The double pentaladder integral to all orders

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.

The Sklyanin bracket and cluster adjacency at all multiplicity

“…This surprising property has been shown to hold in all known six-and seven-point amplitudes [10][11][12][13][14][15][16][17][18], as well as some eight-and nine-point amplitudes [9,10]. In six-particle kinematics it is also consistent with an all-orders analysis of the Landau equations [19], and is obeyed by certain classes of Feynman integrals to all loop orders [20].…”

Cluster adjacency and the four-loop NMHV heptagon

“…It would be very interesting to tackle amplitudes beyond six and seven particle scattering in N = 4 super Yang-Mills where the symbol alpabet is given by a finite cluster algebra. For instance, the applicability of cluster adjacency or extended Steinmann relations to individual Feynman integrals [15,47] strongly suggests that this is a general feature of local quantum field theories. Furthermore, cluster adjacency has an imprint on the amplitude also in special kinematics, such as the multi-Regge limit we studied, implying relations even between functions of different logarithmic order.…”

The cosmic Galois group and extended Steinmann relations for planar $$ \mathcal{N} $$ = 4 SYM amplitudes

“…The adjacent symbol entries of the double pentaladder integrals (which contribute to the six-point amplitude at all loop orders) are also contained within this space [84].…”