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

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 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) 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 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) 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 ⌦ 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 n,k A $ in Momentum twistor space Z

Z =(a,yaa˙ a) 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 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 ) 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 n,m,k I : k-plane in k+m dimensional space Ya n

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 =1, ,k I I ··· Ya = caiZi I =1, ,k+ m

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 i=1 X ···

• ; Positive Grassmannian: matrix mod cai ∈ G+(k, n) k × n GL(k) ca ca > 0 a1 <

⟨Za1, ⋯, Zan⟩ > 0 a1 < ⋯ < an

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

Convex polytope Amplituhedron (Z) 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 n,m,k n generalize n I I I I Y = ciZi Ya = caiZi

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 i=1 AAAConichVFNSxtBGH5cW7Vp1WgvBS/SNKWn8K4UFCEQ9KI9GdPEqNFldx3tkP1idxPQYf9A/4AHT20REX+Gl3q19JCfUHq00EsPfbNZKFVa32Fmnnnmfd55ZsYKHBnFRL0hbfjBw5HRsUe5x0/GJybzU9ONyO+EtqjbvuOHTcuMhCM9UY9l7IhmEArTtRyxYbWX+/sbXRFG0vfexoeB2HHNA0/uS9uMmTLy5c3dVUO10kIqFHuJMpOk3Io6rqFkWU92lZfYhrqdIZMtQ7IyX6ASpTF7F+gZKCCLNT9/hhb24MNGBy4EPMSMHZiIuG1DByFgbgeKuZCRTPcFEuRY2+EswRkms20eD3i1nbEer/s1o1Rt8ykO95CVsyjSVzqnG/pMF/SNfv2zlkpr9L0c8mwNtCIwJt8/q/28V+XyHOPdH9V/PcfYx0LqVbL3IGX6t7AH+u7R8U1tcb2oXtJH+s7+P1CPLvkGXveHfVoV6yfI8Qfot5/7LmjMlXQq6dXXhcpS9hVjmMFzvOL3nkcFK1hDnc/9hCtc44v2QnujVbXaIFUbyjRP8Vdord9/CqQz i=1 X X

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 =1, ,k+ m

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 I =1, ,m+1 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 =1, ,k ··· ··· ··· Z Z Z > 0 i i i 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 > 0 Z Z > 0 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 1 2 m+1 i i c c > 0 i 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 1 k+m 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 i i h ··· i 10 h ··· i h 1 ··· k i Review of the amplituhedron and sign ﬂip

• Canonical form : logarithmic singularities at all boundaries of ( Z ) ⌦ An,k

• The simplest case: Am=4,k=1,n=5

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

• The canonical form is

Y 1234 Y 2345 Y 3451 Y 4512 ⌦ = d log h id log h id log h id log h i Y 5123 Y 5123 Y 5123 Y 5123 h i h i h i h i YdYdYdYdY 12345 4 = h ih i M I J K L Y 1234 Y 2345 Y 3451 Y 4512 Y 5123 ⟨Yijkl⟩ = ϵ Y Z Z Z Z 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 h ih ih ih ih i MIJKL i j k l • Let us rewrite Z i as four-dimensional part and its complement

4 zi 4 ( 1234 ⌘5 + 2345 ⌘1 + + 5123 ⌘4) Zi = z = ⌘ d (Y Y ⇤)⌦ = h i h i ··· h i z i i 1234 2345 3451 4512 5123 ✓ i◆ · Z Z h ih ih ih ih i Y ⇤ =(0, 0, 0, 0, 1) ⌘, : auxiliary grassmann variables 5pt tree amplitude

11 Review of the amplituhedron and sign ﬂip

• To obtain the recursion relation, we need to triangulate the amplituhdron

• Example: Polygon

Yd2Y 123 2 Yd2Y 134 2 Yd2Y 145 2 Yd2Y 156 2 ⌦ = h ih i + h ih i + h ih i + h ih i Y 12 Y 23 Y 31 Y 13 Y 34 Y 41 Y 14 Y 45 Y 51 Y 15 Y 56 Y 61 h ih ih i h ih ih i h ih ih i h ih ih i

• More general case, it is diﬃcult to triangulate.

• How to triangulate more general amplituhedron systematically?

