Hexagon Wilson Loop OPE and Harmonic Polylogarithms

TL;DR

This work establishes that the leading Wilson loop OPE contributions for the hexagon in planar N=4 SYM, both for the MHV remainder $R_6$ and the NMHV ratio $\mathcal{R}_6$, evaluate at any loop order to a basis of harmonic polylogarithms. It provides an explicit, algorithmic reduction of the underlying Fourier-type integrals to residues, via polygamma and $Z$-sum structures, proving that the $\sigma$-dependence is captured by $H_{m_1,\ldots,m_r}(-e^{-2\sigma})$. The authors implement the method in Mathematica and obtain new high-loop predictions: the full $\mathcal{O}(e^{-\tau})$ contributions up to 6 loops for the MHV hexagon and NMHV hexagon, and leading/subleading $\tau$-terms up to 12 loops, with results and code accompanying the arXiv version. The findings reinforce the integrability-based approach to Wilson loops and amplitudes, offering a practical path to non-perturbative insight and potential generalization to higher-point polygons and multi-particle flux-tube states.

Abstract

A recent, integrability-based conjecture in the framework of the Wilson loop OPE for 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.

Figures (4)

• Figure 1: In (a), the dashed null segments connect two non-intersecting edges of the hexagon, crossing from two of its cusps. They break it up into three squares, with every two adjacent ones forming a pentagon. In (b), using conformal transformations we place $O$ at the origin, and $P, F, S$ at null past, null future, and spacelike infinity respectively. The conformal group element $e^{-\tau(D-M_{01})}$ leaves the middle square invariant, and its action on $A$ and $B$ makes them parallel to $x^+$ when $\tau\to\infty$.
• Figure 2: Plot of the MHV hexagon leading OPE contribution at 6 loops, $h^{(6)}\equiv\sum_{n=0}^{5}\tau^n h^{(6)}_n(\sigma)$, as a function of $\tau, \sigma$. Colors of the visible spectrum denote different values of $h^{(6)}$, increasing from blue to red. The function is always positive, and monotonically increasing and decreasing in $\tau$ and $\sigma$ respectively.
• Figure 3: Log-linear plot of the $h^{(l)}_0$ component, yielding the leading OPE contribution at $\tau=0$, as a function of $\sigma$ at different loop orders $l$. Its sign is given by $(-1)^l$, as it varies continuously without vanishing. Increasing $l$ by one increases the magnitude by roughly a factor of 10, maintaining similar shape.
• Figure 4: Plots of the leading OPE contribution $F^{(l)}\equiv\sum_{n=0}^l \tau^n F^{(l)}_n(\sigma)$ to the NMHV ratio function component (\ref{['NMHVratio_final']}) at $l=5, 6$ loops, as a function of $\tau, \sigma$. As the functions change sign, we have also included the $F^{(l)}=0$ plane for comparison.