Pomeron fan diagrams in perturbative QCD

Within QCD reggeon field theory we study the formation of two subsequent triple pomeron vertices in the process P→PP→PPP. We make use of an earlier investigation [1] of the six-reggeon amplitude in deep inelastic scattering and show that in the large-Nc limit pomeron fan diagrams emerge with the same triple pomeron vertex in all places. We thus confirm the BK-equation, but we also find additional terms related to the reggeization of the gluon, and we discuss their potential significance. Our analysis also includes the general pomeron → two odderon vertex: a particular version of this vertex has been included into earlier generalizations the BK equation.


Introduction
Rather long time ago in the study of the interaction of a colorless projectile with two colorless targets the triple-pomeron vertex Γ was constructed, both within QCD reggeon field theory 1 [2,3] and within the dipole picture [4,5]. In figure 1 we illustrate the basic process in which the triple pomeron vertex appears: the scattering of a projectile (virtual photon) on two targets, described by the interaction of three BFKL ladders which are coupled to the triple pomeron vertex.
In momentum space the triple pomeron vertex is given by a somewhat lengthy function V a 1 a 2 a 3 a 4 |b 1 b 2 4←2 (k 1 , k 2 , k 3 , k 4 |q 1 , q 2 ) [1] which depends upon the color and the transverse momenta of the constituent reggeons of the incomjng pomeron and the two outgoing pomerons. Switching via Fourier transform to the space of transverse coordinates and taking the large-N c limit, the expression takes the much simpler simple form [6,7]: d 2 r 1 d 2 r 2 d 2 r 3 P (r 1 , r 2 )P (r 2 r 3 )Γ(r 1 , r 2 , r 3 )P (r 1 , r 3 ) = − g 4 N c 4π 3 d 2 r 1 d 2 r 2 d 2 r 3 P (r 1 , r 2 )P (r 2 r 3 )  Here P (r 1 , r 3 ) denotes the (upper) incoming pomeron, P (r 1 , r 2 ) and P (r 2 , r 3 ) the two (lower) outgoing pomerons. They depend upon the transverse coordinates of their constituent reggeons, and r ik = r i − r k . (we have suppressed the color labels and the rapidity dependence of the pomerons). This triple pomeron vertex plays an important role in applications, since it serves as a fundamental building block in constructing a theory of interacting pomerons, which is expected to describe strong interactions in the Regge kinematics. In particular, it lies at the origin of the structure functions of the heavy nuclei, as given by the Balitski-Kovchegov equation [8,9] which sums pomeron fan diagrams (figure 2). In this nonlinear evolution equation this vertex represents the nonlinear part of the kernel. It is also responsible for pomeron loops, in particular the pomeron the self-mass (figure 3), which was calculated in [10][11][12][13]. In this calculation also the inverted vertex PP→P was introduced.
In the framework of QCD reggeon field theory, reggeon interaction vertices are computed as a sum of several contributions. As an example, the triple pomeron vertex is obtained from the transition vertex: 2 → 4 reggeized gluons. This interaction vertex is the sum of the integral kernel K 2→4 plus also other 'induced pieces' which, at the end, leads to the conformal invariant expression V 4←2 . The derivation of this vertex was an outcome of the calculation of the high energy behavior of the six point function (more precisely: JHEP06(2018)095 from energy discontinuities of the corresponding scattering amplitude). This six point amplitude contains the triple pomeron vertex, as the single splitting of a pomeron in two, illustrated in figure 1. The obtained result is valid in all orders N c , and the equivalence, in the large-N c limit, with (1.1) was proven in [7].
In order to catch up with the fan diagrams of the BK-equation, which have been derived in the color dipole approach, one needs to consider more than one consecutive splitting: the simplest example is shown figure 2. Within QCD reggeon field theory, this requires the study of amplitudes containing al least 6 reggeized gluons at the lower end, i.e. the scattering of a projectile on three color singlet targets. In earlier demonstrations [14] it was tacitly assumed that in consecutive splittings exactly the same vertices Γ appear. An explicit derivation, however, is still missing. Also, it will be interesting to what extent the fan structure is complete and will be valid also beyond the large-N c limit. It is the purpose of this study to fill some of these gaps.
An investigation of amplitudes with up to six reggeized gluons at the lower end has been presented in [1]. The results, in principle, allow to find, in the leading-log approximation, the complete structure of the evolution of the 2,4, and 6 gluon systems. However, the final step of analyzing the transition 6 → 4 gluons has not been completed. In the present paper we apply the large-N c limit and show that, in fact, the fan structure with two consecutive triple pomeron vertices appears, in full agreement with the BK equations. There are, however, two features which do not affect the validity of the fan structure but go beyond the well-known results. First, we find additional contributions related to the reggization of the gluon. We will dicuss their potential significance. Next, our derivation includes the vertex: pomeron → two odderons [1]. As we will discuss in more detail, this vertex is slightly more general than the one used in extensions of the BK equation [15,16].
Our paper is organized as follows. We first (section 2) recapitulate the main steps of the derivation of the triple pomeron vertex in the context of the six point scattering amplitude (i.e. four gluons at the lower end). We will restrict ourselves to the most essential steps and, for details, refer to the literature. We then (section 3) summarize the main results for the eight point reggeon amplitude (i.e.six gluons at the lower end) obtained in [1]. In section 4 we concentrate on the large-N c limit and derive the fan structure where the same triple pomeron vertex describes the consecutive splitting of two BFKL pomerons. In addition, we find reggeizing pieces and discuss their potential significance. In a summarizing section we discuss our results in comparison with the BK equation.
2 The 2 → 4 reggeized gluon transition: the triple-pomeron vertex V 4 To formulate the problem and to fix our notations we first recall the main steps of the derivation of the triple -pomeron vertex V 4←2 in QCD reggeon field theory, the theory of interacting reggeized gluons. It will be useful to recapitulate both the original derivation [2] (which is valid in all orders N c ) and the large-N c (N c 1) derivation in [17] which makes use of the cylinder topology of the pomeron.

