Diffractive scattering on the deuteron projectile in the NLO: triple interaction of reggeized gluons

High-mass diffractive production of protons on the deuteron target is studied in the next-to-leading order (NLO) of the perturbative QCD in the BFKL approach. The non-trivial part of the NLO contributions coming from the triple interactions of the exchanged reggeons is considered. Analytic formulas are presented and shown to be infrared free and so ready for practical calculation.


Introduction
In the perturbative QCD collisions on heavy nuclear targets have long been the object of extensive study. In the BFKL approach the structure function of DIS on a heavy nuclear target is given by a sum of fan diagrams in which BFKL pomerons propagate and split by the triple pomeron vertex. This sum satisfies the well-known Balitski-Kovchegov equation derived earlier in different approaches [1,2]. The corresponding inclusive cross-sections for gluon production were derived in [3,4]. Description of nucleus-nucleus collisions has met with less success. For collision of two heavy nuclei in the framework of the Color Glass Condensate and JIMWLK approaches numerical Monte Carlo methods were applied [5,6,7]. Analytical approaches, however, have only given modest approximate results [8,9,10]. The methods developed so far refer to the collision with heavy nuclei and rely basically on the semi-classical approximation valid in the limit of a very small coupling constant where the amplitudes behave as 1/g 2 . To move to nuclei with the smaller number of nucleons one needs to develop an approach which explicitly uses the composition of the nucleus as a composite of a few nucleons which interact with the projectile in the manner specific to the Regge kinematics. This approach is clearly provided by the framework of the exchange of reggeized gluons which combine into colorless pomerons or higher BKP states [15,16](the BFKL framework). To understand the problem one of the authors (M.A.B) turned to the simplest case of nucleus-nucleus interaction, namely the deuteron-deuteron collisions [11,12]. It was found that in this case the diagrams which give the leading contribution are different from the heavy nucleus case and include non-planar diagrams subdominant in 1/N c where N c is the number of colors. In [13] this technique was applied to the diffractive proton production off the deuteron and it was found that the contribution from the diagrams involving both nucleons in the deuteron is dominant with respect to simple triple pomeron diagrams connected to only one deuteron component. In the reggeized gluon language interaction between two pairs of reggeized gluon dominates over such interaction with only one initial pair.
Higher orders provided by additional BFKL interactions give new terms which at large rapidities have the same order and lead to appearance of fully developed BFKL pomerons and BKP states formed within the deuteron. However, there also appear a true second order terms, which correspond either to second order BFKL interaction or to new elements in the interaction involving not two pairs of reggeized gluons (reggeons) but three pairs of them. Such a NLO interaction was studied in the odderon problem [14], where the corresponding expression were derived and, what is most important, absence of infrared divergence was demonstrated. With that this NLO interaction became ready for the numerical analysis.
In this paper we study the NLO interactions involving three pairs of reggeons for the diffractive production of the protons off the deuteron, the process studied in LO in [13]. This problem is much more difficult than a similar odderon one due to less symmetry and appearance of certain new diagrams which are absent for the odderon just by absence of symmetry. These extra diagrams prevent using the technique of [14] based on the earlier found expression for two gluon emission in [17], which substantially simplified calculation for the odderon. Unfortunately we are bound to use a novel technique and perform rather cumbersome calculations. We shall discover that these extra diagrams individually contain infrared divergency of a very unpleasant character. Our main result is to demonstrate that after summation of all numerous contributions these infrared divergency is canceled and as for the odderon in [14] the final expressions are ready for numerical study.
The NLO contributions to the diffractive production of protons off the deuteron with interaction of three pairs of reggeons involve three types of diagrams shown in Fig 1. The bulk of the contribution comes from diagram in Fig. 1,A in which the intermediate gluon is produced by the process R+R→R+R+G where R stands for reggeon and G for gluon. In Fig. 1,B we show another contribution to the cross-section with two gluons in the intermediate state. Finally Fig. 1,C shows one more possible diagram for this process in which the intermediate gluon arises from two processes R+R→R+G. However, in fact this latter contribution is canceled between the direct and conjugated diagrams and need not be considered.
To conclude this introduction we recall the basic formulas connecting the diagrams with the cross-section itself. We work in the center-of-mass system of the colliding proton and one of the nucleons of the deuteron. The inclusive cross-section of the diffractive proton production d(2k) + p(l) → p(l ′ ) + X is given by where the amplitude A corresponds to Fig. 1. Let the final proton momentum be l ′ = l + λ. The missing mass is then M 2 = −4k + λ − . In terms of the overall and pomeron rapidities Y and y we have Separating the deuteron lines we standardly find (see [18]) where H is the high-energy part of A and κ + is the +-component of the momentum κ transferred to one of the nucleons in the deuteron with κ − = κ ⊥ = 0. For comparison, in the same process with a heavy nucleus projectile, the contribution from the collision with two nucleons is given by (1) with where ρ(b, z 1 ) is the nuclear density normalized to unity. The Glauber approximation corresponds to the contribution which follows when F (z) does not depend on z. Then the square of the deuteron wave function converts into the average < 1/2πr 2 > and in (5) we find integration over the impact parameter b of the square of the profile function T (b). In standard cases the high-energy part contains δ(κ + ): Then for the deuteron and for a large nucleus The paper is organized as follows. In the next section we discuss the main part of the contribution to the high-energy amplitude H corresponding to the transition R+R→R+R+G for the production of the intermediate gluon realized by vertex Γ RR→RRG . Next we discuss the two-gluon intermediate state corresponding to Fig. 1,B. In the last section we make some conclusions. Some long and cumbersome calculations are transferred to the three Appendices.

