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

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

Overview • Planar multi-loop multi-point massless conformal Feynman graphs e.g. Fishnet in 4D [Zamolodchikov ’80] • 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) Yangian symmetry • Cuts (mixed on-shell/off-shell) • Yangian symmetry is NOT broken by loop corrections (at least for correlators) • Generalization: 3D, 6D scalar theory, 4D scalars & fermions

• QFT generating Fishnet graphs (square lattice) • 4D biscalar theory. Complex scalars φ 1 , φ 2 in adj of SU ( N c ) � � L = N c 2 ∂ µ φ 2 + 2 ξ 2 φ † ∂ µ φ † 1 ∂ µ φ 1 + ∂ µ φ † 1 φ † 2 Tr 2 φ 1 φ 2 [Gurdogan, Kazakov ’15] • Double scaling limit of γ -deformed N = 4 SYM d log µ = O ( N − 2 d ξ • ”Almost” conformal in the planar limit. c ) • Integrability (spectrum of anomalous dimensions) [Caetano, Gurdogan, Kazakov ’16] talk by Caetano • Non-unitary. Chiral structure φ † 1 φ † φ 2 2 φ 1 • One Feynman graph per loop order in the planar limit

Fishnet graphs with regular boundary Single-trace correlator � Tr [ χ 1 ( x 1 ) χ 2 ( x 2 ) , . . . χ 2 M ( x 2 M )] � where χ i ∈ { φ † 1 , φ † 2 , φ 1 , φ 2 } Duality transformation p µ i = x µ i − x µ i +1 x -space – correlator graph p -space – loop graph with off-shell legs

Yangian of conformal algebra Conformal algebra so (2 , 4) D = − i ( x µ ∂ µ + ∆) L µν = i ( x µ ∂ ν − x ν ∂ µ ) , , K µ = i ( x 2 ∂ µ − 2 x µ x ν ∂ ν − 2∆ x µ ) P µ = − i ∂ µ , and its infinite-dimensional extension – Yangian [Drinfeld ’85] J , � J , [ � J , � J ] , [ � J , [ � J , � J ]] , [[ � J , � J ] , [ � J , � J ]] . . . where level-zero generators J A ∈ so (2 , 4) and level-one generators � J satisfy � J A , J B � � J B � Jacobi and Serre C J C , J C , J A , � C � = f AB = f AB relations – cubic in J , � J Evaluation representation with evaluation parameters v k � � J A = 1 � 2 f A J C j J B v k J A k + BC k j < k k Yangian symmetry of the Fishnet graphs J A | Fishnet � = � J A | Fishnet � = 0

Yangian of conformal algebra Conformal algebra so (2 , 4) D = − i ( x µ ∂ µ + ∆) L µν = i ( x µ ∂ ν − x ν ∂ µ ) , , K µ = i ( x 2 ∂ µ − 2 x µ x ν ∂ ν − 2∆ x µ ) P µ = − i ∂ µ , and its infinite-dimensional extension – Yangian [Drinfeld ’85] J , � J , [ � J , � J ] , [ � J , [ � J , � J ]] , [[ � J , � J ] , [ � J , � J ]] . . . where level-zero generators J A ∈ so (2 , 4) and level-one generators � J satisfy � J A , J B � � J B � Jacobi and Serre C J C , J C , J A , � C � = f AB = f AB relations – cubic in J , � J Evaluation representation with evaluation parameters v k � � � � P µ = − i � ( L µν + η µν D j ) P k ,ν − ( j ↔ k ) v k P µ + j 2 k j < k k Yangian symmetry of the Fishnet graphs J A | Fishnet � = � J A | Fishnet � = 0

Lax and monodomry matrix • Lax matrix with the spectral parameter u   1 + 1 1 1 1 u J 11 u J 12 u J 13 u J 14    1 1 + 1 1 1  u J 21 u J 22 u J 23 u J 24   L ( u ) =     1 1 1 + 1 1 u J 31 u J 32 u J 33 u J 34   1 1 1 1 + 1 u J 41 u J 42 u J 43 u J 44 consists of so (2 , 4) generators J ij ∈ span { D , P µ , K ν , L µν } • n-point monodromy matrix with inhomogeneities δ 1 , δ 2 , . . . , δ n T ( u ; � δ ) = L n ( u + δ n ) . . . L 2 ( u + δ 2 ) L 1 ( u + δ 1 ) � u − 1 − k J ( k ) T ab ( u ; � δ ) = δ ab + ab k ≥ 0 Quantum spin chain with noncompact representations of so (2 , 4)

