On the low-x NLO evolution of 4 point colorless operators

The NLO evolution equations for quadrupole and double dipole operators have been obtained within the high energy operator expansion method. The corresponding quasi-conformal evolution equations for the composite operators were constructed.


Introduction
This paper develops the Wilson line approach to high energy scattering proposed in [1] to the case of quadrupole and double dipole operators in the next to leading order (NLO). Such operators naturally appear when one studies amplitudes for diffractive processes with the production of 3 or 4 particles in the Regge limit. Moreover, the quadrupole operator enters the definition of the Weizsäcker-Williams gluon distribution [2], [3], [4] which gives the Fock space number density of gluons inside dense hadrons in light-cone gauge. One can find the NLO evolution equation for the operator necessary for the Weizsäcker-Williams gluon distribution differentiating the quadrupole equation obtained in this paper. This result is going to be presented in a future work.
In the Wilson line approach to high energy scattering [1] the amplitudes are convolutions of impact factors and a Green function. The impact factors describe the decomposition of the colliding particles into quarks and gluons while the Green function is responsible for the interaction of these quarks and gluons with the quarks and gluons from the other colliding particle. In this framework such fast-moving partons are depicted as Wilson lines with the path going along their trajectories. Hence, the corresponding Green functions are the operators constructed of the Wilson lines. These operators obey the evolution equations with respect to the rapidity divide. This rapidity divide separates the gluon field into the fast quantum one and the slow external field of the other particle, through which the current quark or gluon is propagating.
In the most thoroughly studied case of a virtual photon splitting into quark antiquark pair, the corresponding Wilson line operator is a color dipole. The evolution equation for this operator is known as the Balitsky -Kovchegov (BK) equation [1], [5]. The NLO corrections to this equation were calculated in [6], [7], [8], [9]. Another interesting case is application of this formalism to a proton. The proton has baryon color structure and can be described as a 3-quark Wilson loop operator (3QWL). The evolution equation for this operator was calculated in the leading order (LO) in [10] and in the NLO in [11]. The latter calculation was based on the NLO hierarchy of the evolution equations for the Wilson lines with open indices [12] and the connected contribution to the 3QWL kernel [13]. These results were also obtained in the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) formalism [14]. The hamiltonian equivalent to the NLO hierarchy was obtained in [15] and the evolution equation for the 3QWL in [16]. The NLO kernel for the evolution Wilson line operators was also constructed in [17].
The quadrupole and the double dipole are 4-particle colorless operators. Their LO linear evolution equations were derived in [4], [9], [18], [19]. Here the results of [12] and [13] are used to construct the NLO evolution equations for these operators and the results of [9] and [11] are used to check these equations.
The paper is organized as follows. The next section contains the definitions and necessary results. Sections 3 and 4 present the NLO evolution equations for the quadrupole and the double dipole operators in the standard and quasi-conformal forms. Section 5 discusses different checks of the results. Section 6 concludes the paper.

