Abstract
We apply a bootstrap procedure to two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory. We argue that the mathematically most complicated part (the $\Lambda^2 B_2$ coproduct component) of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits). We present a concise, closed-form expression which manifests these properties for all n.
Highlights
Some recent investigations [5, 20,21,22] have revealed that the connection between the cluster structure [23] on Confn(P3) and the mathematical structure of amplitudes in SYM theory runs much deeper than merely specifying the appropriate symbol alphabet
We argue that the mathematically most complicated part of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints
A striking and mysterious feature of eq (3.4) is that all of the pairs {v, z} appearing in the formula have Poisson bracket zero. This feature is an output of the bootstrap; the input was much weaker, with the initial ansatz allowing pairs having Poisson bracket {log v, log z} = ±1
Summary
Let us begin by recalling a few relevant facts about the Gr(4, n) Grassmannian cluster algebra. For n > 7 one can mutate indefinitely to produce an infinite number of A- and X - coordinates, but this poses no conceptual obstacle to the bootstrap program since only finitely many can appear in any individual generalized polylogarithm function (i.e., at any finite loop order). These can be enumerated by inspecting the all-n result of [14]: in the notation of that paper, there are n(n − 6) symbol letters of the form 1(23)(n−1 n)(i i+1) (plus all cyclic partners), n 2.
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