Lax and monodomry matrix RTT-relation defines the quadratic algebra for {J ( k ) } R ae , bf ( u − v ) T ec ( u ) T fd ( v ) = T ae ( v ) T bf ( u ) R ec , fd ( u − v ) [Faddeev, Kulish, Sklyanin, Takhtajan,...’79] with Yang’s R-matrix R ab , cd ( u ) = δ ab δ cd + u δ ad δ bc . RTT is compatible with the co-product. Eigenvalue relation for Yangian symmetry the monodromy matrix L n ( u + δ n ) . . . L 2 ( u + δ 2 ) L 1 ( u + δ 1 ) | Fishnet � = λ ( u ; � δ ) | Fishnet � · 1 and expanding this matrix equation in the spectral parameter u , J ( k ) ab | Fishnet � = λ n ( � δ ) δ ab | 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]

Conformal Lax Lax matrix depends on parameters ( u , ∆) ⇔ ( u + , u − ) � � u + · 1 − p · x p L ( u + , u − ) = − x · p · x + ( u + − u − ) · x u − · 1 + x · p where [D.C., Derkachov, Isaev ’12] x = − i σ µ x µ , p = − i 2 σ µ ∂ µ , u + = u + ∆ − 4 , u − = u − ∆ 2 2 • Local vacuum of the Lax L ( u , u + 2) · 1 = ( u + 2) · 1 • Intertwining relation with the scalar propagator x i , j ≡ x i − x j x 2 x 2 L 2 ( • , u + 1) L 2 ( • , u + 1) L 1 ( u + 1 , ∗ ) L 2 ( • , u ) x − 2 12 = = x − 2 12 L 1 ( u , ∗ ) L 2 ( • , u + 1) L 1 ( u , ∗ ) L 1 ( u , ∗ ) x 1 x 1 • Integration by parts ∼ L ( u + 2 , u ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1 ,

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1 L ( u , u + 2) · 1 = ( u + 2) · 1 , x − 2 12 L 1 ( u , ∗ ) L 2 ( • , u + 1) = L 1 ( u + 1 , ∗ ) L 2 ( • , u ) x − 2 12

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1 L ( u , u + 2) · 1 = ( u + 2) · 1 , x − 2 12 L 1 ( u , ∗ ) L 2 ( • , u + 1) = L 1 ( u + 1 , ∗ ) L 2 ( • , u ) x − 2 12

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1 L ( u , u + 2) · 1 = ( u + 2) · 1 , x − 2 12 L 1 ( u , ∗ ) L 2 ( • , u + 1) = L 1 ( u + 1 , ∗ ) L 2 ( • , u ) x − 2 12

Example: Cross integral [ i , j ] ≡ L ( u + i , u + j ) L T ( u + 2 , u ) · 1 = ( u + 2) · 1 L ( u , u + 2) · 1 = ( u + 2) · 1 , x − 2 12 L 1 ( u , ∗ ) L 2 ( • , u + 1) = L 1 ( u + 1 , ∗ ) L 2 ( • , u ) x − 2 12

Example: Cross integral Yangian symmetry of the cross integral L 4 [4 , 5] L 3 [3 , 4] L 2 [2 , 3] L 1 [1 , 2] | cross � = [3][4] 2 [5] · | cross � · 1 where [ i , j ] ≡ L ( u + i , u + j ) and [ i ] ≡ u + i � 1 d 4 x 0 = x − 2 13 x − 2 | cross � = 24 Φ( s , t ) x 2 10 x 2 20 x 2 30 x 2 40 Conformal cross-ratios s , t . The Yangian symmetry implies DE ∂ t + ( s − 1) s ∂ 2 Φ ∂ s 2 + t 2 ∂ 2 Φ ∂ t 2 + 2 st ∂ 2 Φ Φ + (3 s − 1) ∂ Φ ∂ s + 3 t ∂ Φ ∂ s ∂ t = 0

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Example: Double Cross integral [ i , j ] ≡ L ( u + i , u + j )

Fishnet graphs with regular boundary �� � � � � L i [ δ + i , δ − [ δ + i ][ δ − i ] | Fishnet � = i ] | Fishnet � · 1 i ∈C i ∈C out where C = C in ∪ C out is the boundary of the graph

Fishnet graphs with regular boundary �� � � � � L i [ δ + i , δ − [ δ + i ][ δ − i ] | Fishnet � = i ] | Fishnet � · 1 i ∈C i ∈C out where C = C in ∪ C out is the boundary of the graph

Fishnet graphs with irregular boundary Single-trace correlator � Tr [ χ 1 ( x 1 ) χ 2 ( x 2 ) , . . . χ 2 M ( x 2 M )] � where χ i ∈ { φ † 1 , φ † 2 , φ 1 , φ 2 } and some of x i ’s are identified. Composite operators and Lagrangian have the same chiral structure UV finite

Fishnet graphs with irregular boundary �� � L i [ δ + i , δ − | Fishnet � = λ ( u ; � i ] δ ) | Fishnet � · 1 i ∈C