Bootstrapping the three-loop hexagon

By L. Dixon, J. Drummond, J. Henn. 2011

278 citations recorded by the source corpus.

A bstractWe consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar $ \mathcal{N} = 4 $ super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol’s entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 → 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/46d494f092f1ddfbda0763adf0787a26868bfaea

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In Nuclear and High Energy Physics

1. L. Dixon
2. Claude Duhr
3. M. Hippel
4. J. Drummond
5. P. Vieira
6. M. Spradlin
7. A. Sever
8. B. Basso

Citations in the corpus

• Single-valued harmonic polylogarithms and the multi-Regge limit (2012, methodology): “The large number of possible entries, and the fact that the yi variables are defined in terms of square-root functions of the cross ratios (although the ui can be written as rational functions of the yi [28]), complicates the task of identifying the proper function space for this problem.”
• Uniqueness of two-loop master contours (2012, unknown): “In some cases, using this information, it was even possible to guess nontrivial amplitudes [67, 68].”
• Hexagon functions and the three-loop remainder function (2013, methodology): “J H E P 1 2 ( 2 0 1 3 ) 0 4 9 At three loops, the constraint on the leading (double) discontinuity leaves two free parameters in the symbol, α1 and α2 [33].”
• Motivic amplitudes and cluster coordinates (2013, background): “By imposing all physical constraints available to them at the time, they were able to determine the symbol of the two-loop NMHV amplitude exactly [62], and that of the three-loop MHV amplitude up to two parameters [63] which were subsequently determined in [64].”
• Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions (2014, background): “So far, this scrutiny has yielded nothing but perfect agreement between the two, see [4, 9, 10] and [11] for notable recent examples at high loop orders.”
• Space-time S-matrix and flux tube S-matrix II. Extracting and matching data (2013, methodology): “It turns out to be a powerful constraint on the four loops hexagon and, when combined with the techniques of [13] (see also [54]), it allowed the authors of [14] to successfully determine the symbol of the four loops remainder function.”
• Scattering Amplitudes and the Positive Grassmannian (2012, unknown): “A3(h1, h2, h3) ∝ { [12]h1+h2−h3 [23]h2+h3−h1 [31]h3+h1−h2 ∑ ha > 0; 〈12〉h3−h1−h2〈23〉h1−h2−h3〈31〉h2−h3−h1 ∑ ha < 0.”
• From polygons and symbols to polylogarithmic functions (2011, background): “we still need to nd the arguments of these function types. In the context of particle physics it has proved helpful to use guidance from educated guesses, motivated by physical constraints (cf. refs. [64, 65]), to construct the symbol and/or the functions expressing the desired physical quantities. ”
• Looking for a bulk point (2015, unknown): “[15]).”
• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, methodology): “An example of this phenomenon has already been seen at three loops in the MHV hexagon case [16, 18].”
• Bootstrapping an NMHV amplitude through three loops (2014, methodology): “The same basic assumption for the (parity-even) remainder function R (L) 6 [26, 28] results in a consistent solution through four loops [28, 29].”
• Heptagons from the Steinmann cluster bootstrap (2016, methodology): “The correction to the maximally helicity violating (MHV) amplitude has traditionally been expressed in terms of a (BDS) remainder function [10, 12, 25, 26, 30], while the correction to the next-to-maximally helicity violating (NMHV) amplitude has traditionally been expressed in terms of the infrared”
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, background): “In particular, by now the symbols of all two-loop MHV remainder functions [54] and of the two-loop six-point NMHV amplitude [55] are completely known, while at three loops the symbols of the remainder functions for the hexagon in general kinematics [56] and for the octagon in special kinematics [46]”