Sign Flip Triangulation of the Amplituhedron Ryota Kojima - - PowerPoint PPT Presentation

sign flip triangulation of the amplituhedron
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Sign Flip Triangulation of the Amplituhedron Ryota Kojima - - PowerPoint PPT Presentation

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


slide-1
SLIDE 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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

Introduction

  • 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
  • It is possible to calculate all order amplitudes in Planar N=4 SYM

from BCFW.

3

(Britto, Cachazo, Feng, Witten 2005) (Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka 2010)

Feynman diagrams BCFW

g + g → 4g, g + g → 5g, g + g → 6g

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220 6 34300 2485 3 20

slide-4
SLIDE 4
  • 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

  • k-plane in k+m dimensional space
  • Defined as a generalization of convex polytope.
  • m=4 ; physical amplituhedron

4

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

An,m,k(Z)

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Introduction

Z1

Z2 Z3

Z4 Zn

An,k(Z)

slide-5
SLIDE 5
  • For each amplituhedron, we can define a volume form;
  • Calculate the scattering amplitude = Construct the volume form of

the amplituhedron: this is purely geometrical !

5

Introduction

  • point N MHV amplitude

in Momentum twistor space

n

Z

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Canonical form of

y ˙

aa i

− y ˙

aa i+1 = p˙ aa i

Zi = (λa

i , ya˙ a i λa)

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i σa · a μ = pa · a i

= λa

i ˜

λ

· a i

An,k(Z)

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k

Ω has logarithmic singularities at all boundaries of

An,k(Z)

N MHV → amplitude with K+2 negative helicity particles k Canonical Form

slide-6
SLIDE 6

6

Lagrangian

Ω

Canonical form Amplituhedron Feynman diagram Scattering Amplitude

An

Introduction

Geometrization of scattering amplitudes

slide-7
SLIDE 7
  • Efficient calculations of the scattering amplitudes
  • There are another recursion relations
  • What is the relation between these recursion relations?
  • Can we obtain another recursion relation?

7

BCFW recursion relation Local representation, Momentum twistor diagram…

We use the geometrization of scattering amplitudes

Question

Introduction

slide-8
SLIDE 8
  • Question:

Is it possible to obtain new recursion relations from the triangulation of the amplituhedron?

  • In this work, we found new recursion relations of the 2-loop MHV, 1-loop NMHV amplitude.

?

BCFW

Motivation

BCFW Recursion relation One of the triangulation of the amplituhedron

Of cause there are various way of the triangulation

slide-9
SLIDE 9

Introduction Review of the amplituhedron and sign flip New recursion relation of 2-loop MHV and 1-loop NMHV Summary

9

slide-10
SLIDE 10

The definition of the amplituhedron

  • ; Positive Grassmannian:

matrix mod

  • ; Positive Matrix:

matrix

  • This space is the generalization of the interior of the convex polytope.

cai ∈ G+(k, n) k × n GL(k) ZI

i ∈ M+(k + m, n)

(k + m) × n

10

An,m,k(Z)

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I = 1, · · · , k + m

a = 1, · · · , k

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hca1 · · · caki > 0

a1 < · · · < ak

⟨Za1, ⋯, Zan⟩ > 0 a1 < ⋯ < an

Y I

a = n

X

i=1

caiZI

i

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: k-plane in k+m dimensional space YI

a

Y I =

n

X

i=1

ciZI

i

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I = 1, · · · , m + 1

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hZi1Zi2 · · · Zim+1i > 0

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ci > 0

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Convex polytope Amplituhedron I = 1, · · · , k + m

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a = 1, · · · , k

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hZi1 · · · Zik+mi > 0

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hci1 · · · ciki > 0

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An,m,k(Z)

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generalize

Y I

a = n

X

i=1

caiZI

i

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Review of the amplituhedron and sign flip

slide-11
SLIDE 11
  • Canonical form : logarithmic singularities at all boundaries of
  • The simplest case:
  • Boundary
  • The canonical form is
  • Let us rewrite as four-dimensional part and its complement

⟨Y1234⟩, ⟨Y2345⟩, ⟨Y3451⟩, ⟨Y4512⟩, ⟨Y5123⟩,

11

Review of the amplituhedron and sign flip

An,k(Z)

Am=4,k=1,n=5

Zi

Z d4φ Z δ(Y Y ∗)Ω = (h1234iη5 + h2345iη1 + · · · + h5123iη4)4 h1234ih2345ih3451ih4512ih5123i

Zi = ✓ zi δzi ◆

δzi = ηi · φ η, φ : auxiliary grassmann variables

Y ∗ = (0, 0, 0, 0, 1)

5pt tree amplitude

Ω = d log hY 1234i hY 5123id log hY 2345i hY 5123id log hY 3451i hY 5123id log hY 4512i hY 5123i = hY dY dY dY dY ih12345i4 hY 1234ihY 2345ihY 3451ihY 4512ihY 5123i

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⟨Yijkl⟩ = ϵMIJKLYMZI

i ZJ j ZK k ZL l

slide-12
SLIDE 12
  • To obtain the recursion relation, we need to triangulate the

amplituhdron

  • Example: Polygon
  • More general case, it is difficult to triangulate.
  • How to triangulate more general amplituhedron systematically?

