Abstract
We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar $ \mathcal{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 N3LLA. 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.
Highlights
Cross ratios and a nonvanishing remainder function is the six-point case, corresponding to a hexagonal Wilson loop, for which there are three such cross ratios
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
Much of it comes from the operator product expansion (OPE) for Wilson loops, which corresponds to the near-collinear limit of scattering amplitudes
Summary
The BDS ansatz accounts for all of the amplitude’s infrared divergences, or ultraviolet divergences in the case of the Wilson loop interpretation It absorbs the (related) anomaly in dual conformal transformations. [24], a method based on the coproduct on multiple polylogarithms (or, equivalently, a corresponding set of first-order partial differential equations) was developed that allows for the construction of hexagon functions at arbitrary weight. Using this method, the three-loop remainder function was determined as a particular weight-six hexagon function. We extend the analysis and construct the four-loop remainder function, which is a hexagon function of weight eight
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