High-energy amplitudes in N=4 SYM

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1 High-energy amplitudes in N=4 SYM I. Balitsky JLAB & ODU Regensburg 15 July 014 I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

2 Outline Regge limit in a conformal theory. High-energy scattering and Wilson lines. Evolution equation for color dipoles. Leading order: BK equation. NLO BK kernel in N = 4. NLO amplitude in N = 4 SYM Conclusions 1 I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 / 50

3 Outline Regge limit in a conformal theory. High-energy scattering and Wilson lines. Evolution equation for color dipoles. Leading order: BK equation. NLO BK kernel in N = 4. NLO amplitude in N = 4 SYM Conclusions 1 Light-ray operators. BFKL representation of 4-point correlation function in N = 4 SYM. DGLAP representation of 4-point correlation function. Anomalous dimensions from DGAP vs BFKL representations. Conclusions I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 / 50

4 Conformal four-point amplitude A(x, y, x, y ) = (x y) 4 (x y ) 4 N c O(x)O (y)o(x )O (y ) O = Tr{Z } (Z = 1 (φ 1 + iφ )) - chiral primary operator In a conformal theory the amplitude is a function of two conformal ratios A = F(R, R ) R = (x y) (x y ) (x x ) (y y ), R = (x y) (x y ) (x y ) (x y) At large N c A(x, y, x, y ) = A(λ; x, y, x, y ) λ = g N c t Hooft coupling I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

5 Conformal four-point amplitude A(x, y, x, y ) = (x y) 4 (x y ) 4 N c O(x)O (y)o(x )O (y ) O = Tr{Z } (Z = 1 (φ 1 + iφ )) - chiral primary operator In a conformal theory the amplitude is a function of two conformal ratios A = F(R, R ) R = (x y) (x y ) (x x ) (y y ), R = (x y) (x y ) (x y ) (x y) At large N c A(x, y, x, y ) = A(λ; x, y, x, y ) λ = g N c t Hooft coupling Our goal is perturbative expansion and resummation of (λ ln s) n at large energies in the next-to-leading approximation (λ ln s) n (c LO n + c NLO n λ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

6 Regge limit in the coordinate space Regge limit: x + ρx +, x + ρx +, y ρ y, y ρ y ρ, ρ z_ x +, x _, y _ y + z_ z + x, x _ y _, y _ Full 4-dim conformal group: A = F(R, r) R = (x y) (x y ) (x x ) (y y ) ρ ρ x + x +y y (x x ) (y y ) r = [(x y) (x y ) (x y) (x y ) ] (x x ) (y y ) (x y) (x y ) [(x y ) x +y + x +y (x y) + x +y (x y) + x +y (x y ) ] (x x ) (y y ) x +x + y y I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

7 4-dim conformal group versus SL(, C) Regge limit: x + ρx +, x + ρx +, y ρ y, y ρ y ρ, ρ z_ x +, x _, y _ y + z_ z + x, x _ y _, y _ Regge limit symmetry: -dim conformal group SL(, C) formed from P 1, P, M 1, D, K 1 and K which leave the plane (0, 0, z ) invariant. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

8 Pomeron in a conformal theory f + (ω) = eiπω 1 sin πω Ω(r, ν) = A(x, y; x, y ) s = i dν f + (ω(λ, ν))f(λ, ν)ω(r, ν)r ℵ(λ,ν)/ - signature factor sin νρ sinh ρ, cosh ρ = r - solution of the eqn ( H3 + ν + 1)Ω(r, ν) = 0 The dynamics is described by: ω(λ, ν) - pomeron intercept, and F(λ, ν) - pomeron residue. L. Cornalba (007) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

9 Pomeron in a conformal theory A(x, y; x, y ) s = i dν f + (ω(λ, ν))f(λ, ν)ω(r, ν)r ω(λ,ν)/ Pomeron intercept ω(ν, λ) is known in two limits: 1. λ 0 : ω(ν, λ) = λ π χ(ν) + λ ω 1 (ν) +... χ(ν) = ψ(1) ψ( 1 + iν) ψ( 1 iν) - BFKL intercept, ω 1 (ν) - NLO BFKL intercept Lipatov, Kotikov (000). λ : AdS/CFT ω(ν, λ) = ν + 4 λ +... = gravition spin, next term - Brower, Polchinski, Strassler, Tan (006) Cornalba, Costa, Penedones (007) Now: two more terms ( λ 1 and λ 3/ ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

