Non-linear equation in the re-summed next-to-leading order of perturbative QCD: the leading twist approximation

In this paper, we use the re-summation procedure, suggested in Ducloué et al. (JHEP 1904:081, 2019), Salam (JHEP 9807:019 1998), Ciafaloni et al. (Phys Rev D 60:1140361999) and Ciafaloni et al. (Phys Rev D 68:114003, 2003), to fix the BFKL kernel in the NLO. However, we suggest a different way to introduce the non-linear corrections in the saturation region, which is based on the leading twist non-linear equation. In the kinematic region: $$\tau \,\equiv \,r^2 Q^2_s(Y)\,\le \,1$$ τ ≡ r 2 Q s 2 ( Y ) ≤ 1 , where r denotes the size of the dipole, Y its rapidity and $$Q_s$$ Q s the saturation scale, we found that the re-summation contributes mostly to the leading twist of the BFKL equation. Assuming that the scattering amplitude is small, we suggest using the linear evolution equation in this region. For $$\tau \,>\,1$$ τ > 1 we are dealing with the re-summation of $$(\bar{\alpha }_S\,\ln \tau )^n$$ ( α ¯ S ln τ ) n and other corrections in NLO approximation for the leading twist. We find the BFKL kernel in this kinematic region and write the non-linear equation, which we solve analytically. We believe the new equation could be a basis for a consistent phenomenology based on the CGC approach.