Beta function vanishes for N=4 Super Yang-Mills theory

Question

I would like to learn about N=4 Super Yang-Mills theory. It would be very nice if you could recommend me a self complete review of N=4 Super Yang-Mills theory. In specific I would like to see a proof that N=4  Yang-Mills theory has a vanishing Beta function. 

Answer 1

The arguments for  all-orders vanishing of the beta function for $N=4$ SYM theory is discussed in the classic review on supersymmetry by Sohnius (see section 13.2 page 160). However, some fun things to do to complement the arguments there are:

1. See the vanishing of the one-loop beta function for all $\mathcal{N}=4$ vector multiplets by combining the `known' answers for (adjoint) scalar fields and fermions along with that of the gauge field. 
2. Read the paper by Leigh and Strassler, who show that there exists a two-parameter deformation of $\mathcal{N}=4$ theory that has $\mathcal{N}=1$ supersymmetry and preserves the vanishing of the beta function. 
3. Read about the NSVZ (Novikov-Shifman-Vainshtein-Zakhorov) computation of the beta function described in section 4 of this review.
4. Read the paper by Seiberg titled "The power of holomorphy". Also, the review of Intriligator and Seiberg might be useful.

These have been written without too much deep thinking on my part. So any suggestions/modifications to this list are more than welcome.

Answer 2

Well, I am not sure if it can count as a proper proof but you can see that the  β-function vanishes at 1-loop level by knowing how the various fields contribute to it in an \(SU(N) \) theory. You do this by evaluating the (Dynkin) indices of the various representations (Weyl fermions, complex scalars and gauge fields) inside \(b_0\)and see what happens. You can find the Dynkin indices in Terning's "Modern Supersymmetry" Appendix B.

Furthermore, another simple argument on the vanishing of the β-function in N=4 SYM is given in the same book in section 14.4, page 241. 

I hope this helps!