10 Pomeron in a conformal theory A(x, y; x, y ) s = i dν f + (ω(λ, ν))f(λ, ν)ω(r, ν)r ω(λ,ν)/ The function F(ν, λ) in two limits: F 0 (ν) = 1. λ 0 : F(ν, λ) = λ F 0 (ν) + λ 3 F 1 (ν) +... π sinh πν 4ν cosh 3 πν Cornalba, Costa, Penedones (007) F 1 (ν) = see below G. Chirilli and I.B. (009). λ : AdS/CFT ω(ν, λ) = π 3 ν 1 + ν sinh πν +... L.Cornalba (007) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

11 Pomeron in a conformal theory A(x, y; x, y ) s = i dν f + (ω(λ, ν))f(λ, ν)ω(r, ν)r ω(λ,ν)/ The function F(ν, λ) in two limits: F 0 (ν) = 1. λ 0 : F(ν, λ) = λ F 0 (ν) + λ 3 F 1 (ν) +... π sinh πν 4ν cosh 3 πν Cornalba, Costa, Penedones (007) F 1 (ν) = see below G. Chirilli and I.B. (009). λ : AdS/CFT ω(ν, λ) = π 3 ν 1 + ν sinh πν +... L.Cornalba (007) We calculate F 1 (ν) (and confirm ω 1 (ν)) using the expansion of high-energy amplitudes in Wilson lines (color dipoles) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

12 Light-cone expansion and DGLAP evolution in the NLO k > µ k < µ µ - factorization scale (normalization point) k > µ - coefficient functions k < µ - matrix elements of light-ray operators (normalized at µ ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

13 Light-cone expansion and DGLAP evolution in the NLO k > µ k < µ µ - factorization scale (normalization point) k > µ - coefficient functions k < µ - matrix elements of light-ray operators (normalized at µ ) OPE in light-ray operators (x y) 0 T{j µ (x)j ν (y)} = x [ ξ π x α ] s π (ln x µ + C) ψ(x)γµ γ ξ γ ν [x, y]ψ(y) + O( 1 x ) [x, y] Pe ig 1 0 du (x y)µ A µ(ux+(1 u)y) - gauge link I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

14 Light-cone expansion and DGLAP evolution in the NLO k > µ k < µ µ - factorization scale (normalization point) k > µ - coefficient functions k < µ - matrix elements of light-ray operators (normalized at µ ) Renorm-group equation for light-ray operators DGLAP evolution of parton densities (x y) = 0 µ d dµ ψ(x)[x, y]ψ(y) = K LO ψ(x)[x, y]ψ(y) + αs K NLO ψ(x)[x, y]ψ(y) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

15 Four steps of an OPE To factorize an amplitude into a product of coefficient functions and matrix elements of relevant operators. To find the evolution equations of the operators with respect to factorization scale. To solve these evolution equations. To convolute the solution with the initial conditions for the evolution and get the amplitude I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

16 DIS at high energy: relevant operators At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): d k A(s) = 4π IA (k ) B Tr{U(k )U ( k )} B [ ] U(x ) = Pexp ig du n µ A µ (un + x ) Wilson line I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

17 DIS at high energy: relevant operators At high energies, particles move along straight lines the amplitude of γ A γ A scattering reduces to the matrix element of a two-wilson-line operator (color dipole): d k A(s) = 4π IA (k ) B Tr{U(k )U ( k )} B [ ] U(x ) = Pexp ig du n µ A µ (un + x ) Wilson line Formally, means the operator expansion in Wilson lines I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

18 Rapidity factorization Y > Y < η - rapidity factorization scale Rapidity Y > η - coefficient function ( impact factor ) Rapidity Y < η - matrix elements of (light-like) Wilson lines with rapidity divergence cut by η [ ] = Pexp ig dx + A η + (x +, x ) U η x A η µ(x) = d 4 k (π) 4 θ(eη α k )e ik x A µ (k) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

