The Double Pentaladder Integral to All Orders

TL;DR

The authors analyze dual-conformally invariant six-point ladder integrals with pentagon caps in planar ${\cal N}=4$ SYM, deriving all-orders finite-coupling Mellin representations for the double pentaladder family and establishing a rich function space ${\Omega}$ governed by a coproduct structure. They separate the ladders via symmetry-adapted variables $(x,y,z)$, obtaining Mellin-integral solutions with hypergeometric building blocks and identifying auxiliary ladders ${\cal O}$ and ${\cal W}$ that close the system under differentiation. The work connects to the Steinmann hexagon function space by detailing a nonperturbative coproduct/coaction formalism, including a nonperturbative discontinuity analysis and a path-ordered- exponential description, thereby providing a concrete toy model for all-orders structure in hexagon functions. Across weak and strong coupling, they demonstrate explicit perturbative expansions in terms of multiple polylogarithms, exponential suppression at strong coupling, and a scalable coproduct framework that could illuminate the full six-point amplitude space. Overall, the paper offers a complete finite-coupling representation, a robust algebraic framework, and practical tools for constructing and understanding high-loop hexagon functions in ${\cal N}=4$ SYM.

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.

Figures (3)

• Figure 1: The eight- or higher-point $L$-loop ladder integral, labelled by dual coordinates.
• Figure 2: The seven- or higher-point pentabox ladder integral, labelled by dual coordinates, where the ladder is formed out of $L-1$ loops. The dashed line represents a numerator factor that renders the integral DCI.
• Figure 3: The six-point integral $\Omega^{(L)}$, labelled by dual coordinates, where the ladder is formed out of $L-2$ loops. The dashed lines represent numerator factors that render the integral DCI.