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Lattice QCD without topology barriers

  • Martin LüscherEmail author
  • Stefan Schaefer
Open Access
Article

Abstract

As the continuum limit is approached, lattice QCD simulations tend to get trapped in the topological charge sectors of field space and may consequently give biased results in practice. We propose to bypass this problem by imposing open (Neumann) boundary conditions on the gauge field in the time direction. The topological charge can then flow in and out of the lattice, while many properties of the theory (the hadron spectrum, for example) are not affected. Extensive simulations of the SU(3) gauge theory, using the HMC and the closely related SMD algorithm, confirm the absence of topology barriers if these boundary conditions are chosen. Moreover, the calculated autocorrelation times are found to scale approximately like the square of the inverse lattice spacing, thus supporting the conjecture that the HMC algorithm is in the universality class of the Langevin equation.

Keywords

Lattice QCD Lattice Gauge Field Theories Stochastic Processes 

References

  1. [1]
    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [SPIRES].CrossRefGoogle Scholar
  2. [2]
    L. Del Debbio, H. Panagopoulos and E. Vicari, θ-dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [SPIRES].ADSCrossRefGoogle Scholar
  3. [3]
    S. Schaefer, R. Sommer and F. Virotta, Investigating the critical slowing down of QCD simulations, PoS LAT2009 (2009) 032 [arXiv:0910.1465] [SPIRES].
  4. [4]
    S. Schaefer, R. Sommer and F. Virotta, Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B 845 (2011) 93 [arXiv:1009.5228] [SPIRES].ADSCrossRefGoogle Scholar
  5. [5]
    M. Lüscher, Topology, the Wilson flow and the HMC algorithm, PoS Lattice 2010 (2010) 015 [arXiv:1009.5877] [SPIRES].
  6. [6]
    S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 (1987) 216 [SPIRES].ADSGoogle Scholar
  7. [7]
    M. Lüscher and S. Schaefer, Non-renormalizability of the HMC algorithm, JHEP 04 (2011) 104 [arXiv:1103.1810] [SPIRES].ADSCrossRefGoogle Scholar
  8. [8]
    J. Zinn-Justin, Renormalization and stochastic quantization, Nucl. Phys. B 275 (1986) 135 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    J. Zinn-Justin and D. Zwanziger, Ward identities for the stochastic quantization of gauge fields, Nucl. Phys. B 295 (1988) 297 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    M. Lüscher, R. Narayanan, P. Weisz and U. Wolff, The Schrödinger functional: A renormalizable probe for non-Abelian gauge theories, Nucl. Phys. B 384 (1992) 168 [hep-lat/9207009] [SPIRES].ADSCrossRefGoogle Scholar
  11. [11]
    S. Sint, On the Schrödinger functional in QCD, Nucl. Phys. B 421 (1994) 135 [hep-lat/9312079] [SPIRES].ADSCrossRefGoogle Scholar
  12. [12]
    M. Lüscher, The Schrödinger functional in lattice QCD with exact chiral symmetry, JHEP 05 (2006) 042 [hep-lat/0603029] [SPIRES].CrossRefGoogle Scholar
  13. [13]
    K. Symanzik, Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    M. Lüscher and P. Weisz, O(a) improvement of the axial current in lattice QCD to one-loop order of perturbation theory, Nucl. Phys. B 479 (1996) 429 [hep-lat/9606016] [SPIRES].ADSCrossRefGoogle Scholar
  15. [15]
    M. Lüscher, Construction of a selfadjoint, strictly positive transfer matrix for Euclidean lattice gauge theories, Commun. Math. Phys. 54 (1977) 283 [SPIRES].ADSCrossRefGoogle Scholar
  16. [16]
    B. Sheikholeslami and R. Wohlert, Improved continuum limit lattice action for QCD with Wilson fermions, Nucl. Phys. B 259 (1985) 572 [SPIRES].ADSCrossRefGoogle Scholar
  17. [17]
    M. Lüscher, S. Sint, R. Sommer and P. Weisz, Chiral symmetry and O(a) improvement in lattice QCD, Nucl. Phys. B 478 (1996) 365 [hep-lat/9605038] [SPIRES].ADSCrossRefGoogle Scholar
  18. [18]
    Y. Taniguchi, Schrödinger functional formalism with Ginsparg-Wilson fermion, JHEP 12 (2005) 037 [hep-lat/0412024] [SPIRES].ADSCrossRefGoogle Scholar
  19. [19]
    Y. Taniguchi, Schrödinger functional formalism with domain-wall fermion, JHEP 10 (2006) 027 [hep-lat/0604002] [SPIRES].ADSCrossRefGoogle Scholar
  20. [20]
    A.M. Horowitz, Stochastic quantization in phase space, Phys. Lett. B 156 (1985) 89 [SPIRES].ADSGoogle Scholar
  21. [21]
    A.M. Horowitz, The second order Langevin equation and numerical simulations, Nucl. Phys. B 280 (1987) 510 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    A.M. Horowitz, A generalized guided Monte Carlo algorithm, Phys. Lett. B 268 (1991) 247 [SPIRES].ADSGoogle Scholar
  23. [23]
    A.D. Kennedy and B. Pendleton, Cost of the generalised Hybrid Monte Carlo algorithm for free field theory, Nucl. Phys. B 607 (2001) 456 [hep-lat/0008020] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    I.P. Omelyan, I.M. Mryglod, R. Folk, Symplectic analytically integrable decomposition algorithms: classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations, Comp. Phys. Commun. 151 (2003) 272. MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [25]
    K. Jansen and C. Liu, Kramers equation algorithm for simulations of QCD with two flavors of Wilson fermions and gauge group SU(2), Nucl. Phys. B 453 (1995) 375 [hep-lat/9506020] [SPIRES].ADSCrossRefGoogle Scholar
  26. [26]
    M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [SPIRES].ADSCrossRefGoogle Scholar
  27. [27]
    R. Sommer, A New way to set the energy scale in lattice gauge theories and its applications to the static force and αs in SU(2) Yang-Mills theory, Nucl. Phys. B 411 (1994) 839 [hep-lat/9310022] [SPIRES].ADSCrossRefGoogle Scholar
  28. [28]
    S. Necco and R. Sommer, The N f =0 heavy quark potential from short to intermediate distances, Nucl. Phys. B 622 (2002) 328 [hep-lat/0108008] [SPIRES].ADSCrossRefGoogle Scholar
  29. [29]
    A. Ukawa and M. Fukugita, Langevin simulation including dynamical quark loops, Phys. Rev. Lett. 55 (1985) 1854 [SPIRES].ADSCrossRefGoogle Scholar
  30. [30]
    G.G. Batrouni et al., Langevin simulations of lattice field theories, Phys. Rev. D 32 (1985) 2736 [SPIRES].ADSGoogle Scholar
  31. [31]
    N. Madras and A.D. Sokal, The pivot algorithm: a highly efficient Monte Carlo method for selfavoiding walk, J. Statist. Phys. 50 (1988) 109 [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. [32]
    M. Lüscher, Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD, Comput. Phys. Commun. 165 (2005) 199 [hep-lat/0409106] [SPIRES].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

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

Authors and Affiliations

  1. 1.CERN, Physics DepartmentGeneva 23Switzerland

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