19 Spectator frame: propagation in the shock-wave background. Boosted Field Each path is weighted with the gauge factor Pe ig dx µa µ. Quarks and gluons do not have time to deviate in the transverse space we can replace the gauge factor along the actual path with the one along the straight-line path. z z x y Wilson Line [ x z: free propagation] [U ab (z ) - instantaneous interaction with the η < η shock wave] [ z y: free propagation ] I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

20 High-energy expansion in color dipoles Y > Y < The high-energy operator expansion is T{ĵ µ (x)ĵ ν (y)} = d z 1 d z Iµν LO (z 1, z, x, y)tr{ûz η 1 Û z η } + NLO contribution I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

21 High-energy expansion in color dipoles Y > Y < η - rapidity factorization scale Evolution equation for color dipoles d dη tr{uη x U η y } = α s π d (x y) z (x z) (y z) [tr{uη x U y η }tr{ux η U η N c tr{u η x U η y }] + α s K NLO tr{u η x U η y } + O(α s ) (Linear part of K NLO = K NLO BFKL ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50 y }

22 Evolution equation for color dipoles To get the evolution equation, consider the dipole with the rapidies up to η 1 and integrate over the gluons with rapidities η 1 > η > η. This integral gives the kernel of the evolution equation (multiplied by the dipole(s) with rapidities up to η ). α s (η 1 η )K evol I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

23 Evolution equation in the leading order d dη Tr{Û x Û y} = K LO Tr{Û x Û y} +... d dη Tr{Û x Û y} shockwave = K LO Tr{Û x Û y} shockwave x a x * x x x x * * x a x * x b a a b y b (a) (b) (c) (d) b U ab z = Tr{t a U z t b U z } (U x U y) η 1 (U x U y) η 1 + α s (η 1 η )(U x U z U z U y) η Evolution equation is non-linear I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

24 Non linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

25 Non-linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) LLA for DIS in pqcd BFKL (LLA: α s 1, α s η 1) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

26 Non-linear evolution equation Û(x, y) 1 1 N c Tr{Û(x )Û (y )} BK equation d dη Û(x, y) = α sn c π d z (x y) } {Û(x, z) + (x z) (y z) Û(z, y) Û(x, y) Û(x, z)û(z, y) I. B. (1996), Yu. Kovchegov (1999) Alternative approach: JIMWLK ( ) LLA for DIS in pqcd BFKL (LLA: α s 1, α s η 1) LLA for DIS in sqcd BK eqn (LLA: α s 1, α s η 1, α s A 1/3 1) (s for semiclassical) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

27 Conformal invariance of the BK equation Formally, a light-like Wilson line [ p 1 + x, p 1 + x ] = Pexp { ig } dx + A + (x +, x ) is invariant under inversion (with respect to the point with x = 0). I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

28 Conformal invariance of the BK equation Formally, a light-like Wilson line [ p 1 + x, p 1 + x ] = Pexp { ig } dx + A + (x +, x ) is invariant under inversion (with respect to the point with x = 0). Indeed, (x +, x ) = x after the inversion x x /x and x+ x + /x I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

29 Conformal invariance of the BK equation Formally, a light-like Wilson line [ p 1 + x, p 1 + x ] = Pexp { ig } dx + A + (x +, x ) is invariant under inversion (with respect to the point with x = 0). Indeed, (x +, x ) = x after the inversion x x /x and x+ x + /x { [ p 1 +x, p 1 +x ] Pexp ig d x+ x A + ( x+ x, x } x ) = [ p 1 + x x, p 1 + x x ] I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

30 Conformal invariance of the BK equation Formally, a light-like Wilson line [ p 1 + x, p 1 + x ] = Pexp { ig } dx + A + (x +, x ) is invariant under inversion (with respect to the point with x = 0). Indeed, (x +, x ) = x after the inversion x x /x and x+ x + /x { [ p 1 +x, p 1 +x ] Pexp ig d x+ x A + ( x+ x, x } x ) = [ p 1 + x x, p 1 + x x ] The dipole kernel is invariant under the inversion V(x ) = U(x /x ) d dη Tr{V xv y } = α s d z (x y) z 4 π z 4 (x z) (z y) [Tr{V xv z }Tr{V z V y } N c Tr{V x V y }] I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