Sign flip triangulation

Ω = hY d2Y ih123i2 hY 12ihY 23ihY 31i + hY d2Y ih134i2 hY 13ihY 34ihY 41i + hY d2Y ih145i2 hY 14ihY 45ihY 51i + hY d2Y ih156i2 hY 15ihY 56ihY 61i

Review of the amplituhedron and sign flip

slide-13
SLIDE 13
  • ; Interior of this polygon
  • Triangulate

⟨Y12⟩ > 0,⟨Y23⟩ > 0,⟨Y34⟩ > 0,⟨Y45⟩ > 0,⟨Y51⟩ > 0 Z1

Z2

Z3

Z4 Z5

Y

Z1

Z2

Z3

Z4 Z5

⟨Y13⟩ < 0 ⟨Y13⟩ > 0 ⟨Y13⟩ > 0 ⟨Y14⟩ > 0 ⟨Y14⟩ < 0 ⟨Y14⟩ < 0

Each cell is specifies by the signs of ⟨Y1i⟩

Review of the amplituhedron and sign flip

⟨Y12⟩ > 0 ⟨Y12⟩ < 0

hY 12i hY 13i hY 14i hY 15i +

  • (1)

+ +

  • (2)

+ + +

  • (3)
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(1) (2) (3)

⟨Y23⟩ > 0 ⟨Y23⟩ < 0

slide-14
SLIDE 14
  • m=2 k=1 n=6 tree amplituhedron: 4 patterns of signs
  • Generally,
  • Example; m=2 k=2 n=6
  • Sign flip triangulation of the amplituhedron

14

1 sign flip

Each cell of the m=2 k,n amplituhedron is specified as and has precisely k sign flips.

hY ii + 1i > 0

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{hY 12i, · · · , hY 1ni}

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

Review of the amplituhedron and sign flip

hY 12i hY 13i hY 14i hY 15i hY 16i +

  • +

+ + +

  • +

+ +

  • +

+ +

  • +

+ + +

  • +

+ + +

  • +
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6 cells

slide-15
SLIDE 15
  • From each pattern, we can construct the canonical form.
  • Sign flip takes place in ,
  • The canonical form for this amplituhedron is
  • 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.

15

Ω = X

all flips

Ωij

i1

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Y = Z1 + xZi1 + yZi1+1

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Ωi1 = dx x dy y = hY d2Y ih1i1i1 + 1i hY 1i1ihY 1i1 + 1ihY i1i1 + 1i

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Review of the amplituhedron and sign flip

slide-16
SLIDE 16

Introduction Review of the amplituhedron and sign flip New recursion relation of 2-loop MHV and 1-loop NMHV Summary

16

slide-17
SLIDE 17
  • Only m=1,2 case, we can triangulate from sign flip.
  • 1-loop MHV amplituhedron is isomorphic to the m=2 k=2 amplituhedron
  • From this, we can triangulate the 1-loop MHV amplituhedron from sign

flips!

  • 1-loop MHV from sign flips
  • This is corresponding to the BCFW recursion relation.

17

New recursion relation of 2-loop MHV and 1-loop NMHV

A(m = 4, k = 0, n; l = 1) ' A(m = 2, k = 2, n; l = 0)

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ΩP = X

i<j

hABd2AihABd2BihAB(1ii + 1) \ (1jj + 1)i2 hAB1iihAB1i + 1ihABii + 1ihAB1jihAB1j + 1ihABjj + 1i

slide-18
SLIDE 18
  • How about more general case? 2-loop MHV amplituhedron
  • 2-loop MHV from sign flips
  • This is not BCFW recursion relation → New recursion relation.
  • It is not obvious how to derive it from a quantum field theory argument.

