Bootstrapping octagons in reduced kinematics from A2 cluster algebras

Abstract

Multi-loop scattering amplitudes/null polygonal Wilson loops in mathcal{N} = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.