Abstract
We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.
Highlights
Functions drawn from the class of generalized polylogarithms
Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits
These conditions apply at the level of the symbol, but there are important restrictions on the function space that involve multiple zeta value (MZV) constants, which are invisible at the level of the symbol
Summary
Where x2ij ≡ (xμi − xμj ) are squared differences of dual coordinates These cross ratios can be expressed in terms of (planar) two- and three-particle Mandelstam invariants using the translation si,i+1,...,i+n−1 = (ki + ki+1 + · · · + ki+n−1)2 = x2i,i+n. The MHV amplitude corresponds to the leading term in the expansion (2.3), and as such it depends only on the remainder function R6(u, v, w) defined in eq (2.2) This function is expected to be a pure (generalized) polylogarithmic function (to be defined more precisely below) of the cross ratios (2.5) to all loop orders, meaning that the kinematic dependence only appears in polylogarithms and not in any rational prefactors multiplying these functions. (The cross ratios (2.5) are parity even.) In this paper we will often use the following equivalent symbol alphabet, This alphabet makes the Steinmann relations more transparent by isolating the three independent three-particle Mandelstam invariants si,i+1,i+2 in different letters, a, b and c. We describe how we normalize the amplitudes so that they lie in Hhex
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