Wilsonian renormalisation of CFT correlation functions: field theory
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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.
KeywordsRenormalization Group Conformal Field Theory Renormalization Regularization and Renormalons
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- S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics (2017).Google Scholar
- J.M. Lizana and M. Pérez-Victoria, Wilsonian renormalisation of CFT correlation functions: Holography, in preparation.Google Scholar
- 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
- 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
- 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
- J. Glimm and A.M. Jaffe, Quantum physics. A functional integral point of view, Springer-Verlag New York (1987).Google Scholar
- 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