Two and three gluon states coupled to the qq loop
The structure of intermediate states coupled to the upper fermion loop in figure 1 has long been known [17]. We begin with the simplest case, D 2 in figure 4a, which describes a single BFKL pomeron coupled to the fermion loop. The BFKL pomeron is a bound state of two reggeized gluons, and D 2 satisfies the equation Here S 2 is the two-reggeon free Schroedinger operator for energy j 1 = 1 − j where j is the angular momentum, ω(i) the reggeon Regge trajectory with momentum q i , and V 12 the BFKL kernel written as an operator in the transverse momentum space (with the factor g 2 N c included). Both ω(i) and V 12 are well known and can be found in the literature. D 20 (1, 2) ≡ D 20 (q 1 , q 2 ) is the momentum dependent part of the impact factor describing the four different couplings of the two reggeons 1 and 2 to the fermion loop.

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D 2 denotes the full BFKL pomeron coupled to the impact factor (including angular momentum j and both color and transverse momentum).
Here and in the following we adopt, for simplicity, the following notation: the product of two operators includes the integration over transverse momenta, e.g.
Note that we define the operator V 12 in such a way that includes the r.h.s. momentum propagators 1 (k 1 +k) 2 (k 2 −k) 2 . Formally we can solve (2.1) by introducing the BFKL Green's-function G 2 which describes the infinite sum of ladder graphs. Sometimes it is convenient to use, instead of angular momentum j, rapidity y. We put and obtain the rapidity dependent Green's function G 2 (y). We then can write the solution as a function of y: where G 2 (y) satisfies the equation Now consider the coupling of three reggeized gluons to the fermion loop. It is important to note that the three gluons at the lower end may either come from a direct coupling the fermion loop or from transitions from two reggeons coupled to the loop, which afterwards transform into three reggeons by means of the splitting kernel K 3←2 . 2 As a result, the equation for the corresponding amplitude D 3 contains two inhomogeneous terms: Here D 30 denotes the fermion loop with three reggeons coupled to it in all possible ways. Explicitly D a 1 a 2 a 3
Here '12' is our short-hand notation for the sum of the two momenta q 1 +q 2 . The transition from 2 to 3 reggeons, D 3←2 , is accomplished by the kernel K 3←2 where the momentum dependent part of the 2 → 3 kernel (including the momentum propagators for the intermediate two reggeon state) has the form: (2.11) 2 Throughout this paper we find it convenient to orient, in the subscripts of the operators, the arrows from right to left. This is consistent with reading our equations which contain products of operators (beginning with (2.1)) from right to left. In our figures, this corresponds to moving down from top to bottom.

