Exact kinematics in the small x evolution of the color dipole and gluon cascade
Leszek Motyka, Anna M. Stasto
TL;DR
The paper tackles how to incorporate exact kinematic information from tree-level QCD amplitudes into small-x gluon evolution in the large-Nc limit. It presents exact solutions for multi-gluon components of the gluon light-cone wave function in the all-positive-helicity sector and derives a practical, local approximation to the dipole kernel that captures NLL double-log terms and reduces impact-parameter diffusion. The results connect the wave-function fragmentation amplitudes to MHV structures and demonstrate equivalence with Parke–Taylor amplitudes in the high-energy, large-rapidity limit. Overall, the work provides a framework for improved kinematic treatment in dipole evolution and gluon cascades, with implications for more accurate small-x phenomenology.
Abstract
The problem of kinematic effects in the gluon and color dipole cascades is addressed in the large N_c limit of SU(N_c) Yang--Mills theory. We investigate the tree level multi-gluon components of the gluon light cone wave functions in the light cone gauge keeping the exact kinematics of the gluon emissions. We focus on the components with all helicities identical to the helicity of the incoming gluon. The recurrence relations for the gluon wave functions are derived. In the case when the virtuality of the incoming gluon is neglected the exact form of the multi-gluon wave function is obtained. Furthermore, we propose an approximate scheme to treat the kinematic effects in the color dipole evolution kernel. The new kernel entangles longitudinal and transverse degrees of freedom and leads to a reduced diffusion in the impact parameter. When evaluated in the next-to-leading logarithmic (NLL) accuracy, the kernel reproduces the correct form of the double logarithmic terms of the dipole size ratios present in the exact NLL dipole kernel. Finally, we analyze the scattering of the incoming gluon light cone components off a gluon target and the fragmentation of the scattered state into the final state. The equivalence of the resulting amplitudes and the maximally-helicity-violating amplitudes is demonstrated in the special case when the target gluon is far in rapidity from the evolved gluon wave function.