# 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 σ 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 behavior of 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 which creates 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 quantummore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22403442

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 350; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; EINSTEIN FIELD EQUATIONS; GRAVITATION; GROUP THEORY; INFRARED RADIATION; RENORMALIZATION; SCALARS; SINE-GORDON EQUATION; SINGULARITY; SYMMETRY BREAKING

### Citation Formats

```
Nagy, Sandor.
```*Lectures on renormalization and asymptotic safety*. United States: N. p., 2014.
Web. doi:10.1016/J.AOP.2014.07.027.

```
Nagy, Sandor.
```*Lectures on renormalization and asymptotic safety*. United States. https://doi.org/10.1016/J.AOP.2014.07.027

```
Nagy, Sandor. Sat .
"Lectures on renormalization and asymptotic safety". United States. https://doi.org/10.1016/J.AOP.2014.07.027.
```

```
@article{osti_22403442,
```

title = {Lectures on renormalization and asymptotic safety},

author = {Nagy, Sandor},

abstractNote = {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 σ 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 behavior of 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 which creates 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.},

doi = {10.1016/J.AOP.2014.07.027},

url = {https://www.osti.gov/biblio/22403442},
journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = ,

volume = 350,

place = {United States},

year = {2014},

month = {11}

}