Gravituhedron Jaroslav Trnka Center for Quantum Mathematics and - - PowerPoint PPT Presentation

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Gravituhedron Jaroslav Trnka Center for Quantum Mathematics and - - PowerPoint PPT Presentation

Mar 10, 2024 814 likes •1.41k views

Towards the Gravituhedron Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP) University of California, Davis QCD meets Gravity, UCLA, December 2019 What is the scattering amplitude? New picture? ? Various approaches s n o

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SLIDE 1

Gravituhedron

QCD meets Gravity, UCLA, December 2019

Towards the

Jaroslav Trnka

Center for Quantum Mathematics and Physics (QMAP) University of California, Davis

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SLIDE 2

New picture?

?

What is the scattering amplitude?

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SLIDE 3

Various approaches

u n i t a r i t y , r e c u r s i

  • n

r e l a t i

  • n

s

physical principles

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SLIDE 4

Various approaches

u n i t a r i t y , r e c u r s i

  • n

r e l a t i

  • n

s

color-kinematics, CHY,… physical principles unified picture for amplitudes

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SLIDE 5

Various approaches

u n i t a r i t y , r e c u r s i

  • n

r e l a t i

  • n

s

color-kinematics, CHY,… physical principles unified picture for amplitudes

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SLIDE 6

Various approaches

u n i t a r i t y , r e c u r s i

  • n

r e l a t i

  • n

s

color-kinematics, CHY,… positive geometry unified picture for amplitudes physical principles amplitude is form/volume

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SLIDE 7

Various approaches

u n i t a r i t y , r e c u r s i

  • n

r e l a t i

  • n

s

color-kinematics, CHY,… positive geometry unified picture for amplitudes physical principles amplitude is form/volume worldsheet geometry combinatorics of cubic graphs different sort of geometry, D=4 important

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SLIDE 8

Geometry of gluon amplitudes

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SLIDE 9

T ree-level gluon amplitudes

✤ Simplest amplitude: MHV amplitude ✤ General N=4 SYM tree-level amplitude

An = δ(Q) h12ih23ih34ih45i . . . hn1i

dual conformal symmetry momentum twistors geometric interpretation

An,k = An × Rn,k(Zj)

color-ordered amplitudes: cyclic symmetry

(Parke, Taylor) (Nair)

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SLIDE 10

Amplitude as volume

✤ Focus on 6pt NMHV amplitude 1−2−3−4+5+6+

(Hodges 2009)

3-d polytope in momentum twistor space “face” of 4-d polytope

A6 = Z

P

dV

superamplitude

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SLIDE 11

Amplitude as volume

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SLIDE 12

Amplitude as volume

face Z4

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SLIDE 13

Amplitude as volume

face Z6

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SLIDE 14

Amplitude as volume

edge 13 ≡ Z1Z3

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SLIDE 15

Amplitude as volume

h2356i h2345i h1234i h1256i h6123i h1245i

vertices correspond to poles in the amplitude

hii + 1jj + 1i ⇠ si+1...j

“2” is special

1−2−3−4+5+6+

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SLIDE 16

Amplitude as volume

vertices correspond to poles in the amplitude

[61] [12]

s234 s345

[34] [23]

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SLIDE 17

Amplitude as volume

vertices correspond to poles in the amplitude

[61] [12]

s234 s345

[34] [23]

  • nly edges, faces correspond to consistent factorizations / soft limits

s234 = [23] = 0

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SLIDE 18

Amplitude as volume

How to calculate amplitude from this picture?

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SLIDE 19

T riangulation

135

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SLIDE 20

T riangulation

135

Volume of tetrahedron

poles are vertices h1345i3 h1245ih2345ih1234ih1235i

numerator fixed by weights

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SLIDE 21

T riangulation

135

h1356i3 h1256ih6123ih2356ih1235i

Volume of tetrahedron

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SLIDE 22

T riangulation

135

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i

R6,3 =

  • Volume of

polyhedron

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SLIDE 23

T riangulation

135

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i

  • R6,3 =

spurious pole: cancels

Volume of polyhedron

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SLIDE 24

T riangulation

135

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i

  • R6,3 =

In momentum space

h1|2 + 3|4]3 s234[23][34]h56ih61ih5|3 + 4|2]

A6,3 =

+

h3|4 + 5|6]3 s345[61][12]h34ih45ih5|3 + 4|2]

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SLIDE 25

T riangulation

135

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i

  • R6,3 =

In momentum space

h1|2 + 3|4]3 s234[23][34]h56ih61ih5|3 + 4|2]

