Triangulation of 2-loop MHV Amplituhedron from sign ﬂipsIntroduction • Scattering amplitude: Basic objects in QFT. • Study the scattering amplitudes fascinating geometric structures

Triangulation of 2-loop MHV Amplituhedron from sign flips

Ryota Kojima Sokendai, KEK theory centor

Based on

RK “Triangulation of 2-loop MHV Amplituhedron from sign flips” 1812.01822

RK and Jaroslav Trnka(UC Davis, QMAP), in progress

One-day workshop for QFT and string theory 12/4

Introduction• Scattering amplitude: Basic objects in QFT.

• Study the scattering amplitudes fascinating geometric structures underlying scattering amplitudes.

The Amplituhedron

• Loosely speaking, this is defined as a generalization of interior of convex polytope.

• no reference to usual field theory notions: feynman diagrams, Lagrangian…

• Canonical form (related to the volume) of the amplituhedron = Scattering amplitude of Planar N=4 SYM.

2

N. Arkani-Hamed, J. Trnka 2013

• Calculate the scattering amplitude = Construct the canonical form of the amplituhedron: this is purely geometrical problem!

• However it is difficult to construct this form of general amplituhedron.

• One way: Triangulate the general amplituhedron into simple one.

• Sum of these forms = canonical form of the full amplituhedron.

3

has logarithmic singularities on all the boundaries of amplituhedron.

⌦

The definition of canonical form :⌦

Definition of the amplituhedron

Introduction• Question:

Is there a systematic way to triangulate the general amplituhedron?

• In this work, we obtained that

• a systematic way to triangulate the general MHV amplituhedron.

• a new formula for 2-loop MHV amplitude that is not obvious how to derive it from a QFT.

• new geometry: “log amplituhedron”.

• the canonical form of the log amplituhedron = log of the amplitude.

4

　Introduction


Canonical form and triangulation

　Amplituhedron from sign flip and triangulation

Log amplituhedron

Summary

5

Plan

• The amplituhedron : k plane Y in k+m dimensional space

• Positive Grassmannian : k-plane in n-dimensional space

• Positive matrix

6

(N. Arkani-Hamed, J. Trnka 2013)

C = (~c1, · · · ,~cn) 2 G+(k, n)

An,m,k(Z)<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><latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">AAACfHichVHLSuRAFD0d3+2r1YWCG7FHUdTmRkTFlY/NLH21ig+aJJYaOi+S6gYN/QP+wCxmpSIiOl/hxh9w4SeISwU3InOTDojKjLeoqlOn7rl1qkr3LDOQRPcppaa2rr6hsSnd3NLa1p7p6FwL3JJviLzhWq6/oWuBsExH5KUpLbHh+UKzdUus68WFaH+9LPzAdJ1VeeiJHVvbd8w909AkU4VM97atyQNDs8K5SiF0Ru3RYmVoczhdyGQpR3H0fQVqArJIYtHNXGAbu3BhoAQbAg4kYwsaAm5bUEHwmNtByJzPyIz3BSpIs7bEWYIzNGaLPO7zaithHV5HNYNYbfApFneflX0YoDu6pCe6pSt6oNd/1grjGpGXQ571qlZ4hfbjnpWXb1U2zxIH76r/epbYw3Ts1WTvXsxEtzCq+vLRr6eVmeWBcJBO6ZH9n9A93fANnPKzcb4kln8j+gD183N/BWvjOZVy6tJEdnY++YpG9KIfQ/zeU5jFTywiH597hmv8Sb0pP5QRZayaqqQSTRc+hDL5F1vukqY=</latexit>

Y Ia = C↵aZ

I↵

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M+ :positive matrix:positive GrassmannianG+(k, n)

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I = 1, · · · , k +m

a = 1, · · · , k<latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit><latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit><latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit><latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit>

↵ = 1, · · · , n<latexit sha1_base64="A7z7DqZ18uW7vkSotl8BzEVrcqk=">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</latexit><latexit sha1_base64="A7z7DqZ18uW7vkSotl8BzEVrcqk=">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</latexit><latexit sha1_base64="A7z7DqZ18uW7vkSotl8BzEVrcqk=">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</latexit><latexit sha1_base64="A7z7DqZ18uW7vkSotl8BzEVrcqk=">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</latexit>

C = (~c1, · · · , ~cn)/GL(k) hca1 · · · caki > 0 a1 < · · · < ak

G+(k, n)<latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit><latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit><latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit><latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit>

Z = (ZI1 , · · · , ZI

n) 2 M+(k +m,n)<latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit><latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">AAACiXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoRNlqRMUbxnnqxCSn5JcU60TF58V5asZk5in4xmtrZGvn6uRpxgsoG+gZgIECJsMQylBmgIKAfIHlDDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxgyGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRKJoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXDVYafDZ4ITBaoOXBn9wmlUNNgPklkognQTRm1oQz98lEfydoK5cIF3CkIHQhdfNJQxpDBZgt2YC3V4AFgH5Ihmiv6xq+udgqyDVajWDRQavge5faHDT4DDQB3llX5KXBqYGzWbgAkaAIXpwYzLCjPQMDfQMA02UHZygUcHBIM2gxKABDG9zBgcGD4YAhlCgvV0MGxh2Muxi4mYyZLJgsoIoZWKE6hFmQAFMzgCy5ZVA</latexit><latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit><latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit>

