On the highest transcendentality in N=4N=4 SUSY

We investigate the Eden–Staudacher equation for the anomalous dimension of the twist-2 operators at the large spin s   in the N=4N=4 super-symmetric gauge theory. This equation is reduced to a set of linear algebraic equations with the kernel calculated analytically. We prove that in perturbation theory the anomalous dimension is a sum of products of the Euler functions ζ(k)ζ(k) having the property of the maximal transcendentality with the coefficients being integer numbers. The radius of convergency of the perturbation theory is found. It is shown, that at g=∞g=∞ the kernel has an essential singularity. The analytic properties of the solution of the Eden–Staudacher equation are investigated. In particular for the case of the strong coupling constant the solution has an essential singularity on the second sheet of the variable j appearing in its Laplace transformation. Similar results are derived also for the Beisert–Eden–Staudacher equation which includes the contribution from the phase related to the crossing symmetry of the underlying S-matrix. We show, that its singular solution at large coupling constants reproduces the anomalous dimension predicted from the string side of the AdS/CFT correspondence.