Abstract
There has been recent interest in the question of whether four dimensional scale invariant unitary quantum field theories are actually conformally invariant. In this note we present a complete analysis of possible scale anomalies in correlation functions of the trace of the stress-energy tensor in such theories. We find that 2-, 3- and 4-point functions have a non-trivial anomaly while connected higher point functions are non-anomalous. We pay special attention to semi-local contributions to correlators (terms with support on a set containing both coincident and separated points) and show that the anomalies in 3- and 4-point functions can be accounted for by such contributions. We discuss the implications of the our results for the question of scale versus conformal invariance.
Highlights
It has been a long standing conjecture that every unitary scale invariant quantum field theory (SFT) in four spacetime dimensions is automatically conformally invariant
If one were able to show that the 4-point function of the trace of stress energy tensor, including semi-local terms, vanishes in this kinematical limit or that the anomaly cannot be supported by semi-local terms alone one would conclude that the scale anomaly coefficient must vanish and this would imply that the SFT is a CFT
In appendix A we discuss subtleties in the relation between the shortdistance/large momentum limit and OPEs in momentum space, in appendix B we present an alternative derivation of the scale anomaly for the 3- and the 4-point function which does not use the Wess-Zumino action and in appendix C we compute the anomaly in 3- and 4-point functions using a different parametrisation for the dilaton
Summary
It has been a long standing conjecture that every unitary scale invariant quantum field theory (SFT) in four spacetime dimensions is automatically conformally invariant. If one were able to show that the 4-point function of the trace of stress energy tensor, including semi-local terms, vanishes in this kinematical limit or that the anomaly cannot be supported by semi-local terms alone one would conclude that the scale anomaly coefficient must vanish and (as we will argue in detail later) this would imply that the SFT is a CFT. In appendix A we discuss subtleties in the relation between the shortdistance/large momentum limit and OPEs in momentum space, in appendix B we present an alternative derivation of the scale anomaly for the 3- and the 4-point function which does not use the Wess-Zumino action and in appendix C we compute the anomaly in 3- and 4-point functions using a different parametrisation for the dilaton This parametrisation has the feature that the contribution of the Wess-Zumino action vanishes and the entire contribution to the scale anomaly is manifestly due to semi-local terms. In appendix D we discuss a generalisation of our results to the case of the theory containing multiple scalar operators of dimension two and four
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