Contribution from the RR→RRG vertex
The diagram which describes the NLO corrections due to RR→RRG vertex Γ RR→RRG is shown in Fig. 1,A in the introduction. It should be supplemented by a similar diagram with interchanged nucleons in the deuterons and conjugated contributions. The interchange of the nucleons does not change the amplitude. due to the symmetry of the vertex respective to permutations of both the two incoming reggeons and the outgoing reggeons. So it is sufficient to study the diagram in Fig.  1,A and double its contribution. Vertex Γ RR→RRG itself does not depend on impact factors and does not feel evolution. So finding its contribution can be simplified by suppressing evolution and taking some simple impact factors for the pomerons. We take simple quarks for the four scattering centers assuming that their interaction is due to colorless exchange.
The total number of transferred momenta is 7: q 1 , q 2 , q 3 , r 1 , r 2 , r 3 , and q 4 = r 4 . The momentum of the real gluon p is p = q 1 + q 2 − r 1 − r 2 . We choose as independent momenta q 1 , r 1 and p. Then we have So we have 6 longitudinal integrations. There are 5 conditions arising from mass-shell conditions for real intermediate particles and sums of direct and crossed diagrams for the rest, which give: multiplied by 4s 2 . Integration over q 1+ , q 1− and r 1+ are done withe the help of δ-functions, which puts q 1+ = κ + , q 1− = r 1+ = 0. Of the three integrations over p ± and r 1− the δ functions (2π) 2 δ(p 2 )δ(p − + λ − ) allow to integrate over p ± and we are finally left with only one longitudinal integration over Apart from the four pomerons the diagram of Fig. 1,A involves the Lipatov vertex on the left f b 3 a 3 c L(−p, r 3 ) where a 3 , b 3 and c are the color indices of the two reggeons 3 (incoming and out going) and of the real gluon. It does not depend on longitudinal variables and enters only the transversal integral. On the right we meet the RR→RRP vertex of the structure Γ a 2 a 1 c (q 2 , q 1 |r 2 , r 1 ) = f a 2 a 1 c Γ(q 2 , q 1 |r 2 , r 1 ) − Γ(q 1 , q 2 |r 2 , r 1 ) .
It does depend on longitudinal variables and is symmetric in r 1 , r 2 and antisymmetric in q 1 , q 2 .
Doing summation over colors we obtain the imaginary part H 1 coming from Fig. 1,A as where τ ⊥ is the transverse phase volume, y is defined as before via M 2 . We include in each pomeron factor N c g 2 . Then in the final factor just g 4 appears. Note that 2s 2 /p − = 8s 2 k + /M 2 .
In (9) we have used that the momentum part of Γ is antisymmetric under q 1+ ↔ q 2+ , since the color factor is antisymmetric under this exchange. In the vertex Γ we have to take longitudinal variables in accordance with our results q 1+ = κ + , q 1− = 0, q 2+ = p + − κ + , q 2− = 0, r 1+ = 0, r 2+ = 0, and the transverse momenta inside the pomerons are constrained by The interchange of the two projectiles in the deuteron plus complex conjugate contribution multiply (9) by factor 4. Note that in contrast to more or less trivial cases of the scattering off a composite the obtained expression does not contain δ(κ + ), which could lift the integration in (4) and bring the resulting cross-section into the Glauber form. In our case κ + appears in the cross-section via longitudinal momentum q 1+ = κ + . Therefore in (9) after doing the integration over r 1− one has to perform integration over q 1+ with the weight dictated by (4). As we shall see this integration goes over a finite interval of rapidities of the order δ due to the condition that all intermediate gluons should lie at finite rapidity distances from the real intermediate gluon. After integration over q 1+ one finds first terms proportional to δ, which should be dropped, since hopefully they are to be canceled by other terms of the same order, namely terms with pairwise interaction between reggeons, with extra BFKL interactions or with this interaction in the second order. The rest terms are well convergent and do not depend on δ nor on the exponential in (4), since the exponent is small. These terms lead to F (z) independent of z and thus to the same Glauber expression (7) for the amplitude with As mentioned, the left Lipatov vertex in the diagram in Fig. 1,A is on-shell and does not depend on r 1− nor on q 1+ So we have to longitudinally integrate only the vertex RR→RRP. Therefore combining the coefficients in (9) and (2) our result final result for the cross-section from the vertex Γ has the form where M = L(r 3 − q 3 , r 3 )Im dκ + dr 1− 4π 2 Γ(q 2 , q 1 |r 2 , r 1 ) − Γ(q 1 , q 2 |r 2 , r 1 ) .
The total vertex Γ is a sum of 5 pieces Γ = 5 i=1 Γ i . The diagrams with vertices Γ i are shown in Fig. 2. Actual longitudinal integrations in Γ i , i = 1, ...5 are long and tedious. They are described in Appendices 1,2 and 3. To avoid proliferation of notations we denote the resulting integrated Γ with the same letter Γ. Collecting our results derived in the appendices, we find for different pieces the following expressions.
where V 0 is given by (65).
where V 1 is given by (70). Here +(Sym) means addition of The most complicated part comes from Γ 5 = Γ A + Γ B We find where φ 2 is the angle between p and q 2 − r 2 , 0 ≤ φ ≤ π and coefficients T n , U 0 and V 0 are given by (74). where φ 1 is the angle between p and q 1 − r 1 , 0 ≤ φ ≤ π and coefficientsT n ,Ũ 0 andṼ 0 are given by (76).
Separate terms in Γ 3 , Γ 4 and Γ 5 contain non-integrable divergence at q 1 → r 1 , r 2 and q 2 → r 1 , r 2 . However, in Sec. 7.6 it is demonstrated that these singularities cancel in the sum of all Γ i , i = 3, 4, 5.
In (13) the integrated Γ is to be multiplied by the Lipatov vertex The integrated Γ contains products (ae) ⊥ with different a. Summation over polarization transforms So in the end one obtains M as a well defined function of transverse momenta ready for practical evaluations.

