefGoogle Scholar
  256. 256.
    N.H. Fuchs, H. Sazdjian, J. Stern, How to probe the scale of (anti-q q) in chiral perturbation theory. Phys. Lett. B 269, 183–188 (1991) ADSGoogle Scholar
  257. 257.
    J. Stern, H. Sazdjian, N.H. Fuchs, What pi–pi scattering tells us about chiral perturbation theory. Phys. Rev. D 47, 3814–3838 (1993). hep-ph/9301244 ADSGoogle Scholar
  258. 258.
    S. Descotes-Genon, L. Girlanda, J. Stern, Paramagnetic effect of light quark loops on chiral symmetry breaking. J. High Energy Phys. 01, 041 (2000). hep-ph/9910537 ADSCrossRefGoogle Scholar
  259. 259.
    V. Bernard, S. Descotes-Genon, G. Toucas, Chiral dynamics with strange quarks in the light of recent lattice simulations. arXiv:1009.5066 [hep-ph]
  260. 260.
    F.D.R. Bonnet, R.G. Edwards, G.T. Fleming, R. Lewis, D.G. Richards (LHP 04), Lattice computations of the pion form factor. Phys. Rev. D 72, 054506 (2005). hep-lat/0411028 ADSGoogle Scholar
  261. 261.
    D. Brommel et al. (QCDSF/UKQCD 06A), The pion form factor from lattice QCD with two dynamical flavours. Eur. Phys. J. C 51, 335–345 (2007). hep-lat/0608021 ADSCrossRefGoogle Scholar
  262. 262.
    S.R. Amendolia et al., A measurement of the space-like pion electromagnetic form factor. Nucl. Phys. B 277, 168 (1986) ADSCrossRefGoogle Scholar
  263. 263.
    J. Bijnens, N. Danielsson, T.A. Lähde, Three-flavor partially quenched chiral perturbation theory at NNLO for meson masses and decay constants. Phys. Rev. D 73, 074509 (2006). hep-lat/0602003 ADSGoogle Scholar
  264. 264.
    J. Bijnens, Status of strong ChPT. PoS EFT09, 022 (2009). arXiv:0904.3713 [hep-ph] Google Scholar
  265. 265.
    E. Shintani et al. (JLQCD 08A), S-parameter and pseudo-Nambu–Goldstone boson mass from lattice QCD. Phys. Rev. Lett. 101, 242001 (2008). arXiv:0806.4222 [hep-lat] ADSCrossRefGoogle Scholar
  266. 266.
    G.C. Branco, L. Lavoura, J.P. Silva, CP Violation, Int. Ser. Monogr. Phys., vol. 103 (Springer, Berlin, 1999), p. 536 Google Scholar
  267. 267.
    G. Buchalla, A.J. Buras, M.E. Lautenbacher, Weak decays beyond leading logarithms. Rev. Mod. Phys. 68, 1125–1144 (1996). hep-ph/9512380 ADSCrossRefGoogle Scholar
  268. 268.
    A.J. Buras, Weak Hamiltonian, CP violation and rare decays, in Les Houches 1997, Probing the Standard Model of Particle Interactions, pt. 1 (1997), pp. 281–539. hep-ph/9806471 Google Scholar
  269. 269.
    T. Inami, C.S. Lim, Effects of superheavy quarks and leptons in low-energy weak processes \(K_{L}\to\mu\bar{\mu}\), \(K^{+}\to\pi^{+}\nu\bar{\nu}\) and \(K^{0}\leftrightarrow\bar{K}^{0}\). Prog. Theor. Phys. 65, 297 (1981) ADSCrossRefGoogle Scholar
  270. 270.
    C. Aubin, J. Laiho, R.S. Van de Water (Aubin 09), The neutral kaon mixing parameter B K from unquenched mixed-action lattice QCD. Phys. Rev. D 81, 014507 (2010). arXiv:0905.3947 [hep-lat] ADSGoogle Scholar
  271. 271.
    J. Brod, M. Gorbahn, ε K at next-to-next-to-leading order: The charm-top-quark contribution. Phys. Rev. D 82, 094026 (2010). arXiv:1007.0684 [hep-ph] ADSGoogle Scholar
  272. 272.
    U. Nierste, private communication, 2010 Google Scholar
  273. 273.
    K. Anikeev et al., B physics at the Tevatron: Run II and beyond. hep-ph/0201071
  274. 274.
    U. Nierste, Three lectures on meson mixing and CKM phenomenology, in Dubna 2008, Heavy Quark Physics HQP08 (2008), pp. 1–39. arXiv:0904.1869 [hep-ph] Google Scholar
  275. 275.
    A.J. Buras, D. Guadagnoli, Correlations among new CP violating effects in ΔF=2 observables. Phys. Rev. D 78, 033005 (2008). arXiv:0805.3887 [hep-ph] ADSGoogle Scholar
  276. 276.
