Rise of the DIS structure function F_L at small x caused by double-logarithmic contributions

We present calculation of F_L in the double-logarithmic approximation and demonstrate that the synergic effect of the factor 1/x from the \alpha_s^2-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.

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 Sect. II 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 Sect. III. 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 Sect. IV through constructing and solving IREEs. IREEs control both x and Q 2 -evolutions of F L from the starting point. Specifying the input is done in Sect. V. In Sect. VI 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 consider the small-x asymptotics of F L . We also compare the small-x behaviour of F L and the one of the gluon distribution in the hadrons. Then we consider the generalization of our results on F L in region (1) to the small-Q 2 region. Finally, Sect. VII is for concluding remarks.

II. CALCULATING FL 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. It is convenient to represent F (q,g) 1,2 through auxiliary amplitudes A and B which are the convolutions of the tensor W (q,g) µν with g µν and p µ p ν : 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: so that 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 Fig. 1): p p k q q are perturbative components of the structure functions F and F 2 respectively. The superscripts q(g) in Eq, (7) 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): 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. (3,4), one can express F (q,g) 1 and F (q,g) 2 in terms of A (q,g) and B (q,g) so that with Applying (9,10) to W (q,g) µν in the Born and yhe first-loop approximation yields [1]- [11] that F (q) L = 0 in the Born approximation whereas the first-loop results are: Eq. (11) suggests that F L should decrease ∼ x 2 at x → 0. However, the second-order results bring a slower decrease.

III. 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. [10]. 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. Doing so, we account for the leading contributions only. Throughout the paper we use the Feynman gauge for virtual gluons.
in the second-loop approximation. Graphs (a) and (b) correspond to DIS off quarks and graphs (c) and (d) are for DIS off gluons.
In the first place we consider ladder graphs contributing to B, The ladder graphs contributing to W µν in the α 2 sorder are depicted in Fig. 2. Graphs (a) and (b) correspond to DIS off quarks whereas graphs (c) and (d) are for DIS off gluons. Calculations in the small-x kinematics are simpler when the Sudakov variables [26] 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. (13) 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 look as follows in terms of the Sudakov invariants: We have introduced in Eq. (14) dimensionless variables z i,j defined as follows: A. 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 Fig. 2 to B for DIS off quarks. It is given by the following expression: where and We represent it as the sum of N with and In Eqs. (20,21) we have used the quark density matrix ρ(p) = 1 2p (22) and made use of the δ-functions of Eq. (16). 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 (16) over α 1,2 and β 2 and replacing N 2a by N 2a 1 we are left with three more integrations: with η defined as follows: Details of calculation in Eq. (25) 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. (25) at small x are ∼ 1/x. Retaining them only and integrating (25) with logarithmic accuracy, we arrive at with where χ 2 and ρ defined in (17) and with µ being an infrared cut-off. Contribution to B of graph (b) in Fig. 2 is given by the following expression: where Apart from the color factor C F /2, the integrand in Eq. (30) coincides with the integrand of Eq. (16), so we obtain the same leading contribution: where γ (2) is given by Eq. (28). 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. (27) (the same reasoning holds for Eq. (32)). This result stems from the terms in Eq. (18) where momenta k 2 are coupled with the external momenta p and q. The other terms in Eq. (18) (i.e. the ones ∼ k 2 2 , k 1 k 2 ) either cancel k 2 2 in the denominator of Eq. (16), 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. (18) can be reducing the trace down to T r[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.

B. 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 Fig. 2. We calculate their contributions B g to F L . Obviously, where χ (2) is defined in Eq. (17) and C (2) g is the color factor: The term N 2c is defined as follows: with In Eq. (38) 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: The terms ∼ p λ , p σ in (39) 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 (41) 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 (39) in DLA is while 2k 1λ k 1σ brings corrections to it. The first term in (41) contain the longitudinal momenta only and the last term vanishes at λ = σ . Substituting (42) in (37) we obtain N 2c = T r p q +k 2 pk 2p k 1 −k 2 k 1⊥k2 + T r p q +k 2 pk 2k1⊥ k 1 −k 2 pk 2 (43) Retaining in (43) the terms ∼ (pk 2 ) 2 and ∼ (pk 2 ) 3 , we obtain the leading contribution to N DL g : and then integrating over α 2 , we arrive at with z, z 1,2 , l and η defined in Eqs. (103) and (26) respectively. The integral in Eq. (46) coincides with the integral bringing the leading contribution to B (c) q in (25). 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. (28). Now calculate contribution B 2d g to B g of graph (d) in Fig. 2. It is given by the following expression: where χ 2 is defined in Eq. (17) and where we have used the gluon density matrix of Eq. (40). Retaining the terms with pk 2 and neglecting other terms containing k 2 , we obtain substituting Eq. (50) in (48), introducing variables l, z 1,2 , then accounting for the δ-functions, we arrive at with η defined in Eq. (26). Comparison of (51) with Eq. (25) shows that the leading contribution, B 2d Therefore, the total leading contribution B (2) g to B g in the second loop is Eqs. (27,32,47) and (52) demonstrate explicitly that the only difference between leading contributions of all ladder graphs in Fig. 2 is different color factors. Combining Eqs. (33,53) with (10) demonstrate that F L in the α 2 s -order decreases at x → 0 slower than the first-order result (11). Nevertheless, there are no growth of F L in the α 2 s -order and in the α 3 s -order as shown in Ref. [12]. It suggests that only all-order resummations can provide F L with some growth.

