# BFKL equation in the next-to-leading order: solution at large impact parameters

## Abstract

In this paper, we show (1) that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter (\(b^2 \,>\,r^2 \exp ( 2{\bar{\alpha }}_S\eta (1 + 4 {\bar{\alpha }}_S) )\), where *r* denotes the size of scattering dipole and \(\eta = \ln (1/x_{Bj} )\) for DIS), and, therefore, they do not resolve the inconsistency with unitarity; and (2) they lead to an oscillating behaviour of the scattering amplitude at large *b*, in direct contradiction with the unitarity constraints. However, from the more practical point of view, the NLO estimates give a faster decrease of the scattering amplitude as a function of *b*, and could be very useful for description of the experimental data. It turns out, that in a limited range of *b*, the NLO corrections generates the fast decrease of the scattering amplitude with *b*, which can be parameterized as \(N\, \propto \,\exp ( -\,\mu \,b )\) with \(\mu \, \propto \,1/r\) in accord with the numerical estimates in Cepila et al. (Phys Rev D 99(5):051502, https://doi.org/10.1103/PhysRevD.99.051502, arXiv:1812.02548 [hep-ph], 2019).

## 1 Introduction

This paper is motivated and triggered by the result of the numerical solution [1] of the Balitsky–Kovchegov (BK) equation in the next-to-leading order (NLO), in which at large impact parameter, the solution shows an exponential decrease (\(\propto \exp ( - \mu \, b )\)). Since the amplitude decreases at large *b*, the non-linear term in the BK equation is small and can be neglected, reducing the problem of large *b* behaviour, to the solution of the BFKL equation. The large impact parameter behaviour of the scattering amplitude remains the most fundamental problem, which is still unsolved [3, 4, 5] in the frame of the CGC/saturation approach (see Ref. [2] for a review). Indeed, in the CGC/saturation approach, the scattering amplitude decreases as a power of *b* [3, 4, 5] contradicting the Froissart theorem [6, 7]. The intensive attempts to solve this problem and introduce the non-perturbative corrections, which bring the dimensional scale into the problem [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], results in the widely held opinion, that we need to introduce a new non-perturbative dimensional scale in the kernel of the BFKL equation. With this in mind, the result of Ref. [1] looks strange, since the NLO kernel, that has been used in the paper, has conformal symmetry, and no dimensional scale has been introduced.

The goal of this paper is to show, that in NLO we still have power-like behaviour at large values of *b*, as the result of the conformal symmetry of the BFKL kernel. However, we find that there is a kinematic region where the solution has a fast decrease with *b* (\( \propto \,e^{-\mu b}\)) and this falloff can be parameterized as an exponential with \(\mu \, \propto \, 1/r\), where *r* denotes the size of the scattering dipole.

The paper is organized as follows. In the next section we discuss general features of the BFKL Pomeron at large values of the impact parameter. In Sect. 3, we discuss the impact parameter dependence in double log approximation (DLA) of the leading order of the BFKL evolution equation, and show that the scattering amplitude decreases as a power of *b*. Section 4 is the main part of the paper and it deals with the DLA for the next-to-leading (NLA) BFKL evolution equation. We show that the solution for \(b^2\,>\, r^2\exp \left( \frac{1}{2}\eta \right) \), where \(\eta = \ln ( 1/x_{Bj} )\) in DIS, not only has power-like decrease as function of *b*, but leads to an oscillating function, which contradicts unitarity constraints. In Sect. 5 we argue that the main features of the DLA will be preserved in a more general approach. In the conclusion we discuss our findings, and emphasize that we need to introduce the new dimensional scale into the BFKL kernel, which is related to the non-perturbative corrections, that resolve the difficulties at large *b* in the framework of the CGC approach. On the other hand, we note that the NLO corrections suppress the scattering amplitude, and could possibly be useful for the description of the experimental data (see Ref. [1]).

## 2 BFKL Pomeron

*Y*is the rapidity of the scattering dipole and \(\mathbf {b}\) is the impact factor. \({\bar{\alpha }}_S= \alpha _SN_c/\pi \) where \(N_c\) is the number of colours, and \(K\left( \mathbf {x}_{02}, \mathbf {x}_{12}; \mathbf {x}_{10}\right) \) is the kernel of the BFKL equation which in leading order has the following form:

*r*denotes the size of the scattering dipole, while

*R*is the size of the target. For the kernel of the LO BFKL equation (see Eq. (2)) the eigenvalues take the form:

*F*and

*b*) are given in Refs. [26, 27, 28].

*Y*and \(\xi \) we can use the method of stepest descent in calculating the integral of Eq. (10). The equation for the saddle point (\(\gamma = \gamma _{\mathrm{SP}}\)) is

## 3 DLA for LO BFKL equation

*b*, in accord with the general discussion in Refs. [3, 4, 5].

## 4 DLA for NLO BFKL

### 4.1 Generalities

### 4.2 DLA in coordinate representation

**8.402**of Ref. [32]), \(\rho \,\equiv \,\sqrt{L_{x_{02}, x_{01}}L_{x_{12}, x_{01}}}\) and \(L_{ x_{i 2}, x_{01}} \equiv \ln (x_{i 2}^2/x_{01}^2)\). The BFKL equation with the kernel of Eq. (25) is solved in Ref. [1]. It should be stressed, that in the approach of Ref. [30]. the rapidity

*Y*should be replaced by the target rapidity \(\eta = Y - \ln \left( \frac{R^2}{r^2}\right) \,=\,\ln (1/x_{Bj})\) for DIS scattering.

