Rise of the DIS structure function F L at small x caused by double-logarithmic corrections

: We present calculation of F L in the double-logarithmic approximation (DLA) and demonstrate that the synergic eﬀect of the factor 1 /x from the α 2 s -order and the steep x -dependence of the totally resummed double logarithmic contributions of higher orders ensures the power-like rise of F L at small x and arbitrary Q 2 .


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Applying DGLAP to studying F L is model-independent. However according to ref. [37], neither LO DGLAP nor the NLO DGLAP modifications ensure the needed rise of F L at small x and disagree with experimental data at small Q 2 , which sounds quite natural because DGLAP by definition is not supposed to be used in the region of small Q 2 . The modifications of DGLAP in refs. [30,31] are based on treating BFKL as a small-x input for the DGLAP equations. The approach of ref. [32] treats the Pomeron intercept as a parameter fixed from experiment.
In this paper we present an alternative approach to calculate F L : total resummation of double-logarithmic (DL) contributions to F L , accounting for both logarithms of x and Q 2 . The method we use is self-consistent and does not involve any models. We modify the approach which we used in ref. [38] to calculate F 1 in the Double-Logarithmic Approximation (DLA). This approach has nothing in common with the BFKL equation and its ensuing modifications. Indeed, instead of summing leading logarithms, i.e. the contributions ∼ (1/x) (α s ln(1/x)), we sum the DL contributions ∼ α s ln 2 (1/x) as well as the DL of Q 2 . Because of the absence of the factor 1/x such contributions were commonly neglected by the HEP community for a long time. However, it has recently been proved in ref. [38] that the DL contribution to Pomeron is not less important than the BFKL contribution.
We calculate F L in DLA with constructing and solving Infra-Red Evolution Equations (IREEs). As is well-known, the IREE approach was suggested by L.N. Lipatov in refs, [39][40][41][42]. It proved to be a simple and efficient instrument (see e.g. the overviews in refs. [43,44]) for calculating many objects in QCD and Standard Model. Constructing and solving IREEs, we obtain general solutions. In order to specify them one has to define the starting point (input) for IREEs. Conventionally in the IREE technology the Born contributions have been chosen as the inputs. However, F L = 0 in the Born approximation, so the input has to be chosen anew. We suggest that the second-loop expression for F L can play the role of the input and arrive thereby to explicit expressions for perturbative components of F L . We demonstrate that the total resummation of DL contributions together with the factor 1/x appearing in the α 2 s -order provide F L with the rise at small x. We start with considering F L in the large-Q 2 kinematic region with µ being a mass scale. Then we present a generalization of our results to small Q 2 . The scale µ is often associated with the factorization scale. The value of µ is arbitrary 1 except the requirement µ > Λ QCD to guarantee applicability of perturbative QCD. Our paper is organized as follows: in section 2 we introduce definitions and notations, then remind how to calculate F L through auxiliary invariant functions. Calculations of F L in the α 2 s -order are considered in section 3. We represent them in the way convenient for analysis of contributions from higher loops. Then we explain how to realize our strategy: combining the non-logarithmic results from the α 2 s -order with double-logarithmic (DL) contributions from higher-order graphs. Total resummation of DL contributions to F L is done in section 4 through constructing and solving IREEs. IREEs control both x and Q 2 JHEP03(2021)274 -evolutions of F L from the starting point. Specifying the input is done in section 5. In section 6 we present explicit expressions for leading small-x contributions to perturbative components of F L . To make clearly seen the rise of F L at small x we present the smallx asymptotics of F L . After that we compare our results for F L at small-x with the ones predicted by approaches involving BFKL. Then we consider the generalization of our results on F L in region defined by eq. (1.1) to the small-Q 2 region. Finally, section 7 is for concluding remarks.

