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$\to$PP$\to$PPP. We make use of an earlier investigation ~\cite{Bartels:1999aw} of the six-reggeon amplitude in deep inelastic scattering and show that in the large-$N_c$ 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 $\to$ 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]. In Fig. 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.
(1.1) 1 In our definition, QCD reggeon field theory is defined as the theory of interacting reggeized gluons.
The BFKL pomeron appears as a bound state of (at least) two reggeized gluons.  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 [7,8] which sums pomeron fan diagrams (Fig.  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 (Fig. 3), which was calculated in [9][10][11][12]. 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: from  Fig. 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 [6].
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 Fig. 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 [13] 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 4 → 6 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 [14,15].
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 scattering amplitude with a b Figure 4: (a) the amplitudes D 2 and (b) D 3 three or 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 amplitude with five or six gluons at the lower end obtained in [1], and in particular we comment on the pomeron → two odderon vertex. 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 the reggeizing pieces and in a separate subsection discuss their potential significance. In a summarizing section we discuss our results in comparison with the BK equation. 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 [16] 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 Fig. 1 has long been known [16]. We begin with the simplest case, D 2 in Fig.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.
With the color dependence included we have 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 rhs 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 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) It is remarkable that the solution of the equation (2.8) is quite simple, and can totally expressed via the pomeron wave function D 2 [2] (Fig.4b): 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 rhs of (2.12) have a simple interpretation. As an example, the third term, D 2 (12, 3) illustrated in Fig.4b), 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 rhs 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. Fig.4b) 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: (2.16) Formally we could solve (2.15) by This solution is illustrated in Fig.5. However, this is not the final solution. Instead, one first removes a reggeized piece with only two reggeized gluons, i.e. one makes the ansatz ( 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: where on the rhs 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 outgoing pomerons with the substitutions k i → i, (i = 1, ..., 4).The function V 4 (1, 2; 3, 4) is symmetric under permuting (1,2) or (3,4), and under the exchange of the pairs (1, 2) and (34); also, the convolution V 4 (1, 2; 3, 4)D 2 vanishes whenever one of the four transverse momenta goes to zero. For the full vertex one observes Bose symmetry under the exchange of reggeized gluons 1, 2, 3, 4. Considering the configuration where the pairs (1, 2) and (34) are color singlet and taking the large-N c limit we find that in (2.22) only the first term contributes, and integral equation for D I 4 simpifies: 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: 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 proof (for general N c ) of this Fourier tansform, has been presented in [6].
Finally we make a few comments on the reggeizing term D R 4 . As indicated in Fig.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 Fig.4a with two gluons at the lower end. In the scattering on a big nucleus the pomeron in Fig.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 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 Fig.6b, 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 [17].

The topological derivation
As observed in [16], 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: 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 [16]: The inhomogeneous term D 2P 0 is just the projection of the loop D 40 onto the 2-pomeron 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 · G 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 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 Fig. 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:: where d a 1 a 2 a 3 a 4 a 5 a 6 = Tr (t a 1 t a 2 t a 3 t a 4 t a 5 t a 6 ) + Tr (t a 6 t a 5 t a 4 t a 3 t a 2 t a 1 ).
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 rhs 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 rhs 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 rhs D 20 → D 2 . Using the BFKL equation for D R 6 removes, on the rhs 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 rhs 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 Fig.7.
Here the first term denotes the P → 2O pomeron-odderon vertex, first derived and discussed in [1]. It has the form: 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 [18]. 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 which is closely related to the BFKL pomeron. For this special odderon solution, the pomeron→twoodderon vertex becomes similar to the triple pomeron vertex. It is this special odderon solution which appears in the generalizations of the BK-equation [14,15]. Our more general version (3.10), on the other hand, allows to inlcude also other three-gluon solutions, such as the one found in [19,20] 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 Fig.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 Fig.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 Fig.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 Fig.7). . As in Fig.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).
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: with the two terms being illustrated in Fig.8a and b: Fig.8a belongs to the partition (12)(3456), Fig.8b to the partition (123)(456). Both terms contain the vertex V 4←2 which, according to (2.22), consists of three terms. But in the large-N c limit and for our color configuration, only the first term contributes, in which the pairs (ab) and (cd) are color singlet states. Moving further down in Fig.8, we recognize two separate BFKL ladders; this separation is a consequence of the large-N c limit. For the first term, Fig.8a, we have at the bottom the color and momentum structure described in (3.14) - (3.17). The 'L'-term in Fig.8b, 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 rhs of Fig.7. Taking 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. Figure 9: The three terms emerging from the D I 5 term in Fig.7. The first term belongs to the partition (12)(3456), the second and third ones to the partition (123)(456).

