NLO evolution of 3-quark Wilson loop operator

It is well known that 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 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-lines operator with respect to the rapidity of the Wilson lines. We calculate the evolution of the 3-quark Wilson loop operator in the next-to-leading order (NLO) and present a quasi-conformal evolution equation for a composite 3-Wilson-lines operator. We also obtain the linearized version of that evolution equation describing the amplitude of the odderon exchange at high energies.


Introduction
The description of the proton scattering in the framework of k T -factorization can be addressed within the high energy operator expansion developed in [1]. In that paper this method was applied to derive the full leading order (LO) hierarchy of the low-x evolution equations for Wilson lines with arbitrary indices and to the most important case of the color dipole. In the next to leading order (NLO) the evolution equation for the color dipole was derived in [2], [3], [4] and the connected evolution of the 3 Wilson lines was found in [5]. Finally the full NLO hierarchy of the low-x evolution equations was written in [6] and the JIMWLK hamiltonian equivalent to it in [7].
In this 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 other particle. It is well known that the propagation of the fast particle is described by a Wilson line -infinite gauge link ordered along the classical trajectory of the fast particle. For the virtual photon or meson scattering the relevant two-Wilson-lines operator is a color dipole. In the proton case assuming SU(3) symmetry it is the baryon or 3-quark Wilson loop (3QWL) ε i j h ε ijh U i 1i U j 2j U h 3h . Its leading order linear evolution equation was studied in the C-odd case within the JIMWLK formalism and proved equivalent to the C-odd BKP equation [8]- [9] in [10] and its nonlinear evolution equation was derived within Wilson line approach [1] in [11]. The connected contribution to the NLO kernel of the equation was calculated in [5]. In the momentum representation the evolution of this operator was first studied in [12], and the nonlinear equation was worked out in [13]. In the C-odd case the linear NLO evolution equation for the odderon Green function was obtained in [14].
Here the NLO evolution equation for the 3QWL operator is presented. Then as in [3], we construct the composite 3QWL operator obeying the quasi-conformal evolution equation and after that give its linearized kernel in the 3-gluon approximation. In this approximation we also linearize the BK equation and show that it contains the nondipole 3QWL operators.
After completion of this paper we were informed about JIMWLK calculation of the NLO evolution of 3-Wilson-line operator [15]. Both evolution kernels reproduce NLO BK in the dipole limit r 1 → r 2 and survive other checks but the detailed comparison of these two results is beyond the scope of present paper. We also wish to compare our results with the results of S. Caron-Huot [16].
The paper is organized as follows. In Section 2 we remind the general logic of highenergy OPE. Sections 3 and 4 present the derivation of the NLO kernel for 3QWL operator and Section 5 describes the calculation of the quasi-conformal kernel for the composite 3QWL operator. The linearized kernel is given in Section 6. The main results of the paper are listed in Sect. 7 and conclusions in Sect. 8. 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 nucleus. First, we assume that due to saturation the characteristic transverse momenta of exchanged and 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 high-energy OPE we factorize all amplitudes in rapidity. First, we integrate over gluons with 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 gluons with α > σ = e η and leave 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 gluons in our Feynman diagrams, the background field can be taken in the form of a shock wave due to the Lorentz contraction. To derive the expression of a quark (or 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, quarks (or gluons) do not have time to deviate in transverse direction so their 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 3) (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 free propagation to a point of interaction with the shock wave, Wilson line U at the point of interaction, and free propagation to the final point. Thus, the result of the integration over rapidities Y > η gives the proton impact factor proportional to product of two proton wavefunctions integrated over 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. [1,6], these Wilson lines should be connected by appropriate gauge links at infinity making the operator (2.4) and similar many-Wilsonlines 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 gluons with rapidity Y < η give matrix element of triple Wilson-line operator B 123 between target states. The "rapidity cutoff" η is arbitrary (between rapidities of projectile and target) but it is convenient to choose it in such a way that the impact factor does not scale with energy so all energy dependence is shifted to the matrix element of triple Wilson-line operator (see the discussion in Ref. [17]). In the leading order the rapidity evolution of this operator was calculated in Ref. [11], [5] 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 rapidity cutoff we consider matrix element of operators with 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 integration will be ∆η times kernel of the rapidity evolution times Wilson lines with rapidities up to η 2 . As we discussed in Sect. 2, it is convenient to use background-field formalism where gluons with rapidities Y < η 2 form a narrow shockwave. The typical leading-order diagrams are shown in Fig. 2 and it is clear that at this order the evolution equation for 3-line operator can be restored from the evolution of two-line operators since all interactions are either pairwise (Fig. 2b) or self-interactions (Fig. 2a).
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 gluon Wilson line coming from the intersection with the shock wave. (In this paper we set N c = 3 explicitly). The typical diagrams in the next-to-leading order are shown in Fig. 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 self-interaction (Fig.3a) and pairwise-interaction (Fig.3b) diagrams we have also the triple-interaction diagrams of Fig.3c type. It should be emphasized that for the self-interaction and pairwise diagrams one can use the results of Ref. [6] while the tripleinteraction diagrams were already calculated in Ref. [5]. In this paper we will combine these results to get a concise expression for the evolution of three-Wilson-line operator (2.4). Let us start with self-and pairwise-interactions of the type shown in Fig. 3 a-d. At N c = 3 one can use the SU (3) identities to rewrite the result of [6] only through the Wilson lines in the fundamental representation. For the contribution of the the states with 2 gluons crossing the shockwave it reads 3) Note that as discussed in Ref. [6], it is convenient to present some of the terms with one intersection in the two-intersection form with an additional integration over r 4 . In doing so, some U 4 and U 0 factors in Eq.  (3.6) After the convolution with (3.4) 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.
For the elements of SU (3) group one has the identity Taking r 2 = r 1 in (3.12) one can check that it is related to (3.16) via this identity using the other SU (3) identities (B.1) and (B.3). Taking the conjugate of G 1 † , one gets The fully connected "triple" contribution corresponding to the diagrams of Fig. 3 e,f can be taken from [6] or [5] and transformed to the form Here . . . stands for the connected contribution, i.e. G 1 23 gives the contribution of the evolution of line 1 (3.20), with lines 2 and 3 being spectators, G 12 3 -the connected contribution of the evolution of lines 1 and 2 (3.21), with line 3 being intact, and G 123 -the fully connected contribution (3.26). All the rest can be obtained from them by 1 ↔ 2 ↔ 3 transformation. 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 and the 2 ones it goes into after the 1 ↔ 2 ↔ 3 transformations with their symmetric counterparts w.r.t. 0 ↔ 4 exchange. Next we use (B.6) to eliminate 6 such contributions antisymmetric w.r.t. 0 ↔ 4 exchange as Finally, by means of (B.9) we discard the 3 nonconformal terms proportional to and the 2 structures they go into after the 1 ↔ 2 ↔ 3 transformations. Finally, we get (4.8) (4.9) (4.10) (4.14) 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 And G U V is included into the term describing the contribution with one gluon crossing the shockwave in [6]. The contribution of the diagrams with 1 gluon intersecting the shockwave, which are not proportional to the β-function one can take from (5.27) in [5] This term has the correct dipole limit (see (5.28) in [5]). The contribution proportional to β-function reads (from [6]) Or, after some algebra It also has the correct dipole limit and it matches the BFKL kernel [18]. Therefore 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 It differs from (4.27) in the coefficients of B 123 's which turn into 9's; G f inite is defined in (4.20).