Calculating F L through auxiliary amplitudes
The most convenient way to calculate F 1,2 and F L in Perturbative QCD is the use of auxiliary invariant amplitudes. Below we remind how this approach works. The unpolarized part of the hadronic tensor describing the lepton-hadron DIS is and each of F 1 , F 2 depends on Q 2 and x = Q 2 /w, with Q 2 = −q 2 and w = 2pq.
where we use the standard notatons x = −q 2 /w = Q 2 /w, w = 2pq. Neglecting terms ∼ p 2 , we express F 1,2 through A and B:

4)
Each of F 1 , F 2 includes both perturbative and non-perturbative contributions. According to the QCD factorization concept, these contributions can be separated. In scenario of the single-parton scattering, F 1 , F 2 can be represented in any available form of QCD factorization through the following convolutions (see figure 1): are perturbative components of the structure functions F and F 2 respectively. The superscripts q(g) in Eq, (2.6) mean that the initial partons in the perturbative Compton scattering are quarks (gluons). The DIS off the partons is parameterized by the same way as eq. (2.1): with p denoting the initial parton momentum. Throughout the paper we will neglect virtualities p 2 , presuming the initial partons to be nearly on-shell. Introducing the auxiliary amplitudes A (q,g) and B (q,g) similarly to eqs. in terms of A (q,g) and B (q,g) so that (2.9) Applying (2.8), (2.9) to W (q,g) µν in the Born and the first-loop approximation yields (see refs. [1]- [19]) that F Eq. (2.10) suggests that F L should decrease ∼ x 2 at x → 0. However, the second-loop results exhibit a slower decrease.

Leading contributions to B in the second-loop approximation
The second loop brings a radical change to the small-x behaviour of B compared to the first-loop result. Namely, there appear contributions ∼ 1/x in contrast to logarithmic dependence of B in the first loop. Such contributions were calculated in ref. [14]. Nevertheless, we prefer to repeat these calculations in order to represent the results in the way convenient for applying to the total resummation of higher loops in DLA.
In the first place we consider ladder graphs contributing to B, The ladder graphs contributing to W µν in the α 2 s -order are depicted in figure 2. Graphs (a) and (b) correspond to DIS off quarks whereas graphs (c) and (d) are for DIS off gluons. Calculations in the JHEP03(2021)274 small-x kinematics are simpler when the Sudakov variables (see ref. [46]) are used. In terms of them, momenta k i of virtual partons are parameterized as follows: where q and p are the massless (light-cone) momenta made of momenta p and q: In eq. (3.2) q denotes the virtual photon momentum while p is momentum of the initial parton. We remind that we presume that p 2 is small, so we will neglect it throughout the paper. Invariants involving k i looks as follows in terms of the Sudakov invariants: We have introduced in eq. (3.3) dimensionless variables z i,j defined as follows:

Contributions to B for DIS off quarks
We start with calculating the second-loop contribution B (a) q of the two-loop ladder graph (a) in figure 2 to B for DIS off quarks. It is given by the following expression:

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and We represent it as the sum of N : In eqs. (3.9), (3.10) we have used the quark density matrix and made use of the δ-functions of eq. (3.5). They yield that 2k 1 k 2 = k 2 1 + k 2 2 and 2pk 1 = k 2 1 . It turns out that the leading contributions comes from N 2a 1 , so first of all we consider it. Throughout the paper we will use dimensionless variables z 1,2 instead of k 2 1,2⊥ : It is also convenient to use the variable l defined as follows: Using the δ-functions to integrate (3.5) over α 1,2 and β 2 and replacing N (2a) by N we are left with three more integrations: with η defined as follows: Details of calculation in eq. (3.14) can be found in appendix A. Let us remind that throughout this paper we focus on the small-x region. The most important contributions in eq. (3.14) at small x are ∼ 1/x. Retaining them only and integrating (3.14) with logarithmic accuracy, we arrive at where χ 2 is defined in (3.6) and ρ = ln(w/µ 2 ), (3.18) with µ being an infrared cut-off. Contribution to B of graph (b) in figure 2 is given by the following expression: (3.20) Apart from the color factor C F /2, the integrand in eq. (3.19) coincides with the integrand of eq. (3.5), so we obtain the same leading contribution: where γ (2) is given by eq. (3.17) and C (2b) q = C F /2. Our analysis of non-ladder graphs shows that they do not bring the factor 1/x because they do not contain (k 2 2 ) 2 in denominators. Therefore, the total leading contribution B (2) q to B q in the second loop is Now let us consider some important technical details concerning eqs. (3.16) (the same reasoning holds for eq. (3.21)). This result stems from the terms in eq. (3.7) where momenta k 2 are coupled with the external momenta p and q. The other terms in eq. (3.7) (i.e. the ones ∼ k 2 2 , k 1 k 2 ) either cancel k 2 2 in the denominator of eq. (3.5), preventing appearance of the factor 1/x, or cancel 1/k 2 1 , killing ln w. Hence, the first step to calculate the trace in eq. (3.7) can be reducing the trace down to Tr[pk 2pk2 ]. Obviously, it corresponds to neglecting the factor 2pk 1 ink 1pk1 : This observation allows us to develop a strategy to select most important contributions to B in arbitrary orders in α s . In other words, the non-singlet component of F L can be calculated in DLA in the straightforward way, without evolution equations.