Two intermediate gluons
To begin we find that of the two additional contributions shown in Fig. 1, B and C only the first with two intermediate gluons gives non-zero contribution. Indeed the contribution to the unitarian from the diagram in Fig. 1, C cancels between the amplitude and its conjugated term. In this diagram on the right-hand side (rhs) from the cut we find purely imaginary contribution due to the single incoming reggeon q 1 . On the left-hand side (lhs) we have a similar purely imaginary quantity plus an extra gluon q 4 = r 4 , which gives +i. So the total contribution to the unitarian is imaginary and will be canceled by the conjugated one. So we are left only with the contribution from the two-gluon intermediate states, Fig. 1, B. This diagram is quite similar to the diagram in Fig. 2,D but with a different cut. This makes its calculation somewhat different.

Longitudinal integration
The cut separates the diagram into two parts: lhs and rhs. For the lhs we find where we denoted L = L(−p, r 3 ) and e 1 = e(k 1 ) and e 2 = e(k 2 ) are the 4-dimensional polarization vectors with e + = 0. For the rhs we have Integrations over q 1+ and q 2 + are done using the two δ functions. As a result we get factor 1/(4r 1− r 2− ) and q 1,2+ become expressed as The final longitudinal integration is over r 1− with r 2− = λ − − r 1− In the denominator appears Putting So we see that the contribution from diagram Fig. 1,B contains the same denominator as the contribution from Γ 5 (see Appendix). However, as we discuss in Sec. 7.5 the collinear singularity from D = 0 is spurious, since the numerator vanishes. Rhs does not depend on r 1− . Lhs contains factors depending on r 1− Making explicit the xdependence we have in lhs Naturally this expression changes sign if k 1⊥ ↔ k 2⊥ and x ↔ 1 − x. The form (19) is convenient for the study of the limit k 1⊥ → 0.
So lhs contains singular factors 1/x and 1/(1 − x) and grows linearly with x at large x. The singularities at x = 0 and x = 1 are to be integrated in the principal value sense. At large x the integrand reduces to 1/x and the integral over the whole axis converges. The longitudinal integral over r 1− takes the form We define transverse vectors Then after summation over polarizations we get where (L, k 2 − k 1 ) is given by (19). Doing the products in (21) and (22) we rewrite them in terms of transverse products 1 8 The integrals over x are all standard.
I and finally . In these and the following formulas all vectors are 2-dimensional Euclidean.
Using them we finally find for the integrated quantities and