Construction of the kernel: quark part
One can take the quark contribution to the NLO evolution of 3QWL from [6]. The contribution with 2 quarks intersecting the shockwave without subtraction reads 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 [2]. Therefore one can do the same subtraction as in the BK case and G q U V is taken into account in the contribution with one gluon crossing the shockwave in [6]. The latter contribution one can restore from the gluon part via the substitutions As a result the full kernel in QCD reads Here G f inite is defined in (4.20) and G q f inite is defined in (5.7).

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 [4] (see also Ref. [19]) where a is an arbitrary constant. For the conformal 3QWL operator we have the following ansatz     Now we can symmetrize the last 3 lines of this expression w.r.t. 0 ↔ 4 transformation, i.e.
Indeed, in this expression all the nonconformal terms have the SU (3) coefficients independent either of r 4 or of r 0 . Therefore one can integrate them w.r.t. r 4 or r 0 . However, it is easier to transform (4.30) using integral (116) from [4]. We use it 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 (4.9) and (4.10).
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. Finally, the kernel reads In the quark-diquark limit r 3 → r 2 one has Therefore, This is twice the gluon part of the BK kernel (see (67) in [4]).

Linearization
In the 3-gluon approximation We use the following identity to linearize the color structures in (6.21).
Here E is the identity matrix. In the 3-gluon approximation it reads Using identity (B.3) and the fact that in the 3-gluon approximation As a result the coefficient ofL C 12 in (6.21) reads Using integrals (114) and (125) from [4], 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 (6.21) reads 14) The third structure reads Using (B.3) and (7.4),   The color structure in the quark part of the kernel can be linearized via (7.12) Putting things together we have for the linearized kernel The functionsS 123 and I are defined in appendix D (D.16) and (D.12). If we consider the dipole limit r 3 = r 2 and take into account that in this limit we have the linearized BK kernel in the 3-gluon approximation, whose C-even part is the BFKL kernel [18] K (7.31) Let us compare this kernel for B 122 = 2tr(U 1 U † 2 ) with the linearized BK kernel in the 2-gluon approximation from [4]. One can see that their C-even parts coincide as they are fixed by the BFKL kernel [18]. However the 2-gluon approximation is not enough to catch the correct C-odd part of the kernel. But the 3-gluon approximation (7.31) allows one to write it. At once we 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 [11] 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 The linearized equations for C-odd composite 3QWL Green function is the consequence of (7.45) and (C.16) (8.7) 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 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.

Conclusions
In this paper we constructed the NLO evolution equation for the "color triple" -threequark Wilson loop operator ε i j h ε ijh U i 1i U j 2j U h 3h . As in the case of color dipole evolution, for the "rigid cutoff" Y < η of Wilson line the kernel of this equation has non-conformal terms not related to renormalization. We have constructed the composite 3QWL operator (6.2) obeying the NLO evolution equation with 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 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 [20]. . (D.14) is symmetric w.r.t. interchange of its arguments function defined in [22]. Performing inversion and restoring r 0 , we get r 01 2 r 23 2 .

E Decomposition of C-odd quadrupole operator
Here we demonstrate that the C-odd 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. Indeed Therefore