Z = (ZI1 , · · · , ZI

n)<latexit sha1_base64="HueWDe0Z0nPPRyXoO+zEABsa6vQ=">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</latexit><latexit sha1_base64="HueWDe0Z0nPPRyXoO+zEABsa6vQ=">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</latexit><latexit sha1_base64="HueWDe0Z0nPPRyXoO+zEABsa6vQ=">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</latexit><latexit sha1_base64="HueWDe0Z0nPPRyXoO+zEABsa6vQ=">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</latexit> hZI

a1, · · · , ZI

ak+m) > 0

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a1 < · · · < ak+m<latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">AAACd3ichVHLSsNAFD2N7/po1I3gwmJRBKHciKCIC9GNS19VQSUkcdTQvEimRS3+gD/gQhAURMXPcOMPuPATxGUFEVx4mwZERb3DzJw5c8+dMzNm4NiRJHpMKQ2NTc0trW3p9o7Oroza3bMa+aXQEgXLd/xw3TQi4dieKEhbOmI9CIXhmo5YM4tztf21sggj2/dW5EEgtlxj17N3bMuQTOlqj6Fr05vWti+jaUOvFEfdI13NUZ7iyP4EWgJySGLBV6+wiW34sFCCCwEPkrEDAxG3DWggBMxtocJcyMiO9wWOkGZtibMEZxjMFnnc5dVGwnq8rtWMYrXFpzjcQ1ZmMUQPdENVuqdbeqL3X2tV4ho1Lwc8m3WtCPTMcd/y678ql2eJvU/Vn54ldjAZe7XZexAztVtYdX358KS6PLU0VBmmC3pm/+f0SHd8A6/8Yl0uiqVTpPkDtO/P/ROsjuU1ymuL47mZ2eQrWtGPQYzwe09gBvNYQIHP3ccZrnCdelMGlGFlpJ6qpBJNL76Eon0Aep2Q+A==</latexit><latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">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</latexit><latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">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</latexit><latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">AAACd3ichVHLSsNAFD2N7/po1I3gwmJRBKHciKCIC9GNS19VQSUkcdTQvEimRS3+gD/gQhAURMXPcOMPuPATxGUFEVx4mwZERb3DzJw5c8+dMzNm4NiRJHpMKQ2NTc0trW3p9o7Oroza3bMa+aXQEgXLd/xw3TQi4dieKEhbOmI9CIXhmo5YM4tztf21sggj2/dW5EEgtlxj17N3bMuQTOlqj6Fr05vWti+jaUOvFEfdI13NUZ7iyP4EWgJySGLBV6+wiW34sFCCCwEPkrEDAxG3DWggBMxtocJcyMiO9wWOkGZtibMEZxjMFnnc5dVGwnq8rtWMYrXFpzjcQ1ZmMUQPdENVuqdbeqL3X2tV4ho1Lwc8m3WtCPTMcd/y678ql2eJvU/Vn54ldjAZe7XZexAztVtYdX358KS6PLU0VBmmC3pm/+f0SHd8A6/8Yl0uiqVTpPkDtO/P/ROsjuU1ymuL47mZ2eQrWtGPQYzwe09gBvNYQIHP3ccZrnCdelMGlGFlpJ6qpBJNL76Eon0Aep2Q+A==</latexit>

M+

I = 1, · · · , k +m


• Let consider a point inside a convex polygon in projective plane.

• Convexity for the polygon is that matrix made by vertices has all positive minors:

• The interior of the polygon is

• We can generalize this to the Grassmannian and higher dimension.

7

Y = c1Z1 + c2Z2 + · · ·+ cnZn

cj > 0

convex polygon not convex polygon

Zi

hZa1Za2Za3i > 0 a1 < a2 < a3

Y = C · ZZ = (Z1Z2 · · ·Zn) 2 M+(3, n)

C = (c1c2 · · · cn) 2 G+(1, n)

:positive matrixM+


• The amplituhedron is defined as the generalization of the interior of convex polygon.

• m=4 is the physical amplituhedron.

Y Ia = C↵aZ

Ia

C = (~c1, · · · ,~cn) 2 G+(k, n)

Z = (Z1, · · · , Zn) 2 M+(k +m,n)

8

This is the definition of the amplituhedron!

Y = C · ZZ = (Z1Z2 · · ·Zn) 2 M+(3, n)

C = (c1c2 · · · cn) 2 G+(1, n)



• The amplituhedron : k plane Y in k+m dimensional space

Conjecture

　　　　　　　　　N MHV … amplitude with K+2 negative helicity particles

9

C = (~c1, · · · ,~cn) 2 G+(k, n)


Y Ia = C↵aZ

I↵


has logarithmic singularities on all the boundaries of amplituhedron.

⌦

The definition of canonical form :⌦

n-point N MHV tree amplitudeAn,m=4,k(Z)<latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit><latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit><latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit><latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">AAACfXichVHLSiNBFD3pGR8TX3HcCG5kQkQhhNsiRAYGdNzMUhOj4oPQ3ZZJk37RXQnEJn6AP+BiVg4jIuL8hBt/wIWfMLhUcCPoTadBVNRbVNWpU/fcOlWle5YZSKKrhPLpc1d3T++XZF//wOBQavjrSuDWfUOUDNdy/TVdC4RlOqIkTWmJNc8Xmq1bYlWvLbT3VxvCD0zXWZZNT2zZWsUxd0xDk0yVU6ObtiarhmaF861y6GTtHzPZWmtyfaqcSlOOohh/DdQYpBHHops6xia24cJAHTYEHEjGFjQE3DagguAxt4WQOZ+RGe0LtJBkbZ2zBGdozNZ4rPBqI2YdXrdrBpHa4FMs7j4rx5GhSzqhG7qgU/pP92/WCqMabS9NnvWOVnjlof3R4t2HKptnieqT6l3PEjuYjbya7N2LmPYtjI6+sXtwU/xeyIQT9Ieu2f8hXdE538Bp3BpHS6LwG0n+APXlc78GK9M5lXLq0kx67mf8Fb0YwzdM8nvnMYdfWESJz93DX5zhX+JByShZJddJVRKxZgTPQsk/Alcckxc=</latexit>