Contributions to B for DIS off gluons
The second-loop contributions to the DIS off the initial gluon correspond to the ladder graphs (c,d) in figure 2. We calculate their joint contribution B g to F L . Obviously, the contribution of graph (c) is

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where χ (2) is defined in eq. (3.6) and C (2) g = n f N . The numerator N (2c) is defined as follows: In eq. (3.26) the notation H λ σ λσ stands for the ladder gluon rung while ρ λσ denotes the gluon density matrix for the initial gluons which we treat as slightly virtual: (3.27) The terms ∼ p λ , p σ in eq. (3.27) can be dropped because of the gauge invariance. We use the Feynman gauge for the initial gluons: As a result we obtain We have used in the last term of eq. (3.29) that 2pk 1 ≈ k 2 1 . DL contributions to the gluon ladder come from the kinematics where λ ∈ R L , σ ∈ R T or vice versa (the symbols R L and R T denote the longitudinal and transverse momentum spaces respectively). Therefore, the leading term in eq. (3.27) in DLA is while 2k 1λ k 1σ brings corrections to it. The first term in eq. (3.29) contain the longitudinal momenta only and the last term vanishes at λ = σ . Substituting eq. (3.30) in eq. (3.25) we obtain N (2c) = Tr p q +k 2 pk 2p k 1 −k 2 k 1⊥k2 + Tr p q +k 2 pk 2k1⊥ k 1 −k 2 pk 2 ≈ Tr pqpk 2p Retaining in eq. (3.31) the terms ∼ (pk 2 ) 2 and ∼ (pk 2 ) 3 , we obtain the leading contribution to N DL g : . Substituting N c g in eq. (3.24), representing B g as

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and then integrating over α 2 , we arrive at with z, z 1,2 , l and η defined in eqs. (B.2) and (3.15) respectively. The integral in eq. (3.34) coincides with the integral bringing the leading contribution to B (2a) q in eq. (3.14). obtained for the quark ladder graph and calculated in appendix A. So, we arrive at the leading contribution to B: with γ (2) defined in eq. (3.17). Now we calculate contribution B (2d) to B g of graph (d) in figure 2. It is given by the following expression: where we have used the gluon density matrix of eq. (3.28). Retaining the terms with pk 2 and neglecting other terms containing k 2 , we obtain Substituting eq. (3.38) in eq. (3.36), introducing variables l, z 1,2 , then accounting for the δ-functions, we arrive at with η defined in eq. (3.15). Comparison of eq. (3.39) with eq. (3.14) shows that the leading contribution, B Therefore, the total leading contribution B (2) g to B g in the second loop is s -order and in the α 3 s -order as shown in ref. [20]. It suggests that only all-order resummations can provide F L with some growth.

Remark on the scale of α s
The factor γ (2) defined in eq. (3.17) involves the QCD coupling α s treated as a constant because of complexity of the two-loop calculations. However, one cannot implement the expressions for B (2) q,g in eqs. (3.22), (3.41) until the scale of α s has been specified. The adequate parametrization of α s for processes in the Regge kinematics was obtained in ref. [47] but it cannot be used in B (2) q,g because the leading contributions come the kinematics of virtual partons which is closer to the DGLAP one than to the multi-Regge kinematics. For this reason, we suggest using in B (2) q,g the standard DGLAP parametrization α s = α s (Q 2 ).

