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On the discrepancy of the low-x evolution kernels

It is shown that in the case of the forward scattering the most part of the difference between the M\"obius form of the BFKL kernel and the BK kernel in the next-to-leading order (NLO) can be eliminated by the transformation related to the choice of the energy scale in the representation of scattering amplitudes. Change of the nonforward BFKL kernel under this transformation is derived as well. The functional identity of the forward BFKL kernel in the momentum and M\"obius representations in the leading order (LO) is exhibited and its NLO validity in N=4 supersymmetric Yang-Mills theory is proved.


Introduction
The Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework for the description of semihard processes in QCD was developed originally in the momentum representation. In the leading order (LO) it was done in Refs. [1,2]; the next to leading (NLO) corrections to the kernel of the BFKL equation were obtained in Refs. [3] - [7]. The field of applicability of the BFKL approach is quite wide. It includes scattering processes with arbitrary colour exchange.
For scattering of colourless particles an alternative framework to the description of the semihard processes is the colour dipole model [8]. In contrast to the BFKL approach this model is formulated in the impact parameter space. The equation describing in the LO evolution of scattering amplitudes with energy in this model has a simple non-linear generalization, which is called Balitsky-Kovchegov (BK) equation [9]. The NLO corrections to the kernel of the BK equation in the coordinate space have been calculated recently in Refs. [10] - [12].
These two approaches should give the same predictions in the common area. This requirement is definitely fulfilled in the LO. Indeed, for scattering of colourless particles the LO BFKL kernel can be taken in the Möbius representation, which is invariant with respect to the conformal transformations of the transverse coordinates [13]. In the impact parameter space the Möbius representation of the BFKL kernel coincides with the kernel of the colour dipole model [14] (in Ref. [14] it was called therefore dipole form of the BFKL kernel). Moreover, it was shown [15] that the LO BK equation appears as a special case of the nonlinear evolution equation which sums the fun diagrams for the BFKL Green's functions in the Möbius representation.
In the NLO one could expect coincidence of the Möbius form of the BFKL kernel and the kernel of the linearized BK equation. However, the situation is not so simple. First of all, the NLO kernels are not unambiguously defined. The ambiguity of the NLO kernels is analogous to the ambiguity on the NLO anomalous dimensions. It is caused by the possibility to redistribute radiative corrections between the kernels and the impact factors. This freedom in the definition of the kernels allows one to reshape the Möbius form of the BFKL kernel in order to prove its equivalence to the BK one [14].
In QCD the NLO kernels consist of two parts: the quark and the gluon ones. The quark part of the BK kernel was found in Refs. [10] and [11]. The Möbius form of the quark part of the BFKL kernel was obtained in Refs. [14] and [16] from the quark contribution to the BFKL kernel calculated in the momentum representation in Ref. [5]. It was proved [14,16] that with account of the freedom mentioned above this form is equivalent to the quark part of the linearized BK kernel. The Möbius form of the gluon part was obtained in Ref. [17] with use of the gluon correction to the BFKL kernel calculated in Refs. [6] and [7] in the momentum representation. The gluon correction to the BK kernel was found in Ref. [12]. It occured that for the linearized BK equation this correction strongly differs from the Möbius form obtained in Ref. [17].
In this paper we demonstrate that for the case of forward scattering the most part of the difference can be eliminated by the transformation related to the choice of the energy scale in the representation of scattering amplitudes. We also show how the simplest generalizations of this transformation change the nonforward kernel.
The second goal of our work is to exhibit the functional identity of the forward BFKL kernels in the momentum and the Möbius representations and to prove that in the N=4 SUSY Yang-Mills theory this identity remains valid in the NLO. The extension of the BFKL framework to the supersymmetric theories was started in Ref. [18], where the kernel for the forward scattering was found for the N=4 SUSY in the momentum space with the dimension D = 4 + 2ǫ and in the eigenfunction space. This analysis has been recently expanded in Ref. [19], where the nonforward Möbius NLO BFKL kernel was obtained for the SUSY Yang-Mills theories with any N. We continue this line finding the forward kernel in the momentum space for any N, writing it at D = 4, with all singularities cancelled, and demonstrating the functional identity of this kernel to the forward kernel in the Möbius representation for N=4.