k<latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">AAACZHichVHLSsNAFD2Nr1qrVosgCFIsiqtyI4LiqujGZR+2FmopSRw1NE1Ckha0+AO6VVy4UhARP8ONP+CiPyCISwU3LrxNA6Ki3mFmzpy5586ZGdU2dNcjaoeknt6+/oHwYGQoOjwyGhsbL7pWw9FEQbMMyympiisM3RQFT/cMUbIdodRVQ2yqtbXO/mZTOK5umRvevi0qdWXX1Hd0TfGYytaqsSSlyI/ETyAHIIkgMlbsGlvYhgUNDdQhYMJjbECBy60MGQSbuQpazDmMdH9f4BAR1jY4S3CGwmyNx11elQPW5HWnpuurNT7F4O6wMoFZeqAbeqF7uqUnev+1Vsuv0fGyz7Pa1Qq7Ono0mX/7V1Xn2cPep+pPzx52sOx71dm77TOdW2hdffPg7CW/kpttzdElPbP/C2rTHd/AbL5qV1mRO0eEP0D+/tw/QXEhJVNKzi4m06vBV4QxhRnM83svIY11ZFDgcwWOcYLT0KMUleLSRDdVCgWaOL6ENP0B3EyJ6w==</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit>$

<latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">AAACdHichVHLLgRBFD3T3uMxg42EhZiMWE1ui4RYCRtLr0GCTLpbzahMT3enumaEiR/wAxZWhIj4DBs/YOETxJKwsXCnpxNBcDvVderUPbdO1bUDV4aa6CFhtLS2tXd0diW7e3r7Uun+gfXQrypH5B3f9dWmbYXClZ7Ia6ldsRkoYVVsV2zY5YXG/kZNqFD63po+CMROxSp5sigdSzNVSKe2XVHUSpb2tKWUv19IZyhHUYz+BGYMMohjyU9fYRu78OGgigoEPGjGLiyE/G3BBCFgbgd15hQjGe0LHCHJ2ipnCc6wmC3zv8SrrZj1eN2oGUZqh09xeShWjiJL93RNz3RHN/RI77/Wqkc1Gl4OeLabWhEUUsdDq2//qio8a+x9qv70rFHETORVsvcgYhq3cJr62uHJ8+rsSrY+Tuf0xP7P6IFu+QZe7cW5XBYrp0hyA8zvz/0TrE/mTMqZy1OZufm4FZ0Yxhgm+L2nMYdFLCEf9eQUF7hMvBojRsbINlONRKwZxJcwch9lApCN</latexit><latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">AAACdHichVHLLgRBFD3T3uMxg42EhZiMWE1ui4RYCRtLr0GCTLpbzahMT3enumaEiR/wAxZWhIj4DBs/YOETxJKwsXCnpxNBcDvVderUPbdO1bUDV4aa6CFhtLS2tXd0diW7e3r7Uun+gfXQrypH5B3f9dWmbYXClZ7Ia6ldsRkoYVVsV2zY5YXG/kZNqFD63po+CMROxSp5sigdSzNVSKe2XVHUSpb2tKWUv19IZyhHUYz+BGYMMohjyU9fYRu78OGgigoEPGjGLiyE/G3BBCFgbgd15hQjGe0LHCHJ2ipnCc6wmC3zv8SrrZj1eN2oGUZqh09xeShWjiJL93RNz3RHN/RI77/Wqkc1Gl4OeLabWhEUUsdDq2//qio8a+x9qv70rFHETORVsvcgYhq3cJr62uHJ8+rsSrY+Tuf0xP7P6IFu+QZe7cW5XBYrp0hyA8zvz/0TrE/mTMqZy1OZufm4FZ0Yxhgm+L2nMYdFLCEf9eQUF7hMvBojRsbINlONRKwZxJcwch9lApCN</latexit><latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</latexit><latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</latexit>

Canonical form of

Z = (ZI1 , · · · , ZI



k<latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit>

• Consider the space of k-plane Y in k+4 dim space, together with L 2-plane in 4 dim complement of Y

• Consider the space determined by

• And satisfies that all ordered minors of are positive.

• This is the definition of the full amplituhedron

10


An,k,L(Z)

Y I↵ = C↵aZ

Ia , LI

�(i) = D�a(i)ZIa

L(i)<latexit sha1_base64="b/1Sv0RhnMppidPi6Rha7Js+h2g=">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</latexit><latexit sha1_base64="b/1Sv0RhnMppidPi6Rha7Js+h2g=">AAACdnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsIxuQmlmQkJ+ZU+9TGV2tkatbGCygb6BmAgQImwxDKUGaAgoB8geUMMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNZuACRoAhenBjMsKM9AwN9AwDTZQdnKBRwcEgzaDEoAEMb3MGBwYPhgCGUKC95QxzGZYxLGf8wSTHpMqkDlHKxAjVI8yAApgMADFDkNw=</latexit><latexit sha1_base64="b/1Sv0RhnMppidPi6Rha7Js+h2g=">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</latexit><latexit sha1_base64="b/1Sv0RhnMppidPi6Rha7Js+h2g=">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</latexit>