Remark on leading contributions of the ladder graphs in higher loops
Contribution B (n) q of the quark ladder graph to B in the n th order of the perturbative expansion can be written as follows: ( 3.43) and We have used in here the quark density matrix given by eq. (3.11). We are going to calculate B (n) q in DLA. In order to select appropriate contributions in the trace in eq. (3.44), we generalize the approximation of eq. (3.23) to k i , with i = 1, 2, . . . , n − 1: Doing so we arrive at the DL contribution N DL q : Substituting eq. (3.44), we arrive at B (n) q in DLA. The integration region in DLA was found in ref. [48]: Integrations over momenta k 1 , . . . , k n−2 in the region defined by eq. (3.47) yield DL contributions whereas integration over k n , k n−1 yields the factor 1/x. Integration over JHEP03(2021)274 k n , k n−1 is not restricted by eq. (3.47) but runs over the whole phase space. As is known (see ref. [51]), contributions of non-ladder graphs cancel each other in DLA. Such a straightforward approach is comparatively simple for purely quark ladders (e.g., for the non-singlet structure functions) but becomes too complex for calculating singlets where the quark rungs are mixed with gluon ones. It is more practical to implement evolution equations in this case.

Remark on contributions of non-ladder graphs
Our analysis of the non-ladder graphs ∼ α 2 s shows that they do not yield the factor 1/x and because of that they can be neglected. Technically, the reason of their smallness is that they do not contain (k 2 2 ) 2 in denominators. At the same time, non-ladder graphs are essential in higher loops (∼ α n s , with n > 2). They should be accounted for because they bring DL contributions. However, as long as α s is treated as a constant, DL contributions of the non-ladder graphs cancel each other(see ref. [51]) and therefore they are essential at running α s only.

Calculating B q and B g in DLA
We calculate B q and B g with constructing and solving IREEs for it. Constructing IREEs in the DIS context was explained in many our papers. For instance, IREEs for the DIS structure function F 1 can be found in [38]; the overview of the technical details can be found in ref. [43]. The essence of this approach is first to introduce a IR cut-off µ to regulate IR divergences of the graphs contributing to B q,g in higher loops. 2 Once such cut-off has been introduced, amplitudes B q,g become µ-dependent and tracing their evolution with respect to µ allows one to construct IREEs. The IREE technology involves the IR cut-off which restricts from below transverse momenta of virtual partons and exploits the fact that DL contributions of the partons with minimal k ⊥ can be factorized.
The IREEs for B q,g take a simpler form when the Mellin transform has been used. We are going to calculate dependence of B q,g on both w and Q 2 but the standard parametrization B q,g = B q,g (x, Q 2 /µ 2 ) leaves the w-dependence to be µ-independent, so as a result we cannot trace it within the IREE technology. Because of that we replace x with the µ-dependent argument w/µ 2 , arriving at the parametrization B q,g = B q,g w/µ 2 , Q 2 /µ 2 . We stress that this replacement is purely technical detail and the standard parametrization will be restored automatically in final expressions for B q,g . For the present, we write the Mellin transform for B q,g as follows: As usually, the integration line in eq. (4.1) runs to the right of the rightmost singularity of f q,g . The transform inverse to eq. (4.1) is

