Integrability of Conformal Fishnet Theory

Transcription

1 Integrability of Conformal Fishnet Theory Gregory Korchemsky IPhT, Saclay In collaboration with David Grabner, Nikolay Gromov, Vladimir Kazakov arxiv: th Workshop on Non-Perturbative QCD, June 14, p. 1/13

2 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: γ deformed N = 4 SYM A non-unitary chiral (almost) CFT dominated by fishnet graphs Integrable in planar limit, related to conformal SU(,) spin chain This talk: compute exactly correlation functions in γ deformed N = 4 SYM - p. /13

3 Strongly twisted N = 4 SYM L = 1 4 F µν +Dµ φ i D µφ i +i ψ A Dψ A +L int [Leigh,Strassler][Frolov] L int = g ( 1 4 {φ i,φi }{φ j,φj } e iǫijk γ k φ i φ j φi φ j e i γ j ψj φ j ψ4 +e i ) γ j ψ4 φ j ψj +iǫ ijk e i ǫ jkmγ m + ψk φ i ψj +c.c. Twist parameters γ ± 1 = (γ 3 ±γ )/, γ ± = (γ 1 ±γ 3 )/, γ ± 3 = (γ ±γ 1 )/ Is expected to be integrable in the planar limit Double scaling limit: strong twist + weak coupling [Gurdogan,Kazakov] g 0, γ 1, = fixed, γ 3 +i, ξ = g e iγ 3 = fixed Gauge field, fermions and one scalar decouple ] L = tr [ µ φ 1 µφ 1 + µ φ µφ +ξ φ 1 φ φ 1φ Supersymmetry and R symmetry is broken PSU(, 4) SU(,) U(1) U(1) - p. 3/13

4 Bi-scalar chiral CFT L = N c tr [ µ X µ X + µ Z µ Z +(4π) ξ X ZXZ ] Non-unitary theory, chiral vertex Feynman rules: X X Z Z The theory is not complete at the quantum level [Fokken,Sieg,Wilhelm] X X X X 1 ǫ tr(x )tr( X ) 1 ǫ tr(xz)tr( X Z) X X Z Z Double-trace counter terms have to added - p. 4/13

5 Beta functions Quantum corrections induce double-trace interaction vertices L dt /(4π) = α [ tr(x )tr( X )+tr(z )tr( Z ) ] α [ ] 1 tr(xz)tr( X Z)+tr(X Z)tr( XZ) ξ does not run, but new couplings develop beta-functions β i = dα 1 /dlnµ 0 conformal anomaly! β 1 = ( α 1 ξ), β = a(ξ)+α b(ξ)+α 4 c(ξ) Coefficient functions a = ξ 4 +ξ ,b = 4ξ 4 +4ξ ,c = 4 4ξ The theory has two lines of fixed points α 1 = ξ, α = α ± = ± iξ ξ4 3iξ6 4 +ξ 8 ± 65iξ ξ1 10 +O ( ξ 14) In the planar limit, the bi-scalar theory with appropriately tuned double-trace couplings is a genuine non-unitary CFT - p. 5/13

6 Bi-scalar theory at the fixed point tr[x (x)]tr[ X (0)] = x 0 Expected behaviour at the fixed points ξ α ± tr[x (x)]tr[ X (0)] 1 (x ) ± Scaling dimensions of operators at weak coupling Satisfies remarkably simple exact relation ± = iξ ±iξ 6 7i 4 ξ10 +O ( ξ 14) Hint for integrability of the theory ( 4)( ) = 16ξ 4. - p. 6/13

7 Four-point correlation function Exploit the conformal symmetry to compute the four-point correlation function x 1 x 3 tr[x(x 1 )X(x )]tr[ X(x 3 ) X(x 4 )] = x x 4 Is obtained from tr[x (x)]tr[ X (0)] by point splitting the scalar fields G(x 1,x x 3,x 4 ) = G(u,v) x 1 x 34, u = x 1 x 34 x 13 x 4, v = x 14 x 3 x 13 x 4 Admits the conformal partial wave expansion G(u,v) =,S/ Z + C,S u( S)/ g,s (u,v), The sum runs over operators with scaling dimensions and even Lorentz spin S. C,S the OPE coefficient, g,s (u,v) the conformal block For u 0 and v 1, the leading contribution comes from operators O = tr[ X (0)] - p. 7/13

