Triangulation of 2-loop MHV Amplituhedron from sign flips Ryota - - PowerPoint PPT Presentation

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

• n all the boundaries of amplituhedron.

Ω

The definition of canonical form :

Ω

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 Definition of the amplituhedron 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) ∈ G+(k, n)

An,m,k(Z)

Y I

a = CαaZI α

M+ :positive matrix

:positive Grassmannian

G+(k, n)

I = 1, · · · , k + m

a = 1, · · · , k

α = 1, · · · , n

C = (~ c1, · · · , ~ cn)/GL(k)

hca1 · · · caki > 0

a1 < · · · < ak

G+(k, n)

Z = (ZI

1, · · · , ZI n) ∈ M+(k + m, n)

Z = (ZI

1, · · · , ZI n)

hZI

a1, · · · , ZI ak+m) > 0

a1 < · · · < ak+m

M+

I = 1, · · · , k + m

Definition of the amplituhedron

• 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 · Z

Z = (Z1Z2 · · · Zn) ∈ M+(3, n)

C = (c1c2 · · · cn) ∈ G+(1, n)

:positive matrix

M+

Definition of the amplituhedron

• The amplituhedron is defined as the generalization of the interior of

convex polygon.

• m=4 is the physical amplituhedron.

Y I

a = CαaZI a

C = (~ c1, · · · ,~ cn) ∈ G+(k, n)

Z = (Z1, · · · , Zn) ∈ M+(k + m, n)

8

This is the definition of the amplituhedron!

Y = C · Z

Z = (Z1Z2 · · · Zn) ∈ M+(3, n)

C = (c1c2 · · · cn) ∈ G+(1, n)

An,m,k(Z)

Definition of the amplituhedron

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

Conjecture

N MHV … amplitude with K+2 negative helicity particles

9

C = (~ c1, · · · ,~ cn) ∈ G+(k, n)

An,m,k(Z)

Y I

a = CαaZI α

has logarithmic singularities

• n all the boundaries of amplituhedron.

Ω

The definition of canonical form :

Ω

n-point N MHV tree amplitude

An,m=4,k(Z)

k

↔

Canonical form of

Z = (ZI

1, · · · , ZI n) ∈ M+(k + m, n)

Definition of the amplituhedron

k

• 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

Definition of the amplituhedron

An,k,L(Z)

Y I

α = CαaZI a,

LI

γ(i) = Dγa(i)ZI a

L(i)

C ∈ G+(k, n)

D ∈ G+(2, n)

       D(i1) D(i2) . . . D(il) C       

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

ΩP = hY d4Y ih12345i4 hY 1234ihY 2345ihY 3451ihY 4512ihY 5123i

c2 → 0, c3 → 0

Ω = dc2 c2 dc3 c3 = hY d2Y ih123i2 hY 12ihY 23ihY 31i h123i = ✏abcZa

1 Zb 2Zc 3

d2Y = dc2dc3Z2Z3

5-pt NMHV amplitude in momentum twistor space

Canonical form and triangulation

y ˙

aa i

− y ˙

aa i+1 = p˙ aa i

pµ

1 = σµ a˙ aλ1a˜

λ1˙

a

Zi = (λa

i , ya˙ a i λa)

• 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

γ = DγaZI a

Lγ = (ZA, ZB)

D ∈ G+(2, 4)

D = ✓1 x −w y 1 z ◆

x, w, y, z > 0

GL(2)

Ω = dx x dw w dy y dz z = hABd2AihABd2Bih1234i2 hAB12ihAB23ihAB34ihAB14i

Canonical form and triangulation

ZA = Z1 + xZ2 − wZ4 ZB = yZ2 + Z3 + zZ4

(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

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

Triangulation

Canonical form and triangulation

ΩP = 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)

ΩP = X

i,j

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

ΩP = X

i

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

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

a = CαaZI a

C = (~ c1, · · · ,~ cn) ∈ G+(k, n)

Z = (Z1, · · · , Zn) ∈ M+(k + m, n)