C. 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: and We have used in (56) the quark density matrix given by Eq. (22). We are going to calculate B (n) q in DLA. In order to select appropriate contributions in the trace in (56), we generalize the approximation of Eq. (34) to k i , with i = 1, 2, .., n − 1:k Doing so we arrive at the DL contribution N DL q : Substituting (56), we arrive at B (n) q in DLA. The integration region in DLA was found in [27]: Integrations over momenta k 1 , ..., k n−2 in the region (59) yield DL contributions whereas integration over k n , k n−1 yields the factor 1/x. Integration over k n , k n−1 is not restricted by Eq. (59) but runs over the whole phase space. As is known [29], contributions of non-ladder graphs cancel each other in DLA. Such a straightforward approach is comparatively simple for purely quark ladders (e.g., for 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.

D. 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. The technical the reason of their smallness is that they do not yield (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 [29] and therefore they are essential at running α s only.

IV. CALCULATING Bq AND Bg IN DLA
We calculate B q and B g with constructing and solving IREEs for it. The technology of 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 [22]; the thorough overview of the technical details can be found in Ref. [24]. The essence of this approach is to differentiate B q,g with respect to µ and make use of 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. Therefore, we represent B q,g as follows: with ρ defined in Eq. (29)). The transform inverse to (60) is The same transforms we will use for amplitude A in Sect. VI. It is convenient to use beyond the Born approximation the logarithmic variables ρ = ln w/µ 2 , y = ln Q 2 /µ 2 . (62) We have used the mass scale µ of Eq. (1) as an IR cut-off for simplicity reason, in order to avoid introducing extra parameters. One can choose another IR cut-off. Throughout the paper we will address f q,g as Mellin amplitudes. We again remind that the Mellin transforms for amplitudes A q,g are exactly the same as (60, 61). Moreover, IREEs for amplitudes A q,g (x, Q 2 ) and B q,g (x, Q 2 ) are identical. IREEs for A were obtained in Ref. [22]. Because of that we write below IREEs for B q,g in the ω-space without derivation: with h qq , h gq , h qg , h gg being auxiliary amplitudes describing parton-parton scattering in DLA. They can be found in [22]. Besides, explicit expressions for h ik (with i, k = q, g) can be found in Appendix B. Eqs.
where C (±) (ω) are arbitrary factors whereas Ω (±) are expressed through h ik : with In order to specify C (±) (ω), we use the matching: where f q,g correspond again to B q,g but when the photon is (nearly) on-shell. They have to be found independently. Combining Eqs. (67) and (64) lead us to the algebraic system: which makes possible to express C (±) through f 1,2 : The next step is to calculate f 1,2 . They can be found through constructing and solving IREEs for them. The IREEs for f 1,2 are where g q,g stand for the inputs. Solution to Eq. (70) is Substituting (71) in (69) allows us to represent C (±) through h ik and inputs g q,g . We write C (±) in the following form: Combining Eqs. (74), (73) and (64) 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. [24]) and for F 1 as well (see Ref. [22]). The other ingredients in Eq. (73) are inhomogeneous terms g q,g . They first appeared in Eq. (70). Let us compare Eq. (70) for f q,g (ω) and Eq. (63) for f q,g (ω, y). The first difference between them is that Eq. (70) does not contain the derivative ∂/∂y. The reason is that f q,g do not depend on y. The second difference is the presence of terms g q and g g in (70). 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. Similarly to other evolution equations, IREEs evolve inputs. These inputs are usually defined as the Born contributions. For instance, it is true for amplitude A q defined in Eq. (3). Specifying an input g A q for amplitude A q was done in Ref. [22]. However, such choice is not optimal for amplitude B defined in Eq. (4). In what follows we firstly remind how the input g A q was specified and then proceed to specifying inputs g B q,g for amplitude B.
A. Input for amplitude Aq Amplitude A q was calculated with DL accuracy in Ref. [22]. The evolution in this case starts with the Born contribution: where we have neglected the quark mass. A (Born) q in the region (1) does not depend on µ, so it vanishes when differentiated over µ. It is the reason why inhomogeneous terms are absent in IREEs for f A q,g (x, Q 2 ). In the case of smaller Q 2 , where Applying Eq. (61) to A (Born) q , we obtain which leads immediately to

