Abstract
Abstract A recent, integrability-based conjecture in the framework of the Wilson loop OPE for $ \mathcal{N}=4 $ SYM theory, predicts the leading OPE contribution for the hexagon MHV remainder function and NMHV ratio function to all loops, in integral form. We prove that these integrals evaluate to a particular basis of harmonic polylogarithms, at any order in the weak coupling expansion. The proof constitutes an algorithm for the direct computation of the integrals, which we employ in order to obtain the full (N)MHV OPE contribution in question up to 6 loops, and certain parts of it up to 12 loops. We attach computer-readable files with our results, as well as an algorithm implementation which may be readily used to generate higher-loop corrections. The feasibility of obtaining the explicit kinematical dependence of the first term in the OPE in principle at arbitrary loop order, offers promise for the suitability of this approach as a non-perturbative description of Wilson loops/scattering amplitudes.
Highlights
The Operator Product Expansion (OPE) approach to null polygonal Wilson loops is precisely an expansion in terms approaching the collinear limit at different paces, each of which receives contributions at any loop order
We presented a general proof for the exact basis of harmonic polylogarithms, including the dependence of their coefficients and arguments on the kinematical data, which is suitable for describing the leading OPE contribution of the hexagon Wilson loop at any order in the weak coupling expansion g 1
We explored the implications of the recent conjecture [17, 18], which adds significant new ingredients to the OPE approach to null polygonal Wilson loops, for the case of the MHV and NMHV hexagon
Summary
This section serves as a review of the OPE approach for the hexagon Wilson loop, and helps in establishing our notations. In subsection 2.1 we discuss how to take the collinear limit, and outline how at weak coupling the Wilson loop decomposes into terms approaching the limit at different paces, mostly based on [12, 13]. Subsections 2.2 and 2.3 focus on the extension and refinement of this approach for the MHV and NMHV hexagon respectively, as presented in [17, 18], and building on [26,27,28]. The equations which will form the basis of our subsequent analysis are the definition of the conformally invariant, finite Wilson loop observable (2.9), and its leading OPE contribution for the MHV case (2.12) and NMHV case (2.23), in terms of an all-loop integral
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