Our paper is organized as follows. The next section introduces our notation and gives a brief account of the main results of Ref. [17]. Section 3 goes through different transforms of the kernel which do not change observables. Section 4 shows that the difference between the Möbius form of the BFKL kernel and the kernel of the linearized BK equation can be partially eliminated by the suitable transform. Section 5 demonstrates the functional identity of the forward BFKL kernels in the momentum and Möbius coordinate representations. Section 6 discusses how the generalizations of the transformation obtained in section 4 change the nonforward kernel. Section 7 summarizes the main points of our paper. Appendices describe the details of the calculations.

General overview
Our notation is the same as in Refs. [14] and [17]. Thus the Reggeon transverse momenta (and the conjugate coordinates) in initial and final t-channel states are q ′ i ( r ′ i ) and q i ( r i ), i = 1, 2. The state normalization is Here ǫ = (D − 4)/2; D − 2 is the transverse space dimension taken different from 2 to regularize divergences. We will also write for brevity p ij ′ = p i − p ′ j . The s-channel discontinuities of scattering amplitudes for the processes In this expression s 0 is an appropriate energy scale,K is the BFKL kernel, q A = p A ′ A , q B = p BB ′ , and with ω(t) being the gluon Regge trajectory, andK r representing real particle production in Reggeon collisions, where p ± = (p 0 ±p z )/ √ 2; the kernel K r ( q 1 , q ′ 1 ; q) and the impact factors Φ are expressed through the Reggeon vertices according to Ref. [20].
The general form of the Möbius (dipole) kernel in the coordinate representation reads [14,17]: (6) Here r iρ = r i − ρ, and the whole kernel is symmetric with respect to simultaneous 1 ↔ 2 and 1 ′ ↔ 2 ′ substitution.
The quark contribution to the functions g was calculated in Refs. [14] and [16] and after a suitable transform was shown to coincide with the BK result of Ref. [11]. The gluon contribution to these functions was found in the BFKL approach in Ref. [17] and in the dipole one in Ref. [12]. The latter two results are different.
The BFKL framework gives for the gluon contribution where and C = −ψ (1) is the Euler constant; µ is a renormalization scale in the MS-scheme. At once we will emphasize that only the integral of g 0 ( r 1 , r 2 ; ρ) contributes to the kernel. Therefore one can write g 0 in various forms, e.g. in our previous papers we used the equalities to reshape it. But anyway g 0 ( r 1 , r 2 ; ρ) does not coincide with g( r 1 , r 2 ; ρ). For the third function g we have where The functions g( r 1 , r 2 ; r ′ 2 ) and g( r 1 , r 2 ; r ′ 1 , r ′ 2 ) vanish at r 1 = r 2 . In Ref. [17] the latter function was presented in a form where this property was obvious. For integration, however, it is more convenient to use the expression (12) .
At the same time the gluon part of the kernel of the linearized BK equation gives [12] : g BC ( r 1 , r 2 ; r ′ 2 ) = where ln r 2 µ BC = − ln µ 2 − 3 11 g BC ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) = ln 3 Ambiguity in the definition of the kernel To begin with, we are going to discuss the transformations of the kernel which do not affect observables.
First of all, disc s A A ′ B ′ AB in Eq.
(2) remains intact if one changes both the kernel and the impact factors via an arbitrary nonsingular operatorÔ: If the LO kernel is fixed by the requirement that its Möbius form coincides with the kernel of the dipole model, one can shift the NLO contribution usingÔ = 1 −Ô, withÔ ∼ α s , and get where the superscript (B) means the LO kernel. Note that the Möbius form calculated in Refs. [14] and [17] and presented in the previous section corresponds to the kernel obtained by the transformationK where β 0 is the first coefficient of the beta-function, from the kernel defined in Eq. (3). This transformation simplifies the Möbius form and allows to alter the quark part of this form so that it agrees with the result of Ref. [11]. Secondly, there is a freedom in the energy scale s 0 . At first sight, it can lead to an additional ambiguity of the NLO kernel. However, it is not so. Indeed, it was shown [21] that any change of the energy scale can be compensated by the corresponding redefinition of the impact factors. An experienced reader can wonder remembering that in Ref. [3] the scale transformation was associated with the change of the kernel. The matter is that in Ref. [3] one of the impact factors was fixed. Actually, instead of transforming both impact factors one can compensate any change of the scale by transformation of one of the impact factors and the kernel. Evidently, in this case the transformation of the kernel has the form (19) with some specific form of the operator O. Let us discuss this question more in detail.
