[PPT] - Yangian symmetry of fishnet graphs Dmitry Chicherin JGU, Mainz, PowerPoint Presentation

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Yangian symmetry of fishnet graphs

Dmitry Chicherin

JGU, Mainz, Germany IGST, 19 July 2017, Paris

Based on work arXiv:1704.01967 and 1708.xxxxx in collaboration with

V. Kazakov, F. Loebbert, D. M¨

uller, D. Zhong

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Overview

Planar multi-loop multi-point massless conformal Feynman graphs

e.g. Fishnet in 4D

Yangian symmetry

Integrability

Higher order symmetry (extends conformal symmetry)
Set of differential equations
Integrability of the underlying theory
Analytical approach (Bethe ansatz, TBA, QSC, . . . ) and Operator approach

(Baxter Q-operators, transfer matrices, separation of variables) to integrability talk by Korchemsky [Gromov, Kazakov, Korchemsky, Negro, Sizov ’17]

4D biscalar theory (limit of N = 4 SYM)

square fishnet lattice

Single-trace correlators (off-shell legs)
Amplitudes (on-shell legs)
Cuts (mixed on-shell/off-shell)

Yangian symmetry

Yangian symmetry is NOT broken by loop corrections (at least for correlators)
Generalization: 3D, 6D scalar theory, 4D scalars & fermions

[Zamolodchikov ’80]

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QFT generating Fishnet graphs (square lattice)
4D biscalar theory. Complex scalars φ1, φ2 in adj of SU(Nc)

L = Nc 2 Tr

∂µφ†

1∂µφ1 + ∂µφ† 2∂µφ2 + 2ξ2 φ† 1φ† 2φ1φ2

[Gurdogan, Kazakov ’15]
Double scaling limit of γ-deformed N = 4 SYM
”Almost” conformal in the planar limit.

dξ d log µ = O(N−2 c )

Integrability (spectrum of anomalous dimensions)

[Caetano, Gurdogan, Kazakov ’16]

talk by Caetano

Non-unitary. Chiral structure

φ†

1

φ1 φ†

2

φ2

One Feynman graph per loop order in the planar limit

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Fishnet graphs with regular boundary

Single-trace correlator Tr [χ1(x1) χ2(x2), . . . χ2M(x2M)] where χi ∈ { φ†

1, φ† 2, φ1, φ2}

Duality transformation pµ

i = xµ i − xµ i+1

x-space – correlator graph p-space – loop graph with

ff-shell legs

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Yangian of conformal algebra

Conformal algebra so(2, 4) D = −i(xµ∂µ + ∆) , Lµν = i(xµ∂ν − xν∂µ) , Pµ = −i∂µ , Kµ = i(x2 ∂µ − 2xµxν∂ν − 2∆xµ) and its infinite-dimensional extension – Yangian

[Drinfeld ’85]

J , J , [ J, J] , [ J, [ J, J]] , [[ J, J], [ J, J]] . . . where level-zero generators JA ∈ so(2, 4) and level-one generators J satisfy

JA, JB

= f AB

C JC ,

JA,

JB = f AB

C

JC , Jacobi and Serre relations – cubic in J, J Evaluation representation with evaluation parameters vk

JA = 1

2f A BC

j<k

JC

j JB k +

k

vkJA

k

Yangian symmetry of the Fishnet graphs JA|Fishnet = JA|Fishnet = 0

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Yangian of conformal algebra

Conformal algebra so(2, 4) D = −i(xµ∂µ + ∆) , Lµν = i(xµ∂ν − xν∂µ) , Pµ = −i∂µ , Kµ = i(x2 ∂µ − 2xµxν∂ν − 2∆xµ) and its infinite-dimensional extension – Yangian

[Drinfeld ’85]

J , J , [ J, J] , [ J, [ J, J]] , [[ J, J], [ J, J]] . . . where level-zero generators JA ∈ so(2, 4) and level-one generators J satisfy

JA, JB

= f AB

C JC ,

JA,

JB = f AB

C

JC , Jacobi and Serre relations – cubic in J, J Evaluation representation with evaluation parameters vk

Pµ = − i

2

j<k
(Lµν

j

+ ηµνDj)Pk,ν − (j ↔ k)

