On the logarithmic powers of sl(2)sl(2) SYM4

In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the sl(2)sl(2) sector of planar N=4N=4 Super Yang–Mills theory expands as γ(g,s,L)=f(g)lns+fsl(g,L)+∑n=1∞γ(n)(g,L)×(lns)−n+⋯. We find that the sub-logarithmic contribution γ(n)(g,L)γ(n)(g,L) is governed by a linear integral equation, depending on the solution of the linear integral equations appearing at the steps n′⩽n−3n′⩽n−3. We work out this recursive procedure and determine explicitly γ(n)(g,L)γ(n)(g,L) (in particular γ(1)(g,L)=0γ(1)(g,L)=0 and γ(n)(g,2)=γ(n)(g,3)=0γ(n)(g,2)=γ(n)(g,3)=0). Furthermore, we connect the γ(n)(g,L)γ(n)(g,L) (for finite L  ) to the generalised scaling functions, fn(r)(g), appearing in the limit of large twist L∼lnsL∼lns. Finally, we provide the first orders of weak and strong coupling for the first γ(n)(g,L)γ(n)(g,L) (and hence fn(r)(g)).