31 Conformal invariance of the BK equation SL(,C) for Wilson lines Ŝ i (K1 + ik ), Ŝ 0 i (D + im1 ), Ŝ + i (P1 ip ) 1 [Ŝ 0, Ŝ ± ] = ±Ŝ ±, [Ŝ +, Ŝ ] = Ŝ 0, [Ŝ, Û(z, z)] = z z Û(z, z), [Ŝ 0, Û(z, z)] = z z Û(z, z), [Ŝ +, Û(z, z)] = z Û(z, z) z z 1 + iz, z z 1 + iz, U(z ) = U(z, z) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 / 50

32 Conformal invariance of the BK equation SL(,C) for Wilson lines Ŝ i (K1 + ik ), Ŝ 0 i (D + im1 ), Ŝ + i (P1 ip ) 1 [Ŝ 0, Ŝ ± ] = ±Ŝ ±, [Ŝ +, Ŝ ] = Ŝ 0, [Ŝ, Û(z, z)] = z z Û(z, z), [Ŝ 0, Û(z, z)] = z z Û(z, z), [Ŝ +, Û(z, z)] = z Û(z, z) z z 1 + iz, z z 1 + iz, U(z ) = U(z, z) Conformal invariance of the evolution kernel d dη [Ŝ, Tr{U x U y}] = α sn c π dz K(x, y, z)[ŝ, Tr{U x U z }Tr{U z U y}] [ x x + y y + ] z K(x, y, z) = 0 z I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 / 50

33 Conformal invariance of the BK equation SL(,C) for Wilson lines Ŝ i (K1 + ik ), Ŝ 0 i (D + im1 ), Ŝ + i (P1 ip ) 1 [Ŝ 0, Ŝ ± ] = ±Ŝ ±, [Ŝ +, Ŝ ] = Ŝ 0, [Ŝ, Û(z, z)] = z z Û(z, z), [Ŝ 0, Û(z, z)] = z z Û(z, z), [Ŝ +, Û(z, z)] = z Û(z, z) z z 1 + iz, z z 1 + iz, U(z ) = U(z, z) Conformal invariance of the evolution kernel d dη [Ŝ, Tr{U x U y}] = α sn c π dz K(x, y, z)[ŝ, Tr{U x U z }Tr{U z U y}] [ x x + y y + ] z K(x, y, z) = 0 z In the leading order - OK. In the NLO -? I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July 014 / 50

34 Expansion of the amplitude in color dipoles in the NLO Y > Y < The high-energy operator expansion is T{Ô(x)Ô(y)} = d z 1 d z I LO (z 1, z )Tr{Ûz η 1 Û z η } + d z 1 d z d z 3 I NLO (z 1, z, z 3 )[ 1 Tr{T n Ûz η N 1 Û z η 3 T n Ûz η 3 Û z η } Tr{Ûz η 1 Û z η }] c In the leading order - conf. invariant impact factor I LO = x + y + π Z1, Z i (x z i) (y z i) CCP, 007 Z x + y + I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

35 NLO impact factor z z 1 z z z 1 z y z 3 x y z 3 x z z (a) I NLO (x, y; z 1, z, z 3 ; η) = I LO λ z [ 13 π z ln σs 1 z 3 4 Z 3 iπ ] + C The NLO impact factor is not Möbius invariant the color dipole with the cutoff η is not invariant However, if we define a composite operator (a - analog of µ for usual OPE) (b) [Tr{Ûz η 1 Û z η } ] conf + λ π = Tr{Û η z 1 Û η z } d z 1 z 3 z [Tr{T n Û 13 z z η 1 Û z η 3 T n Ûz η 3 Û z η } N c Tr{Ûz η 1 Û z η }] ln az 1 3 z 13 z 3 + O(λ ) the impact factor becomes conformal in the NLO. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

36 Operator expansion in conformal dipoles T{Ô(x)Ô(y)} = + d z 1 d z I LO (z 1, z )Tr{Û η z 1 Û η z } conf d z 1 d z d z 3 I NLO (z 1, z, z 3 )[ 1 N c Tr{T n Û η z 1 Û η z 3 T n Û η z 3 Û η z } Tr{Û η z 1 Û η z }] I NLO = I LO λ π z [ 1 dz 3 z 13 z 3 ln z 1 eη as ] z Z 13 z 3 iπ + C 3 The new NLO impact factor is conformally invariant Tr{Û η z 1 Û η z } conf is Möbius invariant We think that one can construct the composite conformal dipole operator order by order in perturbation theory. Analogy: when the UV cutoff does not respect the symmetry of a local operator, the composite local renormalized operator in must be corrected by finite counterterms order by order in perturbaton theory. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

