[PPT] - Yangian Symmetry for Fishnet Feynman Graphs Deliang Zhong LPTENS PowerPoint Presentation

SLIDE 1

Yangian Symmetry for Fishnet Feynman Graphs

Deliang Zhong LPTENS Paris

With Dmitry Chicherin, Vladimir Kazakov, Florian Loebbert, Dennis Müller arXiv: 1704.01967, 1708.00007 & work in progress

SLIDE 2

❖ Superconformal symmetry ❖ Planar integrability

Origin: mysterious

❖ AdS/CFT correspondence

4D N = 4 Super Yang-Mills

SLIDE 3

γ − twisted 4D N = 4 Super Yang-Mills Bi-scalar limit

[Gürdoğan, Kazakov ’15] ❖ Chiral, non-unitary theory ❖ 1-1 correspondence: correlator and fishnet Feynman diagram

Lφ = NcTr

∂µφ†

1∂µφ1 + ∂µφ† 2∂µφ2 + ξ2 φ† 1φ† 2φ1φ2

.

K(x1, . . . , xn) = hTr[χ1(x1) . . . χn(xn)]i.

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χk ∈ {φ1, φ2, φ†

1, φ† 2}

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⇔

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SLIDE 4

❖ Conformal symmetry for specific choice of couplings ❖ Planar integrability

From first principle
Yangian symmetry for individual Feynman diagrams

❖ Is there an AdS dual?

γ − twisted 4D N = 4 Super Yang-Mills Bi-scalar limit

[Gürdoğan, Kazakov ’15]

[Sieg-Wilhelm ’16] [Grabner, Gromov, Kazakov, Korchemsky ’17]

< l a t e x i t s h a 1 _ b a s e 6 4 = " n z B z l 2 E H H m D T q X d j v n R 2 5 I P C s e I = " > A A A B 7 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S Q i q L e i F 4 8 V j C 2

W

w 2 2 3 b p Z j f s T

Q

S + i O 8 e F D x 6 v / x 5 r 9 x + a g 1 Q c D j / d m m J k X p Y I b 9 L w v p 7 K y u r a + U d 2 s b W 3 v 7 O 7 V 9 w 8 e j M

Z

Q F V Q u l u R A w T X L I A O Q r W T T U j S S R Y J 5 r c F H 7 n k W n D l b z H a c r C h I w k H 3 J K E q d P

V

m t q g 3 v C a 3 h z u X + K X p A E l 2

P

6 Z z 9 W N E u Y R C q I M T 3 f S z H M i U Z O B Z v V + p l h K a E T M m I 9 S y V J m A n z + b k z 9 8 Q q s T t U 2 p Z E d 6 7 + n M h J Y s w i W x n Q n B s l r 1 C / M / r Z T i 8 D H M u w y Z p I t F w y 4 q N z i d z f m m l E U U s I 1 d z e 6 t I x Y S i T a g I w V 9 + + S 8 J z p p X T f / u v N G 6 L t O

w

h E c w y n 4 c A E t u I U 2 B E B h A k / w A q 9 O 6 j w 7 b 8 7 7

r

X i l D O H 8 A v O x z d R O 4 8 X < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " n z B z l 2 E H H m D T q X d j v n R 2 5 I P C s e I = " > A A A B 7 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S Q i q L e i F 4 8 V j C 2

W

w 2 2 3 b p Z j f s T

Q

S + i O 8 e F D x 6 v / x 5 r 9 x + a g 1 Q c D j / d m m J k X p Y I b 9 L w v p 7 K y u r a + U d 2 s b W 3 v 7 O 7 V 9 w 8 e j M

Z

Q F V Q u l u R A w T X L I A O Q r W T T U j S S R Y J 5 r c F H 7 n k W n D l b z H a c r C h I w k H 3 J K E q d P

V

m t q g 3 v C a 3 h z u X + K X p A E l 2

P

6 Z z 9 W N E u Y R C q I M T 3 f S z H M i U Z O B Z v V + p l h K a E T M m I 9 S y V J m A n z + b k z 9 8 Q q s T t U 2 p Z E d 6 7 + n M h J Y s w i W x n Q n B s l r 1 C / M / r Z T i 8 D H M u w y Z p I t F w y 4 q N z i d z f m m l E U U s I 1 d z e 6 t I x Y S i T a g I w V 9 + + S 8 J z p p X T f / u v N G 6 L t O

w

h E c w y n 4 c A E t u I U 2 B E B h A k / w A q 9 O 6 j w 7 b 8 7 7

r

X i l D O H 8 A v O x z d R O 4 8 X < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " n z B z l 2 E H H m D T q X d j v n R 2 5 I P C s e I = " > A A A B 7 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S Q i q L e i F 4 8 V j C 2

