Triangulation of 2-loop - KIASevents.kias.re.kr/ckfinder/userfiles/201811/files/Triangulation of...

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” to appear

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

East Asia Joint Workshop on Fields and Strings 2018 at KIAS

Introduction• Scattering amplitude: Basic objects in QFT.

• Recently there are many progress about the structure of scattering amplitude in Planar N=4 SYM.

• One of the important progress: Geometric structure

The scattering amplitudes can be obtained from some geometrical object : Amplituhedron

• defined as a generalization of the interior of polytope

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

2

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=">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</latexit>

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

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

• Positive matrix

3

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

Introduction

C = (~c1, · · · ,~cn) 2 G+(k, n)


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=">AAACb3ichVHLSsNAFD2Nr1pfVRcKghSLoiDlRgRFEIpuXLbVqmBFkjhqaJqEZFqoxR/wA3ThwgeIiJ/hxh9w4SeIK1Fw48LbNCAq6h1m5syZe+6cmdFdy/Ql0UNEaWpuaW2Ltsc6Oru6e+K9fau+U/YMkTccy/HWdc0XlmmLvDSlJdZdT2gl3RJrenGxvr9WEZ5vOvaKrLpis6Tt2uaOaWiSqYI2r04WjG1H+pPFrXiSUhRE4idQQ5BEGBknfoUCtuHAQBklCNiQjC1o8LltQAXBZW4TNeY8RmawL3CAGGvLnCU4Q2O2yOMurzZC1uZ1vaYfqA0+xeLusTKBUbqna3qhO7qhR3r/tVYtqFH3UuVZb2iFu9VzOLj89q+qxLPE3qfqT88SO5gNvJrs3Q2Y+i2Mhr6yf/yyPJcbrY3RBT2x/3N6oFu+gV15NS6zIneCGH+A+v25f4LVqZRKKTU7nUwvhF8RxRBGMM7vPYM0lpBBns91cYRTnEWelQFlWEk0UpVIqOnHl1AmPgDM+435</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=">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</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=">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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><latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">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</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.

4

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+

Introduction

• 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)

5

This is the definition of the amplituhedron!

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

C = (c1c2 · · · cn) 2 G+(1, n)


Introduction

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

Conjecture

6

Introduction

C = (~c1, · · · ,~cn) 2 G+(k, n)


Y Ia = C↵aZ

I↵

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M+ :positive matrix:positive GrassmannianG+(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=">AAACa3ichVFNLwNRFD0d3/VVuhEsRFMhpLkjEmLVsGCpKE2QZma8Mul8ZWbahMYfsLSxYEMiIn6GjT9g0Z8gYkViY+HOdBJBcCdv3nnn3XPfefepjqF7PlE9JjU1t7S2tXfEO7u6e3oTff0bnl1xNZHXbMN2C6riCUO3RN7XfUMUHFcopmqITbW8GOxvVoXr6ba17h84YsdU9iy9pGuKz1RhqTg5Xp6yJoqJFGUojJGfQI5AClGs2IlrbGMXNjRUYELAgs/YgAKPvy3IIDjM7aDGnMtID/cFjhBnbYWzBGcozJb5v8errYi1eB3U9EK1xqcYPFxWjiBND3RDL3RPt/RI77/WqoU1Ai8HPKsNrXCKvccDa2//qkyefex/qv707KOEudCrzt6dkAluoTX01cPTl7X51XRtjC7pif1fUJ3u+AZW9VW7yonVM8T5AeTv7f4JNqYzMmXk3EwquxA9RTuGMIpx7vcssljGCvJhn09whvPYs5SUBqXhRqoUizRJfAkp/QFiCovt</latexit><latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">AAACa3ichVFNLwNRFD0d3/VVuhEsRFMhpLkjEmLVsGCpKE2QZma8Mul8ZWbahMYfsLSxYEMiIn6GjT9g0Z8gYkViY+HOdBJBcCdv3nnn3XPfefepjqF7PlE9JjU1t7S2tXfEO7u6e3oTff0bnl1xNZHXbMN2C6riCUO3RN7XfUMUHFcopmqITbW8GOxvVoXr6ba17h84YsdU9iy9pGuKz1RhqTg5Xp6yJoqJFGUojJGfQI5AClGs2IlrbGMXNjRUYELAgs/YgAKPvy3IIDjM7aDGnMtID/cFjhBnbYWzBGcozJb5v8errYi1eB3U9EK1xqcYPFxWjiBND3RDL3RPt/RI77/WqoU1Ai8HPKsNrXCKvccDa2//qkyefex/qv707KOEudCrzt6dkAluoTX01cPTl7X51XRtjC7pif1fUJ3u+AZW9VW7yonVM8T5AeTv7f4JNqYzMmXk3EwquxA9RTuGMIpx7vcssljGCvJhn09whvPYs5SUBqXhRqoUizRJfAkp/QFiCovt</latexit>