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It is remarkable that the solution of the equation ( This solution has the same momentum structure as the one-loop impact factor (2.9). Both (2.12) and (2.9) have the cylinder topology. The three terms on the r.h.s. of (2.12) have a simple interpretation. As an example, the third term, D 2 (12, 3) illustrated in figure 4(b)), describes a single BFKL pomeron, where the left constituent reggeon carries the sum of two momenta, q 1 + q 2 , whereas the constituent reggeon on the r.h.s. carries the momentum q 3 . If the q i refer to targets 1,2,3 to which the pomeron couples, the left reggeon splits into two gluons which then couple to target 1 and target 2. Here it is important to note the difference between wavy lines (which correspond to reggeized gluons) and straight lines (which denote elementary gluons): the former ones contain evolution in rapidity whereas the latter ones do not. Figure 4(b)) thus illustrates the nature of reggeization: a reggeized gluon can be viewed as a bound state of two elementary gluons. Later on we will see that a reggeized can also be viewed as a bound state of three or even more gluons.

The triple pomeron vertex
We now extend the procedure to transitions into four reggeized gluons. Let us first follow the derivation described in [2]. The coupling of four reggeized gluons to the fermion loop consists of two groups: where the color tensor is defined as d a 1 a 2 a 3 a 4 = Tr (t a 1 t a 2 t a 3 t a 4 + t a 4 t a 3 t a 2 t a 1 ). (2.14) By drawing all terms on the surface of a cylinder, the two groups differ in the order of t-channel gluons. The integral equation for D 4 can be written as where the momentum dependent part of the kernel K 4←2 has the form: Formally we could solve (2.15) by  Here V 4←2 denotes the triple pomeron vertex as an operator in the color and transverse momentum space. Exhibiting the color structure, this vertex can be written as a sum of three terms:

4←2
(k 1 k 2 k 3 k 4 |q 1 , q 2 )) = δ a 1 a 2 δ a 3 a 4 V 4 (1, 2; 3, 4) + δ a 1 a 3 δ a 2 a 4 V 4 (1, 3; 2, 4) +δ a 1 a 4 δ a 2 a 3 V 4 (1, 4; 2, 3), (2.22) where on the r.h.s. we have suppressed the initial momenta q 1 , q 2 which in the convolution with D 2 are to be integrated, and we have written only the momentum structure of the two JHEP06(2018)095 where we have indicated that, in both color and momentum, the arguments of V 4 D 2 and D 2 are grouped into the pairs (1, 2) and (3,4). Since the rapidity evolution of these pairs proceeds independently, the Green's function G 4 factorizes: G 4 (y; 1234) → G 2 (y; 12)G 2 (y; 34), (2.24) and D I 4 depends upon two independent rapidity variables, y 1 and y 2 : After Fourier transformation to coordinate space, this expression takes the form (1.1) (with suitable substitutions for the three pomerons). A general proof (for general N c ) of this Fourier tansform, has been presented in [7]. Finally we make a few comments on the reggeizing term D R 4 . As indicated in figure 6a. and described in (2.19), the reggeized term comes as a sum of single BFKL pomerons where the lower legs of reggeized gluons split into four elementary gluons. D R 4 represents an additive correction to the single pomeron in figure 4a with two gluons at the lower end. In the scattering on a big nucleus the pomeron in figure 4a, in its simplest form, couples to a single nucleon inside the nucleus, and this coupling can be described by the gluon distribution of the nucleon. Following this simple picture, the corrections contained in D R 4 then can be interpreted as the coupling of a single BFKL pomeron to a pair of JHEP06(2018)095 two nucleons. As an example, consider the last term in (2.19), D 2 (13, 24) (remember that we are considering the configuration where the pairs (12) and (34) are in color singlet states). As illustrated in figure 6(b), in this contribution the BFKL pomeron couples to two different nucleons. Since all the rapidity evolution is contained in the reggeized gluons, this coupling to two nucleons is a 'low energy' interaction. As we will discuss in section 4.4, this new contribution can be interpreted as the second order term in an eikonal initial condition for the nonlinear evolution equations [18].

The topological derivation
As observed in [17], the large-N c limit of the triple pomeron vertex can be obtained also in another way, by making use of the cylinder topology of the pomeron in the large-N c approximation.
To this end we return to D 40 in (2.13) and note that the two groups correspond to two different configurations on the surface of the cylinder, D Here the transition from 2 to 4 reggeons is described by For the transition 3 → 4 we have:

