# O( *N*) and O( *N*) and O( *N*)

## Abstract

Three related analyses of Φ ^{4} theory with O( *N*) symmetry are presented. In the first, we review the O( *N*) model over the p-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context. We demonstrate the existence of a Wilson-Fisher fixed point using an ϵ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point. Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms. In the second analysis, we employ large *N* methods to establish formulas for anomalous dimensions which are valid equally for field theories over the p-adic numbers and field theories on R ^{n}. Results for anomalous dimensions agree between the first and second analyses when they can be meaningfully compared. In the third analysis, we consider higher derivative versions of the O( *N*) model on R ^{n}, the simplest of which has been studied in connection with spatially modulated phases. Our general formula for anomalous dimensions can still be applied. Analogies with two-derivative theories hint at the existence of some interesting unconventional field theoriesmore »

- Authors:

- Princeton Univ., Princeton, NJ (United States). Joseph Henry Lab. of Physics

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1507578

- Grant/Contract Number:
- FG02-91ER40671

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 1/N Expansion; Sigma Models; Renormalization Group

### Citation Formats

```
Gubser, Steven S., Jepsen, Christian, Parikh, Sarthak, and Trundy, Brian. O(N) and O(N) and O(N). United States: N. p., 2017.
Web. doi:10.1007/jhep11(2017)107.
```

```
Gubser, Steven S., Jepsen, Christian, Parikh, Sarthak, & Trundy, Brian. O(N) and O(N) and O(N). United States. doi:10.1007/jhep11(2017)107.
```

```
Gubser, Steven S., Jepsen, Christian, Parikh, Sarthak, and Trundy, Brian. Fri .
"O(N) and O(N) and O(N)". United States. doi:10.1007/jhep11(2017)107. https://www.osti.gov/servlets/purl/1507578.
```

```
@article{osti_1507578,
```

title = {O(N) and O(N) and O(N)},

author = {Gubser, Steven S. and Jepsen, Christian and Parikh, Sarthak and Trundy, Brian},

abstractNote = {Three related analyses of Φ4 theory with O(N) symmetry are presented. In the first, we review the O(N) model over the p-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context. We demonstrate the existence of a Wilson-Fisher fixed point using an ϵ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point. Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms. In the second analysis, we employ large N methods to establish formulas for anomalous dimensions which are valid equally for field theories over the p-adic numbers and field theories on Rn. Results for anomalous dimensions agree between the first and second analyses when they can be meaningfully compared. In the third analysis, we consider higher derivative versions of the O(N) model on Rn, the simplest of which has been studied in connection with spatially modulated phases. Our general formula for anomalous dimensions can still be applied. Analogies with two-derivative theories hint at the existence of some interesting unconventional field theories in four real Euclidean dimensions.},

doi = {10.1007/jhep11(2017)107},

journal = {Journal of High Energy Physics (Online)},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}

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