Combining the small-x evolution and DGLAP for description of inclusive photon induced processes

Interest in studying the inclusive photon induced processes of Deep-Inelastic Scattering (DIS) and Diffractive DIS (DDIS) at high energies includes their experimental investigation and thorough theoretical description. The conventional instrument for theoretical investigation of DIS at large x is DGLAP. It describes the Q^2-evolution. Description of DIS in the small-x region requires accounting for the both Q^2 and x -evolution at the same time. Combining DGLAP with total resummation of double logarithms of x and Q^2, we present a description of structure function F_1 at large Q^2 and arbitrary x. Making use of the necessity of the shift of Q^2 in order to regulate infrared singularities, we obtain an expression for F_1 valid at arbitrary x and Q^2. The obtained expressions coincide with the DLA expressions at small x and at the same time coincides with the DGLAP result at large x and large Q^2. Accounting for virtualities k^2 of the external partons allows us to obtain expressions for F_1 in K_T-Factorization, which are valid at arbitrary Q^2 and arbitrary relations between Q^2 and k^2. Expressions for F_1 in all types of QCD factorization exhibit the small-x asymptotics of the Pomeron type. This Pomeron is generated by the high-energy asymptotics of amplitudes of the 2 ->2 parton-parton forward scattering. These amplitudes are calculated in DLA, so their asymptotics have nothing to do with the BFKL Pomeron. We demonstrate that the parton-parton amplitudes can be used in the DDIS models instead of BFKL Pomeron or combinations of hard and soft Pomerons at non-asymptotic energies. On the other hand, this modification automatically ensures the Pomerons asymptotics, which simplifies and makes more consistent theory of DDIS.


I. INTRODUCTION
Photon induced inclusive processes such as Deep-Inelastic Scattering (DIS), Photoproduction and Diffractive Deep-Inelastic Scattering (DDIS) are the objects of intensive experimental and theoretical investigation. Theoretical studying these processes is based on the concept of QCD factorization which makes possible to separate perturbative and non-perturbative contributions. There are several forms/types of QCD factorization in the literature. Firstly, there is Collinear Factorization [1]. Secondly, there is the more general K T Factorization suggested in Refs. [2,3]. The third, the most general, type of QCD factorization is Basic Factorization suggested in Ref. [4]. In any of those factorizations the approximation of the single-parton photon-hadron interactions is used and as a result, the subjects of consideration are represented through convolutions of perturbative and non-perturbative objects connected by two-parton intermediate states. For instance, representation of the amplitude T of the elastic forward Compton scattering off a hadron in any type of QCD factorization is depicted in Fig. 1 . In the analytical form, QCD factorization of T is generically written as follows: where T q and T g are the perturbative amplitudes of the elastic Compton scattering off a quark and gluon respectively. Non-perturbative blobs Φ q and Φ g denote distributions of those partons in the hadrons. The notations ⊗ refer to integrations over momentum k of any parton in the two-quark and two-gluon intermediate states. The specific form of the integrations as well as a parametrization of momentum k of the intermediate partons depend on the type of chosen factorization 1 . Amplitudes T q,g as well as amplitudes Φ q,g are also given by expressions depending on the type of factorization. In the present paper we focus on the perturbative amplitudes T q,g . In any specific type of factorization, there are kinematic regions where amplitudes T q,g are given by essentially different expressions. Such regions are : Region A. Large x and large Q 2 : (x ∼ 1) (Q 2 ≫ µ 2 ). Region B. Small x and large Q 2 : (x ≪ 1) (Q 2 ≫ µ 2 ). Region C. Small x and small Q 2 : (x ≪ 1) (Q 2 < µ 2 ). Region D. Large x and small Q 2 : (x ∼ 1) (Q 2 < µ 2 ).