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In the next step we consider another toplogical configuration, which is suppressed in 1/N c : the formation of two color singlet cylinders made of reggeons pairs (1,2) and (3,4): we denote this amplitude by D 2P 4 . These two cylinders can start directly from the fermion loop or from a single cylinder formed by 2, 3 or 4 reggeon which then couples to two cylinders. The full amplitude D 2P 4 satisfies the equation [17]: The inhomogeneous term D 2P 0 is just the projection of the loop D 40 onto the 2pomeron state: The inhomogeneous terms D 2P 4←2 , D 2P 4←3 and D 2P 4←4 describe the mentioned transitions into the two-pomeron state from two, three and four evolved reggeons coming from the loop.
and D 2 is the solution of (2.1).
where D 3 is the solution of eq. (2.8). The transition from the single cylinder to the twocylinder configuration is described by where G 2P = (G 2 ) 2 is the Green function for two non-interacting pomerons. For the complete four reggeon amplitude we still have to add the single cylinder contribution D R 4 . We finally want to stress that all results obtained so far do not use the explicit form of function D 20 (1, 2), and they remain valid if one substitutes D 20 by an arbitrary function F 2 (1, 2) with properties All one needs is that the three and four-reggeon contributions come from the impact factors, F 3 and F 4 , related to F 2 in exactly the same manner as D 30 and D 40 are related to D 20 in eqs. (2.9) and (2.13), namely (13,24) .
The transition 2 → 6 Let us now turn to the amplitudes with five and six reggeized gluons at the lower end. For the time being we postpone the large-N c limit and keep the number of colors finite. In [1] an integral equation for D 6 has been derived and analyzed. The final step, however, the analysis of the irreducible piece D I 6 , has not been completed. In section 4 we return to the large-N c limit, and we will show that in this limit D I 6 , contains the fan diagrams entering the Balitsky-Kovchegov (BK) equation.
Let us first review the results obtained in [1]. The structure of the integral equation is illustrated in figure 7. The analysis described in the previous section has illustrated that the functions D i exhibit the following hierarchy structure: D 3 is built from D 2 , D 4 is based upon D 2 and D 3 . Similarly, D 5 contains D 2 , D 3 , and D 4 , and D 6 is obtained from all D i with i < 6. This is illustrated by the integral equation which, in a somewhat simplified form, reads as follows:: with D a 1 a 2 a 3 a 4 a 5 a 6

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One easily verifies that all configurations can be drawn on the surface of a cylinder, and they represent the different orderings of the t-channel gluon lines. On the r.h.s. of (3.1), a new ingredient appears, D 5 , for which the following decomposition has been found:  We remind that the triple pomeron vertex V 4←2 inside D Ia 1 a 2 a 3 a 4 4 has the structure described in (2.22).
Turning to the r.h.s. of (3.1) and using our results for D 2 ,. . . ,D 5 we note that all terms (disregarding, for the moment, the first term D 6;0 ), have a simple structure of multireggeon states: the second term (containing D 3 ) and third term (containing D 4 ) start, at the fermion loop, with a two-reggeon state which then turns into the six-reggeon state. The fourth and fifth terms have both 'R' and 'I' contributions; the 'R'-terms start with two reggeon states and then undergo 2 → 6 transitions. The 'I'-terms begin with a two-reggeon state, then contain the 2 → 4 transition described by the 2 → 4 reggeon vertex discussed in the previous section, and finally end with a transition into the six-reggeon state. This pattern is not altered if we insert, for D 6 , the decomposition with D R 6 being obtained from D 60 in (3.2) by replacing on the r.h.s. D 20 → D 2 . Using the BFKL equation for D R 6 removes, on the r.h.s. of (3.1), the first term, D 60 , and adds new contributions to the transition 2 → 6 vertex. Additional contributions to the 2 → 6 vertex are generated by inserting, in the last term of the r.h.s. of (3.1), D R 6 . The main task therefore is the computation of the complete (and potentially new) 2 → 6 and 4 → 6 vertices. In section 4.4. we will say more about the reggeizing term D R 6 . A very big step was already done in [1]. Namely, starting from (3.1), inserting (3.8) and performing an extensive recombination of terms a modified equation for D I 6 was obtained which has the following structure: This equation is illustrated in figure 7.
Here the first term denotes the P → 2O pomeron-odderon vertex, first derived and discussed in [1]. It has the form: partitions d a 1 a 2 a 3 d a 4 a 5 a 6 (W D 2 ) (1, 2, 3; 4, 5, 6), (3.10) where the sum extends over all partitions of 6 reggeons into two triplets, and d abc are the symmetric structure constants of the SU(3) gauge group. For details of the function W we refer to [1].
Let us say few words about the significance of this vertex. As it is well known, the odderon is a bound state of three reggeized gluons in a C-odd state, and it is natural that the pomeron→two-odderon comes as a part of the 2 gluon → 6 gluon vertex. A special bound state solution of the three gluon system with intercept exactly at one was found in [19]. In this solution, two of the three gluons merge into a single even signature reggeized gluon (degenerate with the 'normal' odd-signature reggeized gluon) and then form a bound state with the third gluon. This results in a quasi-two-gluon bound state