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>

has logarithmic singularities on all the boundaries of amplituhedron.

⌦

The definition of canonical form :⌦

n-point N MHV amplitudeAn,m=4,k(Z)<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><latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit><latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit>

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

<latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</latexit><latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</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

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

• Let consider the simplest case: triangle.

• Boundary of the triangle is

• The canonical form is

• Consider simplex in ,

7

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 31ih123i = ✏abcZ

a1Z

b2Z

c3

d2Y = dc2dc3Z2Z3

Introduction

5-pt NMHV amplitude

• How to obtain canonical form for general amplituhedron?

• More general case, it is difficult to triangulation.

• Triangulation for the amplituhedron ↔ Recursion relation for the amplitude

• Example: BCFW recursion relation

• We can obtain the triangulation of the general amplituhedron from the BCFW.

• However the origin of this triangulation is QFT, not the geometry. 8

⌦ =hY d2Y ih123i2

hY 12ihY 23ihY 31i +hY d2Y ih134i2



hY 15ihY 56ihY 61i

Triangulation

Introduction

(Britto, Cachazo, Feng, Witten 2005)

• Motivation:

Is it possible to triangulate the general amplituhedron

from only geometry?

• Some simple case … straightforward.

• In this work, we consider 2-loop MHV amplituhedron from the sign flip definition.

• We found that

• triangulation of n-point 2-loop MHV amplituhedron.

• formula for n-point 2-loop MHV amplitude.

9

Introduction

• 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)

10

Y = c1Z1 + c2Z2 + · · ·+ cnZn ca > 0


Amplituhedron from sign flip

{[Y 12], · · · , [Y 1n]}

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

11


[Y ii+ 1] > 0

[Y ii+ 1jj + 1] > 0<latexit sha1_base64="O/ri5Q4FBesCtNIwfQ5WbX/m3LY=">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</latexit><latexit sha1_base64="O/ri5Q4FBesCtNIwfQ5WbX/m3LY=">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</latexit><latexit sha1_base64="O/ri5Q4FBesCtNIwfQ5WbX/m3LY=">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</latexit><latexit sha1_base64="O/ri5Q4FBesCtNIwfQ5WbX/m3LY=">AAACcHichVHLSsNAFD2Nr1pfVTeCCx9FEYVyI4LiQkQ3Lm1rfVCLJHHU0TSJSVqoxR/wB7pwo4KI+Blu/AEXfoK4U8GNC2/TgKiod5iZM2fuuXNmRndM6flEDxGlobGpuSXaGmtr7+jsinf3rHp20TVE1rBN213XNU+Y0hJZX/qmWHdcoRV0U6zpB4u1/bWScD1pWyt+2RH5grZryR1paD5T+dyGlBPq/v6Emp+jrXiCkhTE4E+ghiCBMJbt+BU2sQ0bBoooQMCCz9iEBo9bDioIDnN5VJhzGclgX+AYMdYWOUtwhsbsAY+7vMqFrMXrWk0vUBt8isndZeUgRuierumF7uiGHun911qVoEbNS5lnva4VzlbXSV/m7V9VgWcfe5+qPz372MFM4FWydydgarcw6vrSUfUlM5seqYzSBT2x/3N6oFu+gVV6NS5TIn2KGH+A+v25f4LVyaRKSTU1lZhfCL8iin4MY4zfexrzWMIysnzuIao4w3nkWelTBpSheqoSCTW9+BLK+AebLY3V</latexit>

