Advertisement

Topological susceptibility on the lattice and the three-flavour quark condensate

  • Véronique Bernard
  • Sébastien Descotes-Genon
  • Guillaume Toucas
Open Access
Article

Abstract

We reanalyse the topological susceptibility assuming the possibility of a significant paramagnetic suppression of the three-flavour quark condensate and a correlated enhancement of vacuum fluctuations of \( s\overline s \) pairs. Using the framework of resummed χPT, we point out that simulations performed near the physical point, with a significant mass hierarchy between u, d and s dynamical quarks, are not able to disentangle the contributions from the quark condensate and sea \( s\overline s \)-pair fluctuations, and that simulations with three light quark masses of the same order are better suited for this purpose. We perform a combined fit of recent RBC/UKQCD data on pseudoscalar masses and decay constants as well as the topological susceptibility, and we reconsider the determination of lattice spacings in our framework, working out the consequences on the parameters of the chiral Lagrangian. We obtain (Σ(3; 2 GeV))1/3 = 243 ± 12 MeV for the three-flavour quark condensate in the chiral limit. We notice a significant suppression compared to the two-flavour quark condensate Σ(2; 2 GeV)/Σ(3; 2 GeV) = 1.51 ± 0.11 and we confirm previous findings of a competition between leading order and next-to-leading order contributions in three-flavour chiral series.