W

w 2 2 3 b p Z j f s T

Q

S + i O 8 e F D x 6 v / x 5 r 9 x + a g 1 Q c D j / d m m J k X p Y I b 9 L w v p 7 K y u r a + U d 2 s b W 3 v 7 O 7 V 9 w 8 e j M

Z

Q F V Q u l u R A w T X L I A O Q r W T T U j S S R Y J 5 r c F H 7 n k W n D l b z H a c r C h I w k H 3 J K E q d P

V

m t q g 3 v C a 3 h z u X + K X p A E l 2

P

6 Z z 9 W N E u Y R C q I M T 3 f S z H M i U Z O B Z v V + p l h K a E T M m I 9 S y V J m A n z + b k z 9 8 Q q s T t U 2 p Z E d 6 7 + n M h J Y s w i W x n Q n B s l r 1 C / M / r Z T i 8 D H M u w y Z p I t F w y 4 q N z i d z f m m l E U U s I 1 d z e 6 t I x Y S i T a g I w V 9 + + S 8 J z p p X T f / u v N G 6 L t O

w

h E c w y n 4 c A E t u I U 2 B E B h A k / w A q 9 O 6 j w 7 b 8 7 7

r

X i l D O H 8 A v O x z d R O 4 8 X < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " n z B z l 2 E H H m D T q X d j v n R 2 5 I P C s e I = " > A A A B 7 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 B I v g q S Q i q L e i F 4 8 V j C 2

W

w 2 2 3 b p Z j f s T

Q

S + i O 8 e F D x 6 v / x 5 r 9 x + a g 1 Q c D j / d m m J k X p Y I b 9 L w v p 7 K y u r a + U d 2 s b W 3 v 7 O 7 V 9 w 8 e j M

Z

Q F V Q u l u R A w T X L I A O Q r W T T U j S S R Y J 5 r c F H 7 n k W n D l b z H a c r C h I w k H 3 J K E q d P

V

m t q g 3 v C a 3 h z u X + K X p A E l 2

P

6 Z z 9 W N E u Y R C q I M T 3 f S z H M i U Z O B Z v V + p l h K a E T M m I 9 S y V J m A n z + b k z 9 8 Q q s T t U 2 p Z E d 6 7 + n M h J Y s w i W x n Q n B s l r 1 C / M / r Z T i 8 D H M u w y Z p I t F w y 4 q N z i d z f m m l E U U s I 1 d z e 6 t I x Y S i T a g I w V 9 + + S 8 J z p p X T f / u v N G 6 L t O

w

h E c w y n 4 c A E t u I U 2 B E B h A k / w A q 9 O 6 j w 7 b 8 7 7

r

X i l D O H 8 A v O x z d R O 4 8 X < / l a t e x i t >

❖ For : type scalar Feynman diagram α = π 2

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φ4

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[Baxter ’78]

SLIDE 6

Scalar Fishnet Diagrams in General

Dimension d = 3 d = 4 d = 6 Propagator |xij|−1 |xij|−2 |xij|−4 Scalar Fishnet This is the complete set of fishnets introduced as statistical models by

❖ Feynman rules Vertex Propagator ❖ UV/IR finite: unbroken conformal symmetry

Z ddx ⇔

j

⇔ |xij|d−2

[Zamolodchikov ’80]

SLIDE 7

Yangian Symmetry for the Single Box

I4 = Z d4x0 1 x2

10x2 20x2 30x2 40

= = = = = = 1 x2

13x2 24

Φ(u, v) =

4 2 1 3

u = x2

12x2 34

x2

13x2 24

v = x2

14x2 23

x2

13x2 24

❖ We know everything about this building block!

The integral can be expressed in terms of dilogarithms
The integral is conformally invariant

[Ussyukina, Davydychev, ’93]

Ja ∈ {D∆=1, Lµν, Pµ, Kµ}

Dilatation Translation Special Conf. Lorentz

JaI4 = 0, Ja =

4

X

j=1

Ja

j

SLIDE 8

Yangian Symmetry for the Single Box

I4 = Z d4x0 1 x2

10x2 20x2 30x2 40

= = = = = = 1 x2

13x2 24

Φ(u, v) =

4 2 1 3

u = x2

12x2 34

x2

13x2 24

v = x2

14x2 23

x2

13x2 24

❖ We know everything about this building block!