37 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

38 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

39 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) We calculate the matrix element of the r.h.s. in the shock-wave background α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

40 Definition of the NLO kernel In general d dη Tr{Û x Û y} = α s K LO Tr{Û x Û y} + α s K NLO Tr{Û x Û y} + O(α 3 s ) α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) We calculate the matrix element of the r.h.s. in the shock-wave background α s K NLO Tr{Û x Û y} = d dη Tr{Û x Û y} α s K LO Tr{Û x Û y} + O(α 3 s ) Subtraction [ ] of the (LO) contribution (with the rigid rapidity cutoff) 1 v prescription in the integrals over Feynman parameter v Typical integral dv 1 [ 1 ] (k p) v + p (1 v) v = 1 + p ln (k p) p I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

41 Gluon part of the NLO BK kernel: diagrams (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) (XI) (XII) (XIII) (XIV) (XV) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

42 Diagrams for 1 3 dipoles transition (XVI) (XVII) (XVIII) (XIX) (XX) (XXI) (XXII) (XXIII) (XXIV) (XV) (XXVI) (XXVII) (XVIII) (XXIX) (XXX) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

43 Diagrams for 1 3 dipoles transition (XXXI) (XXXII) (XXXIII) (XXXIV) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

44 "Running coupling" diagrams x y (I) (II) (III) (IV) (V) (VI) (VII) (VIII) (IX) (X) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

45 1 dipole transition diagrams a x x a x x x a x x x x a * * a x * * * q q q q q b b b c c b b k k k k k k k k k k c d (a) c d d (b) (c) d c (d) (e) x x x x x * x x x x x * * * * a a a b q q q b a q a k q k k k d d d k d c k k k k c c k c c (f) (g) b (h) b (i) b (j) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

46 Gluino and scalar loops I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

47 Evolution equation for color dipole in N = 4 (I.B. and G. Chirilli) d dη Tr{Ûη z 1 Û η = α s π z } d z { 1 z 3 z 13 z 3 [Tr{T a Ûz η 1 Û z η 3 T a Ûz η 3 Û η α s d z 3 d z 4 z 1 z 34 4π 4 1 α sn c 4π [ π 3 + ln z 13 z 1 z } N c Tr{Ûz η 1 Û z η }] [ 1 + z 1 z 34 ln z ]} 3 z 1 ] ln z 13 z 4 z 14 z 3 z 4 34 z 13 z 4 z 13 z 4 z 3 z 14 Tr{[T a, T b ]Ûz η 1 T a T b Û z η + T b T a Ûz η 1 [T b, T a ]Û z η }(Ûz η 3 ) aa (Ûz η 4 Ûz η 3 ) bb NLO kernel = Non-conformal term + Conformal term. Non-conformal term is due to the non-invariant cutoff α < σ = e η in the rapidity of Wilson lines. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

48 Evolution equation for color dipole in N = 4 (I.B. and G. Chirilli) d dη Tr{Ûη z 1 Û η = α s π z } d z { 1 z 3 z 13 z 3 [Tr{T a Ûz η 1 Û z η 3 T a Ûz η 3 Û η α s d z 3 d z 4 z 1 z 34 4π 4 1 α sn c 4π [ π 3 + ln z 13 z 1 z } N c Tr{Ûz η 1 Û z η }] [ 1 + z 1 z 34 ln z ]} 3 z 1 ] ln z 13 z 4 z 14 z 3 z 4 34 z 13 z 4 z 13 z 4 z 3 z 14 Tr{[T a, T b ]Ûz η 1 T a T b Û z η + T b T a Ûz η 1 [T b, T a ]Û z η }(Ûz η 3 ) aa (Ûz η 4 Ûz η 3 ) bb NLO kernel = Non-conformal term + Conformal term. Non-conformal term is due to the non-invariant cutoff α < σ = e η in the rapidity of Wilson lines. For the conformal composite dipole the result is Möbius invariant I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