C 2 G+(k, n)<latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">AAACcHichVFNLwNBGH66vuuruDRxUBriK827IiFOwoFjiyKhaXbXYNLt7trdNqHxB/yBHlyQiIif4eIPOPgJ4obExcHb7SaC4J3MzDPPvM87z8zojik9n+ghojQ0NjW3tLZF2zs6u7pjPb3rnl1yDZE1bNN2N3XNE6a0RNaXvik2HVdoRd0UG3phsba/URauJ21rzT90RK6o7VlyVxqaz1RucVtaiaX8xGhh0hrLx5KUoiASP4EagiTCSNuxK2xjBzYMlFCEgAWfsQkNHrctqCA4zOVQYc5lJIN9gWNEWVviLMEZGrMFHvd4tRWyFq9rNb1AbfApJneXlQkM0z1d0wvd0Q090vuvtSpBjZqXQ571ulY4+e6T+Orbv6oizz72P1V/evaxi9nAq2TvTsDUbmHU9eWj6svq3MpwZYQu6In9n9MD3fINrPKrcZkRK6eI8geo35/7J1ifSqmUUjPTyfmF8Cta0Y8hjPJ7z2Aey0gjy+ceoIoznEeelbgyoAzWU5VIqOnDl1DGPwBZkI21</latexit><latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">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</latexit><latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">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</latexit><latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">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</latexit>

D 2 G+(2, n)<latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit>

0

BBBBB@

D(i1)

D(i2)...

D(il)

C

1

CCCCCA

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Extending the map Y = C.Z in the obvious way to include the D’s leads us to

define the full amplituhedron.

9. The Amplituhedron An,k,L(Z)

We can now give the full definition of the amplituhedron An,k,L(Z). First, the ex-

ternal data for n � k + 4 particles is given by the vectors ZIa living in a (4 + k)

dimensional space; where a = 1, . . . , n and I = 1, . . . , 4 + k. The data is positive

hZa1 . . . Za4+ki > 0 for a1 < · · · < a4+k (9.1)

The amplituhedron lives in G(k, k + 4;L): the space of k planes Y in (k + 4) di-

mensions, together with L 2-planes L(i) in the 4 dimensional complement of Y ,

schematically

We will denote the collection of (L(i), Y ) as Y .

The amplituhedron An,k,L(Z) is the subspace of G(k, k + 4;L) consisting of all

Y ’s which are a positive linear combination of the external data,

Y = C · Z (9.2)

More explicitly in components, the k-plane is Y I↵ , and the 2-planes are LI

�(i), where

� = 1, 2 and i = 1, . . . , L . The amplituhedron is the space of all Y,L(i) of the form

Y I↵ = C↵aZ

Ia , LI

�(i) = D�a(i)ZIa (9.3)

where as in the previous section the C↵a specifies a k-plane in n-dimensions, and the

D�a(i) are L 2-planes living in the (n � k) dimensional complement of C, with the

positivity property that for any 0 l L, all the ordered (k+ 2l)⇥ (k+ 2l) minors

of the (k + 2l)⇥ n matrix

0

BBB@

D(i1)...

D(il)

C

1

CCCA(9.4)

– 17 –

　Introduction




Log amplituhedron

Summary

11

• Let consider the simplest case: triangle.

• Boundary of the triangle is

• The canonical form is

• Consider simplex in ,

12

Y = Z1 + c2Z2 + c3Z3 c2, c3 > 0

Am=2,k=1,n=3

P4 Am=4,k=1,n=5

⌦P =hY d4Y ih12345i4

hY 1234ihY 2345ihY 3451ihY 4512ihY 5123i

c2 ! 0, c3 ! 0

⌦ =dc2c2

dc3c3

=hY d2Y ih123i2

hY 12ihY 23ihY 31i

h123i = ✏abcZa1Z

b2Z

c3

d2Y = dc2dc3Z2Z3

5-pt NMHV amplitude in momentum twistor space


yaai � yaai+1 = paai

pµ1 = �µaa�1a�1a

Zi = (�ai , y

aai �a)

<latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit><latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit><latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit><latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit>

• 1-loop 4-pt MHV amplituhedron; 2-plane in 4 dimensions

• Use , write the D matrix to

• Positivity:

• Then the canonical form is

13

LI� = D�aZ

Ia L� = (ZA, ZB)D 2 G+(2, 4)

D =

✓1 x 0 �w0 y 1 z

◆

x,w, y, z > 0

GL(2)

