NLO evolution of 3-quark Wilson loop operator

It is well known that the high-energy scattering of a meson from some hadronic target can be described by the interaction of that target with a color dipole formed by two Wilson lines corresponding to the fast quark-antiquark pair. Moreover, the energy dependence of the scattering amplitude is governed by the evolution equation of this color dipole with respect to rapidity. Similarly, the energy dependence of scattering of a baryon can be described in terms of evolution of a three-Wilson-line operator with respect to the rapidity of the Wilson lines. We calculated the evolution of the 3-quark Wilson loop operator in the Next-to-Leading Order (NLO), and we presented a quasi-conformal evolution equation for a composite 3-Wilson-line operator. Futhermore, we obtained the linearized version of that evolution equation describing the amplitude of the odderon exchange at high energies.


Introduction
The behavior of QCD amplitudes in high-energy Regge limit can be described in terms of the rapidity evolution of Wilson-line operators. A well-known general feature of highenergy scattering is that a fast particle moves along its straight-line classical trajectory and the only quantum effect is the eikonal phase factor acquired along this propagation path. In QCD, for the fast quark or gluon scattering off some target, this eikonal phase factor is a Wilson line -the infinite gauge link ordered along the straight line collinear to the particle's velocity. This observation serves as a starting point in the analysis of high-energy amplitudes by the operator expansion (OPE) in the Wilson lines developed in [1]. In a few sentences, the basic outline of the high-energy OPE is the following (for reviews, see refs. [2,3]). First, we introduce a "rapidity divide" η between the rapidity of the projectile Y P and the rapidity of the target Y T and separate all Feynman loop integrals JHEP01(2015)009 over longitudinal momentum (≡ rapidity) into two parts: the coefficient functions (called the impact factors) with Y > η and the matrix elements of the Wilson-line operators with Y < η. As we mentioned above, interaction of the fast particles with the slow ones can be described in the eikonal approximation so the relevant operators are the Wilson lines. Second, we find the evolution equations for these Wilson-line operators with respect to our rapidity factorization scale η. Third, we solve these equations (analytically or numerically) and evolve the Wilson-line operators down to the energies of few GeV at which step we need to convolute the results with the initial conditions for the rapidity evolution. If the target can be described by perturbative QCD (like virtual photon or heavy-quark meson) these initial conditions can be calculated in pQCD. If the target is a proton or a nucleus, the initial conditions are usually taken in the form of Mueller-Glauber model. It should be mentioned that the alternative approach to the high-energy scattering in QCD based on the evolution of the target wave function is described by the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) hamiltonian [4][5][6][7][8][9][10].
The high-energy OPE method is general but originally it was applied to the case of the deep inelastic scattering or meson scattering where the relevant Wilson-line operator is a color dipole. The leading order (LO) evolution of color dipoles was studied in the leading order in the paper [1] and independently by Yu. Kovchegov in refs. [11,12] where it was applied to scattering from large nuclei. This equation is now known as the Balitsky-Kovchegov (BK) equation. In the linear 2-gluon approximation the C-even part of this equation goes into the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [13][14][15] describing the C-even t-channel state known as pomeron. The running-coupling BK evolution of color dipole (rcBK) was obtained in refs. [16] and [17] the full NLO kernel was found in papers [18,19]. The solutions of the rcBK equation are widely used now for pA and heavyion experiments at LHC and RHIC. However, recently it has been realized that many interesting processes are described by the evolution of the more complicated operators such as "color tripoles" (trace of three Wilson lines in the quark representation) and "color quadrupoles" (trace of four Wilson lines). To describe evolution of such operators the NLO BK was generalized to the full hierarchy of Wilson-line evolution in recent paper [20]. (see also the JIMWLK calculation in ref. [21]).
As we briefly mentioned, in the high-energy OPE framework the amplitude in the Regge limit can be written as a convolution of the impact factors and the matrix elements of the Wilson line operators. The impact factors consist of the wavefunctions of the incoming and outgoing particles, which describe their splitting into the quarks and gluons propagating through the shockwave formed by the target particle. The propagation of the fast particle is described by a Wilson line, i.e. an infinite gauge link ordered along the classical trajectory of the particle. As we discussed above, for the high-energy scattering of a virtual photon or a meson the relevant operator is a color dipole whereas in the case of a proton scattering such an operator is the baryon or the 3-Quark Wilson Loop (3QWL) operator ε i j h ε ijh U i 1i U j 2j U h 3h . The rapidity evolution of such a "color tripole" has been frequently discussed in recent years. In the linearized LO approximation it was studied in the C-odd case within the JIMWLK formalism, and it was proved to be equivalent to the C-odd Bartels-Kwiecinski-Praszalowicz (BKP) [22]- [23] equation in [24]. The Green JHEP01(2015)009 function obeying the BKP equation describes the C-odd t-channel state known as odderon. The authors of [24] showed that the the LO linearized evolution equation for the 3QWL operator can be reduced to the BKP equation after the transformation from the coordinate to the momentum space. The full non-linear LO evolution equation for the 3QWL operator was derived within Wilson line approach [1] in ref. [25] and the connected contribution to the NLO kernel of the equation was calculated in [26]. In the momentum representation the evolution of the 3QWL operator was first studied in [27] and the nonlinear equation was worked out in [28]. In the C-odd case the linear NLO evolution equation for the odderon Green function was obtained in [29].
Here we present the full non-linear NLO evolution equation for this 3QWL operator. We calculate the evolution of the 3QWL operator with "rigid cutoff" in the rapidity of the Wilson lines, construct the composite 3QWL operator obeying the quasi-conformal evolution equation similarly to the case of the color dipole discussed in [18], and present its linearized kernel in the 3-gluon approximation. In addition, we linearize the BK equation in the same approximation and show that it contains the non-dipole 3QWL operators. It is worth noting that all the results are written in the M S scheme.
After completion of this paper we were informed about the JIMWLK calculation of the NLO evolution of the 3-Wilson-line operator [30]. Both evolution kernels reproduce NLO BK in the dipole limit r 1 → r 2 and survive other checks but, as it is written, the result of [30] differs from our result since ref. [30] has a much larger basis of operators in the evolution equation (see the discussion in section 9 of this paper).
The paper is organized as follows. In section 2 we remind the general logic of high energy OPE and in section 3 we list all the building blocks necessary for construction of the NLO 3QWL kernel. Sections 4 and 5 present the derivation of the NLO kernel for the 3QWL operator and section 6 describes the calculation of the quasi-conformal kernel for the composite 3QWL operator. The linearized kernel is given in section 7. The main results of the paper are listed in section 8. Conclusions are in section 9. Appendices comprise the necessary technical details.