We begin with the case when s 0 depends only on properties of scattering particles. Just this case was supposed in the definition of the kernel and the impact factors in Ref. [20]. Note that s 0 can be taken as a free parameter. This freedom can be used for optimization of perturbative results [22]. A natural choice is s 0 = Q A Q B , where Q A and Q B are typical virtualities for the impact factors Φ A ′ A and Φ B ′ B correspondingly. Let us consider the transition from such scale to the scale depending on the Reggeon momenta q Ai and q Bi , i = 1, 2, in the impact factors Φ A ′ A and Φ B ′ B respectively: Remind that with the NLO accuracy for any s-independent value c one has Therefore one can write whereF It means that the discontinuity (2) remains unchanged if the change of the scale (23) is accompanied by the change of the impact factors It is possible to leave one of the impact factors (let us take for definiteness A ′Ā |) invariable changing the kernel. Indeed, Therefore instead of the change (25) one can takê where, with the NLO accuracy,K We see that the change of the energy scale can be associated with the transformation of the kernel (19) with the specificÔ. At last, the Möbius kernel (6) is defined with an accuracy to any function independent of r 1 or of r 2 such that after their addition the kernel remains zero at r 1 = r 2 [14,17]. Therefore, one can add to the kernel only the functions which are antisymmetric with respect to the substitution r ′ 1 ↔ r ′ 2 . These functions do not change the symmetric part of the kernel. But this part alone plays a role in the observables. As a result, the third term in the expression (12) in the BFKL kernel, ln and the term ln in the BK kernel (17) give the same contribution to the amplitudes. The ambiguities of the NLO kernels give a hope that the results of Refs. [17] and [12] can be matched.

The kernel for forward scattering in gluodynamics
For a start let us find the gluon contribution to the forward Möbius BFKL kernel and compare it to the BK result obtained in Ref. [12]. We define the matrix element of the forward kernel in the momentum representation as Next, using the state normalization (1) and the denotation r 1 = r + r 2 , r ′ 1 = r ′ + r ′ 2 , we get at physical value D = 4 where The last equality follows from the space uniformity. Thus, the Möbius form of the BFKL kernel for the forward scattering can be obtained from Eq. (33) using the results of Refs. [14], [16] and [17] for In the case of pure gluodynamics it gives Here µ is the renormalization scale in the MS-scheme, The details of the derivation are given in the appendix A.
The result of Ref. [12] for the forward case is It was derived using the relation (33). Note that in the derivation the term was omitted. As we discussed at the end of the previous section, it can be done since this term vanishes at r ′ ↔ − r ′ (i.e. r ′ 1 ↔ r ′ 2 ). Thus, for the difference of the forward kernels one has where r|K (B) The term proportional to 11/3 is related to renormalization (remind that in pure gluodynamics β 0 = 11N c /3). In our opinion, this term arose because the renormalization scheme used in Ref. [12] is not equivalent to conventional MS-scheme. It occurs that the term with logarithms can be eliminated by the transformation applied to the forward BFKL kernel. Indeed, the direct calculation given in the appendix B shows that Here the subscript M means the Möbius representation (i.e. vanishing of the matrix element at r = 0). Comparing with Eq. (28) we see that the transformation (41) corresponds to the change of the energy scale at fixed value of one of the impact factors. Actually, the transformation (41) is of the same type as in Ref. [3]: One can come to the transformation (41) from another side. The differenceK M −K BC has the same eigenfunctions where φ is the azimuthal angle, as the LO dipole kernel. The eigenvalues of the LO dipole kernel coincide with the eigenvalues of the LO BFKL kernel obtained in Ref. [2]: It becomes evident if we write the FBKL kernel for the forward scattering as and compare it with the dipole kernel It is worthwhile to note here that the kernel (46) differs from the usually used symmetric kernel; the former is obtained from the latter by the transformationK →ˆ q −2Kˆ q 2 . For the non-forward case the corresponding transformation isK → (ˆ q 2 1ˆ q 2 2 ) −1/2K (ˆ q 2 1ˆ q 2 2 ) 1/2 . Let us stress that just the transformed kernel can be written in the Möbius form r 1 r 2 |K which is invariant in regard to the conformal transformations of the transverse coordinates [13] and coincides with the kernel of the colour dipole model r 1 r 2 |K d | r ′ 1 r ′ 2 [14]. Moreover, because of this coincidence, in the forward case one can see from Eqs. (46) and (47) the functional identity of the LO BFKL kernel in the momentum and Möbius coordinate representations: q 2 q|K (B) | q ′ / q ′2 is represented by the same function as r ′2 r|K  46) and (47) it is clear that the eigenvalues must be the same as for e inφ r ′ ( q ′2 ) γ−2 , i.e. ω B (n, γ).
