Advertisement

The Yang-Mills gradient flow in finite volume

  • Zoltan Fodor
  • Kieran Holland
  • Julius Kuti
  • Daniel Nogradi
  • Chik Him Wong
Article

Abstract

The Yang-Mills gradient flow is considered on the four dimensional torus T 4 for SU(N) gauge theory coupled to N f flavors of massless fermions in arbitrary representations. The small volume dynamics is dominated by the constant gauge fields. The expectation value of the field strength tensor squared TrF μν F μν (t) is calculated for positive flow time t by treating the non-zero gauge modes perturbatively and the zero modes exactly. The finite volume correction to the infinite volume result is found to contain both algebraic and exponential terms. The leading order result is then used to define a one parameter family of running coupling schemes in which the coupling runs with the linear size of the box. The new scheme is tested numerically in SU(3) gauge theory coupled to N f = 4 flavors of massless fundamental fermions. The calculations are performed at several lattice spacings with a controlled continuum extrapolation. The continuum result agrees with the perturbative prediction for small renormalized coupling as expected.

Keywords

Lattice Quantum Field Theory Lattice Gauge Field Theories 

References

  1. [1]
    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].CrossRefMATHGoogle Scholar
  2. [2]
    R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, JHEP 03 (2006) 064 [hep-th/0601210] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    M. Lüscher, Topology, the Wilson flow and the HMC algorithm, PoS(LATTICE 2010)015 [arXiv:1009.5877] [INSPIRE].
  5. [5]
    M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    R. Lohmayer and H. Neuberger, Continuous smearing of Wilson Loops, PoS(LATTICE 2011)249 [arXiv:1110.3522] [INSPIRE].
  7. [7]
    S. Borsányi et al., High-precision scale setting in lattice QCD, JHEP 09 (2012) 010 [arXiv:1203.4469] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    M. Lüscher, P. Weisz and U. Wolff, A numerical method to compute the running coupling in asymptotically free theories, Nucl. Phys. B 359 (1991) 221 [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Lüscher, Some analytic results concerning the mass spectrum of Yang-Mills gauge theories on a torus, Nucl. Phys. B 219 (1983) 233 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J. Koller and P. van Baal, A rigorous nonperturbative result for the glueball mass and electric flux energy in a finite volume, Nucl. Phys. B 273 (1986) 387 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Koller and P. van Baal, A nonperturbative analysis in finite volume gauge theory, Nucl. Phys. B 302 (1988) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    P. van Baal, The small volume expansion of gauge theories coupled to massless fermions, Nucl. Phys. B 307 (1988) 274 [Erratum ibid. B 312 (1989) 752] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    P. van Baal, Gauge theory in a finite volume, Acta Phys. Polon. B 20 (1989) 295 [INSPIRE].Google Scholar
  15. [15]
    C.P. Korthals Altes, Fluctuations of constant potentials in QCD and their contribution to finite size effects, lecture presented at Gift seminar, CPT-85/P-1806, Jaca Spain (1985) [INSPIRE].
  16. [16]
    A. Coste, A. Gonzalez-Arroyo, J. Jurkiewicz and C. Korthals Altes, Zero momentum contribution to Wilson loops in periodic boxes, Nucl. Phys. B 262 (1985) 67 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A. Coste, A. Gonzalez-Arroyo, C. Korthals Altes, B. Soderberg and A. Tarancon, Finite size effects and twisted boundary conditions, Nucl. Phys. B 287 (1987) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    C. Korthals Altes, Pure QCD in small volumes and the low lying glueball spectrum, Nucl. Phys. Proc. Suppl. 10A (1989) 284 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    ALPHA collaboration, F. Tekin, R. Sommer and U. Wolff, The running coupling of QCD with four flavors, Nucl. Phys. B 840 (2010) 114 [arXiv:1006.0672] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    P. Perez-Rubio and S. Sint, Non-perturbative running of the coupling from four flavour lattice QCD with staggered quarks, PoS(LATTICE 2010)236 [arXiv:1011.6580] [INSPIRE].
  21. [21]
    C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD Phys. Rev. D 69 (2004) 054501 [hep-lat/0311018] [INSPIRE].ADSGoogle Scholar
  22. [22]
    K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and ϕ 4 theory, Nucl. Phys. B 226 (1983) 187 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59 [Erratum ibid. 98 (1985) 433] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  24. [24]
    S. Duane, A. Kennedy, B. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 (1987) 216 [INSPIRE].ADSGoogle Scholar
  25. [25]
    J. Sexton and D. Weingarten, Hamiltonian evolution for the hybrid Monte Carlo algorithm, Nucl. Phys. B 380 (1992) 665 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    T. Takaishi and P. de Forcrand, Testing and tuning new symplectic integrators for hybrid Monte Carlo algorithm in lattice QCD, Phys. Rev. E 73 (2006) 036706 [hep-lat/0505020] [INSPIRE].ADSGoogle Scholar
  27. [27]
    G.I. Egri et al., Lattice QCD as a video game, Comput. Phys. Commun. 177 (2007) 631 [hep-lat/0611022] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Zoltan Fodor
    • 1
    • 2
    • 3
  • Kieran Holland
    • 4
    • 5
  • Julius Kuti
    • 6
  • Daniel Nogradi
    • 3
  • Chik Him Wong
    • 6
  1. 1.University of Wuppertal, Department of PhysicsWuppertalGermany
  2. 2.Jülich Supercomputing Center, Forschungszentrum JülichJülichGermany
  3. 3.Eötvös University, Institute for Theoretical PhysicsBudapestHungary
  4. 4.University of the PacificStocktonU.S.A
  5. 5.Institute for Theoretical Physics, Albert Einstein Center for Fundamental PhysicsBern UniversityBernSwitzerland
  6. 6.University of California, San DiegoLa JollaU.S.A

Personalised recommendations