Symanzik effective actions in the large N limit

  • J. Balog
  • F. Niedermayer
  • P. Weisz


Symanzik effective actions, conjectured to describe lattice artifacts, are determined for a class of lattice regularizations of the non-linear O(N) sigma model in two dimensions in the leading order of the 1/N -expansion. The class of actions considered includes also ones which do not have the usual classical limit and are not (so far) treatable in the framework of ordinary perturbation theory. The effective actions obtained are shown to reproduce previously computed lattice artifacts of the step scaling functions defined in finite volume, giving further confidence in Symanzik’s theory of lattice artifacts.


Field Theories in Lower Dimensions Lattice Quantum Field Theory Sigma Models Nonperturbative Effects 


  1. [1]
    K. Symanzik, Cutoff Dependence In Lattice Φ4 Theory In Four Dimensions, DESY79/76 (Cargèse lecture, 1979), in Mathematical problems in theoretical physics, R. Schrader, R. Seiler, D.A. Uhlenbrock eds., Lect. Notes Phys. 153 (1982) 47.Google Scholar
  2. [2]
    K. Symanzik, Continuum Limit and Improved Action in Lattice Theories. 1. Principles and phi**4 Theory, Nucl. Phys. B 226 (1983) 187 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    K. Symanzik, Continuum Limit and Improved Action in Lattice Theories. 2. O(N) Nonlinear σ-model in Perturbation Theory, Nucl. Phys. B 226 (1983) 205 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    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
  5. [5]
    M. Lüscher and P. Weisz, Computation of the Action for On-Shell Improved Lattice Gauge Theories at Weak Coupling, Phys. Lett. B 158 (1985) 250 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G. Keller, The Perturbative construction of Symanziks improved action for ϕ 4 in four-dimensions and QED in four-dimensions, Helv. Phys. Acta 66 (1993) 453 [INSPIRE].MathSciNetGoogle Scholar
  7. [7]
    J. Balog, F. Niedermayer and P. Weisz, The Puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanziks solution, Nucl. Phys. B 824 (2010) 563 [arXiv:0905.1730] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    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
  9. [9]
    S. Caracciolo and A. Pelissetto, Corrections to finite size scaling in the lattice N vector model for N = infinity, Phys. Rev. D 58 (1998) 105007 [hep-lat/9804001] [INSPIRE].ADSGoogle Scholar
  10. [10]
    U. Wolff, F. Knechtli, B. Leder and J. Balog, Cutoff effects in the O(N) σ-model at large-N, PoS(LAT2005)253 [hep-lat/0509043] [INSPIRE].
  11. [11]
    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
  12. [12]
    M. Lüscher, A New Method to Compute the Spectrum of Low Lying States in Massless Asymptotically Free Field Theories, Phys. Lett. B 118 (1982) 391 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J. Gasser and H. Leutwyler, Spontaneously Broken Symmetries: effective Lagrangians at Finite Volume, Nucl. Phys. B 307 (1988) 763 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    P. Hasenfratz and H. Leutwyler, Goldstone boson related finite size effects in field theory and critical phenomena with O(N) symmetry, Nucl. Phys. B 343 (1990) 241 [INSPIRE].ADSCrossRefGoogle Scholar

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© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, MTA Lendület Holographic QFT GroupBudapest 114Hungary
  2. 2.Albert Einstein Center for Fundamental Physics, Institute for Theoretical PhysicsBern UniversityBernSwitzerland
  3. 3.Max-Planck-Institut für PhysikMunichGermany

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