Sign ﬂip triangulation Review of the amplituhedron and sign ﬂip • ⟨ Y12⟩ > 0,⟨Y23⟩ > 0,⟨Y34⟩ > 0,⟨Y45⟩ > 0,⟨Y51⟩ > 0; Interior of this polygon Z1

⟨Y12⟩ < 0 ⟨Y12⟩ > 0

Z5 Y Z2 ⟨Y23⟩ > 0 ⟨Y23⟩ < 0

• Triangulate Z3 Z4 Z1

Each cell is speciﬁes by Z 2 ⟨Y13⟩ < 0 Z5 ⟨Y13⟩ > 0 the signs of ⟨ Y1i⟩ ⟨Y14⟩ < 0 ⟨Y13⟩ > 0 ⟨Y14⟩ > 0 ⟨Y14⟩ < 0 Y 12 Y 13 Y 14 Y 15 (1) h + i h - i h - i h - i (1) (3) + + - - (2) (2) + + + - (3)

AAADVHichVHNbhMxEJ7NUigLtKFckLhYpI2CUCNv2qqIUwUXjv0hbVFdRV7XTa063tWuE6ksucGFF+DACSSEEI/BhRcAqY+AOHAoEpcemN2s2pIUGMv2+PvmmxnbQaRVYik9dEruhbGLl8Yve1euXpuYLF+fWk/CbixkU4Q6jDcDnkitjGxaZbXcjGLJO4GWG8H+w4zf6Mk4UaF5bA8iud3hbaN2leAWoVb5BwtkW5nU8qCredxPnxFBRmbfY3tZBW+aaW7aWpInfoOwOPenSfUMPHcCn0Xnz0UXTlFCGGMiq5H6swt97251Nh81/84wMaBqjXOIjKrNDRFMmp2T+7XKFVqnuZFRxy+cChS2HJbfA4MdCEFAFzogwYBFXwOHBMcW+EAhQmwbUsRi9FTOS+iDh9ouRkmM4Iju49rG01aBGjxnOZNcLbCKxhmjksAM/UI/0CP6mX6k3+jxX3OleY6slwPcg4FWRq3JlzfXfv1X1cHdwt6p6p89W9iFe3mvCnuPciS7hRjoe09fHa3dX51Jq/Qt/Y79v6GH9BPewPR+incrcvU1ePgB/vBzjzrrjbpP6/7KfGXpQfEV43ALbkMN33sRluARLEMThNN0Uue586L0tXTsuu7YILTkFJob8Ie5E78B4RvVHQ==

Z3 Z4 Review of the amplituhedron and sign ﬂip

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

1 sign ﬂip

• Generally,

Each cell of the m=2 k,n amplituhedron is speciﬁed as Yii+1 > 0 Y 12 , , Y 1n 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 and 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 has precisely k sign ﬂips. h i {h i ··· h i}

• Example; m=2 k=2 n=6 Y 12 Y 13 Y 14 Y 15 Y 16 h + i h - i h + i h + i h + i + - - + + + - - - + 6 cells + + - + + + + - - + + + + - +

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 • Sign ﬂip triangulation of the amplituhedron (N. Arkani-Hamed, H.Thomas, J. Trnka 2017)

14 Review of the amplituhedron and sign ﬂip

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

• Sign ﬂip takes place in 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 , Y = Z + xZ + yZ 1 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 1 i1 i1+1

dx dy Yd2Y 1i i +1 ⌦ = = h ih 1 1 i i1 x y Y 1i Y 1i +1 Yi i +1 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 h 1ih 1 ih 1 1 i • The canonical form for this amplituhedron is

⌦ = ⌦ij allX ﬂips • 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 Introduction

Review of the amplituhedron and sign ﬂip

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

Summary

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

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

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

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

• From this, we can triangulate the 1-loop MHV amplituhedron from sign ﬂips!

• 1-loop MHV from sign ﬂips

ABd2A ABd2B AB(1ii + 1) (1jj + 1) 2 ⌦ = h ih ih \ i P AB1i AB1i +1 ABii +1 AB1j AB1j +1 ABjj +1 i

• This is corresponding to the BCFW recursion relation.