16

Y = c1Z1 + c2Z2 + · · · + cnZn

ca > 0

An,m,k(Z)

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

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

17

Amplituhedron from sign flip and triangulation

hY ii + 1i > 0

{hY 12i, · · · , hY 1ni}

{hY 1234i, · · · , hY 123ni}

hY ii + 1jj + 1i

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

• 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

Amplituhedron from sign flip and triangulation

hY ii + 1i > 0

{hY 12i, · · · , hY 1ni}

hY 12i, hY 23i, · · · , hY 61i > 0

and

C = (~ c1, · · · ,~ cn) ∈ G+(k, n) Y I

a = CαaZI α

Z = (ZI

1, · · · , ZI n) ∈ M+(k + m, n)

• 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

Ω = X

all flips

Ωij

Amplituhedron from sign flip and 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

i1

Y = Z1 + xZi1 + yZi1+1

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

• 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

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

• 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

Amplituhedron from sign flip and triangulation

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

(AB)1, (AB)2

(AB)1, (AB)2

ΩP = 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 < y1z2 z1

Ω1 = 1 x1 ( 1 x2 − 1 x2 − x1 − a) 1 w1 ( 1 w2 − 1 w2 − w1x2

x1

) 1 y1 ( 1 y2 − 1 y2 − y1z2

z1

) 1 z1 ( 1 z2 − 1 z2 − z1 )

Ω =

8

X

i=1

Ωi = x1y1z2 + x2y2z1 + x2w1z1 + x1w2z2 x1x2w1w2y1y2z1z2{(x1 − x2)(y1z2 − y2z1) + (z1 − z2)(w1x2 − w2x1)}

Amplituhedron from sign flip and triangulation

• Example: 5-pt case
• There are 3×3=9 cells
• The sum of these forms correspond to 5pt 2-loop MHV amplitude.

23

Ω = Ω2323 + Ω3434 + Ω2424 + Ω2324 + Ω2334 + Ω2434 + Ω2423 + Ω3423 + Ω3424

Ω2434 = h123A4ih134C4ihABd2AihABd2BihCDd2CihCDd2Di hAB12ihAB13ihAB14ihAB15ihAB23ihAB45ihABCDihCD13ihCD14i2hCD15ihCD34ihCD45i ⇥ ⇢ h123A4i(hAB45ihCD13ihCD14i + hAB15ihCD34ihCD14i) h345A2ihAB14ihCD14ihCD15i + h123C4ihCD14ihAB45ihAB13i

• <latexit sha1_base64="lcX7hweWSfi+hnVezjqIp5+704=">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</latexit><latexit sha1_base64="lcX7hweWSfi+hnVezjqIp5+704=">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</latexit><latexit sha1_base64="lcX7hweWSfi+hnVezjqIp5+704=">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</latexit><latexit sha1_base64="lcX7hweWSfi+hnVezjqIp5+704=">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</latexit>

Amplituhedron from sign flip and triangulation

• 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

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

Amplituhedron from sign flip and triangulation

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

α = CαaZI a,

LI

γ(i) = Dγa(i)ZI a

C ∈ G+(k, n)

D ∈ G+(2, n)

       D(i1) D(i2) . . . D(il) C       

       D(i1) D(i2) . . . D(il) C        < 0

• 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,

(AB)1, (AB)2

(AB)1, (AB)2

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

Log amplituhedron

Ω[log [An-pt 2-loop

MHV

]] = X

i,j,k,l=2,3,··· ,n−1 i<k<l<j

Ω1

ijkl[log] +

X

i<k<j<l

Ω2

ijkl[log] +

X

i<j<k<l

Ω3

ijkl[log] + · · · +

X

k<l<i<j

Ω13

ijkl[log]

• 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 + · · ·

S = log A = gS1 + g2S2 + g3S3 + · · ·

S2 = A2 − 1 2A2

1

S1 = A1

Log amplituhedron

A2

1

A2

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

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

A2

1 − A2

• 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

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

• 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

hABCDi > 0, hABEFi > 0, hCDEFi > 0

k

[w. J. Trnka]