Mass gap and $N=4$ SYM

Question

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!

Answer 1

N=4 SYM is superconformal and so does not have a mass gap (more precisely, this argument applies at the superconformal point: it is also possible to give expectation values to the scalar fields ("to go on the Coulomb branch"), which breaks superconformal invariance, but there is still no mass gap as there are massless abelian gauge bosons).

In general, the infrared behavior of a non-abelian gauge theory (supersymmetric or not) is a non-trivial question.

The mass gap conjecture says only that "pure" non-abelian gauge theory, i.e. with only non-abelian gauge bosons and no extra matter, has a mass gap. A generalization is that non-abelian gauge theory coupled to a "few" quarks in the fundamental representation (like QCD) has a mass gap. But it is not general: for example, for SU(N) gauge theory with N_f quarks, it is expected that in an appropriate scaling limit of N and N_f (search for Banks-Zaks fixed point), the infrared theory is controlled by a non-trivial conformal field theory and so in particular does not have a mass gap.

The infrared behavior of non-abelian gauge theories is much better understood in the supersymmetric case: see the work of Seiberg on N=1 SYM (including topics like Seiberg dualities).