Hermitian Separability of BFKL eigenvalue in Bethe Salpeter approach

We consider the Bethe Salpeter approach to the BFKL evolution in order to naturally incorporate the property of the Hermitian Separability in the BFKL approach. We combine the resulting all order ansatz for the BFKL eigenvalue together with reflection identities for harmonic sums and derive the most complicated term of the next-to-next-to-leading order BFKL eigenvalue in SUSY N=4. We also suggest a numerical technique for reconstructing the unknown functions in our ansatz from the known results for specific values of confomal spin.


Bethe-Salpeter approach to BFKL equation
The Balitsky-Fadin-Kuraev-Lipatov (BFKL) [1] equation is traditionally schematically written in the form of the linear Schrödinger equation for the BFKL Hamiltonian H and the BFKL eigenvalue E which is related to the pomeron intercept. The BFKL eigenvalue depends on two real valued degrees of freedom, the anomalous dimension ν and the conformal spin n emerging through Mellin transform of the two-dimensional transverse momentum. For the singlet BFKL equation the BFKL eigenvalue is a function of a complex variable 1 for continuous ν ranging from −∞ to ∞ and discreet n = 0, ±1, ±2, .... The analytic expressions for the BFKL eigenvalue in the color singlet channel are currently available only for the leading order (LO) and nextto-leading order (NLO) [2] of the perturbative expansion in both QCD and N = 4 SYM theory. There is also some information available for next-tonext-to-leading order (NNLO)in the N = 4 SYM, which follows from modern integrability techniques 2 .
The NLO eigenvalue is expressed through more complicated functions compared to those present at the LO level. In the present paper we focus on one major difference between the LO and the NLO functions, namely the socalled Hermitian separability first discussed by A. Kotikov and L. Lipatov [3,4]. By Hermitian separability one means a possibility of writing a function of complex variable z and its complex conjugatez as a sum two contributions separately dependent on z andz 1 In this paper we follow notation of N. Gromov, F. Levkovich-Maslyuk and G. Sizov [5] and use ν divided by two, instead of the traditional notation of anomalous dimension and conformal spin through iν + |n| 2 . 2 See recent review paper discussing different aspects of integrability techniques applied to the BFKL evolution [6] In our case we restrict F to be the same function for z andz, which reduces any related calculations to much simpler one dimensional problem of computing only one function. In the case of the BFKL eigenvalue the function F (z) is a real single valued function of a complex variable so that f (z,z) is always real for any value of z.
The LO eigenvalue is manifestly Hermitian separable, whereas the NLO eigenvalue [2] is not. It was demonstrated by A. Kotikov and L. Lipatov [3,4] that color singlet NLO eigenvalue in N = 4 SYM can be written as a combination of in terms of a product of two Hermitian separable functions  [3,4] it is natural to consider another evolution where the two degrees of freedom are separated already at the level of the Kernel of the corresponding equation. A natural choice for describing a bound state of two reggeized gluons would be to use the Bethe-Salpeter equation, which was originally constructed to describe bound states.
In the Bethe-Salpeter approach one can represent the BFKL dynamics as pole decomposition in the plane of complex angular momentum j. The leading singularity of j → 1 corresponds to the Regge kinematics in which the original BFKL was derived. It is customary to denote j = 1+ω and make an expansion in powers of ω. The leading-order (LO) contributions would correspond to the simple pole 1/ω, the next-to-leading (NLO) contributions would also include a free term (ω) 0 , the next-to-next-to-leading (NNLO) would account for the first order in ω and so on. Due to recursive structure of the Bethe-Salpeter equation the sum of all those contribution should equal ω itself. This can be written as follows [ where a = αsN C 2π is the coupling constant. We assume that the functions f {i,k} are Hermite separable to any order and can reproduce a structure of the next-to-leading eigenvalue and make prediction for the next-to-next-toleading eigenvalue in the following way. Let us denote the leading order eigenvalue by the next-to-leading order eigenvalue by and so forth.