18

Ωn-pt 2-loop

MHV

= X

i,j,k,l=2,3,··· ,n−1 i<k<l<j

Ω1

ijkl +

X

i<k<j<l

Ω2

ijkl +

X

i<j<k<l

Ω3

ijkl + · · · +

X

k<l<i<j

Ω13

ijkl

hijkAli ⌘ hAB1l + 1ihijkli hAB1lihijkl + 1i = hijk(AB) \ (1ll + 1)i hijAlCki ⌘ hCD1k + 1ihijAlki hCD1kihijAlk + 1i hiAlCkEji ⌘ hEF1j + 1ihiAlCkji hEF1jihiAlCkj + 1i hAlCkEjGii ⌘ hGH1i + 1ihAlCkEjii hGH1iihAlCkEji + 1i.

Ω1

ijkl =

h1ii + 1Aiih1jj + 1ClihABd2AihABd2BihCDd2CihCDd2Di hAB1iihAB1i + 1ihAB1jihAB1j + 1ihABCDihCD1kihCD1k + 1ihCD1lihCD1l + 1i ⇥ hABii + 1ihAjCkCl1i + hAiAjCkCli hABii + 1ihABjj + 1ihCDkk + 1ihCDll + 1i

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New recursion relation of 2-loop MHV and 1-loop NMHV

(RK 2018)

4pt 5pt 6pt 7pt 8 28 68 138 1 9 36 100 BCFW Sign flip

slide-19
SLIDE 19
  • The 1-loop NMHV case
  • However, this amplituhedron is also constructed from m=2, k=3 tree

amplituhedron and convex polygon

  • Polygon is the intersection with the m=4 k=1 tree amplituhedron and

19

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

New recursion relation of 2-loop MHV and 1-loop NMHV

(RK, C. Langer, M. Zheng, J. Trnka, in progress)

An,l=1,k=1(Z) = An,m=4,k=1(Z) × An,l=1,k=0(Z)

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An,l=1,k=1(Z) = An,m=2,k=3(Z) × An,m=2,k=1(Z)

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An,m=2,k=3(Z)

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An,m=2,k=3(Z)

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An,m=4,k=1(Z)

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An,m=2,k=1(Z)

<latexit sha1_base64="N6tKcpaPLKz5hIHMG5UZqvgvnCY=">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</latexit><latexit sha1_base64="N6tKcpaPLKz5hIHMG5UZqvgvnCY=">ACf3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtIxeQmlmQkJ+ZUO9bGV+fp5Noa6WTbGtZqRGnGCygb6BmAgQImwxDKUGaAgoB8geUMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNZuACRoAhenBjMsKM9AwN9AwDTZQdnKBRwcEgzaDEoAEMb3MGBwYPhgCGUKC9DQzLGNYzbGBiZFJn0mMygChlYoTqEWZAUyWAGREkpg=</latexit><latexit sha1_base64="N6tKcpaPLKz5hIHMG5UZqvgvnCY=">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</latexit><latexit sha1_base64="N6tKcpaPLKz5hIHMG5UZqvgvnCY=">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</latexit>
slide-20
SLIDE 20
  • 5-pt case
  • m=2 k=3 n=5 amplituhedron
  • The intersecting polygon is
  • The full form is
  • This is corresponding to the BCFW representation.

20

Ω6 = h12345i2hY ABd2Y ihY ABd2AihY ABd2Bi hY AB12ihY AB23ihY AB34ihY AB45ihY AB51i

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New recursion relation of 2-loop MHV and 1-loop NMHV

̂ 5 ̂ 4 ̂ 3 ̂ 2 ̂ 1

Ω2 = hyd2yi ⇥ hˆ 5ˆ 1ˆ 2i2 hyˆ 5ˆ 1ihyˆ 1ˆ 2ihyˆ 5ˆ 2i + hˆ 5ˆ 2ˆ 3i2 hyˆ 5ˆ 2ihyˆ 2ˆ 3ihyˆ 5ˆ 3i + hˆ 5ˆ 3ˆ 4i2 hyˆ 5ˆ 3ihyˆ 3ˆ 4ihyˆ 5ˆ 4i !

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ˆ i = (Y AB) \ (i 1ii + 1) = ZAhY Bi 1ii + 1i + ZBhY Ai 1ii + 1i

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Ω6 ⇥ Ω2 = h12345i2hY ABd2Y ihY ABd2AihY ABd2Bi hY AB12ihY AB23ihY AB34ihY AB45ihY AB51i ⇥ hyd2yi hˆ 5ˆ 1ˆ 2i2 hyˆ 5ˆ 1ihyˆ 1ˆ 2ihyˆ 5ˆ 2i + hˆ 5ˆ 2ˆ 3i2 hyˆ 5ˆ 2ihyˆ 2ˆ 3ihyˆ 5ˆ 3i + hˆ 5ˆ 3ˆ 4i2 hyˆ 5ˆ 3ihyˆ 3ˆ 4ihyˆ 5ˆ 4i !