B. Inputs for amplitudes Bq, Bg
It is impossible to evolve amplitudes B q,g from their Born values onwards because (B q ) Born = (B g ) Born = 0. Then, one could try to use the first-loop contributions as inputs. However, the IR-evolution, by definition, deals with logarithms, which cannot generate the leading second-loop contributions B (2) q,g which are ∼ 1/x. On the other hand, in Sect. II we demonstrated that these factors, having appeared in the second loop, hold in all higher orders while logarithmic contributions are controlled by the evolution. It prompts us to suggest that B (2) q,g should be used as the inputs in IREEs for B q,g while the Born and the first-loop contributions should be added by hands. Both of B q,g are ∼ 1/x. In order to write it explicitly we write B q,g as follows: g /x. Therefore, we suggest that g /x. Strictly speaking, accepting this suggestion takes us out of the conventional form of DLA, where Born amplitudes have been invariably considered as the starting point of evolution.

VI.
BEHAVIOUR OF Bq,g AT SMALL x Eq. (73) is linear in g q,g . In order to extract the overall factor 1/x we re-define C ± participating in (64): Namely, we introduce C ± so that (82) C ( ) ± obey the system q,g defined in Eq. (74). Using this form for C (±) , substituting Eq. (81) in (73), then proceeding to (64) and (60), we obtain explicit expressions for B q,g . Using Eq. (6), we arrive at the following expressions for F The overall factor 4x at Eq. (84) is the product of the factor 4x 2 of Eq. (9) and the factor 1/x appearing when C (±) are replaced by C (±) . Eq. (84) represents the contributions to F (q,g) L most essential at small x only. For instance, it does not include the first-loop contribution decreasing ∼ x 2 at small x (see (11)). Despite the small factors x at the integrals in Eq. (84), actually both F (q) L and F (g) L 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. (84).
A. Small-x asymptotics of Bq,g At x → 0, F (q,g) 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 results of Ref. [22]. After the asymptotics of F (q,g) L have been calculated and convoluted with the parton distributions Φ q,g (see Eq. (7)), 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 [22] for detail). Then, ω 0 is the Pomeron intercept calculated with DL accuracy. This intercept was first calculated in Ref. [22]. We remind that it has nothing in common with the BFKL intercept. It is convenient to represent ω 0 as follows: Numerical estimates for ∆ (DL) depend on accuracy of calculations. When α s is assumed to be fixed 2 , and when the α s running effects are accounted for. Substituting either (87) or (88) in Eq. (85), one easily finds that F L ∼ x −∆ (DL) at x → 0. The asymptotics of F 1 was calculated in Ref. [22] showed that asymptotically F 1 ∼ x −ω0 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 −ω0 . 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. (85) 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 (85) should be used within its applicability region, otherwise one should use the expressions of Eq. (84). The estimate obtained in Ref. [22] states that (85) can be used at x ≤ 10 −6 .
B. Remark on FL at arbitrary Q 2 The expressions in Eq. (84) are valid in the kinematic region (1) where Q 2 is large. However, it is easy to generalize Eq. (84) to small Q 2 . It was proved in Refs. [22,24] that such a generalization is achieved with replacement of Q 2 by Q 2 + µ 2 . When this shift has been done, F Thus, one can universally apply F L (x,Q 2 ) at both large and small Q 2 .

VII. CONCLUSIONS
Our results predict that F L grows at small x despite the very small factor x 2 at B in Eq. (6). 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. (84) for the partonic components F L of F L . The both these components rise at small x though complexity of expressions in Eq. (84) 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. (6) and ensures the rise of F L at small x, see Eq. (85). Then in Eq. (90) 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. (91) 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. It was presumed in IREEs considered in Sect. V that Q 2 ≥ µ 2 . However, it is easy to extend the expressions in Eq. (84) to the region Q 2 < µ 2 , applying to them the results of Ref. [22] obtained for the structure function F 1 at small Q 2 .

VIII. ACKNOWLEDGEMENT
We are grateful to V. Bertone, N.Ya. Ivanov and Yuri V. Kovchegov for useful communications.

IX. APPENDIX
A. Integration in Eq. (25) We write Eq. (25) in the following form: with I 2a 1,2 defined as integrals over the transverse momenta z 1 : 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. [24] for detail): with and 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. [29]. When α s is running, such compensation is only partial.