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Throughout the paper we will address f q,g as Mellin amplitudes. The same form for Mellin transforms we used in ref. [38] for the structure function F 1 . It is convenient to use beyond the Born approximation the logarithmic variables ρ defined in eq. (3.18) and y defined as follows: IREEs for amplitudes A q,g were obtained in ref. [38] and IREEs for amplitudes B q,g are absolutely the same, so we do not derive them here and only briefly comment on them. IREEs for B q,g in the ω-space look as follows: with h qq , h gq , h qg , h gg being auxiliary amplitudes describing parton-parton scattering in DLA. They can be found in ref. [38]. In addition, explicit expressions for h ik (with i, k = q, g) can be found in appendix B. One can see that eqs.
This similarity is especially clear as one notices that the overall factor e −ωy = µ 2 /Q 2 ω in eq. (4.5) converts the factor w/µ 2 ω of eq. (4.1) into the standard DGLAP factor x −ω , when eq. (4.5) is combined with (4.1). The factors C (±) (ω) are arbitrary factors whereas Ω (±) are expressed through h ik : The next step is to specify coefficient functions C (±) (ω) and we notice that similarity of our approach and DGLAP ends at this point. Indeed, calculating coefficient functions is beyond the scope of DGLAP whereas we continue to apply the IREE approach. Before doing it, let us make use of matching B q,g and amplitudes B q,g which describe the same JHEP03(2021)274 process in the kinematics where the external photons are (nearly) on-shell, i.e. with virtualities Q 2 ≈ µ 2 . It means that B q,g do not depend on y. It is worth mentioning that our strategy here is to some extent similar to the one of the BFKL-induced models where the BFKL Pomeron is used as an input. In the ω-space the matching is where f q,g are related by the Mellin transform (4.1) to amplitudes B q,g which describe the same process, however with the external photons being (nearly) on-shell, i.e. with virtualities Q 2 ≈ µ 2 . It means that f q,g do not depend on y. Combining eqs. (4.8) and (4.5) lead us to the algebraic system: which makes possible to express C (±) through f 1,2 : Now we have to calculate f q,g . We do it with constructing and solving appropriate IREEs. These IREEs are where inhomogeneous terms g q,g stand for the inputs. We remind that, by definition, the inputs cannot be obtained with evolving some simpler objects. We will specify g q,g in the next section Solution to eq. (4.11) is Substituting eq. (4.12) in eq. (4.10) allows us to represent C (±) through h ik and inputs g q,g . We write C (±) in the following form: g , (4.14)

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where Combining eqs. (4.15), (4.14) and (4.5) leads to expressions for f q,g in terms of h ik and g q,g . We remind that explicit expressions for h ik can be found in appendix B. They are known in DLA for both spin-dependent DIS structure function g 1 (see ref. [43]) and for F 1 as well (see ref. [38]). Let us compare eq. (4.11) for f q,g (ω) and eq. (4.4) for f q,g (ω, y). The first difference between them is that eq. (4.11) does not contain the derivative ∂/∂y because f q,g do not depend on y. The second difference is the presence of inhomogeneous terms g q and g g in eq. (4.11). These terms stand for the inputs, i.e. for the starting point of the evolution. Specifying them is necessary for obtaining explicit expressions for f q,g . Below we consider this issue in detail.

Specifying inputs g q and g g for amplitudes B q,g
Specifying inputs g q and g g is the key point of our paper because it is here that we deviate from the routine IREE technology. We remind that throughout the history of the IREE approach the inputs have always been defined as the Born contributions whereas contributions of higher loops were obtained with evolving the Born amplitudes. However, this technology cannot apply to calculating amplitudes B q,g . Indeed, the Born values for both B q and B g are zeros, so substituting them in eq. (4.11) would lead to the system of algebraic homogeneous equations without an unambiguous solution. The next option is to choose the first-loop amplitudes as the inputs. Technically it is possible: they are non-zero (see eq. (2.10)) and evolving them one can obtain B q,g in DLA. However, in this case the important second-loop contributions B (2) q and B (2) g , each ∼ 1/x (see eqs. (3.22), (3.41)), would be left unaccounted because the IR-evolution controls logarithms and cannot generate the factors 1/x. In section 3.4 we presented the scenario where B (2) q and B (2) g were chosen as the inputs and demonstrated that higher loops cannot change this factor. Instead, they can generate DL contributions. Now we implement this scenario in IREEs and choose B q,g as the inputs. To this end, we should express B (2) q,g in the ω-space. In the first place we represent B (2) q,g in the following form: g , with ρ = ln(w/µ 2 ) (see eq. (3.17)). Notice that ρ corresponds to 1/ω 2 in the ω space (see eq. (4.1)). Then, remembering that the Mellin transform does not affect 1/x, we write B (2) q,g
Finally, we specify the inputs g q and g g of eq. (4.14) as follows: We remind that choosing these inputs takes us out of the standard form of DLA, where Born amplitudes were considered as the starting point of evolution.