Rapidity factorization and evolution of Wilson lines
Let us consider the proton scattering off a hadron target like another proton or a nucleus. First, we assume that due to saturation the characteristic transverse momenta of the exchanged and the produced gluons are relatively high (Q s ∼ 2 − 3 GeV for pA scattering at LHC) so the application of perturbation theory is justified. Alternatively, one may think about high-energy scattering of a charmed baryon.
If pQCD is applicable, in accordance with general logic of the high-energy OPE we factorize all amplitudes in rapidity. First, we integrate over the gluons with the rapidity Y close to that of the projectile proton Y p . To this end we introduce the rapidity divide η ≤ Y p which separates the "fast" gluons from the "slow" ones.
It is convenient to use the background field formalism: we integrate over the gluons with α > σ = e η and leave the gluons with α < σ as a background field to be integrated over later. Since the rapidities of the background gluons are very different from the rapidities of the gluons in our Feynman diagrams, the background field can be taken in the form of a shock wave thanks to Lorentz contraction. To derive the expression of a quark (or a gluon) propagator in this shock-wave background we represent the propagator as a path integral over various trajectories, each of them weighed with the gauge factor Pexp(ig dx µ A µ ) ordered along the propagation path. Now, since the shock wave is very thin, the quark (or gluon) does not have time to deviate in the transverse direction. Therefore, its trajectory inside the shock wave can be approximated by a segment of the straight line. Moreover, since there is no external field outside the shock wave, the integral over the segment of straight line can be formally extended to ±∞ limits yielding the Wilson-line gauge factor where A − η is the external shock wave field built from only slow gluons (Our light-cone conventions are listed in the appendix A). The propagation of a quark (or gluon) in the shock-wave background is then described by the free propagation to the point of interaction with the shock wave, the Wilson line U at the interaction point, and the free propagation to the final point.

