Catching integrability with fishnets and instantons Gregory - PowerPoint PPT Presentation

Catching integrability with fishnets and instantons Gregory Korchemsky IPhT, Saclay Part 1: In collaboration with Nikolay Gromov, Vladimir Kazakov, Stefano Negro, Grigory Sizov arXiv:1706.04167 Part 2: In collaboration with Fernando Alday arXiv:1605.06346, 1609.08164, 1610.01425, 1704.00448 IGST 17, July 20, 2017 - p. 1/26

Integrability and fishnet graphs ✔ Fishnet graphs are four-dimensional scalar conformally invariant Feynman diagrams ✔ Define completely integrable lattice model [Zamolodchikov’80] ✔ Appear everywhere in planar N = 4 SYM: ✗ Scattering amplitudes ✗ Correlation functions ✔ What is the relation between integrability of planar N = 4 SYM and fishnet Feynman diagrams? ✔ Simplified model: strongly twisted N = 4 SYM at weak coupling [Joao Caetano, Dima Chicherin talks] ✗ A non-unitary ‘chiral’ (almost) CFT dominated by fishnet graphs ✗ Integrable in planar limit, related to conformal SU (2 , 2) spin chain This talk: use integrability to compute exactly the scaling dimensions of BMN operators - p. 2/26

Strongly twisted N = 4 SYM L = − 1 µν − 1 i D µ φ i + i ¯ ψ A Dψ A + L int 2 D µ φ † 4 F 2 [Leigh,Strassler][Frolov] � 1 j , φ j } − e − iǫ ijk γ k φ † 4 { φ † i , φ i }{ φ † i φ † L int = g 2 j φ i φ j � − e − i i i 2 γ − 2 γ − 2 ǫ jkm γ + ψ j φ j ¯ ψ 4 φ j ¯ ψ k φ i ¯ ψ j + c.c. ¯ ¯ m ¯ ψ 4 + e ψ j + iǫ ijk e j j Twist parameters γ ± 1 = − ( γ 3 ± γ 2 ) / 2 , γ ± 2 = − ( γ 3 ± γ 2 ) / 2 , Is expected to be integrable in the planar limit Double scaling limit: strong twist + weak coupling [Gurdogan,Kazakov] g 2 → 0 , ξ 2 = g 2 e − iγ 3 = fixed γ 1 , 2 = fixed , γ 3 → − i ∞ , Gauge field, fermions and one scalar decouple L = − 1 1 ∂ µ φ 1 − 1 2 ∂ µ φ † 2 ∂ µ φ † 2 ∂ µ φ 2 + ξ 2 φ † 1 φ † 2 φ 1 φ 2 Supersymmetry and R − symmetry is broken PSU (2 , 2 | 4) → SU (2 , 2) × U (1) × U (1) - p. 3/26

Bi-scalar chiral CFT � 1 � 1 ∂ µ φ 1 + 1 2 ∂ µ φ † 2 ∂ µ φ † 2 ∂ µ φ 2 + ξ 2 φ † 1 φ † L = N c 2 φ 1 φ 2 Non-unitary theory, chiral vertex Feynman rules: φ † φ † φ 2 φ 1 2 1 No mass or vertex renormalization in the planar limit Is not complete at quantum level [Fokken,Sieg,Wilhelm’14] ✔ Quantum corrections induce new interaction vertices given by double trace operators ✔ ξ 2 does not run in the planar limit, but new couplings do run – conformal anomaly! ✔ Irrelevant in the planar limit for operator of length J ≥ 3 - p. 4/26

BMN vacuum Operator with SU (2 , 2) × U (1) 2 charges (∆ , 0 , 0 | J, 0) O J ( x ) = tr[ φ J 1 ] Protected in undeformed N = 4 SYM but receive quantum corrections in bi-scalar chiral CFT 1 � O J ( x ) ¯ ∆ J ( ξ 2 ) = J + γ J ( ξ 2 ) O J (0) � ∼ ( x 2 ) ∆ J ( ξ 2 ) , Globe-like Feynman diagrams 0 � O J ( x ) ¯ O J (0) � ∼ γ J ∼ x Anomalous dimension comes from integration in the vicinity of poles (wheel-like graphs) J ξ 2 J + γ (2) J ξ 4 J + . . . γ J = γ (1) Expansion runs in powers of ξ 2 J - p. 5/26