8 Four-point correlation function II Feynman diagrams contributing in the planar limit x 1 x 3 x x 4 Sum of ladder diagrams glued together through double-trace vertices H = V = Integral operators V and H insert quartic vertex and scalar loop, respectively G x 1,x α the double-trace coupling at the fixed point 1 1 α V ξ 4 H x 3,x 4 +(x 1 x ) The operators V and H commute with the generators of the conformal group - p. 8/13

9 Eigenvalues of graph generating kernels V Φ,S (x 1,x ) = δ( )δ S,0 Φ,S (x 1,x ) HΦ,S (x 1,x ) = 1 h,s Φ,S (x 1,x ) Conformal symmetry fixes the form of eigenstates of V and H Φ,S,n (x 10,x 0 ) = tr[x(x 1 )X(x )]O,S,n (x 0 ) = 1 x 1 ( x 1 x 10 x 0 ) ( S)/ ( ) S (n x0 )ln x 0 x 10 All Lorentz indices are projected onto auxiliary light-cone vector n µ The operator O,S,n (x 0 ) carries the scaling dimension = +iν and Lorentz spin S The states Φ,S,n belong to the principal series of the conformal group Eigenvalue of H h,s = 1 ( +S )( +S)( S )( S 4) 16 - p. 9/13

10 Correlation function Decompose the four-point correlation function over the eigenstates x 1 x 3 G(x 1,x x 3,x 4 ) =,S = S 0 x 0 dν h(ν,s) ξ 4 x 4 d 4 x 0 Φ µ 1...µ S ν,s (x 10,x 0 )Φ µ 1...µ S ν,s (x 30,x 40 ) = 1 x 1 x 34 S 0 dν 1 h(ν,s) ξ 4 µ(ν,s) }{{} kinem.factor g +iν,s (u,v) }{{} 4dconf.block The sum runs over the states with = +iν and Lorentz spin S Close the integration contour to the lowest half-plane and pick up residues at h(ν,s) = (ν +S /4)(ν +(S +) /4) = ξ 4, Imν < 0 Two solutions iν = S/+O(ξ 4 ) and iν 4 = (S +)/+O(ξ 4 ) - p. 10/13

11 Exact scaling dimensions x 1 x 3 G(x 1,x x 3,x 4 ) =,S =, 4 S 0 x x 4 Exact scaling dimensions (S) = + (S +1) +1 (S +1) +4ξ 4, 4 (S) = + (S +1) +1+ (S +1) +4ξ 4, Describe conformal operators of twist and 4 Special case: operators with S = 0 (0) = + i ξ 1+ 4ξ 4 +1 = iξ +iξ 6 7i 4 ξ10 +O ( ξ 14) Agrees with the result of explicit calculation at 7 loops! - p. 11/13

12 Exact OPE coefficients The OPE coefficients G(x 1,x x 3,x 4 ) = 1 x 1 x 34 S 0 C,S g,s(u,v)+c 4,S g 4,S(u,v) C,S = πi res ν µ(ν,s) h(ν,s) ξ 4 The residue at the physical pole h(ν,s) = ξ 4 C,S = 4 3 ( 1) S (S +1)Γ ( 1 (S +5)) Γ ( 1 (S + )) [(4 ) +S(S +) ]Γ ( 1 (S +4)) Γ ( 1 (S + 1)) The dependence on the coupling constant enters through the scaling dimensions The operators with zero Lorentz spin ( Γ C,0 = (( 4) +)Γ The exact conformal data for any coupling ξ! ) ( 4 Γ( ) Γ ) ( 1 ) - p. 1/13

13 Conclusions and open questions Strongly γ deformed planar N = 4 SYM has two lines of fixed points The corresponding non-unitary four-dimensional conformal field theory is integrable Closed expression for the four-point correlation function of the simplest protected operators, the exact conformal data Do conformal symmetry and integrability survive in γ deformed planar N = 4 SYM for arbitrary values of the deformation parameters? Does the bi-scalar theory admit a dual AdS description? - p. 13/13