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

## TL;DR

This paper analyzes the impact of next-to-leading order (NLO) corrections on the BFKL equation at large impact parameters, revealing that while they do not resolve unitarity issues, they improve the description of the scattering amplitude's decrease with impact parameter.

## Contribution

It provides a detailed analysis of NLO corrections to the BFKL equation at large impact parameters, highlighting their effects on amplitude behavior and unitarity constraints.

## Key findings

- NLO corrections do not alter the power-like decrease of the amplitude at large impact parameters.
- NLO corrections cause oscillations in the amplitude, conflicting with unitarity.
- In a limited range, NLO corrections lead to a faster exponential decrease of the amplitude with impact parameter.

## Abstract

In this paper, we show (i) that the NLO corrections do not change the power-like decrease of the scattering amplitude at large impact parameter ($b^2 \,>\,r^2 \exp\left( 2\bar(\alpha}_S \eta(1 + 4 \bar{\alpha}_S)\right)$, 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 (ii) they lead to an oscillating behaviour of the scattering amplitude atlarge $b$, 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 Ref.[1].