“CFT and Integrability”

Alexey Zamolodchikov Memorial Meeting CQUEST, Seoul, December 17, 2013

On correlation functions and anomalous dimensions of planar N=4 SYM theory: twist-2 operators and BFKL

Vladimir Kazakov (ENS,Paris) Collaborations with Balitsky Gromov Leurent Sobko Volin

N=4 SYM as a superconformal 4d QFT

Gliozzi,Scherk,Olive’77 -scalar multiplet of R-symmetry, - Weyl spinor • • Global superconformal symmetry PSU(2,2|4) We consider only planar limit. Operators from local fields • 4D Correlators (e.g. for scalars): scaling dimensions structure constants non-trivial functions of ‘tHooft coupling They describe the whole conformal theory via operator product expansion

Correlation functions in weak and strong coupling

• Strong coupling limit (AdS/CFT duality to strings): The problem of computing n-point correlators reduce to finding minimal surface in AdS space Zarembo Janik, Wereszczynski Kazama, Komatsu • Perturbation theory: summing graphs (“spin chain” integrability helps…) Escobedo, Gromov, Vieira, Sever Kostov, Serban V.K., Sobko Vieira, Wang Sobko

• Twist-n operators and BFKL limit in QCD Lipatov Fadin, Kuraev, Lipatov Balitsky, Lipatov Main object is hadron-hadron deep inelastic scattering amplitude related to twist-2 operators of spin • After Mellin transform we get poles: • BFKL limit: twist-3:

Twist-2 operator in N=4 SYM: analytic continuation w.r.t. spin • Naturalness of BFKL limit shows that we should study at any complex value of conformal spin • In N=4 SYM, the most popular twist-2 operator from the closed SL(2) sector qualitative graph of pomeron trajectory Brower, Polchinski, Strassler, Tan • The BFKL LO and NLO dimension of twist-2 operators were extracted from DIS amplitudes (no direct computation in N=4 SYM ! ): 1 -2 -1 0 1 2 Kotikov, Lipatov which gives the typical BFKL pole singularities: • • All these terms checked directly, in perturbation theory and using integrability!

Bajnok, Janik, Lukowski Lukowski, Rej, Velizhanin Our AdS/CFT spectral equations (P-µ-system) should give exact pomeron trajectory in N=4 SYM

Gromov, V.K., Leurent, Volin

Exact spectral P-µ-system (here for twist-2) • • Obtained from the AdS/CFT Y-system for exact anomalous dimensions of planar SYM

Gromov, V.K., Vieira Cavaglia, Fioravanti, Mattelliano, Tateo

Basic functions of spectral parameter and their analyticity: • Asymptotics directly related to dimension and spin: • The satisfy the exact P-µ-system of Riemann-Hilbert equations tilde means monodromy around branchpoint • • (quasi- -periodicity) Ready for study numerically and analytically in various approzimations, W.C., S.C., BFKL But today I will explain direct computation of 2-point correlators in BFKL limit

Conformal operators of twist-N

• Operators of twist-N (Twist=Dimension-Spin) : They form a closed sector under renormalization • They are encoded into Taylor expansion of nonlocal light-ray operators: • The light ray breaks the SO(2,4) symmetry to colinear SO(2,1)~SL(2,R): with conformal spin For scalars, fermions and gluons

Conformal operators of twist=2

• Twist-2 operators (N=2) of conformal spin is constructed as a highest weight of sl(2,R). • For the case of interest given by Gegenbauer polynomial Makeenko, Ohrndorf • In N=4 SYM they are combined into components of a supermultiplet.

• An example: component with R-charge=0: Belitsky, Derkachev,Korchemsky, Manashov

Balitsky, V.K., Sobko

Leading twist=2 light-ray operators

• We want to continue analytically to non-integer spins so that for integer values the correlators were the same.

It is natural to use the the principal series irreps of with • One such operator, generalizing our twist-2, given by light-ray operators where each of 3 terms is a light-ray operator, e.g. for gluons (last term): where • Our goal is to compute 2-point correlator in BFKL limit when only this gluon term survives.

