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 mathcal{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.
Highlights
Despite substantial progress, our understanding of particle scattering in perturbative quantum field theory remains incomplete
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
A similar coaction principle has been observed in the full space of Steinmann hexagon functions, where it constrains the transcendental constants that can appear in the first entry in addition to restricting the symbols of these objects [69, 88]
Summary
Our understanding of particle scattering in perturbative quantum field theory remains incomplete. A great deal is known about the space of functions that can contribute to six- and seven-particle perturbative amplitudes in planar N = 4 SYM When these amplitudes are normalized by the BDS ansatz, they can be written in terms of dual superconformal invariants (that encode the helicity structure) multiplied by multiple polylogarithms that have known kinematic dependence and branch cuts only in physical channels [59,60,61,62,63]. Armed with these differential equations, we consider finite-coupling versions of Ω(L) and Ω (L) by summing over the loop order weighted by (−g2)L, as was done previously for a related box ladder integral [78] While these quantities are not the full finite-coupling six-point amplitude, they do constitute well-defined contributions to it that sum up an infinite class of Feynman integrals. The third, omegacdiscwt0-12.m, gives the c-discontinuity of all the functions in the Ω space through weight 12
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