Bootstrapping octagons in reduced kinematics from $A_2$ 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 $\mathrm{Ad}{\mathrm{S}}_3$ 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 $G(2,n)/T\sim A_{n-3}$. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping $A_2$ functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters $v,1+v,w,1+w$ of $A_1\times A_1$, 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 $A_2$ 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 $A_2$ functions and beyond.

26 pages, several figures and tables, an ancilary file

References in corpus (7)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• An Operator Product Expansion for Polygonal null Wilson Loops
• Bootstrapping an NMHV amplitude through three loops
• Cluster algebras for Feynman integrals
• Analytic Results for MHV Wilson Loops