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which is closely related to the BFKL pomeron. For this special odderon solution, the pomeron→two-odderon vertex becomes similar to the triple pomeron vertex. It is this special odderon solution which appears in the generalizations of the BK-equation [15,16]. Our more general version (3.10), on the other hand, allows to inlcude also other three-gluon solutions, such as the one found in [20,21] with intercept slightly below one.
Returning to the equation (3.9). the second term has the form: Here each sum has a structure similar to D 30 in (2.9). In the second term of figure 7, the summation sign denotes both the summation over partitions (3.11) and the summations contained in (3.12). As it can be read off from figure 7, these L-terms contain a transition described by V 4←2 : a pair of two reggeized gluons → splits into two colorless triplets of reggeized gluons. For each triplet, the initial momentum structure of the L-terms is similar to D 30 in (2.9). Except for the large-N c limit, where each triplet evolves independently in rapidity, in the course of the subsequent evolution the triplets interactwith each other and thus loose their initial triplet structure. Next we discuss the third term, which consists of two terms labeled by 'I' and 'J'. We first list the color structure: Focussing on the subsystem (3456) and comparing the function I with the first line of D 40 in (2.13) one sees the identical structure in the arguments; similarly, the four terms in the function J coincide with the second line in (2.13). Following our discussion of the 'L'-term, we interpret this structure as a transition: a colorless pair of reggeized gluons → a colorless pair + a colorless quartet of reggeized gluons. Initially, at the vertex V 4←2 , the quartet has the same momentum structure as D 40 in (2.13). During the subsequent evolution the separation of pair+quartet will be lost. Finally, the last two terms in figure 7 still contain 'elementary kernels', K 3←2 and K 4←2 which require further recombination. In the terms with K 3←2 and K 4←2 the sum extends over all pairs of gluons inside D I 5 and D I 4 , resp. As we have said before, for general N c the missing next step has not been carried out. In the following we restrict ourselves to the large-N c limit and describe the solution of eq. (3.9) (and figure 7).
4 The large-N c limit of D 6

New reggeizing pieces
As already indicated in [1], the integral equation (3.9) does not represent the final version. Instead, it will be necessary, once more, to separate a reggeizing piece: and rewrite the equation. Our ansatz for the first term, D I;R 6 , will be constructed in such a way that, to lowest order, it removes the L and the I + J terms in (3.11), (3.14),and (3.15). We put:  figure 8, we recognize two separate BFKL ladders; this separation is a consequence of the large-N c limit. For the first term, figure 8a, we have at the bottom the color and momentum structure described in (3.14)- (3.17). The 'L'-term in figure 8(b), described in (3.11)-(3.13), consists of products of two triplets, and in each triplet we have a sum of three terms.
We emphasize that the reggeizing pieces have been constructed in such a way that, to lowest order, they coincide with the second and third terms on the r.h.s. of figure 7. Taking . As in figure 6, the pairs of straight lines at the lower end of reggeized gluons indicate that they depend upon the sum of momenta and finally decay into two gluons. Further details are described in section 4.4. into account the subsequent large-N c evolution, this requirement fixes the reggeizing pieces uniquely. Further below (in section 4.4) we will come back to these reggeizing terms and discuss in more detail their significance.

