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

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 ($$b^2 \,>\,r^2 \exp ( 2{\bar{\alpha }}_S\eta (1 + 4 {\bar{\alpha }}_S) )$$b2>r2exp(2α¯Sη(1+4α¯S)), where r denotes the size of scattering dipole and $$\eta = \ln (1/x_{Bj} )$$η=ln(1/xBj) 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 $$N\, \propto \,\exp ( -\,\mu \,b )$$N∝exp(-μb) with $$\mu \, \propto \,1/r$$μ∝1/r in accord with the numerical estimates in Cepila et al. (Phys Rev D 99(5):051502, 10.1103/PhysRevD.99.051502, arXiv:1812.02548 [hep-ph], 2019).