How to prove that a given correlation function is protected? I would be interested in proving that $2$-point functions made of $1/2$-BPS operators are protected in $\mathcal{N}=4$ SYM (Supersymmetric Yang-Mills), i.e. that the correlator $\langle \mathcal{O}_2(x_1) \mathcal{O}_2(x_2)\rangle$ with
$$\mathcal{O}_2(x)= \text{Tr} \left(\Phi\Phi\right) \tag{1}$$
does not acquire anomalous scaling dimension from the quantum corrections. If I am not mistaken, this means
$$\langle \mathcal{O}_2(x_1) \mathcal{O}_2(x_2)\rangle \propto x_{12}^{-2\Delta} \tag{2}$$
with $\Delta=2$ the scaling dimension of $\mathcal{O}_2$, is protected at all orders in perturbation theory. Is it possible to give a formal proof for all orders? How would this be done (roughly)? Or do you have to do order by order, and assume that it holds at higher orders?
 A: The four-dimensional $\mathcal{N}=4$ SYM contains scalars, fermions and gauge fields and under gauge transformations, the fields and their covariant derivatives transform in a covariant way. Gauge-invariant operators are obtained by taking the trace of a product of such covariant fields at the same spacetime point, i.e
$$\begin{align}
\mathcal{O}(x) = \text{Tr} (\phi^i \cdots \phi^j)(x)
\end{align}$$
and these are the single trace operators of the theory. 
Amongst these single-trace operators consider the following: 
$$\begin{align}
\begin{aligned}
\mathcal{O}(x) = \text{Str} (\phi^{\{i_1} \phi^{i_2} \cdots \phi^{i_k \}})
\end{aligned}
\end{align}$$
where in the above Str denotes the symmetrized trace and is given by summing over all permutations. The above operators correspond to an irreducible representation of the superconformal algebra, and the simplest example to consider is the single-trace, $\Delta=2$ chiral primary operator
$$\text{Str}(\phi^{\{i}\phi^{j\}})=\text{Tr}(\phi^i \phi^j)-\frac{1}{6}\delta^{ij}\text{Tr}(\phi^k \phi^k)$$
These are the $1/2$-BPS states of the algebra. These operators correspond to entries in a unitary multiplet, and they are protected by non-renormalization theorems. [(1)] 
Interestingly -if you are interested- you can find partial non-renormalization theorems for next-to-extremal, i.e (2)
Let me also point out an unprotected single-trace operator, the Konishi. It is the following 
$$K = \text{Tr} (\phi^i \phi^i)$$
The Konishi develops an anomalous dimension that was computed in (3) where the authors show the dependence on the coupling constant of the $\mathcal{N}=4$. 
And finally, are all composite operators made out of the $1/2$-BPS states of the theory protected? No. Because we can consider for example 
$$\cal{O}_{pq} = \cal{O}_p \partial^{\ell} \Box^{\frac{1}{2}(\tau -p-q)} \cal{O}_q$$ 
which are degenerate long double-trace superconformal primary operators of spin $\ell$ and twist $\tau$ and the $\mathcal{O}_{p,q}$ being the protected half-BPS operators in the $[0,k,0]$ of the R-symmetry group and we know that these double-trace guys develop anomalous dimensions. The mixing of these degenerate operators was recently solved in the SUGRA approximation and beyond. 
P.S: Needless to say that the Freedman-Van Proeyen book is an excellent reference. 