Explicit expressions for F L in DLA
Substituting g q,g of eq. (5.3) in (4.12) and combining the result with eqs. (4.10), (4.5), (4.1), we obtain explicit expressions for B q,g . Then, using eq. (2.8) drives us to expressions for F (q,g) L . As the obtained expressions are linear in g q,g , we can factorize from them the overall factor γ (2) /x. To this end we introduce C ± : Using eq. (6.1) allows us to represent expressions for F (q,g) L as follows: The overall factor 4x at eq. (6.2) is the product of the factor 4x 2 of eq. (2.8) and the factor 1/x from the inputs g q,g . Eq. (6.2) includes the contributions to F (q,g) L most essential at small x. It does not include the first-loop contribution and other contributions decreasing at small x (see eq. (2.10)). On the contrary, both F (q) L and F (g) L of eq. (6.2) rise when x is decreasing, albeit this does not look obvious. In order to make it seen clearly we consider below the small-x asymptotics of F (q,g) L , which look much simpler than the parent expressions in eq. (6.2).

Small-x asymptotics of F L
can be approximated by their small-x asymptotics which we denote (F (q,g) L ) AS . Technology of calculating the asymptotics is based on the saddle-point method and the whole procedure is identical to the one for F 1 . So, we can use the appropriate JHEP03(2021)274 results of ref. [38]. After the asymptotics of F (q,g) L have been calculated and convoluted with the parton distributions Φ q,g (see eq. (2.6)), the small-x asymptotics of F L is obtained: where the factor Π includes both numerical factors of perturbative origin and values of the quark and gluon distributions in the ω-space at ω = ω 0 . In any form of QCD factorization Π does not contain any dependency on Q 2 or x (see ref. [38] for detail). Then, ω 0 is the Pomeron intercept calculated with DL accuracy. This intercept was first calculated in ref. [38]. We remind that it has nothing in common with the BFKL intercept. It is convenient to represent ω 0 as follows: when the α s running effects are accounted for. Substituting either eq. (6.5) or eq. (6.6) in eq. (6.3), one easily finds that F L ∼ x −∆ (DL) at x → 0. The asymptotics of F 1 was calculated in ref. [38] showed that asymptotically F 1 ∼ x −ω 0 and therefore F L ∼ 2xF 1 . The growth of F L and xF 1 at small x is caused by the Pomeron behaviour of the parton-parton amplitudes f ik = 8π 2 h ik ∼ x −ω 0 . Amplitudes f gg and f gq , being convoluted with Φ g and Φ q , form the gluon distribution in the initial hadron, which we denote G h : So, at small x Another interesting observation following from eq. (6.3) is that at x → 0. We think that it would be interesting to check this relation with analysis of available experimental data. To conclude discussion of the asymptotics, we notice that the asymptotics as eq. (6.3) should be used within its applicability region, otherwise one should use the expressions of eq. (6.2). The estimate obtained in ref. [38] states that eq. (6.3) can be used at x ≤ 10 −6 . JHEP03(2021)274

Comparison with approaches involving BFKL
Let us start this comparison with considering the second-order graphs (b) and (c) in figure 2, each with a pair of virtual gluons propagating in the t-channel. In section 3 we used the DL configuration, where one of the gluons is longitudinally polarized while polarization of the other gluon is transverse In contrast, contributions to BFKL coming from these graphs involve the kinematics where the both ladder gluons bear longitudinal polarizations. Accounting for these polarizations immediately leads to the following behavior of contributions B with λ = µ 2 /w. Therefore, B (2b,2c) LL are greater than the considered in section 3 contributions B (2) q,g (we remind that B (2) q,g ∼ 1/x). Convoluting graphs (b,c) in figure 2 with a hadron and using appropriate hadron impact factors turns the factor λ into x, so the singular factor in eq. (6.10) is now 1/x 2 . This factor cancels the factor x 2 relating B (2b,2c) LL to F L (see eq. (2.8)). Then, accounting for the impact of higher loops brings the Regge factor x −∆ BFKL , with ∆ BFKL being the intercept of the BFKL Pomeron. Thus we obtain that the BFKL contribution to F L is where ∆ BFKL is used in either LO or NLO. In contrast to eq. (6.11), the contribution (6.3) has the extra factor x and because of it (6.11) may look more important than (6.3). However, the leading singularity ω 0 in eq. (6.3) is large, ω 0 > 1 (see eq. (6.4)), so it cancels the factor x and after that F L ∼ x −∆ DL . Thus the small-x behaviour of F L predicted by eq. (6.3), and the one predicted by eq. (6.11) become very much alike. Indeed, the intercepts of Pomerons in the both approaches are pretty close to each other: ∆ (DL) f ix of eq. (6.5) is close to the intercept of the LO BFKL Pomeron and ∆ (DL) of eq. (6.6) practically coincides with the NLO BFKL Pomeron intercept.
On the contrary, the Q 2 -dependence predicted by eq. (6.3) differs from predictions given by all other approaches: they do not satisfy eq. (6.9). It means that studying the xdependence of experimental data for F L with using Regge fits cannot unambiguously deduce which of these two Pomerons is involved. In order to do it, one should investigate the Q 2dependence of the data. To conclude this section, we once more stress that these approaches deal with different logarithmic contributions and cannot be related to each other.

