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

Vladimir Kazakov (ENS,Paris)

“CFT and Integrability” Alexey Zamolodchikov Memorial Meeting CQUEST, Seoul, December 17, 2013

Collaborations with Balitsky Gromov Leurent Sobko Volin

N=4 SYM as a superconformal 4d QFT

• 4D Correlators (e.g. for scalars):

• We consider only planar limit. Operators from local fields

scaling dimensions

structure constants

They describe the whole conformal theory via operator product expansion

Gliozzi,Scherk,Olive’77

non-trivial functionsof ‘tHooft coupling

• Global superconformal symmetry PSU(2,2|4)

-scalar multiplet of R-symmetry, - Weyl spinor

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

AdS

• Perturbation theory: summing graphs (“spin chain” integrability helps…)

ZaremboJanik, Wereszczynski

Kazama, Komatsu

Escobedo, Gromov, Vieira, SeverKostov,Serban

V.K., SobkoVieira, Wang

Sobko

AdS

4d QFT

• Main object is hadron-hadron deep inelastic scattering amplitude

Twist-n operators and BFKL limit in QCD

• After Mellin transform we get poles:

• BFKL limit:

related to twist-2 operatorsof spin twist-3:

LipatovFadin, Kuraev, Lipatov

Balitsky, Lipatov

• Naturalness of BFKL limit shows that we should study at any complex value of conformal spin

-2 -1 0 1 2

1

Twist-2 operator in N=4 SYM: analytic continuation w.r.t. spin

• In N=4 SYM, the most popular twist-2 operator from the closed SL(2) sector

• The BFKL LO and NLO dimension of twist-2 operators were extracted from DIS amplitudes (no direct computation in N=4 SYM ! ):

qualitative graph ofpomeron trajectory

which gives the typical BFKL pole singularities:

• All these terms checked directly, in perturbation theory and using integrability!

• Our AdS/CFT spectral equations (P-µ-system) should give exact pomeron trajectory in N=4 SYM

Kotikov, Lipatov

Brower, Polchinski, Strassler, Tan

Bajnok, Janik, LukowskiLukowski, Rej, Velizhanin

• Basic functions of spectral parameter and their analyticity:

Exact spectral P-µ-system (here for twist-2)

• The satisfy the exact P-µ-system of Riemann-Hilbert equations

(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

• Asymptotics directly related to dimension and spin:

tilde means monodromy around branchpoint

• Obtained from the AdS/CFT Y-system for exact anomalous dimensions of planar SYM

Gromov, V.K., Leurent, Volin

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

Conformal operators of twist-N

• The light ray breaks the SO(2,4) symmetry to colinear SO(2,1)~SL(2,R):

• Operators of twist-N (Twist=Dimension-Spin) :

• They are encoded into Taylor expansion of nonlocal light-ray operators:

They form a closed sector under renormalization

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).

• In N=4 SYM they are combined into components of a supermultiplet.

• An example: component with R-charge=0:

• For the case of interest given by Gegenbauer polynomialMakeenko, Ohrndorf

Belitsky, Derkachev,Korchemsky, Manashov

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

where

• 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):

• Our goal is to compute 2-point correlator in BFKL limit when only this gluon term survives.

Balitsky, V.K., Sobko

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

where

• Regularization by Wilson “frame”:

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

• 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

Balitsky, V.K., Sobko

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

• 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

• Propagators of gluons in this background can be explicitly calculated

and they depend on the new, “moving” Wilson line

Diagonalization of BFKL evolution

• Projection of dipole on these eigenfunctions:

• The BFKL kernel transforms w.r.t. principal series irrep of SL(2,C) with conformal weights and the eigenfunctions are:

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

• The accurate answer can be motivated from expansion in partial waves in the limit of big conformal ratios:

conf. transf.

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 operatorswith 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 AdS5 ×S5 )

• NLO BFKL corrections…

• Exact wist-2 dimension from P-µ -system

END