The Double Pentaladder Integral to All Orders

TL;DR

This paper computes and analyzes dual-conformally invariant ladder integrals relevant to six-point amplitudes in planar N=4 super-Yang-Mills theory, providing exact formulas, exploring their behavior at different couplings, and describing their function space structure.

Contribution

It introduces exact finite-coupling formulas for double pentaladder integrals and characterizes their function space and algebraic structure, advancing understanding of amplitude integrals in N=4 SYM.

Findings

Exact Mellin integral formulas for double pentaladder integrals.

High-loop weak coupling evaluations in terms of polylogarithms.

Exponential suppression of integrals at strong coupling.

Abstract

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.