To the required next-to-next-to-leading (NNLO) order eq. (5) reads plugging and expanding in the powers of the coupling constant we obtain the first three orders in the perturbation theory as follows. The LO eigenvalue the NLO eigenvalue of order a 2 and finally the NNLO eigenvalue of order a 3 .
The expression in eq. (12) is our master equation for the NNLO BFKL eigenvalue, which we discuss in more details below. In the next two chapters we focus on possible to ways to apply it to the known NNLO results in N = 4 SYM.
Our analysis shows that at the NNLO level the function ω 2 is expressed in terms of three unknown functions f 1,1 , f 0,2 and f 2,0 . The functions f 0,0 , f 0,1 and f 1,0 are known from the previous orders. The functions f i,j are real single valued meromorphic functions of different level of complexity. The most complicated is f 0,2 function and the simplest is f 0,0 . This can be shown by the following arguments. All currently available results show that the BFKL eigenvalue is built of polygamma functions and its generalizations. Those functions are either logarithmically divergent at infinite value of the argument or give a transcendent constants 3 . The transcendentality of constant determines the "transcendentality" of the underlying function. This concept despite not being rigorously proven is very useful and widely used in building the functional basis for different ansätze. Another observation, which is also widely used is that the maximal transcendentality is increased by two units for each order of the perturbations theory. In this notation the function building ω 0 , i.e. digamma function is assigned transcendentality one, the function building ω 1 are assigned maximal transcendentality three and the functions building ω 2 all have maximal transcendentality five. The transcendentality is additive as functions are multiplied. The first term in eq. (12) ω 2 0 f 2,0 has maximal transcendentality five while ω 0 has transcendentality one thus f 2,0 must have maximal transcendentality. Using similar arguments one can see that f 1,1 must have maximal transcendentality four and finally f 0,2 is of maximal transcendentality five. The complexity of the functions increases with maximal transcendentality, which corresponds to maximal number of nested summations used for defining a given function. This also significantly increases a space of functions defining a functional basis for any ansatz.

Recursive analytic solution using roots of LO eigenvalue
In the previous section we have discussed the complexity of the unknown functions building the NNLO eigenvalue. Each of those functions is a sum of two terms where F (z) is single valued real function. A product of two such functions is not reducible to one dimensional problem due to the cross term F (z)G(z). In this section we propose a systematic iterative procedure of reducing the NLO expression to one dimensional problem at each iteration step. Firstly, we note that the expression in eq. (12) can be written as a polynomial of ω 0 = af 0,0 as follows In this representation are left with three unknown functions f 1,1 , f 0,2 and f 2,0 in an explicit way. Three other functions f 0,0 , f 0,1 and f 1,0 are known from the previous orders.
The transcendentality arguments discussed in the previous section apply here as well so that the unknown function f 0,2 is most complicated function of maximal transcendentality five, the unknown function f 1,1 is of maximal transcendentality four and the unknown function f 2,0 is of maximal transcendentality three. It is crucial to know the transcendentality of each function because it defines a number of free coefficients to be fixed in the functional basis for each case.
The analytic continuation of the harmonic sums to the complex plane has being recently widely used to build the functional basis. The transcendentality five implies maximal weight of harmonic sums to be five, the transcendentality implies maximal weight to be four etc. The number of free coefficients to be fixed is directly related to the maximal weight, at weight 3 there are 32 terms in the functional basis, at weight 4 there are 95 terms and finally at the weight 5 there are 288 free coefficients to be fixed.
A it was already mentioned the analytic expression for the full functional dependence of ω 2 on z andz is not known. However, we do have analytic expressions for definite values of ν or n in terms of the analytically continued harmonic sums calculated in recent works N. Gromov, F. Levkovich-Maslyuk and G. Sizov [5], M. Alfimov, N. Gromov and G. Sizov [8] and S. Caron Huot and M. Herraren [9]. Let us consider n = 0, in this case z andz are not independent anymore z +z = −1 (16) and then the NNLO eigenvalue can be written as where F 5 is a real single valued function calculated in Ref. [5]. Having the analytic expression for a particular limit of n = 0 can be used to find the full analytic form of ω 2 in the following way. Firstly, we realize that a rather simple separable form of eq. (17) is a result of pole decomposition of cross terms of type F (z)G(−1 − z), which can be easily found using reflection identities of harmonic sums calculated by the authors in a series of publications [11,12,13,14].