49 Evolution equation for composite conformal dipoles in N = 4 d [ Tr{Û z η dη 1 Û z η } ] conf = α s π α s 4π 4 d z 3 z 1 z 13 z 3 [ 1 α sn c 4π d z 3 d z 4 z 1 z 13 z 4 z 34 π ] [Tr{T a Û η z1 3 Û η z 3 T a Û z3 Û z η } N c Tr{Ûz η 1 Û z η } ] conf ] { ln z 1 z 34 z + 14 z 3 [ 1 + z 1 z 34 ln z 13 z } 4 z 14 z 3 z 13 z 4 z 14 z 3 Tr{[T a, T b ]Ûz η 1 T a T b Û z η + T b T a Ûz η 1 [T b, T a ]Û z η }[(Ûz η 3 ) aa (Ûz η 4 ) bb (z 4 z 3 )] Now Möbius invariant! I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

50 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y (x y) 4 (x y ) 4 T{Ô(x)Ô (y)ô(x )Ô (y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) a 0 = x+y+ (x y), b 0 = x y (x y ) impact factors do not scale with energy all energy dependence is contained in [DD] a0,b0 (a 0 b 0 = R) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

51 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y (x y) 4 (x y ) 4 T{Ô(x)Ô (y)ô(x )Ô (y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) Impact factor (R = (x y) z 1 x +y +Z 1Z - conf. ratio) IF a0 = dν dz 0 R 1 +iν( 1 + α sn c [π π 3 + 4χ(ν) ν π ] ) cosh U a0 (z 0, ν) πν U a (z 0, ν) = 1 d z 1 d z ( z ) 1 1 iν π z 4 1 z U(z 1, z ), U 1 1 Tr{U z1 U 10 z z 0 N } c I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

52 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y (x y) 4 (x y ) 4 T{Ô(x)Ô (y)ô(x )Ô (y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) Dipole-dipole scattering χ(γ) C ψ(γ) ψ(1 γ), γ 1 + iν ( z ) 1 1 [DD] = dν dz +iν ( z ) 1 1 iν 0 z 10 z 0 z D( 1 + iν; λ)rω(ν)/ 10 z 0 Γ( γ)γ(γ 1) { D(γ; λ) = 1 + α [ sn c 4χ(ν) Γ(1 + γ)γ( γ) π 1 + 4ν π 3 + iπ N c 4 ] } Nc + O(g 4 ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

53 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y (x y) 4 (x y ) 4 T{Ô(x)Ô (y)ô(x )Ô (y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) Result : F(ν) = N c 4π 4 αs Nc 1 cosh πν (G.A. Chirilli and I.B.) { 1 + α [ sn c π π π cosh πν ν + iπ N c 4 ]} Nc I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

54 Conclusions High-energy operator expansion in color dipoles works at the NLO level. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

55 Conclusions High-energy operator expansion in color dipoles works at the NLO level. The NLO BK kernel in for the evolution of conformal composite dipoles in N = 4 SYM is Möbius invariant in the transverse plane. The NLO BK kernel agrees with NLO BFKL equation. The correlation function of four Z operators is calculated at the NLO BFKL order. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

56 NLO BFKL and anomalous dimensions of light-ray operators Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

57 NLO BFKL and anomalous dimensions of light-ray operators Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(n, α s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

58 NLO BFKL and anomalous dimensions of light-ray operators Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(n, α s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) ω = n 1 0 corresponds to F i a ω 1 Fai - some non-local operator. I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

59 NLO BFKL and anomalous dimensions of light-ray operators Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(n, α s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) ω = n 1 0 corresponds to F i a ω 1 Fai - some non-local operator. Q: which one? A: gluon light-ray (LR) operator I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

60 Light-ray operators and parton densities Gluon light-ray (LR) operator of twist F a i(x + + x )[x +, x + ] ab F b i (x + + x ) Forward matrix element - gluon parton density 1 z µ z ν p Fµξ(z)[z, a 0] ab Fν bξ (0) p µ z =0 = (pz) dx B x B D g (x B, µ) cos(pz)x B Evolution equation (in gluodynamics) µ d dµ Fa i(x + + x )[x +, x + ] ab F b i (x + + x ) = x + x + dz + z + x + 0 dz + K(x +, x + ; z +, z + ; α s )F a i(z + + x )[z +, z + ] ab F b i (z + + x ) Forward LR operator F(L +, x ) = dx + F i(l a + + x + + x )[L + + x +, x + ] ab F (x bi + + x ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