The cross-section
Apart from factor Z 1 − Z 2 + Z 3 the contribution to the high-energy part will include color, longitudinal and transverse factors, which can be readily read from the diagram. The final longitudinal factor comes from −1/λ − in D and factors 2k + from each projectile quark and 2l − from each target one, which gives the total −16k 2 + l 2 − /λ − . The color factor (1/2)N 4 c is the same as in Fig. 2,D.
The transverse factor T is obtained after transverse integration of the sum Z 1 − Z 2 + Z 3 with the pomerons coupled to the projectiles and targets where t = −λ 2 ⊥ . As before we include in each pomeron factor N c g 2 . Then the final extra factor will be just g 4 .
Note that Z i i = 1, 2, 3 contain singular terms. They first come from function I(k 1 , k 2 ), which is singular when k 1⊥ is parallel to p ⊥ (collinear singularity) and also when one of the 2-dimensional vectors k 1 , k 2 or p goes to zero. The first singularity goes, since the coefficient vanishes as the 4-vectors become lying in the same direction. The second singularity is integrable by itself but it may be accompanied by explicit singularities in Z's which have the structure (kp) ⊥ /k 2 where k is any of the two dimensional vectors k 1⊥ , k 2⊥ or p ⊥ . This combined singularity is canceled after averaging over angles and all the rest singularities turn out to be integrable.
To see this we present expressions for Z i in the limit k 1⊥ → 0. From (26) we find directly For Z 2 we find after making the change k 1⊥ ↔ k 2⊥ Finally Inspecting these expressions we see that in the limit k 1⊥ → 0, apart from integral I, Z 1 and Z 3 remain finite and Z 2 has a singularity proportional to (pk 1 ) ⊥ /k 2 1⊥ . This latter singularity is, as mentioned, liquidated after integration over the angle between k 1⊥ and p ⊥ . So in the end the only remaining singularity is in I and it is integrable.
Dividing by 2 to have the imaginary part we finally find for the diagram The contribution to the cross-section will be given by The total contribution will be given by twice the sum of the contributions given by the diagram in Fig. 2B and the diagram with interchange of gluons 1 ↔ 2. We take into account that the color factor and terms Z i , i = 1, 2, 3 change sign under this interchange. So the net result will be symmetrization of the pomeron part. Thus we find the cross-section as where

Conclusions
We considered the high-mass diffraction on the deuteron in the perturbative QCD reggeon (BFKL-Barters) framework. It has already been shown in [13] that interaction with both components in the deuteron leads to the cross-section which may dominate over the naive triple-pomeron contribution.
In this paper we study the NLO contributions due to the novel structure appearing in the next order and describing the triple interactions between the exchanged reggeons. The corresponding cross-sections are presented in Eqs. (12) and (35). The important result found in relation to these cross-sections is the demonstration that they are free from infrared divergencies and so fit for the practical evaluation. As to these practical calculations we have to stress that the found NLO corrections are not the only one. Another contribution comes from the 2nd order BFKL interaction in the diagrams considered in [13]. Unfortunately one cannot use for them the results found in the study of similar correction to the BFKL equation neither in the vacuum nor octet channels, since the color structure is different in our case. Thus the study of this particular correction requires a new derivation, which, as well-known, is quite long and complicated. Because of this we postpone it to the future separate publication.
Finally we have to note that before attempting to perform practical calculations in the NLO one should try to find corrections to the LO due to appearance of the BKP states in the course of evolution. Their behavior at large energies is known and is subdominant with respect to BFKL pomeron. So one may hope that their influence is also subdominant. However, to make some concrete estimates one should be able to present their wave functions in some possibly approximate form admitting practical use. This point is also to be studied later.