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Thus, the result of the integration over the rapidities Y > η gives the proton impact factor proportional to the product of two proton wavefunctions integrated over the longitudinal momenta. This impact factor is multiplied by a "color tripole" -3QWL operator made of three (light-like) Wilson lines with rapidities up to η: where U i ≡ U ( r i , η). As discussed in refs. [1,20], these Wilson lines should be connected by appropriate gauge links at infinity making operator (2.4) and similar many-Wilsonline operators below gauge invariant. It should be noted that the proton impact factor is non-perturbative, so at this point it can be calculated only using some models of proton wavefunctions. At the second step the integrals over the gluons with the rapidity Y < η give the matrix element of the triple Wilson-line operator B 123 between the target states. The "rapidity cutoff" η is arbitrary (between the rapidities of the projectile and the target) but it is convenient to choose it in such a way that the impact factor does not scale with energy. So all the energy dependence is shifted to the matrix element of the triple Wilson-line operator (see the discussion in ref. [31,32]). In the leading order the rapidity evolution of this operator was calculated in ref. [25,26] while in this paper we present the result for the NLO evolution.

NLO evolution of triple-Wilson-line operator
In this section we outline the calculation of the NLO kernel for the rapidity evolution of 3QWL operator (2.4). In accordance with general logic of high-energy OPE in order to find the evolution of the Wilson-line operators with respect to the rapidity cutoff we consider the matrix element of operators with the rapidities up to η 1 and we integrate over the region of rapidities ∆η = η 1 − η 2 (where η 1 > η 2 > Y target ). Since particles with different rapidities perceive each other as Wilson lines, the result of the integration will be ∆η times the kernel of the rapidity evolution times the Wilson lines with rapidities up to η 2 . As we discussed in section 2, it is convenient to use the background-field formalism where the gluons with rapidities Y < η 2 form a narrow shockwave. As ∆η → 0, we get the evolution equation Here we explicitly write the . . . brackets, which denote that the calculation was performed in the shockwave background. We will often omit them to avoid overloading the notation. The typical leading-order diagrams are shown in figure 2 and it is clear that at this order the evolution equation for the 3-line operator can be restored from the evolution of the two-line operators since all the interactions are either pairwise (figure 2b) or selfinteractions (figure 2a). The result for the LO evolution has the form [25] (In this paper we set N c = 3 explicitly)  where r 1 , r 2 , r 3 are the coordinates of the quark Wilson lines within the 3QWL and r 0 is the coordinate of the gluon Wilson line coming from the intersection with the shock wave; r ij = r i − r j . The notation (i ↔ j) stands for the permutation. It means that we have to change r i ↔ r j and U i ↔ U j . As a result performing (1 ↔ 3), we change B 100 → B 300 , B 320 → B 120 , etc.
The typical diagrams in the next-to-leading order are shown in figure 3 where r 0 and r 4 are the coordinates of intersections with the shock wave. It is clear that at the NLO in addition to the self-interaction (figure 3a) and the pairwise-interaction (figure 3b) diagrams we have also the triple-interaction diagrams of figure 3c type. It should be emphasized that for the self-interaction and the pairwise diagrams one can use the results of ref. [20] while the triple-interaction diagrams were already calculated in ref. [26]. In this paper we combine these results to get a concise expression for the evolution of three-Wilson-line operator (2.4). The building blocks for our work are taken from ref. [20] and the relation JHEP01(2015)009 of our notations to those of ref. [20] is presented in appendix A.
where G 3 is defined in (3.17), and G 9 is defined in (3.25); n f is the number of the quark flavours and µ 2 is the renormalization scale in the M S-scheme.
where H 3 is defined in (3.34), and H 4 is defined in (3.35) and where H 2 is defined in (3.33), and where H 1 is defined in (3.32).
where H 5 is defined in (3.37), and H 6 is defined in (3.38) and It is obvious that expressions (3.3)-(3.11) do not have UV singularities since they have subtractions like U 4 − U i and U 0 − U i , which make them convergent. To construct the evolution equation for the 3QWL operator we have to convolute (3.3)-(3.11) according to (2.4) and simplify the result. The final evolution equation will consist of 3 parts: the part with 2 integrations w.r.t. r 0 and r 4 , the part with one integration w.r.t. r 0 and the purely virtual part. The Wilson line structure in the latter part depends neither on r 0 nor on r 4 . Therefore it seems that we can take both integrals in it. However, we can not do so since this part cancels the UV singularities in the former two parts. In fact, the purely virtual part is unambiguously determined by the first two parts. Indeed, it can be proportional to only one color structure B 123 and it must cancel the other parts if we switch off the shockwave, i.e. when U i = 1. Hence in our paper we will calculate only the former two parts of the evolution equation which come from the diagrams with one and two gluon intersections of the shockwave and then we restore the virtual part from the aforesaid conditions. Throughout the paper (except for section 7 where we discuss linearization of the already constructed UV-safe equation) we work with the parts of the kernels (integrands) and we do not integrate w.r.t. r 0 and r 4 . Therefore we do not need these parts to be UV finite. On constructing all of them we restore the virtual part, which automatically makes the whole kernel UV-convergent.

