Non linear evolution: revisiting the solution in the saturation region

In this paper we revisit the problem of the solution to Balitsky-Kovchegov equation deeply in the saturation domain. We find that solution has the form of Levin-Tuchin solution but it depends on variable $\bar{z} = \ln(r^2 Q^2_s) + \mbox{Const}$ and the value of $\mbox{Const}$ is calculated in this paper. We propose the solution for full BFKL kernel at large $z$ in the entire kinematic region that satisfies the McLerram-Venugopalan initial condition


Introduction
High energy QCD has reached a mature stage [1][2][3][4][5] and has become the common language to discuss high energy scattering where the dense system of partons (quarks and gluons) is produced. The most theoretical progress has been reached in the description of dilute-dense scattering. The deep inelastic scattering of electron is well known example of such process. For these processes the non-linear equations that govern such processes, have been derived and discussed in details [6,7]. The extended phenomenology has been developed based on these equations * which describes the main features of the high energy scattering. For phenomenology the numerical solution to the non-linear equations have been used but it is important to mention that in two limited cases: deeply in the saturation region [9] and in the vicinity of the saturation scale [10,11]; the analytical solutions have been suggested (see Ref. [12] where the procedure to incorporate these analytical solutions are suggested that leads to successful description of HERA data).
In this short paper we re-visit the solution deeply in the saturation region [9]. We have two motivations for this. First, in the semi-classical approach [13] we obtain a different solution with the geometric scaling behaviour [14] than in Ref. [9]. Second, the solution for heavy ions has not been found for the general BFKL kernel [18] in spite of several attempts to find it (see Refs. [15,16]).
Assuming that both x 12 and x 02 are in the saturation region, i.e. x 2 12 Q 2 s (Y, b) > 1 and x 2 02 Q 2 s (Y, b) > 1 we can consider that ∆ ik ≪ 1 and neglect the term proportional to ∆ 02 ∆ 12 in comparison with ∆ 01 . Resulting equation takes the form where we introduce a new variable . One can see that the solution to Eq. (1.5) is It should be stressed that this solution shows the geometric scaling behaviour [14] being function of only one variable: z.
This derivation shows two problems that have been mentioned above: we need to assume that the main contribution in Eq. (1.5) stems from the saturation region; and the answer has a geometric scaling behaviour that contradicts the initial condition for the DIS with nuclei.
Indeed, at Y = Y A for DIS with nuclei we have McLerran-Venugopalan formula for the imaginary part of the dipole-nucleus amplitude, which takes the following form (see Fig. 1)

Equation and geometric scaling solution
We re-write the Balitsky-Kovchegov equation of Eq. (1.1) in the momentum space introducing It takes the form [1,19] 3) The advantage of the non-linear equation in Eq. (2.2) that the non-linear term depends only on external variable and does not contain the integration over momenta. The BFKL kernel: χ − ∂ ∂ξ , can be written as the series over positive powers of ∂/∂ξ except of the first term Differentiating Eq. (2.2) overξ one can see that it can be re-written as where γ = − ∂ ∂ξ . Introducing the variablez instead ofξ and the new function M as we can re-write Eq. (2.5) in the form We are going to find solution inside the saturation region where function M is small at largez. However, we need to re-write Eq. (2.7) replacing it by and neglecting the last term in this equation one can see that we need to solve the following linear equation with γ = ∂ ∂z andz =z + λ First we find the geometrical scaling solution which depends only onz. In this case Eq. (2.10) takes the form The boundary condition for this equation we take where φ 0 (b) is the solution to the linear BFKL equation atz = 0. φ 0 (b) ≤ 1 due to unitarity constraint and should be small to neglect that non-liner term atz = 0.
Eq. (2.11) can be solved using the Mellin transform where m (γ, b) satisfies the equation: (2.14) The solution for m (γ, b) takes the following form and taking into account the explicit form of the BFKL kernel given by Eq. (1.4) one can re-write Eq. (2.15) in the form where forz we use a new definition:z =z + λ − 2 ψ (1).
One can see that in Eq. (2.17) we cannot close the contour of integration in γ neither on the left semi-plane nor on the right one. Introducing γ = iγ we reduce Eq. (2.17) to the form For largez andγ is large aboutz, we can use the approximation which is equal (see formulae 3.462(3), 9.246 of Ref. [20]) is the parabolic cylinder function (see formulae 9.24 -9.25 of Ref. [20]).
Therefore we reproduce the solution of Eq. (1.7). Choosing the coefficient in front of Eq. (2.21) we can easily satisfy the initial condition of Eq. (2.12) which leads to the solution

