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 geometricstructures underlying scattering amplitudes.

The Amplituhedron
N. Arkani-Hamed, J. Trnka 2013

• 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

Deﬁnition of the amplituhedron
The deﬁnition of canonical form ⌦ :⌦ has logarithmic singularitieson all the boundaries of amplituhedron.
•

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

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 toderive it from a QFT.
• new geometry: “log amplituhedron”.

• the canonical form of the log amplituhedron = log of the amplitude.
4

Plan
Introduction
Deﬁnition of the amplituhedron
Canonical form and triangulation
Amplituhedron from sign ﬂip and triangulation
Log amplituhedron
Summary5

Deﬁnition of the amplituhedron•

The amplituhedron An,m,k (Z) : k plane Y in k+m dimensional space<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit>(N. Arkani-Hamed, J. Trnka 2013)

C = (~c1 , · · · , ~cn ) 2 G+ (k, n)

YaI = C↵a Z↵I<latexit sha1_base64="UhQUIfkyiyoi9vsfa8b4nasmyyo=">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</latexit>

Z = (Z1I , · · · , ZnI ) 2 M+ (k + m, n)<latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit>

a = 1, · · · , k

G+ (k, n) :positive Grassmannian
M+ :positive matrix<latexit sha1_base64="hOBD9Dhdu45pGyR/2zsbyAePlvE=">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</latexit>

I = 1, · · · , k + m<latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit>↵ = 1, · · · , n<latexit sha1_base64="A7z7DqZ18uW7vkSotl8BzEVrcqk=">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</latexit>•

Positive Grassmannian G+ (k, n) : k-plane in n-dimensional space<latexit sha1_base64="UkMZ/OpNIHBW813eiGjYS40Vxvc=">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</latexit>

hca1 · · · cak i > 0

a 1 < · · · < akhZaI1 , · · · , ZaIk+m ) > 0

a1 < · · · < ak+m

C = (c~1 , · · · , c~n )/GL(k)•Positive matrix M+

Z = (Z1I , · · · , ZnI )<latexit sha1_base64="HueWDe0Z0nPPRyXoO+zEABsa6vQ=">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</latexit>

I = 1, · · · , k + m<latexit sha1_base64="i1L4ESEd7+5s/KLqH9nWZqOQaC0=">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</latexit>

6<latexit sha1_base64="zPN0q91Zg3CNIXynX0mvrT6/CHw=">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</latexit>

Deﬁnition of the amplituhedron•

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

Y = c1 Z1 + c2 Z2 + · · · + cn Zn

cj > 0convex polygon

not convex polygon•

Convexity for the polygon is that matrix made by vertices Zi has all
positive minors:
hZa1 Za2 Za3 i > 0
a 1 < a2 < a3•

The interior of the polygon is
C = (c1 c2 · · · cn ) 2 G+ (1, n)

Y =C ·Z

Z = (Z1 Z2 · · · Zn ) 2 M+ (3, n)

M+ :positive matrix•

We can generalize this to the Grassmannian and higher dimension.
7

Deﬁnition of the amplituhedron
Y =C ·Z

YaI = C↵a ZaI

C = (c1 c2 · · · cn ) 2 G+ (1, n)
Z = (Z1 Z2 · · · Zn ) 2 M+ (3, n)

C = (~c1 , · · · , ~cn ) 2 G+ (k, n)
Z = (Z1 , · · · , Zn ) 2 M+ (k + m, n)This is the deﬁnition of the amplituhedron! An,m,k (Z)<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit>•

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

•

m=4 is the physical amplituhedron.

8

Deﬁnition of the amplituhedron•

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

YaI = C↵a Z↵I

C = (~c1 , · · · , ~cn ) 2 G+ (k, n)<latexit sha1_base64="UhQUIfkyiyoi9vsfa8b4nasmyyo=">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</latexit>

Z = (Z1I , · · · , ZnI ) 2 M+ (k + m, n)<latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit>

Conjecture
Canonical form of An,m=4,k (Z)<latexit sha1_base64="HWnY4+EcWl9XEx9QcfvtiLEnCng=">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</latexit>$<latexit sha1_base64="Ck+7417GpCiWzl11ZaMzZBz8aSA=">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</latexit><latexit

n-point N k MHV tree amplitude<latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexitk

