Born-Oppenheimer Renormalization group for High Energy Scattering: CSS, DGLAP and all that

In \cite{one}, we have introduced the Born-Oppenheimer (BO) renormalization group approach to high energy hadronic collisions and derived the BO approximation for the light cone wave function of a fast moving projectile hadron. In this second paper, we utilize this wave function to derive the BO evolution of partonic distributions in the hadron -- the gluon transverse momentum and integrated parton distributions (TMD and PDF respectively). The evolution equation for the TMD contains a linear and a nonlinear term. The linear term reproduces the Collins-Soper-Sterman (CSS) equation with a physical relation between the transverse and longitudinal resolution scales. We explain how this equivalence arises, even though the BO and CSS cascades are somewhat different in structures. The nonlinear term in the evolution has a very appealing physical meaning: it is a correction due to stimulated emission, which enhances emission of gluons (bosons) into states with a nonzero occupation. For the evolution of the PDF we again find a linear and nonlinear term. At not very small Bjorken $x$, the linear term recovers the DGLAP equation in the leading logarithmic approximation. At small $x$ however there are contributions from gluon splittings which are in the BFKL kinematics leading to a modification of the DGLAP equation. The nonlinear terms have the same physical origin as in the equation for the TMD -- the stimulated emission corrections. Interestingly the nonlinear corrections are the most important for the virtual terms, so that the net correction to the DGLAP is negative and mimics shadowing, although the physical origin of the nonlinearity is very different.