BFKL equation with running QCD coupling and HERA data

In this paper we developed approach based on the BFKL evolution in $\ln\Lb Q^2\Rb$. We show that the simplest diffusion approximation with running QCD coupling is able to describe the HERA experimental data on the deep inelastic structure function with good $\chi^2/d.o.f. \approx 1.3$. From our description of the experimental data we learned several lessons; (i) the non-perturbative physics at long distances started to show up at $Q^2 = 0.25\,GeV^2$; (ii) the scattering amplitude at $Q^2 = 0.25\,GeV^2$ cannot be written as sum of soft Pomeron and the secondary Reggeon but the Pomeron interactions should be taken into account; (iii) the Pomeron interactions can be reduced to the enhanced diagrams and, therefore, we do not see any needs for the shadowing corrections at HERA energies; and (iv) we demonstrated that the shadowing correction could be sizable at higher than HERA energies without any contradiction with our initial conditions.


The equation
The NLO BFKL equation can be written in the form (see [7,8,14]) with N (r, b : Y ) being the imaginary part of the scattering amplitude of the dipole with size x. and while K N LO (k ⊥ , k ′ ⊥ ) is written in Ref. [14]. One can see that in Eq. (2.1) we do not use the triumvirate structure [16] of the LO BFKL for runninḡ α S which looks as follows: The advantage of this expression that it preserves the bootstrap equations for the reggeized gluon that has been proven in the NLO BFKL approach [15]. On the other hand Eq. (2.5) takes into account part of the NLO corrections of Eq. (2.1) which are not the largest contribution to K N LO in Eq. (2.1). Since the main goal of this paper to clarify some rather qualitative features of the BFKL dynamics with running QCD coupling we feel it is reasonable to use the LO contribution to the simple equation (see Eq. (2.1)) following the example of Refs. [7][8][9].
Finally, in this paper we are going to discuss the following equation:

Green function and the set of Pomerons
In our approach treating the BFKL equation as evolution in k ⊥ we need to find a Green function (G ( Y, r)) which satisfies the following initial condition: Using this function we can find the solution to the BFKL evolution equation (N (r, Y )) with given initial In other word, Eq. (2.8) is a realization of the evolution in r.
We use the Mellin transform to find where χ (f ) is the Mellin transform of the K LO . Solution of Eq. (2.10) was firstly written in Ref. [5] and has been discussed in details (see Refs. [7,8] and references therein).
In this paper we will proceed with the diffusion approximation for χ (f ) for the sake of simplicity. A generalization is simple and straightforward. Therefore The general solution is

For our simplified BFKL kernel
One can see that this solution has a discrete spectrum [6] of states that are determined by the zeros of A (ω, r 0 ) or by the roots of the following equation In Fig. 1 it is plotted function A (ω, r = r 0 ) versus ω. One can see that we have the set of zeros which condenses to zero.
Airy functions have zeros only at the negative values of the argument, and their position can be found with good accuracy from the simple equation:  Figure 3: The distribution of the transverse momenta of gluon in the BFKL Pomeron for BFKL evolution from q 2 = q 2 in (r = r in ) which does not coincide with r = r 0 (Bartels sigar [17]). The solution to the equation Q = Q s (x) is shown in red. The region to the left of this curve is the saturation region of the non linear evolution. The region to the right of the curve is the region where we can trust the linear BFKL equation. The dashed red line shows the case when r 0 = r in which we consider in the paper.
Using Eq. (2.17) we can find the spectrum of the BFKL equation analytically, solving the equation At large n we have a solution ω n = 2 3πbn Finally, the spectrum of the BFKL Pomeron depends only on initial value of r 0 = ln k 2 0,⊥ Λ 2 QCD while the residues depend on the measured r. All features of these poles are the same as in the procedure suggested in Refs. [2,[7][8][9]. The difference of our approach in comparison with the approach of those papers, is in the specific form how we impose the confinement on the BFKL equation. It is well known that the BFKL approach cannot be implemented without introducing the restriction that stem from the confinement region [17]. In Fig. 3 the typical distribution of the gluon momenta in the BFKL Pomeron is presented. For the values of the transverse momenta q ≤ q 0 the unknown mechanism of confinement of quark and gluons plays the dominant role. We took the following approach to introduce the confinement to the BFKL evolution: we put the initial condition at q in = q 0 ( N in in Eq. (2.8)) and consider the BFKL evolution only for the transverse momenta of partons (k ⊥ ≥ q 0 ), This initial condition should be determined from the non-perturbative QCD. The high energy phenomenology [25] as well as N=4 SYM [26] where ∆ IP (∆ IR ) is the Pomeron (secondary Reggeon) intercept, respectively. The physical meaning of the two terms in Eq. (2.20) is clear in the high energy phenomenology based on the Reggeon approach. The first contribution describes the contribution of the soft Pomeron and its intercept will be a parameter of our fit. Function g IP is the residue of the Pomeron contribution in which we include also the ln Y 0 dependence which can stem from the Pomeron interactions. The second term in Eq. (2.20) is responsible for the exchange of the secondary Reggeon. We fix the value of ∆ IR = −0.5 in our fit. For g IP (Y 0 ) and g IR (Y 0 ) we assume the simple form The polynomial in Y 0 reflects the enhanced diagrams for Pomeron interaction shown in Fig. 4.