Keywords

Chiral Lagrangians Lattice QCD QCD 

References

  1. [1]
    G. Colangelo, J. Gasser and H. Leutwyler, The quark condensate from K(e4) decays, Phys. Rev. Lett. 86 (2001) 5008 [hep-ph/0103063] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    S. Descotes-Genon, N. Fuchs, L. Girlanda and J. Stern, Analysis and interpretation of new low-energy pi pi scattering data, Eur. Phys. J. C 24 (2002) 469 [hep-ph/0112088] [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    NA48/2 collaboration, J. Batley et al., New high statistics measurement of K(e4) decay form factors and pi pi scattering phase shifts, Eur. Phys. J. C 54 (2008) 411 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    G. Colangelo, J. Gasser and A. Rusetsky, Isospin breaking in K(l4) decays, Eur. Phys. J. C 59 (2009) 777 [arXiv:0811.0775] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S. Descotes-Genon and M. Knecht, Two-loop representations of low-energy pion form factors and pi-pi scattering phases in the presence of isospin breaking, Eur. Phys. J. C 72 (2012) 1962 [arXiv:1202.5886] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    V. Bernard, S. Descotes-Genon and M. Knecht, Two-loop dispersive representation of \( K_{{e4}}^{ + } \) form factors in the presence of isospin breaking, work in progress.Google Scholar
  7. [7]
    NA48/2 collaboration, J. Batley et al., Determination of the S-wave pi pi scattering lengths from a study of K ± → π ± π 0 π 0 decays, Eur. Phys. J. C 64 (2009) 589 [arXiv:0912.2165] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    NA48/2 collaboration, J. Batley et al., Observation of a cusp-like structure in the pi0 pi0 invariant mass distribution from K ± → π ± π 0 π 0 decay and determination of the ππ scattering lengths, Phys. Lett. B 633 (2006) 173 [hep-ex/0511056] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    N. Cabibbo, Determination of the a 0 − a 2 pion scattering length from K + → π + π 0 π 0 decay, Phys. Rev. Lett. 93 (2004) 121801 [hep-ph/0405001] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    N. Cabibbo and G. Isidori, Pion-pion scattering and the K → 3π decay amplitudes, JHEP 03 (2005) 021 [hep-ph/0502130] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, Cusps in K → 3π decays, Phys. Lett. B 638 (2006) 187 [hep-ph/0604084] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Gasser, B. Kubis and A. Rusetsky, Cusps in K → 3π decays: a theoretical framework, Nucl. Phys. B 850 (2011) 96 [arXiv:1103.4273] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    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
  16. [16]
    S. Descotes-Genon, L. Girlanda and J. Stern, Paramagnetic effect of light quark loops on chiral symmetry breaking, JHEP 01 (2000) 041 [hep-ph/9910537] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    S. Descotes-Genon and J. Stern, Vacuum fluctuations of \( \overline q q \) and values of low-energy constants, Phys. Lett. B 488 (2000) 274 [hep-ph/0007082] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S. Descotes-Genon, Zweig rule violation in the scalar sector and values of low-energy constants, JHEP 03 (2001) 002 [hep-ph/0012221] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev. D 46 (1992) 5607 [INSPIRE].MathSciNetADSGoogle Scholar
  20. [20]
    S. Descotes-Genon and J. Stern, Finite volume analysis of N (f) induced chiral phase transitions, Phys. Rev. D 62 (2000) 054011 [hep-ph/9912234] [INSPIRE].ADSGoogle Scholar
  21. [21]
    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
  22. [22]
    Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE], update online at http://pdg.lbl.gov.ADSCrossRefGoogle Scholar
  23. [23]
    R. Peccei and H.R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D 16 (1977) 1791 [INSPIRE].ADSGoogle Scholar
  24. [24]
    R. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S. Aoki, H. Fukaya, S. Hashimoto and T. Onogi, Finite volume QCD at fixed topological charge, Phys. Rev. D 76 (2007) 054508 [arXiv:0707.0396] [INSPIRE].ADSGoogle Scholar
  26. [26]
    V. Bernard, S.Descotes-Genon and G. Toucas, work in progress.Google Scholar
  27. [27]
    R. Kaiser and H. Leutwyler, Large-N c in chiral perturbation theory, Eur. Phys. J. C 17 (2000) 623 [hep-ph/0007101] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    MILC collaboration, A. Bazavov et al., Topological susceptibility with the asqtad action, Phys. Rev. D 81 (2010) 114501 [arXiv:1003.5695] [INSPIRE].ADSGoogle Scholar
  30. [30]
    RBC, 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
  31. [31]
    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
  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, arXiv:1111.3322 [INSPIRE].
  33. [33]
    S. Dürr, Z. Fodor, C. Hölbling and T. Kurth, Precision study of the SU(3) topological susceptibility in the continuum, JHEP 04 (2007) 055 [hep-lat/0612021] [INSPIRE].CrossRefGoogle Scholar
  34. [34]
    L. Giusti, B. Taglienti and S. Petrarca, Towards a precise determination of the topological susceptibility in the SU(3) Yang-Mills theory, PoS(LAT2009)229 [arXiv:1002.0444] [INSPIRE].
  35. [35]
    TWQCD collaboration, Y.-Y. Mao and T.-W. Chiu, Topological Susceptibility to the One-Loop Order in Chiral Perturbation Theory, Phys. Rev. D 80 (2009) 034502 [arXiv:0903.2146] [INSPIRE].ADSGoogle Scholar
  36. [36]
    RBC, UKQCD collaboration, C. Allton et al., 2 + 1 flavor domain wall QCD on a (2 f m)3 lattice: Light meson spectroscopy with L s = 16, Phys. Rev. D 76 (2007) 014504 [hep-lat/0701013] [INSPIRE].ADSGoogle Scholar
  37. [37]
    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
  38. [38]
    P. Boyle et al., K(l3) semileptonic form-factor from 2 + 1 flavour lattice QCD, Phys. Rev. Lett. 100 (2008) 141601 [arXiv:0710.5136] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    P. Boyle et al., K → π form factors with reduced model dependence, Eur. Phys. J. C 69 (2010) 159 [arXiv:1004.0886] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    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
  41. [41]
    C. Bernard et al., Status of the MILC light pseudoscalar meson project, PoS(LATTICE 2007)090 [arXiv:0710.1118] [INSPIRE].
  42. [42]
    MILC collaboration, A. Bazavov et al., Results from the MILC collaborations SU(3) chiral perturbation theory analysis, PoS(LAT2009)079 [arXiv:0910.3618] [INSPIRE].
  43. [43]
    S. Descotes-Genon, L. Girlanda and J. Stern, Chiral order and fluctuations in multiflavor QCD, Eur. Phys. J. C 27 (2003) 115 [hep-ph/0207337] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    S. Descotes-Genon, N. Fuchs, L. Girlanda and J. Stern, Resumming QCD vacuum fluctuations in three flavor chiral perturbation theory, Eur. Phys. J. C 34 (2004) 201 [hep-ph/0311120] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Descotes-Genon, The role of strange sea quarks in chiral extrapolations on the lattice, Eur. Phys. J. C 40 (2005) 81 [hep-ph/0410233] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S. Descotes-Genon, Low-energy ππ and πK scatterings revisited in three-flavour resummed chiral perturbation theory, Eur. Phys. J. C 52 (2007) 141 [hep-ph/0703154] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    V. Bernard, S. Descotes-Genon and G. Toucas, Chiral dynamics with strange quarks in the light of recent lattice simulations, JHEP 01 (2011) 107 [arXiv:1009.5066] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    M. Kolesar and J. Novotny, πη scattering and the resummation of vacuum fluctuation in three-flavour χPT , Eur. Phys. J. C 56 (2008) 231 [arXiv:0802.1289] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M. Kolesar and J. Novotny, The η decay constant inresummedchiral perturbation theory, Fizika B 17 (2008) 57 [arXiv:0802.1151] [INSPIRE].ADSGoogle Scholar
  50. [50]
    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
  51. [51]
    S.R. Sharpe, Applications of Chiral Perturbation theory to lattice QCD, hep-lat/0607016 [INSPIRE].
  52. [52]
    M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  53. [53]
    D. Becirevic and G. Villadoro, Impact of the finite volume effects on the chiral behavior of f K and B K , Phys. Rev. D 69 (2004) 054010 [hep-lat/0311028] [INSPIRE].ADSGoogle Scholar
  54. [54]
    G. Colangelo and S. Dürr, The pion mass in finite volume, Eur. Phys. J. C 33 (2004) 543 [hep-lat/0311023] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    G. Colangelo and C. Haefeli, An asymptotic formula for the pion decay constant in a large volume, Phys. Lett. B 590 (2004) 258 [hep-lat/0403025] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    B.C. Tiburzi and A. Walker-Loud, Hyperons in Two Flavor Chiral Perturbation Theory, Phys. Lett. B 669 (2008) 246 [arXiv:0808.0482] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    Y. Aoki et al., Non-perturbative renormalization of quark bilinear operators and B K using domain wall fermions, Phys. Rev. D 78 (2008) 054510 [arXiv:0712.1061] [INSPIRE].ADSGoogle Scholar
  58. [58]
    J. Gasser and H. Leutwyler, Spontaneously Broken Symmetries: Effective Lagrangians at Finite Volume, Nucl. Phys. B 307 (1988) 763 [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    J. Gasser and H. Leutwyler, Thermodynamics of Chiral Symmetry, Phys. Lett. B 188 (1987) 477 [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    J. Gasser and H. Leutwyler, Light Quarks at Low Temperatures, Phys. Lett. B 184 (1987) 83 [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA 2012

Authors and Affiliations

  • Véronique Bernard
    • 1
  • Sébastien Descotes-Genon
    • 2
  • Guillaume Toucas
    • 2
  1. 1.Institut de Physique NucléaireOrsay CedexFrance
  2. 2.Laboratoire de Physique ThéoriqueOrsay CedexFrance

Personalised recommendations