N MHV … amplitude with K+2 negative helicity particles<latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit

The deﬁnition of canonical form ⌦ :⌦ has logarithmic singularitieson all the boundaries of amplituhedron.
9ternal data for n
k + 4 particles is given by the vectors ZaI living in a (4 + k)
dimensional space; where a = 1, . . . , n and I = 1, . . . , 4 + k. The data is positiveDeﬁnition of the amplituhedron•hZa1 . . . Za4+k i > 0 for a1 < · · · < a4+k

(9.1)

Consider the
space of
Y in
k+4of dimwith L 2The amplituhedron
lives k-planein G(k, k + 4; L):
the spacek planesspace,
Y in (k + 4)together
dimensions, together with L 2-planes L(i) in the 4 dimensional complement of Y ,plane L(i) in
4 dim complement of Y
schematically<latexit sha1_base64="b/1Sv0RhnMppidPi6Rha7Js+h2g=">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</latexit>•

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,

Consider the space determined by

C 2 G+ (k, n)(9.2)
D 2 G+ (2, n)<latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">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</latexit>

Y↵I = C↵a ZaI ,•I Y =C·Z
L (i) = D a(i) ZaI
I

More explicitly in components, the k-plane is Y↵ , and the 2-planes are LI (i) , where
1L(i) of the form
= 1, 2 and i = 1, . . . , L . The amplituhedron is the space0of all Y,<latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit>D(i1 )
B
D(i2 ) CCY↵I = C↵a ZaI , LI (i) = D a(i) ZaI B(9.3)
B .. C
B . C
B
C
A
(il )
where as in the previous section the C↵a specifies a k-plane@inDn-dimensions,and theC

And satisﬁes that all ordered minors ofare positive.

D a(i) are L 2-planes living in the (n k) dimensional complement of C, with thepositivity property that for any 0  l  L, all the ordered (k + 2l) ⇥ (k + 2l) minorsof the (k + 2l) ⇥ n matrix
01
D(i1 )
n,k,L
B .. C
B . C
(9.4)
B
C
@ D(il ) A
10C<latexit sha1_base64="LJ9bqVeTgAIKcgybcXtffSQebo4=">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</latexit>•

This is the deﬁnition of the full amplituhedron A(Z)Introduction
Deﬁnition of the amplituhedron
Canonical form and triangulation
Amplituhedron from sign ﬂip and triangulation
Log amplituhedron
Summary11

Canonical form and triangulation•

Let consider the simplest case: triangle. Am=2,k=1,n=3
Y = Z1 + c2 Z2 + c3 Z3•Boundary of the triangle isc2 ! 0,•c 2 , c3 > 0

c3 ! 0h123i = ✏abc Z1a Z2b Z3c

The canonical form is
dc2 dc3
hY d2 Y ih123i2
⌦=
=
c2 c3
hY 12ihY 23ihY 31i

d2 Y = dc2 dc3 Z2 Z3

4P• Consider simplex in, Am=4,k=1,n=5

hY d4 Y ih12345i4⌦P =hY 1234ihY 2345ihY 3451ihY 4512ihY 5123i

5-pt NMHV amplitude
in momentum twistor space

Zi = ( ai , yiaȧ a )<latexit sha1_base64="LLrb5iZoYGqAX98+LVL0WNbrfdQ=">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</latexit>

12

pµ1 =

µ
˜ 1ȧ
1a
aȧ

yiȧa

ȧa
yi+1
= pȧai

Canonical form and triangulation• 1-loop 4-pt MHV amplituhedron; 2-plane in 4 dimensions

LI = D a ZaI••

D 2 G+ (2, 4)

Use GL(2), write the D matrix to
✓
◆
1 x 0
w
D=
0 y 1
z

L = (ZA , ZB )ZA = Z1 + xZ2

wZ4

ZB = yZ2 + Z3 + zZ4<latexit sha1_base64="NQB8A726aFxHmqBMiR6OUFzuAhs=">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</latexit>

Positivity:x, w, y, z > 0•

Then the canonical form isdx dw dy dz
hABd2 AihABd2 Bih1234i2⌦==
x w y zhAB12ihAB23ihAB34ihAB14i(N. Arkani-Hamed, J. Trnka 2013)13

Canonical form and triangulation• How to obtain canonical form for general amplituhedron?

