Journal of High Energy Physics

, 2013:175 | Cite as

Chiral condensate from the twisted mass Dirac operator spectrum

  • Krzysztof CichyEmail author
  • Elena Garcia-Ramos
  • Karl Jansen
Open Access


We present the results of our computation of the dimensionless chiral condensate r 0Σ1/3 with N f = 2 and N f = 2 + 1 + 1 flavours of maximally twisted mass fermions. The condensate is determined from the Dirac operator spectrum, applying the spectral projector method proposed by Giusti and Lüscher. We use 3 lattice spacings and several quark masses at each lattice spacing to perform the chiral and continuum extrapolations. We study the effect of the dynamical strange and charm quarks by comparing our results for N f = 2 and N f = 2 + 1 + 1 dynamical flavours.


Lattice QCD Spontaneous Symmetry Breaking 


  1. [1]
    PACS-CS collaboration, S. Aoki et al., 2+1 Flavor Lattice QCD toward the Physical Point, Phys. Rev. D 79 (2009) 034503 [arXiv:0807.1661] [INSPIRE].ADSGoogle Scholar
  2. [2]
    RBC-UKQCD collaboration, C. Allton et al., Physical Results from 2+1 Flavor Domain Wall QCD and SU(2) Chiral Perturbation Theory, Phys. Rev. D 78 (2008) 114509 [arXiv:0804.0473] [INSPIRE].ADSGoogle Scholar
  3. [3]
    RBC Collaboration, UKQCD collaboration, Y. Aoki et al., Continuum Limit Physics from 2+1 Flavor Domain Wall QCD, Phys. Rev. D 83 (2011) 074508 [arXiv:1011.0892] [INSPIRE].ADSGoogle Scholar
  4. [4]
    A. Bazavov et al., Nonperturbative QCD simulations with 2 + 1 flavors of improved staggered quarks, Rev. Mod. Phys. 82 (2010) 1349 [arXiv:0903.3598] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    A. Bazavov et al., Staggered chiral perturbation theory in the two-flavor case and SU(2) analysis of the MILC data, PoS(LATTICE2010)083 [arXiv:1011.1792] [INSPIRE].
  6. [6]
    ETM collaboration, R. Baron et al., Light Meson Physics from Maximally Twisted Mass Lattice QCD, JHEP 08 (2010) 097 [arXiv:0911.5061] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    JLQCD, TWQCD collaboration, H. Fukaya et al., Determination of the chiral condensate from QCD Dirac spectrum on the lattice, Phys. Rev. D 83 (2011) 074501 [arXiv:1012.4052] [INSPIRE].ADSGoogle Scholar
  8. [8]
    V. Bernard, S. Descotes-Genon and G. Toucas, Topological susceptibility on the lattice and the three-flavour quark condensate, JHEP 06 (2012) 051 [arXiv:1203.0508] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    S. Borsányi et al., SU(2) chiral perturbation theory low-energy constants from 2 + 1 flavor staggered lattice simulations, Phys. Rev. D 88 (2013) 014513 [arXiv:1205.0788] [INSPIRE].ADSGoogle Scholar
  10. [10]
    S. Dürr, Topological susceptibility in full QCD: Lattice results versus the prediction from the QCD partition function with granularity, Nucl. Phys. B 611 (2001) 281 [hep-lat/0103011] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    TWQCD collaboration, T.-W. Chiu, T.-H. Hsieh and P.-K. Tseng, Topological susceptibility in 2+1 flavors lattice QCD with domain-wall fermions, Phys. Lett. B 671 (2009) 135 [arXiv:0810.3406] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    T.W. Chiu, T.H. Hsieh and Y.Y. Mao, Topological Susceptibility in Two Flavors Lattice QCD with the Optimal Domain-Wall Fermion, Phys. Lett. B 702 (2011) 131 [arXiv:1105.4414] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    F. Bernardoni, P. Hernández, N. Garron, S. Necco and C. Pena, Probing the chiral regime of N f = 2 QCD with mixed actions, Phys. Rev. D 83 (2011) 054503 [arXiv:1008.1870] [INSPIRE].ADSGoogle Scholar