General rules
Variable r 1− enters one or two Feynman denominators and also may appear in the numerators. One can find (numerically) that at fixed p the on-mass shell vertex multiplied by the polarization vector ǫ with ǫ + = 0 goes as 1/r 2 1− as r 1− → ∞ [19]. This allows to perform the integration over r 1− by closing the contour in the complex plane and taking residues at poles.
In reality the integrand is a sum of terms Γ i with i = 1, ...5 which individually do not go to zero at r 1− → ∞ and contain Feynman poles as well as poles at r 1− = 0 or r 2− = 0. Since we know that sum of these terms goes to zero at large r 1− fast enough, we can forget about the behavior of individual contributions at r 1− → ∞ and and just take the residues in, say, the lower half plane. However, it is important that in all diagrams the residues are to be taken in the same (lower) half plane. Using these rules we can do integrations in terms Γ i with i = 1, ...5 separately. Explicit expressions for the vertex RR→RRP can be taken from our paper [19].

Fig. 2,A
On mass shell multiplied by the polarization vector ǫ the corresponding amplitude Γ 1 is given by Here the denominators are t 2 k 2 They do not depend on r 1− nor on r 2− We finally havē where a 1 = q 1 + r 1 . From this we find We separate the term with a pole at r 1− = 0 presenting where Y 1 does not depend on r 1− . After integration we get Here X 1 = X 1 (r 1− = k 2 1⊥ /2q 1+ ) and the integrals are and I

(2)
A 1 is the contribution of the residue at r 1− = 0 in the lower half plane of the part with Y 1 5.3 Fig. 2

,B
On mass shell multiplied by the polarization vector ǫ the corresponding amplitude Γ 2 is given by Here the color factor is C 2 = N c f a 2 a 1 c and the denominators arē . The coefficientsā, ...ē areā Again we find some terms with poles at r 1− = 0 and can present We get the result 1⊥ /2q 1+ ) and the integrals are and I

Fig. 2,C
This diagram generates two terms with different color factors. The corresponding amplitudes Γ 3 and Γ 4ǫ are given by where C 3 = −C 2 and C 4 = C 1 and the denominator is k 2 1 = −2q 1+ r 1− + k 2 1⊥ + i0. We have Here a i = q i + r i i = 1, 2.
As before we separate terms with poles at r 1− = 0 and r 2− = 0. where The longitudinal integrals are the same for both parts and the same as for Γ 2 . From the pole at r 1− = k 2 1⊥ /2q 1+ we get Γ 3,4 = C 3,4 X
6 Appendix 2. Integration over κ + . Light-cone poles To find the final expression for the diffractive cross-section we have to study the eventual integration over κ + = q 1+ that is over q 1+ or q 2+ with weight exp(iu), u = zmκ + /k + . This integration forms function F (z) according to (4). We separate it in two parts: of the terms which follow from the poles at r 1− = 0 or r 2− = 0 studied in this section and of those which follow from the Feynman poles to be studied in the next section. The characteristic of contributions from light-cone poles is that they do not restrict in any way the region of integration in κ + , which can vary from −∞ to +∞.
We shall study contributions with q 1+ = κ + , so we can use the formulas of the previous subsection and integrate over q 1+ . We call the parts of the amplitudes coming from light-cone poles as Γ (2) i , with i = 1, ...5. In our amplitudes listed in the preceding section there appear the following integrals over q 1+ , which follow from the residues at r 1− = 0 or r 2− = 0. In Γ 1 we have In Γ 2 we find three integrals In Γ 3 we have In Γ 4 there appear two integrals I 3 and I 1 . Finally in Γ 5 we find two integrals , n = 1, 0.
We only need the real parts of these integrals, since they appear with purely imaginary coefficients after taking the residue. In fact they are reduced to just two integrals I 1 and I 3 . Integrals I (n) 2 with n = 1, 0 can be rewritten as If we neglect w 2 in the exponent the two integrals reduce to I 3 and I 1 . Integrals can be presented as a difference

Now the two basic integrals are
which does not contribute to the Glauber approximation, and which is pure imaginary and odd in z. So it will give zero in the amplitude. So light-cone poles do not contribute to the diffractive amplitude.