TriangulationhY d2 Y ih123i2
hY d2 Y ih134i2
hY d2 Y ih145i2
hY d2 Y ih156i2⌦=+++
hY 12ihY 23ihY 31i hY 13ihY 34ihY 41i hY 14ihY 45ihY 51i hY 15ihY 56ihY 61i• More general case, it is diﬃcult to triangulation.

• In some cases the triangulation is known.
(N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Hodges, J. Trnka 2010)• k=1 m=2 : polygon

⌦P =

i

<latexit sha1_base64="ToY37/kvVrc2i7iN8QrMX+E6vXw=">AAAC0HichZHPT9RAFMe/rYq4CKx4MfHSuMGYkGxeCYnGxITgxZvLwgKGkmZaZpfJTn+k7W7ADSFe+Qc8eNLEGOOfwYWrBw78CYYjJFw4+LZbcZUor+nMe5953zdvZrxYqzQjOjbMGzdvjdwevVMauzs+MVm+N7WSRp3Elw0/0lGy5olUahXKRqYyLdfiRIrA03LVa7/sr692ZZKqKFzOdmK5EYhWqJrKFxkjt9x2XgeyJdzaCyftBK5ymonwe44WYUtLy1ZqxracJI92L/GbYWxdUlv9YkNoSO+WK1Sl3Kyrjl04FRRWi8pf4GATEXx0EEAiRMa+hkDK3zpsEGJmG+gxS9hT+brELkqs7XCW5AzBtM1ji6P1goYc92umudrnXTT/CSstTNMRfaVTOqRv9IMu/lmrl9fo97LDszfQytid3H+wdH6tKuA5w9Zv1X97ztDEs7xXxb3HOemfwh/ou2/fny49r0/3HtMnOuH+P9IxHfAJwu6Z/3lR1j+gxA9g/33dV52V2apNVXtxrjK/UDzFKB7iEZ7wfT/FPF6hhgbv+x3nBgzDrJvb5p75bpBqGoXmPv4wc/8nrN2xQg==</latexit>• k=1 m=4 : polytopesX⌦P =<latexit sha1_base64="0HaiBkOR9hKnenTjcs3uZyIe90I=">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</latexit>• 1-loop MHV amplituhedronh1ii + 1i
hY ii + 1ihY 1iihY 1i + 1i

X

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

hABd2 AihABd2 BihAB(1ii + 1) \ (1jj + 1)i2
⌦P =hAB1iihAB1i + 1ihABii + 1ihAB1jihAB1j + 1ihABjj + 1i
i<j

Kermit representation• How to triangulate more general amplituhedron?
Use the Sign ﬂip deﬁnition of the amplituhedron
14

Introduction
Deﬁnition of the amplituhedron
Canonical form and triangulation
Amplituhedron from sign ﬂip and triangulation
Log amplituhedron
Summary

15

Amplituhedron from sign ﬂip and triangulation

An,m,k (Z)<latexit sha1_base64="rtbfKwWuf3iK8FfI4E6z8Bd1xAI=">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</latexit>

YaI = C↵a ZaI

C = (~c1 , · · · , ~cn ) 2 G+ (k, n)
Z = (Z1 , · · · , Zn ) 2 M+ (k + m, n)•

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 = c1 Z1 + c2 Z2 + · · · + cn Zn

16ca > 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(N. Arkani-Hamed, H.Thomas, J. Trnka 2017)

Y is in the m=2 k amplituhedron only if
Y is k plane in k+2 dimensional space which satisﬁes
hY ii + 1i > 0 and {hY 12i, · · · , hY 1ni} has precisely k sign ﬂips.
<latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">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</latexit><latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">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</latexit>