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Let us start with the self-and the pairwise-interactions of the type shown in figure 3 a-d. At N c = 3 one can use the SU(3) identities to rewrite (3.3)-(3.11) only through the Wilson lines in the fundamental representation. We will consider the gluon part without subtractions now. For the contribution of the the states with 2 gluons crossing the shockwave it reads where (3.14) Note that as discussed above and in ref. [20], one can present some of the terms with one intersection in the two-intersection form with an additional integration over r 4 (3.4). In doing so, some U 4 and U 0 factors in eq. (3.14) are replaced by U 4 − U i and U 0 − U i (i = 1, 2 or 3). Such subtractions make this contribution explicitly convergent at r 0,4 = r i . We do not write these subtraction terms here since it is easier to make the subtraction after the color convolution. The functions have the form (3.16) In all these expressions the notation i ↔ j stands for the permutation. As before, it means that we have to change r i ↔ r j . After the convolution with gives the contribution of the 2-gluon states to the evolution of the 3QWL operator U 1 · U 2 · U 3 describing the total interaction of Wilson lines 1 and 2, leaving Wilson line 3 intact.
Taking r 2 = r 1 in (3.22) one can check that it is related to (3.26) using the above identity along with (B.1) and (B.3). Taking the conjugate of G 1 † , one gets The contribution of the evolution of a single line U 1 to the evolution of the 3QWL related to the diagrams with 2 gluons crossing the shockwave reads The connected contribution of the evolution of lines 1 and 2 has the form Here The fully connected "triple" contribution corresponding to the diagrams in figure 3 e can be taken from (3.11) or [26] and transformed to the form The connection of our notations with the notations in [20] is given in the appendix A.