  14. [14]
    ETM collaboration, R. Frezzotti, V. Lubicz and S. Simula, Electromagnetic form factor of the pion from twisted-mass lattice QCD at N f = 2, Phys. Rev. D 79 (2009) 074506 [arXiv:0812.4042] [INSPIRE].ADSGoogle Scholar
  15. [15]
    P. Hernández, K. Jansen and L. Lellouch, Finite size scaling of the quark condensate in quenched lattice QCD, Phys. Lett. B 469 (1999) 198 [hep-lat/9907022] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    P. Damgaard, U.M. Heller, R. Niclasen and K. Rummukainen, Eigenvalue distributions of the QCD Dirac operator, Phys. Lett. B 495 (2000) 263 [hep-lat/0007041] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    T.A. DeGrand and S. Schaefer, Chiral properties of two-flavor QCD in small volume and at large lattice spacing, Phys. Rev. D 72 (2005) 054503 [hep-lat/0506021] [INSPIRE].ADSGoogle Scholar
  18. [18]
    C. Lang, P. Majumdar and W. Ortner, The Condensate for two dynamical chirally improved quarks in QCD, Phys. Lett. B 649 (2007) 225 [hep-lat/0611010] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    JLQCD collaboration, H. Fukaya et al., Two-flavor lattice QCD simulation in the ϵ-regime with exact chiral symmetry, Phys. Rev. Lett. 98 (2007) 172001 [hep-lat/0702003] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    TWQCD collaboration, H. Fukaya et al., Two-flavor lattice QCD in the ϵ-regime and chiral Random Matrix Theory, Phys. Rev. D 76 (2007) 054503 [arXiv:0705.3322] [INSPIRE].ADSGoogle Scholar
  21. [21]
    A. Hasenfratz, R. Hoffmann and S. Schaefer, Low energy chiral constants from ϵ-regime simulations with improved Wilson fermions, Phys. Rev. D 78 (2008) 054511 [arXiv:0806.4586] [INSPIRE].ADSGoogle Scholar
  22. [22]
    K. Jansen and A. Shindler, The ϵ-regime of chiral perturbation theory with Wilson-type fermions, PoS(LAT2009)070 [arXiv:0911.1931] [INSPIRE].
  23. [23]
    K. Splittorff and J. Verbaarschot, The Microscopic Twisted Mass Dirac Spectrum, Phys. Rev. D 85 (2012) 105008 [arXiv:1201.1361] [INSPIRE].ADSGoogle Scholar
  24. [24]
    S. Necco and A. Shindler, Spectral density of the Hermitean Wilson Dirac operator: a NLO computation in chiral perturbation theory, JHEP 04 (2011) 031 [arXiv:1101.1778] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. Necco and A. Shindler, On the spectral density of the Wilson operator, PoS(LATTICE2011)250 [arXiv:1108.1950] [INSPIRE].
  26. [26]
    S. Necco and A. Shindler, Corrections to the Banks-Casher relation with Wilson quarks, PoS(CD12)056 [arXiv:1302.5595] [INSPIRE].
  27. [27]
    F. Burger, V. Lubicz, M. Muller-Preussker, S. Simula and C. Urbach, Quark mass and chiral condensate from the Wilson twisted mass lattice quark propagator, Phys. Rev. D 87 (2013) 034514 [arXiv:1210.0838] [INSPIRE].ADSGoogle Scholar
  28. [28]
    C. McNeile et al., Direct determination of the strange and light quark condensates from full lattice QCD, Phys. Rev. D 87 (2013) 034503 [arXiv:1211.6577] [INSPIRE].ADSGoogle Scholar
  29. [29]
    G. Colangelo et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C 71 (2011) 1695 [arXiv:1011.4408] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    T. Banks and A. Casher, Chiral Symmetry Breaking in Confining Theories, Nucl. Phys. B 169 (1980) 103 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    L. Giusti and M. Lüscher, Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks, JHEP 03 (2009) 013 [arXiv:0812.3638] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    K. Cichy, V. Drach, E. Garcia-Ramos and K. Jansen, Topological susceptibility and chiral condensate with N f = 2 + 1 + 1 dynamical flavors of maximally twisted mass fermions, PoS(LATTICE2011)102 [arXiv:1111.3322] [INSPIRE].
