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

Abstract

In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the $sl(2)$ sector of planar $\mathcal{N}=4$ Super Yang-Mills theory expands as $\gamma (g,s,L)=f(g)\ln s+f_{sl}(g,L)+\sum _{n=1}^{\infty }{\gamma }^{(n)}(g,L)(\ln s{)}^{-n}+...$. 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 $n^{\prime }\le n-3$. We work out this recursive procedure and determine explicitly ${\gamma }^{(n)}(g,L)$ (in particular ${\gamma }^{(1)}(g,L)=0$ and ${\gamma }^{(n)}(g,2)={\gamma }^{(n)}(g,3)=0$). Furthermore, we connect the ${\gamma }^{(n)}(g,L)$ (for finite $L$) to the generalised scaling functions, $f_n^{(r)}(g)$, appearing in the limit of large twist $L\sim \ln s$. Finally, we provide the first orders of weak and strong coupling for the first ${\gamma }^{(n)}(g,L)$ (and hence $f_n^{(r)}(g)$).

15 pages, added references, minor changes in introduction and conclusion

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• Two-loop AdS_5 x S^5 superstring: testing asymptotic Bethe ansatz and finite size corrections
• Large spin behavior of anomalous dimensions and short-long strings duality
• TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT
• Giant Holes
• The high spin expansion of twist sector dimensions: the planar N=4 super Yang-Mills theory