    A.J. Buras, D. Guadagnoli, G. Isidori, On ε K beyond lowest order in the operator product expansion. Phys. Lett. B 688, 309–313 (2010). arXiv:1002.3612 [hep-ph] ADSGoogle Scholar
  277. 277.
    D. Becirevic et al., \(K^{0} \bar{K}^{0}\) mixing with Wilson fermions without subtractions. Phys. Lett. B 487, 74–80 (2000). hep-lat/0005013 ADSGoogle Scholar
  278. 278.
    P. Dimopoulos et al. (ALPHA 06), A precise determination of B K in quenched QCD. Nucl. Phys. B 749, 69–108 (2006). hep-ph/0601002 ADSCrossRefGoogle Scholar
  279. 279.
    P.H. Ginsparg, K.G. Wilson, A remnant of chiral symmetry on the lattice. Phys. Rev. D 25, 2649 (1982) ADSGoogle Scholar
  280. 280.
    M. Della Morte et al. (ALPHA 04), Computation of the strong coupling in QCD with two dynamical flavours. Nucl. Phys. B 713, 378–406 (2005). hep-lat/0411025 ADSCrossRefGoogle Scholar
  281. 281.
    S. Aoki et al. (JLQCD 08), B K with two flavors of dynamical overlap fermions. Phys. Rev. D 77, 094503 (2008). arXiv:0801.4186 [hep-lat] ADSGoogle Scholar
  282. 282.
    J. Kim, C. Jung, H.-J. Kim, W. Lee, S.R. Sharpe (SWME 11), Finite volume effects in B K with improved staggered fermions. arXiv:1101.2685 [hep-lat]
  283. 283.
    Y. Aoki et al. (RBC/UKQCD 10B), Continuum limit of B K from 2+1 flavor domain wall QCD. arXiv:1012.4178 [hep-lat]
  284. 284.
    T. Bae et al. (SWME 10), B K using HYP-smeared staggered fermions in N f=2+1 unquenched QCD. Phys. Rev. D 82, 114509 (2010). arXiv:1008.5179 [hep-lat] ADSGoogle Scholar
  285. 285.
    D.J. Antonio et al. (RBC/UKQCD 07A), Neutral kaon mixing from 2+1 flavor domain wall QCD. Phys. Rev. Lett. 100, 032001 (2008). hep-ph/0702042 ADSCrossRefGoogle Scholar
  286. 286.
    E. Gamiz et al. (HPQCD/UKQCD 06), Unquenched determination of the kaon parameter B K from improved staggered fermions. Phys. Rev. D 73, 114502 (2006). hep-lat/0603023 ADSGoogle Scholar
  287. 287.
    M. Constantinou et al. (ETM 10A), BK-parameter from N f=2 twisted mass lattice QCD. Phys. Rev. D 83, 014505 (2011). arXiv:1009.5606 [hep-lat] ADSGoogle Scholar
  288. 288.
    Y. Aoki et al. (RBC 04), Lattice QCD with two dynamical flavors of domain wall fermions. Phys. Rev. D 72, 114505 (2005). hep-lat/0411006 ADSGoogle Scholar
  289. 289.
    J.M. Flynn, F. Mescia, A.S.B. Tariq (UKQCD 04), Sea quark effects in B K from N f=2 clover-improved Wilson fermions. J. High Energy Phys. 11, 049 (2004). hep-lat/0406013 CrossRefGoogle Scholar
  290. 290.
    A. Hasenfratz, F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking. Phys. Rev. D 64, 034504 (2001). hep-lat/0103029 ADSGoogle Scholar
  291. 291.
    Y. Aoki et al., Non-perturbative renormalization of quark bilinear operators and B K using domain wall fermions. Phys. Rev. D 78, 054510 (2008). arXiv:0712.1061 [hep-lat] ADSGoogle Scholar
  292. 292.
    V. Bertone et al. (ETM 09D), Kaon oscillations in the Standard Model and beyond using N f=2 dynamical quarks. PoS LAT2009, 258 (2009). arXiv:0910.4838 [hep-lat] Google Scholar
  293. 293.
    P. Dimopoulos, H. Simma, A. Vladikas (ALPHA 09), Quenched B K-parameter from Osterwalder-Seiler tmQCD quarks and mass-splitting discretization effects. J. High Energy Phys. 07, 007 (2009). arXiv:0902.1074 [hep-lat] Google Scholar
  294. 294.
    Y. Nakamura, S. Aoki, Y. Taniguchi, T. Yoshie (CP-PACS 08), Precise determination of B K and light quark masses in quenched domain-wall QCD. Phys. Rev. D 78, 034502 (2008). arXiv:0803.2569 [hep-lat] ADSGoogle Scholar
  295. 295.
