Lectures on renormalization and asymptotic safety

Abstract

A short introduction is given on the functional renormalization group method, putting emphasis on its nonperturbative aspects. The method enables to find nontrivial fixed points in quantum field theoretic models which make them free from divergences and leads to the concept of asymptotic safety. It can be considered as a generalization of the asymptotic freedom which plays a key role in the perturbative renormalization. We summarize and give a short discussion of some important models, which are asymptotically safe such as the Gross-Neveu model, the nonlinear $\sigma$ model, the sine-Gordon model, and we consider the model of quantum Einstein gravity which seems to show asymptotic safety, too. We also give a detailed analysis of infrared behavior of such scalar models where a spontaneous symmetry breaking takes place. The deep infrared behaviorof the broken phase cannot be treated within the framework of perturbative calculations. We demonstrate that there exists an infrared fixed point in the broken phase whichcreates a new scaling regime there, however its structure is hidden by the singularity of the renormalization group equations. The theory spaces of these models show several similar properties, namely the models have the same phase and fixed point structure. The quantum Einstein gravity also exhibits similarities when considering the global aspects of its theory space since the appearing two phases there show analogies with the symmetric and the broken phases of the scalar models. These results be nicely uncovered by the functional renormalization group method.

34 pages, 21 figures. Based on the talk presented at the Theoretical Physics School on Quantum Gravity, Szeged, Hungary, 27-31 August 2012. Final version, to appear in Annals of Physics

References in corpus (21)

• Exact evolution equation for the effective potential
• Fixed Points of Higher Derivative Gravity
• Ultraviolet properties of f(R)-Gravity
• Fixed points of quantum gravity in extra dimensions
• Bimetric Renormalization Group Flows in Quantum Einstein Gravity
• Matter Induced Bimetric Actions for Gravity
• The Universal RG Machine
• Flow Equations for the BCS-BEC Crossover
• Asymptotic Safety in Einstein Gravity and Scalar-Fermion Matter
• On fixed points of quantum gravity
• Fixed-Functionals of three-dimensional Quantum Einstein Gravity
• Fixed Points of Nonlinear Sigma Models in d>2
• Fixed points of quantum gravity in higher dimensions
• Propagating gravitons vs. dark matter in asymptotically safe quantum gravity
• Functional renormalization group with a compactly supported smooth regulator function
• Renormalizable parameters of the sine-Gordon model
• On the 2-point function of the O(N) model
• Functional renormalization group for quantized anharmonic oscillator
• On the renormalization of the bosonized multi-flavor Schwinger model
• Supersymmetric Nonlinear O(3) Sigma Model on the Lattice
• Invariant formulation of the Functional Renormalisation Group method for $U(n)\times U(n)$ symmetric matrix models

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