Bootstrapping six-gluon scattering in planar N=4 super-Yang-Mills theory

Drummond

Hippel

et al. 2013

Self Cite

165

398

We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N = 4 superYang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multiRegge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The nearcollinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.

Section: Jhep12(2013)049mentioning

confidence: 79%

Section: Jhep12(2013)049mentioning

confidence: 99%

Hexagon functions and the three-loop remainder function

Drummond

Hippel

et al. 2013

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165

398

“…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.…”

Section: Jhep02(2017)137mentioning

confidence: 99%

Heptagons from the Steinmann cluster bootstrap

Drummond

Harrington

et al. 2017

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165

312

Abstract:We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar N = 4 supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal (Q) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.

“…In this paper we will follow an alternative approach, the hexagon function bootstrap [26][27][28][29][30]. The philosophy of this program is to bypass integrands altogether and focus on infrared-finite quantities from the very beginning.…”

Section: Jhep10(2014)065mentioning

confidence: 99%

“…Many additional properties of hexagon functions, and methods for constructing them, are detailed in refs. [28,30].…”

Section: Jhep10(2014)065mentioning

confidence: 99%

Bootstrapping an NMHV amplitude through three loops

Hippel²

2014

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166

332

Abstract:We extend the hexagon function bootstrap to the next-to-maximally-helicityviolating (NMHV) configuration for six-point scattering in planar N = 4 super-Yang-Mills theory at three loops. Constraints from theQ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function U , which is closely related to the parity-even part of the ratio function V , is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios (u, v, w), as well as on a few two-dimensional planes. Finally, we present an empirical formula for V in terms of elements of the coproduct of the six-gluon MHV remainder function R 6 at one higher loop, which works through three loops for V (four loops for R 6 ).