NLO BFKL and Anomalous Dimensions of Light-Ray Operators§

Ian Balitsky*
Physics Department, ODU, Norfolk VA 23529, USA and Theory Group, Jlab, 12000 Jefferson Ave, Newport News, VA 23606, USA

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© Ian Balitsky; Licensee Bentham Open.

open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.

* Address correspondence to this author at the Theory Group, Jlab, 12000 Jefferson Ave, Newport News, VA 23606, USA; E-mail:
§ Presented at the Low x workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel


The anomalous dimensions of light-ray operators of twist two are obtained by analytical continuation of the anomalous dimensions of corresponding local operators. I demonstrate that the asymptotics of these anomalous dimensions at the “BFKL point”. j→1 can be obtained by comparing the light-cone operator expansion with the highenergy expansion in Wilson lines.

PACS number(s): 12.38.Bx, 12.38.Cy.

Keywords: Anomalous dimension, BFKL, light-ray operators, QCD, Wilson lines.


It is well known that the BFKL pomeron [1] gives the anomalous dimensions of leading-twist gluon operators at all orders near the unphysical point with number of covariant derivatives equal to minus one. The exact statement is that the analytical continuation of anomalous dimension of twist-two gluon operator


to the point is determined by the BFKL equation. The anomalous dimensions are singular at that point and there is a new hierarchy of perturbation theory . In the leading order the relation to the BFKL pomeron was established long ago [2] and for the next-to-leading order BFKL it was done in papers [3]. This constitutes a powerful check for explicit calculations of higher-order anomalous dimensions both in QCD [4] and =4 SYM [5]. However, the explicit meaning of these “local operators at the unphysical point” was somewhat obscure. I argue that these operators should be understood as gluon light-ray operators and BFKL equation gives the anomalous dimensions of these light-ray operators at some specific point.

First, let me remind the standard argument about analytical continuation. It is well known that the moments of structure functions are proportional to matrix elements of twist-2 operators. The behavior of structure functions are governed by anomalous dimensions of these operators


As I mentioned, the BFKL pomeron determines the asymptotics of these anomalous dimension at the “non-physical” point . The standard argument goes like that: in the framework of the BFKL approach the amplitude of, say, virtual scattering has the form

NLO BFKL and Anomalous Dimensions of Light-Ray Operators§ 66

where is the pomeron intercept (the explicit form is given by Eq. (41) below) where and are virtualities of the two photons and . The case of DIS corresponds to and the Regge limit of small-x DIS to

Rewriting this in terms of Bjorken one gets


which turns to


after the shift


The -th moment of the structure function is given by


where . Now let us consider the integral (6) at . The contour goes parallel to imaginary axis but we can close it on the poles of the function . The expansion of at small has the form

so we get






we see that the series (8) giver the analytical continuation of the anomalous dimensions of twist-2 operators (1) at the point which can be formally denoted as .

This method, however, does not tell us the explicit form of this operator and in this paper I will demonstrate that is actually a light-ray operator ( )


and the anomalous dimension of this operator is an analytic continuation of the anomalous dimension (2) of local operators.

For simplicity this will be done in SYM where the conformal invariance simplifies many formulas for the correlation functions. The method to get the anomalous dimensions of the light-ray operator (10) near will be to calculate a 4-point correlation function of certain operators in the double “Regge+DIS” limit . We will compare two calculations of this 4-point CF: with Regge limit first and DGLAP limit second, or vice versa, and demonstrate that the intercept of the BFKL pomeron determines the anomalous dimensions in this region. However, first we need to discuss some properties of light-ray operators.


2.1. Light-Ray Operators in Gluodynamics

For simplicity, let us at first look at gluodynamics. For our purposes it is sufficient to consider “forward” light-ray operator


Evolution equation for “forward” operators has the form (see e.g. Ref. [6])


Note that since the matrix element of the light-ray operator (11) defines gluon parton density

the kernel


is actually a DGLAP kernel (in gluodynamics).

The anomalous dimensions of local twist-2 operators (1) is determined by the kernel (13)


Now consider the anomalous dimensions of LR operator (10). Cobining Eqs. (10) and (12) one obtains

Thus, we see that the anomalous dimension of light-ray operators (11) is an analytical continuation of anomalous dimensions of local operators to non-integer j.

