The cosmic Galois group and extended Steinmann relations for planar $$ \mathcal{N} $$ = 4 SYM amplitudes

Abstract

We describe the minimal space of polylogarithmic functions that is required to express the six-particle amplitude in planar N = 4 super-Yang-Mills theory through six and seven loops, in the NMHV and MHV sectors respectively. This space respects a set of extended Steinmann relations that restrict the iterated discontinuity structure of the amplitude, as well as a cosmic Galois coaction principle that constrains the functions and the transcendental numbers that can appear in the amplitude at special kinematic points. To put the amplitude into this space, we must divide it by the BDS-like ansatz and by an additional zeta-valued constant ρ. For this normalization, we conjecture that the extended Steinmann relations and the coaction principle hold to all orders in the coupling. We describe an iterative algorithm for constructing the space of hexagon functions that respects both constraints. We highlight further simplifications that begin to occur in this space of functions at weight eight, and distill the implications of imposing the coaction principle to all orders. Finally, we explore the restricted spaces of transcendental functions and constants that appear in special kinematic configurations, which include polylogarithms involving square, cube, fourth and sixth roots of unity.

Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

“…On the other hand, just as the momentum twistor space [21], G(4, n), is crucial for the amplituhedron reformulation [22,23] of all-loop integrand in planar N = 4 SYM [24], the space G + (4, n)/T can play a crucial role for all-loop integrated amplitudes (or even non-perturbatively). Recall that for n = 6, 7, all evidence so far [25][26][27][28][29][30][31] is consistent with the conjecture that the alphabet for symbol letters consists of the 9 and 42 cluster variables associated with G + (4, n)/T , respectively; it is thus highly desirable to study the structure of G + (4, n)/T for n ≥ 8. In [12,16,17], it has been shown that polytopes P(4, n) and the associated tropical positive Grassmannian indeed have intriguing applications to the mathematical structures of multi-loop scattering amplitudes in N = 4 SYM.…”

Origin of the Six-Gluon Amplitude in Planar N=4 Supersymmetric Yang-Mills Theory

“…with ρ = ρ(g 2 ) a function of the coupling constant. This function was determined iteratively in [13,33] by demanding that the spaces of functions in which the perturbative amplitudes live obey a coaction principle associated to a cosmic Galois group [48][49][50]. The implementation of this requirement fixes ρ order by order in perturbation theory, ln ρ = 8ζ 2 3 g 6 − 160ζ 3 ζ 5 g 8 + 16(−2ζ 4 ζ 2 3 + 57ζ 2 5 + 105ζ 3 ζ 7 )g 10 + .…”

“…Intriguingly, one of the anomalous dimensions also appears in the light-like limit of the octagon [27][28][29][30][31][32], a correlation function of four operators with large R charge. We also predict the nonlogarithmic term, as well as the coefficient ρ controlling a "cosmic" amplitude normalization [33]. More precisely, we consider the MHV amplitude normalized by the BDS-like ansatz [11,12,34,35], which remains finite as the dimensional regulator = 2− 1 2 D → 0, E(u i ) = lim →0 A 6 (s ij , ) A BDS-like 6 (s ij , ) = exp R 6 + 1 4 Γ cusp E (1) .…”

Rooting out letters: octagonal symbol alphabets and algebraic number theory

“…Feynman) diagrammatic expansion. Examples of such triumphs include the recent determination of certain six-particle amplitudes in planar maximally supersymmetric (N =4) Yang-Mills theory (sYM) through seven loops [21][22][23][24][25][26][27][28][29], and through four loops for seven particles [30][31][32].…”