⌦ =dx

x

dw

w

dy

y

dz

z=

hABd2AihABd2Bih1234i2

hAB12ihAB23ihAB34ihAB14i


ZA = Z1 + xZ2 � wZ4

ZB = yZ2 + Z3 + zZ4<latexit sha1_base64="NQB8A726aFxHmqBMiR6OUFzuAhs=">AAACjHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoR8U7qtlGxRtqV0TFG+mWR8WbxMRwRcU7qdlWAgW0o+KNtauAgvECygZ6BmCggMkwhDKUGaAgIF9gOUMMQwpDPkMyQylDLkMqQx5DCZCdw5DIUAyE0QyGDAYMBUCxWIZqoFgRkJUJlk9lqGXgAuotBapKBapIBIpmA8l0IC8aKpoH5IPMLAbrTgbakgPERUCdCgyqBlcNVhp8NjhhsNrgpcEfnGZVg80AuaUSSCdB9KYWxPN3SQR/J6grF0iXMGQgdOF1cwlDGoMF2K2ZQLcXgEVAvkiG6C+rmv452CpItVrNYJHBa6D7FxrcNDgM9EFe2ZfkpYGpQbMZuIARYIge3JiMMCM9QwM9w0ATZQcnaFRwMEgzKDFoAMPbnMGBwYMhgCEUaG8vw2aGPQx7mfiYTJismWwhSpkYoXqEGVAAkxsA/f2VwQ==</latexit><latexit sha1_base64="NQB8A726aFxHmqBMiR6OUFzuAhs=">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</latexit><latexit sha1_base64="NQB8A726aFxHmqBMiR6OUFzuAhs=">AAACjHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoR8U7qtlGxRtqV0TFG+mWR8WbxMRwRcU7qdlWAgW0o+KNtauAgvECygZ6BmCggMkwhDKUGaAgIF9gOUMMQwpDPkMyQylDLkMqQx5DCZCdw5DIUAyE0QyGDAYMBUCxWIZqoFgRkJUJlk9lqGXgAuotBapKBapIBIpmA8l0IC8aKpoH5IPMLAbrTgbakgPERUCdCgyqBlcNVhp8NjhhsNrgpcEfnGZVg80AuaUSSCdB9KYWxPN3SQR/J6grF0iXMGQgdOF1cwlDGoMF2K2ZQLcXgEVAvkiG6C+rmv452CpItVrNYJHBa6D7FxrcNDgM9EFe2ZfkpYGpQbMZuIARYIge3JiMMCM9QwM9w0ATZQcnaFRwMEgzKDFoAMPbnMGBwYMhgCEUaG8vw2aGPQx7mfiYTJismWwhSpkYoXqEGVAAkxsA/f2VwQ==</latexit><latexit sha1_base64="NQB8A726aFxHmqBMiR6OUFzuAhs=">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</latexit>

(N. Arkani-Hamed, J. Trnka 2013)

• How to obtain canonical form for general amplituhedron?

• More general case, it is difficult to triangulation.

• In some cases the triangulation is known.

• k=1 m=2 : polygon

• k=1 m=4 : polytopes

• 1-loop MHV amplituhedron

• How to triangulate more general amplituhedron?

Use the Sign flip definition of the amplituhedron 14

⌦ =hY d2Y ih123i2

hY 12ihY 23ihY 31i +hY d2Y ih134i2



hY 15ihY 56ihY 61i

Triangulation


⌦P =X

i<j

hABd2AihABd2BihAB(1ii+ 1) \ (1jj + 1)i2

hAB1iihAB1i+ 1ihABii+ 1ihAB1jihAB1j + 1ihABjj + 1i

Kermit representation

(N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Hodges, J. Trnka 2010)

⌦P =X

i,j

h1ii+ 1jj + 1ihY ii+ 1jj + 1ihY i+ 1jj + 11i · · · hY j + 11ii+ 1i

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⌦P =X

i

h1ii+ 1ihY ii+ 1ihY 1iihY 1i+ 1i

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　Introduction




Log amplituhedron

Summary

15

• The amplituhedron is defined as the generalization of the interior of convex polygon.

• There are two definition of the interior of convex polytope:

• vertex-centered description:

• face-centered: Cut out the polytope by a collection of inequalities associated with the boundaries

Y Ia = C↵aZ

Ia

C = (~c1, · · · ,~cn) 2 G+(k, n)

Z = (Z1, · · · , Zn) 2 M+(k +m,n)

16

Y = c1Z1 + c2Z2 + · · ·+ cnZn ca > 0


Amplituhedron from sign flip and triangulation

• Is it possible to define the amplituhedron for the face centered description?

New picture of the amplituhedron: sign flip

Y is in the m=2 k amplituhedron only if

Y is k plane in k+2 dimensional space which satisfies and has precisely k sign flips.

Y is in the m=4 k amplituhedron only if


(N. Arkani-Hamed, H.Thomas, J. Trnka 2017)

17


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• Example: m=2 k=1 n=6 tree amplituhedron

• If Y satisfies these condition, then Y is in the m=2 k=1 n=6 amplituhedron.

• This definition is equivalent to this vertex-centered definition

18


hY ii+ 1i > 0<latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">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</latexit><latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">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</latexit><latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">AAACfXichVHLSiNBFD1pRyeT8RF1I7hpDBFBCbdlIOJCgrNxqcb4wEjobstYWOluujuBGJwPmB+YxaxGRkREf8LN/ICLfILM0oHZCHrTaRAV9TZdde6pe26dqrI8JYOQqJ3Qej709n1Mfkp97h8YHEoPj6wHbt23Rcl2letvWmYglHREKZShEpueL8yapcSGdfC1s77REH4gXWctbHpip2ZWHbknbTNkqpIeKyvTqSqhb+lSTht62Y/SBaqkM5SjKPSXwIhBBnEsu+lTlLELFzbqqEHAQchYwUTA3zYMEDzmdtBizmcko3WBI6RYW+cqwRUmswc8VjnbjlmH807PIFLbvIvi32eljixd0xnd0h86pxu6e7VXK+rR8dLk2epqhVcZ+j5W/P+uqsZziP1H1ZueQ+xhLvIq2bsXMZ1T2F194/DHbXF+NduapGP6y/5/UZuu+ARO4599siJWfyLFD2A8v+6XYH02Z1DOWPmSKSzGT5HEOCYwxfedRwFLWEaJ9/2G37jAZeJey2ozWq5bqiVizSiehJZ/AD1akpA=</latexit><latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">AAACfXichVHLSiNBFD1pRyeT8RF1I7hpDBFBCbdlIOJCgrNxqcb4wEjobstYWOluujuBGJwPmB+YxaxGRkREf8LN/ICLfILM0oHZCHrTaRAV9TZdde6pe26dqrI8JYOQqJ3Qej709n1Mfkp97h8YHEoPj6wHbt23Rcl2letvWmYglHREKZShEpueL8yapcSGdfC1s77REH4gXWctbHpip2ZWHbknbTNkqpIeKyvTqSqhb+lSTht62Y/SBaqkM5SjKPSXwIhBBnEsu+lTlLELFzbqqEHAQchYwUTA3zYMEDzmdtBizmcko3WBI6RYW+cqwRUmswc8VjnbjlmH807PIFLbvIvi32eljixd0xnd0h86pxu6e7VXK+rR8dLk2epqhVcZ+j5W/P+uqsZziP1H1ZueQ+xhLvIq2bsXMZ1T2F194/DHbXF+NduapGP6y/5/UZuu+ARO4599siJWfyLFD2A8v+6XYH02Z1DOWPmSKSzGT5HEOCYwxfedRwFLWEaJ9/2G37jAZeJey2ozWq5bqiVizSiehJZ/AD1akpA=</latexit>