  33. [33]
    ETM collaboration, P. Boucaud et al., Dynamical twisted mass fermions with light quarks, Phys. Lett. B 650 (2007) 304 [hep-lat/0701012] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    ETM collaboration, P. Boucaud et al., Dynamical Twisted Mass Fermions with Light Quarks: Simulation and Analysis Details, Comput. Phys. Commun. 179 (2008) 695 [arXiv:0803.0224] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    R. Baron et al., Light hadrons from lattice QCD with light (u,d), strange and charm dynamical quarks, JHEP 06 (2010) 111 [arXiv:1004.5284] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    European Twisted Mass collaboration, R. Baron et al., Computing K and D meson masses with N f = 2+1+1 twisted mass lattice QCD, Comput. Phys. Commun. 182 (2011) 299 [arXiv:1005.2042] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  37. [37]
    ETM collaboration, R. Baron et al., Light hadrons from N f = 2 + 1 + 1 dynamical twisted mass fermions, PoS(LATTICE2010)123 [arXiv:1101.0518] [INSPIRE].
  38. [38]
    P. Weisz, Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 1., Nucl. Phys. B 212 (1983) 1 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    Y. Iwasaki, Renormalization Group Analysis of Lattice Theories and Improved Lattice Action: Two-Dimensional Nonlinear O(N) σ-model, Nucl. Phys. B 258 (1985) 141 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    Y. Iwasaki, K. Kanaya, T. Kaneko and T. Yoshie, Scaling in SU(3) pure gauge theory with a renormalization group improved action, Phys. Rev. D 56 (1997) 151 [hep-lat/9610023] [INSPIRE].ADSGoogle Scholar
  41. [41]
    Alpha collaboration, R. Frezzotti, P.A. Grassi, S. Sint and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058 [hep-lat/0101001] [INSPIRE].Google Scholar
  42. [42]
    R. Frezzotti and G. Rossi, Chirally improving Wilson fermions. 1. O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    R. Frezzotti and G. Rossi, Chirally improving Wilson fermions. II. Four-quark operators, JHEP 10 (2004) 070 [hep-lat/0407002] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    A. Shindler, Twisted mass lattice QCD, Phys. Rept. 461 (2008) 37 [arXiv:0707.4093] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    R. Frezzotti and G. Rossi, Twisted mass lattice QCD with mass nondegenerate quarks, Nucl. Phys. Proc. Suppl. 128 (2004) 193 [hep-lat/0311008] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    T. Chiarappa et al., Numerical simulation of QCD with u, d, s and c quarks in the twisted-mass Wilson formulation, Eur. Phys. J. C 50 (2007) 373 [hep-lat/0606011] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    F. Farchioni et al., Exploring the phase structure of lattice QCD with twisted mass quarks, Nucl. Phys. Proc. Suppl. 140 (2005) 240 [hep-lat/0409098] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    F. Farchioni et al., The Phase structure of lattice QCD with Wilson quarks and renormalization group improved gluons, Eur. Phys. J. C 42 (2005) 73 [hep-lat/0410031] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    R. Frezzotti, G. Martinelli, M. Papinutto and G. Rossi, Reducing cutoff effects in maximally twisted lattice QCD close to the chiral limit, JHEP 04 (2006) 038 [hep-lat/0503034] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    XLF collaboration, K. Jansen, M. Papinutto, A. Shindler, C. Urbach and I. Wetzorke, Quenched scaling of Wilson twisted mass fermions, JHEP 09 (2005) 071 [hep-lat/0507010] [INSPIRE].Google Scholar