We have used above the conventional notations: Q 2 = −q 2 , with q being the external photon momentum, and x = Q 2 /w, where w = 2pq, with p being the momentum of the incoming/outgoing parton. The parameter µ is associated with the factorization scale. Besides, it plays the role of the infrared (IR) cut-off in Collinear, K T and Basic Factorizations, when the double-logarithmic (DL) contributions to amplitudes T q,g are accounted for. The physical meaning of µ for the parton distributions Φ q,g in K T Factorization and its role in transition from K T Factorization to Collinear Factorization were considered in Ref. [4]. Now let us consider the status of knowledge of amplitudes T q,g in the Regions A-D. The region A is the DGLAP applicability region, so expressions for T q,g in this region are provided by the DGLAP evolution equations [5]. Such expressions can easily be found in the literature. They account for logarithms of Q 2 but neglect total resummations of logarithms of x because such logarithms are unessential in Region A. Description of T q,g in the small-x region B in the double-logarithmic approximation (DLA) in the framework of Collinear Factorization was obtained in Ref. [6]. In particular, it was shown in Ref. [6] that the small-x asymptotics of T q,g exhibits a new, DL contribution to Pomeron. In contrast, double-logarithmic expressions for T q,g in region B in K T Factorization have not been obtained. We do it in the present paper, using the same method as in Ref. [6]: constructing and solving Infrared Evolution Equations (IREE) 2 . Then we obtain expressions for T q,g which combine DL and non-DL contributions available in DGLAP, so these expressions can universally be used at large Q 2 and arbitrary x (i.e. in the region A ⊕ B) in Collinear, K T and Basic Factorizations. Extending expressions for T qg to low Q 2 was suggested in Ref. [11] for Collinear Factorization. In the present paper we generalize this extension to the cases of K T and Basic Factorizations, obtaining thereby expressions which can universally be used at arbitrary x and Q 2 in any form of QCD factorization. Putting q 2 = 0 in Eq. (1), one arrives at the photoproduction amplitudes A γ : Combining it with Eq. (2) leads to the factorized form of A γq : The Optical theorem relates the amplitude A γ to the total cross section of photoproduction. Using the expressions for T qg at low Q 2 allows us to obtain expressions for the perturbative amplitudes A γq , A γg first in Collinear and then in the other forms of QCD factorization.
Our paper is organized as follows: in Sect. II we remind results of Ref. [6] for amplitudes T q,g in Region B in Collinear Factorization and extend T q,g to Regions A,C,D, obtaining explicit expressions for T q,g which can be used at any x and Q 2 . Then we briefly discuss the small-x asymptotics of T q,g , their applicability region the power Q 2corrections to T q,g in the low-Q 2 region C. In Sect. III we study amplitudes T q,g in K T -Factorization. This type of factorization involves dealing with essentially off-shell external partons, which brings a new technical problem which is absent in Collinear Factorization: there is no universal description of T q,g in Region B at at arbitrary virtualities of the external partons. We deal with this problem also by constructing and solving appropriate IREEs. In Sect. IV we consider the photoproduction amplitudes in Collinear, K T and Basic Factorizations. Finally, Sect. V is for concluding remarks.

II. COMPTON AMPLITUDES IN COLLINEAR FACTORIZATION
Eq. (1) for Collinear Factorization takes the following form: where the superscript "col" refers to Collinear Factorization and β is the longitudinal fraction of momentum k in Fig. 1. We start with considering amplitudes T (col) q,g in region B and then extend these expressions to regions A,C,D.
A. Amplitudes T (col) q.g in region B Perturbative amplitudes T (col) q,g in region B were calculated in Ref. [6] in the double-logarithmic approximation (DLA) and the cases of fixed and running α s were investigated separately. Amplitudes T (col) q,g were in Ref. [6] represented as follows: with µ being the factorization scale. The logarithmic variable y in Eq. (5) is related to Q 2 : and ξ (+) (ω) is the positive signature factor: The signature factor ξ (+) (ω) guarantees that T q,g are invariant with respect to permutation of the incoming and outgoing external photons. The integral representation (5) is the asymptotic form of the Sommerfeld-Watson transform but it is frequently addressed as the Mellin transform. Throughout the paper we will name T q,g the Mellin amplitudes. Explicit expressions for the Mellin amplitudes F (col) q,g in the region B are: Obtained in Ref. [6] expressions for Ω (±) , C (±) q and C (±) g can be found in Appendix B where they are expressed through the on-shell parton-parton amplitudes A rr ′ (r, r ′ = q, g). Graphs for amplitudes A rr ′ are depicted in Fig. 2.
