Journal of High Energy Physics

, 2014:119 | Cite as

Topological susceptibility from the twisted mass Dirac operator spectrum

  • The ETM collaboration
  • Krzysztof Cichy
  • Elena Garcia-Ramos
  • Karl Jansen
Open Access


We present results of our computation of the topological susceptibility with N f = 2 and N f = 2 + 1 + 1 flavours of maximally twisted mass fermions, using the method of spectral projectors. We perform a detailed study of the quark mass dependence and discretization effects. We make an attempt to confront our data with chiral perturbation theory and extract the chiral condensate from the quark mass dependence of the topological susceptibility. We compare the value with the results of our direct computation from the slope of the mode number. We emphasize the role of autocorrelations and the necessity of long Monte Carlo runs to obtain results with good precision. We also show our results for the spectral projector computation of the ratio of renormalization constants Z P /Z S .


Lattice QCD Lattice Gauge Field Theories Lattice Quantum Field Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    E. Witten, Current algebra theorems for the U(1) Goldstone boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. Veneziano, U(1) without instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    M. Creutz, Anomalies, gauge field topology and the lattice, Annals Phys. 326 (2011) 911 [arXiv:1007.5502] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    P. Hasenfratz, V. Laliena and F. Niedermayer, The index theorem in QCD with a finite cutoff, Phys. Lett. B 427 (1998) 125 [hep-lat/9801021] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    L. Giusti, G. Rossi, M. Testa and G. Veneziano, The U(A)(1) problem on the lattice with Ginsparg-Wilson fermions, Nucl. Phys. B 628 (2002) 234 [hep-lat/0108009] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    L. Giusti, G. Rossi and M. Testa, Topological susceptibility in full QCD with Ginsparg-Wilson fermions, Phys. Lett. B 587 (2004) 157 [hep-lat/0402027] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M. Lüscher, Topological effects in QCD and the problem of short distance singularities, Phys. Lett. B 593 (2004) 296 [hep-th/0404034] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    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
  9. [9]
    M. Lüscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP 09 (2010) 110 [arXiv:1008.0732] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    S. Schaefer and F. Virotta, Autocorrelations in hybrid Monte Carlo simulations, PoS (LATTICE 2010) 042 [arXiv:1011.5151] [INSPIRE].
  11. [11]
    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
  12. [12]
    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
  13. [13]
    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
  14. [14]
    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
  15. [15]
    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].ADSCrossRefMATHGoogle Scholar
  16. [16]
    ETM collaboration, R. Baron et al., Light hadrons from N f = 2 + 1 + 1 dynamical twisted mass fermions, PoS (LATTICE 2010) 123 [arXiv:1101.0518] [INSPIRE].
  17. [17]
    P. Weisz, Continuum limit improved lattice action for pure Yang-Mills theory. 1, Nucl. Phys. B 212 (1983) 1 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    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
  19. [19]
    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
  20. [20]
    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
  21. [21]
    R. Frezzotti and G. Rossi, Chirally improving Wilson fermions. 1. O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    R. Frezzotti and G. Rossi, Chirally improving Wilson fermions. II. Four-quark operators, JHEP 10 (2004) 070 [hep-lat/0407002] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Shindler, Twisted mass lattice QCD, Phys. Rept. 461 (2008) 37 [arXiv:0707.4093] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    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
  25. [25]
    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
  26. [26]
    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
  27. [27]
    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
  28. [28]
    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
  29. [29]
    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
  30. [30]
    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
  31. [31]
    ETM collaboration, K. Jansen, F. Karbstein, A. Nagy and M. Wagner, \( {\varLambda_{{\overline{M}S}}} \) from the static potential for QCD with N f = 2 dynamical quark flavors, JHEP 01 (2012) 025 [arXiv:1110.6859] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    ETM collaboration, K. Ottnad et al., η and η mesons from N f = 2 + 1 + 1 twisted mass lattice QCD, JHEP 11 (2012) 048 [arXiv:1206.6719] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    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
  34. [34]
    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
  35. [35]
    K. Cichy, K. Jansen and P. Korcyl, Non-perturbative renormalization in coordinate space for N f = 2 maximally twisted mass fermions with tree-level Symanzik improved gauge action, Nucl. Phys. B 865 (2012) 268 [arXiv:1207.0628] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D. Palao, ETMC preliminary result for N f = 4 renormalization constants, private communication.Google Scholar
  37. [37]
    ETM collaboration, B. Blossier et al., Renormalisation constants of quark bilinears in lattice QCD with four dynamical Wilson quarks, PoS (LATTICE 2011) 233 [arXiv:1112.1540] [INSPIRE].
  38. [38]
    ETM collaboration, P. Dimopoulos et al., Renormalization constants for Wilson fermion lattice QCD with four dynamical flavours, PoS (LATTICE 2010) 235 [arXiv:1101.1877] [INSPIRE].
  39. [39]
    G. Colangelo, S. Dürr and C. Haefeli, Finite volume effects for meson masses and decay constants, Nucl. Phys. B 721 (2005) 136 [hep-lat/0503014] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    K. Cichy, E. Garcia-Ramos and K. Jansen, Chiral condensate from the twisted mass Dirac operator spectrum, JHEP 10 (2013) 175 [arXiv:1303.1954] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    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].ADSCrossRefMATHGoogle Scholar
  42. [42]
    K. Cichy, E. Garcia-Ramos, K. Jansen and A. Shindler, Topological susceptibility from twisted mass fermions using spectral projectors, PoS (LATTICE 2013) 129 [arXiv:1312.3535] [INSPIRE].
  43. [43]
    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 [arXiv:1312.3534] [INSPIRE].
  44. [44]
    K. Cichy, E. Garcia-Ramos, K. Jansen and A. Shindler, Short distance singularities and automatic O(a) improvement, in preparation.Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • The ETM collaboration
  • Krzysztof Cichy
    • 1
    • 2
  • Elena Garcia-Ramos
    • 1
    • 3
  • Karl Jansen
    • 1
    • 4
  1. 1.NIC, DESYZeuthenGermany
  2. 2.Adam Mickiewicz University, Faculty of PhysicsPoznanPoland
  3. 3.Humboldt Universität zu BerlinBerlinGermany
  4. 4.Department of PhysicsUniversity of CyprusNicosiaCyprus

Personalised recommendations