## Abstract

Abstract: This paper addresses the question of whether there are 4D Lorentz invariant unitary quantum field theories with scale invariance but not conformal invariance. An important loophole in the arguments of Luty-Polchinski-Rattazzi and Dymarsky-Komargodski-Schwimmer-Theisen is that 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). Despite the invalidity of the general argument of FLP, some of the techniques turn out to be useful in the present context.

Original language | English |
---|---|

Article number | 99 |

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Journal of High Energy Physics |

Volume | 2016 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2016 |

### Bibliographical note

Funding Information:We are very grateful to R. Rattazzi, A. Schwimmer, S. Theisen for useful discussions. A.D. is supported by the grant RFBR 15-01-04217. Z.K. is supported by the ERC STG grant number 335182, by the Israel Science Foundation under grant number 884/11, by the United States-Israel Binational Science Foundation (BSF) under grant number 2010/629, and by the I-CORE Program of the Planning and Budgeting Committee and by the Israel Science Foundation under grant number 1937/12. K.F., M.A.L., and V.P. are supported by the US Department of Energy under grant DE-FG02-91ER40674. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily re ect the views of the funding agencies.

Publisher Copyright:

© 2016, The Author(s).

## Keywords

- Effective field theories
- Space-Time Symmetries

## ASJC Scopus subject areas

- Nuclear and High Energy Physics