+
k

vkPµ

k

Yangian symmetry of the Fishnet graphs JA|Fishnet = JA|Fishnet = 0

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Lax and monodomry matrix

Lax matrix with the spectral parameter u

L(u) =        1 + 1

uJ11 1 uJ12 1 uJ13 1 uJ14 1 uJ21

1 + 1

uJ22 1 uJ23 1 uJ24 1 uJ31 1 uJ32

1 + 1

uJ33 1 uJ34 1 uJ41 1 uJ42 1 uJ43

1 + 1

uJ44

       consists of so(2, 4) generators Jij ∈ span{D, Pµ, Kν, Lµν}

n-point monodromy matrix with inhomogeneities δ1, δ2, . . . , δn

T(u; δ) = Ln(u + δn) . . . L2(u + δ2) L1(u + δ1) Tab(u; δ) = δab +

k≥0

u−1−kJ (k)

ab

Quantum spin chain with noncompact representations of so(2, 4)

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Lax and monodomry matrix

RTT-relation defines the quadratic algebra for {J (k)} Rae,bf (u − v) Tec(u) Tfd(v) = Tae(v) Tbf (u) Rec,fd(u − v)

[Faddeev, Kulish, Sklyanin, Takhtajan,...’79]

with Yang’s R-matrix Rab,cd(u) = δabδcd + uδadδbc. RTT is compatible with the co-product. Yangian symmetry Eigenvalue relation for the monodromy matrix Ln(u + δn) . . . L2(u + δ2) L1(u + δ1) |Fishnet = λ(u; δ) |Fishnet · 1 and expanding this matrix equation in the spectral parameter u, J (k)

ab |Fishnet = λn(

δ)δab |Fishnet

For Jordan-Schwinger representations and scattering amplitudes in N = 4 SYM

[D.C., Kirschner ’13; D.C., Kirschner, Derkachov ’13]

6-vertex model [Frassek, Kanning, Ko, Staudacher ’13], On-shell amplitude graphs [Kanning, Lukowski, Staudacher ’14; Broedel,

de Leeuw, Rosso ’14], scattering amplitudes in ABJM [Bargheer, Huang, Loebbert, Yamazaki ’14], form factors of

composite operators [Bork, Onishchenko ’15; Frassek, Meidinger, Nandan, Wilhelm ’15], form factors of Wilson lines [Bork,

Onishchenko ’16], amplituhedron volume [Ferro, Lukowski, Orta, Parisi ’16], QCD parton evolution kernels [Fuksa, Kirschner ’16], splitting amplitudes [Kirschner, Savvidy ’17]

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Conformal Lax

Lax matrix depends on parameters (u, ∆) ⇔ (u+, u−) L(u+, u−) =

u+ · 1 − p · x

p −x · p · x + (u+ − u−) · x u− · 1 + x · p

where

[D.C., Derkachov, Isaev ’12]

x = −iσµxµ , p = − i

2σµ∂µ , u+ = u + ∆−4 2

, u− = u − ∆

2

Local vacuum of the Lax

L(u, u + 2) · 1 = (u + 2) · 1

Intertwining relation with the scalar propagator

xi,j ≡ xi − xj

L2(•, u + 1) L1(u, ∗)

x2 x1 =

L2(•, u + 1) L1(u, ∗)

x2 x1 L1(u + 1, ∗)L2(•, u) x−2

12

= x−2

12 L1(u, ∗)L2(•, u + 1)

Integration by parts

LT(u + 2, u) · 1 = (u + 2) · 1 ∼ L(u + 2, u)

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Example: Cross integral

[i,j]≡L(u+i,u+j)

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Example: Cross integral

LT(u + 2, u) · 1 = (u + 2) · 1 ,

[i,j]≡L(u+i,u+j)