Here we use the fact that the harmonic sums are meromorphic functions with isolated poles located at negative integers. The product of two harmonic sums F (z)G(−1 − z) have poles at both, positive and negative integers. The reflection identities discussed in the next section separate this product into a sum of two meromorphic functions having poles at either negative integers or positive integers and zero. This pole separation plays a crucial role in our analysis and helps to restore a sum of two pole separated functions (at n = 0 or any other fixed n) into a products of mixed pole structure. The resulting expression is valid for any value of the conformal spin n.
Solving the inverse problem of gathering together separable terms into cross product of different arguments is not an easy task and requires some additional information, say at n = 1 which was calculated by S. Caron Huot and M. Herraren [9]. We will discuss this alternative approach in our further publications.
We note that the problematic cross terms in eq. (12) come from powers of ω 0 , which is proportional to where ψ(z) = d ln Γ(z) dz is digamma function. The function in eq. (18) has infinite number of zero roots z n consistent with eq. (16), where it vanishes as shown in Figure 1.
At those roots the NNLO expression in eq. (15) reduces to Using this procedure we find values of f 0,2 at the zero roots of f 0,0 , namely at z = z n . There are infinitely many of them so that we can fix any finite set of unknown coefficients provided we know the functional basis and it is finite. Note, that the function f 0,2 in eq. (19) has a separable pole structure and thus can be directly compared to the known expression in eq. (17) to fix the required 288 coefficients numerically 4 After all free coefficients for f 0,2 are fixed and f 0,2 is known, one can subtract it from ω 2 and divide by f 0,0 and repeat the same procedure to find a simpler function f 1,1 . At the last iteration we plug the known f 1,1 and write and can finally find the last unknown function f 2,0 . This way we show that it is possible to calculate the full functional form of ω 2 by iterating it at the zero roots of ω 0 . The only possible issue related to this approach is that functions f i,j might have the same roots as ω 0 , which is very unlikely based on the functions building the known NNLO eigenvalue for n = 0 in eq. (17). Suppose we do have some overlap of roots of ω 0 and f i,j , this can be resolved and cross checked by choosing another set roots as there are infinitely many of them.
This procedure being conceptually simple is currently difficult to implement due to small radius of convergence of integral representations of harmonic sums. The small radius of convergence limits the grid and is problematic even for very high precision because the harmonic sums are mostly slowly varying functions inside the radius of convergence. This leads to highly singular matrices for free coefficients, for which it is a computationally challenging problem to find inverted matrix. We believe this technicality will be overcome in the nearest future.
Note, that the proposed iterative procedure allows to restore the full functional form of ω 2 using only n = 0 result by N. Gromov, F. Levkovich-Maslyuk and G. Sizov [5].
Other known cases for n = 0 and ν = 0 in Ref. [5] as well as for n = 1, 2, .. calculated by S. Caron Huot and M. Herraren [9], can be left for cross checking the result.

Comparison with known results in N = 4 SYM
In attempt to compare our ansatz in eq. (12) with the known results we analyze the n = 0 case of the BFKL eigenvalue in N = 4 SYM calculated by N. Gromov, F. Levkovich-Maslyuk and G. Sizov [5]. Their result is presented in terms of the harmonic sums S a 1 ,a 2 ,...,an (z) analytically continued from even integers values of the argument to the complex plane. The nested harmonic sums are defined [15,16,17,18] as nested summation for n ∈ N S a 1 ,a 2 ,...,a k (n) = We discuss the harmonic sums with only real integer values of a i , which build the alphabet of the possible negative and positive indices, which uniquely label S a 1 ,a 2 ,...,a k (n). In eq. (22) k is the depth and w = k i=1 |a i | is the weight of the harmonic sum S a 1 ,a 2 ,...,a k (n).