Using the integral (the calculation is discussed in the Appendix C) where χ ′ means derivative over γ, from Eq. (39) we obtain where ω M (n, γ) − ω BC (n, γ) is the eigenvalue of the differenceK M −K BC corresponding to the eigenfunction e inφ r ′ ( r ′2 ) γ . The first term in Eq. (49) can be written as In the space of the eigenfunctions e inφ r ′ ( r ′2 ) γ we haveK (B) = ω B (n, γ) and ln(ˆ q 2 ) = −∂/∂γ, so that for the forward scattering we obtain Evidently, the first term in Eq. (51) is eliminated by the transformation (41). The second one, as it was already pointed out, in our opinion is related to the difference of the renormalization scheme used in Ref. [12] with conventional MS-scheme and can be eliminated by change of the scheme. 1 Unfortunately, we cannot find the transformation suitable to eliminate the third term. We have to add that in the BFKL approach the term with ζ(3) passed through a great number of verifications. In particular, this term is necessary for the fulfillment of the bootstrap relations. Besides, it is confirmed by the calculation of the three-loop anomalous dimensions in Refs. [23] and [24]. For completeness we present here the characteristic function ω M (n, γ) of the kernel (34), defined by the relation Actually because of running coupling the functions e inφ r ′ ( r ′2 ) γ are not eigenfunctions ofK M anymore, and ω M (n, γ) contains ln r 2 . In Eq. (52) it can be replaced by ∂/∂γ. As the result we have and F (n, γ) = π 2 cos (πγ) The calculation of the integral (52) is discussed in the Appendix C.
The β-dependent terms in ω(n, 1 − γ) are written as α 2 s N c β 0 R/(4π 2 ), where Since − ln q 2 µ 2 χ(n, γ) q|n, γ = χ(n, γ) ln and the Möbius form corresponds to the kernel obtained by the transformation (20), which means we come to the conclusion that in (52) the β-dependent terms in ω M (n, γ) can be written in the form which is exactly the same as in Eq. (53).

SUSY Yang-Mills forward kernel
The extension of the NLO BFKL kernel to supersymmetric theories was performed in Ref. [18] for the forward case in the momentum representation and in Ref. [19] for the nonforward case in the Möbius coordinate representation. Supersymmetric Yang-Mills theories contain gluons and n M Majorana fermions in the adjoint representation of the color group. For N-extended SUSY we have n M = N. For N > 1 besides fermions there are n S scalar particles; n S = 2 at N = 2 and n S = 6 at N = 4. Following Ref. [19] for the Möbius kernel in the SUSY theories we write g SU SY = g Gluon + g F ermion + g Scalar .
One should insert these functions g into the definition of the Möbius kernel in the coordinate representation (6). Using the results of the previous section we can write the forward Möbius kernel in the SUSY case. It reads where µ being the renormalization scale in the MS-scheme.
As it is known, at N = 4 the coupling α s does not run, so that β 0 = 0. Moreover, it is seen from (67) that f SU SY 2 = 0 in this case. Next, let us express our result in the renormalization scheme which preserves the supersymmetry. This scheme is known as the dimensional reduction and it differs from the MS-scheme in the finite charge renormalization (see Ref. [19]) for details). So we get Finally, in the N = 4 case, having β 0 = 0, n S = 6, n M = 4, the kernel simplifies to Now let us consider the forward BFKL kernel in the momentum space. The explicit form of the kernel can be found for QCD in Ref. [3] and for Yang-Mills theory at N=4 in Ref. [18]. In both cases the kernel is written in the space-time dimension D = 4 + 2ǫ to regularize infrared divergencies. Here we solve two problems. First, we found the explicit form of the kernel for SUSY Yang-Mills with any N. Second, we perform explicitly the cancellation of the infrared divergencies and write the kernel at physical space-time dimension D = 4. It permits us to demonstrate that the functional identity of the forward BFKL kernels in the momentum and Möbius coordinate representations exhibited in the previous section in the LO is preserved in the NLO in the N=4 SUSY case.