{hY 12i, · · · , hY 1ni}<latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">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</latexit><latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">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</latexit><latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">AAACl3ichVHLShxBFD12YhwnJrZmo2TTZFCyGIbbg5DgxkFBXDqa8YEtQ3dbjo3VD7prBszgD/gDEbKKICL+g5ss4g9k4SeELBXcuPBOT4skg3qbrjp17j23TlU5kfQSRXTZp7142f9qIDeYfz305u2wPjK6koTN2BU1N5RhvObYiZBeIGrKU1KsRbGwfUeKVWd3rpNfbYk48cLgi9qLxKZvNwJv23NtxVRdL1ttS9pBQwpj3SwbVpziouVuhSopPqSC+5S1X9cLVKI0jF5gZqCALBZD/QQWthDCRRM+BAIoxhI2Ev42YIIQMbeJNnMxIy/NC+wjz9omVwmusJnd5bHBq42MDXjd6Zmkapd3kfzHrDQwQb/plK7ogs7oD90+2qud9uh42ePZ6WpFVB8+GFu+eVbl86yw86B60rPCNj6nXj32HqVM5xRuV9/6+u1qeXppoj1JR/SX/f+gS/rJJwha1+5xVSx9R54fwPz/unvBSrlkUsmsThUqs9lT5PAeH/CR7/sTKljAImq87yHO8QsX2rg2o81rC91SrS/TvMM/oVXvAEvfnJk=</latexit><latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">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</latexit>

hY 12i, hY 23i, · · · , hY 61i > 0<latexit sha1_base64="H/mFKJRiDXXX9zjBZwVYjF6Nzxk=">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</latexit><latexit sha1_base64="H/mFKJRiDXXX9zjBZwVYjF6Nzxk=">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</latexit><latexit sha1_base64="H/mFKJRiDXXX9zjBZwVYjF6Nzxk=">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</latexit><latexit sha1_base64="H/mFKJRiDXXX9zjBZwVYjF6Nzxk=">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</latexit>

and

C = (~c1, · · · ,~cn) 2 G+(k, n)Y Ia = C↵aZ

I↵


Z = (ZI1 , · · · , ZI


• m=2 k=1 n=6 tree amplituhedron: 4 patterns of sign flip

• From each pattern, we can construct the canonical form.

• Sign flip takes place in ,

• The canonical form for this amplituhedron is

19

⌦ =X

all flips

⌦ij


⌦ =hY d2Y ih123i2




hY 15ihY 56ihY 61i

i1<latexit sha1_base64="I9t4JZgxXhV2sGM3zRm5NnBJen0=">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</latexit><latexit sha1_base64="I9t4JZgxXhV2sGM3zRm5NnBJen0=">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</latexit><latexit sha1_base64="I9t4JZgxXhV2sGM3zRm5NnBJen0=">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</latexit><latexit sha1_base64="I9t4JZgxXhV2sGM3zRm5NnBJen0=">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</latexit>

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⌦i1 =dx

x

dy

y=

hY d2Y ih1i1i1 + 1ihY 1i1ihY 1i1 + 1ihY i1i1 + 1i

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• Each sign flip pattern Cell of the triangulated amplituhedron.

• Only m=1,2 case, we can triangulate from sign flip.

• m=4 physical amplituhedron or loop amplituhedron, there isn’t simple relation between a particular cell and one of the flip pattern.

• 1-loop MHV amplituhedron is isomorphic to the m=2 k=2 unphysical amplituhedron

• We can use this isomorphism to triangulate the 1-loop MHV amplituhedron from sign flips.

20

Amplituhedron from sign flip

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• 1-loop MHV from sign flips

• How about more general case? 2-loop MHV amplituhedron

• In the sign flip definition, the 2-loop MHV amplituhedron is defined

• have n cells 2-loop MHV amplituhedron has n^2 cells.

21


Two 2-plane in 4 dimensional space that

are 1-loop MHV amplituhedron And there are further constraint, h(AB)1(AB)2i > 0

(AB)1, (AB)2<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit><latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit><latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit><latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>



⌦P =X

i<j

hABd2AihABd2BihAB(1ii+ 1) \ (1jj + 1)i2

hAB1iihAB1i+ 1ihABii+ 1ihAB1jihAB1j + 1ihABjj + 1i

Kermit representation

• 4-pt case

• The case of

• Another case can be obtained similarly.

