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2021

Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras

He, Song, Li, Zhenjie, Tang, Yichao et al.

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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 $\rm AdS_3$ 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 $v-w, 1- v w$ 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.

Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras · Around

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