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Example: Cross integral

LT(u + 2, u) · 1 = (u + 2) · 1 , L(u, u + 2) · 1 = (u + 2) · 1 x−2

12 L1(u, ∗)L2(•, u + 1) = L1(u + 1, ∗)L2(•, u)x−2 12 [i,j]≡L(u+i,u+j)

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Example: Cross integral

LT(u + 2, u) · 1 = (u + 2) · 1 , L(u, u + 2) · 1 = (u + 2) · 1 x−2

12 L1(u, ∗)L2(•, u + 1) = L1(u + 1, ∗)L2(•, u)x−2 12 [i,j]≡L(u+i,u+j)

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Example: Cross integral

LT(u + 2, u) · 1 = (u + 2) · 1 , L(u, u + 2) · 1 = (u + 2) · 1 x−2

12 L1(u, ∗)L2(•, u + 1) = L1(u + 1, ∗)L2(•, u)x−2 12 [i,j]≡L(u+i,u+j)

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Example: Cross integral

LT(u + 2, u) · 1 = (u + 2) · 1 , L(u, u + 2) · 1 = (u + 2) · 1 x−2

12 L1(u, ∗)L2(•, u + 1) = L1(u + 1, ∗)L2(•, u)x−2 12 [i,j]≡L(u+i,u+j)

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Example: Cross integral

Yangian symmetry of the cross integral L4[4, 5]L3[3, 4]L2[2, 3]L1[1, 2] |cross = [3][4]2[5] · |cross · 1 where [i, j] ≡ L(u + i, u + j) and [i] ≡ u + i |cross =

d4x0

1 x2

10x2 20x2 30x2 40

= x−2

13 x−2 24 Φ(s, t)

Conformal cross-ratios s, t. The Yangian symmetry implies DE Φ + (3s − 1)∂Φ ∂s + 3t ∂Φ ∂t + (s − 1)s ∂2Φ ∂s2 + t2 ∂2Φ ∂t2 + 2st ∂2Φ ∂s∂t = 0

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Example: Double Cross integral

[i,j]≡L(u+i,u+j)

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Fishnet graphs with regular boundary

i∈C

Li[δ+

i , δ− i ]

|Fishnet =

i∈Cout

[δ+

i ][δ− i ]

|Fishnet · 1

where C = Cin ∪ Cout is the boundary of the graph

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Fishnet graphs with regular boundary

i∈C

Li[δ+

i , δ− i ]

|Fishnet =

i∈Cout

[δ+

i ][δ− i ]

|Fishnet · 1

where C = Cin ∪ Cout is the boundary of the graph

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Fishnet graphs with irregular boundary

Single-trace correlator Tr [χ1(x1) χ2(x2), . . . χ2M(x2M)] where χi ∈ { φ†

1, φ† 2, φ1, φ2} and some of xi’s are identified. Composite operators

and Lagrangian have the same chiral structure UV finite

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Fishnet graphs with irregular boundary

i∈C

Li[δ+

i , δ− i ]

|Fishnet = λ(u;

δ) |Fishnet · 1

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Cyclic invariance of the Yangian

Cyclic shift by one site (dual Coxeter number = 0)

LN(uN;∆N)...L1(u1;∆1) |G=λ(u) |G·1

LN−1(uN−1;∆N−1)...L1(u1;∆1)LN(uN−4;∆N) |G=

λ(u) |G·1

Apply cyclicity N times algebraic equations for eigenvalue λ(u) λ(u) λ(u − 4) = P(u) P(u − 2) where deg λ(u) = N, deg P(u) = 2N, P(u) :=

N

i=1

[δ+

i ][δ− i ]

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Scattering amplitudes and cuts of fishnet graphs

Duality transformation xµ

i − xµ i+1 = pµ i . Amplitude p2 i = 0 ⇔ x2 i,i+1 = 0 L2(•, u + 1) L1(u, ∗)

x2 x1 =

L2(•, u + 1) L1(u, ∗)

x2 x1 L1(u + 1, ∗)L2(•, u) δ(x2

12)

= δ(x2

12) L1(u, ∗)L2(•, u + 1)

T(u) |Fishnet = λ(u) |Fishnet · 1 ⇒ T(u) |Fishnetcut = λ(u) |Fishnetcut · 1 for any cut x−2

ij

→ δ(x2

ij ).

Conformal anomaly of on-shell graphs?!