{[Y 1234], · · · , [Y 123n]}<latexit sha1_base64="SEo5phrDuaJBB4qivISPDZIzO8c=">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</latexit><latexit sha1_base64="SEo5phrDuaJBB4qivISPDZIzO8c=">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</latexit><latexit sha1_base64="SEo5phrDuaJBB4qivISPDZIzO8c=">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</latexit><latexit sha1_base64="SEo5phrDuaJBB4qivISPDZIzO8c=">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</latexit>

• 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

Y is k plane in k+4 dimensional space which satisfies and has precisely k sign flips.

[Y ij] ⌘ h(Y1Y2)ZiZji<latexit sha1_base64="UVDIcNrRLmFGotHNcrablRUZsTs=">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</latexit><latexit sha1_base64="UVDIcNrRLmFGotHNcrablRUZsTs=">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</latexit><latexit sha1_base64="UVDIcNrRLmFGotHNcrablRUZsTs=">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</latexit><latexit sha1_base64="UVDIcNrRLmFGotHNcrablRUZsTs=">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</latexit>

• 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 , we can expand Y as

• Then from sign flip, and the canonical form for this flip pattern is

• The canonical form for this amplituhedron is

12


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Y1 = Z1 + x1Zi1 + y1Zi1+1

Y2 = �Z1 + x2Zi2 + y2Zi2+1

x1, y1, x2, y2 > 0<latexit sha1_base64="YNfHGZnRnFpHQOOGjMWD+YFPNN0=">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</latexit><latexit sha1_base64="YNfHGZnRnFpHQOOGjMWD+YFPNN0=">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</latexit><latexit sha1_base64="YNfHGZnRnFpHQOOGjMWD+YFPNN0=">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</latexit><latexit sha1_base64="YNfHGZnRnFpHQOOGjMWD+YFPNN0=">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</latexit>

⌦i1i2 =dx1

x1

dy1y1

dx2

x2

dy2y2

⌦ =X

all flips

⌦ij

• The sum of these forms are the full form of this amplituhedron

• This is the direct triangulation of the amplituhedron.

• Each sign flip pattern Cell of the triangulated amplituhedron.

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

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

13


⌦ =hY d2Y ih123i2




hY 15ihY 56ihY 61i

• 1-loop MHV (k=0) physical amplituhedron is isomorphic to the m=2 k=2 unphysical amplituhedron

• m=2, k=2 tree amplituhedron from sign flip

• it correspond to 1-loop k=0 amplitude.

• This triangulation … direct triangulation of the amplituhedron

14

(Y1, Y2) ! (A,B)

A(m = 4, k = 0, n; l = 1) ' A(m = 2, k = 2, n; l = 0)<latexit sha1_base64="rlkhC14jGWvbcIhW4m0jpUnH9PM=">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</latexit><latexit sha1_base64="rlkhC14jGWvbcIhW4m0jpUnH9PM=">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</latexit><latexit sha1_base64="rlkhC14jGWvbcIhW4m0jpUnH9PM=">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</latexit><latexit sha1_base64="rlkhC14jGWvbcIhW4m0jpUnH9PM=">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</latexit>

