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Sign Flip Triangulation of the Amplituhedron Ryota Kojima SOKENDAI, KEK theory centor Based on RK, JHEP 1904 (2019) 085 arxiv:1812.01822 [hep-th] RK, Cameron Langer, Jaroslav Trnka and Minshan Zheng(UC Davis, QMAP), in progress Strings and

  1. Sign Flip Triangulation of the Amplituhedron Ryota Kojima SOKENDAI, KEK theory centor Based on RK, JHEP 1904 (2019) 085 arxiv:1812.01822 [hep-th] RK, Cameron Langer, Jaroslav Trnka and Minshan Zheng(UC Davis, QMAP), in progress Strings and Fields 2019 at YITP 8/20

  2. Introduction structure of scattering amplitude. in theoretical physics. 2 • Scattering amplitude: Basic objects in QFT. • Recently there are many progress about the understanding of the new • Studying and calculating scattering amplitudes became a new direction • Major motivations: • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory

  3. Introduction (Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka 2010) 3 2485 34300 6 220 <latexit sha1_base64="UNc4SGyPxwXJet7OEnzpsxTFaM=">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</latexit> <latexit sha1_base64="UNc4SGyPxwXJet7OEnzpsxTFaM=">ACpHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvYp2unxRlpmeUJBYV5ZcrmKTrKMQAIZqwKXZhs/R4AWUDPQMwUMBkGEIZygxQEJAvsJwhiGFIZ8hmaGUIZchlSGPoQTIzmFIZCgGwmgGQwYDhgKgWCxDNVCsCMjKBMunMtQycAH1lgJVpQJVJAJFs4FkOpAXDRXNA/JBZhaDdScDbckB4iKgTgUGVYOrBisNPhucMFht8NLgD06zqsFmgNxSCaSTIHpTC+L5uySCvxPUlQukSxgyELrwurmEIY3BAuzWTKDbC8AiIF8kQ/SXVU3/HGwVpFqtZrDI4DXQ/QsNbhocBvogr+xL8tLA1KDZDFzACDBED25MRpiRnqGBnmGgibKDEzQqOBikGZQYNIDhbc7gwODBEMAQCrR3KcNphisMV5nUmHyYgplCIUqZGKF6hBlQAFMcAFY5oLQ=</latexit> <latexit sha1_base64="UNc4SGyPxwXJet7OEnzpsxTFaM=">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</latexit> <latexit sha1_base64="UNc4SGyPxwXJet7OEnzpsxTFaM=">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</latexit> BCFW Feynman diagrams (Britto, Cachazo, Feng, Witten 2005) 3 from BCFW. 20 • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory • Example: BCFW recursion relation • Method for building higher-point amplitudes from lower-point g + g → 4 g, g + g → 5 g, g + g → 6 g • It is possible to calculate all order amplitudes in Planar N=4 SYM

  4. 4 (N. Arkani-Hamed, J. Trnka 2013) Introduction “Amplitude = Volume of a geometric object” Example; Planar N=4 SYM The amplituhedron • k-plane in k+m dimensional space • Defined as a generalization of convex polytope. • m=4 ; physical amplituhedron <latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">ACfHichVHLSuRAFD0d3+2r1YWCG7FHUdTmRkTFlY/NLH21ig+aJYaOi+S6gYN/QP+wCxmpSIiOl/hxh9w4SeISwU3InOTDojKjLeoqlOn7rl1qkr3LDOQRPcpa2r6hsSnd3NLa1p7p6FwL3JviLzhWq6/oWuBsExH5KUpLbHh+UKzdUus68WFaH+9LPzAdJ1VeiJHVvbd8w909AkU4VM97atyQNDs8K5SiF0Ru3RYmVoczhdyGQpR3H0fQVqArJIYtHNXGAbu3BhoAQbAg4kYwsaAm5bUEHwmNtByJzPyIz3BSpIs7bEWYIzNGaLPO7zaithHV5HNYNYbfApFneflX0YoDu6pCe6pSt6oNd/1grjGpGXQ571qlZ4hfbjnpWXb1U2zxIH76r/epbYw3Ts1WTvXsxEtzCq+vLRr6eVmeWBcJBO6ZH9n9A93fANnPKzcb4kln8j+gD183N/BWvjOZVy6tJEdnY+YpG9KIfQ/zeU5jFTywiH597hmv8Sb0pP5QRZayaqQSTRc+hDL5F1vukqY=</latexit> <latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> <latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> <latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory • Geometrization of scattering amplitudes A n,m,k ( Z ) Z 1 Z 2 Z n Z 4 A n,k ( Z ) Z 3

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