Wilsonian renormalisation of CFT correlation functions: field theory

  • J. M. LizanaEmail author
  • M. Pérez-Victoria
Open Access
Regular Article - Theoretical Physics


We examine the precise connection between the exact renormalisation group with local couplings and the renormalisation of correlation functions of composite operators in scale-invariant theories. A geometric description of theory space allows us to select convenient non-linear parametrisations that serve different purposes. First, we identify normal parameters in which the renormalisation group flows take their simplest form; normal correlators are defined by functional differentiation with respect to these parameters. The renormalised correlation functions are given by the continuum limit of correlators associated to a cutoff-dependent parametrisation, which can be related to the renormalisation group flows. The necessary linear and non-linear counterterms in any arbitrary parametrisation arise in a natural way from a change of coordinates. We show that, in a class of minimal subtraction schemes, the renormalised correlators are exactly equal to normal correlators evaluated at a finite cutoff. To illustrate the formalism and the main results, we compare standard diagrammatic calculations in a scalar free-field theory with the structure of the perturbative solutions to the Polchinski equation close to the Gaussian fixed point.


Renormalization Group Conformal Field Theory Renormalization Regularization and Renormalons 


Open Access

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  1. [1]
    G.M. Shore, New methods for the renormalization of composite operator Green functions, Nucl. Phys. B 362 (1991) 85 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    H. Osborn, Derivation of a Four-dimensional c Theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    I. Jack and H. Osborn, Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    I.T. Drummond and G.M. Shore, Conformal Anomalies for Interacting Scalar Fields in Curved Space-Time, Phys. Rev. D 19 (1979) 1134 [INSPIRE].ADSGoogle Scholar
  5. [5]
    H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    J. Polonyi and K. Sailer, Renormalization of composite operators, Phys. Rev. D 63 (2001) 105006 [hep-th/0011083] [INSPIRE].ADSGoogle Scholar
  7. [7]
    J. Polchinski, Scale and Conformal Invariance in Quantum Field Theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Dymarsky, Z. Komargodski, A. Schwimmer and S. Theisen, On Scale and Conformal Invariance in Four Dimensions, JHEP 10 (2015) 171 [arXiv:1309.2921] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1 [arXiv:1302.0884] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].Google Scholar
  12. [12]
    G. Mack, Duality in quantum field theory, Nucl. Phys. B 118 (1977) 445 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics (2017).Google Scholar
  16. [16]
    J.M. Lizana and M. Pérez-Victoria, Wilsonian renormalisation of CFT correlation functions: Holography, in preparation.Google Scholar
  17. [17]
    B.P. Dolan, Covariant derivatives and the renormalization group equation, Int. J. Mod. Phys. A 10 (1995) 2439 [hep-th/9403070] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    H. Sonoda, Operator coefficients for composite operators in the (ϕ 4)4 in four-dimensions theory, Nucl. Phys. B 394 (1993) 302 [hep-th/9205084] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    K. Ranganathan, Nearby CFTs in the operator formalism: The role of a connection, Nucl. Phys. B 408 (1993) 180 [hep-th/9210090] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    K. Ranganathan, H. Sonoda and B. Zwiebach, Connections on the state space over conformal field theories, Nucl. Phys. B 414 (1994) 405 [hep-th/9304053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J.M. Lizana, T.R. Morris and M. Pérez-Victoria, Holographic renormalisation group flows and renormalisation from a Wilsonian perspective, JHEP 03 (2016) 198 [arXiv:1511.04432] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    T.R. Morris, Elements of the continuous renormalization group, Prog. Theor. Phys. Suppl. 131 (1998) 395 [hep-th/9802039] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    P. Glendinning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press (1994).Google Scholar
  26. [26]
    S.K. Aranson, I.U. Bronshtein, V.Z. Grines and Y.S. Ilyashenko, Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems, Springer Science & Business Media (1996).Google Scholar
  27. [27]
    M. Pérez-Victoria, Randall-Sundrum models and the regularized AdS/CFT correspondence, JHEP 05 (2001) 064 [hep-th/0105048] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  28. [28]
    A. Bzowski, P. McFadden and K. Skenderis, Scalar 3-point functions in CFT: renormalisation, β-functions and anomalies, JHEP 03 (2016) 066 [arXiv:1510.08442] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    W. Zimmermann, Convergence of Bogolyubov’s method of renormalization in momentum space, Commun. Math. Phys. 15 (1969) 208 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    W. Zimmermann, Local Operator Products and Renormalization in Quantum Field Theory, in Proceedings, 13th Brandeis University Summer Institute in Theoretical Physics, Lectures On Elementary Particles and Quantum Field Theory, S.D. Deser, M.T. Grisaru and H. Pendleton eds., MIT, Cambridge MA, U.S.A. (1970).Google Scholar
  31. [31]
    J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    J. Glimm and A.M. Jaffe, Quantum physics. A functional integral point of view, Springer-Verlag New York (1987).Google Scholar
  33. [33]
    R. Fernandez, J. Frohlich and A.D. Sokal, Random walks, critical phenomena, and triviality in quantum field theory, Springer-Verlag Berlin Heidelberg (1992).Google Scholar
  34. [34]
    O.J. Rosten, Fundamentals of the Exact Renormalization Group, Phys. Rept. 511 (2012) 177 [arXiv:1003.1366] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: A new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    G.V. Dunne and N. Rius, A comment on the relationship between differential and dimensional renormalization, Phys. Lett. B 293 (1992) 367 [hep-th/9206038] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    F. del Aguila and M. Pérez-Victoria, Constrained differential renormalization and dimensional reduction, hep-ph/9901291 [INSPIRE].
  38. [38]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

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© The Author(s) 2017

Authors and Affiliations

  1. 1.CAFPE and Departamento de Física Teórica y del CosmosUniversidad de GranadaGranadaSpain

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