The large-N c limit of Eq.(3.9)
With the ansatz (4.1) and (4.2) we return to (3.9) (or Fig.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 rhs of (4.3), term by term. We will find that for our color configuration, in the large N c limit the rhs simplifies. For reasons which will become clear soon we begin with the pieces containing D 5 and the 2 → 3 kernel. From (3.7) we see that D 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 rhs of Fig.7 can be written as a sum of different groups of contributions: they are illustrated in Fig.9. In the first group we have the partition (12)(3456), where the subsystem (12) evolves independently, and the 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: As we have said before, the evolution in G 4 factorizes: and we arrive at: 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: As the next term on the rhs (4.3) we consider the triplet states. The first term on the rhs 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 [18][19][20].
A similar discussion applies to the second term on the rhs 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 Fig.9 the kernels for the evolution equation of the reggeizing terms D . (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 Fig.2 we should find a higher order fan diagram where the lower BFKL ladder on the lhs 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 rhs 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 Fig.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: (y 2 − y )(V 4 D 2 (y ))(1, 2; 3 4 ). (4.12) This finishes our discussion of the inhomogeneous terms on the rhs of (4.3), For the partition (12)(3456) we therefore can write the following integral equation for the combination D (4.13) We illustrate this equation in Fig.10.

Two triple pomeron vertices
Let us now analyze the integral equation illustrated in Fig.10. Most strikingly, we notice that the subsystem (12) has been separated from the subsystem (3456): the term D I;R(I+J) 6 on the lhs and on the rhs (illustrated in Fig.8a), as well as the first three terms in the second line of Fig.10 all contain V 4 , and below we have a separate cylinder for the (12) subsystem.  To see this in more detail, we first note that the reggeizing terms D I;R(I+J) 6 ( Fig.8a) in rapidity space can be written as dy G 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 Fig.8a: starting from above, we evolve up to rapidity y where the two-gluon state merges into the four gluon state. Below, the 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).
Next, for the inhomogeneous terms T 1 + T 2 + T 3 we note that -formally -we could solve the integral equation (4.13) by introducing the Green's function G 6 : in rapidity space we put In the large-N c limit there is no interaction between the systems (12) and (3456), such that in the Green's function we can simplify: Consequently, the Green's function G 6 splits into a product: (3,4,5,6), (4.20) where each of the factors satisfies its own equation: We now use this decomposition to evaluate the three terms on the rhs of (4.17). For T 1 we find: dy G 2 (y 1 − y ; 12)G 4 (y 2 − y , 3456)(T I + T J )D 2 (y ). (4.22) Comparison with Fig.10 shows that the transition from 2 to 4 gluons occurs at rapidity y . 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) we write for the T 2 term: (12, 3 4 ). (4.24) Similarly, for the T 3 term dy G 2 (y 1 − y ; 12) dy G 4 (y 2 − y ; 3456)K 4←2 G 2 (y − y ; 3 4 ) · ·(V 4 D 2 (y ))(12, 3 4 ). (4.25) Again, Fig.10 provides an illustration of these two equations.
All three terms have the common Green's function in the subsystem (12). In order to fully separate the evolution of the pair (12), we choose y 1 < y 2 , fix the rapidity y = y 1 , and use the property G 2 (0) = 1, (4.26) i.e. the pair (12) does not evolve in rapidity. This then leads to the following expression for the rhs of (4.17): (12, 3 4 ).
This equation has to be compared with the solution of D 4 in eq.(2.17). On the rhs 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 rhs 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 rhs of (4.27) correspond to the second and third terms on the rhs 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 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) , Figure 11: Two more fan diagrams and has the fan structure illustrated in Fig.2. Therefore we conclude that the rhs of (4.27) which had been obtained from evaluating the rhs 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) (Fig.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 Fig.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.
Let us first return to D R 4 , which we have discussed already at the end of section 2.2. First a word about the order of this term. Compared to D 2 , D R 4 is of a higher order in the coupling constant : it contains an extra α s . Projecting onto the target color configuration where the two pairs (12) and (34) form singlets, it has the order α 2 s N 3 c (1 + n=1 α s N c y) where the sum is provided by evolution. This can be compared to the contribution from the triple pomeron, which has the order α 3 N 4 c y(1 + n=1 α s N c y) where again the sum is given by evolution. As one observes these two contributions are of the same order provided α s N c y ∼ 1 which is assumed in the leading logarithmic approximation. Separating the contribution from the impact factor of the order α s N c one observes that the triple pomeron without evolution has the same order as the reggeized term with a single interaction inside the BFKL chain.
Next we recapitulate our interpretation of D R 4 . Starting from its simplest form, the coupling of a BFKL pomeron to the nucleus is given by the coupling to a single nucleon

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 [19,20] was known, describing a C-odd bound state of three reggeized gluons. Soon afterwards a new solution with intercept one was found in [18] (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 [14,15]. This system was analyzed in rather crude approximation in [25] where the angular dependence was drastically simplified. Recently this very system was investigated in detail in the JIMWLK approach and solved numerically, albeit without the pomeron → two odderon transition [26]. 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 Fig.13. The first example, for the case of four reggeized gluons, was D R 4 ( Fig.6b). For the six-gluon case we first found D R 6 (see (3.8). As illustrated in the first line of Fig.13 (see also Fig.6b and Fig.12a), 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, Fig.8, we encountered new reggeizing pieces, D I;R 6 . As an example, D I;R(I+J) 6 in Fig.8a has the form of the lowest order fan diagram (second line of Fig.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 [8] .
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 [27]. 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.