Regularization by Wilson “frame”

• Light ray operators contain singularities (coinciding local operators) and should be regularized. It will be particularly useful for BFKL limit. • Regularization by Wilson “frame”: where

Correlation function of two light-ray twist-2 operators in BFKL regime

Balitsky, V.K., Sobko • We calculate the correlation function of two such operators placed along the light-cone directions in the BFKL limit • Then we do the OPE in the limit Parameterization:

Reduction to pure Wilson frame

• We could proceed directly with computation of this correlator but, to simplify it, we notice that for the Wilson frame operator: the field strength at the ends can be produced by transverse derivatives: or • So we can simply calculate the correlators of pure Wilson frames!

Correlation function of two Wilson frames

• The main contribution will come from large lengths of frames and, effectively, the frames can be replaced by color dipoles of Balitsky.

The dependence in will reappear in the cut-off of BFKL evolution. • The objects on the r.h.s. are infinite Wilson frames, or color dipoles where • We regularized the gauge field by cutting off high momenta

BFKL evolution of colour dipole

• Evolution of cutoff is possible because in the BFKL kinematics, gluon “ladder” is ordered w.r.t. the rapidities (or cutoffs).

• Cutoff will be related to the shapes of frames and their distances. • In BFKL evolution (renormalization), fast fields with should be intgegrated out, in the background fields with • Due to the boost, the slow background field will be seen by the fast field as a thin “pancake” spread in orthogonal direction.

BFKL evolution of color dipole

• Propagators of gluons in this background can be explicitly calculated and they depend on the new, “moving” Wilson line • Evolution w.r.t. the cutoff can be written in the form of BFKL equation: where the LO BFKL kernel acting on transverse coordinates of dipole is

Diagonalization of BFKL evolution

• The BFKL kernel transforms w.r.t. principal series irrep of SL(2,C) with conformal weights and the eigenfunctions are: • Projection of dipole on these eigenfunctions: inverse transform • Evolution of the Furrier mode of the dipole: where are eigenvalues of BFKL kernel with explicitly know NLO BFKL correction

Correlator of dipoles: from small to any cutoff

• For small cutoffs the correlators of dipoles is given by a one loop calculation: • For arbitrary cutoff, we add the BFKL evolution • The final correlator of dipoles is schematically given by

Choice of cutoff

(the subtlest issue!) • Using orthogonality of BFKL eigenfunctions (characters of SL(2,C) we get for correlators of colour dipoles • In the limit of thin frames, the conform transformation almost does not deform the configuration of frames and cutoffs should depends on two conformal ratios • conf. transf. The accurate answer can be motivated from expansion in partial waves in the limit of big conformal ratios:

Cornalba, Costa, Penedones; Balitski, Cirilli

Correlator of twist-2: final results

• Integrating over lengths of frames and positions and using our observation on relation between empty Wilson frame operator and the light-ray operator with field strengths we obtain in the limit of thin frames: • Inn the numerator ther are standard powers of short distances which should appear in the OPE. The coefficient in front of it is our final result – the correlation function of twist-2 operators in the BFKL limit: • Weak coupling limit it gives

Correlator of twist-2 light-ray operators with explicit field insertions

• For normalization of correlators we now have: • Weak coupling limit it gives

Conclusions

• • We constructed the analytic continuation of twist-2 operators w.r.t. conformal spin in terms of nonlocal light ray operators transforming w.r.t. the collinear conformal algebra SL(2,R) We gave the direct computation of 2-point correlators of a twist-2 operator in N=4 SYM in BFKL approximation. The coordinate dependence (anomalous dimension) is fixed up to NLO and the normalization coefficient -- up to LO. This coefficient is important for fixing the normalization of operators, to use the same regularization scheme for 3-point correlators • AdS/CFT quantum spectral curve (P-µ -system) describes the exact dimension (pomeron Regge trajectory) of twist-2 operator at any N=4 SYM ‘tHooft coupling Future directions • • • • Staring point for computing the 3-point correlators and structure functions in LO BFKL.

The main ingredient – 3-pomeron vertex – is known!

BFKL limit from exact P-µ -system (quantum spectral curve of AdS 5 ×S 5 ) NLO BFKL corrections… Exact wist-2 dimension from P-µ -system

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