Construction of the kernel: gluon part
Taking the contributions of the self-interaction of Wilson lines (3.30) along with the "pairwise" (3.31) and "triple" (3.36) connected contributions from the previous section one can write the full contribution to the evolution of the 3QWL with two gluons intersecting the shockwave in the form Here . . . stands for the connected contribution, i.e. G 1 23 gives the contribution of the evolution of line 1 (3.30) with lines 2 and 3 being spectators, G 12 3 corresponds to the connected contribution of the evolution of lines 1 and 2 (3.31) with line 3 being intact, and G 123 stands for the fully connected contribution (3.36). All the rest can be obtained from them by 1 ↔ 2 ↔ 3 transformation, i.e. by all 5 possible permutations of r 1 , r 2 , and r 3 , which assumes the permutations of U 1 ≡ U ( r 1 ), U 2 , and U 3 as well.
There are several useful SU(3) identities, which help to reduce the number of color structures. They are listed in the appendix B. First, we use (B.5) to get rid of the structure (4.10) Again, here i ↔ j stands for the the permutation. It means that we have to change r i ↔ r j and U i ↔ U j .    Here In the dipole limit, i.e. when the coordinates of 2 quarks in the 3QWL coincide these functions obey the identities (4.15) Using these identities and (B.1) with l = 3, we get the dipole result This expression is twice the corresponding part of the BK kernel for tr(U 2 U † 3 ). The only UV divergent term in (4.8) is the term proportional to L 12 . This term has the same coordinate structure as the corresponding term in the dipole kernel. Therefore we can do the same subtraction as in the dipole case. Using (B.3), we get Therefore we can separate the result into the UV finite and divergent parts where And G U V is included into the term describing the contribution to the kernel with one gluon crossing the shockwave [20], which is proportional to the first coefficient of the β-function.
We take this contribution from (3.3) and (3.10). For the pure gluon field β 0 = 11 where the M S scheme is used. After some algebra one obtains It also has the correct dipole limit and matches the BFKL kernel [33].
There are also diagrams with one gluon intersecting the shockwave which are not proportional to the β-function. They can be taken from (3.3)-(3.11). However, they were already calculated in eq. (5.27) in [26]:

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This term has the correct dipole limit (see (5.28) in [26]). Thus, the real part of the whole kernel reads Finally, from the condition that the kernel must vanish without the shockwave (if all the B = 6) and that the virtual contribution is proportional to B 123 , we get the total kernel where G finite is defined in (4.20) and It differs from (4.27) by the coefficients of B 123 's which turn into 9's to accommodate the condition that the kernel must vanish without the shockwave.

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5 Construction of the kernel: quark part One can take the quark contribution to the NLO evolution of 3QWL from (3.3)-(3.4). The contribution with 2 quarks intersecting the shockwave without subtraction reads where L q 12 is defined in (3.6). Using identity (B.15) one can see that this contribution is conformally invariant, indeed In the dipole limit r 3 → r 2 , one has which is twice the corresponding part of the BK kernel [16], and therefore one can do the same subtraction as in the BK case where

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Similarly to the gluon case one can take G q U V contribution with one gluon crossing the shockwave from the terms proportional to β-function in (3.3) and (3.10). One can restore this contribution from the gluon part via the substitutions As a result, the full kernel in QCD reads Here G finite is defined in (4.20) and G q finite is defined in (5.6).

Evolution equation for composite 3QWL operator
In this section we consider only the gluon part of the kernel since the quark one is quasiconformal. To construct composite conformal operators we will use the prescription [19] (see also ref. [34]) which is exactly the composite dipole operator of [19]. Using the SU (3) Again, i ↔ j stands for the the permutation. It means that we have to change r i ↔ r j and U i ↔ U j . Next, one can use (B.9) to show that all the nonconformal terms have the SU(3) coefficients independent either of r 4 or of r 0 .
To get rid of the non-conformal terms, first we add the symmetrized last 3 lines of (6.7) to the nonconformal part of G finite (4.20), define the result asG (6.9) and work with it to avoid rewriting the conformally invariant parts of (6.7). Taking + ln r 01 4 r 02 2 r 34 4 r 03 4 r 04 2 r 12 2 r 14 2 tr U 4 Indeed, in this expression all the nonconformal terms have the SU(3) coefficients independent either of r 4 or of r 0 . In principle one can integrate them w.r.t. r 4 or r 0 and add to eq. (4.30). However, it is easier to transform (4.30) using integral (116) from [19] in the symmetric form  Using (B.9) to get rid of the terms like it can be transformed to and L 12 andL 12 are the elements of the nonconformal kernel defined in (4.10) and (4.11).
Checking that L C 12 ,L C 12 , M C 12 , and Z 12 have integrable singularities at r 4 = r 0 and that L C 12 , L C 12 , and Z 12 have integrable singularities at r 4 = r 1,2,3 is straightforward. To prove that all the terms with M C have safe behavior at r 4 = r 1,2,3 one has to use SU(3) identity (B.14). Now one can see that the NLO kernel for the evolution equation for the composite 3QWL operator B conf 123 (6.2) is quasi-conformal if one expresses the LO kernel in terms of composite operator (6.5).
The term with Z can be integrated w.r.t. was calculated in the appendix D. Finally, the kernel reads In the quark-diquark limit r 3 → r 2 one has We get which is twice the gluon part of the BK kernel (see (67) in [19]).