The large-N c limit of eq. (3.9)
With the ansatz (4.1) and (4.2) we return to (3.9) (or figure 7) and insert it into the terms containing D I 6 : Our goal is the derivation of an equation for D I;I 6 . As it was said before, in this paper we consider the configuration where the six gluon state consists of three color singlet pairs, and we consider the large-N c limit of the corresponding six gluon amplitude. To be definite, we demand that the pairs (12), (34), and (56) are in color singlet states. This allows for the three partitions (12)(3456), (1234)(56), and (1256)(34).
We proceed by discussing, on the r.h.s. of (4.3), term by term. We will find that for our color configuration, in the large N c limit the r.h.s. simplifies. For reasons which will become clear soon we begin with the pieces containing D I 5 and the 2 → 3 kernel. From (3.7) we see that D I 5 is a sum of 10 terms. Each term contains a V 4←2 vertex. Below this vertex we have the evolution of the four reggeon state, which in the large-N c limit splits into the evolution of two independent two-reggeon states (2.24). At the end of this evolution one of the four reggeized gluons splits into two gluons. Subsequently, we attach a vertex with K 3←2 . This structure implies that the D 5 term on the r.h.s. of figure 7 can be written as a sum of different groups of contributions: they are illustrated in figure 9. In the first group we have the partition (12)(3456), where the subsystem (12) evolves independently, and the JHEP06(2018)095 Figure 9. The three terms emerging from the D I 5 term in figure 7. The first term belongs to the partition (12)(3456), the second and third ones to the partition (123)(456).
2 → 3 transition is inside the subsystem (3456). Second, in the partition (123)(456) there exists the part of D I 5 which contains the splittings (12), (23), and (13), and the 2 → 3 transition sits inside the triplet (456). In the third group, the 2 → 3 transition is inside the triplet (123). Consequently, only the first group contributes to our configuration of three color singlet pairs, whereas the last two terms have triplet structures. As we will discuss below, they have to be combined with the L-terms and are not important for the following discussion. Let us discuss the first group in more detail. Beginning with D I 5 in (3.7), a color singlet of the pair (12) can come only from the last three terms:  where the Green's functions G 2 (1, 2) and G 2 (34, 5) (suppressing the color factors) describe the pomeron evolution in the subsystems (1, 2) and (34, 5) resp. Together with analogous expressions for the other two terms in (4.4) we find:  Apart from the extra Green's function G 2 (1, 2) we have the same structure as D 3 in (2.12). We attach the 2 → 3 vertex: (12, 3 , 4 ). (4.9)

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As the next term on the rhs (4.3) we consider the triplet states. The first term on the r.h.s. of (4.3) contains the pomeron→ 2 odderon vertex. Its structure is explained in (3.10): the two gluon state splits into two triplet states which are characterized by the symmetric d a 1 a 2 a 3 tensors. As an example, we once more chose the partition (123)(456) at the W vertex in (3.10). Below this vertex, the two triplet states, in the large-N c approximation, evolve separately and do not mix. Each of them forms an QCD odderon state, as described for example, in [19][20][21].
A similar discussion applies to the second term on the r.h.s. of (4.3), the L-term (cf. (3.11)-(3.13)): below the vertex V 4←2 two triplet states start, this time accompanied with f -tensors. As written in (3.12)-(3.13), in each of the two triplets the momentum structure is the same as for D 30 in (2.9). These terms represent the inhomogeneous terms, the last two terms in figure 9 the kernels for the evolution equation of the reggeizing terms . (4.10) We note that these triplet terms, at large N c , do not contribute to the color structure we are considering in this paper. However we note that the terms with the f-structure will be crucial when we turn to the next step, the case of 8 gluons. Here we expect new fan diagrams: in addition to the fan diagram in figure 2 we should find a higher order fan diagram where the lower BFKL ladder on the l.h.s. will split into two ladders. This is where the 'L' terms will be needed for building the two lower triple pomeron vertices.
Returning to the r.h.s. of (4.3) we, from now on, disregard the odderon term on both sides of the equation: as an aditive contribution to D where D I;I 6 now is without the two-odderon piece (for simplicity, we continue to use the same notation).
Let us then take a closer look at the remaining pieces. We begin with the terms T I and T J , (3.14)-(3.17) and focus on the partition (12)(3456). . . For this partition, the couplings at the vertex V 4←2 have the color structure δ a 1 a 2 d a 3 a 4 a 5 a 6 . and the momentum structure described in (3.16), (3.17). Considering the four reggeons '3', '4', '5', and '6', the structure of the 7 terms in the sum 'I' and 'J' is exactly the same as for D 40 in (2.13). Reggeons '1' and '2' play the role of spectators: (4.11) Turning to the terms containing D ( 4 I) (the next-to-last term in figure 7), we note that the evolution of the four gluon state in D I 4 is the sum of three states, (12)(34), (13)(24), and (14) (23), and in each of them the evolution is described by two separate BFKL Green's functions. Only the first term belongs to the color singlet (12), and consequently the 2 → 4 kernel has to be inside the quartet (3456). In rapidity space this term can be written in the following form:   (4.13) We illustrate this equation in figure 10.