⌦(A(m = 2, k = 2, n)) = hY d2Y1ihY d2Y2i

⇥X

1<i<j<n

hY1Y2(1ii+ 1) \ (1jj + 1)i2

hY1Y21iihY1Y21i+ 1ihY1Y2ii+ 1ihY1Y21jihY1Y21j + 1ihY1Y2jj + 1i<latexit sha1_base64="kC7eLhtbrYMExc+yaJcU6EtfQEM=">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</latexit><latexit sha1_base64="kC7eLhtbrYMExc+yaJcU6EtfQEM=">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</latexit><latexit sha1_base64="kC7eLhtbrYMExc+yaJcU6EtfQEM=">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</latexit><latexit sha1_base64="kC7eLhtbrYMExc+yaJcU6EtfQEM=">AAADlnichVHdahNBFD7bVVvjT1O9KXizGBo2VMLMIiiSSlVE70xb0x+67TI7naST7B+7k0Bd8gK+QC+8UhARn8ErQX0BL/oI4mUFb7zw7Gap1hA9y+x8853znflmxo08mShCjrQp/czZc9Mz50sXLl66PFueu7KehP2YixYPvTDedFkiPBmIlpLKE5tRLJjvemLD7T3I8hsDEScyDJ6qg0js+KwTyLbkTCHlzGnEfuKLDjNtn6l9zrz03tD0l6wbPRxBrVZdsj0WdDxhbO3tWlsONew4X5+mrRPartpK+iIx7KTvOyltyEa3EQztdsx4eiLCPigyqZSLtGZzFpm0283gqMuuNfyzFkupHNu44BfHLeUZOTFDu5P4SYrMW5EYOuUKqZM8jHFAC1CBIpph+Q3YsAchcOiDDwICUIg9YJDgtw0UCETI7UCKXIxI5nkBQyihto9VAisYsj38d3C1XbABrrOeSa7muIuHI0alAQvkC3lLjsln8o58JT8n9krzHpmXA5zdkVZEzuzz+bUf/1X5OCvY/636p2cFbbide5XoPcqZ7BR8pB88Ozxeu7O6kFbJK/IN/b8kR+QDniAYfOevV8TqCyjhA9C/r3scrFt1Sup05WZl+X7xFDNwDa6Difd9C5bhMTShBVw71N5rH7VP+rx+V3+oPxqVTmmF5iqcCr35C/1D8ns=</latexit>

Triangulation from sign flip

• How about more complicated case?

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

• are 1-loop MHV amplituhedron

• The region of these parameters are further bounded.

Two 2-plane in 4 dimensional space that

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

15

2-loop MHV amplituhedron

h(AB)1(AB)2i > 0

x1, w1, y1, z1 > 0, x2, w2, y2, z2 > 0<latexit sha1_base64="K51LnHveyi2+NaSSx7UEFTPYQN0=">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</latexit><latexit sha1_base64="K51LnHveyi2+NaSSx7UEFTPYQN0=">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</latexit><latexit sha1_base64="K51LnHveyi2+NaSSx7UEFTPYQN0=">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</latexit><latexit sha1_base64="K51LnHveyi2+NaSSx7UEFTPYQN0=">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</latexit>

(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>




• 4-pt case

• The case of

• Another case can be obtained similarly.

• This is 2-loop k=0 4pt amplitude. 16

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.

17

⌦ = ⌦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=">AAADbXichVFNS+NQFD1pnFHrzFgVQVBEpjgzIJSbVlAEQXTjzs+q4EhJMs8aTJOQpAUt7ly5FlyIgoKI+DPc+Adc+BPEhQuF2cxibl8Dop1mXnh5997zzrnnvWd4thWERPdKQm358LG1rT3Z8enzl85UV/dK4JZ9U+RN13b9NUMPhG05Ih9aoS3WPF/oJcMWq8b2TA1frQg/sFxnOdzxxEZJLzrWpmXqIZfc1D4uMIcSBIrQMfkmK6CKLHJy7mGkActhVM5/YVmuZ5tiuVgsTjO+X3Of8RhrFlJpypAcQ42BFgVpRGPeTV3iJ37BhYmylHQQcmyzcMDfOjQQPK5tcBsdPkeWxAUbSTK3zLsE79C5us3/ImfrUdXhvKYZSLbJXWyePjOHMEx3dEXPdEvX9EB/mmpVpUbNyw6vRp0rvELnQd/S7/+ySryG2HplxXoOsYlx6dVi756s1E5h1vmV3aPnpYnF4eo3OqdH9n9G93TDJ3AqL+bFglg8RpIfQHt/3Y3BSjajUUZbGE1PTUdP0YZ+fMUPvu8xTGEW88jDVBzlUDlRThNPaq86oA7WtyaUiNODN0P9/hdhhrjo</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 .