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slide-21
SLIDE 21
  • In this 1-loop NMHV case, there are another recursion relations.
  • We found that these recursion relations are corresponding to the another

triangulations of the pentagon.

21

New recursion relation of 2-loop MHV and 1-loop NMHV

̂ 5 ̂ 4 ̂ 3 ̂ 2 ̂ 1 ̂ 5 ̂ 4 ̂ 3 ̂ 2 ̂ 1

(Y. Bai, S. He, T. Lam 2015) (N. Arkani-Hamed, J. Bourjaily, F. Cachazo, and J. Trnka 2010)

Recursion relation from Momentum twistor diagrams Local representation

slide-22
SLIDE 22
  • 6-pt case
  • m=2 k=3 n=6 amplituhedron … 4 cells
  • For each cell, there is a pentagon
  • This is not the BCFW recursion relation !

22

Ω6 = Ω234

6

+ Ω235

6

+ Ω245

6

+ Ω345

6

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612 123 234 345 456 145 125 123 234 345 456 612 561 612 234 345 456 136 134 561

Ω6 = Ω234

6

× Ω(1)

2

+ (Ω235

6

+ Ω245

6

) × Ω(2)

2

+ Ω345

6

× Ω(3)

2

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(1)

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(2)

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(3)

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New recursion relation of 2-loop MHV and 1-loop NMHV

slide-23
SLIDE 23
  • In general
  • where
  • This is the new recursion relation of n-pt 1-loop NMHV amplitude.

23

Ωn = X

2≤i<j<k≤n

Ω6

ijk × Ω2 +

X

1≤i<j<k<j<m≤n

Ω6

ijklm × Ω(ijk)(jkl)(klm) 2

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Ω6

ijk = hY ABd2Y ihY ABd2AihY ABd2Bi

  • hY A1ii + 1i

hY A1jj + 1i hY A1kk + 1i hAB1ii + 1i hAB1jj + 1i hAB1kk + 1i hBY 1ii + 1i hBY 1jj + 1i hBY 1kk + 1i

  • 2

hY AB1iihY AB1i+1ihY ABii+1ihY AB1jihY AB1j+1ihY ABjj+1i hY AB1kihY AB1k+1ihY ABkk+1i

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Ω(a)(b)(c)

2

= habci2hyd2yi hyabihybcihycai

<latexit sha1_base64="T3Bosd62muBZEf0YEjFyNZnL0lA=">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</latexit><latexit sha1_base64="T3Bosd62muBZEf0YEjFyNZnL0lA=">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</latexit><latexit sha1_base64="T3Bosd62muBZEf0YEjFyNZnL0lA=">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</latexit><latexit sha1_base64="T3Bosd62muBZEf0YEjFyNZnL0lA=">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</latexit>

Ω6

abcde =

hY ABd2Y ihY ABd2AihY ABd2Bihabcdei2 hY ABabihY ABbcihY ABcdihY ABdeihY ABeai

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New recursion relation of 2-loop MHV and 1-loop NMHV

slide-24
SLIDE 24
  • The amplituhedron, generalizing the interior of convex polygon, gives a

complete definition of scattering amplitudes in planar N=4 SYM.

  • From sign flip triangulation, we obtained new recursion relations for
  • 2-loop MHV amplitude
  • 1-loop NMHV amplitude
  • Various recursion relations of the 1-loop NMHV are understood as various

triangulations of the polygon.

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

24

Summary

Ωn-pt 2-loop

MHV

= X

i,j,k,l=2,3,··· ,n−1 i<k<l<j

Ω1

ijkl +

X

i<k<j<l

Ω2

ijkl +

X

i<j<k<l

Ω3

ijkl + · · · +

X

k<l<i<j

Ω13

ijkl

Ωn = X

2≤i<j<k≤n

Ω6

ijk × Ω2 +

X

1≤i<j<k<j<m≤n

Ω6

ijklm × Ω(ijk)(jkl)(klm) 2

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slide-25
SLIDE 25
  • It is possible to generalize this result.
  • 3-loop MHV: There are three 1-loop amplituhedron
  • 1-loop N MHV

25

Summary

A(m = 4, k = 0, n; l = 1) ' A(m = 2, k = 2, n; l = 0)

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An,l=1,k(Z) = An,m=2,k+2(Z) × An,m=2,k(Z)

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k

slide-26
SLIDE 26
  • It is possible to generalize this result.
  • 3-loop MHV: There are three 1-loop amplituhedron
  • 1-loop N MHV

26

Summary

A(m = 4, k = 0, n; l = 1) ' A(m = 2, k = 2, n; l = 0)

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An,l=1,k(Z) = An,m=2,k+2(Z) × An,m=2,k(Z)

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k

Thank you