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.
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Manuscript submitted on 22-11-2013 |
Original Manuscript | NLO BFKL and Anomalous Dimensions of Light-Ray Operators§ |
It is well known that the BFKL pomeron [1(a) Fadin VS, Kuraev EA, Lipatov LN. Multi reggeon processes in the Yang-Mills theory Sov Phys J Exp Theor Phys 1976; 44: 443-50.; (b) Fadin VS, Kuraev EA, Lipatov LN. The Pomeranchuk singularity in nonabelian gauge theories Sov Phys J Exp Theor Phys 1977; 45: 199-204.; (c) Balitsky I, Lipatov LN. The pomeranchuk singularity in quantum chromodynamics Sov J Nucl Phys 1978; 28: 822-9.] 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
(1)
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 [2Jaroszewicz T. Gluonic regge singularities and anomalous dimensions in QCD Phys Lett B 1982; 116: 291-4.] and for the next-to-leading order BFKL it was done in papers [3(a) Fadin VS, Lipatov LN. BFKL pomeron in the next-to-leading approximation Phys Lett B 1998; 429: 127-34.; (b) Camici G, Ciafaloni M. Energy scale(s) and next-to-leading BFKL equation Phys Lett B 1998; 430: 349-0.]. This constitutes a powerful check for explicit calculations of higher-order anomalous dimensions both in QCD [4Vogt A, Moch S, Vermaseren JAM. The three-loop splitting functions in QCD The singlet case Nucl Phys B 2004; 691: 129-81.] and =4 SYM [5Lukowski T, Rej A, Velizhanin VN. Five-loop anomalous dimension of twist-two operators Nucl Phys B 2010; 831: 105-32.]. 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
(2)
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
(3)
which turns to
(4)
after the shift
(5)
The -th moment of the structure function is given by
(6)
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
(7)
where
(8)
Since
(9)
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 ( )
(10)
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.
For simplicity, let us at first look at gluodynamics. For our purposes it is sufficient to consider “forward” light-ray operator
(11)
Evolution equation for “forward” operators has the form (see e.g. Ref. [6Balitsky I, Braun VM. Evaluation equations for QCD string operators Nucl Phys B 1989; 311: 541-84.])
(12)
Note that since the matrix element of the light-ray operator (11) defines gluon parton density
the kernel
(13)
is actually a DGLAP kernel (in gluodynamics).
The anomalous dimensions of local twist-2 operators (1) is determined by the kernel (13)
(14)
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.
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)
(15)
The evolution equations have the form similar to (12)
(16)
where is a function of and . The expansion in powers of gives anomalous dimensions of local operators
(17)
where
(18)
Matrix of anomalous dimensions for these “forward” local operators has the form
(19)
The renorm-invariant combinations were found in Ref. [7Belitsky AV, Derkachov SE, Korchemsky GP, Manashov AN. Superconformal operator in N=4 super Yang-Mills theory Phys Rev D 2004; 70: 045021.]:
(20)
The corresponding anomalous dimensions read
(21)
Now we define
(22)
From Eq. (16) we get the martix of anomalous dimensions for these LR opertaors
(23)
Since the matrix (23) is an analytical continuation of the matrix (16) so are the eigenvectors and eigenfunctions and therefore
(24)
are the multiplicatively renormalized operators with anomalous dimensions
(25)
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
(26)
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
(27)
where is the Konishi operator ( - anomalous dimension). The Regge limit is
(28)
and the DGLAP limit corresponds to . It is convenient to consider “forward” correlation function
(29)
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
(30)
For the CF of “forward” operator (15) and two local operators we get
(31)
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
(32)
Integrating this over the total translation in direction we get
(33)
To get the expansion of as one compares Eq. (33) to the CF of two light-ray operators
(34)
One obtains
(35)
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
(36)
Finally, combining the light-cone expansion (36) with the three-point CF (33) we get the “DGLAP representation” of the 4-point CF (29)
(37)
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
(38)
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
(39)
where is a signature factor and
(40)
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 [8Cornalba L. Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange [arXiv: 0710.5480 [hep-th]]] (see also [9Costa MS, Goncalves V, Penedones J. Conformal Regge theory J High Eenergy Phys 2012; 1212: 091-141.]) 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 [10Balitsky I. Operator expansion for diffractive high-energy scattering Nucl Phys 1996; B463: 99-104.-13Diaknonov D, Balitsky I. High-energy amplitudes in the next-to-leading order Subleties in quantum field theory USA PNPI Publishing Dept 2010; pp: 104-49.].
The pomeron intercept in SYM is known in the leading order and in the NLO [14(a) Kotikov AV, Lipatov LN. NLO corrections to the BFKL equations in QCD and in supersystemic gauge theories Nucl Phys B 2000; 582: 19-61.; (b) Kotikov AV, Lipatov LN. DGLAP and BFKL equations in the N=4 supersymmetric gauge theory Nucl Phys B 2003; 661: 19-61.; (c) Kotikov AV, Lipatov LN. DGLAP and BFKL equations in the N=4 supersymmetric gauge theory Nucl Phys B 2004; 685: 405-7.]
(41)
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. [15Balitsky I, Chirilli GA. High-energy amplitudes in N=4 SYM in the next-to-leading Order Phys Lett B 2010; 687: 204-13.].
Now let us take the DGLAP limit on the top of Regge limit (39). In this limit
and
(42)
so Eq. (39) reduces to
Performing integral over one obtaines the “BFKL” representation of the CF (29) in the double (Regge plus ) limit:
(43)
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
(44)
and
(45)
If we write down the NLO pomeron intercept (41) as a Laurent series near
(46)
we can invert equations (44) and get anomalous dimension near as a series in
(47)
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)
(48)
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.