The Factorized S-Matrix of CFT/AdS

TL;DR

This paper investigates the extent to which the spectra of planar ${\rm N}=4$ SYM and strings on ${\rm AdS}_5\times S^5$ are governed by an exact, factorized S-matrix. By focusing on three two-component sectors at one loop and developing a perturbative asymptotic Bethe ansatz (PABA) to access the three-loop SU(1|1) fermionic sector, it derives explicit three-loop S-matrix data and demonstrates their consistency with both semiclassical string results and near-plane-wave limits. A remarkable relation among the large-tension S-matrices across sectors is exposed, enabling a three-loop SL(2) sector S-matrix to be inferred from known data, and leading to twist-two anomalous dimensions that agree with the results of Kotikov–Lipatov–Onishchenko–Velizhanin. Collectively, these results support the program of encoding the AdS/CFT spectrum in an all-loop, factorized S-matrix, and hint at a unified spin-chain framework that bridges gauge theory and string theory through dressing factors and sector relations.

Abstract

We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.