We discuss possible phase factors for the S-matrix of planar N = 4 gauge theory, leading to modifications at four-loop order as compared to an earlier proposal. While these result in a four-loop breakdown of perturbative BMNscaling, Kotikov-Lipatov transcendentality in the universal scaling function for large-spin twist operators may be preserved. One particularly natural choice, unique up to one constant, modifies the overall contribution of all terms containing odd zeta functions in the earlier proposed scaling function based on a trivial phase. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS 5 × S 5 once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.1 While finalizing our manuscript we were informed that this computation [12] has been completed.

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.

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Abstract:We start with an n−point correlation function in a conformal gauge theory. We show that a special limit produces a polygonal Wilson loop with n sides. The limit takes the n points towards the vertices of a null polygonal Wilson loop such that successive distances x 2 i,i+1 → 0. This produces a fast moving particle that generates a "frame" for the Wilson loop. We explain in detail how the limit is approached, including some subtle effects from the propagation of a fast moving particle in the full interacting theory. We perform perturbative checks by doing explicit computations in N = 4 super-Yang-Mills theory.

We explicitly compute the complete three-loop (O(g 4 )) contribution to the four-point function of chiral primary current-like operators q 2 q 2q2 q 2 in any finite N = 2 SYM theory. The computation uses N = 2 harmonic supergraphs in coordinate space. Dramatic simplifications are achieved by a double insertion of the N = 2 SYM linearized action, and application of superconformal covariance arguments to the resulting nilpotent six-point amplitude. The result involves polylogarithms up to fourth order of the conformal cross ratios. It becomes particularly simple in the N = 4 special case.