Integration regions and problems
Integration over q 1+ should be done taking into account limitations coming from the requirement that the rapidity y 1 of the intermediate gluon with momentum k 1 or k 2 cannot be much different from y. Otherwise the diagram with RR→RRP vertex transforms into the one with R→RP vertex with an extra interaction between the reggeons with momenta q 1 and q 2 . So where one may choose δ >> 1 but much smaller than ln(s/s 0 ). Our integrals go over negative values of q 1+ or positive values of q 2+ . Putting in the first case |q 1+ | = p + x we get the condition In the second case we put q 2+ = p + x and the integration limits in x are the same with |k 1⊥ | → |k 2⊥ |. Actually the integrand in the integration over x contains an exponential factor Note that our study is only valid when M 2 → ∞, so that ξ is small. At fixed δ our integrals are naturally dependent on δ. One expects that this dependence is canceled by other contributions of the same order, which involve extra BFKL interactions between reggeons and correction to the BFKL interaction. Such contributions, also cut at rapidity δ from the fixed rapidity of the intermediate real gluon, hopefully contain terms proportional to the cut integration region in rapidity, that is to δ. Thus in our calculations terms proportional to δ actually should be dropped, since they are to be canceled by similar terms coming from other contributions. This can only be true if the divergence of our integrals at both x → 0 and x → ∞ is logarithmic.
At first sight this poses the first problem in our study. Inspecting our expressions for Γ i , i = 1, ...5 one discovers that after integration over r 1− Γ 2,3,4 contain terms which do not vanish at x → ∞. The sum of them tend to the limit at x → ∞ where color factors correspond to the contribution of diagrams 2,3 and 4. In the sum −C 2 − 2C 3 + C 4 = 1/2, so that the limiting value is different from zero. This generates linear divergence at large x in the limit M 2 → ∞. The corresponding integral is We shall see that after symmetrizing over participating reggeons this linear divergence is canceled. So after integration over q 1+ and dropping terms proportional to δ one finds the cross-section (12). However, this is not the end of the story. In the result of integration over q 1+ one finds various terms which are singular at k 2 1⊥ = 0 or k 2 2⊥ = 0 and so exhibit infrared divergence. Also we find a collinear divergence in the contribution from Γ 5 . As we shall demonstrate, the latter cancels because of the properties of X 5 . Terms singular at say k 2 1⊥ = 0 coming from Γ 3,4,5 behave as badly as 1/|k 1⊥ | 3 or ln |k 1⊥ |/|k 1⊥ | 2 . However, in the sum of all contributions all such singular terms cancel and the remaining expression is free from the infrared divergence.
There are also terms which behave as 1/p 2 ⊥ as p 2 ⊥ → 0, which may also lead to divergence. However, in our case the value of p 2 ⊥ is limited from below by the condition that the Regge kinematics should be valid for the lower pomerons. Fixing their minimal energy square s 1 as s 1 > s 0 we find the condition |p 2 ⊥ | > M 2 s 0 /2s. So if the terms which behave like 1/p 2 ⊥ remain in the total contribution they do not lead to infrared divergence but rather to the behavior ln(s/M 2 ) instead. No other dangerous singularities are found in terms generated by the vertex RR→RRP.
In our calculations we choose δ >> 1 and drop all terms proportional to δ.

Γ 1
We recall that after taking the residue at r 1− = k 2 1⊥ /(2q 1+ ) We find X where T 0,−1 do not depend on q 1+ and explicitly and T −1 = −4(q 2 e) ⊥ r 2 1 + 2k 2 1⊥ (t + 2r 1 , e) ⊥ . Note that T 0 contains a term with a pole at k 2 1⊥ = 0. T −1 does not contain this pole. Integration goes over the negative q 1+ . Putting q 1+ = −p + x we find where ) and u is given by (54). At 0 and has a weak logarithmic singularity at k 2 1⊥ = 0. So it is integrable and does not present difficulties itself. But T 0 contains a pole term in k 2 1⊥ . So the first term in (56) is in fact singular. The second term does not exist at x 1 = 0 and is also singular at k 2 1⊥ = 0. We present where J 0 is given by Here we have left only the real part, which is of interest. As a result where the new functions arê andT Note that at k 1⊥ = 0 we have t ⊥ = q 2⊥ . Therefore at k 2 1⊥ = 0 the pole inT 0 is weakened to 1/|k ⊥ |. This means that the first term in (58) is integrable. Singularities are separated in the second term which contains both the singularities at x 1 = 0 and combined singularities at x 1 = 0 and k 2 1⊥ = 0. Both of them are proportional to δ and have to be discarded.