Two triple pomeron vertices
Let us now analyze the integral equation illustrated in figure 10. Most strikingly, we notice that the subsystem (12)  Here the sum in front of G (34,56) 2 (y 2 − y ) stands for summing,at the lower end of the BFKL evolution, over momentum and color structures, in accordance with (2.13). Interpreting the rapidity evolution in (4.14, we compare with figure 8a: starting from above, we evolve up to rapidity y where the two-gluon state merges into the four gluon state. Below, the JHEP06(2018)095 pair (12) evolves up to y 1 , whereas the other two-gluon state evolves up to y 2 where it splits into the four gluons (3456).
For the other two terms, T 2 and T 3 , the situation is a bit more complicated since they contain already evolution inside the subsystem (3456). Using the identity dy G 2 (y 1 − y )G 2 (y − y 2 ) = G 2 (y 1 − y 2 ) (4.23)
This equation has to be compared with the solution of D 4 in eq. (2.17). On the r.h.s. of (4.27 we keep the momenta of the pair (12) fixed and concentrate on the subsystem (3456) below the upper triple pomeron vertex, i.e. V 4 D 2 (y 1 ). Then we see, term by term, that the r.h.s. of (4.27) is the same as of (2.17), provided we substitute D 20 → (T I + T J )D 2 . The second and third lines on the r.h.s. of (4.27) correspond to the second and third terms on the r.h.s. of (2.17), once we take into account that K 3←2 D 3 in (2.17) really means the same as K 3←2 G 2 in (4.27). We then make use of the discussion given after (2.17). For the subsystem (3456) below V 4 D 2 (y 1 ) we have the same structure as for D 4 , i.e. we have the sum of reggeizing piece and an irreducible one. The former one has the form: G (34,56) 2 (y 2 − y 1 ) (V 4 D 2 (y 1 )) (1, 2; 3 4 ), (4.28) where the symbol Σ reminds that the Green's function G (34,56) 2 (y 2 − y 1 ) at its lower end has the same momentum structure as D R 4 . The second term, the analogue of D I 4 , has the JHEP06(2018)095 , Figure 11. Two more fan diagrams. form: dy G 2 (y 2 − y ; 34)G 2 (y 2 − y ; 56)V 4 G 2 (y − y 1 ; 3 4 )(V 4 D 2 (y 1 ))(12, 3 4 ). (4.29) Finally we re-introduce the integration of y and the Green's function for the pair (12). The first part then takes the form dy G in (4.14). The second term becomes: D I;I 6 (y 1 , y 2 , y 3 ) = dy G 2 (y 1 − y ; 12) dy G 2 (y 2 − y ; 34)G 2 (y 3 − y ; 56) ·V 4 G 2 (y − y ; 3 4 )(V 4 D 2 (y ))(12, 3 4 ) (4.31) and has the fan structure illustrated in figure 2. Therefore we conclude that the r.h.s. of (4.27) which had been obtained from evaluating the r.h.s. of (4.17) is, in fact, the sum in (4.14) and D I;I 6 in (4.31). The fan structure in eq. (4.31) (figure 2) represents the main result of this paper.
For completeness a few words need still to be said about the remaining two partitions, (1234)(56) and (1256)(34). By going through the same sequence of arguments as described for the partition (12)(3456) one arrives at the fan diagrams shown in figure 11. It is only after adding these two additional fan diagram that the Bose symmetry of pomeron diagrams is restored. . In our derivation, they appear as additive corrections to the 'simple' fan diagrams, and they require an interpretation. needs some further investigation), both approaches seem to arrrive at the consistent result: -our derivation (performed in perturbation theory in momentum space) leads to contributions which arise from the reggization of the gluon but can be reformulated as specific corrections in the couplings of the pomerons to the nuclei; -the derivation based upon the color dipole picture does not see such 'reggeized' contributions, but by including the nontrivial form of couplings of pomerons to the nuclei it leads to the same result.
Clearly, a more detailed analysis of this equivalence is highly desirable and we hope to come back to this in a future publication.