Y is in the m=4 k amplituhedron only if
Y is k plane in k+4 dimensional space which satisﬁes

hY ii + 1jj + 1i and {hY 1234i, · · · , hY 123ni} has precisely k sign ﬂips.
<latexit sha1_base64="aVJljgoLvoFGAimhUgr3HU4OYjo=">AAACm3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuYxVTH5CTmpeekKkQaGhmbKMQUgXk6Mckp+SXFOsiSeTDJmNp4AWUDPQMwUMBkGEIZygxQEJAvsJwhhiGFIZ8hmaGUIZchlSGPoQTIzmFIZCgGwmgGQwYDhgKgWCxDNVCsCMjKBMunMtQycAH1lgJVpQJVJAJFs4FkOpAXDRXNA/JBZhaDdScDbckB4iKgTgUGVYOrBisNPhucMFht8NLgD06zqsFmgNxSCaSTIHpTC+L5uySCvxPUlQukSxgyELrwurmEIY3BAuzWTKDbC8AiIF8kQ/SXVU3/HGwVpFqtZrDI4DXQ/QsNbhocBvogr+xL8tLA1KDZDFzACDBED25MRpiRnqGBnmGgibKDEzQqOBikGZQYNIDhbc7gwODBEMAQCrR3DsMhhtMMZ5hkmZyZvJh8IEqZGKF6hBlQAFMoALSAnY0=</latexit><latexit sha1_base64="RQVr5W81XIkN6DNUXMiTMku+V/M=">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</latexit>

17

Amplituhedron from sign ﬂip and triangulation

Y is k plane in k+2 dimensional space which satisﬁes
hY ii + 1i > 0 and {hY 12i, · · · , hY 1ni} has precisely k sign ﬂips.
<latexit sha1_base64="FaumjL3s8tPWKdODZNr3UAzdY6A=">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</latexit><latexit sha1_base64="1K932jzVvgyP2+ZrBP0E73sgs8w=">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</latexit>• Example: m=2 k=1 n=6 tree amplituhedronhY 12i, hY 23i, · · · , hY 61i > 0

and<latexit sha1_base64="H/mFKJRiDXXX9zjBZwVYjF6Nzxk=">AAACp3ichVHLSiNBFD224yu+orMR3ASDoiDhdhQVF4Poxp3PxIiR0N2WsbHS3XRXAhr8AX/AhasZEBU/w82s3InjJwwuFdy48KbTKCrqbbrq1Ln33DpVZXrSDhTRbYPW+KOpuaW1Ldbe0dnVHe/pzQZu2bdExnKl6+dMIxDSdkRG2UqKnOcLo2RKsWbuztXyaxXhB7brrKo9T2yWjKJjb9uWoZgqxGfz0nCKUiTW9XTeD+HoC5Uee6GsLVcFr5kJPcokflEhnqQUhZH4CPQIJBHFohs/RR5bcGGhjBIEHCjGEgYC/jagg+Axt4kqcz4jO8wLHCDG2jJXCa4wmN3lscirjYh1eF3rGYRqi3eR/PusTGCQrumc7ukvXdB/evq0VzXsUfOyx7NZ1wqv0H3Yt/L4rarEs8LOq+pLzwrbmAq92uzdC5naKay6vrJ/dL8yvTxYHaI/dMf+f9MtXfIJnMqDdbIklo8R4wfQ31/3R5BNp3RK6UvjyZnZ6Cla0Y8BDPN9T2IG81hEhvc9wxVu8E8b0Ra0rJarl2oNkeYn3oRmPAMZxaJN</latexit>• 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

YaI = C↵a Z↵I<latexit sha1_base64="UhQUIfkyiyoi9vsfa8b4nasmyyo=">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</latexit>

C = (~c1 , · · · , ~cn ) 2 G+ (k, n)

Z = (Z1I , · · · , ZnI ) 2 M+ (k + m, n)<latexit sha1_base64="1ASzJgmPJImGhMbzQlavxPUGq0k=">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</latexit>

18

Amplituhedron from sign ﬂip and triangulation•

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 i1 ,<latexit sha1_base64="I9t4JZgxXhV2sGM3zRm5NnBJen0=">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</latexit>dx dy
hY d2 Y ih1i1 i1 + 1i
⌦i1 =
=
x y
hY 1i1 ihY 1i1 + 1ihY i1 i1 + 1i

Y = Z1 + xZi1 + yZi1 +1<latexit sha1_base64="DYV0WAffibblXhdtrOuUqcotOfA=">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</latexit><latexit sha1_base64="qh5KiSEkRcIowp+FrWg1+mfSztE=">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</latexit>•

The canonical form for this amplituhedron is ⌦ =X

all flips

hY d2 Y ih123i2
hY d2 Y ih134i2
hY d2 Y ih145i2
hY d2 Y ih156i2⌦=+++
hY 12ihY 23ihY 31i hY 13ihY 34ihY 41i hY 14ihY 45ihY 51i hY 15ihY 56ihY 61i

19

⌦ij

Amplituhedron from sign ﬂip•

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 isnʼt simple
relation between a particular cell and one of the ﬂip pattern.