Γ 2
After taking the residue at r 1− = k 2 1⊥ /(2q 1+ ) One finds where T i do not depend on q 1+ and are given by Here In T 1 the four lines correspond to contributions fromāA, (b +c)B,c(C − B) andēE respectively. All of them contain poles in k 2 1⊥ = 0. Integration over q 1+ gives In contrast to Γ 1 these integrals are have a finite limit at k 2 1⊥ = 0, although I −1 is singular at x 1 = 0. To separate the singularities we present where J 1 is given by (55) to finally obtain with a new functionT Remarkably inT 1 , which is a sum of functions each having a pole at k 2 1⊥ = 0, the leading singularity at k 2 1⊥ = 0 cancels. Indeed the terms proportional to q 2 1 /k 2 1⊥ in (63) are (modulo At k 2 1⊥ = 0 we havet ⊥ = −r 2⊥ and r 2⊥ − q 2⊥ = −p ⊥ . Term p 2 ⊥ T 2 /t 2 ⊥ cancels the first term in T 1 . The second term in T 1 becomes and cancels the last term in T 1 . Finally witht 2 ⊥ = r 2 2 the term p 2 and cancels the remaining third term in T 1 . So the factor multiplying q 2 1 /k 2 1⊥ in (63) vanishes at k 2 1⊥ = 0. Numerical studies show that in fact at k 1⊥ → 0 this term behaves as |k 1⊥ |. This means that the last term in (62) is integrable and so regular. The contributions singular at x 1 → 0, k 2 1⊥ → 0 or simultaneously in these two limits are contained in the first two terms. As we shall see the term with J 1 will be eventually canceled. The second term is proportional to δ and should be dropped according to our approach. So the meaningful part of Γ 2 is contained in the third term in (62).

Γ 3
After taking the residue at r 1− = k 2 1⊥ /(2q 1+ ) we have Note that at r 1− = k 2 1⊥ /(2q 1+ ) we have We find where Integration over q 1+ = −xp + gives Here J 1 and J 0 are the old integrals which do not depend on any variables and I −1 is new: .
So we finally have The integral I 0 converges at x 1 = 0 and has a weak (logarithmic) singularity at k 2 1⊥ = 0. It is integrable over k 1⊥ by itself but V 0 contains a pole term in k 2 1⊥ . So dropping the term with J 1 , apart from the trivial singular term with J 0 we get a strongly singular term which is not integrable in k 1⊥ .

