Bootstrapping a Five-Loop Amplitude Using Steinmann Relations

Abstract

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude.

The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

“…A leading-order strong-coupling analysis is also possible [37,38], but even more remarkably, the building blocks in question can also be obtained to all loops [39] by means of analytic continuation from a collinear limit where the dynamics is governed by an integrable flux tube [40][41][42][43][44][45][46][47][48][49][50][51], see also [52][53][54]. These developments render the MRK as one of the best sources of 'boundary data' [54][55][56][57] for determining the six-gluon amplitude in general kinematics through five loops, by exploiting its analytic structure with the help of the bootstrap method [30,[58][59][60][61][62].…”

“…The perturbative coefficients (6.1) will be a linear combination of the respective coefficients of all the terms in the right-hand side of (3.51). However for the hexagon amplitudes R h 1 h 2 they have already been obtained up to at least 8 loops to NLLA [55,62], and more generally the holomorphic part may evaluated in terms of harmonic polylogarithms with the method of [84], see also [54,57]. So we only need to focus on the last term in (6.1), that contains the genuine heptagon contributions.…”

Heptagons from the Steinmann cluster bootstrap

“…However, the most powerful applications to date have been to the planar limit of N = 4 super-YangMills (SYM) theory in four dimensions [8,9]. Fueled by an increased understanding of the classes of analytic functions appearing in amplitudes in general quantum field theories, as well as the stringent constraints obeyed by amplitudes in planar N = 4 SYM, it has been possible to advance as far as five loops [10][11][12][13][14][15]. These results in turn provide a rich mine of theoretical data for understanding how scattering amplitudes behave.…”

“…Both helicity configurations of the six-point amplitude have been determined through five loops [15], while the MHV seven-point amplitude has been determined at symbol level through three loops [12]. The seven-point NMHV amplitude has not yet received attention in the bootstrap program, but it has been calculated through two loops using slightly different methods [57].…”

“…[38,39], the bootstrap approach developed in refs. [10][11][12][13][14][15] assumes that the MHV and NMHV amplitudes at each loop order belong to a particular class of iterated integrals, or generalized polylogarithms. More specifically, the L-loop contribution to the remainder and ratio functions is expected to lie within the space spanned by polylogarithms of weight 2L [40] whose symbols can be written in terms of cluster A-coordinates.…”