To solve the first problem we must change the quark contribution in the kernel of Ref. [3] for the gluino one and add the scalar particle contribution. It is known (see Refs. [18] and [19]) that the gluino contribution to the"real" kernel can be obtained from the quark one by the change of the coefficients, n f → n M N c for the "non-Abelian" (leading at large N c ) part (including the trajectory) and n f → −n M N 3 c for the "Abelian" (suppressed at large N c ) part, so that this contribution can be found quite easy. To obtain the scalar contribution is a more subtle task. According to Ref. [19], the scalar contribution also can be divided into "non-Abelian" and "Abelian" parts. Both in the integral representation of the trajectory and in the "non-Abelian" part of the "real" kernel the scalar contribution can be obtained from the corresponding quark contribution by the substitution n f → n S N c and the change of the fermion polarization operator with the scalar one which differs from the former by the factor 1/(4(1 + ǫ)). We are interested in the kernel expanded in powers of ǫ. It is clear that since the factor mentioned above depends on ǫ this kernel can be obtained applying substitutions, which are different for the different terms in the expansion of the polarization operator. It is not difficult to see (for details see Ref. [25]) that these substitutions must be The n-th equality here corresponds to the n-th term in the expansion. Therefore from Eq. (6) of Ref. [3] for the trajectory (in the MS scheme) we obtain where The "Abelian" part of the scalar contribution to the non-forward kernel is given by Eq. (28) of Ref. [19]. For the forward case, using the Feynman parametrization and performing integration over k 1 , for D = 4 (for details see Ref. [25]) we obtain where f 2 is defined in Eq. (37). Taking into account that q|K| q ′ differs from the symmetric kernel presented in Ref. [3] by the factor q ′ 2 / q 2 , for the "real" part we obtain where k = q − q ′ , f 1 is defined in Eq. (37) and f SU SY 2 in Eq. (67). Eqs. (72) and (76) solve the first problem raised in the beginning of this section. But they contain the infrared divergencies which complicate their use. One can cancel the divergencies and write the kernel at physical space-time dimension D = 4 following Refs. [14] and [17] and using the integral representation for the trajectory The quark contribution to f ω is defined by Eqs. (76) and (77) of Ref. [14]. The gluino and scalar contributions can be obtained from the quark one by the substitutions discussed above. The gluon contribution is given, with the required accuracy, by Eq. (23) of Ref. [17]. As the result, we obtain with this accuracy Just like Refs. [14] and [17], in the limit ǫ → 0 we introduce the cut-off λ → 0 keeping ǫ ln λ → 0. Then in the regions k 2 ≤ λ 2 we have and in the region ( k − q) 2 ≤ λ 2 the same expression with the substitution k 2 → ( k − q) 2 . Comparing Eq. (79) with Eq. (76) we see that, when the kernel K( q, q ′ ) in Eq. (75) acts on any function nonsingular at q = q ′ , the contribution of the region k 2 ≤ λ 2 in the "real" part cancels almost completely the contributions of the regions k 2 ≤ λ 2 and ( k − q) 2 ≤ λ 2 in the doubled trajectory ω(− q 2 ). The only piece which remains uncancelled is Outside the regions k 2 ≤ λ 2 and ( k − q) 2 ≤ λ 2 one can put ǫ = 0. Thus we come to the representation of the symmetric kernel which solves the second problem: presentation of the kernel in the physical space-time dimension D = 4 with explicit cancellation of the infrared divergencies.
To compare the BFKL kernel in the Möbius representation (66) and in the momentum representation, we have to take into account that the expression (66) corresponds to the kernel obtained from the symmetric one by the transformation (20) withˆ q 2 1 =ˆ q 2 2 =ˆ q 2 and that q|K| q ′ = q ′ 2 K( q, q ′ ) q −2 , so that Comparing this expression with Eq. (66) one can see that at β = 0 they are functionally identical up to the normalization factors: Note that with these normalization factors the kernels are symmetric with respect to r ↔ r ′ or q 1 ↔ q ′ 1 substitution. Actually the identity relation can be expected, because at β = 0 the kernelsˆ r −2K Mˆ r 2 andˆ q 2Kˆ q −2 have the same eigenvalues corresponding to the eigenfunctions connected by the replacement r ↔ q.