General solution and initial condition at Y = Y A
As has been mentioned we are not able to find the geometric scaling solution that satisfy both initial and boundary conditions given by Eq. (1.8) and Eq. (2.12). We need to solve a general Eq. (2.10) to find such a solution. We start with re-writing boundary condition of Eq. (1.8) for function M (z, b; Y ) in momentum representation.
where function I (ω) has to be found from Eq. (2.23). At Y = Y A Eq. (2.23) can be written as One can see that Sinceξ < 0 we can take the integrals over γ in I 1 Y,ξ and in I 2 Y,ξ closing contours of integrations on the left semi-plane. In I 1 ξ and I 2 ξ we have two sets of poles: γ = −n − 1 from Γ (1 + γ) and is the integer part or floor function of Y , from Γ (−γ − Y ). These sets lead to the following contributions to I 1 Y,ξ and to I 2 Y,ξ : ‡ For simplicity we use Y = Y − YA to the end of this section. We hope that using the same letter Y for both variables, will not cause any inconvenience. (2.31) In Eq. (2.30) -Eq. (2.33) 1 F 1 (α, β, t) is the confluent hypergeometric function (another notation is Φ (α, β, t), see formulae 9.2 of Ref. [20]) and 2 F 1 (α, β, γ, t) is the hypergeometric function (see formulae 9.1 of Ref. [20]).
where I 1 (t) is the modified Bessel function of the first kind (see formulae 8.445 -8.451 of Ref. [20]). Using their asymptotic behaviour at large values of the argument we obtain We replace Eq. (2.31) by the integral, i.e.
The steepest decent method leads to the following contribution at the saddle point t SP which can be found from the equation The large Y behaviour of Eq. (2.32) can be obtained using the transformation 2 F 1 (α, β, γ, t) = (1 − t) γ−α−β 2 F 1 (γ − α, γ − β, γ, t) (see formulae 9.131 of Ref. [20]) and knowing the asymptotic representation of Bessel function we get Finally, we replace Eq. (2.33) by the integral to estimate the large Y dependence of this term, i.e.
Taking the integral by the steepest decent method in the same way as in Eq. (2.39) we obtain the equation for the saddle point t SP :  it is a function ofz. One can see that the equation for first correction takes the following form after is the solutions to Eq. (2.11) that takes the form which we will use below. It should be noted that A < 0 due to unitarity constraints; and Eq. (2.22) for A was derived from the matching of this solution atz = 0. Here, we wish to suggest a better procedure for matching.
Using Eq. (3.2) we can re-write the equation for the first correction as follows After taking the integral the last term of this equation can be reduced to the form Using the Mellin transform of Eq. (2.13) we obtain Eq. (3.3) in the form: The solution to Eq. (3.5) takes the form (see Ref. [21]) where m (0) is given by Eq. (2.16). As has been discussed in the integral of Eq. (2.13) we expect that large γ's will be essential. In Eq. (3.6) the typical dγ ′ ∼ 1/ √ κ ≪ γ and we can replace this integral by (3.7) See Ref. [20]: formula 3.352(2) for the last integration and formula 8.21 for the exponential integral E (x). For large γ Eq. (3.7) takes the form where D p (z) is the parabolic cylinder function (see formulae 9.24 -9.25 of Ref. [20]).

Corrections at smallz
In the region of smallz we can solve Eq. (2.2) for the amplitude N (z) noting that the geometric scaling solution to the BFKL equation occurs at γ = γ cr . In the vicinity of the saturation scale the linear BFKL equation can be simplified and replaced by One can see that this equation leads to N (z) ∝ exp ((1 − γ cr )z). The solution to the BFKL equation with full kernel in the vicinity of the saturation scale takes the form [10][11][12] Therefore, we can trust Eq. (3.10) only for For suchz the non-linear equation (see Eq. (2.2)) can be re-written in the form The solution to this equation that satisfies the initial condition N (z = 0) = N 0 takes the following form As we will discuss in the next subsection the scattering amplitude in the coordinate space is equal to We will use this equation in the matching procedure described below in the next section.

Matching procedure
In this subsection we would like to discuss matching of two solutions at small and largez. First, we calculate the scattering amplitude of two dipoles in the coordinate representation where we know that at large z this amplitude N (z) → 1.
From Eq. (2.1) we see that The main contribution to the integral over x stems from k ⊥ x ≤ 2.4, where 2.4 is the zero of J 0 (k ⊥ x) (J 0 (k ⊥ x = 2.4) = 0), and therefore, we can replace the integral by For the solution with the geometric scaling behaviour Eq. (3.18) takes the form Bearing Eq. (3.19) in mind we can formulate the matching procedure in the following way  In general we consider this matching as a strong argument for the procedure suggested in Rev. [12] for finding the approximate solution of the non-linear equation which is suited for phenomenology.

Conclusions
In this paper we show that at large z the solution to Balitsky-Kovchegov equation takes the following form which is the same as solution given in Ref. [9] at z ≫ 1. However, the asymptotic behaviour of the solution depends on different variablez = z − A κπ/2 + 2ψ(1) while the solution at small z in the vicinity of the saturation scale is determined by z. This observation, we believe, is essential for understanding the matching of the solutions at small and large z and for searching the solution for intermediate z.
We found the solution in the entire kinematic region at large z which satisfies the McLerran-Venugopalan initial condition. This problem has been discussed in Refs. [15,16] in the case of simplified kernels but here we give the solution for the full BFKL kernel.