On the logarithmic powers of $sl(2)$ SYM$_4$

Understand

In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the $sl(2)$ sector of planar ${\cal N}=4$ Super Yang-Mills theory expands as $\gamma(g,s,L)=f(g) \ln s + f_{sl}(g,L) + \sum \limits_{n=1}^\infty \gamma^{(n)}(g,L) (\ln s)^{-n} + ...

• $.
• We find that the sub-logarithmic contribution $\gamma^{(n)}(g,L) $ is governed by a linear integral equation, depending on the solution of the linear integral equations appearing at the steps $n'\leq n-3$.
• We work out this recursive procedure and determine explicitly $\gamma^{(n)}(g,L) $ (in particular $\gamma^{(1)}(g,L)=0$ and $\gamma^{(n)}(g,2)=\gamma^{(n)}(g,3)=0$).