**6.621(2)**of Ref. [32] and

**15.3.19**of Ref. [31]

### 4.3 Difficulties present in the method of steepest descent

In the LO to evaluate the integral of Eq. (10) we use the method of steepest descent. We now attempt to use it for the case of the NLO.

*N*is the imaginary part of the scattering amplitude, which is positive, we expect, that we will have some difficulties with this method.

### 4.4 Expansion in series

*n*/ 2, from \(\phi _{in}(\gamma )\) (see Eq. (12)) and also every term with even

*n*has singularities: the branch point from \( -\, i \,2\, \sqrt{{\bar{\alpha }}_S}\) to \( i \,2\, \sqrt{{\bar{\alpha }}_S}\). For \(\xi - \frac{1}{2}Y \,>\,0\) we can move the contour \(C_1\) to the left and integrate each term with the contour \(C_2\). Note, that for \(\xi - \frac{1}{2}Y \,<\,0\) we can close the contour on the singularities of the initial conditions, or make an analytical continuation of the scattering amplitude from the region \(\xi - \frac{1}{2}Y \,>\,0\). For large \(\xi (Y - \xi )\), we can use the method of steepest descend to obtain the answer in this kinematic region.

**3.771(8)**of Ref. [32]:

**8.335(1)**of Ref. [32]): \(\Gamma \left( 2(n + 1)\right) \,=\,\left( 2^{2 n + 1}/\sqrt{\pi }\right) \Gamma \left( n + 1 \right) \,\Gamma \left( n+3/2\right) \).

**5.7.6.1**in Ref. [33]. Hence, we obtain the explicit form of the solution

*b*the solution decreases as the power of

*b*; (2) in the limited range of \(\xi \) we can parameterize this decrease as \({\tilde{N}} \propto \exp \left( - \mu ^2 b^2 \right) \) with \(\mu ^2 =\mathrm{Const}/(r R)\) for sufficiently small values of \(\mathrm{Const}\); and (3) at large

*b*we have oscillating behaviour, which is in contradiction to \({\tilde{N}} \,>\,0\), that follows from the unitarity constraints.

*b*, which leads to the violation of the Froissart theorem [3, 4, 5].

### 4.5 Numerical estimates

*n*in Eq. (37) we can re-write the solution in the form:

*N*, which comes from the numerical calculation for Eq. (44), choosing \(B_1 = \frac{1}{2}\) and \(B_2 = 1\) in Eq. (13), taking \({\bar{\alpha }}_S= 0.2\) and fixing \(Y=10\). The logarithmic plot in this figure shows, first, that at large

*b*we have the power-like decrease, as we have discussed, and, second, that we can reproduce the solution which decreases as \(e^{ - 1.06 b/\sqrt{r\,R}}\) in the region of \(\xi = \) 4–10. It should be stressed that such fast decrease cannot be achieved in the LO BFKL, for which, \({\tilde{N}}\) increases at large

*b*. We will discuss this in detail in the conclusions below. It is interesting to note that the slope \(1.06/\sqrt{r \,R}\) is close to one, that has been found in Ref. [1] for \(r = R = 10\) GeV\(^{-1}\).

*Y*and on the initial conditions. One can see that the range of

*b*in which we can trust the exponential parameterization also depends on the values of

*r*and

*Y*, reproducing the main pattern of the solution given in Ref. [1].

Numerical value for the slope \(\mu \) in GeV versus \(Y = \ln (1/x)\)

r (GeV\(^{-1}\)) | Y = 10 | Y = 3 |
---|---|---|

1 | 0.079 | 0.394 |

10 | 0.058 | 0.092 |

25 | 0.043 | 0.082 |

63 | 0.026 | 0.038 |

Numerical value for the slope \(\mu \) in GeV for \(Y =10\) and for different values of \(B_1\) and \( B_2\)

r (GeV\({=1}\)) | \(B_1{=}1/2\), \(B_2 {=}1\) | \(B_1{=}1/2\), \(B_2 {=}2\) | \(B_1{=}1/3\), \(B_2 {=}1\) |
---|---|---|---|

1 | 0.079 | 0.067 | 0.076 |

10 | 0.058 | 0.053 | 0.057 |

25 | 0.043 | 0.036 | 0.039 |

63 | 0.026 | 0.023 | 0.022 |

## 5 Beyond DLA

**3.711**of Ref. [32] continuing it analytically for imaginary

*A*.

The \({\bar{\alpha }}_S\) dependence of all parameters in Eq. (49) are shown in Fig. 6.

## 6 Conclusions

In this paper, we show that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter and, therefore, they cannot resolve the contradiction with the unitarity [3, 4, 5]. On the other hand, in a limited range of *b*, the NLO corrections lead to a fast decrease of the scattering amplitude with *b*, which can be parameterized as \(N\, \propto \,\exp ( -\,\mu ^2\,b^2 )\) with \(\mu ^2 \propto 1/r^2\), in accord with the numerical estimates in Ref. [1].

We demonstrate that the NLO correction leads to an oscillating behaviour of the scattering amplitude as function of *b*. Such oscillations contradict the unitarity constraints, as *N*, being the imaginary part of the scattering amplitude, should be positive (\(N\,>\,0\)).

*b*(see Fig. 8) and could be useful in the description of the experimental data (see Ref. [1]).

We believe that we need to introduce non-perturbative corrections with an additional dimensional scale to the BFKL kernel, and that their influence will be much more important than that of the NLO BFKL kernel that we have discussed here.

## Notes

### Acknowledgements

We thank our colleagues at Tel Aviv university and UTFSM for the discussions. The special thanks go to Asher Gotsman his encouraging support. This research was supported by Proyecto Basal FB 0821(Chile), Fondecyt (Chile) Grants 1180118 and 1191434.

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