• This is 2-loop k=0 4pt amplitude. 22

ZA = Z1 + x1Z2 + w1Z3, ZB = �Z1 + y1Z3 + z1Z4,

ZC = Z1 + x2Z2 + w2Z3, ZD = �Z1 + y2Z3 + z2Z4,

hABCDi = h1234i{(x1 � x2)(y1z2 � z1y2) + (z1 � z2)(w1x2 � x1w2)} > 0

(y1z2 � z1y2), (z1 � z2), (w1x2 � x1w2) > 0

0 < x1, 0 < x2 < x1 +(z1 � z2)(w1x2 � x1w2)

y1z2 � z1y2,

0 < w1, 0 < w2 <w1x2

x1

0 < z1, 0 < z2 < z1,

0 < y1, 0 < y2 <y1z2z1

⌦1 =1

x1(1

x2� 1

x2 � x1 � a)1

w1(1

w2� 1

w2 � w1x2x1

)1

y1(1

y2� 1

y2 � y1z2z1

)1

z1(1

z2� 1

z2 � z1)

⌦ =8X

i=1

⌦i =x1y1z2 + x2y2z1 + x2w1z1 + x1w2z2

x1x2w1w2y1y2z1z2{(x1 � x2)(y1z2 � y2z1) + (z1 � z2)(w1x2 � w2x1)}


• Example: 5-pt case

• There are 3×3=9 cells

• The sum of these forms correspond to 5pt 2-loop MHV amplitude.

23

⌦ = ⌦2323 + ⌦3434 + ⌦2424 + ⌦2324 + ⌦2334 + ⌦2434 + ⌦2423 + ⌦3423 + ⌦3424<latexit sha1_base64="OwH72Cq4dAd7Z864qkAqYZ42FA4=">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</latexit><latexit sha1_base64="OwH72Cq4dAd7Z864qkAqYZ42FA4=">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</latexit><latexit sha1_base64="OwH72Cq4dAd7Z864qkAqYZ42FA4=">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</latexit><latexit sha1_base64="/1bSdlR0dPrqWobOKM384JGg3Ho=">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</latexit>

⌦2434 =h123A4ih134C4ihABd2AihABd2BihCDd2CihCDd2Di

hAB12ihAB13ihAB14ihAB15ihAB23ihAB45ihABCDihCD13ihCD14i2hCD15ihCD34ihCD45i

⇥⇢h123A4i(hAB45ihCD13ihCD14i+ hAB15ihCD34ihCD14i)

� h345A2ihAB14ihCD14ihCD15i+ h123C4ihCD14ihAB45ihAB13i�

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• n-pt case, there are cells from the way to chose i,j,k,l .

• This is the triangulation of n-pt 2-loop MHV amplituhedron from sign flip.

• We checked that the sum of these forms is corresponding to the 2-loop amplitude up to 22-pt numerically.

• This looks like a kermit representation of 2-loop MHV amplitude.

• It is not obvious how to derive it from a quantum field theory argument. 24

[1

2(n� 3)(n� 2)]2

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⌦n-pt 2-loop

MHV=

X

i,j,k,l=2,3,··· ,n�1

i<k<l<j

⌦1

ijkl +X

i<k<j<l

⌦2

ijkl +X

i<j<k<l

⌦3

ijkl + · · ·+X

k<l<i<j

⌦13

ijkl

hijkAli ⌘ hAB1l + 1ihijkli � hAB1lihijkl + 1i = hijk(AB) \ (1ll + 1)ihijAlCki ⌘ hCD1k + 1ihijAlki � hCD1kihijAlk + 1i

hiAlCkEji ⌘ hEF1j + 1ihiAlCkji � hEF1jihiAlCkj + 1ihAlCkEjGii ⌘ hGH1i+ 1ihAlCkEjii � hGH1iihAlCkEji+ 1i.

⌦1ijkl =

h1ii+ 1Aiih1jj + 1ClihABd2AihABd2BihCDd2CihCDd2DihAB1iihAB1i+ 1ihAB1jihAB1j + 1ihABCDihCD1kihCD1k + 1ihCD1lihCD1l + 1i

⇥ hABii+ 1ihAjCkCl1i+ hAiAjCkClihABii+ 1ihABjj + 1ihCDkk + 1ihCDll + 1i

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　Introduction




Log amplituhedron

Summary

25

• Loop amplituhedron

• And satisfies that all ordered minors of are positive: Positivity

• What the geometry which has “Negativity”?

26

Log amplituhedron

Y I↵ = C↵aZ

Ia , LI

�(i) = D�a(i)ZIa

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D 2 G+(2, n)<latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">AAACcHichVFNLwNBGH66vuuruEgclIb4SvOuSIiTIOGoKBJtmt01mNjurt1tExp/wB9wcEEiIn6Giz/g0J8gbkhcHLzdbiI0eCcz88wz7/POMzO6Y0rPJypHlLr6hsam5pZoa1t7R2esq3vDswuuIdKGbdrulq55wpSWSPvSN8WW4wotr5tiUz9YqOxvFoXrSdta948ckc1re5bclYbmM5VdzEgrvpQbH5mcsEZzsQQlKYh4LVBDkEAYK3bsBhnswIaBAvIQsOAzNqHB47YNFQSHuSxKzLmMZLAvcIIoawucJThDY/aAxz1ebYesxetKTS9QG3yKyd1lZRxD9Ei39EoPdEdP9PFrrVJQo+LliGe9qhVOrvO0d+39X1WeZx/7X6o/PfvYxUzgVbJ3J2AqtzCq+uLx2eva7OpQaZiu6Jn9X1KZ7vkGVvHNuE6J1XNE+QPUn89dCzYmkyol1dRUYm4+/Ipm9GEQI/ze05jDMlaQ5nMPcYYLXEZelF6lXxmopiqRUNODb6GMfQLo4o19</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit><latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit>

0

BBBBB@

D(i1)

D(i2)...

D(il)

C

1

CCCCCA

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Extending the map Y = C.Z in the obvious way to include the D’s leads us to

define the full amplituhedron.