61 Light-ray operators and parton densities Expansion in ( forward ) local operators F(L +, x ) = n= Evolution equation for F(L +, x ) L n + (n )! Og n(x ), O g n dx + F i a n Fai (x +, x ) µ d dµ F(L +, x ) = γ n (α s ) = u 1 K gg - DGLAP kernel du K gg (u, α s )F(uL +, x ) du u n K gg (u, α s ) µ d dµ Og n = γ n (α s )O g n u 1 K gg (u) = α ( [ sn c 1 ūu + π ūu] + 11 ) + 1 δ(ū) + higher orders in α s I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

62 Light-ray operators Conformal LR operator (j = 1 + iν) F µ (L +, x ) = F µ j (x ) = 0 dν π (L +) 3 +iν F µ 1 dl + L 1 j + F µ (L +, x ) +iν(x ) Evolution equation for forward conformal light-ray operators µ d dµ F j(z ) = 1 γ j (α s ) is an analytical continuation of γ n (α s ) 0 du K gg (u, α s )u j F j (z ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

63 Supermultiplet of LR operators Gluino and scalar LR operators Λ(L +, x ) = i dx + [ λ a (L + + x + + x )[x + + x +, x + ] ab σ λ b (x + + x ) + c.c] Φ(L +, x ) = dx + φ a,i (L + + x + + x )[x + + x +, x + ] ab φ b,i (x + + x ) Λ j (x ) = 0 SU 4 singlet LR operators. dl + L j+1 + Λ(L +, x ), Φ j (x ) = 0 dl + L j+1 + Φ(L +, x ) S 1j (x ) = F j (x ) + j 1 8 Λ j(j 1) j(x ) Φ j (x ) 8 S j (x ) = F j (x ) 1 8 Λ j(j + 1) j(x ) + Φ j (x ) 4 S 3j (x ) = F j (x ) j + 4 Λ (j + 1)(j + ) j(x ) Φ j (x ) 8 All operators have the same anomalous dimension γ S 1 j (α s ) γ j (α s ) = α s π N c[ψ(j 1) + C] + O(αs ), γ S j = γ S 1 j+1, γs 3 j = γ S 1 j+ I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

64 Three-point CF of LR operator and two locals Three-point correlation function of local operators (O = φ a I φa I - Konishi operator) (µ z 3) γ K S 1n (z 1 )O(z )O(z 3 ) = c n [1 + ( 1) n ] z 1 z 13 z 3 ( z1 z 1 z 13 z 13 x3 (µ x3 ) γ K dx 1+ S 1n (x 1+, x 1 )O(x, x )O(x 3, x 3 ) ) n ( µ z 3 z 1 z 13 ) 1 γ(n,αs) = πi c n[1 + ( 1) n ] Γ(1 + n + γ n ) ( x3 ) ( 1+ γn x 1 x3 4 Γ (1 + n + 1 γ + x ) 13 1 n γn n) x x 3 x x 3 CF of light-ray operator and two local operators x 3 (µ x 3 ) γ K S 1j (x 1 )O(x, x )O(x 3, x 3 ) = πi c j[1 + e iπj ] Γ(1 + j + γ j ) ( x3 ) γ 1+ j ( x x3 4 Γ (1 + j γ + x ) 13 1 j γj j) x x 3 x x 3 I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

65 Light-cone OPE CF of light-ray operator and two local operators in forward kinematics x3 (µ x3 ) γ K dx 3 S µ j (x 1 )O(L + x 3, x )O(x 3, x 3 ) = 1 0 du c(j, α s)[1 + e iπj ]L j [ (ūu)j ( ūuµ x ) 1 3 γ(j,αs) x 1 u + x13 ū] 1+j [x1 u + x13 ū] I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