• There are 13 patterns that

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

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

18

[1

2(n� 3)(n� 2)]2

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i, j, k, l = 2, 3, · · · , n� 1, i < j, k < l<latexit sha1_base64="jz57gcdtm503UHy0zDYhl66kGIo=">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</latexit><latexit sha1_base64="jz57gcdtm503UHy0zDYhl66kGIo=">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</latexit><latexit sha1_base64="jz57gcdtm503UHy0zDYhl66kGIo=">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</latexit><latexit sha1_base64="J+QSMRr52VInSt9QIIIXxLd8APU=">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</latexit>

⌦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.


i < k < l < j · · · (1), i < k < j < l · · · (2), i < j < k < l · · · (3), i = k < l < j · · · (4),i = k < j = l · · · (5), i = k < j < l · · · (6), i < k < j = l · · · (7), i < j = k < l · · · (8),k < i = l < j · · · (9), k < i < l < j · · · (10), k < i < j = l · · · (11), k < i < j < l · · · (12),

k < l < i < j · · · (13).<latexit sha1_base64="XrwAzbQCJOmf1xp3VyqmDedA2ww=">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</latexit><latexit sha1_base64="XrwAzbQCJOmf1xp3VyqmDedA2ww=">AAADkHichVHNbtNAEB7XQEv4aVouSFwioloNiqJ1WmhAtihwQZz6Q9pKdRXZ223ZeP0j24lUIl6AGycOnEBCCPEAXEFceAEOfQTEsUhceuis4yRuK9pZ2Tv7zXzfzOw6oeBxQsi+MqZeuHhpfOJy4crVa9cni1PTa3HQiShr0kAE0YZjx0xwnzUTngi2EUbM9hzB1h33iYyvd1kU88B/nuyFbMuzd32+w6mdINSaUu5wwzWE0bbodpDEs3qlqmkSahsig+qVasnCxRFzh+jcADXz9PlK1fIDv+M5LLKsggy2zQHlbqpt5rXvDbXziQuo0i9ojgo2jku7BjdHde9LaYTyk5BMW8IjbV3Pw0N1vX5SXsj4IDpXqWlaq1gmNZJa6bSjZ04ZMlsKip/Agm0IgEIHPGDgQ4K+ABtiXJugA4EQsS3oIRahx9M4g1dQQG4Hsxhm2Ii6+N/F02aG+niWmnHKplhF4BchswQz5Bf5TA7IT/KF/CaH/9XqpRqylz3cnT6Xha3J1zdX/53L8nBP4MWIdWbPCexAI+2VY+9hisgpaJ/fffn2YPXBykxPIx/IH+z/PdknP3ACv/uXflxmK++ggA+gn7zu085avaaTmr48X158nD3FBNyC2zCL970Ai/AUlqAJVHmjfFW+Kd/VabWhPlQf9VPHlIxzA46Z+uwIaujoQQ==</latexit><latexit sha1_base64="XrwAzbQCJOmf1xp3VyqmDedA2ww=">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</latexit><latexit sha1_base64="XrwAzbQCJOmf1xp3VyqmDedA2ww=">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</latexit>

⌦1ijkl =

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

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

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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.

• Is there any recursion relation related to this representation?

19

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

• It is possible to generalize this result.

• 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 direct triangulation of the L-loop integrand from this triangulation?

20

Summary

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