9. The Amplituhedron An,k,L(Z)

We can now give the full definition of the amplituhedron An,k,L(Z). First, the ex-

ternal data for n � k + 4 particles is given by the vectors ZIa living in a (4 + k)

dimensional space; where a = 1, . . . , n and I = 1, . . . , 4 + k. The data is positive

hZa1 . . . Za4+ki > 0 for a1 < · · · < a4+k (9.1)

The amplituhedron lives in G(k, k + 4;L): the space of k planes Y in (k + 4) di-

mensions, together with L 2-planes L(i) in the 4 dimensional complement of Y ,

schematically

We will denote the collection of (L(i), Y ) as Y .

The amplituhedron An,k,L(Z) is the subspace of G(k, k + 4;L) consisting of all

Y ’s which are a positive linear combination of the external data,

Y = C · Z (9.2)

More explicitly in components, the k-plane is Y I↵ , and the 2-planes are LI

�(i), where

� = 1, 2 and i = 1, . . . , L . The amplituhedron is the space of all Y,L(i) of the form

Y I↵ = C↵aZ

Ia , LI

�(i) = D�a(i)ZIa (9.3)

where as in the previous section the C↵a specifies a k-plane in n-dimensions, and the

D�a(i) are L 2-planes living in the (n � k) dimensional complement of C, with the

positivity property that for any 0 l L, all the ordered (k+ 2l)⇥ (k+ 2l) minors

of the (k + 2l)⇥ n matrix

0

BBB@

D(i1)...

D(il)

C

1

CCCA(9.4)

– 17 –

0

BBBBB@

D(i1)

D(i2)...

D(il)

C

1

CCCCCA< 0

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• We can construct the canonical form from this negative amplituhedron.

• The canonical form = Log of the 2-loop MHV amplitude !!

• We call this: log amplituhedron.

27

Two 2-plane in 4 dimensional space that

are 1-loop MHV amplituhedron And there are further constraint,



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Log amplituhedron

⌦[log [An-pt 2-loop

MHV]] =

X

i,j,k,l=2,3,··· ,n�1

i<k<l<j

⌦1

ijkl[log] +X

i<k<j<l

⌦2

ijkl[log] +X

i<j<k<l

⌦3

ijkl[log] + · · ·+X

k<l<i<j

⌦13

ijkl[log]

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• The loop expansion of the amplitude is

• The log amplitude is

• : two 1-loop amplituhedron

• : two 1-loop amplituhedron with positivity

28

A = 1 + gA1 + g2A2 + g3A3 + · · ·<latexit sha1_base64="H1E3F+7/WFD6K9V+1LnL40D/UtY=">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</latexit><latexit sha1_base64="H1E3F+7/WFD6K9V+1LnL40D/UtY=">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</latexit><latexit sha1_base64="H1E3F+7/WFD6K9V+1LnL40D/UtY=">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</latexit><latexit sha1_base64="H1E3F+7/WFD6K9V+1LnL40D/UtY=">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</latexit>

S = logA = gS1 + g2S2 + g3S3 + · · ·<latexit sha1_base64="EQ0Jy+DZKGDV2v2XQWwnQVLUf50=">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</latexit><latexit sha1_base64="EQ0Jy+DZKGDV2v2XQWwnQVLUf50=">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</latexit><latexit sha1_base64="EQ0Jy+DZKGDV2v2XQWwnQVLUf50=">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</latexit><latexit sha1_base64="EQ0Jy+DZKGDV2v2XQWwnQVLUf50=">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</latexit>

S2 = A2 �1

2A2

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Log amplituhedron

A21

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h(AB)1(AB)2i > 0<latexit sha1_base64="9ZkdJOzPi0G9ArI4TGWoa6Zc4/w=">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</latexit><latexit sha1_base64="9ZkdJOzPi0G9ArI4TGWoa6Zc4/w=">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</latexit><latexit sha1_base64="9ZkdJOzPi0G9ArI4TGWoa6Zc4/w=">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</latexit><latexit sha1_base64="9ZkdJOzPi0G9ArI4TGWoa6Zc4/w=">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</latexit>

h(AB)1(AB)2i < 0<latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">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</latexit><latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">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</latexit><latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">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</latexit><latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">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</latexit>

A21 �A2

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• The amplituhedron, generalizing the interior of convex polygon, gives a complete definition of scattering amplitudes in planar N=4 SYM.

• From sign flip definition, we can obtain the canonical form of n-point 2-loop MHV amplituhedron from only geometry.

• This representation of the 2-loop MHV amplitude is completely new.

• It is not obvious how to derive it from a quantum field theory argument.

• From log amplituhedron, we can obtain the log amplitude from geometry.

29

Summary

⌦n-pt 2-loop

MHV=

X

i,j,k,l=2,3,··· ,n�1

i<k<l<j

⌦1

ijkl +X

i<k<j<l

⌦2

ijkl +X

i<j<k<l

⌦3

ijkl + · · ·+X

k<l<i<j

⌦13

ijkl

• To generalize this sign flip triangulation.

• MHV higher loop case

• 3-loop: There are 3 1-loop amplituhedron and

• In general, there are L 1-loop amplituhedron and L(L-1)/2 constraints.

• Is it possible to obtain the L-loop MHV amplitude from this triangulation?

• Triangulate the N MHV loop amplituhedron from sign flip .

30

Future work

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Documents

Triangulation of 2-loop MHV Amplituhedron from sign ﬂipsIntroduction • Scattering amplitude: Basic objects in QFT. • Study the scattering amplitudes fascinating geometric structures