A6,3 =

+

h3|4 + 5|6]3 s345[61][12]h34ih45ih5|3 + 4|2]

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SLIDE 26

Amplituhedron

✤ This volume picture does not generalize ✤ Amplituhedron: generalization to Grassmannians, beyond ✤ Search for “dual Amplituhedron” still ongoing

dual A = dlog form A = volume

(Arkani-Hamed, JT 2013) (Arkani-Hamed, Thomas, JT 2017) (Herrmann, Langer, Zheng, JT, in progress) (Arkani-Hamed, Hodges, JT 2015)

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SLIDE 27

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 28

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 29

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 30

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 31

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 32

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 33

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 34

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 35

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 36

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 37

BCFW recursion relations

✤ Calculate 6pt amplitude using BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

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SLIDE 38

✤ Calculate 6pt amplitude using BCFW recursion relations

etc

BCFW recursion relations

1−2−3−4+5+6+

many different shifts λ e

λ

We always get one of two formulas:

R6,3 = (1) + (3) + (5) = (2) + (4) + (6)

R-invariants

R[a, b, c, d, e] = (habcdiηe + hbcdeiηa + · · · + heabciηd)4 habcdihbcdeihcdeaihdeabiheabci

make manifest dual conformal (and also Yangian) symmetry

where (1) = R[2, 3, 4, 5, 6]

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SLIDE 39

R-invariants

R6,3 = (1) + (3) + (5) = (2) + (4) + (6)

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i + h3456i3 h2345ih2356ih2346ih2456i + h1456i3 h1245ih1256ih2456ih1246i + h1346i3 h1234ih6123ih1246ih2346i

1−2−3−4+5+6+

For helicity amplitude

(Drummond, Henn, Korchemsky, Sokatchev) (Arkani-Hamed, Cachazo, Cheung, Kaplan) (Mason, Skinner)

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SLIDE 40

R-invariants

R6,3 = (1) + (3) + (5) = (2) + (4) + (6)

h1345i3 h1245ih2345ih1234ih1235i h1356i3 h1256ih6123ih2356ih1235i + h3456i3 h2345ih2356ih2346ih2456i + h1456i3 h1245ih1256ih2456ih1246i + h1346i3 h1234ih6123ih1246ih2346i

For helicity amplitude 1−2−3−4+5+6+

(Drummond, Henn, Korchemsky, Sokatchev) (Arkani-Hamed, Cachazo, Cheung, Kaplan) (Mason, Skinner)

Two different triangulations

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SLIDE 41

Gluon recap

Amplitude = volume of geometry

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SLIDE 42

Gluon recap

Amplitude = volume of geometry

Triangulation: spurious vertices BCFW recursion: spurious poles rigid formulas = fixed by geometry manifest dual conformal symmetry

=

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SLIDE 43

Graviton amplitudes

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SLIDE 44

Graviton amplitudes

✤ No momentum twistors, use spinor helicity variables ✤ MHV amplitude: non-trivial, does not factorize ✤ Natural proposal: study BCFW recursion relations

An = det H (abc)(def)

Problem: good large-z behavior

I dz A(z)(a + bz) z = 0

Other formulas

(Hodges 2012) (Mason, Skinner 2008) (Spradlin, Volovich, Wen 2009) (Berends, Giele, Kuijf 1988)

An(z) ∼ 1 z2 Many different formulas, no rigidity, they all look different

(Bern, Dixon, Perelstein, Rozowsky 1999

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SLIDE 45

Example of BCFW formula

✤ Explicit example: (34) shift

| ↔ D3 = ⟨13⟩8[14][56]7 ⟨23⟩⟨56⟩[62]⟨1|3 + 4|5] + ⟨35⟩[56]⟨62⟩⟨1|3 + 4|2]

  • ⟨14⟩[25][26]⟨34⟩2p2

134⟨1|3 + 4|2]⟨1|3 + 4|5]⟨1|3 + 4|6]⟨3|1 + 4|2]⟨3|1 + 4|5]⟨3|1 + 4|6]

+ (1 ↔ 2).

| ↔ D1 = ⟨23⟩⟨1|2 + 3|4]7 ⟨1|2 + 3|4]⟨5|3 + 4|2][51] + [12][45]⟨51⟩p2

234

  • ⟨15⟩⟨16⟩[23][34]2⟨56⟩p2

234⟨1|3 + 4|2]⟨5|3 + 4|2]⟨5|2 + 3|4]⟨6|3 + 4|2]⟨6|2 + 3|4]

+ (1 ↔ 2). | D2 = − ⟨13⟩7⟨25⟩[45]7[16] ⟨16⟩[24][25]⟨36⟩p2

245⟨1|2 + 5|4]⟨6|2 + 5|4]⟨3|1 + 6|5]⟨3|1 + 6|2]

+ (1 ↔ 2) + (5 ↔ 6) + (1 ↔ 2, 5 ↔ 6).

| D6 = ⟨12⟩[56]⟨3|1 + 2|4]8 [21][14][24]⟨35⟩⟨36⟩⟨56⟩p2

124⟨5|1 + 2|4]⟨6|1 + 2|4]⟨3|5 + 6|1]⟨3|5 + 6|2].

