Scale invariance, conformality, and generalized free fields

Anatoly Dymarsky, Kara Farnsworth, Zohar Komargodski, Markus A. Luty, Valentina Prilepina

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29 Citations (Scopus)


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 languageEnglish
Article number99
Pages (from-to)1-16
Number of pages16
JournalJournal of High Energy Physics
Issue number2
Publication statusPublished - 1 Feb 2016


  • Effective field theories
  • Space-Time Symmetries


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