Gradient flow of O(N) nonlinear sigma model at large N

  • Sinya Aoki
  • Kengo Kikuchi
  • Tetsuya Onogi
Open Access
Regular Article - Theoretical Physics


We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N. We parameterize solution of the field at flow time t in powers of bare fields by introducing the coefficient function X n for the n-th power term (n = 1, 3, ··· ). Reducing the flow equation by keeping only the contributions at leading order in large N, we obtain a set of equations for X n ’s, which can be solved iteratively starting from n = 1. For n = 1 case, we find an explicit form of the exact solution. Using this solution, we show that the two point function at finite flow time t is finite. As an application, we obtain the non-perturbative running coupling defined from the energy density. We also discuss the solution for n = 3 case.


Lattice Quantum Field Theory Field Theories in Lower Dimensions Nonperturbative Effects 


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]
    M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [arXiv:1006.4518] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    M. Lüscher, Chiral symmetry and the Yang-Mills gradient flow, JHEP 04 (2013) 123 [arXiv:1302.5246] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, PoS(LATTICE 2013)016 [arXiv:1308.5598] [INSPIRE].
  5. [5]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The gradient flow running coupling scheme, PoS(LATTICE 2012)050 [arXiv:1211.3247] [INSPIRE].
  6. [6]
    S. Borsányi, S. Dürr, Z. Fodor, C. Hölbling, S.D. Katz et al., High-precision scale setting in lattice QCD, JHEP 09 (2012) 010 [arXiv:1203.4469] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A. Hasenfratz, D. Schaich and A. Veernala, Nonperturbative β-function of eight-flavor SU(3) gauge theory, arXiv:1410.5886 [INSPIRE].
  8. [8]
    P. Fritzsch and A. Ramos, The gradient flow coupling in the Schrödinger Functional, JHEP 10 (2013)008 [arXiv:1301.4388] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  9. [9]
    P. Fritzsch and A. Ramos, Studying the gradient flow coupling in the Schrödinger functional, PoS(Lattice 2013)319 [arXiv:1308.4559] [INSPIRE].
  10. [10]
    A. Ramos, The gradient flow in a twisted box, PoS(Lattice 2013)053 [arXiv:1308.4558] [INSPIRE].
  11. [11]
    J. Rantaharju, The Gradient Flow Coupling in Minimal Walking Technicolor, PoS(Lattice 2013)084 [arXiv:1311.3719] [INSPIRE].
  12. [12]
    P. Fritzsch, A. Ramos and F. Stollenwerk, Critical slowing down and the gradient flow coupling in the Schrödinger functional, PoS(Lattice 2013)461 [arXiv:1311.7304] [INSPIRE].
  13. [13]
    Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C.H. Wong, The Yang-Mills gradient flow in finite volume, JHEP 11 (2012) 007 [arXiv:1208.1051] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  14. [14]
    Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi et al., The lattice gradient flow at tree-level and its improvement, JHEP 09 (2014) 018 [arXiv:1406.0827] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    L. Del Debbio, A. Patella and A. Rago, Space-time symmetries and the Yang-Mills gradient flow, JHEP 11 (2013) 212 [arXiv:1306.1173] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    H. Suzuki, Energy-momentum tensor from the Yang-Mills gradient flow, PTEP 2013 (2013) 083B03 [arXiv:1304.0533] [INSPIRE].Google Scholar
  17. [17]
    H. Makino and H. Suzuki, Lattice energy-momentum tensor from the Yang-Mills gradient flow - a simpler prescription, arXiv:1404.2758 [INSPIRE].
  18. [18]
    A. Shindler, Chiral Ward identities, automatic O(a) improvement and the gradient flow, Nucl. Phys. B 881 (2014) 71 [arXiv:1312.4908] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    C. Monahan and K. Orginos, Finite volume renormalization scheme for fermionic operators, PoS(Lattice 2013)443 [arXiv:1311.2310] [INSPIRE].
  20. [20]
    FlowQCD collaboration, M. Asakawa, T. Hatsuda, E. Itou, M. Kitazawa and H. Suzuki, Thermodynamics of SU(3) gauge theory from gradient flow on the lattice, Phys. Rev. D 90 (2014) 011501 [arXiv:1312.7492] [INSPIRE].ADSGoogle Scholar
  21. [21]
    O. Bär and M. Golterman, Chiral perturbation theory for gradient flow observables, Phys. Rev. D 89 (2014) 034505 [arXiv:1312.4999] [INSPIRE].ADSGoogle Scholar
  22. [22]
    M. Dalla Brida and D. Hesse, Numerical Stochastic Perturbation Theory and the Gradient Flow, PoS(Lattice 2013)326 [arXiv:1311.3936] [INSPIRE].
  23. [23]
    K. Kikuchi and T. Onogi, Generalized Gradient Flow Equation and Its Application to Super Yang-Mills Theory, JHEP 11 (2014) 094 [arXiv:1408.2185] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A.M. Polyakov, Interaction of Goldstone Particles in Two-Dimensions. Applications to Ferromagnets and Massive Yang-Mills Fields, Phys. Lett. B 59 (1975) 79 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A.A. Migdal, Gauge Transitions in Gauge and Spin Lattice Systems, Sov. Phys. JETP 42 (1975) 743 [INSPIRE].ADSGoogle Scholar
  26. [26]
    J. Balog, F. Niedermayer, M. Pepe, P. Weisz and U.-J. Wiese, Drastic Reduction of Cutoff Effects in 2 − D Lattice O(N) Models, JHEP 11 (2012) 140 [arXiv:1208.6232] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    H. Makino and H. Suzuki, Renormalizability of the gradient flow in the 2D O(N) non-linear σ-model, PTEP 2015 (2014) 033B08 [arXiv:1410.7538] [INSPIRE].MATHGoogle Scholar
  28. [28]
    G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].ADSGoogle Scholar
  29. [29]
    H. Makino, F. Sugino and H. Suzuki, Large-N limit of the gradient flow in the 2D O(N) nonlinear σ-model, arXiv:1412.8218 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
  2. 2.Department of PhysicsOsaka UniversityToyonakaJapan

Personalised recommendations