Graphs for the parton-parton amplitudes.
The Mellin transform for the parton-parton amplitudes is similar to the one in Eq. (5): Technically, it is more convenient to use the Mellin amplitudes h rr ′ which are proportional to f rr ′ : Explicit expression for amplitudes h rr ′ can be found in Appendix A. Let us discuss Eq. (8). It is easy to see that its structure exhibits a distinct similarity to the structure of the DGLAP description of F (col) q,g , which is especially obvious when the approximation of fixed α s is used: Indeed, the factors Ω (±) in the exponents of Eq. (8)  in Eq. (11) while the factors C q,g . However in contrast to DGLAP, the coefficient functions and the anomalous dimensions in (8) are calculated with the same means and they include the total resummations of the DL contributions. The both coefficient functions and anomalous dimensions in (8) are made of amplitudes h rr ′ . When the partonic amplitudes h rr ′ are replaced by their Born values, the integrands in Eq. (5) coincide with the integrands of the LO DGLAP expressions in which the most singular in ω terms are retained. Further expansions of h rr ′ and substituting them in Eq. (8) lead to the NLO (as well as to NNLO DGLAP, etc), where the most singular terms only are accounted for in each order in α s .
B. Extending the small-x Eqs. (5,8) to region A Expressions in Eqs. (5,8) are defined in the region B. Now we are going to obtain an interpolation formulae for amplitudes T (col) q,g , which would reproduce the DGLAP expressions at large x and at the same time would coincide with Eqs. (5,8) at small x. Such extension can be obtained with the four steps: Step 1: Subtract the most singular in ω terms (i.e. the terms ∼ α s /ω in the case of LO DGLAP; α 2 s /ω 3 in the case of NLO DGLAP, etc.) from the DGLAP anomalous dimensions γ the result of such amputation.
Step 2: Add γ DGLAP (±) to Ω (±) . The new anomalous dimensions contain the total resummations of the double-logarithmic contributions. They are essential at small x and unimportant at large x. At the same time, (12) contains less singular in 1/ω n DGLAP contributions which are dominant at large x.
Step 3: Subtract the LO DGLAP term = 1 and the most singular in ω terms (i.e. the term 1 ∼ α s /ω 2 in the case of NLO DGLAP, α 2 s /ω 4 in the case of NNLO DGLAP, etc.) from the DGLAP expressions for the coefficient functions. We denote ∆Ĉ (±) q,g the result of such amputation.
Step 4: Add the results obtained in Step 3 to the DL expressions C (±) q,g , arriving thereby at new coefficient functions C (±) q,g : The coefficient functions C (±) q,g include both DL contributions and the less singular DGLAP terms. The subtractions in Steps 1,3 are necessary to avoid the double counting.
Replacing Ω (±) and C (±) q,g in Eq. (5) by Ω (±) and C (±) q,g , we obtain the interpolation expressions T (col) q,g for the Compton amplitudes : Eq. (14) combines the small-x evolution in DLA with the DGLAP results for the coefficient functions and anomalous dimensions, which are important at large x. These expressions can universally be used as the interpolation formulae for T (col) q,g in the region A ⊕ B.
