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I am confused about the role of the mass gap conjecture for $N=4$ SYM. The beta function is zero at all loops so how can the theory develop an energy scale? Does the mass gap conjecture only applies to non-supersymmetric theories? Thanks!

Because $\mathcal{N}=4$ is conformal, it doesn't have a mass gap. The mass-gap is about pure Yang-Mills without matter fields, but it's expected that with many kind of matter fields it will be gapped in any case in which the theory of Yang-Mills plus matter fields isn't conformal anyway. Something related to it is the fact that the Gribov-Zwanziger form of the action don't have a ghost condensate in vacuum so gluons don't develop mass ## https://arxiv.org/abs/1404.7163 ##.

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