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We address 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 existing arguments is that
trace of the energy-momentum tensorĀ TĀ could be a generalized free field.
We rule out this possibility. This is achieved by using the structure of
four-dimensional anomalies. In addition, we emphasize that short
distance physics is not the same as large momentum physics and explain
the relation to the problem of scale vs conformal invariance.
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