Born-Oppenheimer Renormalization group for High Energy Scattering: the Modified BFKL, or where did it all go?

We continue exploring the Born-Oppenheimer renormalization group generating evolution in frequency of physical observables. In this paper we study the evolution of the total cross section for dilute-dilute scattering retaining only eikonal emissions. We derive and analyze the analog of the BFKL equation in this framework. The frequency evolution has a very strong effect on the solutions of the BO-BFKL equation, slowing down the evolution of the scattering amplitude in a spectacular fashion: the intercept of the Pomeron is decreased by about a factor of three relative to the canonical LO BFKL result. The anomalous dimension is also modified significantly - from the BFKL value of one it goes down to the negative value of $\approx-0.2$. Introducing saturation boundary as a proxy for the full saturation dynamics, we find that the dependence of the saturation momentum on rapidity $\eta$ becomes quite weak with $Q^2_s\sim e^{a\bar\alpha_s\eta}$ with $a\approx 0.784$ as opposed to the BFKL value $a=4.88$. Our results underscore the necessity to take into account the DGLAP effects in the high energy evolution. This is left for future work.