    P. Dimopoulos et al. (ALPHA 07), Flavour symmetry restoration and kaon weak matrix elements in quenched twisted mass QCD. Nucl. Phys. B 776, 258–285 (2007). hep-lat/0702017 ADSCrossRefGoogle Scholar
  296. 296.
    S. Aoki et al. (JLQCD 97), Kaon B parameter from quenched lattice QCD. Phys. Rev. Lett. 80, 5271–5274 (1998). hep-lat/9710073 ADSCrossRefGoogle Scholar
  297. 297.
    K.G. Wilson, Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974) ADSCrossRefGoogle Scholar
  298. 298.
    M. Lüscher, P. Weisz, On-shell improved lattice gauge theories. Commun. Math. Phys. 97, 59 (1985) ADSCrossRefMATHGoogle Scholar
  299. 299.
    Y. Iwasaki, Renormalization group analysis of lattice theories and improved lattice action: two-dimensional nonlinear O(N) sigma model. Nucl. Phys. B 258, 141–156 (1985) ADSCrossRefGoogle Scholar
  300. 300.
    T. Takaishi, Heavy quark potential and effective actions on blocked configurations. Phys. Rev. D 54, 1050–1053 (1996) ADSGoogle Scholar
  301. 301.
    P. de Forcrand et al., Renormalization group flow of SU(3) lattice gauge theory: numerical studies in a two coupling space. Nucl. Phys. B 577, 263–278 (2000). hep-lat/9911033 ADSCrossRefGoogle Scholar
  302. 302.
    G.P. Lepage, P.B. Mackenzie, On the viability of lattice perturbation theory. Phys. Rev. D 48, 2250–2264 (1993). hep-lat/9209022 ADSGoogle Scholar
  303. 303.
    M. Lüscher, S. Sint, R. Sommer, P. Weisz, U. Wolff, Non-perturbative O(a) improvement of lattice QCD. Nucl. Phys. B 491, 323–343 (1997). hep-lat/9609035 ADSCrossRefGoogle Scholar
  304. 304.
    L. Susskind, Lattice fermions. Phys. Rev. D 16, 3031–3039 (1977) ADSGoogle Scholar
  305. 305.
    K. Orginos, D. Toussaint, R.L. Sugar (MILC 99), Variants of fattening and flavor symmetry restoration. Phys. Rev. D 60, 054503 (1999). hep-lat/9903032 ADSGoogle Scholar
  306. 306.
    E. Follana et al. (HPQCD 06), Highly improved staggered quarks on the lattice, with applications to charm physics. Phys. Rev. D 75, 054502 (2007). hep-lat/0610092 ADSGoogle Scholar
  307. 307.
    M. Creutz, Why rooting fails. PoS LAT2007, 007 (2007). arXiv:0708.1295 [hep-lat] Google Scholar
  308. 308.
    P. Hasenfratz, V. Laliena, F. Niedermayer, The index theorem in QCD with a finite cut-off. Phys. Lett. B 427, 125–131 (1998). hep-lat/9801021 ADSGoogle Scholar
  309. 309.
    M. Lüscher, Exact chiral symmetry on the lattice and the Ginsparg–Wilson relation. Phys. Lett. B 428, 342–345 (1998). hep-lat/9802011 ADSGoogle Scholar
  310. 310.
    D.B. Kaplan, A Method for simulating chiral fermions on the lattice. Phys. Lett. B 288, 342–347 (1992). hep-lat/9206013 MathSciNetADSGoogle Scholar
  311. 311.
    V. Furman, Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions. Nucl. Phys. B 439, 54–78 (1995). hep-lat/9405004 ADSCrossRefGoogle Scholar
  312. 312.
    H. Neuberger, Exactly massless quarks on the lattice. Phys. Lett. B 417, 141–144 (1998). hep-lat/9707022 MathSciNetADSGoogle Scholar
  313. 313.
    P. Hasenfratz et al., The construction of generalized Dirac operators on the lattice. Int. J. Mod. Phys. C 12, 691–708 (2001). hep-lat/0003013 MathSciNetADSCrossRefGoogle Scholar
  314. 314.
    P. Hasenfratz, S. Hauswirth, T. Jorg, F. Niedermayer, K. Holland, Testing the fixed-point QCD action and the construction of chiral currents. Nucl. Phys. B 643, 280–320 (2002). hep-lat/0205010 ADSCrossRefGoogle Scholar
  315. 315.
    C. Gattringer, A new approach to Ginsparg–Wilson fermions. Phys. Rev. D 63, 114501 (2001). hep-lat/0003005 ADSGoogle Scholar
  316. 316.