A(1−, 2−, 3−, 4+, 5+, 6+) = D1 + D

flip 1

+ D2 + D3 + D

flip 3

+ D6.

(Cachazo, Svrcek)

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SLIDE 46

Reminder: Yang-Mills formula

A6,3 = X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2]

(123, 456) → (321, 654)

relabeling sum multiparticle anti-holomorphic holomorphic spurious

Only adjacent indices appear

h1|2 + 3|4]3 s234[23][34]h56ih61ih5|3 + 4|2]

h3|4 + 5|6]3 s345[61][12]h34ih45ih5|3 + 4|2]

+

Rewrite:

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SLIDE 47

✤ Formula for graviton amplitude

New graviton formula

1−2−3−4+5+6+

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

in comparison to YM:

(JT, in progress)

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SLIDE 48

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

S3 × S3

(123, 456) → (321, 654)

relabeling sum

(JT, in progress)

1−2−3−4+5+6+

New graviton formula

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SLIDE 49

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

multi-particle factorization channel

(JT, in progress)

New graviton formula

1−2−3−4+5+6+

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SLIDE 50

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

anti-holomorphic factorization channel

(JT, in progress)

New graviton formula

1−2−3−4+5+6+

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SLIDE 51

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

holomorphic factorization channel

(JT, in progress)

New graviton formula

1−2−3−4+5+6+

slide-52
SLIDE 52

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

spurious pole

(JT, in progress)

New graviton formula

1−2−3−4+5+6+

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SLIDE 53

✤ Formula for graviton amplitude

= X

S

h1|2 + 3|4]3 s234 · [23][34] · h56ih61i · h5|3 + 4|2] X

S

h1|2 + 3|4]6 · h3|4 + 5|6] s234 · [23]2[34][24] · h56i2h61ih15i · h5|3 + 4|2]+ s7

123

[12]2[23]2[13]2 · h45i2h56i2h64i2

extra term: respects complete label symmetry

(JT, in progress)

New graviton formula

1−2−3−4+5+6+

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SLIDE 54

Second formula

✤ The other Yang-Mills formula: (1) + (3) + (5) ✤ Gravity: A6,3 = X

S

h23i6[56]6 · h6|1 + 2|3] s234 · [61][51] · h34ih24i · h2|3 + 4|5] · h4|5 + 6|1]2 + X

S

s3

123 · h12ih23i[46][56]

[12][23] · h45ih56i · h4|5 + 6|1]h4|5 + 6|3] · h6|4 + 5|3]h6|4 + 5|1] A6,3 = X

S

h23i3[56]3 s234 · [61] · h34i · h2|3 + 4|5] · h4|5 + 6|1] + s3

123

[12][23] · h45ih56i · h4|5 + 6|1] · h6|4 + 5|3]

(JT, in progress)

Structure of poles again analogous!

5

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SLIDE 55

Outlook

✤ Very early stage, so far only have some nice formulas

reminiscent of Amplituhedron geometry

✤ Challenge 1: What is the singularity structure? ✤ Challenge 2: What is the positive space?

it can not be only logarithmic gravity amplitudes have double poles

An

λ3=αλ2

− − − − − → Fn

α=0

− − − → O ✓ 1 α2 ◆ . . . . . . h12ih23ih13i

needs to be formulated in momentum space must capture all properties of gravity amplitudes

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SLIDE 56

Outlook

✤ Very early stage, so far only have some nice formulas

reminiscent of Amplituhedron geometry

✤ Challenge 1: What is the singularity structure? ✤ Challenge 2: What is the positive space?

it can not be only logarithmic gravity amplitudes have double poles

An

λ3=αλ2

− − − − − → Fn

α=0

− − − → O ✓ 1 α2 ◆ . . . . . . h12ih23ih13i

needs to be formulated in momentum space must capture all properties of gravity amplitudes apart from geometry new symmetry in tree-level graviton amplitudes?

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SLIDE 57

Thank you!