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Generalizations: tri-scalar theories in 3D, 6D, and scalar-fermion theories in 4D

L4D,int = LYukawa + Lφ4 scalar and fermion fields L3D,int = ξNcTr Y1Y †

4 Y2Y † 1 Y4Y † 2

L6D,int = NcTr

ξ1 φ†

1φ2φ3 + ξ2 φ1φ† 2φ† 3

[Mamroud, Torrents ’17]

[Caetano, Gurdogan, Kazakov ’16] [Gurdogan, Kazakov ’15]

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Scalar-fermion theories in 4D

Lint

φψ = NcTr

ξ2

1φ† 3φ† 1φ3φ1 + ξ2 2φ† 2φ† 1φ2φ1

+

ξ1ξ2( ¯

ψ1φ1 ¯ ψ4 − ψ1φ†

1ψ4)

Lax matrix for representations (∆, ℓ, ˙

ℓ) L(u) = 1 x 1 u+ · 1 + S p u− · 1 + S 1 −x 1

[D.C., Derkachov, Isaev ’12]

where ρα and ˜ ρ ˙

α are auxiliary spinors

S = 1 2 σi (ρ σi ∂ρ) , S = 1 2 σi (˜ ρ σi ∂˜

ρ)

Chiral and anti-chiral fermions Lf ↔ ( 1

2, 0)

, L

¯ f ↔ (0, 1 2)

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Scalar-fermion theories in 4D

Fermionic propagators

P ˜

ρj ρi = ρi|xij|˜

ρj] x4

ij

are intertwining operators, i.e. Lf

1(u + 3 2, • ) L2( ∗ , u) P ∂ ˜

ρ2

ρ1

= P

∂ ˜

ρ2

ρ1 L1(u, • ) L ¯ f 2( ∗ , u + 3 2)

No local vacuum for fermionic Lax. It is replaced by

Lf

1(u, u + 1 2) ˆ

L

¯ f 2(u + 1 2, u) P ˜ ρ2 ρ1 = P ˜ ρ2 ρ1

where Lax ˆ L¯

f is obtained from L¯ f by a sequence of involutions.

Build the monodromy out of L, Lf , L¯

f , ˆ

Lf , ˆ L¯

f

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Yangian in the momentum representation. Off-shell

D-dimensional scalar theory. Lax matrix in the momentum representation L(u+, u−) = 1 ∂ ∂ ∂ 1 u+ · 1 p u− · 1 1 −∂ ∂ ∂ 1

where

x = σµ∂pµ , p = 1

2σµ∂µ , u+ = u + ∆−D 2

, u− = u − ∆

2

Intertwining operator
S(α)

ij

f

(pi, pj) =
dDp

(p2)

D 2 −α f (pi − p, pj + p)

L1(u + α, •) L2(∗, u) S(α)

12 = S(α) 12 L1(u, •) L2(∗, u + α)

Local vacuum

L(u, u + D

2 ) δ(D)(p) = δ(D)(p) 1

= S

( 1 2 ) 45 S ( 1 2 ) 34 S ( 3 2 ) 23 S ( 3 2 ) 12 S ( 1 2 ) 17 S ( 3 2 ) 67 S ( 3 2 ) 56 S ( 1 2 ) 78 S ( 1 2 ) 18 Ω

where Ω = 8

i=1 δ(3)(pi)

3D Example. Construct monodromy eigenvector

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Yangian in the momentum representation. On-shell

Consider 3D theory. Light-cone momenta p2 = 0 are parametrized by spinors pµ = λα(σµ)α

βλβ

Lax matrix for on-shell representation of the conformal algebra L(u) =

u δβ

α − λα ∂ ∂λβ

λαλβ

∂2 ∂λα∂λβ

u δβ

α + ∂ ∂λα λβ

Off-shell Lax (at ∆ = 1 + D

2 ) and on-shell Lax are consistent

L(u, u − 1) f (λ ⊗ λ) = L(u) f (λ ⊗ λ) Replace L → L and pµ → λαλβ Toff-shell(u) |Goff-shell = λ(u) |Goff-shell ⇒ Ton-shell(u) |Gon-shell = λ(u) |Gon-shell

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Conclusions

Conformal Yangian symmetry of single-trace correlators and amplitudes
RTT-realization of the Yangian algebra
Propagators – intertwining operators of the Laxes and monodromy
Yangian invariance is a consequence of (bi)local operator relations
Lasso proof of the Yangian invariance
The approach also works for Fishnet graphs in 3D, 6D, and for scalar-fermion

theories in 4D

4D theory with three fermions? Several graphs at given order
Fishnet with fermions in 3D and 6D?
Relation to 4-point Fishnet correlator graphs from [Basso,Dixon ’17]? Soft limit
Conformal anomalies on shell