Definitions and building blocks
The light cone vectors n 1 and n 2 are defined as n 1 = (1, 0, 0, 1) , n 2 = 1 2 (1, 0, 0, −1) , n + 1 = n − 2 = n 1 n 2 = 1 (2.1) and any vector p can be decomposed as For brevity the following notation for traces is used where and b − η is the external shock wave field built from only slow gluons The parameter η separates the slow gluons entering the Wilson lines from the fast ones in the impact factors. The field b µ (r) = b − (r + , r)n µ 2 = δ(r + )b ( r) n µ 2 . (2.8) The coordinates r 1,2,3,4 denote the quarks, and r 0 , r 5 are the coordinates of the gluons. In intermediate formulas the coordinates r 6,7 will also be used. The SU (N c ) identities are necessary to rewrite the SU (N c ) operators only through the Wilson lines in the fundamental representation. For a generic operator O the rapidity evolution equation has the form where K LO ∼ α s and K N LO ∼ α 2 s . The ... brackets were explicitly written to denote that the calculation was performed in the shockwave background. Hereafter they will be often omitted to avoid overloading the notation. The BK equation in this notation reads [1] where r ij = r i − r j . The LO quadrupole evolution equation reads [4] Here (0 → 1 ≡ 0 → 4) stands for the substitution U 0 → U 1 or U 0 → U 4 , which gives the same result. In addition (1 ↔ 3, 2 ↔ 4) means that one has to change r 1 ↔ r 3 , r 2 ↔ r 4 and U 1 ↔ U 3 , U 2 ↔ U 4 . We will also need the LO evolution equations for the double dipole, sextupole and the dipole-quadrupole product. All these equations follow from the LO hierarchy [1] directly. (2.13) (2.14) (2.15) Here 1 → 3 → 5 → 1 stands for permutation, i.e. one has to change r 1 → r 3 , r 3 → r 5 , For the self and the pairwise NLO interactions one can take the results of [12] while the triple-interaction diagrams were already calculated in [13]. The results of these papers were derived using sharp cutoff on the rapidity variable. Since this paper is devoted to color singlet operators one can drop the kernels which vanish acting on the colorless operators, as was shown in [15]. The rest reads (2.19) Here where L ij ≡ L( r i , r j ) and L q ij ≡ L q ( r i , r j ) were introduced in this form in [11] L 12 = 1 r 01 2 r 25 2 − r 02 2 r 15 2 −  (2.27)

24)
We will also need the following functions. The function It was also introduced as M 2 in [20]. It has the property The functionsL ij ≡L( r i , r j ) and M ij ≡ M ( r i , r j ) were introduced in [11] as well (2.32) HereL ij is conformally invariant. Moreover,L ij is antisymmetric w.r.t. both 5 ↔ 0 and i ↔ j transformations while M ij is antisymmetric w.r.t. 5 ↔ 0. One can also combine all (2.33) The NLO BK kernel reads [9] K N LO ⊗ U 12 † = α