    A. Hasenfratz, R. Hoffmann, S. Schaefer, Hypercubic smeared links for dynamical fermions. J. High Energy Phys. 05, 029 (2007). hep-lat/0702028 ADSCrossRefGoogle Scholar
  317. 317.
    C. Morningstar, M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD. Phys. Rev. D 69, 054501 (2004). hep-lat/0311018 ADSGoogle Scholar
  318. 318.
    S. Dürr et al. (BMW 08A), Scaling study of dynamical smeared-link clover fermions. Phys. Rev. D 79, 014501 (2009). arXiv:0802.2706 [hep-lat] Google Scholar
  319. 319.
    S. Capitani, S. Dürr, C. Hoelbling, Rationale for UV-filtered clover fermions. J. High Energy Phys. 11, 028 (2006). hep-lat/0607006 ADSCrossRefGoogle Scholar
  320. 320.
    R. Sommer, A new way to set the energy scale in lattice gauge theories and its applications to the static force and α s in SU(2) Yang-Mills theory. Nucl. Phys. B 411, 839–854 (1994). hep-lat/9310022 ADSCrossRefGoogle Scholar
  321. 321.
    C.W. Bernard et al., The static quark potential in three flavor QCD. Phys. Rev. D 62, 034503 (2000). hep-lat/0002028 ADSCrossRefGoogle Scholar
  322. 322.
    R. Arthur, P.A. Boyle (RBC Collaboration), Step scaling with off-shell renormalisation. arXiv:1006.0422 [hep-lat]
  323. 323.
    C. Bernard et al. (MILC 07), Status of the MILC light pseudoscalar meson project. PoS LAT2007, 090 (2007). arXiv:0710.1118 [hep-lat] Google Scholar
  324. 324.
    G. Colangelo, S. Dürr, C. Haefeli (CDH 05), Finite volume effects for meson masses and decay constants. Nucl. Phys. B 721, 136–174 (2005). hep-lat/0503014 ADSMATHCrossRefGoogle Scholar
  325. 325.
    G. Herdoiza, private communication, 2011 Google Scholar
  326. 326.
    R. Brower, S. Chandrasekharan, J.W. Negele, U. Wiese, QCD at fixed topology. Phys. Lett. B 560, 64–74 (2003). hep-lat/0302005 ADSMATHCrossRefGoogle Scholar
  327. 327.
    O. Bär, S. Necco, S. Schaefer, The epsilon regime with Wilson fermions. J. High Energy Phys. 03, 006 (2009). arXiv:0812.2403 [hep-lat] CrossRefGoogle Scholar
  328. 328.
    T. Bunton, F.-J. Jiang, B. Tiburzi, Extrapolations of lattice meson form factors. Phys. Rev. D 74, 034514 (2006). hep-lat/0607001 ADSCrossRefGoogle Scholar
  329. 329.
    B. Borasoy, R. Lewis, Volume dependences from lattice chiral perturbation theory. Phys. Rev. D 71, 014033 (2005). hep-lat/0410042 ADSCrossRefGoogle Scholar
  330. 330.
    S. Aoki, H. Fukaya, S. Hashimoto, T. Onogi, Finite volume QCD at fixed topological charge. Phys. Rev. D 76, 054508 (2007). arXiv:0707.0396 [hep-lat] ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2011

Authors and Affiliations

  • FLAG working group of FLAVIANET
  • G. Colangelo
    • 1
  • S. Dürr
    • 2
    • 3
  • A. Jüttner
    • 4
  • L. Lellouch
    • 5
  • H. Leutwyler
    • 1
  • V. Lubicz
    • 6
  • S. Necco
    • 4
  • C. T. Sachrajda
    • 7
  • S. Simula
    • 8
  • A. Vladikas
    • 9
  • U. Wenger
    • 1
  • H. Wittig
    • 10
  1. 1.Albert Einstein Center for Fundamental Physics, Institut für Theoretische PhysikUniversität BernBernSwitzerland
  2. 2.Bergische Universität WuppertalWuppertalGermany
  3. 3.Jülich Supercomputing CentreForschungszentrum JülichJülichGermany
  4. 4.Physics Department, TH UnitCERNGeneva 23Switzerland
  5. 5.Centre de Physique ThéoriqueMarseilleFrance
  6. 6.Dipartimento di FisicaUniversità Roma Tre, and INFNRomeItaly
  7. 7.School of Physics and AstronomyUniversity of SouthamptonSouthamptonUK
  8. 8.INFN, Sezione di Roma TreRomeItaly
  9. 9.INFN, Sezione di Tor Vergata, c/o Dipartimento di FisicaUniversità di Roma Tor VergataRomeItaly
  10. 10.Institut für Kernphysik and Helmholtz Institute MainzUniversity of MainzMainzGermany

Personalised recommendations