•

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

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>•

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

20

Amplituhedron from sign ﬂip and triangulation
1-loop MHV from sign ﬂips•

X

hABd2 AihABd2 BihAB(1ii + 1) \ (1jj + 1)i2
⌦P =hAB1iihAB1i + 1ihABii + 1ihAB1jihAB1j + 1ihABjj + 1i
i<j

Kermit representation•

How about more general case?

2-loop MHV amplituhedron•

In the sign ﬂip deﬁnition, the 2-loop MHV amplituhedron is deﬁned
Two 2-plane in 4 dimensional space (AB)1 , (AB)2 that<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>(AB)1 , (AB)2 are 1-loop MHV amplituhedron<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>

And there are further constraint, h(AB)1 (AB)2 i > 0•(AB)1 , (AB)2 have n cells

2-loop MHV amplituhedron has n^2 cells.

<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>

21

Amplituhedron from sign ﬂip and triangulation•

4-pt case
ZA = Z1 + x1 Z2 + w 1 Z3 , Z B =Z 1 + y 1 Z 3 + z1 Z 4 ,ZC = Z 1 + x2 Z 2 + w 2 Z 3 , Z D =Z 1 + y 2 Z 3 + z2 Z 4 ,hABCDi = h1234i{(x1•x2 )(y1 z2

z1 y2 ) + (z1The case of(y1 z2z2 )(w1 x2

0 < x1 ,z1 y2 ), (z1z2 ), (w1 x2

x1 w 2 ) > 0

0 < x2 < x1 +0 < w1 ,0 < z1 ,0 < y1 ,

⌦1 =

•

1 1
(
x1 x2

x2

1
x1

1 1
(
a w1 w2)1
w2

w1 x 2 )x1

1 1(y 1 y2

1y2

x1 w2 )} > 0

w 1 x2x1
0 < z2 < z1 ,y 1 z20 < y2 <
z1

1 1
(
z1 z2

1
z2

z1

)

Another case can be obtained similarly.
8X

x 1 y 1 z2 + x 2 y 2 z1 + x 2 w 1 z1 + x 1 w 2 z2⌦=⌦i =x1 x2 w1 w2 y1 y2 z1 z2 {(x1 x2 )(y1 z2 y2 z1 ) + (z1 z2 )(w1 x2
i=1•z2 )(w1 x2 x1 w2 ),y 1 z2 z1 y 2

0 < w2 <y1 z 2 )z1(z1

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

w2 x1 )}

Amplituhedron from sign ﬂip and triangulation•

Example: 5-pt case•

There are 3 3=9 cells⌦ = ⌦2323 + ⌦3434 + ⌦2424 + ⌦2324 + ⌦2334 + ⌦2434 + ⌦2423 + ⌦3423 + ⌦3424<latexit sha1_base64="/1bSdlR0dPrqWobOKM384JGg3Ho=">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</latexit>
sha1_base64="OwH72Cq4dAd7Z864qkAqYZ42FA4=">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</latexit>

h123A4 ih134C4 ihABd2 AihABd2 BihCDd2 CihCDd2 Di⌦2434 =
hAB12ihAB13ihAB14ihAB15ihAB23ihAB45ihABCDihCD13ihCD14i2 hCD15ihCD34ihCD45i⇢⇥ h123A4 i(hAB45ihCD13ihCD14i + hAB15ihCD34ihCD14i)<latexit sha1_base64="lcX7hhweWSfi+hnVezjqIp5+704=">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</latexit>•

h345A2 ihAB14ihCD14ihCD15i + h123C4 ihCD14ihAB45ihAB13i

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

23

Amplituhedron from sign ﬂip and triangulation•1[n-pt case, there are 2 (n3)(n2)]2 cells from the way to chose i,j,k,l .