Linearization
In the 3-gluon approximation

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We use the following identity to linearize the color structures in (6.22).
Here E is the identity matrix. In the 3-gluon approximation the previous expression reads Using identity (B.3) and the fact that in the 3-gluon approximation we get As a result, the coefficient ofL C 12 in (6.22) reads Using integrals (114) and (125) from [19], The second structure reads Again, applying identity (B.3) and equality (7.4) one gets in the 3-gluon approximation Finally, the coefficient of L C 12 in (6.22) reads and therefore The third structure reads Using (B.3) and (7.4) we get As a result, one obtains The color structure in the quark part of the kernel can be linearized using (7.12) Finally, we get the linearized kernel in the form Here δ ij = 1, if r i = r j and δ ij = 0 otherwise;μ 2 and β are defined in (5.9); F 0 and F 140 are defined in (7.18) and (7.19); L q 12 is defined in (3.6) and L C 12 is defined in (6.17) and . Now, if we consider the dipole limit r 3 = r 2 and take into account that in this limit ((F 140 + (0 ↔ 4)) + (2 ↔ 3)) | r 3 = r 2 = 0, (7.30) ((F 340 + (0 ↔ 4)) + (2 ↔ 1)) | r 3 = r 2 = 0, (7.31) ((F 240 + (0 ↔ 4)) + (1 ↔ 3)) | r 3 = r 2 = 0, (7.32) we obtain the linearized BK kernel in the 3-gluon approximation whose C-even part is the BFKL kernel [33]  Let us compare this kernel for B 122 = 2tr(U 1 U † 2 ) with the linearized BK kernel in the 2-gluon approximation from [19]. One can see that their C-even parts coincide as they are fixed by the BFKL kernel [33]. However the 2-gluon approximation is not enough to figure out the correct C-odd part of the kernel. Only the 3-gluon approximation (7.33) allows JHEP01(2015)009 one to write it. It is easy to see that even for the color dipole the C-odd part of the kernel in the 3-gluon approximation can not be expressed through dipoles only. One necessarily needs to introduce the 3QWL operators as is clear from the second line of this expression. One can check it by direct calculation via (7.12), indeed As in [25] to separate the C-even and C-odd contributions we introduce C-even (pomeron) and C-odd (odderon) Green functions and where B123 is the 3-antiquark Wilson loop operator The NLO kernel for the C-even Green function in the 3-gluon approximation reads      From these expressions one can see that terms with L ij , L C ij , which comprise the BFKL kernels, contribute only to the evolution of the C-even part of the Green function while terms with F 0 ,L ij ,L C ij contribute only to the evolution of the C-odd one. The BK equation for the color dipole B 122 = 2tr(U 1 U † 2 ) in the 3-gluon approximation reads (see (7. This equation contains the nondipole 3QWL operators in its quark part. The question how to choose the operator basis for the evolution equation is nontrivial. We tried to find the basis with the minimal number of the operators. The part of the kernel with one integration does not present a problem since all the operators in it can be reduced to products of the 3QWLs B. Next, the quark contribution to the part of the kernel with 2 integrations can be reduced to one operator up to 3 permutations (8.1), (8.4). It is obviously a minimal choice here. The corresponding gluon contribution depends on 3 operators up to permutations (8.1), (8.4): (1 ↔ 2)) − (0 → 4) + (0 ↔ 4), ( U 0 U 4 † U 2 · U 1 U 0 † U 4 · U 3 − (1 ↔ 2)) − (0 ↔ 4), and There are 3 operators of the first type, 3 of the second type, and 12 ones of the third type. The operators of the first and the second type are independent because of the different symmetry w.r.t. (0 ↔ 4) and (i ↔ j) permutations, where i, j = 1, 2, 3. In the dipole limit the part of the kernel containing the operators of the third type reduces to the antisymmetric w.r.t. (0 ↔ 4) structure (6.23). If this part of the kernel could be expressed through the operators of the first two types, it would reduce to the same operator in the dipole limit as the part of the kernel containing the operators of the second type (6.24). Plainly, (6.23) and (6.24) depend on different operators, which can not be expressed through each other. There remains a question how many of the 12 third type operators are independent. In fact they are not all independent. They obey identity (B.14), which ensures the UV-safety of evolution equation (6.22) as we discussed above. Hence, we could rewrite the evolution equation using only any 11 of the 12 third type operators. However, such rearrangement makes the equation much more cumbersome and blurs its symmetry w.r.t. permutations. Therefore we left all the 12 operators in the final formulae. Using identities (B.1)-(B.4) with l = 0, 1, 2, 3, 4 we were unable to express 11 third type operators via one another.