  51. [51]
    ETM collaboration, B. Blossier et al., Average up/down, strange and charm quark masses with N f = 2 twisted mass lattice QCD, Phys. Rev. D 82 (2010) 114513 [arXiv:1010.3659] [INSPIRE].ADSGoogle Scholar
  52. [52]
    ETM collaboration, K. Ottnad et al., η and ηmesons from N f =2+1+1 twisted mass lattice QCD, JHEP 11 (2012) 048 [arXiv:1206.6719] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    ETM collaboration, M. Constantinou et al., Non-perturbative renormalization of quark bilinear operators with N f = 2 (tmQCD) Wilson fermions and the tree-level improved gauge action, JHEP 08 (2010) 068 [arXiv:1004.1115] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos and F. Stylianou, Renormalization constants of local operators for Wilson type improved fermions, Phys. Rev. D 86 (2012) 014505 [arXiv:1201.5025] [INSPIRE].ADSGoogle Scholar
  55. [55]
    D. Palao, ETMC preliminary result for N f = 4 renormalization constants, private communication.Google Scholar
  56. [56]
    ETM collaboration, P. Dimopoulos et al., Renormalization constants for Wilson fermion lattice QCD with four dynamical flavours, PoS(LATTICE2010)235 [arXiv:1101.1877] [INSPIRE].
  57. [57]
    ETM collaboration, B. Blossier et al., Renormalisation constants of quark bilinears in lattice QCD with four dynamical Wilson quarks, PoS(LATTICE2011)233 [arXiv:1112.1540] [INSPIRE].
  58. [58]
    European Twisted Mass collaboration, C. Alexandrou et al., Light baryon masses with dynamical twisted mass fermions, Phys. Rev. D 78 (2008) 014509 [arXiv:0803.3190] [INSPIRE].ADSGoogle Scholar
  59. [59]
    K. Cichy, E. Garcia-Ramos, K. Jansen and A. Shindler, Computation of the chiral condensate using N f = 2 and N f = 2 + 1 + 1 dynamical flavors of twisted mass fermions, PoS(LATTICE 2013)128.
  60. [60]
    C. Bernard et al., Status of the MILC light pseudoscalar meson project, PoS(LAT2007)090 [arXiv:0710.1118] [INSPIRE].
  61. [61]
    Y. Aoki et al., The QCD transition temperature: results with physical masses in the continuum limit II., JHEP 06 (2009) 088 [arXiv:0903.4155] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    HPQCD collaboration, R. Dowdall et al., The Upsilon spectrum and the determination of the lattice spacing from lattice QCD including charm quarks in the sea, Phys. Rev. D 85 (2012) 054509 [arXiv:1110.6887] [INSPIRE].ADSGoogle Scholar
  63. [63]
    K. Jansen and C. Urbach, tmLQCD: A Program suite to simulate Wilson Twisted mass Lattice QCD, Comput. Phys. Commun. 180 (2009) 2717 [arXiv:0905.3331] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  64. [64]
    ALPHA collaboration, U. Wolff, Monte Carlo errors with less errors, Comput. Phys. Commun. 156 (2004) 143 [Erratum ibid. 176 (2007) 383] [hep-lat/0306017] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • Krzysztof Cichy
    • 1
    • 2
    Email author
  • Elena Garcia-Ramos
    • 1
    • 3
  • Karl Jansen
    • 1
    • 4
  1. 1.NIC, DESYZeuthenGermany
  2. 2.Faculty of PhysicsAdam Mickiewicz UniversityPoznanPoland
  3. 3.Humboldt Universität zu BerlinBerlinGermany
  4. 4.Department of PhysicsUniversity of CyprusNicosiaCyprus

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