The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory

TL;DR

The authors bootstrap the four-loop six-point MHV remainder function $R_6^{(4)}(u,v,w)$ in planar ${ m cal N}=4$ SYM from analytic constraints, using hexagon functions, symbol/ coproduct techniques, and data from the near-collinear OPE and multi-Regge limit. This yields a unique weight-eight function whose MRK limit furnishes the NNLLA BFKL eigenvalue and N$^3$LLA impact factor, and allows exploration of the function’s quantitative behavior along several kinematic lines. The work confirms consistent factorization in MRK through NNLLA, reveals deep structure connecting BFKL data to flux-tube excitations, and demonstrates remarkable stability in ratios of successive loop orders across large regions of cross-ratio space, hinting at convergence properties and finite-coupling structure. The results pave the way for extensions to higher loops and to broader kinematics, and they provide a robust framework for connecting perturbative amplitudes to nonperturbative aspects via integrability and Wilson-loop OPE data.

Abstract

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar N=4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at NNNLLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.

Figures (5)

• Figure 1: The ratio $R^{(L)}_6(u,u,w)/R^{(L-1)}_6(u,u,w)$ for $L=3$ (blue) and $L=4$ (green) in Region I. The solid red line represents the curve $\Delta(u,u,w)=0$. At small values of $(u,w)$, the plot is cut off at $u=0.06$ or $w=0.06$.
• Figure 2: The successive ratios $R_6^{(L)}/R_6^{(L-1)}$ on the line $(u,u,1)$.
• Figure 3: The successive ratios $R_6^{(L)}/R_6^{(L-1)}$ on the line $(u,1,1)$.
• Figure 4: The successive ratios $R_6^{(L)}/R_6^{(L-1)}$ on the line $(u,u,u)$.
• Figure 5: The remainder function on the line $(u,u,u)$ plotted at two, three, and four loops and at strong coupling. The functions have been rescaled by their values at the point $(1,1,1)$.