# Scale invariance plus unitarity implies conformal invariance?

What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?

• Sep 20 '13 at 6:26

I cannot claim to speak for "the community" (whoever they might be), but so far I have only heard positive replies from knowledgeable people. Of course, people will need to read the paper in close detail, there will be discussions in seminars etc. so it'll take at least a couple of months before there will be a serious consensus.

Let me give a few brief comments on the proof. First of all, much of the machinery feeds off the a-theorem proof by Komargodski and Schwimmer (http://arxiv.org/abs/arXiv:1107.3987) and its refinements ("LPR" http://arxiv.org/abs/arXiv:1204.5221). The idea is that you can turn on a "dilaton" background and then probe the theory using these dilatons. Before people studied 4-pt "on-shell" (in a technical sense) amplitudes of these dilatons, in the new paper they look at 3-pt functions that are off-shell. These dilaton amplitudes are connected to (essential Fourier transforms of) matrix elements of the trace $T$ of the stress tensor. If $T = 0$ then the theory is conformal invariant.

The new idea is to note that $T$ has integer scaling dimension ($\Delta = 4$) which means that there will be logs $$e \ln \frac{-p^2}{\mu^2}$$ in these amplitudes, and in a unitary theory $e \geq 0.$ If $e = 0$ you can show (using unitarity) that $T = 0$ identically so you would be done.

There is a final step (which I haven't internalized yet) where they say that in a scale-invariant theory this $e > 0$ anomaly isn't allowed because unitary gives bounds on operator dimensions, which in turn control the small-distance or large-momentum behaviour of the amplitude.