Γ 4
After taking the residue at r 1− = k 2 1⊥ /(2q 1+ ) we have where X (1) 4 = X 4 (r 1− = k 2 1⊥ /(2q 1+ )). So we find X where Integration over q 1+ = −xp + gives Dropping the term with J 1 we find a trivial term with J 0 and a highly singular term similar to that in Γ 3 It differs from a similar term in Γ 3 only by the sign. However, since C 3 /C 4 = 2 these terms do not cancel in the sum. 1. Case A: r 2− = k 2 2⊥ /2q 2+ (q 2+ > 0). We express q 1+ = p + − q 2+ . We then have The common denominator is Putting q 2+ = xp + we rewrite it as As before we present where In these formulas 2. Case B: r 1− = k 2 1⊥ /2q 1+ (q 1+ < 0). We express q 2+ = p + − q 1+ . We then have The denominator is Putting q 1+ = xp + with x < 0 we rewrite the denominator as Presenting again we findT In these formulas We pass to the final integration of Γ 5 over q 1+ or q 2+ .
Integration of X A 5 . We have to integrate over the transferred "+"-momentum κ + , which is equivalent to integration over q 1+ or q 2+ . In case A it is convenient to integrate over q 2+ = xp + with x > 0. Before integration The denominator can be written in Euclidean metric as Presenting X (A) 5 according to Eq. (73) we shall have the integrated Γ A 5 as where , and all vectors are to be taken in Euclidean 2-dimensional metric (E2DM) with k 2 2 , p 2 ≥ 0. All integrals are well convergent at small x so that we can safely put x 1 = 0. At large x I A n with n = 1, 2 diverge and we have to separate the diverging terms. All integrals are standard and apart from these divergent terms they are expressed via integral We start from T 1 . Terms of order 1/k 2 2 come from coefficient α 2 = 1 − 2q 2 2 /k 2 2⊥ . Collecting them we find in the limit k 2 → 0 (in E2DM) Next we find that terms proportional to 1/k 2 2 are in fact absent in T 0 and U 0 . We are left with T −1 and V 0 . In the limit k 2 → 0 (also in E2DM) The singular terms from T 1 , T −1 and V 0 give in the sum 4 (pk 2 )q 2 2 k 2 2 p 2 (q 1 e) − q 2 1 (pe) + 4 At k 2 = 0 we have q 2 = r 2 so that all power-like singularities cancel. So the only dangerous terms at k 2 → 0 are the logarithmic ones. They come only from the contribution proportional to V 0 and lead to the singular term (in Lorentz metric) To finish with case A we consider behavior at k 1 → 0. Obviously in this case only terms with U 0 and V 0 may lead to non-integrable singularities. So we have to calculate values of U 0 and V 0 at k 1 → 0 and so k 2 → p. Elementary calculations give in E2DM As a result the singular terms proportional to (pk 1 )/k 2 1 cancel, so that Γ A is integrable at k 1 = 0. Its only singularity is at k 2 → 0 and given by (81) Integration of X B 5 . Case B is considered quite similarly. Now we integrate over q 1+ = xp + with x < 0. Before integration The denominator can be written in Euclidean metric as Presenting X according to Eq. (75) we shall have the integrated Γ B 5 as where tildes mean case B. The integrals are 1 )(k 1 − xp) 2 and all vectors are taken in E2DM. All integrals are well convergent at small x i → 0 so that we van safely put x 1 = 0. At large |x| I B n with n = 1, 2 diverge and we have to separate the diverging terms. All integrals are expressed via integral where φ 1 is the angle between k 1 and p, 0 < φ 1 < π. All the rest integrals are given as follows The denominator in I B 0 is the same as in I A 0 (see (18)), the difference between them is in the numerator. As in case A the basic integral I B 0 contains both a collinear singularity at φ 1 = π and comparatively weak (integrable) singularities at k 1 → 0 and k 2 → 0 where it behaves as 1/k 1 or 1/k 2 . As mentioned, the collinear singularity is in fact absent, since the whole X B vanishes when k 1 and k 2 are parallel to p. Dropping the divergent term with J 1 on the reasons discussed earlier we present the integrated Γ B as a sum of two terms, one of which is proportional to ln k 2 1 and the other proportional to I B 0 : We first analyze possible non-integrable singularities as k 1 → 0. The power-like singularities come again from singularities of integrals I B n , K B 1 and K B 2 and from the coefficientsT n ,Ũ 0 and K 0 . As in case A term withT 2 is non-singular due to factor (pk 1 ) 2 . Also non-singular terms inT 1 , T 0 andŨ 0 do not lead to non-integrable singularity as k 1 → 0. So as in case A, dangerous terms include terms of the order 1/k 2 1 inT 1 ,T 0 and terms non-vanishing as k 1 → 0 inT −1 ,Ũ 0 andṼ 0 We start fromT 1 . Terms of order 1/k 2 1 come from coefficient α 1 = 1 − 2q 2 1 /k 2 1⊥ . Collecting them we find in the limit k 1 → 0 (in E2DM) T 1 = 4 q 2 1 k 2 1 q 2 2 (pe) − p 2 (q 2 e) .
Recalling that C 3 = −1, C 4 = −1/2 and C 5 = 1/2 we find that this singularity is canceled. Thus the only singular term in the whole integrated Γ are those proportional either to J 1 or to δ. Hopefully the latter can safely be discarded due to cancelations with similar terms from the diagrams with pairwise interregional interaction. Terms with J 1 actually go after symmetrization, as will be demonstrated below.

Terms with J 1
These terms are present in linear diverging expressions in Γ i . i = 2, ...5 The corresponding contributions from Γ 2,3,4 are 2 = −(pe) ⊥ p 2 ⊥ α 1 , 1 , T 1 = −2(pe) ⊥ α 1 , 1 , T where α 1 = 1 − 2q 2 1 /k 2 1⊥ . In the sum we get We have to add to this contribution the three others obtained by interchanges (q 1 ↔ q 2 ) and (r 1 ↔ r 2 ). At these exchanges both longitudinal and transversal momentum components are to be exchanged. Exchange q 1 ↔ q 2 does not change the momentum part since one has to integrate both over q 1,2+ and q 1,2⊥ . However, this exchange gives a minus sign due to change of the color factor. Exchange r 1 ↔ r 2 does not change the color factor but we have to take residues in the lower half-plane of r 1− . This means upper half plane of variable r 2− . Therefore simultaneously with changing r 1⊥ ↔ r 2⊥ after taking the residue one should change −iθ(−q 1+ ) ↔ iθ(q 2+ ), which is equivalent to taking x → −x in our formulas plus the overall minus sign. In practice this means that in terms with J 1 we have to take the minus sign. Denoting the result of adding the interchanged contribution as Sym we get This expression should be integrated with the pomeron symmetric in r 1⊥ and r 2⊥ . Then the result of integration vanishes. So in the end no terms proportional to J 1 come from Γ 2+3+4 . This means that the sum of these vertices has no linear divergency after symmetrization. From Γ 5 we have two contributions: In the sum we find Symmetrization can be done either by changing q 1 ↔ q 2 or changing r 1 ↔ r 2 . In both cases we have the sign minus. So