Sign flip triangulations of the amplituhedron
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Springer
Received: March 26, 2020 Accepted: May 1, 2020 Published: May 25, 2020
Sign flip triangulations of the amplituhedron
a
Department of Particle and Nuclear Physics, SOKENDAI (The Graduate University for Advanced Studies), Tsukuba, Ibaraki, 305-0801, Japan b KEK Theory Center, Tsukuba, Ibaraki, 305-0801, Japan c Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA, U.S.A.
E-mail: [email protected], [email protected] Abstract: We present new triangulations of the m = 4 amplituhedron relevant for scattering amplitudes in planar N = 4 super-Yang-Mills, obtained directly from the combinatorial definition of the geometry. Using the “sign flip” characterization of the amplituhedron, we reproduce the canonical forms for the all-multiplicity next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (N2 MHV) tree-level as well as the NMHV one-loop cases, without using any input from traditional amplitudes methods. Our results provide strong evidence for the equivalence of the original definition of the amplituhedron [1] and the topological one [2], and suggest a new path forward for computing higher loop amplitudes geometrically. In particular, we realize the NMHV one-loop amplituhedron as the intersection of two amplituhedra of lower dimensionality, which is reflected in the novel structure of the corresponding canonical form. Keywords: Scattering Amplitudes, Supersymmetric Gauge Theory ArXiv ePrint: 2001.06473
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)121
JHEP05(2020)121
Ryota Kojimaa,b and Cameron Langerc
Contents 1 Introduction
1
2 The amplituhedron 2.1 Definition(s) of the amplituhedron
4 4 5 6 6 7 8 9 9 10 17
4 6 × 2 representation of the one-loop NMHV amplituhedron 4.1 NMHV one-loop as a product of m = 2 amplituhedra 4.2 Five point case 4.3 Six point case 4.4 All multiplicity generalization
18 18 20 22 25
5 Conclusion
26
A 6 × 2 representation of the six-point integrand
27
1
Introduction
Scattering amplitudes have been a continuous source of insight into the hidden structure and simplicity underlying perturbative quantum field theory. Recent years have revealed an unexpected and surprising connection between the S-matrix in an increasingly wide variety of theories and a broad notion of “positive geometry” [1–10]. The complete geometric reformulation of the Feynman diagram approach to calculating amplitudes was accomplished in N = 4 supersymmetric-Yang-Mills (sYM) in the planar limit with the definition of the amplituhedron. This remarkable generalization of polytopes and the positive Grassmannian is conjectured to contain all of the complexities of tree-level amplitudes and loop-level integrands of the theory by associating to the geometry a canonical differential form, defined by having logarithmic singularities on all boundaries (of all co-dimensionality). More broadly, the key idea that scattering amplitudes can be understood as differential forms on
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JHEP05(202
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