## Scale invariance, conformality, and generalized free fields

## Abstract

This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum fi eld theories with scale invariance but not conformal invariance. We present an important loophole in the arguments of Luty-Polchinski-Rattazzi and Dymarsky-Komargodski-Schwimmer-Theisen that is the trace of the energy-momentum tensor T could be a generalized free field. In this paper we rule out this possibility. The key ingredient is the observation that a unitary theory with scale but not conformal invariance necessarily has a non-vanishing anomaly for global scale transformations. We show that this anomaly cannot be reproduced if T is a generalized free field unless the theory also contains a dimension-2 scalar operator. In the special case where such an operator is present it can be used to redefine ("improve") the energy-momentum tensor, and we show that there is at least one energy-momentum tensor that is not a generalized free field. In addition, we emphasize that, in general, large momentum limits of correlation functions cannot be understood from the leading terms of the coordinate space OPE. This invalidates a recent argument by Farnsworth-Luty-Prilepina (FLP). Finally, despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in themore »

- Authors:

- Skolkovo Institute of Science and Technology, Moscow (Russia)
- Univ. of California, Davis, CA (United States)
- Weizmann Inst. of Science, Rehovot (Israel)

- Publication Date:

- Research Org.:
- Univ. of California, Davis, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1327254

- Grant/Contract Number:
- FG02-91ER40674

- 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: 2016; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Space-Time Symmetries; Effective field theories

### Citation Formats

```
Dymarsky, Anatoly, Farnsworth, Kara, Komargodski, Zohar, Luty, Markus A., and Prilepina, Valentina. Scale invariance, conformality, and generalized free fields. United States: N. p., 2016.
Web. doi:10.1007/JHEP02(2016)099.
```

```
Dymarsky, Anatoly, Farnsworth, Kara, Komargodski, Zohar, Luty, Markus A., & Prilepina, Valentina. Scale invariance, conformality, and generalized free fields. United States. doi:10.1007/JHEP02(2016)099.
```

```
Dymarsky, Anatoly, Farnsworth, Kara, Komargodski, Zohar, Luty, Markus A., and Prilepina, Valentina. Tue .
"Scale invariance, conformality, and generalized free fields". United States. doi:10.1007/JHEP02(2016)099. https://www.osti.gov/servlets/purl/1327254.
```

```
@article{osti_1327254,
```

title = {Scale invariance, conformality, and generalized free fields},

author = {Dymarsky, Anatoly and Farnsworth, Kara and Komargodski, Zohar and Luty, Markus A. and Prilepina, Valentina},

abstractNote = {This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum fi eld theories with scale invariance but not conformal invariance. We present an important loophole in the arguments of Luty-Polchinski-Rattazzi and Dymarsky-Komargodski-Schwimmer-Theisen that is the trace of the energy-momentum tensor T could be a generalized free field. In this paper we rule out this possibility. The key ingredient is the observation that a unitary theory with scale but not conformal invariance necessarily has a non-vanishing anomaly for global scale transformations. We show that this anomaly cannot be reproduced if T is a generalized free field unless the theory also contains a dimension-2 scalar operator. In the special case where such an operator is present it can be used to redefine ("improve") the energy-momentum tensor, and we show that there is at least one energy-momentum tensor that is not a generalized free field. In addition, we emphasize that, in general, large momentum limits of correlation functions cannot be understood from the leading terms of the coordinate space OPE. This invalidates a recent argument by Farnsworth-Luty-Prilepina (FLP). Finally, despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in the present context.},

doi = {10.1007/JHEP02(2016)099},

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

number = 2,

volume = 2016,

place = {United States},

year = {2016},

month = {2}

}

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