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The fact that a QFT in 3+1d is scale invariant does not automatically imply that the QFT is also invariant under the full conformal group, cf. e.g. this Phys.SE post. Counterexamples are known, but as far as I know are all non-supersymmetric QFTs.

I would like to ask if considering a supersymmetric QFT in 3+1d, scale invariance implies conformal invariance. If yes, what is the argument? If no, what is a known counterexample?

I don't know if the question is studied, or is an open problem. Second question would be if the answer changes with the ammount of supersymmetry we choose.

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    $\begingroup$ See this paper: arxiv.org/pdf/1302.0884.pdf $\endgroup$ – Ryan Thorngren May 20 '18 at 18:35
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    $\begingroup$ Actually, for non-supersymmetric weakly coupled theories in 3+1d you can prove the IR and UV limits are conformal invariant fixed point; that is scale invariant is inevitably enhanced to conformal. arxiv.org/pdf/1204.5221.pdf I don't know if this paper might be useful $\endgroup$ – apt45 May 22 '18 at 6:29

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