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Reduction of the number of the operators used in the evolution equation simplifies the equation and helps to solve it. The best example of this fact is the LO equation for the 3QWL. By construction it contains several different operators [25] ∂B 123 ∂η = α s 2π 2 d r 4 After application of identity (B.3) it turns into (3.2). The latter equation depends only on products of B and has the closed form. In the NLO the reduction of the color structures helped to find the quasi-conformal form and simplified the equation, which will aid numerical solution.

Conclusions
In this paper we constructed the NLO evolution equation for the "color triple" -three-quark Wilson loop operator ε i j h ε ijh U i 1i U j 2j U h 3h . As in the case of the color dipole evolution, for the "rigid cutoff" Y < η of the Wilson lines the kernel of this equation has nonconformal terms not related to renormalization. We have constructed the composite 3QWL operator (6.2) obeying the NLO evolution equation with the quasi-conformal kernel. We linearized the quasi-conformal equation in the 3-gluon approximation. It is worth noting that our results have correct dipole limit in the case when the coordinates of the two lines coincide. We also constructed the 3-gluon approximation of the BK equation and showed that it contains non-dipole 3QWL operators (8.8), (8.10).
The 3QWL operator may have many phenomenological applications. First, it is a natural SU(3) model for a baryon Green function in the Regge limit. Also, it is the irreducible operator describing C-odd (odderon) exchange. For example as shown in the appendix E, the odderon part of the quadrupole operator tr(U 1 U † 2 U 3 U † 4 ) in the 3-gluon approximation in SU(3) can be decomposed into a sum of 3QWLs Moreover, even the NLO evolution equation for the dipole C-odd Green function in the 3-gluon approximation (8.10) in QCD can not be written without the introduction of the 3QWL operator. The evolution equation for the C-odd part of the 3QWL operator is the generalization of the BKP equation for odderon exchange to the saturation regime. However, it is valid for the colorless object, i.e. for the function B − ijk = B − ( r i , r j , r k ) , which vanishes as r i = r j = r k . The linear approximation of the equation for the C-odd part of the 3QWL should be equivalent to the NLO BKP for odderon exchange acting in the space of such functions. One may try to restore the full NLO BKP kernel from our result via the technique similar to the one developed for the 2-point operators in [35].
The result for the evolution of the 3QWL operator was also presented in [30] (which was put on arXiv the same day as our paper). As we mentioned above, both evolution