<latexit sha1_base64="4azmlmTOY/oZBmK+1CkIsYoD+Bg=">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</latexit>
sha1_base64="HI9/+KXEKLvMdmB5Ro3vV9InWNQ=">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</latexit>

2-loop
⌦n-pt
=
MHV

X

i,j,k,l=2,3,··· ,n 1
i<k<l<j⌦1ijkl +X⌦2ijkl +i<k<j<lh1ii + 1Ai ih1jj + 1Cl ihABd2 AihABd2 BihCDd2 CihCDd2 Di
hAB1iihAB1i + 1ihAB1jihAB1j + 1ihABCDihCD1kihCD1k + 1ihCD1lihCD1l + 1i
hABii + 1ihAj Ck Cl 1i + hAi Aj Ck Cl i⇥
hABii + 1ihABjj + 1ihCDkk + 1ihCDll + 1i

⌦1ijkl =<latexit sha1_base64="Hmerqh0Abg7Btk/l1THglYJu87Y=">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</latexit>X

i<j<k<l⌦3ijkl + · · · +hijkAl i ⌘ hAB1l + 1ihijkli

hijAl Ck i ⌘ hCD1k + 1ihijAl ki

X

⌦13
ijkl

k<l<i<jhAB1lihijkl + 1i = hijk(AB) \ (1ll + 1)i

hiAl Ck Ej i ⌘ hEF 1j + 1ihiAl Ck ji

hAl Ck Ej Gi i ⌘ hGH1i + 1ihAl Ck Ej ii

hCD1kihijAl k + 1i

hEF 1jihiAl Ck j + 1i

hGH1iihAl Ck Ej i + 1i.

•

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

Introduction
Deﬁnition of the amplituhedron
Canonical form and triangulation
Amplituhedron from sign ﬂip and triangulation
Log amplituhedron
Summary25

hZa1 . . . Za4+k i > 0 for a1 < · · · < a4+k

(9.1)

Log amplituhedron•

The amplituhedron lives in G(k, k + 4; L): the space of k planes Y in (k + 4) dimensions, together with L 2-planes L(i) in the 4 dimensional complement of Y ,schematicallyLoop amplituhedron

C 2 G (k, n)

We will denote the collection of (L(i) , Y ) as Y.
+
I The amplituhedron
I An,k,L (Z)I is the subspace of G(k,Ik + 4; L) consisting of all↵aa(i)↵ which area linear combination
a data,(i)Y’s
a positiveof theexternal

Y =C

Z ,

L

=D

Z

Y =C·Z•<latexit sha1_base64="R8CcWWs4AcytWSgwgVTmWkYassI=">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</latexit>

D 2 (9.2)
G+ (2, n)<latexit sha1_base64="xdcr9Es3+PC6MC736CJrT+EBwQ0=">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</latexit>1
D(i1 )C are LI (i) , where
More explicitly in components, the k-plane is Y↵I , and theB2-planes
B D(i2 ) C
CL of the form.. Y,
= 1, 2 and i = 1, . . . , L . The amplituhedron is the spaceB
Bof all
C (i).
BC@ D(i ) AlY↵I = C↵a ZaI , LI (i) = D a(i) ZaI(9.3)
C

And satisﬁes that all ordered minors of0

are positive: Positivity<latexit sha1_base64="LJ9bqVeTgAIKcgybcXtffSQebo4=">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</latexit>•

where as in the previous section the C↵a specifies a k-plane in n-dimensions, and theD a(i) are L 2-planes living in the (n k) dimensional complement of C, with thepositivity property that for any 0  l  L, all the ordered (k + 2l) ⇥ (k + 2l) minorsof the (k + 2l) ⇥ n matrix01
01
D(i1 )D(i1 )B B.. DC(i2 ) C
B B. CC(9.4)
B B C.
C@ B
D(il ) A.
C<0
BC . C

What the geometry which has “Negativity”?

@ D(i ) AlC<latexit sha1_base64="kYCFwP/1JhRyFAOIwfZMCzMzEac=">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</latexit>– 17 –

26

Log amplituhedron•

We can construct the canonical form from this negative amplituhedron.

Two 2-plane in 4 dimensional space (AB)1 , (AB)2 that<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>(AB)1 , (AB)2 are 1-loop MHV amplituhedron<latexit sha1_base64="W6L7+49MpBqG26hyHbv3jH8OEnc=">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</latexit>

And there are further constraint, h(AB)1 (AB)2 i < 0<latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">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</latexit>•

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

2-loop⌦[log [An-pt]] =MHV<latexit sha1_base64="+Ax3c3Z2tLW+VnkGgt3nIkys+Nk=">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</latexit>•X⌦1ijkl [log] +i,j,k,l=2,3,··· ,n 1
i<k<l<jXi<k<j<lWe call this: log amplituhedron.