Remark on F L at arbitrary Q 2
The expressions in eq. (6.2) are valid in the kinematic region (1.1) where Q 2 is large. However, it is easy to generalize eq. (6.2) to small Q 2 . It was proved in refs. [38,43] and used for the structute function F 1 in ref. [52] that such a generalization is achieved with replacement of Q 2 by Q 2 + µ 2 . When this shift has been done, F

JHEP03(2021)274 7 Conclusions
Our results predict that F L grows at small x despite the very small factor x 2 at B in eq. (2.5). First, we re-calculated with logarithmic accuracy the available in the literature second-loop contributions B (2) q and B (2) g , each contains the large power factor 1/x in contrast to the Born and first-loop contributions. This calculation allowed us to conclude that 1/x will be present in higher-loop expressions and cannot disappear or be replaced by another power factor. We demonstrated that most important contributions coming from higher orders are double logarithms. Accounting for DL contributions to all orders in α s , we calculated the x and Q 2 -evolution of B (2) q,g in DLA. This evolution proved to be similar to the evolution of the structure function F 1 . Eventually we obtained eq. (6.2) for the partonic components F L of F L . The both these components rise at small x though complexity of expressions in eq. (6.2) prevents to see the rise. To make the rise be clearly seen, we calculated the small-x asymptotics of F L , which proved to be of the Regge type. The asymptotics make obvious that the synergic effect of the factor 1/x and the total resummation of double logarithms overcomes smallness of the factor x 2 at B in eq. (2.5) and ensures the rise of F L at small x, see eq. (6.3). Then in eq. (6.8) we noticed that the rise of F L and the gluon distributions in the hadrons at small x are identical. We also suggested in eq. (6.9) the simple relation between derivatives of logarithm of F L . This relation could be checked with analysis of experimental data, so such check could test correctness of our reasoning. The explicit expressions for F L obtained in section 5 are valid at Q 2 ≥ µ 2 . In section 6 we obtained the extension of those expressions to the region Q 2 < µ 2 . Confronting our results on the asymptotics of F L with the ones based on BFKL Pomeron, we demonstrated that they predicted the similar small-x behavior and widely different Q 2 -dependence.

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and W = (ω 2 − 2(b qq + b gg )) 2 − 4(b qq − b gg ) 2 − 16b gq b qg , (B.4) where the terms b rr include the Born factors a rr and contributions of non-ladder graphs V rr : b rr = a rr + V rr . (B.5) The Born factors are (see refs. [43,44] for detail): where A and A stand for the running QCD couplings as shown in ref. [47]: with η = ln µ 2 /Λ 2 QCD and b being the first coefficient of the Gell-Mann-Low function. When the running effects for the QCD coupling are neglected, A(ω) and A (ω) are replaced by α s . The terms V rr approximately represent the impact of non-ladder graphs on h rr (see ref. [43] for detail): Let us note that D = 0 when the running coupling effects are neglected. It corresponds the total compensation of DL contributions of non-ladder Feynman graphs to scattering amplitudes with the positive signature as was first noticed in ref. [51]. When α s is running, such compensation is only partial.