17 New recursion relation of 2-loop MHV and 1-loop NMHV • How about more general case? 2-loop MHV amplituhedron

• 2-loop MHV from sign ﬂips (RK 2018)

⌦n-pt 2-loop = ⌦1 + ⌦2 + ⌦3 + + ⌦13 MHV ijkl ijkl ijkl ··· ijkl i,j,k,l=2,3, ,n 1 i

ijkAl AB1l +1 ijkl AB1l ijkl +1 = ijk(AB) (1ll + 1) 2 2 2 2 h i⌘h ih ih ih i h \ i 1 1ii +1Ai 1jj +1Cl ABd A ABd B CDd C CDd D ijA C CD1k +1 ijA k CD1k ijA k +1 ⌦ijkl = h ih ih ih ih ih i h l ki⌘h ih l ih ih l i AB1i AB1i +1 AB1j AB1j +1 ABCD CD1k CD1k +1 CD1l CD1l +1 iA C E EF1j +1 iA C j EF1j iA C j +1 h ih ih ih ih ih ih ih ih i h l k ji⌘h ih l k ih ih l k i ABii +1 AjCkCl1 + AiAjCkCl h ih i h i AlCkEjGi GH1i +1 AlCkEji GH1i AlCkEji +1 . ABii +1 ABjj +1 CDkk +1 CDll +1 h i⌘h ih ih ih i 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 ⇥ h ih ih ih i

• This is not BCFW recursion relation → New recursion relation.

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

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

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

• The 1-loop NMHV case (RK, C. Langer, M. Zheng, J. Trnka, in progress)

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

• However, this amplituhedron is also constructed from m=2, k=3 tree

amplituhedron and convex polygon (N. Arkani-Hamed, H.Thomas, J. Trnka 2017)

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

• Polygon is the intersection with the m=4 k=1 tree amplituhedron and (Z) 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 n,m=2,k=3 (Z) 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 n,m=2,k=1 (Z) 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 n,m=2,k=3

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

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

• 5-pt case

• m=2 k=3 n=5 amplituhedron

12345 2 Y ABd2Y Y ABd2A Y ABd2B ⌦ = h i h ih ih i 6 Y AB12 Y AB23 Y AB34 Y AB45 Y AB51 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 h ih ih ih ih i • The intersecting polygon is 5̂ ˆi =(Y AB) (i 1ii + 1) \ = Z YBi 1ii +1 + Z Y Ai 1ii +1 4̂ 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 Ah i Bh i 1̂ ˆˆˆ 2 ˆˆˆ 2 ˆˆˆ 2 2 512 523 534 ⌦2 = yd y h i + h i + h i h i⇥ y5ˆ1ˆ y1ˆ2ˆ y5ˆ2ˆ y5ˆ2ˆ y2ˆ3ˆ y5ˆ3ˆ y5ˆ3ˆ y3ˆ4ˆ y5ˆ4ˆ ! 2̂ 3̂ 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 h ih ih i h ih ih i h ih ih i

• The full form is 12345 2 Y ABd2Y Y ABd2A Y ABd2B ⌦ ⌦ = h i h ih ih i 6 ⇥ 2 Y AB12 Y AB23 Y AB34 Y AB45 Y AB51 h ih ih ih ih i 5ˆ1ˆ2ˆ 2 5ˆ2ˆ3ˆ 2 5ˆ3ˆ4ˆ 2 yd2y h i + h i + h i ⇥h i y5ˆ1ˆ y1ˆ2ˆ y5ˆ2ˆ y5ˆ2ˆ y2ˆ3ˆ y5ˆ3ˆ y5ˆ3ˆ y3ˆ4ˆ y5ˆ4ˆ ! AAAFFHicjVFLb9NAEJ4mBkp4NAUhIXGxiBoVIUVrOwEEQmoDlbjRB+lDdRvZ7iax6pfsTSRj+Q/wA+DACSSEUCV+AbcKiRM3Dv0JiGORuHBgYjtJkzoNa3ln5pv5dr7ZVR1D9xghR1OZLHfu/IXpi7lLl69cncnPXlv37Lar0ZpmG7a7qSoeNXSL1pjODLrpuFQxVYNuqPtPuvmNDnU93bZeMN+hO6bStPSGrikMofps5rP83KRNpX5PZrpJvSQS+eJjueEqWiAbitU0KC+IUrnCy24U7Yo9eGuxurcrbvUSw/BiOlztwWFwIiGIKdWilAJK5RRwIO4EWBH6rWS5mEyYpH1U4vcptMHmhweWWwoLKmFkhNiIYX/+gXR/pHBYhJ/KT+X2s+Hd8UrE2EgTlYzpNcpP5Ur/o0SKTXmikjG9Rvmp3H4Wrd5ssTv1fIGUSLT4046QOAVI1rKd/wgy7IENGrTBBAoWMPQNUMDDbxsEIOAgtgMBYi56epSnEEIOuW2solihILqPexOj7QS1MO6e6UVsDbsY+LvI5GGO/CCfyDH5Rg7IT/J37FlBdEZXi49WjbnUqc+8urn2ZyLLRMugNWCdqZlBAx5EWnXU7kRIdwot5ndevjlee7g6FxTJe/IL9b8jR+QQJ7A6v7UPK3T1LeTwAYTR6z7trIslgZSElXJhoZo8xTTcgtswj/d9HxbgGSxDDbTsjeyj7NPsEvea+8Idcl/j0sxUwrkOQ4v7/g9/lI0j h ih ih i h ih ih i h ih ih i