Discussion
In our study we have used the framework of QCD field theory of reggeized gluons for discussing, in the large-N c limit, diagrams with up to six reggeized gluons, using the results of [1]. Our main interest was the formation of two consecutive triple pomeron vertices. We have found that in diagrams with higher splitting kernels K n←2 with n =5 and 6 all those contributions are present which are needed to form the triple pomeron vertex at each splitting. This result agrees with the conclusions derived in the framework of the dipole model, the Balitski-Kovchegov equation.
In the course of our analysis a few other terms appeared, which require some attention. First, our analysis contains the pomeron → two odderon vertex (see the discussion in section 3, after (3.10)). This vertex was found as early as in 1999, when only the Janik-Wosiek solution (JW) for the odderon [20,21] was known, describing a C-odd bound state of three reggeized gluons. Soon afterwards a new solution with intercept one was found in [19] (the BLV odderon). This solution has a simple quasi-two-gluon structure corresponding to the fusion of two of the three reggeized gluons into a single object (the zero range "digluon"). Since the structure of this odderon is practically identical to the pomeron, it became possible to radically simplify vertices for both pomeron →two odderon and odderon → odderon+pomeron transitions, which actually reduce to the triple pomeron form. This allowed to easily derive the coupled system of evolution of the pomeron and odderon including both transitions [15,16]. This system was analyzed in [26] where the angular dependence introduced by the pomeron→two odderon transition (POO) was drastically simplified. Recently this very system was investigated in detail and at finite N c [27]. Calculating the pomeron-odderon (PO) evolution the authors neglected the POO term and limited themselves to the lowest azimuthal harmonic. A posteriori, by calculating with the previously obtained solution the POO term the authors then verified that the latter was small. In [27] the full JIMWLK equation was solved numerically by Langevin simulations on the lattice, assuming a certain choice of azimuthally unsymmetric initial conditions which generate odderon contibutions. It was found that the results were in very good agreement with those obtained form the previously studied coupled PO system of equations. It might very well be (although not shown explicitly) that the full JIMWLK system contains the POO transitions not only for the BLV odderon but also the JW odderon solutions. Compared to these important numerical studies, we find it important to stress that the pomeron → two odderon vertex found in [1] and in our present analysis, describing the full transition from 2 to 6 reggeized gluons, is not restricted to the BLV form of the odderon but is valid also for a more general odderon consisting of three gluons, such as the JW odderon. It would be interesting to study the coupled evolution of pomerons and odderon with the full angular dependence and the JW odderon as well.
Second, our investigation has led us to separate, from the pomeron fan diagrams, an extra class of contributions which result from the reggeization of the gluon ('reggeized terms'). A detailed discussion has been given in section 4.4 and the main results are summarized in figure 13. The first example, for the case of four reggeized gluons, was D R 4 ( figure 6(b)). For the six-gluon case we first found D R 6 (see (3.8). As illustrated in the JHEP06(2018)095 first line of figure 13 (see also figure 6(b) and figure 12(a)), both D R 4 and D R 6 , contain the two gluon state of the pomeron, D 2 , which at the lower end splits into four or six gluons, resp. To be complete we have to include D 3 (defined in (2.12)) and D R 5 (given in (3.5)), where D 2 splits into three or 5 gluons, resp. Within a nucleus, all these contributions can be interpreted as couplings of a single pomeron to pairs, triplets,. . . of nucleons inside the nucleus. Somewhat later, figure 8, we encountered new reggeizing pieces, D I;R 6 . As an example, D I;R(I+J) 6 in figure 8(a) has the form of the lowest order fan diagram (second line of figure 13), with one of the two lower pomerons splitting into 4 gluons. It thus seems as if the reggeizing pieces, being fully compatible with the fan structure, start to 'dress' the lower order fan diagrams, by introducing nontrivial structures into the coupling of pomerons to the nucleus and thus leading to a result compatible with [9] .
Finally, our investigation of the appearance of more than one triple pomeron vertex within the framework of QCD reggeon field theory based upon reggeized gluons also might serve other purposes. First, it would be interesting to analyze the six gluon system also for finite N c . Most likely, a 2 → 4 reggeon vertex for non-singlet color quantum numbers should exist; there is also the possibility of a 4 → 6 reggeon vertex. Our large-N c study might provide help in such an analysis. More general, experience shows that reggeon field theory interaction vertices -either interactions between reggeized gluons or interactions between bound state fields such as the pomeron or the odderon -cannot directly be read off from high energy Feynman diagrams. In the derivation based upon energy discontinuities several steps of 'reduction' have to be performed, as we have demonstrated in this paper. A very attractive way is the use of Lipatov's effective action where the reggeized gluon appears as a fundamental degree of freedom. But also in this framework interactions between reggeized gluons arise only after a careful analysis of several contributions. Examples are the transition vertex 1 → 3 reggeized gluons which has been derived in [28]. It has also been shown that, in accordance with signature conservation, the vertex 1 → 2 reggeized gluons disappears.
An exciting open problem is the appearance of two triple pomeron vertices in the scattering of two nuclei on two nuclei. We hope to address this problem in a future investigation.