Construction of the kernel
First, one has to discuss the singularities of the building blocks from the previous section. All the ultraviolet (UV) singularities in (2.16), (2.20), and (2.28) were removed by the renormalization. It means that these expressions converge at r 0 = r 1,2,3,4,5 and r 5 = r 0,1,2,3,4 . In particular, the functions J in A 2 (2.24), A 3 (2.25), and (2.28) are convergent at these points, which ensures UV-safety of these expressions. However, the function J 11105 in the first line of (2.16), has the UV singularity at r 0 = r 5 = r 1 . As in (2.22) this singularity is removed by the subtraction in the color structure. Nevertheless, these expressions have infrared (IR) singularities, which appear as both r 0,5 → ∞. Indeed, changing the variables e.g. as r 0 = ut, r 5 = ut,t = 1 − t, one faces a logarithmic singularity integrating w.r.t. u first (3.1) Hence this double integral is ill-defined and requires either regularization or definition in terms of the iterated integrals. To understand how to correctly treat the IR singularities one can either return to the diagrams and keep the regularization, or calculate the known dipole equation and fix the definition from there. The latter way is attempted here. Assembling BK kernel (2.34) from (2. 16-2.20), one can see that all the β-functional terms go to M β 12 , A 1 (2.21) reshapes to the terms ∼ L 12 , L q 12 , the Wilson line operators from (2.16), (2.24-2.25) depending on both r 5 and r 0 give the term ∼L 12 after the symmetrization Next, B 1 (2.26) gives one half of the double logarithm contribution. All the remaining terms are to be equal to the other half of the double logarithm contribution. They read This term is IR safe. The second line is the product of 2 expressions symmetric w.r.t. 0 ↔ 5 permutation. Therefore one can set U 5 → U 0 there. In the first line there is a product of 2 expressions antisymmetric w.r.t. 0 ↔ 5 permutation. Hence, one could add and subtract N c U 2 † 1 in the first brackets and write One could understand the latter integral as an iterated one. Then, using the integrals one could get the other half of the double logarithm term in the BK kernel Although such treatment gives the correct result, it does not take into account the IR singularity of J. Indeed if one introduces the dimensional regularization into (3.3) then one gets However, in the dimensional regularization the integral d d r 5 J ijj05 would be ∼ rather than 0 and the double integral because the second integral w.r.t. r 0 has an IR divergence as r 0 → ∞ and starts from 1 . Therefore if one wants to integrate the coefficient of Thus, the result depends on the regularization. Such an ambiguity is the consequence of the fact that the initial expressions do not have the IR regularization. To avoid this ambiguity one needs the evolution equations for Wilson lines (2.24-2.25) with the IR regularization. Alternatively, one can write them in the form where the terms which do not depend on both U ab 5 and U ab 0 are integrated w.r.t. the coordinate of the other gluon. In this paper the procedure discussed in (3.2-3.7) is used. Technically it means that for the terms ∼ U ab 5 U a b 0 , the gluons are treated equally and the kernel is represented in the form of symmetrized sum (3.2). In the terms depending only on U ab 5 or only on U ab 0 , the integration order is fixed as d r 0 d r 5 or d r 5 d r 0 correspondingly and the integrals are understood as iterated. As a result, one can take the inner integral via (3.6). The terms independent of U ab 5 and U ab 0 are also symmetrized according to (3.2) and in them the substitution J ijj05 → J jji05 is made. This substitution can be understood as follows. First one drops the terms with d r 5 J ijj05 . They vanish (3.6) if one treats the integrals as iterated. Next, one adds the totally antisymmetric w.r.t. (5 ↔ 0) terms J jji05 . These terms vanish if they are integrated w.r.t. r 0 and r 5 in the double integral. After that the first integral in (3.6) is enough to calculate all the integrals. Again, I stress that although such treatment gives the correct dipole result (as well as the evolution equation for the baryon operator coinciding with [11]) it involves the cavalier treatment of the IR singularities.
Taking the contributions of the self-interaction of one Wilson line (2.16), the connected contributions of 2 Wilson lines (2.20) and the connected contributions of 3 Wilson lines (2.28) with the appropriate charge conjugation, and using the integration procedure described above, one can write the full NLO evolution of the quadrupole operator can write its full NLO evolution equation as Here the NLO dipole kernel is written in our notation in (2.34),G s (G a ) is the product of the coordinate and color structures (anti)symmetric w.r.t. 0 ↔ 5 transposition,G β is proportional to β-function andG is the remaining contribution with 1 gluon crossing the shockwave.

Quadrupole
We start from the product of the symmetric structures where L was introduced in (2.22). It is a conformally invariant contribution.
Here L q is defined in (2.23).
where M jk i is defined in (2.29). Using property (2.30) one can show that G s2 vanishes without the shockwave, i.e. when all the U → 1. Indeed, it is clear from the representation The contribution which is the product of the antisymmetric w.r.t. 5 ↔ 0 parts reads Here the functionsL and M ij are defined in (2.31) and (2.32).
From (2.31) and (2.32) one can see that it is possible to express G a in terms of only one function M jk i (2.29). The β-functional part of 1-gluon contribution G β (3.11) has the same structure as LO kernel (2.12) Here M β is defined in (2.33). The 1-gluon term without beta function reads In G one can pick the terms independent of r 0 and integrate them out if they are convergent. We call these terms G 0 . In fact the choice of G 0 is not unique. We have (3.25) All the integrals with the functions G s , G a , G β and G are convergent. It is clear from the explicit expressions for G β and G. For G s and G a one can see it recalling that L (q) ij has unintegrable singularity at r 0 = r 5 and M ij k has unintegrable singularity at r 0 = r 5 = r k . In all expressions in this section these singularities cancel.