Anomalous dimensions from wheel graphs J ξ 2 J + γ (2) J ξ 4 J + . . . γ J = γ (1) Anomalous dimension at M wrappings γ ( M ) = wheel graph with M frames J � 2 J − 2 � γ (1) γ (2) = = − 2 ζ 2 J − 3 , = J J J − 1 [Broadhurst’85] Double wrapping: [Ahn,Bajnok,Bombardelli,Nepomechie’13] � 189 ζ 7 − 144 ζ 32 � γ (2) J =3 = [Panzer’15] � � γ (2) 309 ζ 11 + 16 ζ 3 , 8 + 20 ζ 5 , 6 − 4 ζ 6 , 5 + 40 ζ 8 , 3 − 8 ζ 3 , 3 , 5 + 40( ζ 3 , 5 , 3 + ζ 5 , 3 , 3 ) − 200 ζ 2 J =4 = 4 5 γ (2) = Sum of multiple zeta values ζ i 1 ,i 2 ,... [Gurdogan,Kazakov’15] J This talk: use integrability to find γ J ( ξ ) for J = 3 and arbitrary ξ - p. 6/26

Wheel graphs and Heisenberg SU (2 , 2) spin chain Wheel graphs are generated by integral operator [Gurdogan,Kazakov’15] J � 1 H J ( x , y ) = ( x i − y i ) 2 ( y i − y i +1 ) 2 = x J x 2 x 1 i =1 y 2 y J y 1 T J,M ( x , y ) = H J ◦ H J ◦ · · · ◦ H J = � �� � M Correlation function 0 � 1 ξ 2 JM � 0 |H M � O J ( x ) ¯ O J (0) � ∼ J | x � = � 0 | | x � = 1 − ξ 2 J H J M ≥ 0 - p. 7/26 x

Wheel graphs and Heisenberg SU (2 , 2) spin chain II Fundamental transfer matrix for noncompact SU (2 , 2) spin chain � � T J ( u ) = tr 0 R 01 ( u ) R 02 ( u ) . . . R 0 J ( u ) R 12 ( u ) − integral operator, solution to Yang-Baxter equation for the principal series of the 4D conformal group [Derkachov,GK,Manashov’01], [Chicherin,Derkachov,Isaev’13] x 1 x 1 c ( u ) x 2 R u ( x 1 , x 2 | y 1 , y 2 ) = y 2 12 ) − u +1 = ( x 2 12 ) − u − 1 [( x 1 − y 2 ) 2 ( x 2 − y 1 ) 2 ] u +2 ( y 2 y 1 Diagrammatic representation of the transfer matrix x 2 x J x 2 x J x 1 x 1 x 1 x 1 u = − 1 ... ... T J ( u ) = − − − − → y J y J y 1 y 2 y 1 y 2 y J H J ∼ T J ( − 1) Fishnet generating operator is the special case of the transfer matrix - p. 8/26

Baxter equation for length J = 3 ✔ Baxter equation for the SU (2 , 2) spin chain � � � � u + i u − i A ( u + i ) q ( u + 2 i ) − B q ( u + i )+ C ( u ) q ( u ) − B q ( u − i )+ A ( u − i ) q ( u − 2 i ) = 0 2 2 A ( u ) , B ( u ) , C ( u ) transfer matrices, sum over local integrals of motion ✔ For the BMN operators of length J = 3 , it factorizes into the product of 2nd order Baxter eqs � (∆ − 1)(∆ − 3) � − m u 3 − 2 q ( u ) + q ( u + i ) + q ( u − i ) = 0 4 u 2 ∆ − scaling dimension, m − integral of motion ✔ The quantization conditions for ∆ and m follow from the double scaling limit of Quantum Spectral Curve for twisted N = 4 SYM [Gromov,Kazakov,Leurent,Volin’15] m 2 = − ξ 6 , q 4 (0 , m ) q 2 (0 , − m ) + q 2 (0 , m ) q 4 (0 , − m ) = 0 in terms of ‘pure’ solutions q 2 ( u, m ) and q 4 ( u, m ) with large u asymptotics q 2 ( u, m ) ∼ u ∆ / 2 − 1 / 2 , q 4 ( u, m ) ∼ u − ∆ / 2+3 / 2 , u → ∞ - p. 9/26

✂ ✁ ✂ ✁ Numerical solution for J = 3 The scaling dimension ∆ 3 of the BMN operator O 3 = tr( φ 3 1 ) for arbitrary coupling ξ 3 Re[ ( ^3)] Im[ ( ^3)] 3.0 1.5 2.5 1.0 2.0 0.5 1.5 0.1 0.2 0.3 0.4 0.5 1.0 - 0.5 - 1.0 0.5 - 1.5 0.1 0.2 0.3 0.4 0.5 The scaling dimension becomes imaginary at ξ 3 ⋆ ≈ 0 . 2 � ξ 3 ⋆ − ξ 3 ∆ 3 ( ξ ) = 2 + c Weak coupling expansion has finite convergency radius Nonunitary CFT = ⇒ complex scaling dimensions, no unitary bounds on ∆ ’s At ξ = ξ ⋆ the operator O ∆ collides with its shadow O 4 − ∆ - p. 10/26