C. Extending Eq. (14) to the small-Q 2 region C ⊕ D Extension of the expressions (14) to describe the amplitudes T (col) q,g to the region C ⊕ D is also not rigorous. The standard approach suggested in Ref. [9] and used since that in many papers (see e.g. Ref. [10]) to describe DIS at low Q 2 is to make a shift of Q 2 : where the mass scale m was fixed on basis of certain physical considerations, depending on specific situation. In contrast to preceding papers, we proved in Ref. [11] (see also the overview [12]) that the scale m with DL accuracy can be unambiguously specified: the IR cut-off µ plays the role of the scale m. Our proof was based on the well-known fact that DL contributions arise from integrations ∼ dk 2 i⊥ /k 2 i⊥ over the transverse momenta k i⊥ of virtual partons, which requires introducing an IR cut-off µ. It can be introduced through the shift k 2 i⊥ → k 2 i⊥ + µ 2 , which eventually leads to the shift (15), with The proof of (16) in Refs. [11,12] was done in the context of the spin-dependent structure function g 1 but it holds for amplitudes T (col) q,g either. Applying the Principle of Minimal Sensitivity [13], we estimated in Ref. [6] the value of µ of Eq. (16): So, the universal expression for the Compton amplitudes T at arbitrary values of x and Q 2 ) in the framework of Collinear Factorization is where we have denoted D. Small-x asymptotics of the Compton amplitudes and comparison with the DGLAP asymptotics Asymptotics of T (col) q,g at x → 0 and Q 2 > µ 2 were considered in detail in Ref. [6]. In the present paper we briefly remind results of Ref. [6] and compare these asymptotics with the asymptotics of the same amplitudes obtained in the DGLAP approach. The standard mathematical tool to calculate the small-x asymptotics of T (col) q,g is Saddle Point method. Applying this method to T (col) q,g , we obtain that the small-x asymptotics of amplitudes T (col) q,g are of the Regge type. They both have the same intercepts ω 0 : The factors Π q,g (ω 0 ) in Eq. (20) are called the impact factors 3 They are the only difference between the asymptotics of T maximal intercept ω DL H corresponded to the roughest approximation where the quark contributions were neglected and α s was fixed while the minimal intercept ω DL S corresponded to the most accurate calculation where both gluon and quark contributions were accounted for and α s was running: As the both intercepts > 1, the Reggeons of Eq. (20) with the intercepts (21) are the DL contributions to Pomeron, or DL Pomerons. In accordance with the conventional Pomeron terminology we call them hard (with the subscript H) and soft (with the subscript S) DL Pomerons respectively. It is interesting fact that ω DL H is close to the LO BFLK Pomeron intercept and ω DL S almost coincides with the NLO BFLK Pomeron intercept despite that BFKL have nothing in common with resummation of DL contributions. The asymptotics in Eq. (20) are given by much simpler expressions than the parent amplitudes in Eq. (18). However, the asymptotics should not be used outside their applicability region. It was shown in Ref. [6] that the asymptotics should be used at x < x max and estimates of x max at various values of Q 2 were obtained. In particular, at Q 2 = 10 GeV 2 x max = 10 −6 .
It was shown in Ref. [6] that the greater is Q 2 , the less is x max . When x ≥ x max the parent amplitudes T q,g should be used instead of the asymptotics.

E. Remark on Asymptotic Scaling
Eq. (20) can be written in such a way: Although both T (col) q and T (col) g by definition depend on two independent variables x and Q 2 , Eq. (23) manifests that T (col) q,g ln 3/2 (1/x) asymptotically depend on the single variable ζ only. We name this remarkable property the Asymptotic Scaling. This property is absent in the DGLAP approach. Indeed, DGLAP predicts that the x-and Q 2dependence of T (col) q,g at x → 0 are unrelated: where γ DGLAP is the anomalous dimension and b is the first coefficient of the β-function. The intercept a has the phenomenological origin: it is generated by the terms x −a in the fits for the initial parton densities.

F. Perturbative power Q 2 -contributions
We begin studying the power Q 2 -corrections with considering T (col) q at the large-Q 2 region, where Q 2 ≫ µ 2 and where the logarithmic variable y defined in (19) can be expanded in the series as follows: Being substituted in Eq. (18), it allows us to write T The first term in the r.h.s. of Eq. (27) describes the Compton amplitude in the region A ⊗ B. At large x it is given by the DGLAP formulae whereas the second and third terms can wrongly be interpreted as contributions of the higher twists, probably of the non-perturbative origin though actually they are altogether perturbative: This example demonstrates that ignoring the shift (15) and interpretating all terms inversely proportional to Q 2 as an impact of higher twist could be misleading. On the other, at Q 2 < µ 2 the expansion (26) is replaced by another one: It leads to the small-Q 2 expression for T (col) q : which decreases down to zero when Q 2 → 0. It explains very well why the terms ∼ 1/(Q 2 ) n are never seen at small Q 2 where, naively, their impact could be great. We remind that our estimate of µ is µ ≈ 1 GeV (see Eq. (17)). Disappearing of the terms ∼ 1/(Q 2 ) n at Q 2 1 GeV obtained from analysis of experimental data could confirm correctness of our reasoning.