Double dipole
The symmetric contribution reads The β-functional contribution has the form where M β is introduced in (2.33). The remaining contribution reads (3.33)  As for the quadrupole, it is straightforward to check that none of the functionsG s ,G a , G β ,G has unintegrable singularities.

Quasi-conformal evolution equation for composite operators
To construct composite conformal operators we use the prescription [9] (see also [21]) where a is an arbitrary constant. The conformal dipole reads [9] U conf 12 † = U 2 † 1 + The evolution equation for this operator [9] is quasi-conformal where M β 12 is defined in (2.33); L C ij ≡ L C ( r i , r j ) andL C ij ≡L C ( r i , r j ) were introduced in this form in [11] L C 12 = L 12 + The conformal double dipole operator reads The evolution equations for the conformal quadrupole and double dipole operators in the conformal basis have the general form As in the previous section, the individual NLO evolution of the dipoles here is taken out of the functionsG Therefore one can rewrite (4.9) Plainly, G β andG β are the same as in (3.11) and (3.12). The other functions G conf will be given below.
To obtain these functions one has to calculate the evolution equations for conformal operators (4.6, 4.7) using (2.11-2.15) and express the results in terms of conformal operators via (4.1). Technically, it means that one has to add to the kernels of the evolution equations from the previous section the corrections in the form of double integrals w.r.t. r 0 and r 5 [9]. To get the conformally invariant results one has to symmetrize these corrections according to (3.2). Then, the terms with color operators independent of r 0 (or r 5 ) can be integrated w.r.t. r 0 (or r 5 ) via the integrals from appendix A of [22]. Finally, the terms with color operators independent of both r 0 and r 5 can be integrated with respect to both r 0 and r 5 . In addition to the integrals from appendix A of [22], one needs the following integral (4.12)

Quadrupole
For the symmetric contribution G conf s we have where G q did not change. It is defined in (3.15) . (4.14) where L C is defined in (4.4).
r 02 2 r 05 2 r 12 2 r 35 2 . In fact, there is freedom in the definition of the functions M Cjk i , R ijk and R ijkl since one can redistribute terms between them. For example, one can try to redefine M Cjk i so that to make the functions R zero.

Discussion and conclusion
This paper presents the evolution equations for the double dipole and quadrupole operators in the standard (3.11), (3.12) and quasi-conformal forms (4.8), (4.9). They have correct dipole limits and in SU(3) obey group identity (5.1) with the corresponding evolution equations for the 3QWL operator obtained in [11]. This fact ensures the correctness of all the 3 results. To construct the composite operators, prescription (4.1) was used. It was proposed in [9] for the dipole and proved in [21]. Here it gave the quasi-conformal kernels for both double dipole and quadrupole operators, thus being checked by the specific calculation of the evolution of the 4-point operators.
Unlike the dipole and the 3QWL operators, the evolution of the quadrupole and the double dipole ones generates several operators in the virtual part. Indeed, the virtual gluons do not change the color structure of a dipole or a baryon. New color structures appear in the evolution of these operators only when the gluons cross the shockwave. Therefore, one can write the virtual part of the evolution equations for them without calculation. The double dipole and the quadrupole, on the contrary, mix in the virtual part with each other and with the double dipoles and quadrupoles with the other order of the Wilson lines. Therefore they had to be calculated explicitly. Using the evolution equations for Wilson lines from [12] in this calculation, one encounters ill-defined integrals which were treated here so as to obtain the known result for the dipole and the 3QWL operators. Although such treatment gave the equations satisfying all the checks, it is important to have the initial expressions with the regularization of the IR singularities and to check the results of this paper. Such checks may be performed starting from the evolution equations found in [16] and [17].
The equations obtained in this paper may be used to derive the NLO evolution equation for Weizsäcker-Williams gluon distribution. This work is in progress. They can also be important in the analysis of higher than dipole Fock components of the virtual photon in the diffractive processes.