We also have non-local hidden symmetry
Hidden symmetry gives two second order differential equation

b PµI4 = 0, b Pµ = − i

2 4

X

j<k=1

⇥ (Lµν

j +⌘µνDj)Pk,ν−(j ↔ k)

⇤ −

4

X

j=1

jPµ

j

Depend on the graph Bilocal piece: universal

0 = Φ + (3u − 1)@Φ @u + 3v @Φ @v + (u − 1)u@2Φ @u2 + v2 @2Φ @v2 + 2uv @2Φ @u@v 0 = Φ + (u ↔ v)

SLIDE 9

Yangian Symmetry for Scalar Fishnets

❖ The general fishnet can be obtained by connecting single boxes

= (~ u)

[1,2] [1,2] [1,2] [2,3] [1,2] [1,2] [2,3] [2,3] [4,5] [3,4] [2,3] [3,4] [3,4] [3,4] [3,4] [4,5] [4,5] [4,5]

❖ The hidden symmetries come from integrability ! ❖ Fully encoded in the RTT relation

R12(u − v)T1(u)T2(v) = T2(v)T1(u)R12(u − v)

SLIDE 10

❖ What is R and T?

R-matrix: Yang’s R-matrix
T-matrix: product of Lax operator with inhomogeneities
Lax operator: spinorial Lax operator

Yangian Symmetry for Scalar Fishnets

− − −

}) = Ln[δ+

n , δ− n ]Ln−1[δ+ n−1, δ− n−1] · · · L1[δ+ 1 , δ− 1 ]

T(~ u)

[Chicherin, Derkachov Isaev ’13]

Lk,αβ(u+

k , u− k ) = uk 1k,αβ + 1 2Sab αβ J∆k k,ab,

Inhomogeneities Differential conformal generator Spin generator

u+

k := uk + ∆k − 4

2 , u−

k := uk − ∆k

2

[δ+, δ−] := (u+, u−) ≡ (u + δ+, u + δ−) ), [δ] := u + δ

SLIDE 11

Yangian Symmetry for Scalar Fishnets

❖ T includes all symmetry generators

Higher level generators

T(u) = un 1 +un−1 J + un−2 b J + · · ·

Level 0 generators (Conformal symmetry) Level 1 generators (Dual conformal symmetry)

( J ˆ J

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❖ Yangian symmetry algebra [Drinfeld ’85]

[Ja, Jb] = f c

abJc

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[Ja, ˆ Jb] = f c

abˆ

Jc

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[ˆ Ja, [ˆ Jb, Jc]] − [Ja, [ˆ Jb, ˆ Jc]] = O(J3)

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Level 0 Level 1 Serre relation

SLIDE 12

Yangian Symmetry for Scalar Fishnets

T(~ u) In = (~ u) In 1, (~ u) = Y

j∈out

+

j − j = [3]5[4]9[5]4

), [δ] := u + δ

❖ Yangian symmetry: eigenvalue equation for Feynman integral In

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= (~ u)

[1,2] [1,2] [1,2] [2,3] [1,2] [1,2] [2,3] [2,3] [4,5] [3,4] [2,3] [3,4] [3,4] [3,4] [3,4] [4,5] [4,5] [4,5]

SLIDE 13

Yangian Symmetry for Scalar Fishnets

❖ Proof of Yangian symmetry: intertwining relations

Intertwiners
Action on the interaction vertex
Vacuum

[δ, δ + 1] [δ + 1, δ + 2]

= [δ + 2]×

[δ + 1, δ + 1] [δ, δ + 1]

=

[δ + 1, δ]

[ ? , ] [ + 1, • ]

[ ? , + 1] [, • ]

=

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1 x2

12

L2[, • ]L1[ ? , + 1] = L2[ + 1, • ]L1[ ? , ] 1 x2

12

.

⇔

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Lαβ[, + 2] · 1 = LT

αβ[ + 2, ] · 1 = [ + 2]αβ. Integration by parts

SLIDE 14

Yangian Symmetry for Scalar Fishnets

❖ Proof of Yangian symmetry: example

[1,2] [1,2] [1,2] [2,3] [2,3] [2,3] [2,3] [3,4] [3,4] [3,4] [4,5] [4,5] [4,5] [4,5] [1,2] [1,2] [2,2] [2,3] [2,3] [3,4] [3,4] [3,4] [4,5] [4,5] [4,5]

→

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❖ The end of the story.