2.2. Singlet light-ray operators in N = 4 SYM

In SYM, in addition to the gluon operators (1) there are twist-2 local gluino and scalar operators (as usual, we consider only “forward” highest weight local operators). Similarly to Eq. (11) we can define gluino and scalar light-ray operators (of even parity)


The evolution equations have the form similar to (12)


where is a function of and . The expansion in powers of gives anomalous dimensions of local operators




Matrix of anomalous dimensions for these “forward” local operators has the form


The renorm-invariant combinations were found in Ref. [7]:


The corresponding anomalous dimensions read


Now we define


From Eq. (16) we get the martix of anomalous dimensions for these LR opertaors


Since the matrix (23) is an analytical continuation of the matrix (16) so are the eigenvectors and eigenfunctions and therefore



are the multiplicatively renormalized operators with anomalous dimensions


Note that at high energy ( ) only the contribution of survives because at small the anomalous dimension of is negative while those of and are non-negative and hence we omit contributions of and operators in what follows.

For future use we need also the operators with the light rays going along the direction defined as


and similarly for , and operators.


To get the anomalous dimensions of light-ray operators (24) near we consider the correlation function of four Konishi operators in the double BFKL + DGLAP limit. Define


where is the Konishi operator ( - anomalous dimension). The Regge limit is


and the DGLAP limit corresponds to . It is convenient to consider “forward” correlation function


in the double limit: Regge ( ) plus DGLAP ( ).

First we look at the CF (29) in the DGLAP limit. In this limit we need to expand the product two Konishi operators near the light cone with the leading term being twist-two light-ray operators (24). To get the coefficient functions in this expansion it is convenient to discuss first the three-point

correlation function of a LR operator with two local operators. The correlation function of three local operators is fixed by conformal invariance. In our case it yields


For the CF of “forward” operator (15) and two local operators we get


Since a light-ray operator is an “analytical continuation” of a local operator to non-integer the CF of light-ray operator and two local operators has similar form


Integrating this over the total translation in direction we get


To get the expansion of as one compares Eq. (33) to the CF of two light-ray operators


One obtains


It is easy to see that the substitution of Eq. (34) to the r.h.s. of this equation reproduces the 3-point CF (33). Similar result with exchange of and directions reads


Finally, combining the light-cone expansion (36) with the three-point CF (33) we get the “DGLAP representation” of the 4-point CF (29)


In next Section we compare this formula with the “BFKL” representation of the same CF (29).


The 4-point CF (27) is a function of two conformal ratios. In the Regge limit (28) it is convenient to choose them as


It is easy to see that R increases with “energy” ( ) while is energy-independent.

It was demonstrated that a 4-point CF in SYM in the Regge limit and at large can be parametrized as a contribution of a Regge pole with


where is a signature factor and


is a solution of the Laplace equation in hyperboloid . The dynamics is described by the pomeron intercept and the “pomeron residue” . This formula was proved in [8] (see also [9]) by considering the leading Regge pole in a conformal theory. Also, it was demonstrated up to the NLO level that the structure (39) is reproduced by the high-energy OPE in Wilson lines [10-13].

The pomeron intercept in SYM is known in the leading order and in the NLO [14]


We will not need the explicit form of the “pomeron residue” but it can be easily restored from the NLO result for the CF of four protected operators and calculated in Ref. [15].

Now let us take the DGLAP limit on the top of Regge limit (39). In this limit



so Eq. (39) reduces to

Performing integral over one obtaines the “BFKL” representation of the CF (29) in the double (Regge plus ) limit:


where .


Let us compare integrals (37) and(43) for the correlation function (27). In both cases, the integrals run parallel to the imaginary axis but the singularities of the integrands lie on the negative real axis. We can close the integrals around negative real axis and compare the integrands near and . We see that the integrands coincide if one makes an identification




If we write down the NLO pomeron intercept (41) as a Laurent series near


we can invert equations (44) and get anomalous dimension near as a series in



In principle, the inversion of Eqs. (44) gives all orders in but in practice one gets the first few terms since the analytical form of the inversion is not known.

It is worth noting that the second of Eqs. (44) corresponds to the shift (5)


in the integral (43). In the momentum space this shift comes from the change of the energy scale from the symmmetric to non-symmetric while in the coordinate space it comes directly from the symmetric “energy scale” R, see Eq. (38).


The author confirms that this article content has no conflict of interest.


This work was supported by contract DE-AC05-06OR23177 under which the Jefferson Science Associates, LLC operate the Thomas Jefferson National Accelerator Facility.


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