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 Introduction • Scattering amplitude: Basic objects in QFT. • Recently there are many progress about the understanding of the new structure of scattering amplitude. • Studying and calculating scattering amplitudes became a new direction in theoretical physics. • Major motivations: • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory 2 Introduction • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory • Example: BCFW recursion relation (Britto, Cachazo, Feng, Witten 2005) • Method for building higher-point amplitudes from lower-point <latexit sha1_base64="UNc4SGyPxwXJet7OEnzpssxTFaM=">AAACpHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvYp2unxxRlpmeUJBYV5ZcrmKTrKMQAIZqwKXZhs/R4AWUDPQMwUMBkGEIZygxQEJAvsJwhhiGFIZ8hmaGUIZchlSGPoQTIzmFIZCgGwmgGQwYDhgKgWCxDNVCsCMjKBMunMtQycAH1lgJVpQJVJAJFs4FkOpAXDRXNA/JBZhaDdScDbckB4iKgTgUGVYOrBisNPhucMFht8NLgD06zqsFmgNxSCaSTIHpTC+L5uySCvxPUlQukSxgyELrwurmEIY3BAuzWTKDbC8AiIF8kQ/SXVU3/HGwVpFqtZrDI4DXQ/QsNbhocBvogr+xL8tLA1KDZDFzACDBED25MRpiRnqGBnmGgibKDEzQqOBikGZQYNIDhbc7gwODBEMAQCrR3KcNphisMV5nUmHyYgplCIUqZGKF6hBlQAFMcAFY5oLQ=</latexit> g + g 4g, g + g 5g, g + g 6g ! ! ! Feynman diagrams 220 2485 34300 BCFW 3 6 20 • It is possible to calculate all order amplitudes in Planar N=4 SYM from BCFW. (Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka 2010) 3 Introduction • Efficient calculations of the scattering amplitudes • Use amplitudes as a probe to explore quantum field theory • Geometrization of scattering amplitudes “Amplitude = Volume of a geometric object” Example; Planar N=4 SYM The amplituhedron (Z) (N. Arkani-Hamed, J. Trnka 2013) A<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> n,m,k Z1 • k-plane in k+m dimensional space • Defined as a generalization of convex polytope. Z2 Zn Z4 • m=4 ; physical amplituhedron n,k(Z) A Z3 4 Introduction • For each amplituhedron, we can define a volume form; Canonical Form ⌦ has logarithmic singularities at all boundaries of (Z) ⌦ An,k n -point N k MHV amplitude Canonical form of (Z) <latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</latexit> ⌦ <latexit sha1_base64="seuqwV9vlEHPhS7NYNRX4KJB8gw=">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</latexit> n,k A $ in Momentum twistor space Z Z =(λa,yaa˙ λa) <latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit> i i i k · · · N MHV → amplitude with K+2 negative pμσaa = paa = λaλ˜a i μ i i i helicity particles yaa˙ yaa˙ = paa˙ i − i+1 i • Calculate the scattering amplitude = Construct the volume form of the amplituhedron: this is purely geometrical ! 5 Introduction Geometrization of scattering amplitudes Ω What happensCanonical during form the scattering Amplituhedron process of elementary particles? An Scattering Amplitude ℒ Lagrangian Feynman diagram 6 Introduction • Efficient calculations of the scattering amplitudes BCFW recursion relation • There are another recursion relations Local representation, Momentum twistor diagram… Question • What is the relation between these recursion relations? • Can we obtain another recursion relation? We use the geometrization of scattering amplitudes 7 Motivation One of the triangulation of the BCFW Recursion relation amplituhedron Of cause there are various way of the triangulation • Question: Is it possible to obtain new recursion relations from the triangulation of the amplituhedron? BCFW ? • In this work, we found new recursion relations of the 2-loop MHV, 1-loop NMHV amplitude. Introduction Review of the amplituhedron and sign flip New recursion relation of 2-loop MHV and 1-loop NMHV Summary 9 Review of the amplituhedron and sign flip The definition of the amplituhedron ( Z ) A<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> n,m,k I : k-plane in k+m dimensional space Ya n a<latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit> =1, ,k I I ··· Ya = caiZi I =1, ,k+ m <latexit sha1_base64="JSscDD1R4t/ZDAonHGmCavgQP8I=">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</latexit> i=1 X ··· • ; Positive Grassmannian: matrix mod cai ∈ G+(k, n) k × n GL(k) ca ca > 0 a1 < <ak h 1 ··· k i ··· • I ; Positive Matrix: matrix Zi ∈ M+(k + m, n) (k + m) × n ⟨Za1, ⋯, Zan⟩ > 0 a1 < ⋯ < an • This space is the generalization of the interior of the convex polytope. Convex polytope Amplituhedron (Z) A<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit> n,m,k n generalize n I I I I Y = ciZi Ya = caiZi <latexit sha1_base64="ll6T7Cv+ii63Mdhwco8k/zHMEY8=">AAACgHichVHLLgRBFD3ae7wGG4nNxIRYjdsiIRKJsGHnNd6m090KFf1Kd88kdGZh6wcsrEhExIZvsPEDFj5BLElsLNzp6UQQ3EpVnTp1z61TVYZnySAkeqxRauvqGxqbmlMtrW3tHenOruXALfqmyJuu5fqrhh4ISzoiH8rQEqueL3TbsMSKsT9d2V8pCT+QrrMUHnhiy9Z3HbkjTT1kSkv3rhVmJzaDoq1FckItFyKnnDE1ua7JwqyWzlKO4sj8BGoCskhizk1fYhPbcGGiCBsCDkLGFnQE3DagguAxt4WIOZ+RjPcFykixtshZgjN0Zvd53OXVRsI6vK7UDGK1yadY3H1WZtBPD3RFL3RP1/RE77/WiuIaFS8HPBtVrfC0juOexbd/VTbPIfY+VX96DrGDsdirZO9ezFRuYVb1pcOTl8Xxhf5ogM7pmf2f0SPd8Q2c0qt5MS8WTpHiD1C/P/dPsDycUymnzo9kJ6eSr2hCL/owyO89iknMYA55PvcIl7jBraIog8qQolZTlZpE040voYx/AKCGk6o=</latexit>