Multiple zeta functions and double wrapping in planar N=4 SYM
Sébastien Leurent, Dmytro Volin
TL;DR
<3-5 sentence high-level summary> The paper advances the analytic solution of the AdS/CFT spectral problem for the Konishi operator by applying the FiNLIE to the Y-system, enabling a weak-coupling expansion up to the leading double-wrapping order ($8$ loops). It shows that all perturbative quantities can be expressed in terms of multiple Hurwitz zeta (eta) functions and Euler-Zagier sums, and provides a Mathematica toolkit to manipulate these functions. A key result is the eight-loop anomalous dimension, which features a non-reducible Euler-Zagier sum $\zeta(1,2,8)$, and the authors conjecture that Euler-Zagier sums suffice at any order. They also derive the leading transcendental contributions to all orders and elucidate how exact Bethe equations relate to pole-regularity constraints in the Y-system, with broader implications for the arithmetic structure of planar ${\cal N}=4$ SYM.
Abstract
Using the FiNLIE solution of the AdS/CFT Y-system, we compute the anomalous dimension of the Konishi operator in planar N=4 SYM up to eight loops, i.e. up to the leading double wrapping order. At this order a non reducible Euler-Zagier sum, zeta(1,2,8), appears for the first time. We find that at all orders in perturbation, every spectral-dependent quantity of the Y-system is expressed through multiple Hurwitz zeta functions, hence we provide a Mathematica package to manipulate these functions, including the particular case of Euler-Zagier sums. Furthermore, we conjecture that only Euler-Zagier sums can appear in the answer for the anomalous dimension at any order in perturbation theory. We also resum the leading transcendentality terms of the anomalous dimension at all orders, obtaining a simple result in terms of Bessel functions. Finally, we demonstrate that exact Bethe equations should be related to an absence of poles condition that becomes especially nontrivial at double wrapping.