Instantaneous stochastic perturbation theory

Open Access
Regular Article - Theoretical Physics

Abstract

A form of stochastic perturbation theory is described, where the representative stochastic fields are generated instantaneously rather than through a Markov process. The correctness of the procedure is established to all orders of the expansion and for a wide class of field theories that includes all common formulations of lattice QCD.

Keywords

Lattice QCD Lattice Gauge Field Theories Lattice Quantum Field Theory Stochastic Processes 

Notes

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.

References

  1. [1]
    G. Parisi, Y.-S. Wu, Perturbation theory without gauge fixing, Sci. Sin. 24 (1981) 483 [INSPIRE].MathSciNetGoogle Scholar
  2. [2]
    P.H. Damgaard and H. Hüffel, Stochastic quantization, Phys. Rept. 152 (1987) 227 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    F. Di Renzo, G. Marchesini, P. Marenzoni and E. Onofri, Lattice perturbation theory on the computer, Nucl. Phys. Proc. Suppl. 34 (1994) 795 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    F. Di Renzo, E. Onofri, G. Marchesini and P. Marenzoni, Four loop result in SU(3) lattice gauge theory by a stochastic method: Lattice correction to the condensate, Nucl. Phys. B 426 (1994) 675 [hep-lat/9405019] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    F. Di Renzo and L. Scorzato, Numerical stochastic perturbation theory for full QCD, JHEP 10 (2004) 073 [hep-lat/0410010] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    M. Brambilla, M. Dalla Brida, F. Di Renzo, D. Hesse and S. Sint, Numerical stochastic perturbation theory in the Schrödinger functional, PoS(Lattice 2013)325 [arXiv:1310.8536] [INSPIRE].
  7. [7]
    M. Dalla Brida and D. Hesse, Numerical stochastic perturbation theory and the gradient flow, PoS(Lattice 2013)326 [arXiv:1311.3936] [INSPIRE].
  8. [8]
    G.G. Batrouni, G.R. Katz, A.S. Kronfeld, G.P. Lepage, B. Svetitsky and K.G. Wilson, Langevin simulations of lattice field theories, Phys. Rev. D 32 (1985) 2736 [INSPIRE].ADSGoogle Scholar
  9. [9]
    A. Ukawa and M. Fukugita, Langevin simulation including dynamical quark loops, Phys. Rev. Lett. 55 (1985) 1854 [INSPIRE]ADSCrossRefGoogle Scholar
  10. [10]
    G.S. Bali, C. Bauer, A. Pineda and C. Torrero, Perturbative expansion of the energy of static sources at large orders in four-dimensional SU(3) gauge theory, Phys. Rev. D 87 (2013) 094517 [arXiv:1303.3279] [INSPIRE].ADSGoogle Scholar
  11. [11]
    A.S. Kronfeld, Dynamics of Langevin simulations, Prog. Theor. Phys. Suppl. 111 (1993) 293 [hep-lat/9205008] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D.E. Knuth, The art of computer programming. Volume 3. Sorting and serching, Addison-Wesley (1973).Google Scholar
  14. [14]
    M. Lüscher, R. Narayanan, P. Weisz and U. Wolff, The Schrödinger functionala renormalizable probe for non-Abelian gauge theories, Nucl. Phys. B 384 (1992) 168 [hep-lat/9207009] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S. Sint, On the Schrödinger functional in QCD, Nucl. Phys. B 421 (1994) 135 [hep-lat/9312079] [INSPIRE].
  16. [16]
    M. Lüscher, Step scaling and the Yang-Mills gradient flow, JHEP 06 (2014) 105 [arXiv:1404.5930] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Lüscher, Selected topics in lattice field theory, in E. Brézin and J. Zinn-Justin, Fields, strings, critical phenomena: proceedings, North-Holland, Amsterdam The Netherlands (1990), [Conf. Proc. C 880628 (1988) 451] [INSPIRE].
  18. [18]
    C. van Loan, Computational frameworks for the fast Fourier transform, Frontiers in applied mathematics (Book 10), Society for Industrial and Applied Mathematics, Philadelphia U.S.A. (1992).Google Scholar
  19. [19]
    Y. Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia U.S.A. (2003) and online at http://www-users.cs.umn.edu/∼saad/.
  20. [20]
    M. Lüscher, Lattice regularization of chiral gauge theories to all orders of perturbation theory, JHEP 06 (2000) 028 [hep-lat/0006014] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.CERN, Physics DepartmentGeneva 23Switzerland
  2. 2.Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsBernSwitzerland

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