Comparison of the kernels: nonforward case
In this section we present the results of the simplest generalizations for the nonforward case of the transformation used in section 4 for the elimination of the discrepancy between the results of Refs. [17] and [12] for the forward scattering. We start from the commutator K (B) ,K (B) 1 4 ln (q 2 1q 2 2 ) . The calculation of this commutator is described in the appendix B. The result is (84) The subscript M means that the operator acts in the Möbius representation, i.e. the matrix element (84) vanishes as r 1 → r 2 . One can see that the above expression has ultraviolet singularities which cancel in the convolution with Möbius impact factors. But the structure of these singularities is different from the structure of singularities in the gluon contribution to the kernel given in Eqs. (7) and (8). Therefore it is obvious that this commutator cannot make g 0 and g( r 1 , r 2 , r ′ 2 ) coincident with the corresponding g 0 BC and g 0 BC ( r 1 , r 2 , r ′ 2 ). As for g( r 1 , r 2 ; r ′ 1 , r ′ 2 ), we have g( r 1 , r 2 ; r ′ 1 , r ′ 2 ) − g BC ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) = ln and again we see that this commutator does not help to eliminate the discrepancy. Here we have used the equality ( q 2 ) γ ln ( q 2 ) = ∂( q 2 ) γ /(∂γ) to perform the integration. The coordinate representation of K (B) , 1 4 ln (q 2 M is also given in the appendix B and it also fails to resolve the difference between the BFKL and BK kernels. Thus, the simplest generalizations are unable to eliminate the discrepancy in the nonforward case.

Summary
In this paper we studied the properties of the next-to-leading BFKL kernel in gluodynamics and SUSY Yang-Mills theories. In particular our study is connected with the discrepancy between the gluon contribution to the BFKL kernel in the Möbius representation found in Ref. [17] and the kernel of the linearized BK equation calculated in Ref. [12].
We analyzed the ambiguity of the next-to-leading kernel, in particular in its connection with the energy scale. Broadly speaking, the ambiguity is related to the rearrangement of the radiative corrections between the kernel and the impact factors, and the one connected with the energy scale is not an exception. It is shown that the change of the energy scale can be associated with the specific form of the general transformation of the NLO kernel discussed in Refs. [14] and [17].
The ambiguity can be used to remove the discrepancy between the results of Refs. [17] and [12]. It was explicitly demonstrated in the case of forward scattering. We found the Möbius kernel for this case and showed that the major part of the difference between this kernel and the corresponding kernel of Ref. [12] can be eliminated by the suitable transformormation which can be associated with the change of the energy scale, so that the difference is reduced to two terms. One of them is proportional to the first coefficient of the β-function. In our opinion, this term is connected with the difference of the renormalization scheme used in Ref. [12] from the conventional MS-scheme. This term can be eliminated by the change of the renormalization scheme. Unfortunately, we cannot eliminate the third term, proportional to ζ(3). In the BFKL approach this term passed through a great number of verifications. It is also confirmed by the calculation of the three-loop anomalous dimensions in Refs. [23] and [24]. 2 We calculated the characteristic function ω M (n, γ) describing the action of the forward Möbius kernel on the eigenfunctions of the leading order kernel and compared it with the corresponding function of the BFKL kernel in the momentum representation found in Ref. [18]. The forward Möbius kernel was found using the results of Refs. [14], [16] and [17] whereas the calculation of Ref. [18] is based on the results of Ref. [3]. Therefore the comparison serves as the cross-check of the results of these papers. The coincidence of the characteristic functions gives a strong argument in favour of rightness of the used results.
We studied also the forward BFKL kernel in supersymmetric Yang-Mills theories for any N-extended SUSY both in the momentum and in the Möbius coordinate representations and demonstrated the functional identity (up to the normalization factors) of the form of the kernel in these representations for N = 4. We calculated the kernel in the Möbius representation in the impact parameter space using the results of Refs. [14], [16], [17] and [19], and found the kernel for any N in the momentum space using the results of Refs. [3] and [18]. Performing explicit cancellation of the infrared divergencies and writing the kernel at physical space-time dimension D = 4 we demonstrated the functional identity mentioned above, that confirms the correctness of the used result.
At last, we checked how the simplest generalizations of the transformation, used for the elimination of the discrepancy between the results of Refs. [17] and [12] for the forward scattering, work in the general (nonforward) case. Unfortunately, these generalizations are not effective. This does not mean that the transformation eliminating the discrepancy (apart from the difference in the renormalization scheme and in the ζ(3) term) does not exist. Moreover, we hope that it exists; but in this case the generalization from the forward case is more refined than that we used. = − − exactly coincides with the result (118). Unfortunately it is clear that this commutator does not eliminate the discrepancy between the kernels.