  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.