Main formulae
In ω-representation Eq. (2.19) and Eq. (2.21) can be written in the form Using Eq. (2.14),Eq. (2.8),Eq. (2.9) and Eq. (2.10) we can re-write the dipole-target amplitude: All above formulae have been written in the momentum representation. For calculating F 2 (Q, Y ) it is more convenient to use the coordinate representation going from the dipole transverse momentum to the size of the dipole. Such a transformation it is easy to do in Eq. (2.23) by just replacing r = ln k 2 For calculating the amplitude for the deep inelastic scattering we need to recall that this process happens through the virtual photon fluctuating into a qq pair(dipole) with the qq pair proceeding to interact with the target [18][19][20]. The cross section for the DIS process in this dipole picture can be written as follows [21][22][23][24] σ T,L tot γ * + proton|Y = ln(1/x Bj ), (2. 25) where |Ψ T,L | 2 is the probability to find the dipole with size x ⊥ into virtualphoton with transverse or longitudinal polarization; and σ tot is the total cross section of qq (dipole) interaction with the proton. The wave function of the virtual photon are known [19,24] |Ψ γ * →qq where

27)
α em is the fine-structure constant and Z f is the fraction of the electron charge that carries by the quark(antiquark) with flavour f and mass m f .
Finally, we need to recall that

Description of the HERA data
Using formulae of the previous subsection we describe the HERA data on the deep inelastic structure function F 2 . This set of data was published in Ref. [13] and presents the combined data set of ZEUS and H1 collaborations. The experimental errors are small and to describe these data is a challenge for any theoretical approach. In our procedure of the description we see two sets of the phenomenological parameters: the intercept of the soft Pomeron λ 1 and two functions g IP (Y 0 ) and g reg (Y 0 ), which are characterized the initial non-perturbative function of x Bj ( Y = ln(1/x Bj )) at Q 2 = Q 2 0 (r = r 0 ); and two inputs for the Q 2 evolution: the initial val;ue of Q = Q 0 from which we start the evolution in ln(Q 2 ) ( Q > Q 0 ) and the mass of the quarks (m f ). It turns out that the value of m f in all our fits ≤ 10M eV and, therefore, we are dealing with current quarks as it should be in our approach.
As far as the fit of the initial function of Y 0 , it turns out that we have a set of fits with different values of the parameters(see Table 1). One can see from this table that we found the set of solutions which have in common the fact that λ where ′ denotes the sum without the term with n = k. It should be stressed that in spite of the fact that the largest contribution stems from one term in sum in Eq. (2.23), we have to sum up to n = N max ≈ 200 to obtain the accuracy of our calculation smaller than the experimental errors. All these solutions lead to good χ 2 /d.o.f and the reason why we have them is clear from Fig. 6-a in which we plotted the values of N in (Y 0 , r 0 ) in Eq. (2.8). One can see that in the HERA kinematic range ( to the left from the vertical line in Fig. 6-a) all solutions give the same N in and the difference started to be visible only for larger values of Y 0 .
It should be stressed that our initial condition cannot be describe by the contribution of only two Regge poles: Pomeron and the secondary trajectory. We need to take into account the interaction of the Pomerons. On the other hand in our parameterization we restrict ourselves by contribution of the enhanced diagrams (see Fig. 4), In other words it looks that we do not need to take into account the screening corrections. x Figure 7: The deep inelastic structure function F 2 versus x. The data are taken from Ref. [13]. r 0 = 1.83. All other parameters in Table 2 for λ 1 .
However this conclusion is premature since the simple formula with screening corrections: is able to describe the initial condition in the HERA kinematic region and leads to qualitatively reasonable values of the total cross sections at large Y (see Fig. 6-b) * . It worthwhile mentioning that ∆ that gives the description, is rather large (∆ = 0.25) in agreement with the recent outcome from high energy Regge phenomenology [25].
The quality of the fit one can see from   Table 1 for λ Different solutions give the same descriptions: see Fig. 9 in which we compare the solution with λ and λ (7) 1 . In Fig. 10 we plot the calculated value of d ln F 2 x Bj , Q 2 /d ln(1/x Bj ) at different values of x Bj . The solid lines corresponds to the kinematic region in which we fit the data. The dashed curves can be considered as predictions. One can see that we predict the dependence of this observable on x Bj but this dependence is rather mild in the HERA kinematic region.

Conclusions
In this paper we developed approach based on the BFKL evolution in ln Q 2 . We show that the simplest diffusion approximation with running QCD coupling is able to describe the HERA experimental data on the deep inelastic structure function with good χ 2 /d.o.f. ≈ 1.3. We consider this result as the strong argument against the wide spread opinion that the BFKL dynamics has not been seen experimentally at HERA. This result confirms the outcome of Refs. [7][8][9], in which the BFKL equation was considered as the theory of the reggeons.
From our description of the experimental data we learned several lessons:

•
The non-perturbative physics at long distances started to show up at Q 2 = 0.25 GeV 2 ; • The scattering amplitude at Q 2 = 0.25 GeV 2 cannot be written as sum of soft Pomeron and the secondary Reggeon but the Pomeron interactions should be taken into account;

•
The Pomeron interactions can be reduced to the enhanced diagrams and, therefore, we do not see any needs for the shadowing corrections at HERA energies; 1 (dashed line) and λ 1 (solid line).

•
We demonstrated that the shadowing correction could be sizable at higher than HERA energies without any contradiction with our initial conditions.
We believe that these lessons as well as the fact that we can reach a good description of the HERA data in the framework of the BFKL dynamics, can be useful for future attempts to understand the interface between long (soft) and short(hard) distance physics.    Table 1: The value of the fitted parameters for the initial condition in Q 2 evolution.