SLIDE 15

… wait a minute!

SLIDE 16

Scalar Fishnets in Momentum Space

❖ The dual Feynman diagram in momentum space

Problem 1: different kind of interaction vertices

Split into two

SLIDE 17

Scalar Fishnets in Momentum Space

❖ The dual Feynman diagram in momentum space

Solution: joining external points

SLIDE 18

Scalar Fishnets in Momentum Space

❖ The dual Feynman diagram in momentum space

Problem 2: All external legs are off-shell
Solution: adding delta-functions

p8 p3 p7 p4 p1 p6 p2 p5 x8 x4 x2 x6 p8 p3 p7 p4 p1 p6 p2 p5 x1 x3 x5 x7 x8 x4 x2 x6

Off-shell external legs On-shell external legs

SLIDE 19

Scalar Fishnets in Momentum Space

❖ The dual Feynman diagram in momentum space

Direct way of doing: Lax operator in momentum space
We can obtain it by using Fourier transformation
Similar intertwining relations

L(x)(u+, u−) Z dDp eipxf(p) ≡ Z dDp eipxL(p)(u+, u−)f(p)

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SLIDE 20

❖ New ingredients: Yukawa vertices & fermion line (dashed line) ❖ “Brick wall” diagrams ❖ Lax operator with spin ❖ Still have Yangian symmetry!

Fishnets with Fermions in 4D

4 3 00 ¯

[Chicherin, Derkachov Isaev ’13]

SLIDE 21

Summary

❖ Yangian symmetries (PDEs) for Feynman diagrams ❖ Integrability for non-vanishing dual Coxeter number ❖ New kind of fishnet diagrams: brick wall

Fishnet diagrams

N = 4 SYM

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Bi-scalar limit

γ − twisted N = 4 SYM

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SLIDE 22

Outlook

Yangian symmetry for the full γ-deformed theory (3 bosons and 3 fermions) ?
Yangian symmetry in 3d and 6d for models containing both bosons and fermions?
For on-shell massless scattering processes, how to understand Yangian symmetry if

we have anomaly-like behaviour arising from collinear particles?

Using Yangian PDEs to compute the Feynman integrals?
Deriving Yangian symmetry of correlators and amplitudes directly from the

Lagrangian ?

[Chicherin, Sokatchev ’17]

SLIDE 23

❖ For special cases of Fishnet (4pt case), integrability gives integral representations for fishnets.

How to find their Yangian symmetry?
For m, n large but their ratio fixed, one can solve the saddle point equation

exactly.

Continuum limit hints at AdS dual?

φ†m

1

φm

1

φn

2

φ†n

2

Outlook

[Basso, Dixon ’17] [Basso, Dixon, Kosower, D-l.Z ’In progress]

SLIDE 24

Yangian Symmetry for Fishnet Feynman Graphs

Deliang Zhong LPTENS Paris

4D N = 4 Super Yang-Mills

γ − twisted 4D N = 4 Super Yang-Mills Bi-scalar limit

⇔

γ − twisted 4D N = 4 Super Yang-Mills Bi-scalar limit

Scalar Fishnet Diagrams in 4D

:

:

:

:

:

:

:

:

:

:

:

:

· · ·

· · ·

· · ·

· · ·

φ4

Scalar Fishnet Diagrams in General

Dimension d = 3 d = 4 d = 6 Propagator |xij|−1 |xij|−2 |xij|−4 Scalar Fishnet This is the complete set of fishnets introduced as statistical models by

Yangian Symmetry for the Single Box

Yangian Symmetry for the Single Box

Yangian Symmetry for Scalar Fishnets

Yangian Symmetry for Scalar Fishnets

Yangian Symmetry for Scalar Fishnets

[Ja, Jb] = f c

[Ja, ˆ Jb] = f c

Jc

[ˆ Ja, [ˆ Jb, Jc]] − [Ja, [ˆ Jb, ˆ Jc]] = O(J3)

Yangian Symmetry for Scalar Fishnets

Yangian Symmetry for Scalar Fishnets

=

⇔

Yangian Symmetry for Scalar Fishnets

→

… wait a minute!

Scalar Fishnets in Momentum Space

Scalar Fishnets in Momentum Space

Scalar Fishnets in Momentum Space

Scalar Fishnets in Momentum Space

Fishnets with Fermions in 4D

Summary

Outlook

Outlook

Thank you!