III. CALCULATING PERTURBATIVE COMPTON AMPLITUDES IN KT FACTORIZATION
Originally, K T Factorization was introduced in Refs. [2,3] to make possible applying BFKL for description of hadronic reactions. It implied the kinematic region of very high, or asymptotic, energies, i.e. at x ≪ 1. However, in the cases when BFKL is not involved one can use K T Factorization at x ∼ 1 as well. In the present Sect. we generalize DL expressions for T (col) q,g to K T Factorization. In the framework of K T Factorization Eq. (1) takes the following form: where the superscript "KT " refers to K T -Factorization, µ is the factorization scale, β is the longitudinal fraction of the external partons (see Fig. 1) and k 2 ⊥ stands for their transverse momentum. Φ (KT ) q,g denote initial parton distributions and amplitudes T (KT ) q,g are perturbative. The subscripts "q, g" refer to quarks and gluons in the same way as in Sect. II. We are going to calculate T (KT ) q,g in DLA. DL contributions to them arrive from the kinematic region However, amplitudes T (KT ) q,g in DLA cannot be described by a single universal expression at different virtualities k 2 ⊥ in (32). There are two regions, where they are given by different expressions: Region F: Deeply-virtual k 2 ⊥ , where In order to avoid confusions, we will denote T (E) q,g and T (F) q,g the amplitudes T (KT ) q,g in the regions E and F respectively. We will use the Mellin transform for T (E,F) q,g in the following form: where y is given by Eq. (6) In terms of the logarithmic variables regions E and F defined in Eqs. (33,34) look as follows: Region E: ρ > y + z, Region F: ρ < y + z.

A. Calculating the off-shell Compton amplitudes in region E
IREEs for ϕ (E) q,g have the following form: ∂ϕ The l.h.s. of IREEs in (38) are obtained with applying the differential operator to Eq. (35). As a result, the l.h.s.of Eq. (38) contain explicitly the derivatives over y and z while the factor ω corresponds to differentiation of the Mellin factor (w/µ 2 ) ω . The convolutions in r.h.s. of (38) are similar in structure to the convolutions in the DGLAP evolution equations. They involve amplitudes F (col) q which were considered in Sect. II and amplitudes H rr ′ which are still unknown. We will calculate amplitudes H rr ′ in the next Sect. In order to specify general solutions to Eq. (38), we will use the matching with T (col) q,g defined in Eqs. (5). The matching (40) implies the following hierarchy between y and z: The requirement (41) is temporary. The final expressions for T (E) q,g will be written in the form free of this hierarchy. Solving Eq. (38) goes easier when y and z are replaced by new variables ξ, η: Obviously, the inverse transform is In terms of ξ, η Eq. (38) looks simpler: ∂ϕ Solution to Eq. (44) is: with the variables y ′ , z ′ defined as follows: In order to lift the relation between y and z of Eq. (41) we replace η with |η| everywhere save Eq. (46). Doing so and substituting Eq. (45) in (35), we arrive at expressions for amplitudes T (E) q,g in region E: B. Calculating the off-shell Compton amplitudes in region F IREEs for T (F) q,g are very simple: We do not use the Mellin transform for T (F) q,g and because of it the l.h.s. of (48) contain derivatives over ρ, y and z. Integration over momenta of all virtual partons in the deeply-virtual region F do not bring any dependence 4 of T (F) q,g on µ. As a result, the r.h.s. of Eq. (48) are zeros. A general solution to Eq. (48) is with Ψ q and Ψ g being arbitrary functions. In order to specify them we use the following matching: Amplitudes T (F) q,g coincide with amplitudes T (E) q,g on the border surface between regions E and F, where ρ = y + z: with the new amplitudesT (E) q,g defined as follows: According to Eq. (49) the expression for T in the whole region F can be obtained from amplitudesT (E) q,g by the simple change of their arguments: Both ρ − y and ρ − z do not depend on the IR cut-off µ, so amplitudes T and the following definition (cf. (10)): A. Photoproduction amplitudes in Collinear kinematics According to Eq. (58), putting