There are two different analytic continuations of the harmonic sums [19] a) the analytic continuation from the even integer values of the argument and b) the analytic continuation from the odd integer values to the complex plane. Following the notation of Ref. [5] we use the analytic continuation of the harmonic sums from the even integer values of the argument. The argument of the harmonic sums is a complex number where ν is continuous and real valued anomalous dimension and n is a conformal spin which takes integer values. In our analysis we use the reflection identities for harmonic sums recently calculated by the authors up to weight of five [11,12,13,14]. The harmonic sums are defined for positive integer argument n and require an analytic continuation to the complex plane if one wishes to use them as a general functional basis. The reflection identities allow to decompose a product of two harmonic sums of different arguments z and −1 − z into a sum of two sets of harmonic sums each of whose separately depends either on z or −1 − z. The reflection identity can be schematically written as follows where {a}, {b}, {c} and {d} are sets of letters building the indices of the harmonic sums. There is fixed number of reflection identities at any given weight,for example, there are 216 irreducible reflection identities at weight w = 5. All of the reflection identities up to weight of five were calculated by the authors in Ref. [11,12,13,14]. The simplest reflection identity at weight w = 2 reads where S 1,1 (z) can be written as using the quasi shuffle identities of harmonic sums. We use the reflection identities and apply our ansatz in eq. (12) to the result of Ref. [5]. Our analysis shows that the most complicated part of the BFKL eigenvalue for arbitrary values of ν and n takes the following form (27) +256 (S 1,1,−2,1 (z) + S 1,1,−2,1 (z)) + simpler functions By simpler functions we mean the harmonic sums of lower depth 5 , for example S 1,1,−2,1 (z) has depth four, whereas S 1,−2,1 (z) has depth three and S −2,1 (z) has depth two. The term S 1 (z)S 1,−2,1 (z) is the most complex in the sense that it has the highest depth compared to any other function or any other product emerging in the final result. The depth is additive for the functions in the product. For example, a term S 1 (z)S 1,−1,1,1 (z) would be more complex since S 1,−1,1,1 (z) has depth four, but according to our analysis such terms are absent as well as S 1 (z)S −2,1,1 (z), S 1 (z)S 1,−2,1 (z) and S 1 (z)S 1,1,−2 (z) terms.
For n = 1 the relation between z andz is slightly different and reads z = −z, which shifts the argument of the harmonic sum by unity resulting into the "one over the argument" terms. In this case we have −128 S −2,1,1 (z) z + 128 S 1,−2,1 (z) z −128 S −2,1,1 (z) z + 128 S 1,−2,1 (z) z +simpler functions The corresponding term for n = 1 calculated by Caron Huot and Herraren [9] (see eq. (C.5) of their paper) reads two variables into a product of two much simpler one-variable functions. We extend this analysis to higher orders of the perturbation theory in eq. (5). The next-to-next-to-leading (NNLO) eigenvalue of the BFKL equation is of particular interest because the recent progress made using integrability techniques by N. Gromov, F. Levkovich-Maslyuk and G. Sizov [5], M. Alfimov, N. Gromov and G. Sizov [8] and S. Caron Huot and M. Herraren [9] for specific values of either conformal spin n or anomalous dimension ν in N = 4 SYM. The full analytic form of the NNLO BFKL eigenvalue is still to be found.
We argue that the most complicated terms in the proposed ansazt for the NNLO eigenvalue is a product of real functions of one complex variable. We also show that the unknown terms are either hermitian separable on their own or hermitian separable functions multiplied by powers of the leading order (LO) eigenvalue ω 0 or other known functions. In Chapter 2 we suggest an recursive approach for calculating the unknown functions based on the zeros of ω 0 . This approach should be also applicable to higher orders in the perturbative expansion.