• This is corresponding to the BCFW representation.

20 New recursion relation of 2-loop MHV and 1-loop NMHV • 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.

Recursion relation from ̂ ̂ 5 Momentum twistor diagrams 5 (Y. Bai, S. He, T. Lam 2015) ̂ ̂ 1̂ 4̂ 1 4

2̂ 3̂ 2̂ 3̂ Local representation

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

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

• 6-pt case

• m=2 k=3 n=6 amplituhedron … 4 cells

⌦ = ⌦234 + ⌦235 + ⌦245 + ⌦345 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 6 6 6 6 6

• For each cell, there is a pentagon

612 612 612 123 136 125 123 561 561

234 145 134 456 234 456 345 456 (1) 345 (2) 234 (3) 345

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

⌦ = ⌦234 ⌦(1) +(⌦235 + ⌦245) ⌦(2) + ⌦345 ⌦(3) 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 6 6 ⇥ 2 6 6 ⇥ 2 6 ⇥ 2

• This is not the BCFW recursion relation !

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

• In general

⌦ = ⌦6 ⌦ + ⌦6 ⌦(ijk)(jkl)(klm) n ijk ⇥ 2 ijklm ⇥ 2

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 2 i

• where

⌦6 = Y ABd2Y Y ABd2A Y ABd2B ijk h ih ih i 2 YA1ii +1 YA1jj +1 YA1kk +1 hAB1ii +1ihAB1jj +1ihAB1kk +1i h ih ih i BY 1ii +1 BY 1jj +1 BY 1kk +1 h ih ih i ⇥ Y AB1i Y AB 1i+1 Y ABii+1 Y AB1j Y AB1j+1 Y ABjj+1 h ih Y ABih1k Y ABih1k+1 Y ABkkih +1 ih i 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 h ih ih i

Y ABd2Y Y ABd2A Y ABd2B abcde 2 ⌦6 = h ih ih ih i abcde Y ABab Y ABbc Y ABcd Y ABde Y ABea 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 h ih ih ih ih i

abc 2 yd2y ⌦(a)(b)(c) = h i h i 2 yab ybc yca 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 h ih ih i • This is the new recursion relation of n-pt 1-loop NMHV amplitude.

23 Summary

• 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

⌦n-pt 2-loop = ⌦1 + ⌦2 + ⌦3 + + ⌦13 MHV ijkl ijkl ijkl ··· ijkl i,j,k,l=2,3, ,n 1 i

• 1-loop NMHV amplitude

⌦ = ⌦6 ⌦ + ⌦6 ⌦(ijk)(jkl)(klm) n ijk ⇥ 2 ijklm ⇥ 2

• 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

• It is possible to generalize this result.

• 3-loop MHV: There are three 1-loop amplituhedron

• 1-loop N k MHV

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

25 Summary

• It is possible to generalize this result.

• 3-loop MHV: There are three 1-loop amplituhedron

• 1-loop N k MHV

Thank you