Weak and strong coupling expansion at J = 3 ✔ At weak coupling ξ < 1 the Baxter equation can be solved perturbatively in m 2 = − ξ 6 + ξ 18 � ∆ 3 − 3 = − 12 ζ 3 ξ 6 + ξ 12 � 189 ζ 7 − 144 ζ 32 � − 1944 ζ 8 , 2 , 1 − 3024 ζ 33 − 3024 ζ 5 ζ 32 + 6804 ζ 7 ζ 3 � + 198 π 8 ζ 3 + 612 π 6 ζ 5 + 270 π 4 ζ 7 + 5994 π 2 ζ 9 − 925911 ζ 11 + O ( ξ 24 ) 175 35 8 where ζ i 1 ,...,i k = � 1 . . . n i k n 1 > ··· >n k > 0 1 / ( n i 1 k ) are multiple zeta functions ✔ At strong coupling ξ ≫ 1 the Baxter equation can be solved semiclassically (see below) � √ (16 ξ 3 ) + 55 1 (16 ξ 3 ) 2 + 2537 1 1 2 ξ 3 / 2 ∆ 3 − 2 = i 2 1 + (16 ξ 3 ) 3 + 2 2 � + 830731 (16 ξ 3 ) 4 + 98920663 1 (16 ξ 3 ) 5 + 31690179795 1 1 (16 ξ 3 ) 6 + O ( ξ − 21 ) . 8 8 16 Expansion coefficients grow factorially, asymptotic Borel nonsummable series Receives nonperturbative corrections ∆ 3 ∼ e − cξ 3 (similar to the cusp in N = 4 SYM) ✔ Excellent agreement with numerical results ✔ Generalization to tr( φ J 1 ) with J > 3 (in progress) - p. 11/26

Excited states with the U (1) charge J = 3 Operators with the same SU (2 , 2) × U (1) 2 charge (∆ , 0 , 0 | J = 3 , 0) but higher length � 2 ) n 2 � 1 ( φ 1 φ † 1 ) n 1 ( φ 2 φ † φ 3 O n 1 ,n 2 = tr + permutations + derivatives ‘Length’ of the operator ∆(0) = 3 + 2 n 1 + 2 n 2 � 1 ) n 1 � 1 ( φ 1 φ † φ 3 Protected operators O n 1 , 0 = tr Operators of length 5 O 1 = tr( φ † O 2 = tr( φ † O 3 = tr( φ † O 4 = tr( φ † 2 φ 3 2 φ 2 2 φ 1 φ 2 φ 2 2 φ 2 φ 3 1 φ 2 ) , 1 φ 2 φ 1 ) , 1 ) , 1 ) φ † 2 φ 1 → φ 1 φ † φ † 2 φ 1 φ 2 → φ 1 → φ 2 φ 1 φ † φ 1 φ 2 → φ 2 φ 1 , Mixing to leading order 2 , 2 φ † 2 φ 1 φ 2 φ 1 φ 1 O 3 → O 4 O 3 → O 4 O 3 → O 2 - p. 12/26

Logarithmic multiplet Mixinig matrix µ d dµ O i ( x ) = V ij O j ( x ) for operators of length 5     4 ξ 2 O ( ξ 4 ) O ( ξ 6 ) 0 0 1 0 0     4 ξ 2 O ( ξ 4 ) 0 0 0 0 0 0      = U − 1 V =  U     − ξ 4 4 ξ 2 − 2 iξ 3   0 0 0 0 0 2 iξ 3 0 0 0 0 0 0 0 ⇒ Nonunitary CFT = Nonhermitian mixing matrix Lower block describes two conformal operators ∆ ± = 5 ± 2 iξ 3 Upper block is Jordan cell of rank r = 2 Logarithmic multiplet � � 1 0 1 � ˜ O i ( x ) ˜ O j (0) � = ln( x 2 µ 2 ) ( x 2 ) 5 1 The same pattern takes place for operators of higher length Chiral bi-scalar CFT = Logarithmic CFT 4 [Caetano’16] The Baxter equation + QSC allows us to predict the scaling dimensions for arbitrary ξ 2 - p. 13/26