q 2 = 0 in Eqs. (5,8) (8) were obtained in Ref. [6] for Q 2 ≫ µ 2 only and they cannot be used at Q 2 < µ 2 . In order to describe A (col) q,g at low Q 2 , including Q 2 = 0, we use the prescription in the Sect. II and replace x, y in Eqs. (5,8) by x,y defined in Eq. (19). Then, putting Q 2 = 0 in x, y, we obtain Replacement of x, y by x 0 , y 0 in Eqs. (5,8) allows us to obtain photoproduction amplitudes in DLA in Collinear Factorization: Combining Eqs. (5,8) and (59), we obtain: where a γq is the averaged photon-quark coupling a γq =ē 2 q , x 0 = µ 2 /w and Accounting for non-DL radiative correction in A We will use the Mellin transform for A (KT ) γq and A (KT ) γg in the following form: IREE for the Mellin amplitudes F γq (ω, z), F γg (ω, z) are similar to Eq. (55):  : We remind that expressions for amplitudes f

V. CONCLUSION
In this paper we have studied first the perturbative amplitudes T q,g of elastic Compton scattering off quarks and gluons and then perturbative components A γq , A γg of the photoproduction amplitudes. T q,g were calculated in Ref. [6] in DLA at small x and large photon virtualities Q 2 (i.e. in kinematic region B) in the framework of Collinear Factorization . We converted these results in expressions which can be used at arbitrary x and Q 2 in Collinear, K T and Basic Factorizations. Extension to Region A was done with combining the total resummation of DL contributions and DGLAP description of T q,g . By doing so we obtained in Eq. (14) the interpolation expressions reproducing T q,g in DLA at small x and coinciding with the DGLAP description of T q,g when x are not small. Using the shift of Eq. (15), we extended description T q,g to small Q 2 . As a result, we arrived at Eq. (18) to the expressions for T q,g in Collinear Factorization, which can be used at arbitrary x and Q 2 .
In contrast to Collinear Factorization, T q,g are the essentially off-shell in K T Factorization and they cannot be described in DLA by a single expression valid at arbitrary values of virtualities k 2 ⊥ of the external partons. It made us consider separately regions of moderate (Eq. (33)) and deep (Eq. 34) virtualities and obtain expressions Eqs. (47) and (52) for T q,g in those regions. We obtained them by constructing appropriate IREEs and solving them.
After transition from Region B to the to low Q 2 Region C has been studied, we became able to obtain explicit expressions (60) and (67) for the photoproduction amplitudes A γq , A γg in Collinear and K T Factorizations respectively.
Small-x asymptotics of amplitudes T q,g are of the Regge type with the same intercept in any form of QCD factorization. Because of it we considered such asymptotics in Collinear Factorization (see Eq. (20)) and discussed dependence of its intercept on accuracy of the calculations: the higher is the accuracy, the lesser is the intercept. In other words, the hard Pomeron becomes the soft one, when the accuracy grows. We think that further increasing the accuracy can lead to vanishing supercritical Pomeron(s) and restoration of the Unitarity.
The next interesting point is that when T q,g are calculated in DLA, their asymptotics depends on the single variable ζ = Q 2 /x 2 as shown in Eq. (23). Neither DGLAP nor BFKL cause such dependence.
It is important to use the Regge asymptotics within their applicability region, i.e. at x < x max , with x max given by Eq. (22). When x > x max the asymptotic is considerably less than the parent amplitudes, so such amplitudes should be used instead of the asymptotics. Ignoring this point leads to various misconceptions: for instance, appearance of artificial/model (hard) Pomerons and spin-dependent Pomerons.
To conclude, let us notice that DL Pomeron can play an important role for description of various hadronic reactions where Pomerons are used and the diffractive DIS in the first place, see e.g. [15].
The factors C (±) are also expressed through the parton-parton amplitudes: Appendix C: Expressions for the factors C1,2 and C ′ 1,2 of Eq. (57) The terms where R is given by Eq. (B2).