On the highest transcendentality in N=4 SUSY

TL;DR

The paper tackles the twist-2 anomalous dimension at large spin in $\mathcal N=4$ SUSY by solving the Eden-Staudacher (ES) integral equation and its Beisert-Eden-Staudacher (BES) extension, linking weak and strong coupling via integrability and AdS/CFT. It develops an inverse Laplace representation to convert the problem into a tractable linear-algebra framework with explicitly computable kernels, enabling detailed analysis of analytic properties and convergence. A central result is the maximal transcendentality of perturbative coefficients, expressed as sums of products of $\zeta$-values with integer coefficients, and the exact strong-coupling behavior obtained from singular solutions, which, for BES, reproduces the leading string-theory predictions. The work thus provides a unified, analytic bridge between gauge-theory perturbation theory and string-theory expectations, while highlighting remaining subleading corrections and the need for further refinement of the singular-solution analysis.

Abstract

We investigate the Eden-Staudacher equation for the anomalous dimension of the twist-2 operators at the large spin s in the N=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 zeta(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=infty 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.