Integrability and transcendentality

We derive the two-loop Bethe ansatz for the sl(2) twist operator sector of N = 4 gauge theory directly from the field theory. We then analyse a recently proposed perturbative asymptotic all-loop Bethe ansatz in the limit of large spacetime spin at large but finite twist, and find a novel all-loop scaling function. This function obeys the Kotikov-Lipatov transcendentality principle and does not depend on the twist. Under the assumption that one may extrapolate back to leading twist, our result yields an all-loop prediction for the large spin anomalous dimensions of twist 2 operators. The latter also appears as an undetermined function in a recent conjecture of Bern, Dixon and Smirnov for the all-loop structure of the maximally helicity violating n-point gluon amplitudes of N = 4 gauge theory. This potentially establishes a direct link between the worldsheet and the spacetime S matrix approach. A further assumption for the validity of our prediction is that perturbative BMN (Berenstein-Maldacena Nastase) scaling does not break down at four-loop level or beyond. We also discuss how the result gets modified if BMN scaling does break down. Finally, we show that our result qualitatively agrees at strong coupling with a prediction of string theory.