27⌦2ijkl [log] +X

i<j<k<l⌦3ijkl [log] + · · · +X

k<l<i<j⌦13
ijkl [log]Log amplituhedron•

The loop expansion of the amplitude is

A = 1 + gA1 + g 2 A2 + g 3 A3 + · · ·<latexit sha1_base64="H1E3F+7/WFD6K9V+1LnL40D/UtY=">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</latexit>•

The log amplitude isS 1 = A1
<latexit sha1_base64="4KG1lRZLJGs6/o+wpYSBtiwpmHo=">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</latexit><latexit sha1_base64="EQ0Jy+DZKGDV2v2XQWwnQVLUf50=">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</latexit>

S = log A = gS1 + g 2 S2 + g 3 S3 + · · ·S 2 = A2
<latexit sha1_base64="uFBgGG/zzHHg8LGDpYIoPWmoQAI=">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</latexit>

•

1 2
A1
2

A21 : two 1-loop amplituhedron<latexit sha1_base64="kUgEisZCD3pBfFBJI2ES+Nc/lgI=">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</latexit>•

A2 : two 1-loop amplituhedron with positivity h(AB)1 (AB)2 i > 0<latexit sha1_base64="3Hj00t4mBG89by+THlJwZw6fq8A=">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</latexit><latexit sha1_base64="9ZkdJOzPi0G9ArI4TGWoa6Zc4/w=">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</latexit>

A21<latexit sha1_base64="XuZRMicFZs9ivKmLATFdn45/tpE=">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</latexit>

A2h(AB)1 (AB)2 i < 0<latexit sha1_base64="J0scVD9//SRJY+8kfd0mV6cRnPw=">AAAChHichVHLSsNAFD2Nr1pfVTeCC4tFUZBy4wNFRHxsXPqqClZKEscanCYhSQu1uHTjD7hwpSAi3eoXuPEHXPgJ4lLBjQtv04CoqDdk7pkz99w5M6M70vR8oseIUlff0NgUbY61tLa1d8Q7uzY8u+AaIm3Y0na3dM0T0rRE2jd9KbYcV2h5XYpN/WCxur5ZFK5n2ta6X3LETl7LWeaeaWg+U9l4X0ZqVk6KxND8wnBWraXRRMYN2BnKxpOUoiASP4EagiTCWLbjV8hgFzYMFJCHgAWfsYQGj79tqCA4zO2gzJzLyAzWBY4QY22BqwRXaMwe8Jjj2XbIWjyv9vQCtcG7SP5dViYwQA90TS90TxV6ovdfe5WDHlUvJc56TSucbMdJz9rbv6o8Zx/7n6o/PfvYw1Tg1WTvTsBUT2HU9MXD05e16dWB8iBd0DP7P6dHuuMTWMVX43JFrJ4hxg+gfr/un2BjNKVSSl0ZT84thE8RRS/6McT3PYk5LGEZad73GBXc4FZpVEaUMWWiVqpEQk03voQy+wHMPJMX</latexit>

28

Summary•

The amplituhedron, generalizing the interior of convex polygon, gives acomplete 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.
2-loop
⌦n-pt
=
MHV

X

i,j,k,l=2,3,··· ,n 1
i<k<l<j⌦1ijkl +X

i<k<j<l⌦2ijkl +X

i<j<k<l⌦3ijkl + · · · +

X

⌦13
ijkl

k<l<i<j•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 fromgeometry.

29

Future work•

To generalize this sign ﬂip triangulation.

•

MHV higher loop case•3-loop: There are 3 1-loop amplituhedron andhABCDi > 0, hABEF i > 0, hCDEF i > 0<latexit sha1_base64="0vinGb9c46qnQdzBYhJVAWLNY3Q=">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</latexit>•

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 thistriangulation?

•k

Triangulate the N MHV loop amplituhedron from sign ﬂip [w. J. Trnka] .
<latexit sha1_base64="lgePgv8Rj7cvyx5HnnvpNH2qTIg=">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</latexit><latexit

30