66 Light-cone OPE CF of light-ray operator and two local operators in forward kinematics x3 (µ x3 ) γ K dx 3 S µ j (x 1 )O(L + x 3, x )O(x 3, x 3 ) = 1 0 du c(j, α s)[1 + e iπj ]L j [ (ūu)j ( ūuµ x ) 1 3 γ(j,αs) x 1 u + x13 ū] 1+j [x1 u + x13 ū] Limit x 3 0 (µ ) γ K dx 3 S j + (x 1 )O(L + x 3, x )O(x 3, x 3 ) = Γ( 1 + j + γ ) c(j, α s )[1 + e iπj ]L j Γ( + j + γ) (x1 ) 1+j (µ x1 (µ )γ(j,αs) ) γ(j,αs) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

67 Light-cone OPE Light-ray operators in (+) and (-) directions: Φ + (L +, x ) = dx + φ a,i (L + + x + + x )[x + + x +, x + ] ab φ b,i (x + + x ) Φ + j (x ) = dl + L j+1 + Φ(L +, x ) 0 Φ (L, x ) = dx + φ a,i (L + x + x )[x + x, x ] ab + φ b,i (x + x ) Φ j (x ) = 0 dl L j+1 + Φ(L, x ) and similarly for Λ j s and G j s CF of two LRs : S + j= 3 +iν(x 1 )S j = 3 +iν (x 3 ) = δ(ν ν )a(j, α s ) (x 13 ) j+1 (x 13 µ ) γ(j,αs) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

68 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = 1 +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

69 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = 1 +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

70 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = 1 +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = 1 +i dj C(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) 1 γ(j,αs) S µ j (x )O(L + x 4, x 3 )O(x 4, x 4 ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

71 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = 1 +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = = 1 +i dj C(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) 1 1 +i dj C(j, α s ) 1 i 1 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j γ(j,αs) S µ j (x )O(L + x 4, x 3 )O(x 4, x 4 ) x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

72 Light-cone limit of Cornalba s formula Cornalba s formula as x 1 0 [x 1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π L + L 1 0 tanh πν ( x dudv dν 1 x 34 ūu vv ) 1 +iν ( L + L ūu vv ν cosh πν [x13 v + x 14 v] x 1 x 34 ) ℵ(ν)/f+ = iα s 8 π 1 1 +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] ℵ(ξ) (L + L ) 1+ℵ(ξ) = BFKL representation of the amplitude in the (Regge) + (x1 0) limit I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

73 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = 1 +i dj C(j, α s ) 1 i 1 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

74 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = 1 +i dj C(j, α s ) 1 i 1 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) BFKL result (ℵ(ξ, α s ) = αsnc π χ(ξ) pomeron intercept) [x1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π 1 1 +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] ℵ(ξ) (L + L ) 1+ℵ(ξ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

75 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = 1 +i dj C(j, α s ) 1 i 1 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) BFKL result (ℵ(ξ, α s ) = αsnc π χ(ξ) pomeron intercept) [x1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π 1 1 +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] We compare these results around ξ j 1 0 ℵ(ξ) ℵ(ξ)ℵ (ξ) α sn c πξ +ζ(3)α s ξ 1 + ℵ(ξ, α s ) = j and γ(j, α s ) = ξ ℵ(ξ) +... γ j = α sn c π(j 1) ℵ(ξ) (L + L ) 1+ℵ(ξ) s N c α + [0 + ζ(3)(j 1)]( π(j 1) )) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

76 Result γ j = α sn c π(j 1) s N c α + [0 + ζ(3)(j 1)]( π(j 1) )) is an anomalous dimension of light-ray operator F j F(x ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

77 Conclusion and Outlook 1 Conclusion: NLO BFKL gives anomalous dimensions of light-ray operators F +i ω 1 F+ i as ω 0 (in all orders in α s ) I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50

78 Conclusion and Outlook 1 Conclusion: NLO BFKL gives anomalous dimensions of light-ray operators F +i ω 1 F+ i as ω 0 (in all orders in α s ) Outlook In QCD 3-point CF of F ω1 1 F(x 1 ) F ω 1 F(x ) F ω3 1 F(x 3 ) as ω 1, ω, ω 3 0 (joint project with V. Kazakov and E. Sobko): I. Balitsky (JLAB & ODU) High-energy amplitudes in N=4 SYM Regensburg 15 July / 50